Systems, Decision and Control in Energy III (Studies in Systems, Decision and Control, 399) 3030876748, 9783030876746

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Studies in Systems, Decision and Control 399

Artur Zaporozhets   Editor

Systems, Decision and Control in Energy III

Studies in Systems, Decision and Control Volume 399

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

More information about this series at https://link.springer.com/bookseries/13304

Artur Zaporozhets Editor

Systems, Decision and Control in Energy III

Editor Artur Zaporozhets Institute of General Energy National Academy of Sciences of Ukraine Kyiv, Ukraine

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-030-87674-6 ISBN 978-3-030-87675-3 (eBook) https://doi.org/10.1007/978-3-030-87675-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Energy continues to play a leading role in the worldwide economy. Today, it is the main driver of many economies in the world. The development of technologies in the field of energy determines the future of virtually all countries, continents and even the entire Earth. Despite the importance of the development of breakthrough technologies, the energy sector is not devoid of certain trends which are caused by a number of reasons. Undoubtedly, one of these trends is the development of “green energy”, as a replacement for traditional fossil fuels the use of which has significantly worsened the ecological situation and—according to many respected scientists and scholars—has played a decisive role in climate changes observed in recent years. This situation leads to an urgent need to look for new methods and means to optimize the operation of existing power plants that work on fossil fuels. Today, many countries need such optimization, and Ukraine is no exception. To take Ukraine as a good end representative situation, the installed power of power plants in Ukraine is 54.365 GW including: • • • • •

nuclear power plants—13.835 GW (25.4%); thermal power plants—21.842 GW (40.2%); hydroelectric power plants—6.307 GW (11.6%); solar power plants—5.062 GW (9.3%); and wind power plants—1.071 GW (1.9%).

Despite a significant increase in the potentials of power plants using renewable energy sources, in Ukraine, like in many developing countries, it is unlikely to be able to quickly abandon the use of power plants working on fossil fuels. But even these power plants are obsolete and have reached their technical potential. This situation stimulates the development of approaches to the optimization of functioning of the existing equipment in terms of not only an increase of power but also a reduction of harmful emissions. This book demonstrates the results of research in the field of energy not only from Ukraine but also from neighboring countries. It consists of six thematic sections:

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1. 2. 3. 4. 5. 6.

Preface

Electric Power Engineering; Heat Power Engineering; Nuclear Power Engineering; Fossil Fuels; Cybersecurity and Computer Science; and Environmental Safety.

As before, the main role in the preparation of this book has played the Council of Young Scientists at the Department of Physical and Technical Problems of Energy of the National Academy of Sciences of Ukraine, and personally Volodymyr Artemchuk, as well as by the State Institution “The Institute of Environmental Geochemistry of National Academy of Sciences of Ukraine”, and personally Alexander Popov, Valeria Kovach and Andrii Iatsyshyn. Kyiv, Ukraine July 2021

Artur Zaporozhets

Contents

Electric Power Engineering On Applicability of Model Checking Technique in Power Systems and Electric Power Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vadym V. Shkarupylo, Ihor V. Blinov, Alexander A. Chemeris, Valentyna V. Dusheba, and Jamil A. J. Alsayaydeh Method of Regulating the Operating Modes of Main Electrical Systems in Terms of Voltage and Reactive Power . . . . . . . . . . . . . . . . . . . . . Vladislav Kuchanskyy and Volodymyr Tereshchuk

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Heat Power Engineering Creation of High-Speed Methods for Solving Mathematical Models of Inverse Problems of Heat Power Engineering . . . . . . . . . . . . . . . . . . . . . . Artur Zaporozhets, Vladyslav Khaidurov, and Tamara Tsiupii

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Mathematical Model of Optimal Support of Thermal Energy with Coal Products Taking into Account Environmental Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vitaliy Makarov, Mykola Makortetskyi, Mykola Perov, Tetiana Bilan, and Nataliia Ivanenko

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Modelling the Impact of Energy-Saving Technological Changes on the Market Capitalization of Companies . . . . . . . . . . . . . . . . . . . . . . . . . . Olexandr Yu. Yemelyanov, Tetyana O. Petrushka, Anastasiya V. Symak, Liliia I. Lesyk, and Oksana B. Musiiovska

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Mathematical Approaches to Forecasting and Researching the Technical State of Cylindrical Shells of Energy Objects’ Elements Based on Vibration Monitoring Systems . . . . . . . . . . . . . . . . . . . . 107 Viktoria Dzyuba and Artur Zaporozhets

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Contents

Nuclear Power Engineering Optical Fiber in Nuclear Power Plants: Applications to Improve the Reliability, Safety and Work Stability of Fault Control Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Ievgen Zaitsev, Anatolii Levytskyi, Kromplyas Bogdan, and Rybachok Pavlo Actual Issues on Radiological Assessment for Events with Liquid Radioactive Materials Spills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Yurii Kyrylenko, Iryna Kameneva, Oleksandr Popov, Andrii Iatsyshyn, Volodymyr Artemchuk, and Valeriia Kovach Formation of Radiation Doses of Ukraine’s Population in Areas Contaminated by Radionuclides After the Accident at the Chernobyl Nuclear Power Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Iryna Matvieieva, Yurii Rudyak, Yurii Zabulonov, Andrii Iatsyshyn, Dmytro Taraduda, and Kachur Taras Fossil Fuels Prospects for the Rational Use of Waste from Uranium Mining Enterprises of Ukraine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Maryna Yaroshchuk, Yurii Fomin, Oleksandra Buglak, Oleksandr Vaylo, and Yurii Demikhov Iterative Solution of the Inverse Problem of Resistivity Logging of Oil and Gas Wells: Testing and Examples . . . . . . . . . . . . . . . . . . . . . . . . . 187 Mykyta Myrontsov, Oleksiy Karpenko, Oleksandr Trofymchuk, Stanislav Dovgyi, and Yevheniia Anpilova Mineralogical-Geochemical Properties of Bentonite Clays of the Cherkasy Deposit to Increase the Environmental Safety of Radwaste Disposal at the Vektor Storage Complex . . . . . . . . . . . . . . . . . 203 Borys Shabalin, Konstiantyn Yaroshenko, Serhii Buhera, Nataliia Mitsiuk, and Oleg Myroshnyk Metal–carbon Nanocomposite for Purification of Natural and Technogenicly Polluted Water from Oil Pollutants . . . . . . . . . . . . . . . . 221 Yurii Zabulonov, Vadim Kadoshnikov, Tetyana Melnychenko, Valeriia Kovach, and Liudmyla Sydorchuk Cybersecurity and Computer Science Method of Improving the Security of 5G Network Architecture Concept for Energy and Other Sectors of the Critical Infrastructure . . . 237 Maksim Iavich, Giorgi Akhalaia, and Sergiy Gnatyuk

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The Method of Determining the Elements of Urban Infrastructure Objects Based on Hough Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Hennadii Khudov, Vladyslav Khudov, Iryna Yuzova, Yuriy Solomonenko, and Irina Khizhnyak Application of Discriminant Analysis in the Interpretation of Well-Logging Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Oleksiy Karpenko, Mykyta Myrontsov, and Yevheniia Anpilova Noise Immunity of Devices of Automated Systems for Technological Control of Energy Facilities in the Almaty Region . . . . . . . . . . . . . . . . . . . . 277 B. R. Kangozhin, O. A. Baimuratov, M. S. Zharmagambetova, S. S. Dautov, and D. B. Kangozhin Environmental Safety Ecological Aspects of the Assessment of Safety Limits of the Near Surface of Radioactive Wastes in the Chornobyl Exclusion Zone . . . . . . . 293 Yuriy Olkhovyk, Sergey Mikhalovsky, and Andrew B. Cundy Environmental Hazards of the Donbas Hydrosphere at the Final Stage of the Coal Mines Flooding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 Yevheniia Anpilova, Yevhenii Yakovliev, Oleksandr Trofymchuk, Mykyta Myrontsov, and Oleksiy Karpenko Complex Oxygen Regimes of Water Objects Under the Anthropogenic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 Inna Skurativska, Sergii Skurativskyi, Oleksandr Popov, Deineka Viktoriia, Eduard Mykhliuk, and Maksym Dement Mathematical Software for Estimation of the Air Pollution Level During Emergency Flowing of Gas Well for Education and Advanced Training of Specialists in the Oil and Gas Industry . . . . . . 335 Oleksandr Popov, Teodoziia Yatsyshyn, Anna Iatsyshyn, Yulia Mykhailiuk, Yevhen Romanenko, and Valentyna Kovalenko Environmental Assessment of Recreational Territories of Ukraine . . . . . . 353 Nataliia Ridei, Tetiana Khitrenko, Valeriia Kovach, Oleg Karagodin, Hrushchynska Natalia, and Oleksii Mykhalchenko Legal Aspects of the Use of Renewable Energy Sources and the Implementation of the Concept of “Green Economy” in Ukraine in the Context of Sustainable Development Strategy . . . . . . . . 373 Volodymyr Yermolenko, Olena Hafurova, Maryna Deineha, Tamara Novak, and Yuliia Shovkun

Electric Power Engineering

On Applicability of Model Checking Technique in Power Systems and Electric Power Industry Vadym V. Shkarupylo , Ihor V. Blinov , Alexander A. Chemeris , Valentyna V. Dusheba , and Jamil A. J. Alsayaydeh

Abstract The chapter is devoted to the analysis of model checking technique applicability in energetics scenarios. A review of model checking technique application taking place in safety–critical scenarios has been conducted. Proven and widely adopted TLC checker has been considered as an instrument for design solutions verification. To this end, the Temporal Logic of Actions and corresponding formalisms— TLA + and PlusCal—have been applied to formally specify the functional properties provided as block-diagrams. Power systems management scenarios taking place in electricity markets have been approached as promising problem domain. Stratified representation of standardized verification and validation process, with a pointed out place of model checking technique, has been provided. Communicating Sequential Processes formalism, Hoare Triples and corresponding rules have been applied to synthesize the formal specifications of safety–critical system functional properties. Corresponding stratified technique has been introduced and discussed. Electric power industry scenario has been approached as a case study. With respect to named scenario, two alternative variations of TLC checker—by way of depth-first- and breadth-first search—have been numerically estimated in terms of multithreading adoption.

V. V. Shkarupylo (B) · A. A. Chemeris · V. V. Dusheba G.E. Pukhov Institute for Modelling in Energy Engineering of the National Academy of Sciences of Ukraine, Kyiv, Ukraine V. V. Dusheba e-mail: [email protected] V. V. Shkarupylo National University of Life and Environmental Sciences of Ukraine, Kyiv, Ukraine I. V. Blinov The Institute of Electrodynamics of the National Academy of Sciences of Ukraine, Kyiv, Ukraine J. A. J. Alsayaydeh Center for Advanced Computing Technology, Universiti Teknikal Malaysia Melaka, Melaka, Malaysia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Zaporozhets (ed.), Systems, Decision and Control in Energy III, Studies in Systems, Decision and Control 399, https://doi.org/10.1007/978-3-030-87675-3_1

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Keywords Energetics · Formal method · Formal specification · Model checking · Safety–critical system · Verification · TLA · TLC

1 Introduction Modern society in diverse spheres of its activity significantly relies on safety–critical systems (SCS) functioning. Those are the systems, bound with strict safety requirements, i.e., when unexpected scenario of system functioning can lead to critical outcomes, e.g., avionics systems and related DO-178C document specifying the aspects of corresponding embedded software certification [1]. The following problem domains can also be considered as the demonstrative ones: system for controlling orientation of a spacecraft, systems maintaining the processes taking place in energetics, etc. Modern representatives of named systems development process organization is commonly built on model-based approach [2]. The distinctive feature of corresponding model checking methods is its applicability to automated usage [3]. The process of SCS engineering is approached as the sequence of the following steps: requirements analysis, design, implementation, validation (either by way of simulation or by way of testing). It is known that design faults discovery and fixing at late stages of named process is coupled with significant time and material costs [4]. While considering the model checking techniques, the notion of “formal verification” is assumed. To this end, the viewpoint and constraints described in IEEE 1012-2016 standard are applied [5]. Verification procedure is considered here as a constituent of more complex V&V (Verification and Validation) process covering all the aforementioned stages of engineering process. Verification is applied as a tool for discovery the correspondence between the artifacts obtained and requirements specifications provided. Meanwhile, validation is approached as a mechanism for proving the applicability of obtained results of engineering process to the intended use.

2 Stratification of the Concepts 2.1 Constituents of Verification and Validation Process To proceed further, a differentiation between the concepts used needs to be conducted first (Fig. 1). In Fig. 1, the aggregation relation is depicted with diamond arrow, implementation relation—with dotted arrow. With respect to UML-notation (Unified Modeling Language), this means that verification and validation procedures form the V&V process. At the same time, Model Checking (MC) and Runtime Verification (RV) are the ways of verification implementation. Similarly, Simulation and Testing are the

On Applicability of Model Checking Technique …

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Fig. 1 Schematic stratified representation of V&V process

variations of validation implementation. Moreover, Stochastic MC is the extension of MC technique providing the ability to diminish the effect of state space explosion problem taking place while exhaustive state space search during the conventional MC.

2.2 Safety–Critical Applications of Model Checking Technique While MC is typically applied at design time, the distinctive feature of RV is its execution at a runtime [6]. Thus, to conduct the RV, SCS needs to be implemented first. This implies that design stage has already been completed. Thereby, specified aspect makes sense to consider the MC and RV techniques as complementary ones. There are another alternatives though, e.g., deductive verification, equivalence checking. The reason these techniques are not depicted in Fig. 1 is their complexity in terms of automation. This peculiarity significantly harms the aspect of practical applicability in industrial scenarios. On contrary, MC technique encompasses a plethora of

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industry-related domains, e.g., TAS Control Platform as a computing platform for railway control applications—a fault-tolerant solution satisfying SIL 4 level (Safety Integrity Level), i.e. failure rate up to 10−9 per hour [7]. To do so, the TLC (TLA checker) model checker, the Temporal Logic of Actions (TLA) and corresponding formalisms—TLA + and PlusCal—have been brought to the use [8–10]. A systematic approach to software SCSs engineering is described in CENELEC—EN 50128 standard, where the usage of formal methods is recommended [11]. With respect to energetics, a demonstrative example is the Finnish nuclear industry, where MC methods have been successfully applied since 2008 until present days [12]. It has been stressed that, in accordance with IEC 61508-3 standard, the use of formal methods is highly recommended to correspond to SIL 4 requirements, and is recommended with respect to SIL 2 and SIL 3 [13]. To this end, MC technique has been integrated into nuclear I&C (Instrumentation and Control) process. To automate this process, the MODCHK tool, aimed at application logics verification, has been applied [12]. Moreover, the need for tools automating the process of formal specifications synthesis has been stressed. Here comes the aspect of choosing the right atomicity level of specification, sufficiently satisfying the goal of verification by way of model checking. In this context, it has been shown previously that the growth of specification state variables number prompts the exponential growth of related time costs [14]. Named peculiarity is even more significant with respect to specifications with concurrency, e.g., when concurrency is represented as interleaving [15]. As an outcome, available random access memory resources of computing platform have been run out during the model checking process. To complement the picture, the embedded software engineering process also need to be mentioned as demonstrative safety–critical domain, e.g., OpenComRTOS real time operating system, where TLA, TLA+ and TLC have been fruitfully applied [16, 17]. At the same time, to encompass also the non-functional properties, e.g., time constraints, the well-established UPPAAL integrated tool has been utilized [18]. With respect to nuclear power systems, the defined interpretation of functional safety concept takes place. In the Order of the State Inspectorate for Nuclear Regulation of Ukraine No. 140, (dated: 22.07.2015), amended in accordance with the Order of the State Inspectorate for Nuclear Regulation No. 508 (dated: 25.11.2019), the following definition of functional safety is provided: “functional safety is a property of a nuclear power plant system, which consists in the ability to perform all the necessary functions important for safety, to maintain the required properties and to meet the specified characteristics in all modes and operating conditions provided by the project” [19]. At the same time, the directions of Ukrainian electricity market transformation have been described in The Law of Ukraine on Electricity Market [20]. This, in particular, raises the need for the analysis of influence of renewables on the electricity market price in Ukraine, which can be met with appropriate model creation [21]. Previous contribution on this topic has been carried out by way of simulation model of the new electricity market of Ukraine creation and proving [22, 23]. During the transformation of the electricity market, new processes and models of information exchange can be described using UMM (UN/CEFACT Modeling

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Methodology), as well as creating role models of the electricity market [23]. However, the implementation of new rules and algorithms for market segments management systems or power systems, e.g., the balancing market or ancillary services market [24, 25], requires prior verification. The processes of integration of European power systems [26] and the electricity markets coupling [27, 28] require the use of new operating algorithms. Inaccuracies in their functioning can lead to serious economic consequences and disruption of the stable operation of power systems. Control technologies in the power system, e.g., energy storage systems [29, 30] and the development of algorithms for their operation [31, 32], require preliminary verification of created solutions. To this end, the MC technique is proposed to be applied. To do so, the formal specification needs to be synthesized first to be utilized as the input data for the MC.

3 Description of Introduced Formalization Technique 3.1 Problem Domain Specification Drawing the parallels with spacecraft domain, the accent of current chapter is depicted in Fig. 2. Decomposition, provided in Fig. 2, is made with respect to positions of ECSSE-00A standard [33], which then has been substituted with ECSS-S-ST-00C Rev.1 version [34]. With respect to system engineering process, design stage has been considered. At the same time, software plane has been stepped up.

Fig. 2 System approach to the analysis of SCS designing artifacts

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Fig. 3 Place of model checking technique application

To show the accent of work with respect to engineering process as a sequence of steps, more detailed diagram is depicted in Fig. 3. In Fig. 3, with dashed box, the accent of work is delimited, where input and output are the artifacts: input—textually represented requirements, output—blockdiagrams. An artifact is the formalized entity, with structure and content [35]. An outcome, with structure and content, of certain step of designing stage of engineering process is defined as the artifact.

3.2 Concepts Utilized In presented work, the following artifacts are considered: algorithm block-diagram, corresponding formal specification as the input data for model checking technique. Conceptual representation of the approach applied is depicted in Fig. 4. In Fig. 4, two planes of specification perception have been distinguished—analytical plane and implementation plane. Analytical plane represents developer’s perception of system under analysis. Kripke structure represents the transition system. CSP (Communicating Sequential Processes) formalism provides the mechanism to transform the Kripke structure based concepts into the action-based concepts taking place

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Fig. 4 Approach to specification synthesis

in the TLA. To this end, the notion of a “protocol” has been utilized [36]. In addition, Hoare triples are used to compact the specification, e.g., the composition rule [37]. On the bottom plane, the concepts uncovering the aspects of specification implementation take place. PlusCal algorithmic language is applied as a preliminary step to TLA+ specification creation—to set the structure of specification and clarify its perception [38]. At the same time, the resulting TLA+ specification is treated as an input data for model checking technique applied—TLC (TLA Checker).

3.3 Concepts Formalization By applying the aforementioned “artifact” notion, with respect to the structure, depicted in Fig. 4, four artifact types have been distinguished: • Kripke structure. • CSP-representation of Kripke structure, with Hoare triples and corresponding rules applied. • Specification metamodel (prototype)—on the basis of PlusCal formalism—encapsulating the algorithmic constituent of yet to be synthesized TLA+ specification. • Resulting specification on the basis of TLA+ formalism. Proposed technique provides the mechanism to obtain the resulting specification by synthesizing named artifacts in a sequential manner. To this end, with respect to Fig. 4, three subsequent transitions need to be made.

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To formalize the approach represented in Fig. 4, the following structure has been introduced: A, T ,

(1)

where A = {a1 , a2 , a3 , a4 }—set of the artifact types involved: ∀ai ∈ A (i = 1, 2, .., 4) there is a formal system uniquely identifying corresponding artifact type: a1 ∈ A —Kripke structure; a2 ∈ A —CSP-formalism; a3 ∈ A —PlusCal-formalism; a4 ∈ A —TLA+ formalism; T ⊂ A2 —set of transitions between the artifacts: T = {(a1 , a2 ), (a2 , a3 ), (a3 , a4 )}; a2 = T (a1 ), a3 = T (a2 ) = T (T (a1 )), a4 = T (a3 ) = T (T (a2 )) = T (T (T (a1 ))). The following assumptions have been made [39]: • Kripke structure on the basis of a set of atomic prepositions AP is applied to analytically interpret the specification: M = S, {s0 }, R, L,

(2)

where S—finite set of states; {s0 } ⊂ S—set of initial states of transition system; R ⊂ S 2 —set of transitions: ∀s ∈ S ∃s ∈ S : s, s ∈ R; L : S → 2AP —states labeling function. • Specification on the basis of TLA+ formalism is considered as an outcome of the technique proposed. Implementation of the approach depicted in Fig. 4 is provided below. In total, two steps have been introduced—targeting both planes distinguished (Fig. 4). Step 1. Analytical Plane Elaboration Behaviors Formalization with Kripke Structure. Let functional properties of SCS are represented as the behaviors on the basis of Kripke structure concepts (2) as a set: B = {bi }, i = 1, 2, ..., m ∈ N ,

(3)

where bi ∈ B—i-th functional property of a system as a sequence of states: bi = s0 , s1 , ..., sl ,

(4)

where s0 ∈ S—initial state of transition system, s1 = R(s0 )—subsequent state, s2 = R(s1 ) = R(R(s0 )), and so forth; l ≤ (n · p − 1), n, p ∈ N , n—number of transition system state variables, p—number of allowable values of state variables. Shifting to CSP Notation. To make a shift to TLA, CSP formalism has been adopted first: a step from the concept of “behavior” (4) to the concept of “process protocol” (as a tuple of events prompting corresponding actions that can be represented with the TLA+ formalism) has been made. Such a transition can be specified as follows:

On Applicability of Model Checking Technique … Table 1 Correspondence Between the Constituents of Kripke Structure and CSP Formalism

Stratas

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Mathematical Apparatus Kripke structure

CSP formalism

1

2

3

1

Behavior (4)

Process protocol (6)

2

State, transition (2)

Event

q : B → P,

(5)

where P = {pi }—set of process protocols: |P| = |B|, i.e., q(bi ) = pi , where bi ∈ B, pi ∈ P: pi = e1 , e2 , ..., en  ∈ P,

(6)

where pi ∈ P—protocol of i-th computing process obtained on the basis of corresponding behavior bi ∈ B, ej (j = 1, 2, ..., n)—j-th event prompting the transition   s, s ∈ R between the adjacent states s, s ∈ S of a transition system represented with the structure (2). Thus, the event is approached as a change of state variable(s) value(s) taking place at certain transition. Parallels between the concepts of (4) and (6) are drawn in Table 1. In Table 1, the elements of strata 1 represent the upper abstraction level, of strata 2—the lower one. With respect to aforementioned notion of artifact, the elements of strata 1 define the structure, while the elements of strata 2 form the context. The latter are the building blocks forming the elements of strata 1. In general, Table 1 provides the mechanism to conduct (a1 , a2 ) ∈ T transition (1), Fig. 4. Hoare Triples Forming. With respect to the event, a conjunction over the elements of L(s) ⊂ AP is approached as a precondition to the event to take place. At the same time,   i.e., the post-condition, is obtained through a transition   an outcome of the event, s, s ⊂ R: L(R(s)) = L s ⊂ AP. To join the events with respect to a protocol (6), Hoare triples are applied [40]: 

   φj−1 ej φj ,

(7)

where φj−1 and φj —pre- and post-conditions for j-th event respectively [41]. This means that, in case of φj−1 takes place, an ej event arises, prompting corresponding outcome, i.e., the φj post-condition. Composition Rule Application. Expression (7) complements the content of the right bottom cell of Table 1 representing the elements of the lower hierarchical layer in terms of the analytical plane interpretation on the basis of the SCP formalism—by representing the tools for events “gluing”—through the pre- and post-conditions. To join the events in the compact manner, a composition rule has been utilized:

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{φ0 }e1 {φ1 }, {φ1 }e2 {φ2 }, ..., {φn−1 }en {φn } , {φ0 }e1 ; e2 ; ...; en {φn }

(8)

where φ0 —initial precondition constructed with the elements of L(s0 ) as a conjunction; φ1 —post-condition of e1 event and, at the same time, precondition to the subsequent e2 event, etc. Here comes the duality of conditions for j = 1, 2, ..., n − 1. To diminish specification redundancy, the aforementioned duality property has been addressed with the composition rule (8), where the outcome of named rule application is placed as the denominator of the expression. At the same time, the triples taking place in the numerator, are approached as a whole—the sequence of triples (numerator) is reduced to the sequence of events (denominator). The precondition to the sequence to take place is the initial one—φ0 , the post-condition—a final one—φn . The key idea here is to directly approach the ej−1 event as a precondition to the subsequent ej event. The outcome of sticking to (8) rule is the element of the upper strata of the CSP-interpretation (Table 1), i.e. the process protocol as the sequence of events— CSP-representation of the Kripke concept of the behavior (4). It can be seen in (8) that the denominator is the representation of (6) process protocol, complemented with φ0 pre- and φn post-condition. Named conditions are brought to the use to subsequently join process protocols together with respect to (3) on the basis of TLA + formalism (Fig. 4). Conditional Rule Application. At the same time, to specify the events at the analytical plane, the conditional rule of Hoare logic has been applied:    φ  ∧ apj,k ej {ξ }, φ  ∧ ¬apj,k ej {ξ }   ,     {φ  }if apj,k then ej else ej endif {ξ } 

(9)

 n∈N where apj,k ∈ AP = V × D: V = vj j=1 —set of state variables, D—set of variable values: D = ran(s), where s : V → D and L(s) ⊂ AP;   φ  ∧ apj,k ≡ φ  ∧ vj , dk ≡ φ,

(10)

  φ  ∧ apj,z ≡ φ  ∧ vj , dz ≡ ξ,

(11)

where dz = dk .   Events Formalization. An event is constructed over the elements of L(s)L s ⊂ AP set as an implication augmented with the “next”—X —temporal operator: ej ≡ (vj , dk ) → X (vj , dz ) ≡ ¬(vj , dk ) ∨ X (vj , dz ),

(12)

Expression (12) means that at certain current state s ∈ S of a transition system (1), constructed through the model checking process, vj , dk ∈ L(s) ⊂ AP atomic

On Applicability of Model Checking Technique … Table 2 Correspondence Between the CSP and PlusCal Constructs

Stratas

13

Mathematical Apparatus CSP formalism

PlusCal formalism

1

2

3

1

Process protocol (6)

2

Event (12)

Algorithm   vj := op vj

      preposition is true and vj , dz ∈ L s ⊂ AP is false. By way of s, s ∈ R transition,    a subsequent state  s = R(s) ∈ S of a transition system is reached: vj , dz ∈ L s ⊂ AP is true, and vj , dk ∈ L(s) ⊂ AP is false.     To specify that, with respect to a current state s ∈ S, vj , dk ∧¬ vj , dz constituent  of a precondition φ is true, but, with respect to a subsequent  state s = R(s) ∈ S,  named constituent is replaced with a true-valued ¬ vj , dk ∧ vj , dz expression, X operator is utilized.     If approach ej as a function, dom ej = L(s)L s ⊆ AP:   ej : L(s)L s → {0, 1},

(13)

where 0 and 1—Boolean values representing “false” and “true” respectively. Step 2. Implementation Plane Elaboration Shifting From the Analytical to Implementation Plane. Similar to the way described above (Table 1), the constituents of PlusCal metamodel are obtained. To this end, with respect to (6), the sequence of events is approached as the composition of functions: el ◦ el−1 ◦ ... ◦ e1 .

(14)

In other words, according to (14), e1 ≺ e2 ≺ ... ≺ el , where ≺—partial order operator. Parallels between the CSP- and PlusCal-constructs are drawn in Table 2. In Table 2, vj —representation of vj ∈ V state variable with respect to a subsequent state s = R(s) ∈ S—as an outcome of X temporal operator application; op— an operation, either changing or not the value of given state variable with respect to a subsequent state. If op does not change the value of vj ∈ V , then vj := vj   assignment takes place. Let us specify this assignment as follows: vj := u vj , where u—“unchanged” operator. Its functioning is prescribed with an empty statement rule:     vj , dk skip vj , dk .

(15)

With respect to (13), j-th event can either occur or not. To encompass also the latter scenario, (15) rule has been brought to the use. Table 2 provides the description of (a2 , a3 ) ∈ T transition (1), Fig. 4.

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Fig. 5 Approach to specification synthesis

With respect to TLA+ formalism, to specify the elements of the lower hierarchical plane, the concept of an “action” has been applied (8). To specify an action, “if–thenelse”-condition construct has been adopted. To do so, both variations of ej (13) need to be covered. Here comes the “duality” property of the event, depicted in Fig. 5. In Fig. 5, in accordance with (15), j-th “empty” event is as follows: ej0 ≡

         vj , dk → X vj , dk ≡ ¬ vj , dk ∨ X vj , dk .

(16)

With respect to implementation plane, TLA+ specification of the event encompasses both (12) and (16):        ej ≡ vj := if φ  ∧ apj,k then vj := dz else vj := dk .

(17)

Targeting a4 ∈ A artifact (1), expression (17) represents lower strata of implementation plane. To specify the elements of the upper strata, the implementation of the aforementioned “action” concept is as follows:     aj ≡ u(v1 ) ∧ u(v2 ) ∧ ... ∧ u vj−1 ∧ ej ∧ u vj+1 ∧ ... ∧ u(vn ).

(18)

With respect to (6), process protocol implementation is specified as a conjunction: ψi ≡ a1 ∧ X a2 ∧ X 2 a3 ∧ ... ∧ X l−1 al , where X 2 a3 ≡ XX a3 , and so forth.

(19)

On Applicability of Model Checking Technique … Table 3 Correspondence Between the PlusCal and TLA+ Constructs

Stratas

15

Mathematical Apparatus PlusCal formalism

TLA+ formalism

1

2

3

1

Algorithm   vj := op vj

ψ(20)

2

aj (18)

The alternativeness (3) of the behaviors is formalized in the resulting TLA+ temporal formula as a disjunction:  ψ ≡ φ0 ∧ G ψ1 ∨ ψ2 ∨ ... ∨ ψm ,

(20)

where G—“globally” temporal operator, φ0 —specification of the initial state of a transition system: φ0 ≡ (v1 , 0) ∧ (v2 , 0) ∧ ... ∧ (vn , 0).

(21)

Description of the resulting (a3 , a4 ) ∈ T transition (1), (Fig. 4), is provided in Table 3. In Table 3, resulting temporal formula ψ (20) represents the resulting artifact the MC technique is applied to.

4 Case Study 4.1 Multithreaded TLC Implementation Description Preconditions Applied. A technique described above is used to synthesize the TLA+ specifications, the TLC technique is applied to. Alternative variations of the TLC technique have been examined in terms of the speedup factor obtained through multithreaded implementation: TLC by way of breadth-first search (BFS) and by way of depth-first search (DFS). Speedup factor calculation:

α = t 1 t tc ,

(22)

where t 1 —average time spent to conduct the TLC in a single threaded manner, t tc —in a multithreaded manner, tc—number of threads executed simultaneously. Description of the Input Data. A block-diagram of control algorithm of the initial state of the registers of input/output unit has been approached as the representation of the functional property of a system under development (Fig. 6).

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Fig. 6 Fragment of the block-diagram considered

In Fig. 6, to obtain the complete block-diagram, the external rectangular region needs to be duplicated 14 times, but with different state variables. The following sub-diagram has been considered: with 15 state variables and the depth of search space of corresponding transition system—42. Four scenarios have been numerically estimated—for tc = 1, 2, 4, 8 (22).

On Applicability of Model Checking Technique …

17

4.2 Obtained Results Experimentation has been conducted on the following platform: Java Runtime Environment—64 bit, build 1.8.0_251-b08; TLC version—2.14; CPU (Core frequency— 3.8 GHz, 4 cores, 8 threads); amount of available random-access memory—16 GB. Obtained results are provided in Table 4. It can be seen in Table 4 that, despite being significantly more effective in terms of corresponding time costs, BFS-variation of the TLC model checker lags behind in terms of speedup factor α noticeably, comparing to alternative BFS-variation [42]. Graphical representation of obtained results is provided in Fig. 7. Table 4 Speedup Factors for TLC Technique Variations No

tc

t DFS /t BFS

TLC Variation BFS t BFS , sec

DFS α

t DFS , sec

α

1

2

3

4

5

6

7

1

1

3.552

1.000

44.575

1.000

12.549

2

2

3.072

1.156

28.384

1.570

9.240

3

4

2.966

1.198

19.989

2.230

6.739

4

8

2.815

1.262

20.526

2.172

7.292

Fig. 7 Obtained results

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In Fig. 7, it can be seen that positive effect of multithreading utilization with respect to DFS-variation surpasses the one for the alternative BFS-variation significantly. Generalized estimation of the positive effect out of multithreading implementation is as follows: β = max(αXFS ),

(23)

where αXFS ∈ {αBFS , αDFS }. By encompassing both technique variations, the following estimation has been obtained: β ∈ [1.262, 2.230],

(24)

for tc = 1, 2, 4, 8.

5 Conclusions Thus, in given chapter, the following results have been presented and thoroughly discussed: • By reviewing the standardized verification and validation process, a recommendation to apply the model checking technique during the design stage of safety–critical system engineering process has been made. • Expediency of model checking technique usage with respect to power systems in particular and electric power industry in general has been grounded. • Stratification-based technique providing the transparent unified mechanism of functional properties formal specifications synthesis has been introduced and practically applied to generate the input data for broadly utilized TLC model checker. • As a case study, the electric power industry domain has been approached. Two variations of TLC checker have been investigated—model checking by way of the depth-first search of a transition system and by way of breadth-first search. It has been demonstrated that the first variation is significantly more fruitful in terms of a speedup of multithreaded implementation—from 1.570 to 2.230 (with respect to DFS-variation); from 1.156 to 1.262 (with respect to alternative BFS-variation). At the same time, BFS-variation is significantly more effective in general—from 7.292 to 12.549 times. Further research is aimed at developing the instruments to apply the introduced technique with respect to the algorithms implemented in the decentralized energy market coupling mechanisms. Acknowledgements Paper has been prepared within the 0121U110615 “Development of methods and means for safety-critical systems designing process artifacts verification” research work, carried

On Applicability of Model Checking Technique …

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out in the G. E. Pukhov Institute for Modelling in Energy Engineering (PIMEE) of the National Academy of Sciences of Ukraine, with a support of the Institute of Electrodynamics of the National Academy of Sciences of Ukraine.

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Method of Regulating the Operating Modes of Main Electrical Systems in Terms of Voltage and Reactive Power Vladislav Kuchanskyy

and Volodymyr Tereshchuk

Abstract To meet the needs of consumers in reactive power and to cover reactive power losses in power lines and power transformers (autotransformers), the following reactive power sources are used: (1) the charging power of the power line, with its loads less than the natural power; (2) generators at the transmitting end of the line; (3) synchronous compensators or banks of static capacitors installed at the receiving end of the power transmission or directly at the points of consumption of electrical energy. With an increase in the load of the power transmission line, the inductive losses of reactive power increase, which leads to a decrease in the volumes of the capacitive charging power of the line flowing into the receiving power system. On the other hand, the full use of the charging power of the power transmission, as well as the transmission through the line of reactive power generated by the generators at the transmitting end of the line, are limited by additional losses of electrical energy and an unacceptable decrease in voltage at the receiving end of the power transmission. Thus, when designing long-distance power transmission, the problem arises of choosing a rational ratio of the use of reactive power generated by various sources. The choice of the type of reactive power source is determined on the basis of the results of a technical and economic comparison of several options for the use of reactive power sources of various types. Keywords Conception smart grid · Open phase mode · Power transfer capability · STATCOM · Controlled shunt reactor · FACTS · Charging power · Secondary arc current · Single phase auto reclose · Operating modes optimization

V. Kuchanskyy (B) The Institute of Electrodynamics of the National Academy of Sciences of Ukraine, Kyiv, Ukraine V. Tereshchuk Polissya National University Ukraine, Zhytomyr, Ukraine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Zaporozhets (ed.), Systems, Decision and Control in Energy III, Studies in Systems, Decision and Control 399, https://doi.org/10.1007/978-3-030-87675-3_2

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1 Introduction To solve the problems of control of operating modes electrical power system (EPS) and power flows both in existing and in new or modernized power transmission lines in order to ensure the reliability and stability of the operation of separate and interconnected power systems, as well as the EPS as a whole, modern software and hardware tools are widely used and high-speed communication channels (fiber-optic, satellite, etc.). However, even with their use, the control efficiency in many cases is limited by the insufficient speed of the power control circuits with mechanically switchable devices that are widely used in modern EPS. Another side of the problem related to this is that the control commands for such devices cannot be generated at a high frequency, since mechanical devices have a lower wear resistance compared to static devices. Therefore, the possibilities of effective application of the flexible power lines (FACTS) technology are largely associated with the development and improvement of contactless electrical devices, power electronics elements and their control systems. In particular, the use of modern high-speed converting and executive devices in FACTS regulators will allow the latter to effectively perform the following functions: reactive power control; voltage regulation, changing the direction and magnitude of power flows; functions of active or hybrid filters, etc. To study and evaluate the effectiveness of a new generation of FACTS regulators, it is necessary to modernize a significant part of the tools for calculating and analyzing power flows used in planning modes and in the process of EPS functioning. Currently, one of the main methods for solving the problem of calculating flows, power, including, in the presence of FACTS elements, is the Newton–Raphson method. However, with a sequential search for solutions by this method, at the end of each iteration, the subproblem of redefining the state variables of the controlled power devices arises, and as a result of such redefinitions, the iterative process loses the property of quadratic convergence. Therefore, the improvement and development of methods and tools for calculating modes taking into account the interaction of various FACTS devices is also an urgent task. Modern methods of synthesis of FACTS devices are largely related to the functions performed by these devices. To minimize undesirable interaction (mutual influence) of FACTS devices, the methods and procedures used for their synthesis should take into account the fact of such interaction. Nonlinearity of power system models, their parametric uncertainty and unpredictable changes in operating modes in emergency situations significantly complicate the task of coordinated control synthesis. The tasks of adaptive control (regulation) of EPS, damping of fluctuations in power flows and a number of others require the development of special methods for the synthesis of FACTS devices [1–20]. Traditional reactive power compensation devices have a number of significant disadvantages. In addition, the problem of improving the quality of electricity is currently very urgent. The upward trend in nonlinear loads in the overall composition of electricity consumers, which are sources of higher harmonics, aggravates this

Method of Regulating the Operating Modes of Main Electrical Systems …

25

problem. Passive AC filters used to improve the quality of power have high losses and do not effectively reduce higher harmonics. Therefore, research and development of principles for controlling active filtration based on elements of power electronics are relevant for solving problems of improving the quality of electricity in EPS. New technologies for the creation and development of power transmission networks with high-speed electronic control of their modes of operation necessitate other methods for the design and construction of power plant equipment, changes in approaches and procedures for planning the operation of networks for the transmission and distribution of electricity. These technologies can also change the nature of business operations in the energy market due to the emergence of the possibility of high-speed control of electricity flows. Due to its many inherent and promising economic and technical advantages, FACTS technology is consciously supported by manufacturers of electrical equipment, power supply systems, and research organizations around the world.

2 Devices for Controlled Compensation of Reactive Power in Main Electric Networks Technological progress in the modern world is inextricably linked with consumption electricity. The analysis shows that the growth dynamics of the consumed of electricity in different countries of the world depends not only on growth population, but also on the rates of economic growth of countries, the level their technical development and other factors. For example, in developing countries the rate of growth of energy consumption at the initial stage of electrification lead to a doubling of consumption every 10 years, while in in developed countries, this growth is stable at 1–2% per year, According to preliminary estimates over the past 20 years in developing countries energy consumption is expected to grow by 220%, and in industrialized countries it is will amount to only 37%. Subject to differences in geographic conditions and availability natural energy resources ways to solve electrification problems are essential differ in technical terms and in the required investment. Therefore, the most feasible solution for the global energy sector is the integration power systems of different regions with subsequent distribution electricity based on market mechanisms [1–10]. Historically, the priority place in the electric power industry was given to variable current. The main advantage of AC is the ability its consumption in almost any places where the lines pass power transmission. A significant factor limiting for a long period development of direct current transmission lines, was the lack of elemental» Bases for effective interconversion of constant and variable currents. As a result, AC systems took over the dominant position, despite the following shortcomings [11–16]: • the need for voltage control depending on the load lines; • the presence of reactive power and the need to control its value;

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• synchronization problems between two independent systems; limitation on stability in steady-state and transient regimes. Note that the first DC transmissions were characterized by additional difficulties: the need for reactive power to ensure switching valves; generation of higher harmonics; high cost of conversion equipment. With the development of basic elements, the cost of power transmission at constant current has decreased, and many problems have been eliminated. At present DC transmission times are more cost effective over distances over 800 km, if there is no power take-off along the route of their passage (for example, when laying gears on the ocean floor, etc. [17– 28]). In the world, long distances, uneven distribution energy resources and centers of their consumption determined the development of energy transmission on alternating current. As a result of power transmission over long distances there are restrictions on bandwidth and provision reliable power supply between the countries [17–20]. There are restrictions for the issuance of “locked” capacities of a number of power plants, suboptimal distribution power flows through parallel power lines and etc. The solution of these problems depends on the development of FACTS [1–3]. These lines include devices capable of control voltage, reactive, power, compensate for power, generated by a non-linear load. Using voltage converters, it is possible to integrate systems that differ in power, voltage and the direction of energy flows. Modern lockable thyristors are switched off by applying a negative pulse to the control electrode. They are calculated for operation in circuits up to 4.5 kV and with currents up to 3 kA in the absence of snubbers and up to 5 kA—when using various types: snubbers, consisting capacitor with a capacity of 4 µF. Simultaneity, shutdown segments provided by drivers of low inductance. The novel design allows to reduce time accumulation charge and decay-current up to 3 µs. Such devices are sometimes called also devices with “hard” shutdown. Other advantages of these devices are new lockable thyristors with buffer layer connection and thin anode emitter. It should be noted that the rapid uniform achieved by inclusion and short edge control due to the driver with a low inductance allowed to make serial and parallel connection of these devices [21–27]. Thanks to new technologies, great strides have been made in creating a powerful high voltage transistor. Thus, the high power density modules (HVIGBT) type transistor is designed for low-current switching of a device with a voltage of 4.5 kV and a current of 900 A. Its turn-off time does not exceed 6 µs. The main advantage is relatively simple handling in the absence of snubbers. In this respect, its peripheral currents can be significantly reduced, but in terms of currents, its area of application is inferior to the area of application of devices of the type [22–30]. But the key devices, being the main elements in the converter circuits, do not fully determine their technical and economic indicators. To a large extent, the latter depend on the specific indicators of capacitors and transformer-reactor equipment. Specific indicators of the latter largely determine the mass and dimensions of filters of constant and variable currents. On the other hand, the specific indicators of magnetic cores are improved reactors by creating a distributed gap. In this case, use magnetic circuit made of pressed iron

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27

powder and alloys with included bonding materials with low magnetic permeability. Modular design and manufacturing deserves special mention. Already, leading companies create keys in a set consisting of a driver and the body of the main key. For serial production of high-voltage or current inverter modules can be expected to appear actually converters for different voltages and currents, filters, etc. The principle of creating power electronic devices consists in a set of discrete elements. Therefore, modular design of these devices is promising [30–42]. The technical progress of the element base had a direct impact on the circuitry of converting devices. First, it became possible operation of voltage and current converters in four quadrants of complex planes of network parameters: rectification and inversion as with capacitive, and with the inductive nature of the current. As a result, pulse-width modulation (PWM) allows to track a given waveform of voltage or current at the output of the converter (depending on its type). Now you can form a harmonic voltage or current given by the shape of the curve and the sign and, therefore, provide generation or consumption of sinusoidal reactive currents. In addition, fully manageable electronic keys allow to control voltage and current with a non-sinusoidal curve for elimination of distortions created by non-linear loads.

3 Fixed Parameter Method and Basic Method Relative Gains for Calculating the Economy Mode Power Systems In this chapter provides a method called method fixed parameters, in which balancing mode is performed in the same way as indicated in [42–50], but another way of optimization is given regime. It is visual and gives immediate communication with methods based on the principle equality of relative gains. Let us assume that at all nodal points of the network, the number which is equal to n can be present as generating sources of active and reactive power, and the loads of consumers and that the maximum and the minimum limits limiting permissible values of stresses in the nodal points and capacities of generating sources. At the nodal point taken as the base point, the vector voltage U b is directed along the real axis [51–58]. In each given mode m nodal points with indices h, active power of sources with indices a, reactive powers m2 , sources with indices r will have a fixed value equal to the upper or lower limit. The latter, in particular, refers to the cardinalities of the nodal points at which there are only loads. Static load characteristics and damage at consumers associated with voltage deviation at the nodal points from the optimal one, we will take into account according to the method described in [50–65]. Let’s imagine active and reactive generating power and voltage of any nodal points j in the form of some functions of n’ = 2n components currents I i and parameter U b : Pg j = ϕ P j (U B , Ii );

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Q g j = ϕ Q j (U B , Ii );

(1)

U j = ϕ Q j (U B , Ii ).

(2)

In this case, the equations expressing the equality to zero of active and reactive loads at the base point must be observed: ϕB P = UB

n 



I j = 0;

j=1

ϕB Q = UB

n 



I j = 0,

(3)

j=1

and equations expressing the constancy of fixed voltage and power sources: ϕU h (U B Ii ) = Uh∗ ;

(4)

ϕa (U B , Ii ) = Pg∗ ; ϕg (U B , Ii ) = Q ∗g , 



(5)

where I j and I j are real and imaginary components nodal currents, and the totality of   all I j and I j gives all currents Ii . The superscript (*) in the Eqs. (4) and (5) indicates that this is the parameter value set fixed. So that each arbitrary mode located within the specified limits, could be balanced and fixed voltages and powers had set value, in this mode can be arbitrarily taken active powers n-1-m1 sources and reactive power n-1-m-m2 sources. These powers will be called free. The capacities of the remaining sources will have either a given fixed value, or serve to balance active and reactive capacities and compliance with the specified voltages. The latter will be determined in the process calculation of the regime [65–79]. Voltage at the reference point when balancing mode U B will be considered constant and equal a given value. This mode, i.e., the values of all n  currents Ii , will be determined by the solution of Eqs. (3), (4) and (5), as well as (1) for the accepted values of free capacities. To optimize the mode in this method, we will consider n = 2n constraint equations ϕs (U B Ii ) = A∗s , which are the same as were taken when balancing the regime, including Eqs. (1) expressing the constancy of the adopted values of free capacities, i.e., we will consider free capacities fixed in some values. Here s = l, 2, 3, …, n  , and each s corresponds to a certain index of functions ϕ in Eqs. (3), (4) and (5). Let us compose the following n’ equations for all i:

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29

n]

∂G  ∂ϕs = 0, + λs ∂ Ii ∂ s=1

(6)

where G—overall total costs in the power system, expressed as a function of Ii ; λs —some multipliers. The values of the currents of the original balanced mode, we substitute it into (6) and from the obtained system of linear equations, we define all the values of Ks. Since the number of functions ϕs is equal to the number of variables Ii , the assignment of the quantities ϕs determines all Ii , moreover, in the field of possible practical modes this system of values Ii for li will be unique. Therefore, the function G can be considered a function on the values of ϕs , as independent variables; the quantities ϕs , in turn, are functions from variables Ii . Then, considering Eq. (6) as a result differentiation of a complex function, we obtain that the quantities—λs are the partial derivatives function G on ϕs −λs =

∂G . ∂ϕs

(7)

According to Eq. (7), the total differential costs will be equal to: 

dG =

n 

−λs dϕs .

(8)

s=1

Therefore, if we take some small increments of ϕs parameters proportional to values of λs , i.e., ϕs = θ λs where θ is some small positive value, the same for all s, then, in accordance with (8), the change mode will go in the direction of decreasing  G normal to the equipotential flow surface in n -dimensional space. Taking successively such increments for free parameters ϕs , balancing each once the mode and defining new values of λs , we will all the time to approach the optimum, since costs will decrease all the time. Wherein, if the change in ϕs of any parameter ϕs will be have such a sign that the parameter ϕs will have to go beyond the limits set for him, then the value of this parameter for the next step calculation must be left equal to its limiting value. On the contrary, if for some function ϕs , which had the maximum or minimum limit values, sign λs , and therefore ϕs will be such that this parameter should deviate from the limit value into the acceptable for zone, then for the next step of calculation you need accept this change ϕs . So this parameter will no longer be fixed in the limit meaning and can be considered free. In the process of performing the calculation, you need to follow for the values of generating capacities selected to compensate for active and reactive losses and maintaining fixed voltages at the nodal points, as well as behind the stresses at the nodal points where they are not fixed. If some power will go beyond the permissible limits for them, you need to take their values fixed in the limiting value and take as unknown capacities any other power from the number of free. If some voltages go

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beyond the permissible limits, then there should also be fixed at the limit value and take unknowns some additional reactive power from among those that were free. The quantity θ should be chosen sufficiently small, so that at each step does not happen too much a large change in the values of λs and that they are opportunities have not changed their sign. With another hand, it should not be too small to the optimization process did not go too slowly. As we approach the optimum, some the values λs of generating capacities will keep nonzero absolute values will be have signs corresponding to the fixing of the parameter at the upper or lower limit. Other λs generating sources will decrease all the time, approaching zero. These λs will match free capacity. When all λs are free capacities will be zero, we will have an optimal mode, and since the increment costs for any change in free capacity will be equal to zero. At the same time, Eq. (6), if λs in them refer only to parameters, fixed at a given limit meaning, and will be observed corresponding to this the constraint equation. Thus, we get the correspondence with equations defining the optimum by the method of Lagrange multipliers. Signs of multipliers λs for parameters fixed in limiting values will correspond to the consolidation of each given parameter in the maximum or minimum limit values. Therefore, as shown in [50–60], there will be the optimum also for the case when the limiting conditions are set to maximum and minimum limits of possible change of parameters mode, i.e. when the constraints of the variables are given by inequalities. At the same time, in the optimal mode, it does not matter, which of the generating sources from among not fixed in the limit value were chosen to compensate for changes in losses and maintain voltages and which were considered free. If we were chosen as free other powers, then for them in optimal mode we would get λs = 0. Relatively large absolute values of λs for some fixed parameters corresponding to given regime restrictions, they say that the limitation of these parameters in the limiting value significantly impairs efficiency regime. Above, we considered the base point voltage U B given and considered it in all equations as a constant parameter. But on condition of the problem, it may be unknown and it is necessary find its economic value. Then it is necessary to introduce U B into the number of independent variables along with the currents Ii and calculate dG subject to the constraints of the regime, which will determine the derivative dU B required direction of change of value U B to approach the optimum. For this, it is necessary to draw up an additional equation connection ϕU B (U B ) = U B = U B∗ , and introduce an additional multiplier λU B , which will be equal to the required derivative ∂ϕ (U ) with the opposite sign. Insofar as in this case U∂B Ii B , all will be equal to zero, the Eq. (6), from which the values are determined λs , will not change. Now let’s establish the connection of the above method with the method of calculating the optimal mode using the principle of equality of relative gains. Currently, when calculating the optimal mode taking into account losses in networks are often used the optimum condition, which is expressed in the fact that ε for all power plants, the values or values μ = 1− j∂π of μ and ν determined by ∂P broad sense, introducing consideration of all specified restrictions regime, and thus get independent, a fairly simple calculation method, not based on the method of

Method of Regulating the Operating Modes of Main Electrical Systems …

31

Lagrange multipliers when solving the problem of economic distribution capacity without taking into account the restrictions of the regime. These quantities are given a physical interpretation as increases in expenses for individual power plants when the base load changes points due to the loading of these power plants. However, you can, on the contrary, accept the equality these relative gains as the initial criterion of the optimal regime, to give it more broad sense, introducing consideration of all specified restrictions regime, and thus get independent, a fairly simple calculation method, not based on the method of Lagrange multipliers. Indeed, if these increases in costs for power plants will be equal, then, by transferring the load from one power plant to a reference point and then from a base point to some other power plant, we end up with that increment costs in the energy system when redistributing loads between any power plants will be equal to zero. This is a sign of optimal distribution of capacities.

4 Criterion for Optimizing the Mode of the Bulk Electrical Power Network So, as a criterion for the optimal distribution powers, the equality of relative increases in costs in the energy system at change only active or only reactive base point loads by changing the active and reactive power of individual power plants, participating in the regulation of the regime, and subject to the specified mode restrictions. Let’s call them basic relative gains and denote ε B Pi and ε B Q i . This criterion is applied in [5], where there is in view of the observance of the constancy of the given voltages at several nodal points. When solving some problems, by definition the optimal mode of the power system is necessary to know the value of the basic relative gains system in optimal mode. She may be obtained directly when determining the optimal mode by the method of fixed parameters. The quantities −λ B P and −λ B Q in this method for optimal mode, according to what was said in [3] represent a change in the flow rate in the system dG with an infinitesimal change in the active or reactive load of the reference point by ∂ PB or ∂ Q B while respecting the specified mode restrictions and, therefore, are equal to the relative increments of interest to us: dG = εB P; d PB dG = = εB Q . d QB

−λ B P = −λ B Q

(9)

In contrast to the conditions corresponding to the Eq. (7), for the optimum regime according to [3] functions ϕs for free capacities can take any infinitesimal increments.

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When this power, selected for balancing losses and maintaining tensions, must take some increments meeting compliance given conditions. The increment of the function dG in this case is not depends on whether, due to the change in power, the from power plants involved in the regulation mode, subject to the specified constant voltages and powers received increments d PB or d Q B . Therefore, in the optimal mode these relative gains should be equal for all power plants that may participate in the regulation of the regime, i.e. free and allocated to balance losses and maintaining tensions. This is also evidence the validity of the criterion of equality of basic relative increments of individual power plants. Using the fixed parameter method you can determine the basic relative gains separate power plants also for non-optimal regime at given power plants, balancing mode and maintaining fixed voltage. As you know, the principle of equality of relative cost increments is used in the method of determining optimal mode by calculating variations (small increments) active and reactive capacities P j and Q j of individual power plants when balancing the mode through the basic point [61–70]. The above definition of basic relative increments allows to clarify the calculation formulas for this method. When calculating by the method of mode variations taking into account the constancy of the specified voltages in several nodal points, changes in loads on static characteristics and damage to consumers the equality of the basic relative increase in the costs of idle power plants ε B Pi and ε B Q i . They are determined by the following formulas referring respectively to variations P j when Q B = 0 and Q j when PB = 0: ε B Pi

ε B Qi

ε B Pi + dG = = d PB

ε Q j + ε pz G = = Q B



n

π Q j

Di i=1 P j + π 1 − P j

+

1−

m

Q h h=1 ε Qh P j n Pnl − j=1 P j

n

q Q j

Pnl j=1 Q j

−

n



+

n

Q nl j=1 Q j

z + ε Qz Q P j

Di i=1 Q j

+

m

+

m

Q h h=1 Q j

h=1

;

(10)

h ε Qh lQ Q j

,

(11) where ε Qh , ε Qz ε B Pi ε Q j ε pz ε Qh —relative gains costs for active and reactive capacities at power plants j, at power plant z dedicated to balanced active or reactive losses, and at power plants h allocated for regulation stresses at m points; π and q— increments of active and reactive network losses; Pnl and Q nl —increments of load of nodal points in accordance with the static characteristics; Q h —increments of reactive power plants allocated for regulation voltage; D B —incremental damage to consumers, voltage deviation nodal points from the optimal for consumers. The denominators in formulas (10) and (11) represent are the relative increments of the active and reactive loads of the base point with variations active or reactive power of power plants; they can be directly obtained when calculating the balance of the regime.

Method of Regulating the Operating Modes of Main Electrical Systems …

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5 Conclusions The chapter studies the optimization matters in terms of power systems, such as optimal active and reactive power distribution between the generators of power stations, optimal condition calculation of power and voltage losses, finding optimal power system equipment configuration. The ways of solution of the set tasks are introduced using mathematical methods adapted to the electric power systems conditions. It is concluded that the approaches to the optimization problems solution are aimed at the control of combined heat and power plants, and very few research under the conditions of industrial power systems have been conducted so far. The above method for determining the factors and their corresponding gradient components costs in the power system implemented in developed program for calculating the optimal power system mode, and its application gives positive results. The developed program for calculating the optimal power system mode compiled for power systems with the number of nodes up to 1000, the number of branches up to 2000 and the number of power plants up to 500. Optimization produced either by minimizing losses in the network, or at a minimum fuel consumption at power plants, or at a minimum cost consumed fuel. The regime equations are solved by Newton’s method. The features of the development of power electronics are considered, the main characteristics of power semiconductor devices and their comparative assessment was carried out. It is shown that on the basis of power semiconductor devices an element base for powerful converters without compulsory switching thyristors, which does not cause distortion of the waveform generated current at the operating frequency. This element base is the basis construction of converters for electric power purposes, including the number of used in “flexible” power lines, in the inserts of constant current, static compensators of inactive power, flow regulators energy.

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41. Dias, O., Tavares, M.C., Magrin, F.: Hardware implementation and performance evaluation of the fast adaptive single-phase auto reclosing algorithm. Electr. Power Syst. Res. 168, 169–183 (2019) 42. Ortega, J.S., Tavares, M.C.: Transient analysis and mitigation of resonant faults on halfwavelength transmission lines. IEEE Trans. Power Delivery 35(2), 1028–1037 (2020). https:// doi.org/10.1109/TPWRD.2019.2935870 43. Mestas, P., Tavares, M.C.: Effects of arc extinction time reduction during three-phase reclosing. In: IEEE PES Transmission & Distribution Conference and Exhibition - Latin America (T&DLA), Lima, 2018, pp. 1–5 (2018). https://doi.org/10.1109/TDC-LA.2018.8511721 44. Acosta, J.S., Tavares, M.C.: Enhancement of overhead transmission line capacity through evolutionary computing. In: IEEE PES Transmission & Distribution Conference and Exhibition - Latin America (T&D-LA), Lima, 2018, pp. 1–5 (2018). https://doi.org/10.1109/TDC-LA. 2018.8511761 45. de Mattos, L.M.N., Tavares, M.C., Mendes, A.M.P.: A new fault detection method for singlephase autoreclosing. IEEE Trans. Power Delivery 33(6), 2874–2883 (2018). https://doi.org/10. 1109/TPWRD.2018.2855105 46. Dias, O., Magrin, F., Tavares, M.C.: Comparison of secondary arcs for reclosing applications. IEEE Trans. Dielectr. Electr. Insul. 24(3), 1592–1599 (2017). https://doi.org/10.1109/TDEI. 2017.006188 47. Sarmiento, J.S.A., Tavares, M.C.: Enhancement the overhead transmission lines’ capacity by modifying the bundle geometry using heuristics algorithms. In: 2016 IEEE PES Asia-Pacific Power and Energy Engineering Conference (APPEEC), Xi’an, 2016, pp. 646–650 (2016). https://doi.org/10.1109/APPEEC.2016.7779583 48. Tavares, M.C., Talaisys, J., Camara, A.: Voltage harmonic content of long artificially generated electrical arc in out-door experiment at 500 kV towers. IEEE Trans. Dielectr. Electr. Insul. 21(3), 1005–1014 (2014). https://doi.org/10.1109/TDEI.2014.6832243 49. Garcia, A., Tavares, M.C.: Long arcs in free air: stationary parameters for secondary arc current range. In: Power Systems Computation Conference, Wroclaw, 2014, pp. 1–6 (2014). https:// doi.org/10.1109/PSCC.2014.7038501 50. Djuric, M., Terzija, C.: A new approach to the arcing faults detection for fast autoreclosure in transmission systems. IEEE Trans. Power Delivery 10(4), 1793–1798 (1995) 51. Bo, Z., Johns, A., Aggarwal, R., Goody, J., Gwyn, B.: Digital simulation of an EHV transmission system for design and real time testing of new protection. In: First International Conference on Digital Power System Simulators, p. 57. IEEE, New York (1995) 52. Boisseau, A.C., Wyman, B.W., Skeats, W.F.: Insulator flashover deionization times as a factor in applying high-speed reclosing circuit breakers. Trans. Am. Inst. Electr. Eng. 68(2), 1058–1067 (1949) 53. Giesbrecht, J., Ouellette, J., Henville, J.: Secondary arc extinction and detection real and simulated. In: 2008, IET 9th International Conference on Developments in Power System Protection, pp. 138–143 (2008) 54. Luxenburger, R., Schegner, P.: Determination of secondary arc extinction time and characterization of fault conditions of single-phase autoreclosures. In: 2005 International Conference on Future Power Systems, 2005, pp. 5–10 (2005) 55. Milne, K.: Single-pole reclosing tests on long 275-Kv transmission lines. IEEE Trans. Power Appar. Syst. 82(68), 658–661 (1963) 56. Kizilcay, M., Pniok, T.: Digital simulation of fault arcs in power systems. Euro. Trans. Electr. Power 1(1), 55–60 (1991) 57. Tavares, M., Talaisys, J., Portela, C., Camara, A.: Harmonic content and estimation of length variation of artificially generated electrical arc in out-door experiments. In: IEEE Electrical Power and Energy Conference, pp. 346–351. IEEE, New York, Oct. 2011 (2011) 58. Camara, A.S., Portela, C.M., Tavares, M.C.: Single-phase autoreclosure studies considering a robust and reliable secondary arc model based on a gray-box model. In: International Conference on High Voltage Engineering and Application, pp. 486–489. IEEE, New York, Nov. 2008 (2008)

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59. Camara, A.S.B., Portela, C.M.J.C., Tavares, M.C.: Single-phase auto-reclosure studies: some basic aspects on main elements representation. In: International Conference on High Voltage Engineering and Application, pp. 482–485, Nov. 2008 (2008) 60. Santos, G., Tozzi, C., Tavares, M.: Visual evaluation of the length of artificially generated electrical discharges by 3D-snakes. IEEE Trans. Dielectr. Electr. Insul. 18(1), 200–210 (2011) 61. Levytskyi, A.S., Zaitsev, I.O., Kobzar, K.O.: Measuring the stroke of cone disk springs in power accumulators of the turbogenerator stator core using a capacitive sensor. Devices Methods Measurements 9(2), 121–129 (2018). https://doi.org/10.21122/2220-9506-2018-9-2-121-129 62. Zaitsev, Ie., Shpylka, A., Shpylka, N.: Output signal processing method for fiber Bragg grating sensing system. In: Proceedings of the 15th International Conference on Advanced Trends in Radioelectronics, Telecommunications and Computer Engineering (TCSET-2020). LvivSlavske (Ukraine), pp. 152–155 (2020). https://doi.org/10.1109/TCSET49122.2020.235412 63. Zaitsev, Ie.O., Levytskyi, A.S., Sydorchuk, V.E..: Air gap control system for hydrogenerators. Devices Methods Measurements 8(2), 122–130 (2017). https://doi.org/10.21122/2220-95062017-8-2-122-130 64. Zaitsev, Ie.O., Levytskyi, A.S., Novik, A.I., Bereznychenko, V.O., Smyrnova, A.M.: Research of a capacitive distance sensor to grounded surface. Telecommun. Radio Eng. 78(2), 173–180 (2019). https://doi.org/10.1615/TelecomRadEng.v78.i2.80 65. Levytskyi, A.S., Zaitsev, I.O., Bereznychenko, V.O., Sukhorukova, O.E.: Measuring transducer for air gap capacitive sensor in hydrogenerator. Devices Methods Measurements 11(1), 33–41 (Rus) (2020). https://doi.org/10.21122/2220-9506-2020-11-1-33-41 66. Zaitsev, Ie., Panchyk, M.V.: Physical processes and their influence on the development of defects in the stator core of powerful generators. Sci. Educ. New Dimension. Natural Tech. Sci. 224, 81–84 (2020). https://doi.org/10.31174/SEND-NT2020-224VIII27-20 67. Zaitsev, Ie.,Levytskyi, A., Kromplyas, B., Panchyk, M., Bereznychenko, V.: Study influence industrial frequency magnetic field on capacitive pressing sensor for large turbogenerator core clamping system. In: Proceedings of the 2019 IEEE Ukraine International Conference on Electrical and Computer Engineering (UKRCON-2019), 2 – 6 Jule, Lviv (Ukraine), pp. 566– 569 (2019). https://doi.org/10.1109/UKRCON.2019.8879949 68. Zaitsev, Ie.O., Levytskyi, A.S., Kromplyas, B.A.: Capacitive distance sensor with coplanar electrodes for large turbogenerator core clamping system. In: Proceedings of the 2019 IEEE 39th International Conference on Electronics and Nanotechnology (ELNANO), April 16–18, Kiev (Ukraine), pp. 644–647 (2019). https://doi.org/10.1109/ELNANO.2019.8783916 69. Zaitsev, I.O., Levytskyi, A.S., Kromplyas, B.A.: Hybrid capacitive sensor for hydro- and turbo generator monitoring system. In: Proceedings of the International conference on modern electrical and energy system (MEES-17) November 15–17, Kremenchuk (Ukraine), pp. 288–291 (2017). https://doi.org/10.1109/MEES.2017.8248913 70. Zaitsev, I.O., Levytskyi, A.S.: Determination of response characteristic of capacitive coplanar air gap sensor. In: Proceedings of the 2017 IEEE Microwaves, Radar and Remote Sensing Symposium (MRRS-2017), August 29–June 30, Kyiv (Ukraine), pp. 85–88 (2017). https://doi. org/10.1109/MRRS.2017.8075034 71. Zaitsev, Ie.O, Levytskyi, A.S.: Hybrid electro-optic capacitive sensors for the fault diagnostic system of power hydrogenerator. In: Ebrahimi, A. (ed.) Clean Generators - Advances in Modeling of Hydro and Wind Generators, pp. 25–42. Intechopen (2020). https://doi.org/10. 5772/intechopen.77988 72. Levytskyi, A., Zaitsev, Ie.: Hybrid fiber-optic measuring tools for control and diagnostic parameters of hydrogenerators. Hydropower Ukraine 3–4, 32–33 (Ukr) (2016); Zaitsev, I.O., Levytskyi, A.S., Sydorchuk, V.E.: Air gap control system for hydrogenerators. Devices Methods Measurements 8(2), 122–130 (Rus) (2017). https://doi.org/10.21122/2220-9506-2017-8-2122-130 73. Bereznychenko, V.O., Zaitsev, I.O.: Contactless capacitive sensor of the system for monitoring the parameters of the beating of the powerful electrical machines shafts Works of the Institute of Electrodynamics of the National Academy of Sciences of Ukraine 57, 81–88 (Ukr) (2021). https://doi.org/10.15407/publishing2020.57.081

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74. Zaitsev, I., Levytskyi, A., Bereznychenko, V.: Development shaft run-out measurement transducers for powerful generators fault control system with capacitive coplanar concentric sensor. In: Proceedings of the 1st International Scientific and Practical Conference Theory and Practice of Science: Key Aspects. Rome (Italy), pp. 1014–2021 (2021). https://doi.org/10.51582/interc onf.19-20.02.2021.103 75. Zaitsev, Ie., Levytskyi, A.S., Kromplyas, B.A., Panchyk, M.V., Bereznychenko V.O.: Study industrial frequency magnetic field influence on STM32F051K8T6 microcontroller functioning stability. Works of the Institute of Electrodynamics of the National Academy of Sciences of Ukraine 52, 80–86 (Ukr) (2019). https://doi.org/10.15407/publishing2019.52.080 76. Baranov, G., Komisarenko, O., Zaitsev, Ie.O., Chernytska, I.: S.M.A.R.T. technologies for transport tests networks, exploitation and repair tools. In: Proceedings of the International Conference Artificial Intelligence and Smart Systems (ICAIS) 25–27, March 2021, Pichanur (India), pp. 621–625 (2021). https://doi.org/10.1109/ICAIS50930.2021.9396055 77. Zaitsev, Ie., Levytskyi, A., Bereznychenko, V.: Hybrid diagnostics systems for power generators faults: systems design principle and shaft run-out sensors. In: Power Systems Research and Operation: Selected Problems. Springer, Berlin (2021) 78. Rezinkina, M.M., Sokol, Y.I., Zaporozhets, A.O., Gryb, O.G., Karpaliuk, I.T., Shvets, S.V.: Physical modeling of the electrophysical processes of the formation of the corona during the operation of electric power facilities. In: Sokol, Y.I., Zaporozhets, A.O. (eds.) Control of Overhead Power Lines with Unmanned Aerial Vehicles (UAVs). Studies in Systems, Decision and Control, vol. 359. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-69752-5_8 79. Rezinkina, M.M., Sokol, Y.I., Zaporozhets, A.O., Gryb, O.G., Karpaliuk, I.T., Shvets, S.V.: Mathematical models of electric fields of electric transmission lines. In: Sokol, Y.I., Zaporozhets, A.O. (eds.) Control of Overhead Power Lines with Unmanned Aerial Vehicles (UAVs). Studies in Systems, Decision and Control, vol. 359. Springer, Cham (2021). https:// doi.org/10.1007/978-3-030-69752-5_5

Heat Power Engineering

Creation of High-Speed Methods for Solving Mathematical Models of Inverse Problems of Heat Power Engineering Artur Zaporozhets , Vladyslav Khaidurov , and Tamara Tsiupii

Abstract Experimental methods for studying thermophysical processes and their systems are the most reliable source of information about the thermal state of an object. The theory and practice of solving inverse problems acquire special significance in our time, when, due to the current circumstances, energy facilities, industry, transport, economy, communications, buildings and structures have worked out their planned resource and need to prevent man-made disasters, namely, they need urgent updating, modernization and reconstruction. The main problems in solving applied inverse heat conduction problems include the correctness of the formulation of the problem, the time of solving systems of algebraic equations for a discrete analogue of the heat conduction equation, the amount of calculations in the optimization of a quadratic functional in inverse problems (the number of calculations per iteration), stability and convergence of unconstrained optimization methods. In this chapter, a numerical analysis of series of methods of unconstrained optimization on nonlinear problems is carried out, which make it possible to reduce the search time for a numerical solution to a large class of nonlinear inverse heat conduction problems. Also, significant modifications of the classical Newton’s method are proposed, which make it possible to obtain the desired numerical solution of nonlinear problems much faster than the classical Newton’s method. Modifications of Newton’s method with a variable step give a much faster convergence and make it possible to obtain fairly quickly solutions to nonlinear inverse heat conduction problems. Keywords Optimization mathematical models · Newton type methods · Inverse heat conduction problems · Simulation · MATLAB A. Zaporozhets (B) · V. Khaidurov Institute of General Energy of NAS of Ukraine, Kyiv, Ukraine e-mail: [email protected] A. Zaporozhets State Institution “The Institute of Environmental Geochemistry of National Academy of Sciences of Ukraine”, Kyiv, Ukraine T. Tsiupii National University of Life and Environmental Sciences of Ukraine, Kyiv, Ukraine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Zaporozhets (ed.), Systems, Decision and Control in Energy III, Studies in Systems, Decision and Control 399, https://doi.org/10.1007/978-3-030-87675-3_3

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1 Introduction Many important questions of heat transfer are reduced to solving inverse problems of heat conduction. Obviously, to solve the non-stationary classical problem of heat transfer, it needs to know the initial conditions, boundary conditions, physical parameters of the medium, etc. [1–4]. Having all the necessary values, it is possible to calculate the temperature field in the investigated area. Such problem is called direct. Another situation is also possible. The temperature field is known, but the values of certain defining parameters that characterize the specific environment in which the technical thermal process takes place are not known. In this case, it is necessary to restore the values of the undefined parameters of the medium using the temperature field and the known values of some of the important parameters. Such problems are inverse. For example, in a non-stationary problem, it is necessary to find the initial conditions using the temperature field measured after some time of the experiment’s start, or to find the missing boundary conditions using the stationary temperature field. Experience shows that inverse heat conduction problems are very resourceintensive problems, their solution requires a lot of computing power and significant processor time [5–9]. This chapter is devoted to the study of possible ways to speed up the procedure for obtaining solutions to inverse heat conduction problems.

2 Formulation of the Inverse Heat Conduction Problem Let the heat equation is given in the area D × [0, τ ], D ⊂ Rn [5, 8–13]: ρC

∂T = ∇(k∇T ). ∂t

(1)

Complete solution to the heat conduction problem contains the following data: • Temperature value at all internal points of the computational area of the problem, that is ∀x ∈ D,∀t ∈ [0, τ ],T (x, t)—known distribution. This means that the known distributions at the initial moment of time and are: Tinit (x) = T (x, 0),

(2)

and at the final moment of time T f in (x) = T (x, τ ); • Boundary conditions: ∂ D = ∂ DD + ∂ DN , x ∈ ∂ DD ;

(3)

Creation of High-Speed Methods for Solving Mathematical …

T = T (x, τ ); x ∈ ∂ D N : ∂ T /∂ n = p(x, τ ),

43

(4)

where ∂ D—boundary of the computational area; ∂ D D —part of the boundary (possibly empty) on which a condition of the first kind is specified (Dirichlet condition); ∂ D N —part of the boundary on which a condition of the second kind is → specified (Neumann condition); − n —normal to the boundary of the computational area of the problem; • Dependence of the problem parameters on coordinates and temperature: ρ = ρ(x, T ), C = C(x, T ), k = k(x, T ),

(5)

where ρ—object’s density; C—object’s specific heat; k—thermal diffusivity. Taking into account the above data on the complete solution of the heat conduction problem, it is possible to formulate a direct problem for the heat conduction equation in the D × [0, τ ] area. Equation (1), initial temperature distribution (2), boundary conditions (4), dependence of parameters on coordinates and temperature (5) are given. It is necessary to find the final temperature distribution (3) and, as an intermediate result, the temperature value at all internal points of the computational area for Eq. (1) at ∀t ∈ [0, τ ]. The inverse problem of thermal conductivity is considered solved if it is found such initial distribution of the temperature Tinit (x) = T (x, 0), that  thesolution of the direct problem using the initial condition of this distribution T Tinit coincides with the desired final distribution T f in . Therefore, it is necessary to find the initial temperature distribution Tinit such that       J Tinit = T Tinit − T f in  → min .

(6)

The L 2 norm is used as the norm in functional (6), that is,    x =  x 2 d D, D

or in the discrete case x =



xi2 .

  Obviously, for a given initial distribution Tinit the value T Tinit is obtained as a result of solving the differential Eq. (1). To shorten the entries, we use the notation for Eq. (1): L(T, θ ) = 0.

(7)

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In (7), it is emphasized that the solution was obtained using the sought-for parameter L(T, θ ) = 0, in our case with the initial condition θ = T (x, 0) = Tinit (x). It could be said that differential Eq. (7) is a constraint for (6). That is, taking into account the modern classification of optimization problems, we are dealing with the problem of finding the global minimum of functional (6) with a constraint in the form of a differential Eq. (7). The problems of finding an extremum with constraints are written using Lagrange multipliers, that is, the problem is posed as a problem for an unconditional extremum 



2 T (θ ) − T f in d D + λL(T, θ ) → min,

D

therefore, it is necessary to find the initial temperature distribution θ = Tinit such that the T (θ ) distribution gives a minimum to the functional:  J (θ ) =



2 T (θ ) − T f in d D → min,

(8)

D

and for arbitrary initial conditions θ = Tinit the value T (θ ) in (8) is found as a result of solving the differential Eq. (7). The second problem in the chapter is the problem of identifying the boundary condition. Let Eq. (1), initial temperature distribution (2), final temperature distribution (3), boundary conditions on a part of the boundary in the form (4), dependence of parameters on coordinates and temperature (5) be given. It is necessary to find the boundary conditions (4) on that part of the boundary of the computational area, on which they are not specified (unknown) and, as an intermediate result, the temperature value at all internal points of the area. The inverse problem of thermal conductivity is considered solved if such a temperature distribution or temperature flow is found at the area’s boundary that the solution of the direct problem using the found boundary conditions (T (θbound ) distribution) coincides with the required finite distribution T f in .

3 Methods for Solving Conditional Optimization Problems As it was noted earlier, the general formulation of inverse heat conduction problems provides for finding the extremum of the functional under the condition of constraints in the form of differential equations of mathematical physics. Before solving such problems, we describe methods for solving problems with constraints. Let a simple conditional extremum problem be given. It is necessary to find x0 , y0 : F(x0 , y0 ) → max

(9)

Creation of High-Speed Methods for Solving Mathematical …

45

under the condition L(x0 , y0 ) = 0.

(10)

Let us offer a solution to problems (9) and (10) in different ways. First method. If condition (10) can be transformed to the form y = ϕ(x),

(11)

then we substitute (11) into Eq. (9). As a result of these transformations, the problem of minimizing a function of one variable is obtained. It can be formulated as follows: x0 : F(x0 , ϕ(x0 )) → max .

(12)

Then the solution to the posed problem (9) and (10) is the equation: ∂ F dϕ ∂F (x0 ) + (x0 ) = 0. ∂x ∂ϕ d x

(13)

So, one constraint reduces the number of independent variables in (9) by one unit. But in most real scientific and technical problems, condition (10) cannot be solved with respect to y. Therefore, we describe the second method of the procedure for finding the extremum of a function with a constraint in the form of equations. Second method. Let’s find ∂ϕ/∂ x as a derivative of an implicit function ∂y ∂ L/∂ x ∂ϕ = =− . ∂x ∂x ∂ L/∂ y Then Eq. (13) will be rewritten as follows: ∂F ∂ F ∂ L/∂ x (x0 , y0 ) − (x0 , y0 ) = 0. ∂x ∂ y ∂ L/∂ y

(14)

Now we will describe the third method to solve the problem. Third method. We rewrite (13) as follows: 

 ∂F ∂ F ∂φ ∂ F dy ∂F (x0 ) + (x0 ) = 0 = (x0 , y0 ) + (x0 , y0 ) = 0 ∂x ∂φ ∂ x ∂x ∂y dx  ∂F ∂F = (x0 , y0 )d x + (x0 , y0 )dy = 0 . ∂x ∂y

From Eq. (10) obviously follows the equation:

d L(x0 , y0 ) =

 ∂L ∂L dx + dy (x0 , y0 ) ∂x ∂y

(15)

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=

∂L ∂L (x0 , y0 )d x + (x0 , y0 )dy = 0. ∂x ∂y

(16)

Let’s calculate (15) + λ (16). As a result, we get: 

∂F ∂F (x0 , y0 )d x + (x0 , y0 )dy = 0 ∂x ∂y  ∂L ∂L +λ (x0 , y0 )d x + (x0 , y0 )dy = 0 ∂x ∂y  ∂L ∂F = (x0 , y0 ) + λ (x0 , y0 ) d x ∂x ∂x  ∂F ∂L + (x0 , y0 ) + λ (x0 , y0 ) dy = 0. ∂y ∂y

The increments dx and dy are arbitrary. Therefore, expressions in parentheses must equal zero. Let us find λ, so that ∂F ∂L (x0 , y0 ) + λ (x0 , y0 ) = 0. ∂y ∂y

(17)

Then we get the equation: ∂F ∂L (x0 , y0 ) + λ (x0 , y0 ) = 0. ∂x ∂x

(18)

Obviously, Eqs. (17) and (18) are the conditions for the extremum of the function: F(x, y) + λL(x, y). So, we got the Lagrange multipliers. It should be noted that all three methods described above are equivalent. During solving problems for a conditional extremum, it can be used any of them. Now let’s look at some examples. It is necessary to find the maximum of the function: max F(x, y) = x y

(19)

L(x, y) = x 3 + y 3 − 2 = 0.

(20)

with condition:

First method. It follows from Eq. (20): 1/3  . y = ϕ(x) = 2 − x 3

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47

Substitute the resulting equation into Eq. (19). We get the following problem: it 1/3  . According to the condition for the is necessary to find max F(x) = x 2 − x 3 existence of an extremum, it follows:     1 −3x 2 1 − x3 3 1/3 F = 2−x +x  (21) 2/3 = 2  2/3 = 0 → x = 1. 3 2 − x3 2 − x3  1/3 This means that the maximum of the y = 2 − x 3 = 1 is found at x = 1, that is, the function F(x, y) is also equal to 1. It should be noted that if we consider the function F(x) = xϕ(x), where ϕ(x) = 1/3  , then Eq. (21) will have the form 2 − x3     dϕ 1 −3x 2 ∂F 3 1/3 ∂ F = ϕ(x) = 2 − x = x, and =  ,  . ∂x ∂ϕ dx 3 2 − x 3 2/3 The obtained result fully corresponds to the solution of the problem and is similar to expression (13). Second method. In the first method, it was believed that constraint L(x, y) = 0 can be represented in the form y = ϕ(x). Then one variable of the problem can be excluded. Now we believe that it is impossible to solve this restriction with respect to y, as before, so we have   dy 1 −3x 2 dϕ = =   . dx dx 3 2 − x 3 2/3 Let’s find the derivative of the last equation as the derivative of an implicitly given function:  ∂L ∂x dy =−  . dx ∂ L ∂y We have L(x, y) = x 3 + y 3 − 2. Then ∂L ∂L = 3x 2 , = 3y 2 . ∂x ∂y   Given the fact that ∂ F ∂ x = y or ∂ F ∂ y = x, Eq. (18)  ∂ F ∂L ∂x ∂F  =0 − ∂x ∂y ∂ L ∂y will take the form

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y−x

3x 2 = 0. 3y 2

From this we have y 3 = x 3 or y = x. Substituting this relation into Eq. (20), we obtain 3x 3 = 3 or x = y = 1. The difference with the previous method is that at first we removed one variable and, thus, lost the constraint. Further, the problem was solved for an unconditional extremum of a function of one variable. In the second method, you cannot get rid of one variable. This means that the constraints remain in the problem. By constraint, the derivative of one variable with another is found as the derivative of an implicit function. Having found this derivative, we substitute it into the objective function and use the result for obtaining one variable through another. The final values of the variables are found by the constraint of the problem. That is, the system is solved in this way. Third method. Lagrange multipliers are a convenient formalization of the second method—the informal actions of representing one variable through another are removed with the subsequent substitution of the result in the objective function.

4 Problems with Constraints in the Form of Differential Equations Let the problem be formulated – it is necessary to find max F(x, y) = x y.

(22)

We write the condition L(x, y) = x + y − 2 = 0 or the same y = 2 − x in the form of a differential equation. Taking into account y = −1, y = 0, we will have y − y = x − 2. The boundary conditions have the form: y(−2) = −4, y(2) = 0. Similarly, we believe that it is impossible to find an analytical solution to the problem. Here again the problem is solved: it is necessary to find the max F(x, y) = x

x y. Previously, the analytic expression y = y(x) was substituted. Now it’s necessary to solve the differential equation, memorize the result and find y from the given x. The extremum will be found numerically, for example, by Newton’s method.

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4.1 Parameter-Dependent Differential Equations It is necessary to find a parameter of the differential equation a such that the definite integral of the solution of this equation has an extreme value: 1 y(x, a)d x → extr emum.

a : F(a) = 0

Let the solution to the differential equation have the form:   y(x, a) = ax(1 − x)(a − x) = a x − x 2 (a − x)     = a ax − ax 2 − x 2 + x 3 = a x 3 − x 2 (a + 1) + ax . Then 1



  a2 + a a2 1 1 1 a 2 1 + =a − −a − a x − x (a + 1) + ax d x = − 4 3 2 2 3 3 4 

3

2

0

=

a a2 − . 6 12

1 Extremum is: 2a − 12 = 0 → a = 41 . 6 Let’s take an arbitrary equation that has the indicated solution. The boundary conditions are as follows: y(0, a) = y(1, a) = 0. Then

  y = a 3x 2 − 2x(a + 1) + a , y = a(6x − 2(a + 1)). Then, 3y x y − y − x 

 3   = a x(6x − 2(a + 1)) − 3x 2 − 2x(a + 1) + a − x 3 − x 2 (a + 1) + ax x   2 2 2 = a 6x − 2x(a + 1) − 3x + 2x(a + 1) − a − 3x + 3x(a + 1) − 3a   = a x 2 (6 − 3 − 3) + x(−2(a + 1) + 2(a + 1) + 3(a + 1)) − a − 3a = a(3x(a + 1) − 4a) = 3xa(a + 1) − 4a 2 . As a result, we get a differential equation that depends on the parameter: x y − y −

3y − 3xa(a + 1) + 4a 2 = 0. x

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The final formulation of the optimization problem. It is necessary to find such a value for a, at which: 1 a : F(a) =

y(x, a)d x → extr emum,

(23)

0

where y(x, a)—solution of the differential equation: 3y − 5xa(a + 1) + 4a 2 = 0, moreover x ∀a y(0, a) = y(1, a) = 0.

x y − y −

(24)

4.2 Solving Technique by Newton’s Method Instead of (24), we solve the equation: ϕ(a) =

dF = 0. da

(25)

We carry out linearization (25) by Newton’s method: ϕ(ai+1 ) = ϕ(ai + ai ) = ϕ(ai ) + ai

dϕ d2 F dF (ai ) = (ai ) + ai 2 (ai ) = 0. da da da

From here we get:  

dF 2 (a ) i d F ai = − da , 2 (ai ) da

ai+1 = ai + ai . Derivatives are calculated as follows: dF F(ai ) − F(ai−1 ) , (ai ) = da ai−1

 F(ai ) − F(ai−1 ) d2 F F(ai−1 ) − F(ai−2 ) 2 . − = (a ) i da 2 ai−1 + ai−2 ai−1 ai−2 Now let’s move on to solving an extremal problem with a constraint in the form of a differential equation in two parameters.

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51

Dependence on two parameters. The problem for an extremum of two parameters can be formulated as follows: it is necessary to find 1 y(x, a, b)d x → extr emum,

a, b : F(a, b) =

(26)

0

where y(x, a, b)—solution of the differential equation: L(y, a, b) = 0.

(27)

Instead of (27), we solve a system of equations of the following form: dF = 0, da dF ϕ2 (a, b) = = 0. db ϕ1 (a, b) =

(28)

Next, we linearize system (28) according to Newton’s method as follows: ϕ1 (ai+1 , bi+1 ) = ϕ1 (ai + ai , bi + bi ) ∂ϕ1 ∂ϕ1 = ϕ1 (ai , bi ) + ai (ai , bi ,) + bi (ai , bi ) ∂a ∂b ∂2 F ∂2 F ∂F = (ai , bi ) + ai 2 (ai , bi ) + bi (ai , bi ) = 0. ∂a ∂a ∂a∂b Similarly, for the second Eq. (28) we have: ∂2 F ∂2 F ∂F (ai , bi ) + ai (ai , bi ) + bi 2 (ai , bi ) = 0. ∂b ∂b∂a ∂b Now we have the following system of equations: ∂2 F ∂2 F ∂F , b + b (a ) (ai , bi ) = − (ai , bi ), i i i 2 ∂a ∂a∂b ∂a d2 F ∂2 F ∂F ai (ai , bi ) + bi 2 (ai , bi ) = − (ai , bi ). ∂b∂a ∂b ∂b

ai

The resulting system can be written in matrix form as follows: ⎛ ⎝

⎞⎛

∂2 F ∂2 F , bi ) ∂a∂b a (ai , bi ) da 2 (ai ⎠⎝ i

2 ∂2 F , bi ) ∂∂bF2 (ai , bi ) ∂b∂a (ai

Then we get

bi

⎞

⎛

⎠ = −⎝

⎞

∂F , bi ) ∂a (ai ⎠ ∂F , bi ) ∂b (ai

.

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⎛ ⎝

ai bi

⎞

⎛

⎠ = −⎝

⎞−1 ⎛ ⎞ ∂2 F ∂2 F ∂F , b , b , b (a ) (a ) (a ) i i i i i i 2 ∂a ∂a∂b ⎠ ⎝ ∂a ⎠

2 ∂2 F , bi ) ∂∂bF2 (ai , bi ) ∂b∂a (ai

∂F , bi ) ∂b (ai

.

Let’s take into account that ai+1 = ai + ai , bi+1 = bi + bi . Derivatives of the first and second orders with respect to the required parameters are as follows: ∂F F(ai , bi ) − F(ai−1 , bi ) , (ai , bi ) = ∂a ai−1 ∂F F(ai , bi ) − F(ai , bi−1 ) , (ai , bi ) = ∂b bi−1

F(ai , bi ) − F(ai−1 , bi ) d2 F 2 , b = (a ) i i 2 da ai−1 + ai−2 ai−1  F(ai−1 , bi ) − F(ai−2 , bi ) , − ai−2 ∂2 F (ai , bi ) ∂a∂b

 F(ai , bi ) − F(ai , bi−1 ) 1 F(ai−1 , bi ) − F(ai−1 , bi−1 ) = − ai−1 bi−1 bi−1 1 = (F(ai , bi ) − (F(ai , bi−1 ) + F(ai−1 , bi )) + F(ai−1 , bi−1 )), ai−1 bi−1 ∂2 F (ai , bi ) ∂b2

 F(ai , bi ) − F(ai , bi−1 ) 2 F(ai , bi−1 ) − F(ai , bi−2 ) , = − bi−1 + bi−2 bi−1 bi−2 ∂2 F (ai , bi ) ∂b∂a

 F(ai , bi ) − F(ai−1 , bi ) 1 F(ai , bi−1 ) − F(ai−1 , bi−1 ) = − bi−1 ai−1 ai−1 1 = (F(ai , bi ) − (F(ai−1 , bi ) + F(ai , bi−1 )) + F(ai−1 , bi−1 )). ai−1 bi−1 From the last transformations it can be seen that the mixed derivatives coincide, that is, the Hessian matrix is symmetric. To calculate all derivatives, it’s necessary to have a set of 6 values: F(ai , bi ), F(ai−1 , bi ), F(ai−2 , bi ), F(ai , bi−1 ),F(ai−1 , bi−1 ), F(ai , bi−2 ). In a new iteration step, it’s necessary to have

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a set of 6 values:F(ai+1 , bi+1 ), F(ai , bi+1 ), F(ai−1 , bi+1 ), F(ai+1 , bi ),F(ai , bi ), F(ai+1 , bi−1 ).

5 Effective Methods for Solving Inverse Problems Numerous solutions of the problems considered in the section are obtained by the methods of finite differences and finite elements [9, 14–20]. We will minimize functional (8) by solving the system of equations: 

∂J = 0 , i = 1, N , ∂(θi )

(29)

where θi —the required temperature value at the i-th node of the computational grid; N—total number of nodes in which it’s necessary to find the temperature value. System (29) is solved by Newton’s method and its modifications. Then the optimization problem can be posed as follows. Among the numerous existing optimization methods of Newton’s type, it is necessary to choose the most effective ones for finding solutions to inverse problems and to reduce the number of calls in them to the procedure for solving the direct problem of heat conduction. Example Let f (x) → min, x ∈ R. Let us first apply the method of steepest descent to the unconstrained optimization problem. In the direction of descent, it can be chosen to move along the antigradient, for example, like this [5, 9–11, 17–20]:   x (k+1) = x (k) − h k f x (k) , h k > 0, k = 0, 1, 2, ... . The problem is to find at each step such a h k value that minimizes the function for a certain value of x (k) , which is taken from the previous iteration. The search for such h k value can be implemented using any one-dimensional unconstrained optimization methods. The value of the h k ∈ R 1 parameter is taken from the condition of the minimum of the function f (x) in the direction of movement of the antigradient’s movement:       f x (k) − h k · ∇ f x (k) = min f x (k) − h · ∇ f x (k) , h > 0. Taking into account the strategy of finding the global minimum in the onedimensional case, we can write an iterative formula for finding the global minimum of a functional of type (8):       (k) (k) (k) (k) = min J Tinit , h > 0. − h k · ∇ J Tinit − h · ∇ J Tinit J Tinit So, using approximations to the initial condition, it can be refined using method of the steepest descent.

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5.1 Modifications of the Classical Newton Method Methods of thistype  replace the procedure for solving a nonlinear system of equations → x = Fi (x1 , ..., xn ) = 0 by a procedure for finding a solution to a of the form Fi − sequence of systems of algebraic equations of the form: ⎛  ⎞ −  ⎞⎛ k+1 ⎞ → → F1 − x1 − x1k xk xk ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎟ = −⎜ ... ⎟. ⎜ ... ... ⎟⎜ ... ... ⎝ ⎠ ⎝ ⎠⎝ ⎠  k  →k  −  → ∂ Fn ∂ Fn − k+1 k k x  ... x x − x x F n n n ∂ x1 ∂ xn ⎛

∂ F1 ∂ x1

 k x ...

∂ F1 ∂ xn

(30)

Note that it is necessary to find a solution to the system of the form (29), where the unknown parameters are the values of the initial condition at the nodes of the computational grid. So, the matrix of system (30) will be the Hessian of the original system, that is, the matrix consisting of the second derivatives of the equations of the system: ⎛ ⎜ ⎜ ⎜ ⎜ ⎝

∂ 2 J1 2 ∂ Tinit

1



k Tinit

... ∂ 2 Jn ∂ Tinitn ∂ Tinit1





...

∂ 2 J1 ∂ Tinit1 ∂ Tinitn

...

 k Tinit ...



⎞ ⎟ ⎟ ⎟. ⎟  ⎠

k Tinit

...  k 2 ∂ Jn Tinit ∂T 2

(31)

initn

Since for calculating the Hessian in (31), differentiation is decoupled with respect to the initial condition, then, in the general case, it is simply impossible to find the derivative analytically in such problems. It must be found numerically using the truncated Taylor series. For the numerical search for the second derivative, it is necessary to have the value of the differentiated function at three points, and for the search for the mixed derivative—at four points. Let’s consider a simple case. Let it be required to solve a nonlinear equation of the F(x) = 0 form. The classical Newton’s method, which is used to solve this equation, has the following form: xk+1 = xk −

F(xk ) . F (xk )

(32)

The accuracy of the method (32) can be improved by using this approach. Let it be required to solve a differential equation of the y = f (x) form. Integrating both parts of it, we get the expression:

Creation of High-Speed Methods for Solving Mathematical …

⎧x ⎨ k+1 ⎩

y d x =

xk

xk+1 f (x)d x xk

=

⎧ ⎨ ⎩

⎫ ⎬ ⎭

=

⎧ ⎨ ⎩

55

xk+1 yk+1 − yk =

xk+1 yk+1 = yk + xk

⎫ ⎬

f (x)d x xk

⎫ ⎬ ⎭ (33)

y (x)d x . ⎭

Obviously, the procedure for finding the integral is closely related to the accuracy of the calculations. Using the right rectangle formula, Eq. (33) can be rewritten as follows: ⎫ ⎧ xk+1 ⎬  ⎨ y (x)d x = yk+1 = yk + hy (xk ) yk+1 = yk + ⎭ ⎩ xk  (34) = yk+1 = yk + (xk+1 − xk )y (xk ) . These transformations are reduced to the classical Newton’s method if we take yk+1 = 0 in (34) and solve the equation for xk+1. It is known that the accuracy of one iteration step of the polyline method is O h 2 . The classical Newton’s method has a similar calculation accuracy. It is possible to increase the accuracy of one step during solving an equation using the predictor–corrector method. During applying the trapezoid method to find the integral, we will have: ⎧ ⎨ ⎩

xk+1 yk+1 = yk +

y (x)d x

xk

⎫ ⎬

   = yk+1 = yk + 0.5h y (xk ) + y (xk+1 ) ,

⎭



  yk+1 = yk + 0.5(xk+1 − xk ) y (xk ) + y (xk+1 ) . (35)

In Eq. (35), the value of y (xk+1 ) is unknown. It is being evaluated. Evaluation can be done using the polyline method as follows: z = xk − β

yk 2yk . , x k+1 = x k − yk y (xk ) + y (z)

(36)

The most popular variation (36) is the use of β = 1. Methods can also be proposed in which, in the trapezoidal rule, they take not the arithmetic mean at the ends, but the geometric mean. As a result, we get: ⎧ ⎨ ⎩

xk+1 yk+1 = yk +



y (x)d x xk

⎫ ⎬ ⎭

! =

 " y (xk )y (xk+1 ) = yk + h . (37) 0.5(y (xk ) + y (xk+1 )) 

yk+1

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An assessment of the unknown value in the same way is carried out in (37) for y (xk+1 ). As a result we have:   y (xk ) + y (z) yk . z = xk − β , xk+1 = xk − yk yk 2y (xk )y (z)

(38)

Another variant that follows the rule of averages can be written like this: ⎧ ⎨ ⎩

xk+1 yk+1 = yk +

y (x)d x

xk

⎫ ⎬

  xk + xk+1 = yk+1 = yk + hy ⎭ 2

  xk + xk+1 . = yk+1 = yk + (xk+1 − xk )y 2 (39)

In (39) there is a factor that, as in the previous expressions, requires an evaluation. As a result, we will have: z = xk − β

yk yk   , x k+1 = x k − xk +z . yk y 2

(40)

Obviously, the last method has the third order of computational accuracy. Solving the main classes of inverse heat conduction problems, as was indicated earlier, the derivatives of the functional behind each of its parameters must be calculated numerically, since the solution of such problems can only be obtained numerically, that is, in the form of tables of values. Before writing the formulas for the above methods (36), (38), and (40), we write the classical Newton’s method for the formulated inverse problem of heat conduction: G k θ k = −R k , θ k+1 = θ k + θ k ,

(41)

     where G = G i j = ∂ 2 J ∂θi ∂θ j —Hesse matrix; θ = ( θ1 , ..., θn )T — = column vector of parameter increments; R = (R1 , ..., Rn )T  T   ∂ J ∂θ1 , ..., ∂ J ∂θn —the column vector of the derivatives of the objective functional. Using methods (36), (38) and (40) to modify method (32) leads to such methods. Recall that the inverse problem of heat conduction is being solved, which is reduced to finding the minimum of a functional of type (8). In this case, the formulas of method (36) for minimizing the functional are written as follows: z=θ

(k)

 (k)   (k)  J θ − β · H −1 θ

Creation of High-Speed Methods for Solving Mathematical …

θ

(k+1)

=θ

(k)

  (k)  −1  (k)  + H (z) . −2· H θ J θ

57

(42)

Similarly, the formulas of the method (39) for solving the nonlinear problem are as follows:  (k)   (k)  (k) J θ , z = θ − β · H −1 θ      (k)     (k)  −1 1 (k+1) (k) (k) · H (z) H θ + H (z) J θ . (43) θ =θ − · H θ 2 The formulas of method (40) for the problem posed are as follows:  (k)   (k)  J θ , − β · H −1 θ # (k) $  (k)  θ +z (k) . = θ − H −1 J θ 2

z=θ θ

(k+1)

(k)

(44)

(k)

where θ —vector of unknowns of the problem at the k-th iteration; H —Hesse matrix. It should be noted that method (44) has the third order of accuracy. To speed up the operation of methods (42), (43) and (44) on nonlinear inverse problems in general, (k+1) . Then the formulation the method of steepest descent is used to find the vector θ of method (42) is finally written as follows:  (k)  (k)   (k)   (k) = min J θ − β · ∇ J θ , β > 0, J θ − βk · ∇ J θ     (k) (k) (k) J θ , z = θ − β · H −1 θ J (z − h k · ∇ J (z)) = min J (z − h · ∇ J (z)), h > 0,   (k)  −1  (k)  (k+1) (k) + H (z) . θ = θ − 2h k · H θ J θ

(45)

Similarly, the formulation of method (43) is finally written in the form:  (k)  (k)   (k)   (k) = min J θ − β · ∇ J θ , β > 0, J θ − βk · ∇ J θ     (k) (k) (k) J θ , z = θ − β · H −1 θ J (z − h k · ∇ J (z)) = min J (z − h · ∇ J (z)), h > 0, −1   (k)    (k)  h k   (k)  (k+1) (k) · H θ · H (z) H θ + H (z) J θ . θ =θ − 2 The formulation of the method (44) of the problem will look like this:

(46)

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 (k)  (k)  (k)   (k)  = min J θ − β · ∇ J θ , β > 0, J θ − βk · ∇ J θ     (k) (k) (k) J θ , z = θ − β · H −1 θ J (z − h k · ∇ J (z)) = min J (z − h · ∇ J (z)), h > 0,  (k) %   (k)  (k+1) (k) . θ = θ − h k H −1 θ + z 2 J θ

(47)

Obviously, during finding solutions to the problem by the classical Newton method, it is necessary to find the gradient of the functional (8). To do this, it is necessary to find numerous solutions 2n of direct problems of heat conduction, where n—umber of parameters. During calculating each off-diagonal element of the Hessian matrix, it is necessary to solve more 4n direct problems. Taking into account the symmetry of the Hesse matrix, the number of solutions to direct problems of heat conduction per iteration step of the classical Newton’s method is as follows [13–16, 20]: N f = 1 + 4(1 + 2 + ... + n) + 2n = 1 + 2n(n + 1) + 2n = 2n 2 + 4n + 1. The number of calls to the procedure for solving a direct problem per one iteration step of solving similar problems by the classical Newton’s method has a quadratic dependence. In this case, acceleration is possible here. One of the most effective ways to speed up is to perform an interpolation procedure.

5.2 Interpolation Method for Calculating Hessian Elements It is known that all second-order partial derivatives in the Hessian are carried out only numerically for nonlinear inverse problems. Consider the function of two variables F(x, y), for which it is necessary to find the global minimum in the space R2 . That is, the task is posed as follows: F(x, y) → min, (x, y) ∈ R2 .

(48)

The classical Newton method for problem (48) has the form: ⎛

⎞

⎛

⎞

⎛

 x (k) , y (k) ⎝ ⎠=⎝ ⎠−⎝  (k) (k)  ∂2 F x ,y y (k+1) y (k) ∂ y∂ x x (k+1)

x (k)

∂2 F ∂x2



 ⎞−1 ⎛ ∂ F  (k) (k)  ⎞ x (k) , y (k) x ,y ⎠ · ⎝ ∂x ⎠     . ∂2 F ∂F (k) (k) (k) (k) x x , y , y ∂ y2 ∂y (49)

∂2 F ∂ x∂ y



Let us write the derivatives of the first and second orders, which are calculated in (49) numerically using the difference method:

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59

   ∂ F  (k) (k)    (k) x ,y = F x + x, y (k) − F x (k) , y (k) x, ∂x    ∂ F  (k) (k)    (k) (k) x ,y = F x , y + y − F x (k) , y (k) y, ∂y       F x (k) − x, y (k) − 2F x (k) , y (k) + F x (k) + x, y (k) ∂ 2 F  (k) (k)  , x ,y = ∂x2 ( x)2  (k) (k)   (k) (k)    (k) (k) + F x , y + y F x , y − y − 2F x , y ∂ 2 F  (k) (k)  x ,y = , 2 2 ∂y ( y)     F x (k) + x, y (k) + y + F x (k) − x, y (k) − y ∂ 2 F  (k) (k)  x ,y = ∂ x∂ y 4 x y     (k) (k) F x − x, y + y + F x (k) + x, y (k) − y . − 4 x y (50)  (k+1) (k+1)  So, for calculating a new pair of numbers , it’s necessary to have  (k) x(k)  , y 8 points, in addition to the central point x , y , of which 4 points are necessary for calculating mixed second derivatives—for Hessian elements that do not stand on its main diagonal. The  main idea of using the interpolation method is to obtain a pair of x (k+1) , y (k+1) numbers using 6 points on which a second-order surface is constructed. The scheme of the interpolation idea is as shown below. Let’s show the operation of the scheme, which is shown in Fig. 1. For example, let’s take the point (1; 1), at which we need to find the partial  of the F(x,  y) func derivative tion, which has the following form: F(x, y) = cos y 2 + 1 (x y + 4) . Let’s find the second-order mixed derivative in three ways: numerically using the last formula in (50), numerically using interpolation, and analytically (to check and analyze the results). First method. Increment values are x = 0.01; y = 0.01. ∂2 F F(1 + 0.01; 1 + 0.01) + F(1 − 0.01; 1 − 0.01) (1; 1) = ∂ x∂ y 4 · 0.01 · 0.01 F(1 − 0.01; 1 + 0.01) + F(1 + 0.01; 1 − 0.01) ≈ 0.036. − 4 · 0.01 · 0.01 f ( x, y ) = a1 x 2 + a2 y 2 + a3 xy + + a4 x + a5 y + a6

f xx′′ = 2a1 , f yy′′ = 2a2 , f xy′′ = a3 . Fig. 1 General interpolation scheme for finding mixed derivatives in hessian

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Second method. The second-order interpolation surface in two spatial variables has the following form: Finterp (x, y) = a1 x 2 + a2 y 2 + a3 x y + a4 x + a5 y + a6 . To unambiguously determine the six coefficients of the surface equation, it is necessary to have the coordinates of six points. Let there be given six points with coordinates (1; 1), (1 + 0.01; 1), (1 − 0.01; 1), (1; 1 + 0.01), (1; 1 − 0.01), (1 + 0.01; 1 + 0.01). Let’s compose a system of linear algebraic equations (SLAE) in matrix form, which has the form: ⎞⎛

⎛ x12 y12 x1 y1 x1 y1 1

⎜ ⎜ 2 ⎜ x2 ⎜ ⎜ ⎜ x2 ⎜ 3 ⎜ ⎜ 2 ⎜ x4 ⎜ ⎜ 2 ⎜ x5 ⎝ x62

⎞

⎛

a1

F(x1 , y1 )

⎞

⎟⎜ ⎟ ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎜ a2 ⎟ ⎜ F(x2 , y2 ) ⎟ y22 x2 y2 x2 y2 1 ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎟ 2 ⎜ ⎜ ⎟ ⎟ y3 x3 y3 x3 y3 1 ⎟⎜ a3 ⎟ ⎜ F(x3 , y3 ) ⎟ ⎟ ⎟⎜ ⎟ = ⎜ ⎟. ⎟⎜ ⎟ ⎜ ⎟ 2 y4 x4 y4 x4 y4 1 ⎟⎜ a4 ⎟ ⎜ F(x4 , y4 ) ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎟ 2 ⎜ ⎜ ⎟ ⎟ y5 x5 y5 x5 y5 1 a F(x5 , y5 ) ⎟ ⎠⎝ 5 ⎠ ⎝ ⎠ y62 x6 y6 x6 y6 1 a6 F(x6 , y6 )

Substituting 6 points and solving the SLAE, we obtain the value of the coefficient a3 ≈ 0.036. Third method. Let us find analytically for verification of the mixed second-order derivative for the function F(x, y).  

y2 − 4 x+y x+y x y + 4 − y(x + y) = . sin sin xy + 4 xy + 4 (x y + 4)2 (x y + 4)2    2y(x y + 4)2 − 2x(x y + 4) y 2 − 4 x+y = Fx y = sin xy + 4 (x y + 4)4 

y 2 − 4 x y + 4 − x(x + y) x+y + cos xy + 4 (x y + 4)2 (x y + 4)2  2   2y(x y + 4) − 2x y − 4 x+y = sin xy + 4 (x y + 4)3  2   

2 y −4 4−x x+y + cos xy + 4 (x y + 4)4     2 

y − 4 4 − x2 8(x + y) x+y x+y = + . sin cos xy + 4 xy + 4 (x y + 4)3 (x y + 4)4

Fx (x, y) = − Fyx

Thus, the result is Fyx (1; 1) = Fx y (1; 1) ≈ 0.036.

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61

Fig. 2 Result of constructing a second-order surface and using the interpolation procedure in the process of calculating mixed derivatives

Now let’s visualize the error surface during interpolation to find the mixed secondorder derivative in the Hessian. This approach, as shown in Fig. 2 makes it possible to effectively solve multidimensional nonlinear inverse problems. At each iteration of the inverse problem of heat conduction for two-dimensional objects under study, during finding the elements of the Hessian, only 6 points out of 9 are used. During studying thermal processes, which are reduced to the analysis of threedimensional objects, the second-order surface will look like: Finterp (x, y, z) = a1 x 2 + a2 y 2 + a3 z 2 + a4 x y + a5 x z + a6 yz + a7 x + a8 y + a9 z + a10 .   For example, let’s take the function F(x, y, z) = sin yz − x y + z 2 . Let’s find all the partial derivatives of the second order for this function: ∂F ∂x ∂F ∂z 2 ∂ F ∂x2 ∂2 F ∂z 2 ∂2 F ∂ x∂ y

   ∂F  = −y cos yz − x y + z 2 , = (z − x) cos yz − x y + z 2 , ∂y   = (y + 2z) cos yz − x y + z 2 .  ∂2 F    = −y 2 sin yz − x y + z 2 , 2 = −(z − x)2 sin yz − x y + z 2 , ∂y     = 2 cos yz − x y + z 2 − (y + 2z)2 sin yz − x y + z 2 .     = − cos yz − x y + z 2 + y(z − x) sin yz − x y + z 2 ,

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  ∂2 F = y(y + 2z)sin yz − x y + z 2 , ∂ x∂z     ∂2 F = cos yz − x y + z 2 − (z − x)(y + 2z)sin yz − x y + z 2 , ∂ y∂z The calculation results show that to find all the numerous derivatives, it is necessary to use 27 points. Only 10 points are used to construct the interpolation surface. Therefore, the value of mixed derivatives can be calculated for multivariate nonlinear inverse problems using the interpolation method. It should also be noted that 6/9 > 10/27. In this case, with an increase in the dimension of the problem, it is necessary to decrease the values of the increments of the variables during constructing the interpolation surface to find the elements of the Hessian. For the n-dimensional case, the equation of the interpolation surface in short form is as follows: Finterp (x1 , x2 , ..., xn ) =

n

n

  ai, j xi x j . i=1 j=i

Interpolation by surfaces of the second order gives a number of advantages, especially as the dimension of the inverse problem grows. It should be noted that it is advisable to use interpolation for nonlinear problems.

5.3 Fourier Method for Constructing Initial Approximations Let’s consider a simple direct problem of heat conduction in a homogeneous rod of length, which is described by a parabolic equation of the form [20–23]: ' & ∂2T ∂T = , x ∈ [0; L], t ∈ 0, t f . 2 ∂t ∂x

(51)

The initial conditions are as follows: T (x, 0) = θ (x).

(52)

The boundary conditions for (51) can be arbitrary. As an example of boundary conditions, we take the Dirichlet conditions, which have the form: T (0, t) = 0, T (L , t) = 1.

(53)

Having solved the direct problem (51)–(53), the temperature distribution at the moment of the t f is found:

Creation of High-Speed Methods for Solving Mathematical …

  T x, t f = (x).

63

(54)

Then the inverse problem is formulated as follows: knowing Eqs. (51), boundary conditions (52), and the final temperature distribution (54), restore the initial temperature distribution (53). Let’s write down one of the most popular mathematical formulations for the problem at hand – it’s necessary to find the distribution of θ (x) such that L J (θ (x)) =

   2 T θ, x, t f − (x) d x → min,

(55)

0

  where T θ, x, t f —temperature field, which is a solution to the problem with the desired θ (x): ∂T ∂2T , = ∂t& ∂' x 2 x ∈ [0; L], t ∈ 0, t f , T (x, 0) = θ (x).

(56)

The boundary conditions of problem (55) at (56) are conditions of the form (53). In fact, problem (55) and (56) is a minimization problem with a constraint in the form of differential equations. The minimization problem is solved in discrete form. On the segment [0; L] n nodes of xi , i = 1, n are introduced. So, to solve problem (55), it is necessary to find the θi = θ (xi ) = T (xi , 0),i = 1, n at these nodes, that is, the solution depends on n parameters. The known solution of problem (56) at a fixed time instant is expanded in a Fourier series. It is necessary to find the temperature distribution at the initial moment of time. The general solution of the homogeneous non-stationary heat conduction problem can be represented as: T (x, t) =

∞

Ck e−k t sin(kx). 2

k=1

Obviously, at the initial moment of time, the condition is written in the form: T (x, 0) =

∞

Ck sin(kx).

k=1

Let’s introduce the notation: Dk = Ck e−k t . 2

Then, at any fixed moment in time, the solution can be represented as follows:

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T (x, t) =

∞

Dk sin(kx).

k=1

The temperature distribution obtained experimentally is expanded in a series and the unknown Dk are found. Since the moment of time is known, the unknown Ck 2 values can be easily found from the Dk = Ck e−k t equation, thereby finding an approximation to the initial condition. This approximation is a truncated series. The general solution of any differential equation is represented as the sum of the general solution of the homogeneous equation and some particular solution of the inhomogeneous equation. The described method is based on the fact that the solution of the heat equation at a certain point in time of the t f can be expanded in a Fourier series by taking only a few terms of this schedule. After the process of expansion into a series, the process of refinement of the approximation is carried out. First, we describe below the method for the one-dimensional case. First, we will consider the standard direct one-dimensional problem of heat conduction in a homogeneous medium, which has the form (51)–(53). Further, the method of separation of variables [20–23] is applied to the heat conduction equation. If L = π, then the solution of this equation with zero boundary conditions of the first kind will have the following form: T (x, t) =

∞

Ck e−k t sin(kx). 2

(57)

k=1

Problem (51)–(53), making a change of variables in the heat equation, can always be reduced to a problem with boundary conditions with values 0 and 1. In this case, the solution to problem (57) at time t f can be written as [20–23]: ∞ 

 x 2 Ck e−k t f sin(kx) + . T x, t f = π k=1

(58)

Using the Fourier series, it can be obtained a general solution of problem (51)–(53) with the boundary conditions T (0, t) = 0,T (π, t) = 1. Then the initial problem, taking into account (58), can be written in the form: T (x, 0) =

∞

k=1

Ck sin(kx) +

x . π

(59)

Since it is necessary to find the initial condition (59) of the inverse problem (55), (51), (54), (53), the constants Ck are obtained by the schedule  in a series decoupling  of the direct heat conduction problem at time t f . The T x, t f value in real technical problems can only be obtained experimentally. It is obvious that in (59) several terms of the decomposition of the initial condition in a series are taken. In any case, a qualitative approximation to the initial condition is obtained. It can be taken N first

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65

terms in (59). Then we will have T (x, 0) ≈

N

Ck sin(kx) +

k=1

x . π

(60)

Equation (60) is only an approximation to the initial condition. In connection with this, the error function f err (x) is introduced. Then we will have: T (x, 0) =

N

Ck sin(kx) +

k=1

x + f err (x). π

(61)

In (61), the Ck , k = 1, N values are known, then the solution to the formulated inverse problem consists in finding the f err (x). The same functional is used to solve the problem (55). The process of minimization (55) will already be carried out by the method of steepest descent for the obtained approximation to the initial condition. Thanks to this approach, the number of calls to the function for solving the direct problem is significantly reduced. To use this tool, the gradient of functional (55) is found numerically. The first derivatives with respect to the unknown arguments of the functional, taking into account that it depends on the unknown correction function, are calculated using (61). To find the minimum of the functional, a movement is performed in the direction of the antigradient. The step length is determined by any one-dimensional unconstrained optimization method, for example, the Newton method. The situation is the same in the two-dimensional inverse problem of heat conduction. The solution of the two-dimensional heat equation is represented as the sum of the general solution of the corresponding homogeneous equation and some particular solution of the inhomogeneous equation. The solution to an inhomogeneous equation is found by any numerical method. Let in equation 

∂T ∂T ∂T =k + 2 ∂τ ∂x2 ∂y

(62)

' & solution be in the : [0; L x ] × 0; L y area. At the boundaries of the computational area, for example, zero conditions of the first kind are specified. Applying the method of separation of variables for (62), the sought solution is in the form of T (x, y, τ ) = F1 (x)F2 (y)F3 (t). Then a system of equations is obtained: ⎧ ⎪ F (x) = C1 cos λ1 x + C2 sin λ1 x, ⎪ ⎨ 1 F2 (y) = C3 cos λ2 y + C4 sin λ2 y, ⎪ ⎪ ⎩ F (t) = C e−λ2 t , λ2 = k λ2 + λ2 . 3

5

1

2

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Using the boundary conditions, the general solution to the homogeneous problem is F3 (t) = e T (x, y, t) =

−k



πn Lx

∞

∞

m=1 n=1

2  2  t + πm Ly

Amn sin

,

π nx π my sin F3 (t). Lx Ly

(63)

Further, it is using the procedure which is similar to the one-dimensional case. Now the error function will have the form f err (x, y). By enumerating all unknown parameters, it can be written derivatives with respect to unknown arguments of functional (55), which are calculated using (63). The described method was tested on a number of linear and nonlinear problems. It should be noted that the described method is also effective for problems in which the boundary conditions change over time. For example, consider several different inverse problems. First task. It is necessary to find the global minimum of the functional 1 1 J (θ (x, y)) = 0

   2 T θ, x, y, t f − (x, y) d xd y → min,

(64)

0

under the condition ' & ∂T ∂2T ∂2T = + , (x, y) ∈ [0; 1]2 , t ∈ 0, t f , t f = 0.1, 2 2 ∂t ∂x ∂x T (x, y, 0) = θ (x, y),

(65)

  where θ (x, y)—unknown function; T θ, x, t f —desired solution to the problem; (x, y)—given temperature distribution at time t f . The boundary conditions of the problem are as follows: T (x, 0, t) = 0.5sin(π x), T (x, 1, t) = 0.5sin(2π x), T (0, y, t) = 0.5sin(2π y), T (1, y, t) = 0.5sin(π y).

(66)

In problems (64)–(66), T (x, y, t f ) distribution is known, and it’s shown in Fig. 3. Obviously, the problem is incorrectly posed. Solution of the problem will seek using the methods described above. Figure 4 shows the results of the method on the two problems considered above, which are programmatically implemented in MATLAB. Figures 5 and 6 show the main results of the convergence of methods for solving the problem (64)–(66). The construction of an approximation to the desired initial condition makes it possible to obtain a numerical solution of problem (64)–(66) in four times faster

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Fig. 3 Final temperature distribution of problem (64)–(66): on the left—temperature field; on the right—graph of the temperature field level lines

Fig. 4 Reconstructed initial condition of the problem: on the left—temperature field; on the right— graph of the temperature field level lines

with a computational accuracy 10–9 . The initial approximation reduced the number of calls to the procedure for solving the direct problem of heat conduction by almost 3.5 times. Figure 6 shows a graph of the error surface of the found solution to the problem. The obtained methods were tested on a number of problems, including nonlinear. Mathematical description of some nonlinear problems is given below. Also, the foundations of this method can be applied to solve inverse problems in environmental problems [22, 24–26].

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Fig. 5 Comparison of the convergence of the method with zero approximation to the initial condition and with the initial condition based on the Fourier series (accuracy 10–9 )

Fig. 6 Absolute deviation of the specified temperature field relative to calculated for the found initial condition

Problem 1 It’s necessary to find the temperature field at the initial moment of time for the nonlinear heat equation: ρ(T )C(T )



 ∂ ∂T ∂ ∂T ∂T = a(T ) + a(T ) , ∂t ∂x ∂x ∂y ∂y

(x, y) ∈ [0; 1]2 , t ∈ [0; 0.02], a(T ) = T 2 + 0.1. ρ(T ) = 1 + 0.15T + 0.25T 2 , C(T ) = 0.25T 2 − 0.01T + 0.15 T (x, 0, t) = 0.5esin(π(0.5−x)+0.5π ) , T (x, 1, t) = 0.5esin(π(0.5−x)−0.5π ) , T (0, y, t) = 0.5esin(π(0.5−y)+0.5π ) , T (1, y, t) = 0.5esin(π(0.5−y)−0.5π ) . Figure 7 shows the temperature distribution at time t = 0.2. Figure 8 shows the found temperature distribution of problem (67).

(67)

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69

Fig. 7 Temperature field of the problem at time t = 0.2

Fig. 8 Temperature field at the initial moment of time

Problem 2 It is necessary to find the initial temperature distribution for the nonlinear heat equation:



 ∂ ∂T ∂ ∂T ∂T = a(T ) + a(T ) , ρ(T )C(T ) ∂t ∂x ∂x ∂y ∂y (x, y) ∈ [0; 1]2 , t ∈ [0; 0.02], a(T ) = T 2 + 0.1. 1 + T + 2T 2 , C(T ) = 0.5 + (0.1 − T )2 , 10 T (x, 0, t) = 0.5esin(π(0.5−x)+0.5π ) , T (x, 1, t) = 0.5esin(π(0.5−x)−0.5π ) ,

ρ(T ) =

T (0, y, t) = 0.5esin(π(0.5−y)+0.5π ) , T (1, y, t) = 0.5esin(π(0.5−y)−0.5π ) .

(68)

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The temperature field of problem (68) at time t = 0.02 is shown in Fig. 9. The reconstructed temperature distribution of problem (68) is shown in Fig. 10. Tables 1 and 2 contain the main results of the convergence of the obtained methods on grids with different characteristics. Now let us show in tabular form the results of all methods, before using which the procedure for constructing a qualitative approximation to the initial condition of problems (67) and (68) based on the Fourier method was applied. During constructing the initial condition in the problems under consideration, 4 terms of the series expansion were used. Testing of two classical methods (Newton’s method and the method of steepest descent) and three developed methods (45), (46) and (47) were carried out on a number of nonlinear inverse problems, in particular, Eqs. (67) and (68). The problems

Fig. 9 Temperature field at a fixed time t = 0.02

Fig. 10 Reconstructed temperature distribution at the initial moment of time

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71

Table 1 Results of the work of methods on triangulation with 1241 nodes, 2360 elements (120 of which are limiting) without the use of the Fourier method (accuracy 10–11 ) Method

Task 1 (51)

Task 2 (52)

Method of steepest descent

350 iterations/10,718,750 calls to 387 iterations/9,660,100 calls to the function of solving the direct the function of solving the direct problem (100%) problem (100%)

Classical Newton’s method

15 iterations/55.26%

18 iterations/73.03%

First modification of Newton’s method (45)

7 iterations/25.35%

7 iterations/29.72%

Second modification of Newton’s method (46)

8 iterations/29.18%

6 iterations/25.38%

Third modification of Newton’s method (47)

5 iterations/18.35%

5 iterations/21.17%

Table 2 Results of the work of methods on triangulation with 4841 nodes, 9440 elements (240 of which are limiting) without the use of the Fourier method (accuracy 10–11 ) Method

Task 1 (51)

Task 2 (52)

Method of steepest descent

193 iterations/1,821,805 calls to 298 iterations/3,656,021 calls to the function of solving the direct the function of solving the direct problem (100%) problem (100%)

Classical Newton’s method

12 iterations/54.17%

16 iterations/60.72%

First modification of Newton’s method (45)

7 iterations/16.28%

9 iterations/20.25%

Second modification of Newton’s method (46)

9 iterations/16.31%

7 iterations/11.28%

Third modification of Newton’s method (47)

5 iterations/8.58%

6 iterations/9.35%

were solved using different triangulations. All obtained results of methods calculations are without constructing of approximation to the initial condition (Tables 1 and 2). The computational accuracy for all methods on different grids is equal to 10–11 . It is shown experimentally that for solving the nonlinear problems described in the chapter, the method of steepest descent has the largest number of calls to the procedure for solving the direct problem of heat conduction. For each problem, the total number of calls to the procedure for solving the direct problem during finding a solution by this method was taken as 100%. The number of calls to the procedure for solving the direct problem of heat conduction, which was obtained by other methods (Tables 1 and 2), are listed in relation to the method of steepest descent.

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6 Conclusions The chapter contains a description of the developed modifications of the main actual numerical methods for solving the main classes of multidimensional linear and nonlinear inverse heat conduction problems. The main part of the chapter is devoted to modifications of classical methods in order to reduce the number of calls to the procedure for solving the direct problem of heat conduction in order to obtain numerous solutions of specific inverse problems. The main practical results obtained in the chapter include the method of interpolation by second-order surfaces, which is actively used for approximate calculations of the elements of the Hesse matrix. This method makes it possible to reduce the number of calls to the procedure for solving direct problems to obtain a numerical solution to one inverse problem. Also in this chapter, the main modifications of the classical Newton method (45), (46), and (47) are obtained for minimizing the quadratic functional in the classical formulation of the main classes of inverse problems. Methods (45)–(47) are stable in solving inverse heat conduction problems due to the introduction of a variable step and the predictor–corrector method. Rational methods for solving nonlinear inverse problems are modifications of Newton’s method, and methods of linear problems are the method of steepest descent. Due to the developed computational methods for solving inverse problems of heat power engineering, the total number of calculations was reduced by 8–10 times compared to the total number of calculations that are required for classical methods, in particular, the classical Newton’s method.

References 1. Babak, V.P., Babak, S.V., Myslovych, M.V., Zaporozhets, A.O., Zvaritch, V.M.: Principles of construction of systems for diagnosing the energy equipment. In: Diagnostic Systems for Energy Equipments, pp. 1–22. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-444 43-3_1 2. Zaporozhets, A.O.: Methods and means for the control of the fuel combustion process. In: Control of Fuel Combustion in Boilers, pp. 1–33. Springer, Cham (2020). https://doi.org/10. 1007/978-3-030-46299-4_1 3. Zanoli, S.M., Pepe, C.: Two-layer linear MPC approach aimed at walking beam billets reheating furnace optimization. J. Control Sci. Eng. (2017). https://doi.org/10.1155/2017/5401616 4. Zaporozhets, A.O.: Experimental research of a computer system for the control of the fuel combustion process. In: Control of Fuel Combustion in Boilers, pp. 89–123. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-46299-4_4 5. Andreas, K.: An Introduction to the Mathematical Theory of Inverse Problems. Second edition. Springer Science+Business Media, LLC (2011). https://doi.org/10.1007/978-1-4419-8474-6 6. Roosta-Khorasani, F., Mahoney, M.W.: Sub-sampled Newton methods. Math. Program. 174(1– 2), 293–326 (2019). https://doi.org/10.1007/s10107-018-1346-5 7. Martin, J., Wilcox, L.C., Burstedde, C., Ghattas, O.: A stochastic Newton MCMC method for large-scale statistical inverse problems with application to seismic inversion. SIAM J. Sci. Comput. 34(3), A1460–A1487 (2012). https://doi.org/10.1137/110845598

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Mathematical Model of Optimal Support of Thermal Energy with Coal Products Taking into Account Environmental Constraints Vitaliy Makarov , Mykola Makortetskyi , Mykola Perov , Tetiana Bilan , and Nataliia Ivanenko Abstract The subject of the research is the directions and amount of technological re-equipment of the coal industry to ensure the optimal fuel structure of thermal power companies and reduce the negative impact on the environment. The aim of the article is to develop a mathematical model of the optimal structure of coal products for coal-fired power plants with minimal total financial costs while ensuring the design characteristics of the quality of coal products for consumers, taking into account environmental constraints. Research methods: system analysis to determine many indicators of energy, economic and environmental efficiency; linear programming to develop a model for optimizing the structure of coal products; multicriteria optimization and comparative analysis to form variants of the coal structure; expert assessments to form of the information base. The developed model combines detailed consideration of technical and economic indicators of technological equipment of mines and concentrators with algorithms of coordination of fuel flows that allow to forecast structure of finished coal production with maintenance of necessary indicators of its quality on all sites of production. The software implementation of the model allows performing multivariate calculations of the national electricity supply with coal products with design quality characteristics and in the required volumes. The optimal level of enrichment of extracted coal and optimal variants of the forecasted fuel structure for coal-fired power plants of Ukraine for the period up to 2040 have been determined. Keywords Coal industry · Coal products · Thermal energy · Mathematical model · Ecology

1 Introduction In Ukraine, coal is the only energy resource with enough reserves for hundreds years, which determines its leading role in ensuring energy security requirements. Various V. Makarov · M. Makortetskyi · M. Perov · T. Bilan (B) · N. Ivanenko Institute of General Energy of NAS of Ukraine, Kyiv, Ukraine e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Zaporozhets (ed.), Systems, Decision and Control in Energy III, Studies in Systems, Decision and Control 399, https://doi.org/10.1007/978-3-030-87675-3_4

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aspects of the efficient functioning of power systems while ensuring energy security of the country were discussed in many papers [1–8]. However, the current technical and technological state of coal production is extremely unsatisfactory and needs a thorough upgrade. Currently, the process of national fuel supply to thermal energy plants needs special attention, taking into account the objectively unstable structure of the fuel base o in the country, as well as the significant uncertainty of the sectors of consumption of coal products. The urgency of the work lies in the objective need for an adequate assessment of the role of coal in the fuel balance of Ukraine, due to many factors. First, it is the presence and availability of deposits, forecasts of the product structure of coal products, increasing the share of renewable energy sources in the fuel balance, infrastructural constraints of the import subsystem, and the shortage of certain branded groups of coal in the global coal market. The need for systematic consideration of these factors is a prerequisite for the development of qualitative forecasts of technological development of the coal industry. In this regard, it is important to create mathematical models and software to optimize the technological development of the coal industry, taking into account up-to-date environmental requirements [5, 9–12]. Ukrainian and foreign scientists addressed different methodological issues in the field of development of the coal industry: Kiyashko Yu. I. (assessment of the efficiency of mines in different options for the use of treatment equipment) [13], Kulik M. M., Alaverdyan L. M. (optimization of coal industry development) [14, 15], Pavlenko I. I. (forecasting the development of the coal industry with limited investment) [16], Yashchenko Y. P., Kosarev I. V. (technical development of mines) [17], Henderson J. (model of supply and demand in the coal markets) [18], Suwala W. (model of restructuring of the coal industry) [19] and others. However, these studies did not take into account the relationship of the mine fund with processing plants and energy facilities, and therefore their results are fragmentary. Studies of this problem by foreign experts concern the peculiarities of the functioning of the coal mining and processing industries of other countries, and do not take into account the conditions of energy supply of Ukraine’s economy, the gradual reorganization of the coal industry. Expected trends in fuel supply of coal-fired thermal power plants (CTPP) and strengthening of environmental requirements determine, firstly, the need for changes in the technological structure of electricity production, and secondly—improving the consumer quality of coal products, which can be optimized by coal supply to concentrators. Research on optimizing the structure of coal products of different technological purposes for its quality, taking into account quality indicators, current and future requirements of consumers of thermal generation, taking into account environmental constraints is currently relevant. In particular, work [20–22] is devoted to the issue of reliability of supply and efficiency of fuel resources used at energy facilities. The aim of the study is to develop a mathematical model of the optimal structure of coal products for CTPP with minimal total financial costs while ensuring the design

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characteristics of the quality of coal products for consumers, taking into account environmental constraints. This goal involves the following tasks: development of a mathematical model for forecasting the optimal structure of coal products for CTPP, taking into account the up-to-date environmental constraints; improvement of software and information tools for model implementation; determination of the optimal structure of coal products for CTPP. The following research methods are used in the work: system analysis to determine the set of indicators of energy, economic and environmental efficiency; linear programming to develop a model for optimizing the structure of coal products; multicriteria optimization and comparative analysis to form the variants of the structure of coal fuel; expert assessments to form the information base.

2 Mathematical Model of Optimization of Finished Coal Production The total cost of coal products supplied to CTPP is calculated in accordance with the established base price for coal products, taking into account the thermophysical characteristics of its components and transportation costs, which can be expressed by the formula [21]: N  M  L G   k

S=

k

k

sikjl · xikjl ,

(1)

k=1 i=1 j=1 l=1

where S—total cost of coal products for all CTPP, UAH; G—amount of CTPP; N k —number of suppliers of coal products on the k-th CTPP; M k —number of grades of thermal coal coming to the k-th CTPP; L k —number of types of coal products supplied to the k-th CTPP; sikjl —price of 1 ton of finished coal products supplied to the k-th CTPP, UAH per ton; xikjl —volumes of finished coal products coming to the k-th CTPP, thousand tons. Therefore the total cost of coal products S depends on the volumes received from suppliers and the wholesale price of these products, which, in turn, largely depends on its quality indicators, i.e.   sikjl = f Aikjl , Wikjl ,

(2)

where Aikjl and Wikjl —the percentage of ash and moisture in the components of coal products, respectively. The quality of finished coal products in turn depends on the volume of processing of ordinary coal in concentrators, which significantly affects its cost:

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  sikjl = f Aikjl (θi ), Wikjl (θi ) ,

(2’)

where θi is a parameter that characterizes the level of ordinary coal sent to concentrators. The parameter θi is a variable that takes the value 0 ≤ θi ≤ 1. In addition, the efficiency of CTPP significantly depends on the costs associated with environmental constraints arising from the requirements of climate change international agreements. Therefore, Eq. (1) should take into account the financial costs of carbon and pollutants taxes [23], emitted during coal combustion, as well as the cost of installing gas treatment equipment, which is proportional to the electric capacity of CTPP and depends on the level of emission treatment in accordance with environmental constraints. Therefore, the mathematical model of supplying CTPP with coal products with specified thermophysical properties at minimum total cost (S) can be formulated as follows: ⎧ ⎫

Mk  Lk G ⎨ Nk  H H ⎬       cm · z mk , (3) d mk + sikjl · xikjl + k S = min ⎭ ⎩ m=1

k=1

m=1

i=1 j=1 l=1

with restrictions: • on the weighted average caloric content of fuel reserves of each CTPP N  M  L  k

δ1k

·

q kp

≤

k

k

N  M  L  k

qikjl

·

xikjl /

i=1 j=1 l=1

k

k

xikjl ≤ δ2k · q kp ; k = 1, ..., G;

(4)

i=1 j=1 l=1

• on the total needs of coal products of each CTPP, tons of conventional fuel per year N  M  L  k

k

k

k

x ikjl ≥ X ; k = 1, ..., G;

(5)

i=1 j=1 l=1

• on the volumes of the m-th emission of pollutants on the k-th CTPP M  L N  

mk z imk ; m = 1, ..., H, jl < Z

(6)

i=1 j=1 l=1

where cm —specific value of the tax on the m-th emission into the atmosphere, UAH/t; z imk jl —emissions of greenhouse gases (GHG) and pollutants; H—the number of GHG and pollutants; d mk = ηmk · d0mk —specific cost of treatment equipment for m-th emission of the k-th CTPP, UAH/kW, value d0mk —maximum allowable value of specific cost of treatment; ηmk —a variable value that characterizes the depth of treatment of the installed equipment, 0 < ηmk < 1; k —capacity of the k-th CTPP, mW; x ikjl —volumes of coal production in terms of conventional fuel, i.e. x ikjl =

Mathematical Model of Optimal Support of Thermal Energy …

79

qi jl · xikjl /Q u ; q kp —designed lower calorific value for k-th CTPP, mJ/kg;δ1k , δ2k — constant values that set the lower and upper design limits of calorific value of fuel for the k-th CTPP, 0 < δik < 1. The carbon equivalent of Qu for conversion of natural fuel into conventional equals to the value of 29.33 MJ/kg. Caloric content of coal products in the restriction (4) is calculated by the formula [24]:    da f  qikjl = qi jl 1 − Aikjl (θi ) · 1 − Wikjl (θi )     − 2, 442 Wikjl (θi ) + υi jl 1 − Aikjl (θi ) · 1 − Wikjl (θi ) ; k = 1, ..., G; i = 1, ..., N k ; j = 1, ..., M k ; l = 1, ..., L k ,

(7)

da f

where qikjl —specific calorific value of coal products, MJ/kg; qi jl —average values of higher calorific value for dry ashless fuel, MJ/kg; υi jl —coefficient that takes into account the hydrogen content in coal (average values: 0.46 for brown and hard coal other than anthracite and 0.21—for anthracite). The main types of emissions from the coal combustion on CTPP are carbon monoxide, nitrogen dioxide, sulfur dioxide, benzo(a)pyrene, solid particles, carbon dioxide. These emissions were calculated on the base [25–31], which describes in detail the methods of emission calculation. Therefore, the total amount of Z mk pollutant m emitted in the atmosphere with flue gases (g/s, t/year) is determined by equation: Z mk = cm · V k · x k · b,

(8)

where cm —mass concentration of the GHG and pollutant m in dry flue gases at a standard coefficient of excess air a0 = 1.4 and normal conditions (temperature 273 K and pressure 101.3 kPa), mg/m3 ; V k -volume of dry flue gases formed during the complete combustion of 1 kg (1 m3 ) of fuel, at a0 = 1.4, nm3 /kg of fuel (m3 /m3 of fuel); x k —estimated fuel consumption at the enterprise, t/year (thousand m3 /year); b—conversion factor. Estimated emissions in grams per second totals b = 0,278 · 10–3 , in tons per year - b = 10–6 . The algorithm of determining the volumes of dry flue gases V k is described in detail in [22], after which the total amount of pollutant emissions is unambiguously determined from Eq. (8). It should be noted that the amount of emission tax largely depends on the level of their treatment. Mathematical dependences for determining the volume of flue gases V k take into account the level of treatment of emissions of pollutants by a factor of ηmk , the value of which for each power plant depends on the capacity of the installed treatment equipment. However, in practical calculations, due to the lack of these data, it is proposed to consider the parameter ηmk as an unknown value and determine it depending on the established emission limits [32, 33]. This significantly affects the overall cost of CTPP, taking into account the high cost of treatment equipment.

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Table 1 Potential of thermal power industry of Ukraine consuming gas coal Indicator

2025

2030

2035

2040

Volume of gas coal production, thousand tons

25,877.6

21,507.6

17,992.6

17,992.6

Volume of gas coal production, thousand tons of conventional fuel

15,544.8

13,065.9

10,986.6

10,986.6

370.2

355.7

350.7

334.1

42.0

36.7

31.3

32.9

Volume of gas coal production, thousand tons

37,950.0

36,705.0

35,090.0

35,090.0

Volume of gas coal production, thousand tons of conventional fuel

22,836.9

22,266.1

21,292.7

21,292.7

370.2

355.7

350.7

334.1

61.7

62.6

60.7

63.7

Existing mining equipment

Specific consumption of conventional fuel, g/kWh Electricity generation, TWh New mining equipment

Specific consumption of conventional fuel, g/kWh Electricity generation, TWh

3 Potentially Possible Volumes of Coal Production The volume and structure of fuel required for CTPP largely depends on the design characteristics of electricity generating enterprises, as well as on the volume of electricity needs of the national economy. The maximum possible volumes of generation of the electric power are calculated based on the forecast of gas coal production and design characteristics of CTPP (Table 1). In turn, the volume of extracted coal depends on the technological condition of the mines. Therefore, the volumes of extracted coal are calculated both for production on actually existing equipment and under the condition of using new advanced mining equipment.

4 Forecast Calculations of the Coal Products Structure The forecast of providing the optimal fuel structure to the thermal power companies of Ukraine, which consume coal of the gas group, were developed using the model described in the paragraph 3. The calculations use the forecast of electricity production developed by the Institute of General Energy of the National Academy of Sciences of Ukraine [5], which identifies the electricity production by enterprises that consume gas coal. Four optimal variants of the fuel structure forecast for CTPP for the period up to 2040 were considered. They based on the forecast of coal production until 2040 taking into account the optimal levels of coal enrichment, namely:

Mathematical Model of Optimal Support of Thermal Energy …

(1)

(2)

81

First option does not take into account restrictions on caloric value of fuel (4), as well as emissions of GHG and pollutants (6). The mines use existing coal mining equipment. Table 2 presents the results of the calculations; Second option does not take into account restrictions on caloric value of fuel (4), as well as emissions of GHG and pollutants (6). The mines use advanced coal mining equipment. Table 3 presents the results of the calculations;

Table 2 Forecast of the optimal structure of coal products under option 1 Characteristic Forecast indicators

Optimal forecast of needs

2025

2030

2035

2040

Electricity generation at 37.74 CTPP, TWh

47.95

59.69

68.59

Total coal production, thousand tons

25,877.6

21,507.6

17,992.6

17,992.6

Volumes of coal sent for 20,033.9 processing, thousand tons

13,595.8

9944.4

8640.0

Volumes of enrichment, 11,551.7 thousand tons

8020.6

6060.5

5247.3

Total finished coal 17,395.5 products, thousand tons

15,932.4

14,108.7

14,599.9

Total volumes of finished coal products, thousand tons of conventional fuel

15,544.8

13,065.9

10,986.6

10,986.6

Needs of coal products 13,970.8 for CTPP, thousand tons of conventional fuel

17,056.2

20,934.5

22,918.4

Volumes of finished 18,230.7 products, thousand tons

Needs extension the total coal production

Lower heat value, kcal / kg

5367.5

Ash content of marketable products, %

18.1

Sulfur content, %

1.9

Cost of marketable products, UAH million

50,387.8

Costs for installation of treatment equipment, UAH million

0.0

Emission taxes, UAH million

12,143.3

Total costs, UAH million

62,531.1

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Table 3 Forecast of the optimal structure of coal products under option 2 Characteristic Forecast indicators

Optimal forecast of needs

2025

2030

2035

2040

Electricity 37.74 generation at CTPP, TWh

47.95

59.69

68.59

Total coal production, thousand tons

37,950.0

36,705.0

35,090.0

35,090.0

Volumes of coal 27,926.0 sent for processing, thousand tons

25,409.8

25,068.4

22,537.2

Volumes of enrichment, thousand tons

16,145.3

14,818.0

15,187.6

13,798.8

Total finished coal products, thousand tons

26,169.3

26,113.2

25,209.2

26,351.6

Total volumes of finished coal products, thousand tons of conventional fuel

22,836.9

22,266.1

21,292.7

21,292.7

Needs of coal products for CTPP, thousand tons of conventional fuel

13 970.8

17 056.2

20 934.5

22 918.4

Volumes of finished 18,114.6 products, thousand tons

22,086.8

27,050.7

Needs extension the total coal production

5401.1

5407.4

5421.7

20.7

21.3

21.5

1.9

2.0

2.1

48,491.9

59,004.5

72,686.8

0.0

0.0

0.0

12,258.1

15,551.3

19,616.8

Lower heat value, kcal / kg Ash content of marketable products, % Sulfur content, % Cost of marketable products, UAH million Costs for installation of treatment equipment, UAH million Emission taxes, UAH million

(continued)

Mathematical Model of Optimal Support of Thermal Energy …

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Table 3 (continued) Characteristic Total costs, UAH million

(3)

(4)

2025

2030

2035

60,750.0

74,555.9

92,303.6

2040

Third option takes into account restrictions (4)–(6). The restrictions on the calorific value of fuel are within ± 10% of the designed. The mines use existing coal mining equipment. Table 4 presents the results of the calculations; Fourth option takes into account restrictions (4)–(6). The restrictions on the calorific value of fuel are within ±10% of the designed. The mines use advanced coal mining equipment. Table 5 presents the results of the calculations.

The calculations for the first and second options evidence that the implementation of new technologies in mines and concentrators of Ukraine in the short term will save 1781 million UAH of thermal power plants total cost. Calculations for the third and Table 4 Forecast of the optimal structure of coal products under option 3 Characteristic Forecast indicators

Optimal forecast of needs

2025

2030

2035

2040

Electricity generation at 37.74 CTPP, TWh

47.95

59.69

68.59

Total coal production, thousand tons

25,877.6

21,507.6

17,992.6

17,992.6

Volumes of coal sent for 20,424.0 processing, thousand tons

14,182.5

13,463.4

7961.3

Volumes of enrichment, 11,843.0 thousand tons

8375.6

8048.2

4901.7

Total finished coal 17,296.6 products, thousand tons

15,700.7

12,577.4

14,933.0

Total volumes of finished coal products, thousand tons of conventional fuel

15,544.8

13,065.9

10,986.6

10,986.6

Needs of coal products 13,970.8 for CTPP, thousand tons of conventional fuel

17,056.2

20,934.5

22,918.4

Volumes of finished 17,778.1 products, thousand tons

Needs extension the total coal production

Lower heat value, kcal / kg

5435.3

Ash content of marketable products, %

18.9 (continued)

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Table 4 (continued) Characteristic

2025 Sulfur content, %

2030

Cost of marketable products, UAH million

49,201.6

Costs for installation of treatment equipment, UAH million

17,460.5

Emission taxes, UAH million Total costs, UAH million

2035

2040

1.8

1487.7 68,149.8

Table 5 Forecast of the optimal structure of coal products under option 4 Characteristic Forecast indicators

2025

2030

2035

2040

Electricity generation at CTPP, TWh

37.74

47.95

59.69

68.59

Total coal production, thousand tons

37,950.0 36,705.0 35,090.0

35,090.0

Volumes of coal sent for processing, thousand tons

32,506.2 26,120.1 22,454.9

24,769.3

Volumes of enrichment, thousand tons

19,367.7 15,741.2 13,896.7

15,084.6

Total finished coal products, thousand tons

24,811.5 26,326.1 26,531.7

25,405.3

Total volumes of 22,836.9 22,266.1 21,292.7 finished coal products, thousand tons of conventional fuel

21,292.7

Needs of coal 13,970.8 17,056.2 20,934.5 products for CTPP, thousand tons of conventional fuel

22,918.4

(continued)

Mathematical Model of Optimal Support of Thermal Energy …

85

Table 5 (continued) Characteristic

2025

Optimal forecast Volumes of of needs finished products, thousand tons

17,708.8 21,720.3 The ratio of Needs extension needs and the total coal volumes of coal production does not provide 5576.0 5505.0 restrictions on particulate emissions 18.1 19.2

Lower heat value, kcal / kg Ash content of marketable products, % Sulfur content, % Cost of marketable products, UAH million

2030

2.0

2035

2040

1.9

48,020.4 58,707.9

Costs for 10,690.0 installation of treatment equipment, UAH million

7859.9

Emission taxes, UAH million

4111.8

2927.4

Total costs, UAH 61,637.8 70,679.6 million

fourth options evidence that the savings from the implementation of new technologies can be 6512 million UAH. The results of calculations show that without the modernization of the coal industry, the supply of domestic generating enterprises with domestic fuel is possible only until 2030, both in the case of restrictions on calorific value of fuel and restrictions on emissions and without taking them into account. To provide CTPP with fuel in the required volumes and with acceptable thermophysical characteristics for forecasted electricity production, it is necessary to modernize the mines by new high-performance coal mining equipment, as well as commissioning new benches at existing or new mines. In addition, it is extremely important to send to concentrators the optimal amount of coal produced, which is a guarantee of providing power plants with fuel of the required quality with minimal financial costs. From the analysis of the calculated data it is possible to draw one more conclusion—it is not profitable for the electric generating enterprises to install the refining equipment at the enterprises. It is cheaper to pay a tax on emissions of GHG and pollutants.

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5 Conclusions 1.

2.

3.

4.

A mathematical model of technological re-equipment of production capacities of the coal industry in accordance with the demand for thermal power plants of the UES of Ukraine has been developed. Unlike the known ones, the model combines detailed consideration of technical and economic indicators of technological equipment of mines and coal processing plants with algorithms for coordination of flows of all types of fuel, which made it possible to forecast the structure of finished coal products with the necessary quality indicators at all production sites. In turn, this allowed within a single model to obtain a technologically complete and closed representation of the processes of production, conversion and consumption of coal products in the production sequence “mine-factory-TPP” taking into account the costs of purchasing coal products by thermal power plants, gas cleaning equipment and tax payment for emissions of GHG and pollutants from coal combustion. Model calculations of the volumes of heat energy supply with domestic coal products have confirmed the possibility of meeting the needs of power plants by 2030 at the current level of development of the coal industry. Outside this period, the projected growth of electricity and heat production will require coordinated technological modernization of its coal and enrichment sectors, and in the longer term—the import of coal fuel in volumes that depend on the level of development of alternative technologies for electricity generation, including renewable energy sources. This conclusion remains relevant both in the case of taking into account the restrictions on the caloric value of fuel and emissions of GHG pollutants and without taking them into account. Calculations of the forecast structure of marketable coal products for the energy sector of Ukraine on the criterion of minimum costs and its quality indicators proved that the optimal level of enrichment of extracted coal is currently in the range of 70–85%. The results of the calculations of the first and second options showed that the coordinated introduction of new technologies in mines and concentrators of Ukraine in the short term will save thermal power plants 1.8 billion UAH, the third and fourth options—6.5 billion UAH. The problem of meeting environmental requirements is the lack of economic incentives for thermal power plants to use treatment equipment due to the higher cost of its installation and operation. Existing taxes on emissions are not sufficient to form these incentives. This problem may be partially solved by mutually agreed implementation of new technologies in coal production and enrichment. However, according to the results of the calculations, these measures are insufficient to ensure full compliance with EU and world environmental restrictions, in particular on particulate matter emissions due to the high ash content of domestic coal. It is obvious that the creation of incentives for the installation of treatment equipment lies in the modernization of tax legislation in the field of environmental protection.

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Modelling the Impact of Energy-Saving Technological Changes on the Market Capitalization of Companies Olexandr Yu. Yemelyanov , Tetyana O. Petrushka , Anastasiya V. Symak , Liliia I. Lesyk , and Oksana B. Musiiovska

Abstract The authors performed modelling of the impact of energy-saving technological changes on the market capitalization of companies. The regularities of the influence of energy-saving technological changes on the financial condition of enterprises are considered. Partial mechanisms of such influence are allocated. Factors influencing the market value of enterprises, such as the size and degree of fluctuations in the profitability of their activities, the volume of these activities, and the level of financial stability of enterprises are identified. Regarding the factors directly affecting the market capitalization of enterprises, they include the expected amount of profit of enterprises and the level of risk of their activities. In order to assess the market value of companies’ equity, it is proposed to capitalize their net profit at a risk-free rate with a further decrease in the result obtained in proportion to the level of riskiness of companies. An evaluation indicator of this level has been constructed, which has a transparent economic meaning and provides a reasonable assessment of the degree of fluctuation of expected profits of companies. A method for decomposing the growth of market capitalization of companies due to introducing energy-saving technologies is proposed. According to the analysis of a sample of one hundred Ukrainian companies, a significant impact of introducing technologies to reduce natural gas consumption on the financial condition and market capitalization of these companies is established. It is established that with the increase in the relative level of investment in introducing energy-saving technologies at the surveyed enterprises there was an increase in the growth rate of their market capitalization. It was also found that introducing energy-saving technologies at enterprises mainly led to the growth of market capitalization of the studied companies. The use in the practice of companies the methodological principles of modelling the impact of energy-saving technological changes on the market capitalization of enterprises, proposed by the authors, will increase the validity of decisions on implementing projects of such changes.

O. Yu. Yemelyanov · T. O. Petrushka · A. V. Symak · L. I. Lesyk · O. B. Musiiovska (B) Lviv Polytechnic National University, Lviv, Ukraine e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Zaporozhets (ed.), Systems, Decision and Control in Energy III, Studies in Systems, Decision and Control 399, https://doi.org/10.1007/978-3-030-87675-3_5

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Keywords Energy-saving technological change · Company · Modelling · Market capitalization · Financial condition · Impact · Natural gas

1 Introduction Reducing unemployment [1], improving social security [2] and overall welfare [3] are among the main challenges currently facing the governments of many countries. Significant and long-term economic growth is needed to achieve these goals. This need contradicts the energy strategies of most countries, which provide for a gradual reduction in fossil energy consumption to preserve their reserves for future generations [5], increase the energy independence of countries [6], reduce the cost of extraction of non-renewable energy carriers [7] taking into account their negative impact on the environment [8]. The solution to this contradiction should be based on increasing energy efficiency [9] and the development of green energy [10]. In turn, increasing the efficiency of consumption of non-renewable energy sources requires, among other things, largescale implementation of energy-saving technological changes in the economy. Such changes should take place primarily in those sectors of the economy that are characterized by significant consumption of fossil energy sources, in particular, coal, natural gas and crude oil. However, for companies to be interested in energy-saving technological changes, their owners and managers must have a clear idea of the positive economic results of such changes. These results may be reflected, in particular, in reduced costs, increased revenues and profits, but the most generalized economic result of introducing energy-saving technologies is an increase in the market value of companies. It is the confidence in such an increase that can serve as a powerful incentive for business owners and managers to take measures to save energy resources. At the same time, forecasting the impact of energy efficiency measures on companies’ market value is an extremely difficult task. This is due to the fact that the value of companies, in particular the market value of their equity, is influenced by many factors, between which there are complex relationships. Therefore, to solve this problem, it is necessary to pre-model the impact of energy-saving technological changes on the market capitalization of companies. Many scientists from different countries are focused on the problem of the economical consumption of energy resources, its technical, economic and social aspects. In particular, scientists have identified the main factors that affect the consumption of energy resources [11], identified the main obstacles to energy efficiency [12], substantiated various ways to overcome them [13], assessed the prospects for green energy [14] etc. Special attention should be paid to the attempts made by some scientists to assess the economic consequences of the introduction of energy-saving technologies and other energy-saving measures. In particular, the possibilities of ensuring economic growth against the background of reducing the consumption of certain types of

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energy resources are considered. However, the data obtained by different scientists for different countries are quite contradictory. Thus, in [15], a positive relationship between energy consumption and economic growth is established. In [16], for the case of natural gas, this connection was established only in the long run. However, in [17], no such connection was found. The studies that show the possibility of ensuring sustainable economic well-being while reducing energy consumption [18] and proved the possibility of a long-term increase in income of enterprises with a simultaneous increase in energy efficiency [19] should also be noted. In addition, some scientific papers are devoted to the so-called rebound effect of energy consumption, in which an increase in energy efficiency causes an increase in energy consumption [20]. However, some scientists have not found this effect [21] or found that this effect is quite small [22]. The growth of production volumes can occur with the simultaneous replacement of fossil energy sources with alternative types. However, as noted in [23], this trend is not typical for all countries. A similar conclusion was obtained in [24] (for the countries of the European Union) and in [25] (for the countries of the Balkan and the Black Sea regions). A separate group of publications consists of those in which the authors identify the reasons that cause barriers to introducing energy-saving technologies and other measures to improve energy efficiency. In particular, such reasons include lack of information [26], low profitability of energy-saving projects [27], lack of financial resources [28], etc. At the same time, among the ways to overcome these barriers, scientists highlight, first of all, credit support [29] and raising awareness of business owners and managers of the expected results of energy-saving measures [30]. Such an increase, in turn, requires the owners and managers of enterprises to have competence in forecasting the consequences of implementing measures to improve energy efficiency, in particular, energy-saving technologies. Currently, there is a number of publications on assessing the impact of such measures on the environment [31], product competitiveness [32], the cost of production [33], profits of companies [34], and their financial condition [35]. However, the issue of the impact of energysaving technological changes on the market capitalization of companies, which is one of the most generalizing indicators of their activities [36], has been insufficiently studied. At the same time, many companies when making investment decisions use the increase in market capitalization as a criterion for the feasibility of implementing these decisions [37]. Given the above, it seems necessary to perform modelling of the impact of energysaving technological changes on the market capitalization of companies. For this purpose, in particular, it is necessary to establish the factors influencing the market value of the enterprise; determine the place occupied by implementing energy-saving technological changes; build a model for estimating the market value of companies considering such changes; test this model on the example of enterprises.

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2 Theoretical Aspects of the Impact of Energy-Saving Technological Changes on the Financial Condition of Enterprises and Their Market Value Implementing energy-saving technological changes at enterprises, provided that these changes are quite large-scale, can significantly affect some characteristics of these enterprises. In particular, such characteristics include the financial condition of enterprises, primarily their profitability, financial stability and business activity. In turn, the financial condition of enterprises largely determines their market value. Considering the patterns of the impact of energy-saving technological changes on the financial condition of the company, we should identify three partial mechanisms of such impact, namely: (1)

(2)

(3)

The mechanism of influence caused by the reduction of the enterprise’s unit cost of production as a result of its implementing energy-saving measures. The magnitude of this decrease is caused by the following main factors: the norms of consumption of certain types of energy resources for manufacturing products in accordance with and after implementing energy-saving measures at the enterprise; energy prices, the specific consumption of which is expected to be reduced; the amount of additional costs associated with implementing energy-saving measures at the enterprise (such costs may include, in particular, additional costs for depreciation of purchased energy-efficient equipment, costs for the purchase of other types of energy resources, if one of their types is to be replaced by another, etc.). The mechanism of such influence provides that the reduction of the specific cost of production at constant prices and natural volumes of its production leads to an increase in profits. This, in turn, causes an increase in product profitability. Also, reducing the company’s unit cost of production may allow it to slightly reduce the level of prices for these products and due to price competition to increase sales of its products. Under such conditions, the company’s profit will increase based on the total volume of its products; The mechanism of influence caused by the growth of physical volumes of production and sale of the enterprise’s products owing to implementing measures on energy saving; In turn, this growth can be caused by two main reasons. First, implementing energy-saving projects may involve a simultaneous increase in the value of the production capacity of the enterprise to manufacture certain types of products. Secondly, the growth of physical sales of the enterprise may be caused by increasing demand for these products due to lower prices, which, in turn, is due to increased energy efficiency. If the company had a certain reserve of production capacity before the growth of demand for its products, then after such growth, this reserve can be fully or partially used; The mechanism of influence caused by the increase in the size of the enterprise’s capital owing to the realization of investment measures on energy saving. This increase is a consequence of the investment costs incurred by the company

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to implement energy- efficiency measures. In this case, investment should be considered as a basic factor in changing the financial condition of the enterprise because such investment ultimately leads to a reduction in unit costs of energy resources, and growth in the physical volume of its production and marketing. As a result, implementing investment measures for energy saving may significantly change the enterprise’s capital structure. This occurs if the structure of sources of funding these measures differs significantly from the capital structure of the company before their implementation. Accordingly, the volume and structure of funding sources for energy-saving projects implemented at the enterprise can have a significant impact on the level of its financial stability. It should also be noted that the level of financial stability of the enterprise may also be affected by an increase in its profits due to implementing energy-saving investment projects. This effect is due to the fact that the growth of corporate profits at a constant rate of capitalization causes an increase in the market value of equity of economic entities, which, in turn, has a positive impact on their financial stability. Thus, the partial mechanisms described above, which determine the impact of investments in energy efficiency on business activity, profitability and financial stability of enterprises, are interrelated. Therefore, it is possible to combine them into one general mechanism of influence of implementing energy-saving investment projects on the financial condition of the enterprise, as shown in Fig. 1. It should also be noted that business activity, profitability and financial stability of the enterprise are assessed mainly by relative indicators. Accordingly, in addition to the absolute parameters of energy-saving investment projects (investment volumes, energy consumption rates and physical volumes of products), it is advisable to highlight the relative parameters. In this case, these project characteristics should be linked to the initial (i.e. before implementing energy-saving projects) values of similar characteristics of the enterprise. Then it is possible to distinguish the following parameters of energy-saving investment projects at the enterprise which directly affect the indicators of its financial condition: • the relative level of consumption norms of certain types of energy resources for projects, which is calculated by comparing the value of these norms after implementing investment measures for energy saving and basic (before implementing these measures) values of the relevant norms; • the relative level of physical sales of projects, which is calculated by comparing the value of these volumes after implementing energy-saving investment measures and basic (before implementing these measures) volumes of relevant products of the enterprise; • the relative level of investment in energy-saving projects, which is estimated by the ratio between the volume of investment in energy-saving projects and the value of the initial (before implementing the projects) total assets of the enterprise. The selection of mechanisms of the influence of implementing energy-saving measures on the financial condition of enterprises is the initial stage of determining

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Change in the company's profit based on the total volume of its products

Increasing demand for products

Reduction of product prices

Indicators of energy-saving investment projects at the enterprise

Decrease in specific cost of production

Volumes of investments in projects

Volumes of financing from own sources Change in the amount of equity of the enterprise

Change in the level of profitability of the enterprise The current level of profitability of the enterprise

Increase in physical sales volumes

Change in operating income of the enterprise

Volumes of financing from loan sources Structure of project funding sources

Change in the level of financial stability of the enterprise

Change in business activity of the enterprise

The current level of financial stability of the enterprise

The current level of business activity of the enterprise

Fig. 1 The general mechanism of influence of implementing energy-saving investment projects on the financial condition of the enterprise

the role that these measures play in shaping the market value of companies. At the same time, since the market value of enterprises is an absolute indicator, in addition to the relative parameters of their financial condition, the scale of energy-saving measures that are implemented should be taken into account. Thus, it is possible to identify the following factors influencing the market value of enterprises:

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(1)

(2) (3)

95

the level of profitability of their activities, which largely depends on the business activity of enterprises. Not only the average expected value of this level is important but also the degree of fluctuation of possible values of the level of profitability relative to its average value; the volume of economic activity of enterprises, in particular, revenues from sales of products; the level of financial stability of enterprises.

These factors have an indirect impact on the market value of enterprises. Regarding the factors of direct influence, they should, first of all, include: (1)

(2)

the expected amount of profit of enterprises. This indicator is determined by the expected values of the level of profitability of enterprises and the volume of such activities; the level of risk of enterprises’ activity. This indicator depends on the degree of fluctuations in the profitability of companies and their financial stability.

It should be noted that estimating the expected amount of profit of enterprises from a methodological point of view is not a difficult task (for this purpose, you can use the indicator of mathematical expectation). However, the assessment of the level of risk of companies’ activities requires a preliminary design of the indicator by which such an assessment will be carried out.

3 Designing an Indicator for Assessing the Riskiness of the company’s Activities First consider the case when the company’s profit can take only two values—zero (when the company’s activities do not bring any profit) and some positive value. Let us assume we know the probability of obtaining this positive value of profit. Then the mathematical expectation of the company’s profit will be determined by the following formula: Pc2 = P · I,

(1)

where Pc2 is the mathematical expectation of the company’s profit if this profit can take only two values - zero and a positive number; P is a positive value of the company’s profit; I is the probability of obtaining this profit. Under these conditions, the level of risk of the enterprise will be 1 − I. The same result can be obtained by calculating the coefficient of variation on the average linear deviation of the company’s profit: l=

|0 − Pc2 | · (1 − I ) + |P − Pc2 | · I P · I · (1 − I ) + (P − Pc2 )I = = 2 · (1 − I ), Pc2 P·I

(2)

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where l is the coefficient of variation by the average linear deviation of the company’s profit if this profit can take only two values—zero and a positive number. For non-negative values of a random variable, the coefficient of variation of the linear deviation may not exceed 2. Therefore, for the case under consideration, the level of risk of the company can be represented as the result of dividing expression (2) by 2. Now, turn to a more complex but much more realistic case when the company’s profits can acquire not two but an arbitrary number of values. We assume that the probability of occurrence of each of these values is known. In other words, we know the probability of occurrence of each situation in which the company may find itself and to which a certain amount of its profit will correspond. The values of such probabilities and the corresponding profit values can be obtained, in particular, using the results of an expert survey. As an additional condition, we introduce the same probability of obtaining each value of profit (i.e. the same probability of each situation in which the company may find itself). This condition is not fundamental, as it can always be achieved by increasing the number of situations under consideration (i.e. by highlighting additional situations with the same probability and the same profit). Now, place all the situations in ascending order of the company’s profits (or at least its non-decline, if there are situations with the same amount of profit), i.e.: P1 ≤ P2 ≤ ... ≤ Pi ≤ ... ≤ Pn−1 ≤ Pn ,

(3)

where Pi is the company’s profit in the i-th situation; n is the number of situations in which the company may find itself. Consider an ordered set of such quantities: P1 ; P2 −P1 ; . . . ; Pi −Pi−1 ; . . . Pn−1 −Pn−2 ; Pn −Pn−1 .

(4)

Each element of this set can be associated with an element of the set of values of the probability of not receiving a certain amount of profit: 0; 1/n; . . . ; (i−1)/n; . . . ; (n−2)/n; (n−1)/n.

(5)

Then the set of values of the mathematical expectation of profit will look like this: P1 ; (P2 −P1 )(1−1/n); . . . ; (Pi −Pi−1 )(1−(i−1)/n); . . . (Pn−1 −Pn−2 )(1−(n−2)/n); (Pn −Pn−1 )(1−(n−1)/n).

(6)

Given the above, the value of the riskiness of the company can be estimated by finding the weighted average of the elements of the set (5). The weights will be the fractions of the values of the elements of the set (6) in the total value of these values: n R=

  · (Pi − Pi−1 ) · 1 − i−1 n   , n  i−1 i=1 (Pi − Pi−1 ) · 1 − n

i=1

 i−1  n

(7)

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where R is the level of riskiness of the company’s economic activity. It should be noted that the value of indicator (7) cannot exceed 1. In this case, the closer this value is to 1, the more risky is the activity of a particular firm.

4 Building a Model of the Market Capitalization of the Company First, assume that the company’s profit is the same in all situations. Then the risk of the company’s activity is absent, and the market value of its equity will be determined as the ratio of profit to the risk-free discount rate (i.e. the rate that does not include a premium for risk). If the company’s profit differs in different situations, then under such conditions the result of the obtained ratio should be reduced in proportion to the level of risk calculated by formula (7). Taking into account these considerations, we obtain the following expression to determine the value of the market capitalization of the company:  C=

Pcn r

 · (1 − R),

(8)

where C is the value of the market capitalization of the company; Pcn is the value of the mathematical expectation of the company’s profit; r is the risk-free discount rate. The Pcn index is calculated by the following formula: Pcn =

n 1  · Pi . n i=1

(9)

In this case, the following equality is valid:   n n   i −1 1  · . Pi = (Pi − Pi−1 ) · 1 − n i=1 n i=1

(10)

Considering this, formula (8) can be represented as follows:   n  i−1    i−1 · − P · 1 − (P ) i i−1 i=1 n  · 1− C= n n i−1 i=1 (Pi − Pi−1 ) · 1 − n    n n  i −1 1  1  i −1 · · (Pi − Pi−1 ) · 1 − . Pi − · = n · r i=1 r i=1 n n 

Pcn r

(11)

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Let us consider the possibility of taking into account energy-saving technological changes as a factor of changing the market capitalization of the company. For this purpose, for each situation in which the company may find itself, we match an additional indicator. This indicator will be the average expected cost of purchasing the type of energy resources, the reduction of consumption of which is expected to be achieved due to introducing energy-saving technologies. Then the magnitude of the increase in profits of the enterprise in a particular situation due to introducing energy-saving technologies can be determined by the following formula: Pi = (E i · α − Co1 ) · β − Co2 ,

(12)

where ΔPi is the value of profit growth of the enterprise in the i-th situation due to introducing energy-saving technologies; E i is average expected costs for the acquisition of the type of energy resources, the reduction of consumption of which is expected to be achieved due to introducing energy-saving technologies; α is the expected share of E i reduction after introducing energy-saving technologies; C o1 is additional costs associated with introducing energy-saving technologies, which depend on the volume of production using these technologies; β is the expected rate of profit growth due to the increase in sales of the enterprise due to the reduction of its cost after introducing energy-saving technologies; C o2 is additional costs associated with introducing energy-saving technologies that do not depend on the volume of production. By calculating the expected values of indicator (12) for each situation in which the company may find itself after introducing energy-saving technologies, you can calculate a new value of the level of risk by formula (7). Then, according to formula (11), you can calculate the increase in market capitalization of the company due to introducing energy-saving technologies, as well as decompose this increase: C = C1 − C0 = C1 + C2 + C3 ,

(13)

where C is an increase in the market capitalization of the company due to introducing energy-saving technologies; C 1 , C 0 is the market capitalization of the company, calculated by formula (11), before and after implementing energy-saving technological changes; C 1 is an increase in the market capitalization of the company due to introducing energy-saving technologies, because of changes in the mathematical expectation of the company’s profit; C 2 is an increase in the market capitalization of the company due to introducing energy-saving technologies, because of changes in the level of risk of the company; C 3 is an increase in the market capitalization of the company due to introducing energy-saving technologies, because of the joint change of the mathematical expectation of the company’s profit and the level of risk of its activities. The indicators C 1 , C 2 and C 3 can be calculated by the following formulas: C1 = (Pcn1 − Pcn0 ) · (1 − R0 );

(14)

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C2 = Pcn0 · (R0 − R1 );

(15)

C1 = (Pcn1 − Pcn0 ) · (R0 − R1 ),

(16)

where Pcn1 , Pcn0 is a mathematical expectation of the company’s profit, calculated by formula (9), respectively, after and before introducing energy-saving technologies by the company; R1 , R0 is the level of risk of the company, calculated by formula (7), respectively, after and before introducing energy-saving technologies. Using formulas (14) and (15), it is possible to make a relative assessment of the level of impact on the growth of market capitalization of the company, which occurred due to introducing energy-saving technologies, two factors—the mathematical expectation of the company’s profit and risks. For this purpose, it is advisable to use the following expressions: m1 =

C1 (Pcn1 − Pcn0 ) · (1 − R0 ) ; = C1 + C2 (Pcn1 − Pcn0 ) · (1 − R0 ) + Pcn0 · (R0 − R1 )

(17)

m2 =

C2 Pcn0 · (R0 − R1 ) , = C1 + C2 (Pcn1 − Pcn0 ) · (1 − R0 ) + Pcn0 · (R0 − R1 )

(18)

where m1 is the relative level of influence of changes in the mathematical expectation of the company’s profit on the growth of its market capitalization, which occurred due to introducing energy-saving technologies; m2 is the relative level of impact of changes in the company’s riskiness on the growth of its market capitalization, which occurred due to introducing energy-saving technologies. Thus, the market value of the company should be assessed at least twice using formula (11): for the case when the company will not make energy-saving technological changes, and for the case when such changes in the company will occur. However, one should consider the fact that the actual change in the market value of the enterprise (obtained according to calculations or empirical data) may not correspond to the change in this value, which is due to introducing energy-saving technologies in the company. This discrepancy may be due to the fact that, in addition to energysaving technological changes, some other changes may occur in the company and in its external environment during the period of implementing energy-saving technologies. Then, to assess the impact of energy-saving technological changes on the change in the market capitalization of the company, you can use the following indicator: C1 (C1 /C0 ) = , Iec =  Cf1 C f 1 /C0

(19)

where I ec is an indicator for assessing the impact of energy-saving technological changes on changes in the market capitalization of the company; C f 1 is the actual value of the company’s market capitalization after the introduction of energy-saving technologies.

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Let C 1 exceed C 0, . Also, assume that the value of (19) exceeds 1. Then it will mean that in the internal or external environments of the company there have been some negative changes, which to some extent offset the positive effect of the introduction of energy-saving technologies. If the value of indicator (19) is less than 1, it will mean that in the internal or external environments of the company there have been some positive changes that have complemented the positive effect of the introduction of energy-saving technologies.

5 Assessing the Impact of Introducing by Ukrainian Companies Technologies to Reduce Natural Gas Consumption on the Financial Condition and Market Capitalization of These The Assessing the impact of energy-saving technological changes on the market capitalization of companies requires a preliminary analysis of the impact of these changes on the financial condition of these companies. In order to perform such an analysis, 100 enterprises from the western region of Ukraine, which belong to three types of economic activity, were selected. For each type of economic activity, the surveyed enterprises were distributed according to the relative level of investment in projects for introducing technologies to reduce natural gas consumption, as shown in Table 1. The gradation of this level is as follows: a high relative level of investment (exceeds 0.1); an average relative level of investment (from 0.05 to 0.1); a low relative level of investment (greater than zero, but less than 0.05); a zero relative level of investments (there were no investments in energy-saving technologies in the reporting period). Based on statistical and accounting data of the surveyed enterprises, the values of the following indicators of their financial condition were calculated: capital intensity of products (defined as the ratio of operating income of enterprises to their total Table 1 Distribution of surveyed enterprises by the relative level of investment in projects for introducing technologies to reduce natural gas consumption The relative level of investment, unit share

High

Number of enterprises by types of economic activity, units Manufacture of metal products

Manufacture of glass and glass products

Production of bricks, tiles and other building materials from clay

6

5

4

13

6

8

Low

7

9

6

Zero

16

7

13

Average

Source Compiled by the authors

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assets), product profitability, return on total capital of enterprises on their net profit and average annual autonomy ratio. The latter ratio characterizes the share of companies’ equity in the total amount of their capital. The calculations of these indicators were carried out according to the aggregate data of all enterprises from each of their groups. Comparing the values of the relevant indicators for 2019 with their value for the previous year, the growth rate of the considered indicators of the financial condition of the surveyed enterprises was calculated. Information on these indices is given in Table 2. As follows from the data presented in Table 2, for all types of economic activity, the introduction of energy-saving technologies by the surveyed enterprises had a positive impact on most indicators of the financial condition of companies. In particular, there are tendencies to an increase in the profitability of products and capital of the surveyed enterprises with an increase in the relative level of investment. Thus, implementing technologies to reduce natural gas consumption had a positive impact on the profitability of products and capital of the surveyed enterprises. If we analyze Table 2 Average growth rates of individual indicators of the financial condition of the surveyed enterprises by type of economic activity during 2018–2019, times Names of indicators of financial condition of enterprises

The relative level of investment, unit share

Manufacture of metal products

Manufacture of glass and glass products

Production of bricks, tiles and other building materials from clay

The capital intensity of products

High

0.99

0.97

0.94

Average

0.96

0.95

0.91

Low

0.91

0.92

0.87

Zero

0.90

0.89

0.85

High

1.13

1.07

1.09

Average

1.14

1.05

1.05

Low

1.09

1.06

1.03

Zero

1.06

1.03

1.02

1.19

1.20

1.11

Product profitability

Profitability of High total capital Average

Coefficient of autonomy

1.16

1.19

1.09

Low

1.17

1.13

1.08

Zero

1.13

1.09

1.04

High

0.96

0.95

0.93

Average

1.04

1.02

1.01

Low

1.03

1.01

1.04

Zero

1.02

1.01

1.02

Source Calculated by the authors based on the data on the activities of the surveyed enterprises for the period of 2018–2019

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the dynamics of the coefficient of autonomy, we can see that introducing energysaving technologies had a positive impact on the financial stability of enterprises with an average level of investment and negatively, if enterprises had a high level. Regarding the direct impact of introducing energy-saving technologies on the change in the market value of the surveyed enterprises, the results of the relevant calculations are presented in Table 3. As follows from the data given in Table 3, for all types of economic activity, the increase in the relative level of investment in introducing energy-saving technologies is accompanied by an increase in the growth rate of the estimated capitalization of enterprises due to introducing energy-saving technologies. Table 3 also presents the results of calculating the relative level of influence of two factors (mathematical expectation of company profits and the level of risk of Table 3 Averaged indicators of the impact of introducing energy-saving technologies by groups of surveyed enterprises on changes in the market capitalization of these enterprises Names of indicators

The Manufacture Manufacture relative of metal of glass and level of products glass investment, products unit share

Production of bricks, tiles and other building materials from clay

(1) The growth rate of the estimated capitalization of enterprises due to introducing energy-saving technologies, times

High

1.23

1.16

1.21

Average

1.16

1.15

1.17

Low

1.11

1.11

1.15

(2) The relative level of influence of High changes in the mathematical expectation Average of companies’ profits on the growth of Low their market capitalization due to introducing energy-saving technologies (3) The relative level of impact of High changes in the riskiness of companies on Average the growth of their market capitalization Low due to introducing energy-saving technologies

0.43

0.38

0.53

0.47

0.42

0.49

0.51

0.41

0.42

0.52

0.57

0.42

0.48

0.53

0.45

0.45

0.54

0.53

(4) The actual growth rate of the estimated capitalization of enterprises due to introducing energy-saving technologies, times

High

1.25

1.14

1.27

Average

1.19

1.19

1.19

Low

1.09

1.17

1.20

(5) Indicator for assessing the impact of energy-saving technological changes on the change in market capitalization of enterprises, times

High

0.984

1.018

0.953

Average

0.975

0.966

0.983

Low

1.018

0.949

0.958

Source Calculated by the authors based on the data on the activities of the surveyed enterprises for the period of 2018–2019

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Table 4 Actual values of the F-criterion for assessing the impact of introducing energy-saving technologies on the change of market capitalization of the surveyed enterprises Names of indicators

Manufacture Manufacture Production of metal of glass and of bricks, products glass tiles and products other building materials from clay

(1) The growth rate of the estimated capitalization of enterprises due to introducing energy

7.03

6.54

6.85

(2) The actual growth rate of the estimated 6.46 capitalization of enterprises due to introducing energy

5.75

6.02

Source Calculated by the authors

their activities) on the growth of market capitalization of the surveyed companies due to introducing energy-saving technologies. It turned out that for most groups of the studied enterprises there is no definite dominant factor, i.e. the influence of each of these two factors is approximately the same. In addition, as evidenced by the data in Table 3, for all groups of surveyed enterprises, the values of the indicator for assessing the impact of energy-saving technological changes on the change in the market capitalization of companies are close to 1. Thus, introducing energy-saving technologies in enterprises has mainly led to an increase in the market capitalization of the surveyed companies. In order to obtain more reliable results of the assessment of the impact of introducing energy-saving technologies on the change of market capitalization of the studied enterprises, variance analysis was applied. Its results are presented in Table 4. All values in Table 4 exceed the critical value of the F-criterion with the significance level α = 0.05. Thus, the relative level of investment in projects to implement technologies to reduce natural gas consumption affects the market capitalization of the studied enterprises. It follows that for these enterprises there is a significant impact of energy-saving technological changes on the market value of equity. Forecasting the impact of energy efficiency measures on companies’ market value is an extremely difficult task. This is because the value of companies, in particular, the market value of their equity, is influenced by many factors, between which there are complex relationships. Therefore, to solve this problem, it is necessary to pre-model the impact of energy-saving technological changes on the market capitalization of companies. It is possible to identify such factors influencing the market value of enterprises as the size and degree of fluctuations in the profitability of their activities, the volume of economic activity of enterprises and the level of their financial stability. These factors have an indirect impact on the market value of enterprises. Concerning the factors of direct influence, first of all, they should include the expected amount of profit of enterprises and the level of risk of their activities.

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In order to assess the market value of companies’ equity, it is advisable to capitalize their net profit at a risk-free rate with a further decrease in the result got in proportion to the level of riskiness of companies. The proposed evaluation indicator of this level has a transparent economic meaning and provides a reasonable assessment of the degree of fluctuation of companies’ expected profits. According to the analysis of a sample of one hundred Ukrainian companies, a significant impact of introducing technologies to reduce natural gas consumption on the financial condition and market capitalization of these companies has been found. In particular, there is a tendency to increase the profitability of products and the capital of the surveyed enterprises with an increase in the relative level of investment in the technological renewal of production. It was also found that with the increase in the relative level of investment in introducing energy-saving technologies at the surveyed enterprises, there was an increase in the growth rate of their estimated capitalization due to introducing energy-saving technologies. In addition, for all groups of surveyed enterprises, the value of the indicator for assessing the impact of energy-saving technological changes on the market capitalization of companies is close to 1. Thus, introducing energy-saving technologies at the enterprises mainly led to the growth of the market capitalization of the surveyed companies.

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Mathematical Approaches to Forecasting and Researching the Technical State of Cylindrical Shells of Energy Objects’ Elements Based on Vibration Monitoring Systems Viktoria Dzyuba

and Artur Zaporozhets

Abstract The chapter is devoted to the problem of accident-free operation of the shell elements of power facilities, from the point of view of existing mathematical approaches to diagnostics of their technical condition. A review of literary sources on the subject of the chapter is carried out, which shows that cylindrical structures are one of the most common classes of combined structures, which are successfully used in a wide variety of fields of modern technology, in particular in the design of energy objects. This is due to the fact that, due to their curvilinear structure, they allow the most rational distribution of material in diverse structures while maintaining the properties of strength and stability. The study of existing monitoring systems is carried out and the need for the formation of an integrated approach to the study of the technical state of spatial energy objects is substantiated. The analysis of the known criteria of permissible fatigue damageability of materials is carried out, which make it possible to determine the fatigue durability at different load amplitudes. In general, the wear curve is presented taking into account the fatigue damage of the material. The individual cases of damage that cannot be investigated using fatigue fracture criteria and strength calculations are considered. Based on this, it is proposed to use specialized computer systems to predict the values of the mechanical characteristics of the stress–strain state. It is assumed that this will increase the accuracy of quality control of the production of thin-walled cylindrical shells in real time. The main stages of an integrated approach to monitoring the technical condition of cylindrical shells of elements of energy objects are formulated. It is assumed that this will lead to a decrease in further destruction and an increase in the serviceability of energy facilities, which, in turn, will allow maintaining an appropriate level of environmental safety. V. Dzyuba Bohdan Khmelnytsky National University of Cherkasy, Cherkasy, Ukraine A. Zaporozhets (B) Institute of General Energy of NAS of Ukraine, Kyiv, Ukraine e-mail: [email protected] State Institution “The Institute of Environmental Geochemistry of National Academy of Sciences of Ukraine”, Kyiv, Ukraine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Zaporozhets (ed.), Systems, Decision and Control in Energy III, Studies in Systems, Decision and Control 399, https://doi.org/10.1007/978-3-030-87675-3_6

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Keywords Mathematical model · Cylindrical shell · Monitoring of technical condition · Energy facilities · Vibration diagnostic systems · Computer system · Environmental safety

1 Introduction Spatial energy objects in the form of cylindrical shells are one of the most common classes of modern combined structures. This is due to the fact that they have important properties, such as strength, stiffness, stability, and damage to which can lead to negative consequences both for the environment and the life of people in general. Structural elements of energy objects that relate to plates and shells include: flanges, piston bottoms, bellows, turbine blades, boilers, cylinders, chemical devices, combustion chambers of engines, rotors, drums and hulls (aircraft, helicopters, missiles, ships, nuclear reactors), versatile building structures and tanks, etc. In most cases, a mathematical model of this kind of structures is represented by large-scale nonlinear systems of differential equations with distributed parameters [1]. Their detailed analysis provides for the creation of software systems based on specialized computer systems, which allow to calculate the parameters of the physical and technical characteristics in the tasks of the theory of shells in real time [2]. The destruction of structural elements during operation can have a different nature of occurrence—from non-observance of the appropriate design conditions in production and during transportation to the presence of a negative impact of external factors. To diagnose the current state of shell-type structures, non-destructive testing methods are used, in particular, vibration monitoring provides data for the formation of integral characteristics of the object under study, which in the future allow to avoid the emergency conditions [3, 4]. Thus, in order to prevent the functional unsuitability of energy infrastructure facilities and preserve their residual resource, it is necessary at the initial stage to determine negative changes in the structure of cylindrical elements and predict their fatigue life (crack initiation) and limiting state.

2 Analysis of Sources The search for effective methods and technologies for diagnosing the current state of cylindrical elements of energy objects is an urgent task among scientists. Significant progress in this direction is impossible without the use of complex intelligent monitoring systems that implement the new concept of Structural Health Monitoring (SHM). The operation of such systems is based on built-in or attached sensor elements that continuously diagnose the objects under study and ensure their productive operation [5–7].

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The work [8] analyzes the existing types of sensor equipment, which are used during operating of intelligent monitoring systems for diverse spatial objects. However, the practical use of such monitoring systems is not always possible, which is associated with a number of specific requirements for the operation of these sensors. This direction of research is very relevant and promising, especially from the point of view of supporting environmental safety, as evidenced by the recent appearance of a large number of publications on this topic, in particular, a number of new results can be found in works [9–11]. In particular, a summary of many years of research on the structural monitoring of storage tanks for environmentally hazardous substances is presented in the book [12]. General characteristics of the existing methods of non-destructive diagnostics are presented in works [6, 11, 12]. It should be noted that vibration monitoring has recently significantly expanded the existing range of measured characteristics due to the automation of diagnostic processes [13]. At the same time, the issue of the accuracy of diagnostics of vibration parameters for complex structures of the shell type remains unresolved in full. New research results regarding the main approaches to monitoring shell structures can be found in the works of N. I. Burau, N. I. Gud, S. A. Tsybulnik, D. A. Shevchuk, S. Kandasamy, A. A. Lakis, S. Haldar, K. M. Liew, X. Zhao, A. J. M. Ferreira and others. Modern research of scientists in the direction of increasing the efficiency of production and operation of shell structures is closely related to the development and improvement of computer systems [1, 2, 4, 14]. Computer systems of this kind are designed for specialized work in a specific field of science or technology. In this regard, considerable attention of scientific works is paid to the issue of creating specialized computer systems for modeling closed shells of variable thickness and calculating their limiting parameters [1, 2, 14–16]. Some another computer systems for environmental monitoring are described in these works [17–19]. A great contribution to the development of specialized computer systems was made by such scientists as V. M. Rudnitsky, P. N. Goncharov, M. I. Gordeichuk, A. A. Melnik, Ya. M. Nikolaychuk, J. Axelson, M. Barr, J. Gannsle, T. Hill, R. Rajsuman et al. Based on this, there is a need for a relatively comprehensive approach for monitoring the technical state of the spatial elements of energy objects: (1) by predicting the values of the parameters of the stress–strain state in the process of manufacturing thin-walled cylindrical structures, which will make it possible to identify manufacturing defects at early stages; (2) using vibration diagnostics systems that assess the technical condition of the object and monitor the damage that has occurred for their further safe operation. Solving the problems of integrated monitoring will increase the durability of shell products, which, in turn, will create new prospects for the development of environmental safety of energy infrastructure facilities.

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3 Purpose and Tasks The study is carried out with the aim of increasing the durability of the operational properties of the cylindrical shells of the elements of energy objects by timely detecting changes in the stress–strain state using vibration monitoring systems and further prediction of the values of the physical and technical characteristics of thin-walled cylindrical shells. To achieve this purpose, it is proposed to solve the following tasks: 1. 2.

3.

to consider the technical requirements for the operation of the vibration monitoring system in real time; to analyze and evaluate a number of existing mathematical models of permissible fatigue damage of material to determine the residual life of cylindrical elements of energy objects; to investigate the physical, technical and geometric characteristics of the parameters of the stress–strain state of thin-walled cylindrical shells.

4 Methods In the study of monitoring systems for the technical state of cylindrical shells of elements of energy objects, an analysis of publications of Ukrainian and foreign scientists was carried out. In the process of considering the mathematical models for predicting the permissible fatigue damage of the material, the method of comparative analysis was used with mathematical apparatus of the mechanics of a deformable body, mathematical analysis, and differential geometry. To predict and evaluate the limiting parameters of thin-walled shells, the following theories were used: differential equations, elasticity, linear algebra using computer modeling methods.

5 Results 5.1 Features of the Operation of Intelligent Vibration Monitoring Systems for the Technical Condition of Shell Structures Analyzing the rapid development of engineering technologies, it becomes clear that the existing diagnostic approaches (preventive maintenance, visual inspection of elements, etc.) are losing their relevance. But new design ideas are emerging based on the general principles of SHM for observing the technical condition of cylindrical structures [6, 7]. Integrated intelligent monitoring systems are a set of sensors that are part of a multi-channel information network, which, in turn, is connected and integrated

Mathematical Approaches to Forecasting and Researching … Table 1 General requirements for vibration monitoring systems

111

No.

Parameters

Criteria

1

Number of measuring coordinates

3 (X, Y, Z)

2

Sensor type

Differential

3

Working frequency range, Hz

0.4–500

4

Storage capacity

>107 indications

5

Time synchronization

Yes

6

Duration of autonomous work

>5 h

7

Working temperature

−30 to 50 °C

with control modules and calculating technical parameters. The implementation of SHM in the design of energy facilities opens up prospects for the creation of “smart structures” with the use of innovative eco materials [9, 10]. However, despite significant achievements in this direction, it is not always possible to coordinate diagnostic methods and methods of fixing sensor sensors with the real conditions of the monitoring objects functioning. One of the most optimal methods for diagnosing the technical state of structural elements is vibration diagnostics, where the measuring devices are vibration velocity and vibration acceleration sensors. It is advisable to carry out vibration monitoring in order to establish the individual integral properties of the object under study, and to identify deviations in the mechanical characteristics of the stress–strain state at the initial stage. Along with this, there are generally accepted technical requirements for vibration monitoring systems, which are given in Table 1 [12]. Since the vibration parameters depend on the existing external and internal operational factors that can affect the final monitoring result, the process of studying a vibration signal is rather complicated, and its accuracy largely depends on the adequacy of the mathematical model. It should be noted that recently there has been a tendency towards a decrease in the cost of electronic components. This can significantly reduce the cost of processing the results of vibration monitoring and expand the analysis of vibration parameters [14, 16].

5.2 Assessment of the Technical Condition of Cylindrical Elements with Using the Fatigue Curve The assessment of the residual life of cylindrical products consists in detecting irreversible changes in the structure of the material and identifying a fatigue crack as a result of cyclic loads. From the moment of fatigue crack formation and its growth to the critical point, the crack length is chosen as the measure of damage assessment [12].

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In the study of structural elements of energy objects, a significant place is occupied by cylindrical metal products, which are subjected to cyclic loads, which provoke the appearance of cracks. If we consider the range of cracks from 0.1 to 0.5 mm, then such loads, depending on the number of cycles N, at which complete destruction will occur, are referred to one of three groups [11]: • I group—quasi-static N = 2…102 ; • II group—low-cycle N = 102 …104 ; • III group—multi-cycle N = 104 …1012 , where vibration loads are a special case— N = 109 … 1012 . It can be noted that for the case of mixed load amplitudes, the total damage is expressed in the sum of the above groups of loads. If we assume that with each i-th cycle the object of research receives some damage, then its destruction occurs after the condition [13] is fulfilled: a=

r  ni = 1, Ni i=1

(1)

where r—number of load levels, ni —number of loading cycles of a structural element at a stress amplitude Hai ; N i —limiting number of cycles of the test sample at the stress amplitude Hai . The sum (1) of the relative fatigue damage varies in the range 0.1 ≤ a ≤ 1. The Weller’s curve (fatigue curve) [3] is used to visually assess the durability of the operational object. The construction of the fatigue curve consists in a graphical representation of the relationships between existing stresses (deformations) and the number of cycles before their failure on the verge of critical values. Let’s illustrate in the general case the wear curve (Fig. 1) with taking into account the summation of fatigue damages (1) at high-cycle loads N = 106 . In Fig. 1: H o —fatigue limit, which is the largest stress at which the material of the studied sample can withstand an unlimited number of cycles; H max —tensile strength of the material of the test sample; H(a)—amplitude of operating tension; N max —the largest number of cyclic tensions to failure of the sample; N i —number of load cycles; N(H o )—inflection point of the wear curve. It follows that the determination of the durability of a structural object consists in analyzing the time interval for which the accumulation of damage is traced until the onset of the crack and the time of the cyclic growth of the crack until the critical length is reached. Exceeding the allowable range for crack length leads to destruction of structural elements.

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Fig. 1 Wear curve with taking into account fatigue damage

5.3 Mathematical Models of Permissible Fatigue Damage of Material Testing of laboratory samples of cylindrical products, as a rule, occurs with a symmetric load cycle, but practical use makes its own adjustments, since cyclic loads with a changed amplitude occur [12]. In this regard, it becomes necessary to coordinate the results of laboratory tests with the real operating conditions of cylindrical objects. The general model, in the context of deformation criteria for predicting durability under a symmetric load cycle, is represented by the Coffin-Manson equation [4]: 

Hf  b c ε pl εe ε   = + = 2N f + ε f 2N f , 2 2 2 E 

where ε—range of elastic–plastic deformation; H f —fatigue factor; b—exponent of fatigue strength. The use of this model makes it possible to establish the relationship between deformation and fatigue durability.

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Scientific novelty of the results associated with the study of the effect of medium stress on fatigue durability, provided that the object of study is in the permissible range of cyclic elastic–plastic deformation [8, 16]. In the case of high-cycle fatigue N = 104 … 1012 , the variation of the range of the average cycle stress to the endurance limit is based on the force conditions of destruction (classic models Gerber, Goodman, Soderberg, Morrow) [11]. During studying real shell-type objects under the influence of an asymmetric load, the need for predicting the accuracy with respect to the occurrence of a crack increases. To do this, it can be used J. Morrow model [12]: 

H f − Hm  b c εeq   = 2N f + ε f 2N f , 2 E where εeq —equivalent range of deformations; c—exponent of fatigue plasticity;  ε f —coefficient of fatigue plasticity; H m —local average stress. The J. Morrow model calculates the average stress of the load cycle due to the modified fatigue strength coefficient of the elastic component. Consider the fatigue fracture model proposed by Smith, Watson and Topper (SWT) [13]:  Har = Hmax

1− R = Ha 2



2 , 1− R

where H ar , H max —amplitudes of the voltages during the reversible and pulse load cycles, R—ratio of the highest and the lowest loads. This model allows to investigate the influence of the asymmetric load cycle coefficient on the material endurance limit. During the study of the average voltage at N = 104 … 1012 , a modification of the SWT model is used:  Hmax εa =



Hf E

2  2b  b+c  2N f + ε f H f 2N f ,

which provides that the parameter R will be stable with an arbitrary admissible combination of model components, since R is the total value of the local mean load and the load amplitude during the pulse load cycle. A generalization of the SWT and J. Morrow models is a fatigue fracture model according to the Ince, Glinka variant: 

e p + εa,eq = εa,eq = εa,eq p

Hf  2b c   2N f + ε f 2N f , E

e where εa,eq , εa,eq —equivalent amplitudes of local elasticity and deformation.

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Note that the use of models SWT, J. Morrow and their various modification options is possible under the following conditions: ε p Hmax εe + > 0. ·  2 2 Hf Thus, the main feature of the model proposed by Ince, Glinka is the possibility of correcting the average stress and taking into account its effect on fatigue durability, which in practice emphasizes the priority of choosing this particular model.

5.4 Prediction of the Values of the Parameters of the Stress–Strain State to Ensure the Durability of Thin-Walled Cylindrical Structures The use of fatigue fracture criteria and strength calculations does not allow to fully investigate all the existing causes of damage in the cylindrical elements of energy objects. A significant part of the unsuitability of cylindrical products is associated with errors at the stage of their design and production. In this regard, there is a need for an integrated approach to monitoring the objects under study, including with the involvement of specialized computer systems (SCS) for the production of thin-walled shell elements [1, 2, 15]. With regard to the shell elements of energy infrastructure facilities, there is a tendency of increasing requirements for efficiency, durability, reliability and a decrease in their mass in general. This results in complex calculations that cannot be performed using existing approximate methods and systems, especially during using boundary conditions with sufficient accuracy. The complexity of calculations is associated with the use of many characteristics of such components as: displacement fields, deformations, stresses, velocities, natural frequencies, vibration modes, and the like. Note that in some cases, it is possible to use application packages (ANSYS, MSC Nastran, Abaqus FEA, COMSOL, SCAD Office) or to calculate unknown parameters by numerical methods, but such approaches are not always effective, since the final algorithm is reduced to solving systems of linear equations large order. In this regard, it becomes necessary to use combined methods based on iterative processes. By combining several numerical-analytical methods, it is possible to achieve an increase in the accuracy of calculating the required parameters without losing the existing connections in the mathematical model. One of the possible results can be found in [1]. It is assumed that predicting the values of the mechanical characteristics of the stress–strain state of thin-walled cylindrical elements, directly during the production process, will avoid further unfavorable changes in their structure associated with imperceptible manufacturing defects.

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The work [15] presents SCS, where one of the structural units is a production process control subsystem, which in real time, due to functionally dependent blocks, provides continuous control over the central program control. The use of such an SCS makes it possible to increase the accuracy of quality control of the production of cylindrical elements in real time, since there is a decrease in the number of elementary operations at each iteration.

6 Discussion The study of existing materials within the scope of this chapter has shown that each individual cylindrical element or complex structural objects requires a detailed study of their technical state and functioning features, which will make it possible to establish individual characteristics for a successful selection of methods and diagnostic tools. During the study of the durability of cylindrical elements, it is necessary to compare, in a time frame, the initial aggregate of damage without disruption and the cyclical increase in the crack to the maximum allowable length. In this regard, the main criteria for the fatigue damageability of the material are considered, which make it possible to coordinate laboratory testing with the real conditions of the functioning of the samples under study. It is analyzed that the model of fatigue fracture according to the Ince, Glinka variant is convenient to use and has important applied value, since it allows changing the value of the average stress and investigating its effect on fatigue durability. For clarity of the wear boundary of the material, it can be used the geometric interpretation, namely, it is necessary to establish the ordinate of the asymptote to which the wear curve approaches. It has been established that the use of vibration diagnostics systems is the optimal method for studying the deviations of the parameters of the stress–strain state of cylindrical elements. However, analyzing the quality of a vibration signal is a complex process that partly depends on the adequacy of the mathematical model. Because the adequacy of the model, in most cases, is tracked only in a limited area of the adequacy of changes in external parameters. One of the ways to solve the problem of monitoring the technical state of cylindrical shells of elements of energy objects is the development of mathematical approaches to the study of the stress–strain state using SCS. In [15], it is shown that the practical implementation of a SCS allows organizing a process control in real time with an increased accuracy of quality control of about 5%. In addition, it can be predicted that in the near future the development of SCS will progress due to the latest advances in microprocessor systems. Thus, in order to form an integrated approach to the study of the technical state of cylindrical shells of elements of energy objects, it is necessary to take into account the following factors:

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• predicting the values of the stress–strain state coefficients during the production process; • ensuring the appropriate technical conditions during transportation; • compliance with operating rules in real operating conditions; • analysis of mathematical models for predicting the permissible fatigue damage to the material; • regular monitoring using existing diagnostic systems.

7 Conclusions The chapter provides a rationale for the relevance of the use of cylindrical shells of elements of energy facilities and the need for regular monitoring of their technical condition. According to the results of the studies carried out, the advantages and disadvantages of the existing diagnostic systems were shown. In particular, a new concept of SHM for monitoring the technical condition of cylindrical elements and the prospects for the development of “intelligent structures” are considered. With using vibration monitoring systems, it is possible to establish hidden defects, initial damage, at the same time, the accuracy of the vibration signal analysis depends on the adequacy of the mathematical model. Analysis of mathematical models that allow studying the permissible fatigue damage of materials has shown that it is appropriate to take into account the effect of medium stress on fatigue durability in a modified version of Ince, Glinka. The main factors contributing to the occurrence of damage and destruction of materials are considered, in addition, solutions to the causes that cannot be taken into account using fatigue calculations are found. Proceeding from this, it is proposed to use specialized computer systems that allow to increase the accuracy of quality control of the production of thin-walled cylindrical shells in real time. This will avoid further irreversible changes in the structure of the cylindrical structures associated with manufacturing defects. And also, it is assumed that the use of such systems will allow organizing the production process with minimal raw material costs. The studies carried out allow us to state that the durability of structural elements of energy objects is ensured when the values of the stress–strain state coefficients and the calculated efforts do not go beyond the limit norms. So, in order to avoid emergency situations during the operation of cylindrical elements of power facilities, it is necessary to use an integrated approach in diagnosing their technical condition. This will allow monitoring their development and predicting the values of the limiting parameters. The above approach opens up new prospects in the trouble-free operation of energy infrastructure facilities, which means that an improvement in the ecological state of the environment can be predicted.

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References 1. Grigorenko, A.Ya., Vlaikov, G.G.: Some problems of elasticity theory for anisotropic bodies of cylindrical form. Kyiv, National Academy of Sciences of Ukraine, S.P. Timoshenko Institute of Mechanics, Technical Center, 217p (2002) 2. Nikolaychuk, Ya., M., Pituh, I.R., Vozna, N.Ya.: Design of specialized computer systems. Ternopil: Terno-graf, 392 p (2010) 3. Li, H., Pang, F., Chen, H., Du, Y.: Vibration analysis of functionally graded porous cylindrical shell with arbitrary boundary restraints by using a semi analytical method. Compos. B Eng. 164, 249–264 (2019). https://doi.org/10.1016/j.compositesb.2018.11.046 4. Najafizadeh, M.M., Isvandzibaei, M.R.: Vibration of functionally graded cylindrical shells based on different shear deformation shell theories with ring support under various boundary conditions. J. Mech. Sci. Technol. 23, 2072–2084 (2009). https://doi.org/10.1007/s12206-0090432-2 5. Staszewski, W., Boller, C., Tomilson, G.: Health Monitoring of Aerospace Structures: Smart Sensor Technologies and Signal Processing, 288 p. Wiley (2004) 6. Bouraou, N., Rupich, S., Tsybulnik, S.: Problems of intellectualizing in the SHM systems: estimation, prediction, multi-class recognition. Sci. J. Ternopil Natl. Tech. Univ. 88(4), 135–144 (2017) 7. Khan, A.A., Zafar, S., Khan, N., Mehmood, Z.: History current status and challenges to structural health monitoring system aviation field. J. Space Techn. 4(1), 67–74 (2014) 8. Rupich, S.: Multi-class recognition of objects technical condition by a classifier based on Neural Network. Thesis for a candidate degree (PhD) in speciality 05.11.13 “Instruments and methods for controlling and determining the composition of substances”. – National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv (2018) 9. Babak, V.P., Babak, S.V., Myslovych, M.V., Zaporozhets, A.O., Zvaritch, V.M.: Methods and models for information data analysis. In: Diagnostic Systems For Energy Equipments. Studies in Systems, Decision and Control, vol. 281, pp. 23–70. Springer, Cham (2020). https://doi.org/ 10.1007/978-3-030-44443-3_2 10. Eremenko, V., Zaporozhets, A., Babak, V., Isaienko, V., Babikova, K.: Using Hilbert transform in diagnostic of composite materials by impedance method. Periodica Polytech. Electr. Eng. Comput. Sci. 64(4), 334–342 (2020). https://doi.org/10.3311/PPee.15066 11. Zaycev, V.M., Shevchuk, D.V.: Study of the channel for measuring the deformation of the monitoring system of buildings and structures in operation. Bullet. Eng. Acad. Ukraine 1, 217–221 (2017) 12. Burau, N.I., Lukyanchenko, O.O., Kostina, O.V., Tsybulnik, S.O.: Structural monitoring of vertical steel tanks. Kyiv, Center for educational literature, 160 p (2019) 13. Srinivasa, C.V, Suresh, Y.J., PremaKumar, W.P.: Finite element studies on free vibration of laminated composite cylindrical skew panels BT. Adv. Mech. Eng. 2014, 174085 (2015). https:// doi.org/10.1155/2014/174085 14. Xia, M., Sun, Q.: Thermomechanical responses of nonlinear torsional vibration with NiTi shape memory alloy – Alternative stable states and their jumps. J. Mech. Phys. Solids 102, 257–276 (2017). https://doi.org/10.1016/j.jmps.2016.11.015 15. Dzyuba, V.A., Zazhoma, V.M., Rudnytskyi, V.M., Steblyanko, P.O.: Automation of processes for calculating the stress-strain state of cylindrical shells. Bullet. Eng. Acad. Ukraine 4, 100–104 (2019) 16. Xie, K., Chen, M.: An analytical method for free vibrations of functionally graded cylindrical shells with arbitrary intermediate ring supports. J Braz. Soc. Mech. Sci. Eng. 43, 100 (2021). https://doi.org/10.1007/s40430-021-02829-5 17. Iatsyshyn, A., et. al.: Application of Open and Specialized Geoinformation Systems for Computer Modelling Studying by Students and Ph.D. Students. CEUR Workshop Proceedings, vol. 2732, pp. 893–908 (2020). http://ceur-ws.org/Vol-2732/20200893.pdf

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18. Zabulonov, Y., Popov, O., Burtniak, V., Iatsyshyn, A., Kovach, V., Iatsyshyn, A.: Innovative developments to solve major aspects of environmental and radiation safety of Ukraine. In: Zaporozhets, A., Artemchuk, V. (eds.) Systems, Decision and Control in Energy II. Studies in Systems, Decision and Control, vol. 346, pp. 273–292 (2021). https://doi.org/10.1007/978-3030-69189-9_16 19. Popov, O.O., et al.: Immersive technology for training and professional development of nuclear power plants personnel. CEUR Workshop Proceedings, vol. 2898, pp. 230–254 (2021). http:// ceur-ws.org/Vol-2898/paper13.pdf

Nuclear Power Engineering

Optical Fiber in Nuclear Power Plants: Applications to Improve the Reliability, Safety and Work Stability of Fault Control Instrumentation Ievgen Zaitsev , Anatolii Levytskyi , Kromplyas Bogdan , and Rybachok Pavlo Abstract The paper presents a design method, which is aimed at reducing the effect of ionizing radiation on fault control instrumentation. It is shown that the use of optical fiber in the structure of fault control instrumentation improves the reliability, safety and work stability of fault control instrumentation. The use of optical fiber provides an opportunity to improve the metrological characteristics of information and measurement channels for valves control system. In most cases the control must be carried out in rather harsh internal working environment of the diagnostic object, which is characterized by the presence of strong magnetic fields, high temperature, high humidity, ozone, cross-guidance between communication lines and ionizing radiation of NPPs. Using the advantages of using valet fault control instrumentation, fault control instruments were implemented for valves control system with optical fiber, which can function in the pressurized zone of NPP reactors. Scheme of the valves control system controller with the fiber-optic sensor proposed. The conducted theoretical researches allowed developing optoelectronic control systems with higher technical characteristics, which allows improving the quality of control of valet system operation in the pressurized zone of NPP reactors. Keywords Nuclear power plants · Valet · Fault control · Optical fiber · Instrumentation · Reactors · Radiation · Shaft

1 Introduction In recent years, Ukraine is increasingly faced with a significant percentage of depreciation of equipment of power plants, almost worn out. Ensuring the reliability and safety of operation as the main equipment of NPPs, TPPs, CHPs, HPPs and PSPs, which today remain the basis of electricity production is an urgent problem. Complete replacement of equipment with a new one in a short time is not a problem because of the need to attract significant funds. An alternative to e-cue replacement is a partial I. Zaitsev (B) · A. Levytskyi · K. Bogdan · R. Pavlo Institute of Electrodynamics of the National Academy of Sciences of Ukraine, Kyiv, Ukraine e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Zaporozhets (ed.), Systems, Decision and Control in Energy III, Studies in Systems, Decision and Control 399, https://doi.org/10.1007/978-3-030-87675-3_7

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replacement of equipment based on its actual technical condition through the use of monitoring and control systems, which ensure, first of all, the reliability of operation of power plants. The safety of NPP operation, especially after the accident at Fukushima-1, set even more stringent requirements for the reliability of the equipment before the designers and manufacturers of control valves for NPPs [1]. In this regard, electronic control and measuring instruments must have a significant drawback, which mechanical means do not have, namely: the inability to generate all standard signals when the mains power is turned off, even if there are internal backup battery sources. The solution lies in the creation and development of low-power units measuring and control means, is primarily concerned with specialized means of switching electrical circuits, because the main and direct commands to control, measuring tools, among which is the adjustment and shut-off valve formed switches type “dry contact”. To ensure the readiness of measuring instruments to perform all functions after a long stay without power supply, it is necessary to provide for the presence in all nodes of the mode of controlled in a state of inactivity and the ability to quickly exit it by commands of the device controller, mode for changing the angle of rotation used only. So, the control and measuring devices of modern controlled-shaft-stop valves at nuclear power plants are not just switches, but complex programming devices with data exchange, therefore, to call them road swings is to narrow their functions. Some manufacturers, in particular in [2], call fault control instrumentation as intelligent control units.

2 Problems of Fault Control Instrumentations Realization for NPP Valves Control System The problem of creating valves control system is the elevated radioactive background and other environmental factors (especially high temperatures), as well as strict requirements for maintenance of valves control system in the reactor shell, significantly limiting the use of electronics, including control and measuring means of electric drives valves control system. One of the options for solving this problem is the creation of such control, measuring devices, in which in the hermetic shell of the reactor are directly only sensors of valves control system, and all equipment for processing, and analysis of sensor signals, the formation of control signals, is located outside the shell. To implement this option, it is necessary that: • sensors and connection lines met all the requirements for equipment in the pressurized area; • number of connection lines was minimal, taking into account the strict design limit of the number of cable branches and lines from the pressurized zone, and the transmission of signals along the lines—especially highly reliable and noisetolerant;

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• length of communication lines with the sensor was in the range of 100–600 m. Analysis of fault control instrumentations for NPP valves control system has shown that the most promising for their functional and technical characteristics are those of electronic equipment. It is possible to adjust the functionality of the valves control system, to improve the performance and adjust the procedure, in order to avoid the need for the necessary and precise automation of information systems. Therefore, all fault control instrumentations for NPP valves control system, to which there are no special requirements for resistance to external influences, completed with electronic fault control instrumentations. The trend that observed is the creation of specialized, structurally and in terms of functional characteristics, oriented to a specific type of valves control system, cannot considered as positive way of development. This path leads to an increase in their nomenclature at such large facilities as nuclear power plants, complicates maintenance, requires an increase in stocks of individual units and the products themselves. The creation of a small number of fairly unified, built on the same ideology, structurally and programmatically related control and measuring instruments is quite persistently required by the corresponding operating services of the NPP, and is confirmed in the relevant regulatory documents [3, 4]. The introduction of modern fault control instruments for NPP valves control system in operation faces additional difficulties. The most important of these is the requirement to introduce new tools without upgrading or changing the design of electrical connections. It is not difficult constructively agree on new developments of fault control instrumentations for NPP valves control system of previous generations, as there is a rather strict unification for them. However, the implementation of data transmission, including promising digital interfaces, requires additional connections of new communication lines, which is problematic at NPP. Therefore, for modern fault control instrumentations for NPP valves control system, which should replace the previous ones, it is necessary to provide communication coordination with the standard means of automated NPP control system, for this purpose it is necessary to develop methods of data transmission reliability of functioning of all complex. For effective control of the valves control system, an informational signal about the actual position of these valves with appropriate accuracy is required. In the previous chapter, there were some difficulties with the unification of fault control instruments for NPP valves control system, due to the discrete samples of the shaft rotation angle for different types of rotation angle sensors. Development sensors of shaft rotation with angle resolution of 0.1% can increase the level of unification of fault control instruments if it satisfies the requirements mentioned above. In addition, all fault control instrumentations are characterized by operation in a small number of ranges of normalized currents—(1–2), when in practice used up to 9 ranges. Different interchangeable modifications of fault control instrumentations for NPP valves control system produced for different ranges, which significantly expands the range of applied means, reduces their interchangeability and increases the required repair stock. In addition, the circuit design solutions of the output current do not allow obtaining a bipolar signal proportional to the deviation from a certain fixed

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state in both directions, which for some systems would allow more efficient and less costly control. According to modern safety standards [3], for new fault control instrumentations for NPP valves control system, torque control on the drive shaft is also provided, which was not available before. The use of spring signaling devices of the set torque level requires a special design of the drive reducer, which is impossible for control and shut-off valves of previous years of production. At the same time, controlling this parameter can significantly extend the life of the entire valves control system. One of the best ways to solve this problem may be to create such fault control instruments that would determine the moment on the shaft by indirect methods, and, for example, by the parameters of energy consumption, especially since work in this direction is underway. In [3], among the requirements for the valves control system was also the requirement to have in the fault control instruments for NPP valves control system or built-in diagnostic tools, or to be adapted to connect external technical diagnostic tools for continuous or periodic technical inspection. Electronic fault control instrumentations can perform these functions [5–15]. The main task is to develop methods and techniques for diagnosing control valves, which would allow maximum use of information obtained from fault control instruments for NPP valves control system about the angle of rotation of the shaft. Considering requirements above, it is often extremely problematic to implement instrumentation based on magnetic or capacitive shaft rotation alarms. For signaling devices with magnetic modules, the requirements for temperature and radiation level for electronic field sensors are critical. For capacitive sensors there are enough basic solutions for remote measurement of their informative parameter, even when its capacity is commensurate with the capacity of the supply conductors [16], but this requires multi-line connection of the sensor to the measuring unit and individual shielding of these lines. The option is impossible without significant modernization of the unit due to the control line in the reactor zone, which is under pressure. In addition, the introduction of multi-line harnesses with a signal frequency of 10–100 kHz introduces an additional element of unreliability due to cross-noise, electromagnetic interference in the screens and the need to create special ground circuits. A solution to this problem may be the use of fiber-optic sensors with the transmission of an informative signal over the fiber-optic cable. Prospects for the use of fiber-optic information-measuring systems and hybrid fiber-optic systems for control, diagnosis, control and emergency protection of power equipment, including nuclear power plants, are analyzed in detail in [17–19], the analysis of industrial fiber-optic system in [20–23]. Construction of control and measuring instruments for valves control system with the use of fiber-optic sensors allows [17–23]: • eliminate the influence of electromagnetic fields; • eliminate indirect electromagnetic radiation; • eliminate cross-barriers in communication lines;

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• eliminate the problems connected with contour grounding and bias voltages in places of connection of dissimilar conductors; • eliminate the problems of arc formation and sparks; • provide high resistance to extreme parameters, including high levels of radiation; • use a thinner, and therefore lighter and stronger, than the electric cable, or increase the number of cores of the cable without changing the cable seal in the reactor area; • due the fact that optical cables are made of dielectric materials, they have no paths of electric current. The factor that hinders the use of fiber-optic sensor and devices based on them for specialized valves control system of NPPs is the absence of industrial samples of fiber-optic sensor and opto-electronic components for them. Therefore, consider the possible options for fiber-optic sensor, which have the prospect of mass production. Fiber-optic sensors made according to the following structures [17–23]: • with change in the characteristics of the fiber (including special fibers); • with change in radiation parameters; • with sensitive element at the end of the fiber. On an industrial scale, fiber-optic sensors of rotation parameters based on the Signac effect [22, 23] and optical gyroscopes produced. However, they cannot used in valves control system controllers for the following reasons: complex electronics, low level of informative parameter, and most importantly—the impossibility of territorial diversity of the electronic unit from the actual fiber-optic sensor and the transmission of information signal by fiber -optical cable. In addition, the fiber-optic sensor on this effect will be very difficult to constructively couple with the already established design solutions of valves control system drives.

3 Design Principle Fault Control Instrumentations for NPP with Optical Fiber The operability of modern complex objects of the power complex ensured by means of control, monitoring, fault diagnostic and emergency protection systems [10, 12, 25–31]. At present, these systems mainly use electronic measuring technologies, which use the conversion of the measured parameter into an electrical signal with its subsequent processing. In their composition, these systems contain a large number of primary converters (sensors) of various physical quantities, which supply these systems with information about the state of the machine. The principle of operation of the sensors used is based on various measurement methods, the main ones are the following [25–84]: • resistive (temperature of the components of the structure) eddy current (beats of the shaft and displacement of the components of the structure);

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• capacitive (air gap between the stator and the rotor, shaft run-out vibration of the stator winding rods); • optical (air gap between the stator and the rotor, shaft run-out). Sensors of physical quantities of modern information-measuring systems in many cases are developed taking into account their application in order to meet the reliability requirements and provide the necessary metrological characteristics [33, 40, 42, 49, 50, 72, 79]. In information-measuring systems, measured information is transmitted via communication lines from sensors to signal processing devices. Communication lines are mainly shielded cables (with the exception of optical sensors), which are assembled during installation in multi-wire bundles. The length of these lines, as a rule, cannot exceed a certain value, and the use of harnesses introduces an element of unreliability due to crosstalk of the measurement channels, electromagnetic interference in the screens and the need to create special ground loops. The use of fiber-optic information and measurement systems for measuring physical quantities, in which a parameter measured and converted into an optical signal using a fiber-optic sensor and transmitted for registration and processing using a fiber-optic cable, will eliminate the disadvantages inherent in traditional electronic measurement systems. In addition, the use of a fiber-optic cable will significantly increase the length of the communication lines between sensors and secondary converters. Intensive developments in the field of creating fiber-optic information-measuring systems for various objects, providing a more efficient transmission of information about the state of the object in comparison with traditional collection systems, are underway in many countries of the world. There is an improvement of existing and creation of new fiber-optic devices, systems, their components and the technology of manufacturing the optical fibers themselves [74–77]. The first attempts to create sensors based on optical fibers attributed to the mid-1970s. Publications about more or less acceptable designs and experimental designs of such sensors appeared in the second half of the 70 s. However, it believed that this type of sensor formed as one of the areas of technology only in the early 80 s. At the same time, the term “optical-fiber sensors” also appeared [77]. Nowadays fiber-optic sensor for information-measuring systems allows measuring many physical quantities [47, 51, 74, 78–92]: pressure, temperature, distance, rotation speed, linear velocity, acceleration, mass, liquid level, deformation, etc. Many publications have been devoted to the creation and theoretical studies of fiber-optic sensors. One of the classifications of fiber-optic sensors, taking into account the form of the measured physical quantity and the physical phenomena used in the optical fiber, is given in [78–81]. In Fig. 1 was shown block diagram for fault control instrumentation with optical fiber. The operation of the circuit is based on the fact that the measured parameter affects the primary measuring transducer, in which the light beam entering through the fiber-optic cable from the source (laser or light-emitting diode LED) changes its characteristics. Through another cable, the light beam with the changed

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6 2 1

2 3

4

5 Fig. 1 Block diagram for fault control instrumentation with optical fiber: 1—an optical source (a laser or light-emitting diode); 2—an optical fiber; 3—transducer; 4—an optical detector; 5—an electronic processing unit; 6—measured parameter

parameters enters the detector, where these changes are measured. The processor performs further data registration and processing. Methods of realization fault control instrumentations for NPP, goal at reducing the impact of radiation on the component characteristics. For obtaining this, necessary do next: 1. 2. 3. 4. 5.

6.

correctly select and place electronic components, make wider use of ceramic insulators in the parts of switches, connector-connectors, sockets, etc. use fiberglass and other inorganic materials for cuffs and cable insulation. use film and metal film resistors. to protect against gamma rays, use screens containing lead, uranium, thorium, and tungsten, bismuth, gold and platinum. to protect against neutrons, use screens made of a mixture of light and heavy elements (concrete with a high water content), fought (an alloy of boron carbide with aluminum), lithium, beryllium, iron, copper, tungsten, bismuth. to ensure impact of radiation and other influences, maximally use nonsusceptible materials, for example: fiber optic and other optical elements.

4 Fault Control Instrumentations for Valves Control System with Optical Fiber Optical fiber sensors structures, which work, based on the dependence of the light reflectance in the touch unit of the distance from the surface of the reflector and the reflectivity of the reflector was detail considered by author in [22]. The developed electronic unit of control and measuring devices of optical fiber based on processing and analysis of the signal from the optical detector determines the degree of external influence and generates appropriate solutions or data. Changing state of the external environment causes modulation of the reflected light flux, which can serve as an informative parameter changes the position of the

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Fig. 2 Fiber-optic alarm device of fault control instrumentations structure for valves control system

reflector. In Fig. 2 was shown fiber-optic alarm device of fault control instrumentations structure for valves control system. Device was realization on the principle of modulation of light flux due to reflection of rays [23]. The sensor consists of a toothed disk 1, a fiber-optic cable 2, through which the end of the disk 1 is supplied with light from a source whose role is performed by the LED 4. Part of the light flux through the optical splitter 3 is fed to the photodetector 5 of generates electric current. A reflective coating applied to the end face of the flat teeth of the disc 1. The disc is mounted on a shaft that is connected to the signal (or output) shaft of the valves control system. During the rotation of the disk in the fiber-optic cable, the light flux modulated due to its reflection from the tooth surface, which transformed into modulation of the output signal of the photodetector 5. In the selection and counting device 6, from this signal, pulses are formed, the number of which is directly proportional to the angle of rotation of the disc 1, and, therefore, to the drive shaft of the valves control system. It can see from the optical scheme that in order to achieve the design distinctiveness, the areas of the teeth and depressions must be the same, and their linear dimensions (width) are larger than the diameter of the optical fiber. This very significantly limits the resolution of such a signaling device, since its increase due to an increase in the number of teeth with the same dimensions is possible only due to an increase in the diameter of the disc, and increases the size of the signaling device. In addition, the operation of applying a reflective surface to the teeth is a complex technological operation and, therefore, expensive.

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Fig. 3 Signaling device for valves control system

Fig. 4 Signaling device for valves control system

In Fig. 3 shown version of the signaling device is more technological, because here the coding element is a flat disk 9 with uniformly applied in a circle reflective pads 9.1. In addition, due to the placement of the fiber-optic cable core perpendicular to the plane of the disk, the signaling device can be more compact. In Fig. 4 shown block diagram of the signaling device. In the signaling device, the fixed flat mirror 8 functions as a reflector of light flux. An element sensitive to the angular displacement of the shaft is a flat disc 7 arranged in a circle coding holes. In this signaling device the light beam from the cable 2 when passing through the hole in the disk 7 is reflected from the mirror 8, which when rotating the wheel 7 also causes modulation of the light flux into the fiber optic cable and accordingly at the photo detector input. With the correct choice of the size of the sensor elements, in this embodiment, you can achieve greater resolution by the angle of rotation of the shaft than in the previous one. In all considered embodiments of fiber-optic sensors to achieve the resolution required for single-speed valves control system, it is very difficult, so their main goal its alarms for multi-speed valves control system.

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Fig. 5 The scheme of the valves control system controller with the fiber-optic sensor

The benefit of all the options considered is that only one core fiber-optic cable is required to transmit the signal from the fiber-optic sensor to the processing equipment. Given also the large distances over which the signal transmitted, this can lead to a significant reduction in the ratio of the amplitude of the modulated signal to the background level. In Fig. 5 shown valves control system with fiber-optic sensor controller. The functional diagram of which is the above disadvantages are significantly reduced by dividing the cores of the fiber-optic cable on the transmitting and receiving, and the modulation of the luminous flux is carried out by completely masking the excitation flux of the photodetector [23]. On the shaft of the signaling device of the angle of rotation of the shaft, connected to the shaft by electric drive of valves control system, is attached to the coding disk 1 with holes placed in a circle. The luminous flux of the light source 4, which is excited by the luminous flux generator, through the transmission core of the fiberoptic cable. 2 constantly enters the input of the photodetector 5, and the receiving core of the fiber-optic cable 2—only when passing the hole of the coding disk 1 opposite the end of the transmitting core. The lens 3 is used to focus the flow from the hole on the receiving core. When the disk rotates, current pulses are formed at the output of the photodetector without a constant component, which greatly simplifies the construction of the counting pulse shaper for the electronic track controller. In most cases, a regular threshold element, such as a Schmidt trigger, will suffice. The electronic track controller on the basis of counting the number of pulses determines the angle of rotation of the shaft and generates the appropriate commands for the

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switching circuit, from the outputs of which power is supplied to the drive motor valves control system. In the proposed scheme, the attenuation of the modulated signal is determined only by the parameters of the fiber-optic cable, and it is significantly less than in previous versions. In addition, the amplitude of the modulated signal is also significantly higher, which allows you to transmit the fiber-optic sensor signal over much greater distances, which can meet the design requirements for the construction of track controllers for specialized NPP valves control system. For the encoder of the angle of rotation to the lumen significantly reduces the requirements for the ratio of the size of the mask and the hole, so in this embodiment, the fiber-optic sensor can implement a much higher resolution of about 0.5°, which may well meet the requirements for shaft angle meters NPP reactor pressurized system.

5 Conclusions The fiber-optic sensor has a simple structure and allows you to maintain it in the containment of the NPP reactor. The need to develop tools for valet NPPs is due to the fact that the existing means of control do not meet the modern requirements of the relevant regulations governing the safety of NPP operation. Such requirements are the reliability of operation, the accuracy of the control commands and the ability to diagnose this valve. The chapter formulates the concept of building a new generation of control and measuring instruments, which includes the following principles, structural unification of secondary equipment, the ability to diagnose the characteristics of the controlled parameter, design and communication coordination with automated standard control systems and diagnostics of NPPs. The implementation of these principles in the general structure of fault control instrumentations for valves control system, in the designs of angle sensors and structures of secondary transducers is shown. The optimal conditions for the use of fiber optic sensors are determined. Based on the developed fiber-optic sensors, control means can be created for NPP systems with a territorial distance of electronic units and sensors. Such facilities can operate in extreme conditions under the cap of the NPP reactor.

References 1. Yastrebenetsky, M.A., Rosen, Yu.V., Gromov, G.V., Inushev, V.V., Nosovsky, A.V., Gashev, M.Kh., Stolyarchuk, B.V.: Requirements for information and control systems of Ukrainian NPPs based on the analysis of the accident at the Fukushima-1 NPP. Nucl. Radioactivity Safety 4(52), 3–10 (2011) (In Russian) 2. Electric drives multi-turn for nuclear power plants, 55 p (2012) (In Russian) 3. NP-068–05. General technical requirements, 67 p (2005) (In Russian) 4. GOST 26843-86. Nuclear power reactors. General requirements for the control and protection system, 15 p (2005)

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Actual Issues on Radiological Assessment for Events with Liquid Radioactive Materials Spills Yurii Kyrylenko , Iryna Kameneva , Oleksandr Popov , Andrii Iatsyshyn , Volodymyr Artemchuk , and Valeriia Kovach

Absrtact According to the International Scale of Nuclear and Radiological Events INES, accidents involving liquid radioactive material (LRM) spills, depending on the magnitude of the atmospheric release and the corresponding radiological consequences, can theoretically be assigned different levels of danger (from level “0” Deviation” “to “7” Major accident”). This study was conducted mainly on the publications of leading scientists dealing with LRM spills, available incident databases, and descriptions of relevant software. An analysis and classification of events with LMR spill that occurred at nuclear facilities in different countries have been carried out. This made it possible to identify the main causes of such emergencies, the characteristic thermodynamic processes that occur during these situations, the routes of distribution of radioactive substances, the exposure conditions, and the extent of potential contamination. After reviewing the existing tools, approaches, and requirements for radiological impact assessment for the group of accidents with LMR spills, it can be stated that they do not comprehensively cover the features and have a number of shortcomings in modeling the course of accidents with spills of liquid radioactive media. Analysis and systematization of modern scientific approaches show that the problems of estimating the radiological impact of emissions in such accidents remain relevant and require further research.

Y. Kyrylenko (B) State Scientific and Technical Center for Nuclear and Radiation Safety, Kyiv, Ukraine Y. Kyrylenko · O. Popov · A. Iatsyshyn · V. Artemchuk · V. Kovach State Institution “The Institute of Environmental Geochemistry of National Academy of Sciences of Ukraine”, Kyiv, Ukraine Y. Kyrylenko · I. Kameneva · O. Popov · A. Iatsyshyn · V. Artemchuk G.E. Pukhov Institute for Modelling in Energy Engineering of NAS of Ukraine, Kyiv, Ukraine O. Popov Interregional Academy of Personnel Management, Kyiv, Ukraine V. Kovach National Aviation University, Kyiv, Ukraine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Zaporozhets (ed.), Systems, Decision and Control in Energy III, Studies in Systems, Decision and Control 399, https://doi.org/10.1007/978-3-030-87675-3_8

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Keywords Liquid radioactive material · Radiological consequences · Safety analysis · Nuclear facilities

1 Introduction In order to improve the safety of nuclear power plants, it is necessary to sophisticate the methodological and instrumental base for the analysis and safety assessment of currently operating nuclear power units, as well as the planned power units. The development of new computers and mathematical models for the estimation of radiation consequences is aimed at solving a lot of problems in this area. First of all, these are tasks such as [1]: • minimizing of radiological impact on the public, personnel and the environment, taking into account social and economic factors (ALARA principle), in accordance with the International Atomic Energy Agency (IAEA) safety requirements; • analysis of normal operation, design basis accidents and late phases of beyonddesign basis accidents at nuclear power plants (NPPs) within the framework of the design documentations of the operating organization and environmental impact assessments; • expert assessment of safety analysis reports in accordance with current nuclear and radiation safety regulations, rules and standards; • probabilistic safety analysis level 3; • emergency preparedness and response. Today abnormal operation of NPP (events the frequency of which may exceed 10–2 event/year) require more realistic and accurate modelling of such events at NPPs. After analysis of the obtained results of a probabilistic safety analysis for Ukrainian NPPs, abnormal operation includes events involving the spillage of liquid radioactive materials (LRMs).

2 Methods This study was conducted mainly on the publications of leading scientists dealing with LRM spills, available incident databases, and descriptions of relevant software. The following methods were used in this work: the method of comparative analysis; generalization; analysis of experience on the impact of LRM spills; software analysis. There are many works dedicated for safety analysis provided for abnormal operation modes of nuclear area facilities, medical centres, milling and mining facilities etc. In particular, various problems in this area are covered in previous articles [1– 11] by the authors of this study. In paper [12], among others, presented motivating factors towards a willingness to contribute to collaborative tasks and probabilistic

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Fig. 1 Factors influencing probability of a traffic accident [15]

analysis for the handling of radioactive material. In technical report [13], the research results about basic experimental data and the method for safety assessment of fire and explosion incidents of nuclear fuel facilities were summarized. The article [14] shows ways of handling streams from the storage at sanitary centers containing I125 , classified as radioactive waste of low activity. Author of paper [15] writes that it is estimated that several hundred traffic collisions, involving vehicles carrying hazardous goods, including radioactive material, are being registered. Therefore, this article presents the factors influencing the probability of a traffic accident (Fig. 1) and the method of selection model of road transportation of dangerous goods. Related problems are described in [16]. The solubility of radon into NAPL (Non-Aqueous Phase Liquid) vapors is investigated in papers [17, 18], either as growth of radon exhalation from a material contaminated with increasing volumes of kerosene or as radon partition between liquid kerosene, water, and total air, considered as the sum of kerosene vapors plus air. Related problems are described in [19]. In paper [20], authors discuss the results of a Partitioning Interwell Tracer Test performed in a large scale experiment with a well-defined TCE (trichloroethylene) spill, and present a novel combined analytical–numerical inverse modeling approach using measured concentration profiles within a TCE plume to predict the distribution of the DNAPL (Dense Non-Aqueous Phase Liquids) in a virtual vertical plane of the source (Fig. 2). Authors [21–23] describe, that oil and gas exploration and production generate considerable amounts of ’waste’ which, if not taken care of properly, will cause extensive environmental pollution. Wastes in the upstream operations include produced water, drilling wastes, and sludge and solids, including naturally occurring radioactive materials (NORM). The book [24] presents current research on serious environmental and health issues from radioactive contamination, which is, typically, the result of a spill or accident during the production or use of a radionuclide. In the article [25] are discussed the problems of emergency responders. Shown, that emergency responders are a segment of the general population and share some of the same fears of radioactive materials as the whole population. Radioactive material

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Fig. 2 Principle of locating the contaminant phase [20]

incidents are not a common 911 call type. Radiation training has been included in emergency responder training standards for several decades and covers a broad range of topics from simple awareness and recognition to technical knowledge of the materials, detection and identification capabilities, self-protection, medical effects, and countermeasures to overall public and environmental safety and health. In works [26, 27], a miniature radiation portal monitor and an operation program were developed and tested for monitoring radioactive contamination (Fig. 3) of water caused by nuclear power plant accidents or unplanned release of radioactive materials. Papers [27, 28] describe the safe use of radioisotopes. The characteristics of isotopes most commonly used in a molecular biology laboratory are detailed, as well as the safety precautions and monitoring methods peculiar to each one. Detection and imaging methods used in the experimental analysis are reviewed. Finally, an outline of an orderly response to a spill of radioactive material is presented. Authors in [28, 29] show the consensus derived from the data in the United States from 1980 to 1989 is that the number of incidents where natural and technological disasters interact is rising while preparations, which recognize the complications inherent in such combined events, remain cursory. An analysis in paper [30] is presented of the possible radiological consequences of various potential contamination events involving 223 Ra dichloride (Xofigo), the

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Fig. 3 Monitoring system for flowing water radioactive contamination [26]

FDA-approved therapeutic agent used in the treatment. The authors conclude that the medical use of Xofigo does not pose any significant radiation safety issue with respect to potential contamination events, even if multiple incidents might occur during the course of a year since all worst-case potential contamination events considered in this study will not result in significant radiation exposures to workers. Paper [23] describes a novel Gum Karaya stabilized magnetite for the efficient removal of radioactive phosphorus 32 P from liquid radioactive waste. Authors also agree that liquid radioactive waste is a common by-product when using radioactive isotopes in research and medicine and efficient remediation of such liquid waste is crucial for increasing safety during the necessary storage of the material. However, no generalized and comprehensive study has been found, which helps understand the scale of existing issues in this area and how to address them. The study was conducted to analyze actual issues on radiological assessment for events with liquid radioactive materials spills and software in this area. To achieve this goal, the following tasks were studied: 1. 2.

Analysis of known incidents with liquid radioactive materials spills. Exploring the capabilities of existing software to address the tasks associated with minimizing the risks of incidents involving liquid radioactive materials spills, in particular for the health of personnel and the public.

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3 General Context 3.1 Liquid Radioactive Materials LRM are liquid solutions, which include impurities of radioactive elements (possibly bound in high-molecular complexes). The isotopic composition of LRM is determined primarily by the source of radioactive impurities. The main sources of LRM at nuclear power plants and nuclear complexes are as follows: • primary coolant that is discharged for operational reasons; • water that is used to back flush filters and ion exchangers; • floor drains that collect water that has leaked from the active liquid systems and fluids from the decontamination of the plant and fuel flasks; • leaks of secondary coolant; • laundries and changing room showers; • and chemistry laboratories. LRM can be located: both under containment of NPP units and beyond (for example, in an auxiliary building). At Ukrainian power units, temperatures of the LRM can reach 320 °C (under pressure), fluctuations in the range from 40 to 100 °C are possible in pipelines and tanks, depending on the ways of discharge of radioactive effluents. Accidents involving LRM spills are characterized by intense heat transfer due to the evaporation of the liquid—the formation of vapor-aerosol forms, which are subsequently localized on the materials of treatment or localization systems— for example, on drops of a sprinkler system or on gas-aerosol filters of ventilation equipment. In case of disability of localizing systems, significant emissions of radionuclides can be considered due to the leakage from the emergency rooms. The isotopic composition and activity of the LRM at the NPP varies greatly. E.g., the primary coolant and the water of NPP spent fuel pool holding with light water reactors makes a collection of fragments of forced separation 235 U and 238 U, isotopes of corrosion metals, neutron activation products, etc. Within an auxiliary building, radioactive media can be maintained for a long time and include only long-lived radionuclides (60 Co, 134 Cs, 137 Cs, 54 Mn etc.) in the isotope composition. A similar case is observed in research reactors. Heavy water, which is used as a moderator on liquid reactors, also has high activity due to the presence of tritium in it. World nuclear complexes produce and process radioactive materials (e.g. NPP fuel, isotopic mixtures, industrial and medical sources). At nuclear complexes, as well as at nuclear power plants, reactor installations of different capacity are used. Nuclear complexes, chemical plants and research centers also work with the LRM. A distinctive feature of these enterprises in comparison with the NPP, in terms of the characteristics of the LRM used, is the large range of radioactive solutions involved in the technological processes of the enterprise. At nuclear power plants and nuclear complexes, LRM are mainly aqueous solutions of decay products of nuclear fuel according to IAEA safety standards and FSUE’s materials [31]. Pilot plants and chemical plants may contain a full range of

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isotopes and solvents. LRM can be found on the site for processing the radioactive liquids. Issue is a significant problem for radiotherapy medical hospitals. The problem is to analyze the emergency situations related to the special reservoir systems in the underground rooms of hospital (DTS systems—Decay Tank Systems containing 99m Tc, 131 I, 18 F, shown in Fig. 4).

Fig. 4 DTS (Decay Tank Systems)

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3.2 Events with LRM Spills According to the International Scale of Nuclear and Radiological Events INES [23], accidents involving LRM spills, depending on the magnitude of the emission and the corresponding radiological consequences, can theoretically be assigned different levels of danger (from level “0” Deviation to “7” Major accident). This approach reflects the design features reflects the design features of heat removal from the core of reactors operating on liquid coolant. In the event of a severe accident with melting of the reactor core, the products of nuclear fuel fission come into direct contact with the liquid coolant and water of the emergency cooling systems, which will lead to the further formation of LRM. It should also be noted that a significant contribution to the emission activity is caused by the neutron activation of the coolant during the campaign at the reactor units. In both the first and the second case, the activity of the liquid media represents a small part of the total emission activity. Therefore, when assessing the consequences of severe accidents, the emission activity due to evaporation from liquid spills is often neglected. However, if the accident involves the release solely due to the evaporation of LRM from open surfaces, depending on the concentration of radionuclides in the liquid and the conditions of the accident, the release can pose a significant threat to personnel, the public and the environment. Table 1 shows events with spills of liquid radioactive material. Pictures taken from the scene of the event in Lithuania, Ignalina NPP (October 2010) are shown in Fig. 5. At least one of the related incident is described in [40]. The British Nuclear Group (BNG), on 22 April 2005, announced the leakage of radioactive liquid from a pipe in the feed clarification cell at the THORP reprocessing plant at Sellafield. The liquor comprised of nitric acid and dissolved nuclear fuel, spilled into a secondary containment vessel. BNG expects that the pipe began to fail in August 2004 after changes to the restraint mechanisms for suspended tanks, putting more pressure on the connected pipework. A detailed engineering review was carried out throughout Sellafield to assess the potential for stress-induced fatigue and that the testing, maintenance, and reliability of plant instrumentation be improved. Paper [41] concludes that the Ballinger correlation based on the small volume, limited height experimental data is reasonably conservative (approximately a factor of three) for spill heights from 3 to 5 m, but that excessive conservatism may result with application to heights greater than 5–7 m, depending on the density of the aqueous solution being spilled. Incident occurred in late January 2002 at Washington State University when a 22 Na sealed source was breached, and a small portion of the contents accidentally migrated to various locations in the very large laboratory, is described in paper [42]. Damage control provided a challenge to radiation safety personnel. Resources of the Radiation Safety Office were severely taxed during both the immediate reaction and the subsequent several months of decontamination prior to the release of the laboratory for unrestricted use of radioactive materials once again. Salient features of this

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Table 1 Events with spills of liquid radioactive material (worldwide experience) [23, 31–38] No. Date

Location

INES level Short description

1

December 1952

Canada, research reactor «NRX»

5

Partial core melting, leakage of around 4000 m3 radioactive water of total activity 10,000 Ci

2

June 1973

USA, Hanford complex

3

Leakage of 430 m3 radioactive liquid material from radioactive waste tank

3

May 1984

USSR, Kalinin NPP

N/A

Pressure decreasing in reactor, leakage of 200 m3 radioactive liquid material

4

August 1984

USSR, nuclear complex «Mayak»

N/A

Leakage of primary coolant, 13 persons from personnel were exposed

5

December 1987

Germany, Biblis NPP

1

Due to personnel failure, spill of primary coolant was happened

6

February 1991

Japan, Mihama NPP (Unit 2)

2

As result of pipe failure, radioactive coolant and steam were released outside of secondary circuit

7

March 1991

India, NPP KALPAKKAM-1

1

3H

8

May 1991

Slovakia, Bohunice NPP (Unit 1)

3

Due to personnel failure, 100 L of high-radioactive solution were spilled on the floor of reactor building. Total activity is ~ 1011 Bq

9

August 1992

Czech Republic, Dukovany NPP (Unit 1)

1

Due to personnel failure, radioactive boric solution were spilled in the volume of 1,6 m3 . Specific activity of solution was 783E−3 Bq/L

10

September 1992 Russia Kola NPP (Unit 3)

0

Spill of 400 m3 of water with specific activity 2E−5 Ci/L, which lead to dose rate elevation in the emergency room up to 3 μR/s. Contamination of open surfaces was around 2E+3 β-parts. /(cm2 *min)

11

December 1992

2

15 m3 was spilled with the total activity 6 Ci

Russia, Beloyarsk NPP

concentration decreasing in the air of technological room due to spillage of heavy water

(continued)

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Table 1 (continued) No. Date

Location

12

August 1993

USA, Browns Ferry Nuclear N/A Power Plant

INES level Short description Pipe failure due to corrosion and spill of radioactive coolant

13

May 1994

Russia, nuclear complex «Mayak»

N/A

Release of radioactive materials through a ventilation system into the atmosphere due to spills of liquid radioactive material

14

May 1995

Canada, Bruce Nuclear Generating Station

N/A

As results of the accident 60 tons of heavy water was lost. During 3 days the release was observed

15

January 1996

India, Kalpakam NPP (Unit 0 1)

Heavy water spillage in room of reactor building. It was successful drainage of spillage liquid

16

May 1996

Bulgaria, Kozloduy NPP

1

As result of liquid radioactive material spillage from technological circuit of auxiliary building, around 8 m2 of the wall and 6 m2 of ground surface was contaminated. The event happened during pipe deactivation procedures. Maximal gamma dose rate in a place of spill localization has been achieved 20 mR/h

17

June 1999

Russia, nuclear complex «Mayak»

N/A

Leakage of the coolant due to corrosion in pipe

18

April 2000

India, Narora NPP (Unit 2)

1

Exposure of 2 person from personnel due to 3 H inhalation as result of 7 m3 heavy water spillage from moderator system. Doses did not exceed of the national permissible exposure levels

19

November 2000 Russia, Federal State 2 Unitary Enterprise “Russian Research Center” Applied Chemistry "

71 m3 of liquid radioactive waste was spilled onsite

20

April 2004

Release radioactive water during planed testing

Germany, Philippsburg NPP N/A (Unit 1)

(continued)

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Table 1 (continued) No. Date

Location

INES level Short description

21

December 2004

Czech Republic, Temelin NPP

0

As result of incident 20 m3 of liquid radioactive materials were spilled onsite. Specific activity was evaluated around 1E6−1E5 Bq/l. Contamination of walls surfaces ~8 Bq/cm2 . Spill was successfully localized using drainage system

22

April 2005

UK, Sellafield NPP

3

Spillage of the high-enriched U–Pu solution on the floor of the room

23

March 2006

US, Nuclear Fuel Services Erwin plant

N/A

Spillage of 35 L high-enriched uranium solution onsite. As result of the event more than 100 persons received doses exceeded annual permissible level in 5 times according to US standards

24

February 2002

Ukraine, Khmelnytskyi NPP 1

As a result of the leak, 3 m3 of liquid of 2.84E−6 Ci/L (0.105 GBq/m3 ) specific activity was released to the soil. A maximal dose rate at 1-m distance from the soil was 7 mR/hour. 30 m2 of the NPP territory was fenced off and decontaminated

25

October 2010

Lithuania, Ignalina NPP

N/A

Spillage of ~300 tons of radioactive decontamination solution contained 1% nitric acid and potassium permanganate

26

October 2011

Pakistan, Karachi NPP (Unit 1)

1

Alerting signal due to 3 H concentration exceeding in the air of room. During the event 1200 kg of heavy water was spilled (continued)

incident also are described in this paper in conjunction with a portrayal of measures taken during early damage control and the following deliberate remediation. Analysis of about 30 worldwide events at nuclear facilities over the past 65 years (photos of one of the incidents are presented in Fig. 5) showed that most events related to the spill of liquid radioactive material are classified as relatively low levels

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Table 1 (continued) No. Date

Location

INES level Short description

27

Pakistan, Karachi NPP (Unit 1)

2

August 2017

Overexposure of 4 persons from personnel as a result of radioactive heavy water spillage liquidation. Total amount of radioactive water was 1589 kg. Doses of exposure during the spill localization reached 20.8, 24.2, 30.9 and 36.2 mSv (dose limit for personnel—20 mSv)

Fig. 5 Spillage of ~300 tons of radioactive decontamination solution contained 1% nitric acid and potassium permanganate [39] (Lithuania, Ignalina NPP, October 2010)

by INES scale (levels from “0” to “3”). The preferred sources of radionuclides emissions are the coolant of the first circuits of reactor units, liquid radioactive waste, decontamination solutions, high-radioactive solutions of uranium and plutonium, liquid moderator on heavy water reactors. During the analysis of the radiological consequences of incidents and accidents with LRM spills, a critical group of exposed persons was identified—the personnel of facilities, and critical routes of exposure: external from the spill and internal due to inhalation of radionuclides. In particular, accidents with heavy water spills are also characterized by β-exposure of the surface layer of the skin. In the vast majority of events, personnel are exposed at the stage of actions on equipment restoring and spills localization. During some events, overexposure of personnel above the dose limits set by the IAEA and relevant national regulations has been reported. Although in most cases the information on environmental contamination is not provided, currently the requirements of the regulatory framework of many countries (including Ukraine [43]) require justification for not exceeding the established exposure limits for all reference groups as a result of normal operation, which are classified as events with LRM spills in safety analysis.

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3.3 Calculation Tools Overview Modern integrated codes of engineering level (MELCOR, MAAP, CONTAIN and others [44]) are successfully used to estimate radioactive release into the atmosphere for most NPP design and beyond design basis accidents, but do not have the means to model processes typical of LRM spill events. The main disadvantages of these codes include: • lack of models to take into account the process of radioactive decay of doseforming radionuclides; • inadequacy and uncertainty in the simulation of the middle and late phases of the accident (in the conditions of quasi-stationary thermodynamic balance) or in the flooding of LRM with temperatures close to the air phase temperatures; • no description of the dynamics of evaporation of radioactive substances from the free surfaces of liquids, neglect of thermo- and hydrodynamic processes in the boundary layer of both liquid and air phases (due to the lack of detailed spatial grid). Computational fluid dynamics (CFD) modelling using industry-wide software packages such as ANSYS (FLUENT and CFX modules), OpenFOAM, SolidWorks (Fluid Simulation) and others [45], is widely used in various countries to describe thermo-hydraulic processes associated with the elements of the reactor core and technological circuits of nuclear power plants. However, due to the lack of models describing the transport of multicomponent radioactive air mixtures, this package of tools can be used indirectly in the assessment of radiological consequences. Although, today FLUENT and CFX are not used for quantitative and qualitative modeling of aerodynamic parameters of air of technological premises of Ukrainian NPPs, it should be mentioned that they can be used to provide air velocity fields, concentrations and humidity both indoors and on local terrain. Existing software for dose assessment MicroShield, MCNP, PENELOPE, GEANT, EasyQAD (a brief description of which is presented in [39]) allow to partially estimate the parameters of exposure and can be used to describe dynamic tasks only to calculate the dose of external exposure. Their use is limited to determining the derivatives of the radiation situation in accordance with the geometry of the affected area, the instantaneous values of concentrations in the spill and air. The inhalation pathway of exposure is currently described only by the existing methodological basis, in accordance with the IAEA methodologies of the International Commission on Radiation Protection, which are currently not adapted to solve dynamic problems. Among the leading software tools for assessing the radiological impact on the population and the environment are the following: European decision support systems RODOS and ARGOS, COSYMA, fast-running assessment programs HotSpot, InterRAS and probabilistic analysis tool MACCS [46]. These computer tools are used to predict radiation exposure both in real time and at the emergency planning stage. RODOS and ARGOS decision support systems integrate Gaussian, Lagrangian

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and Euler models of atmospheric transport, covering estimated distances from units to thousands of kilometers. A significant disadvantage of the above means when used in the simulation of accidents with RRS spills is the inability to conduct comprehensive CFD-modeling and analysis of radiation exposure in near range (tens/hundreds of meters). This restriction does not allow to take into account the effect of aerodynamic shading of the emission source and the effects of atmospheric dispersion, which are characteristic of conditions of uneven terrain height.

4 Results An analysis and classification of events with LMR spill that occurred at nuclear facilities in different countries have been carried out. This made it possible to identify the main causes of such emergencies, the characteristic thermodynamic processes that occur during these situations, the routes of distribution of radioactive substances, the exposure conditions, and the extent of potential contamination. After reviewing the existing tools, approaches, and requirements for radiological impact assessment for the group of accidents with LMR spills, it can be stated that they do not comprehensively cover the features and have a number of shortcomings in modeling the course of accidents with spills of LRM. Analysis and systematization of modern scientific approaches show that the problems of estimating the radiation impact of emissions in such accidents remain relevant and require further research.

5 Discussion The authors have already developed a general mathematical model of source term for accidents with LMR spill in order to analyze and predict radiological consequences of of radioactive release on technological premises and adjacent areas, which unlike other models takes into account the parameters of radioactive liquids and design conditions, and can be integrated into DSS RODOS, taking into account the requirements of related means of assessing the radiation impact and the specifics of accidents with LMR spills, which solves the problem of a comprehensive assessment for such events. Research, identification, verification, and validation of this model will be shown in future publications. At the same time, the verification of the obtained model is one of the most difficult tasks, as there are few accurately described such incidents, and the possibility of conducting appropriate field experiments is quite a debatable issue.

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6 Conclusions Thus, the analysis of worldwide experience in accidents and incidents with spills of LRM shows that the issues related to radiological impact assessment of atmospheric releases in such events remain relevant and require further research. Existing software for assessing radiation exposure have a number of shortcomings in modeling the course of accidents involving the spillage of LRM. To take into account the peculiarities of modeling accidents of this type, it is proposed: • to identify possible scenarios of emergency processes using events experience and actual results of probabilistic safety analysis for NPPs; • to determine the general conditions and characteristics of atmospheric release, source term, features of remediation strategies, and reduction of radiological impact in such events; • to develop a holistic mathematical model of transport of radioactive substances in emergency area, in order to assess the levels of onsite as well as offsite contamination and the quantitative characteristics of the release into the environment; • to adapt the developed model to the existing software tools for estimating potential radiological consequences for personnel and public; • providing a number of demonstration calculations according to representative emergency scenarios to describe the limits and conditions of application of the obtained set of computer tools.

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Formation of Radiation Doses of Ukraine’s Population in Areas Contaminated by Radionuclides After the Accident at the Chernobyl Nuclear Power Plant Iryna Matvieieva , Yurii Rudyak , Yurii Zabulonov , Andrii Iatsyshyn , Dmytro Taraduda , and Kachur Taras Abstract The paper describes the set of the indicators that led to the growing technogenic impact on natural ecosystems and biota due to the rapid increase of natural and artificial radionuclides in the environment. The study states that there is a need to create system to assess reliability of ecological systems and biota taking into account content of artificial pollutants as far as living conditions and peculiarities of radiation doses formation of population of contaminated areas of Ukraine changed drastically and contamination level of many foods exceeds acceptable levels even 35 years after the Chernobyl catastrophy. The biological objects have an extremely high reliability which far exceeds reliability of any technical system that can be shown through the definition of biosystems reliability are described. The new sensitivity indicator is offered—the factor of radiocapacity to assess the impact on the state of the plant ecosystem to radiation exposure. A new radioecological concept is described in the paper along with the specific mathematical modelling methods. The study presents the development and application of methodology to assess state of ecological systems of different types and levels during radiation damage based on the use of mathematical chamber models and reliability theory as well as the consideration and determination of negative impact of radionuclides on the state of ecological systems. Development of reliability model of radionuclide transport and substantiate application of proposed method to study distribution and redistribution I. Matvieieva National Aviation University, Kyiv, Ukraine Y. Rudyak I. Horbachevsky Ternopil State Medical University, Ternopil, Ukraine Y. Zabulonov · A. Iatsyshyn (B) State Institution “The Institute of Environmental Geochemistry of National Academy of Sciences of Ukraine”, Kyiv, Ukraine A. Iatsyshyn G.E. Pukhov Institute for Modelling in Energy Engineering of NAS of Ukraine, Kyiv, Ukraine D. Taraduda · K. Taras National University of Civil Defence of Ukraine, Kharkiv, Ukraine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Zaporozhets (ed.), Systems, Decision and Control in Energy III, Studies in Systems, Decision and Control 399, https://doi.org/10.1007/978-3-030-87675-3_9

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of radionuclides in the environment and in assessing dose loads on biota, humans and environmental safety on the basis of developed modified mathematical chamber models of agroecosystems is described. Was shown that modern radioecology lacks methods and models suitable for assessing and forecasting of local ecosystems state for specific settlements of Ukraine. Therefore, was offered a method of operative creation of environmental safety model for some settlements with binding to concrete conditions of any settlement. Such model will allow to minimize scope and detail of monitoring and to predict critical situations in ecosystem under study. Chamber models of real ecosystems affected by the Chernobyl accident were developed and analyzed. Keywords Radionuclides · Radiocapacity factor · Chamber models · Environmental safety · Ecological control

1 Introduction Determining of distribution and redistribution ways of the main dose-forming radionuclide 137 Cs in the environment and ecological control of territories is an important task. As far as it determines state and dynamics of Chernobyl contamination of natural ecosystems. The radioecological situation that developed after the accident at the Chernobyl Nuclear Power Plant (ChNPP) radically changed living conditions and peculiarities of radiation doses formation of population of contaminated areas of Ukraine. Contamination level of many foods exceeds acceptable levels even 35 years after the Chernobyl disaster. The aim of the chapter—development and application of methodology to assess state of ecological systems of different types and levels during radiation damage based on the use of mathematical chamber models and reliability theory; consideration and determination of negative impact of radionuclides on the state of ecological systems; development of reliability model of radionuclide transport and substantiate application of proposed method to study distribution and redistribution of radionuclides in the environment and in assessing dose loads on biota, humans and environmental safety on the basis of developed modified mathematical chamber models of agroecosystems. The research methods—factual materials and literature data; own field research, dosimetric and radiation measurements of objects in selected ecological systems; mathematical modeling of the studied ecosystems.

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2 Chamber Model Development of Radionuclides Migration in the Ecosystem One of the most important and prior tasks of our time is to ensure population‘ livelihood, especially residents of areas located near nuclear power plants, their protection from radiation and nuclear accidents. Radiation and environmental situation became especially acute after the Chornobyl catastrophe. The accident at the fourth unit of ChNPP led to need for comprehensive analysis and clarification of distribution, composition and amount of radioactive products emissions in time and space. Analysis of radiation consequences of such scale led not only to use of existing methods at that time, but also to development of new approaches to protect the population, as described in [1–13]. Estimation results of the total release of radioactive substances into the environment after the accident at the fourth unit of ChNPP are known. Knowing, determined the Specific activity of fission products was defined knowing changes dynamics in the temperature of nuclear fuel over time. Those radionuclides were released into the environment after the accident [14–18]. Peculiarity of the Chornobyl catastrophe is unevenness, contamination “spotting” of huge areas and living organisms. Biological objects have an extremely high reliability which far exceeds reliability of any technical system. This follows primarily from the time of biological systems existence. It is much longer than time of trouble-free existence of technical systems. We can offer following as a definition of biosystems reliability: reliability is a fundamental property of biological objects, which determines their effective existence and functioning in randomly varying environmental conditions and over time [19, 20]. Measure of reliability is probability of systems failure, which can vary from 0 to 1. It is proposed to use a sensitive indicator—the factor of radiocapacity to assess the impact on the state of the plant ecosystem of radiation exposure. This factor idea is basis of new radioecological concept. It should be noted that radio capacity of ecosystems is defined as limit of radionuclide deposition in ecosystem and its elements, above which there is suppression and death of ecosystem biota. Chamber models of different types are used to describe ecological processes occurring in ecosystems. Method of chamber models is the simplest and adequate mathematical method for describing radioecological processes in ecosystems of different complexity [21–23]. Method of structuring ecological systems consists of four consecutive procedures: (1) development of chamber radioecological model of corresponding defined ecological system; (2) determination of its configuration (serial, combined or parallel); (3) determination of parameters and rates of radionuclide transition; (4) reliability calculation of structural elements and the ecological system as a whole by models. Let’s consider a simplified ecosystem “environment – biota” to analyze parameters used in radioecology (Fig. 1). Method of chamber models in the simplest two-chamber form is used: “environment”—“biota” based on the application of the software product MAPLE-5 for theoretical analysis. Chambers contain stock of radionuclides Y (x) and Z(x) with

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Fig. 1 Block scheme of two chambers ecosystem model

time—x; a12 is absorption rate of radionuclides, and a21 is rate of their outflow from biota into the environment. System of two differential equations is formed under given initial conditions: Y (0) = 1, Z(0) = 0. That is at the beginning of radioecological process the entire stock of radionuclides (let it be—137 Cs) is in the chamber “environment” for simplicity. Initial content of radionuclides in environment is taken as 1. Parameters a12 and a21 set rate of radionuclides transition from chamber to chamber and determine what proportion of the stock passes per unit time: dy(x) dz(x) = a21 z(x) − a12 y(x); = a12 z(x) − a21 y(x). dx dx

(1)

Initial conditions: Y (0) = Y 0, Z(0) = 0. Solution is: a21 a12 e(−(a12 +a21 )x) y(x) = + , z(x) = a12 + a21 a12 + a21

a12 e(−(a12 +a21 )x ) a21 a12 +a21

a21

+

a12 a21 a12 +a21

.

(2)

Let’s analyze formulas that determine dynamics of radionuclide supply in each chambers. Assume that radionuclides content in the chambers reached its equilibrium at X = ∞. Then for such time the exponent is equal to zero. For radionuclides stock that will be in the chamber “environment” (F S ) and in the “biota” chamber (F b ): FS =

a21 a12 , Fb = . a12 + a21 a12 + a21

(3)

Value determining proportion of radionuclides is called the radiocapacity factor. The radioonuclides are deposited in particular chamber (component) of the ecosystem. Developed models and theory of ecosystems radiocapacity made it possible to introduce radiocapacity factor to determine state of the ecosystem’s biota. Factor of ecological and radiation capacity of particular element oflandscape F j or ecosystem is determined as follows:  ai j  , (4) Fj =  ( ai j + a ji )  where aij —sum of contaminats transfer rate from different components of the ecosystem to particular element of landscape or ecosystem—j, according to chamber models, and aji —the sum of velocities of pollutants from the studied chamber— j—to other components of ecosystem associated with them.

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Therefore, based on theoretical studies we can assess the reliability of the ecosystem component as an element of the system of transport of radionuclides through the chambers by formula (4) using of radionuclide exchange rates between chambers αij and α ji . Thus, determining reliability of ecological systems involves development chamber radioecological model of ecological system with determining its configuration [24– 26], setting parameters, radionuclides transition rates and calculating reliability of structural elements and ecological system as a whole. Comparing Eqs. (1, 2 and 3), you can get the expression: Fb 1 − Fs a12 = = = Z. a21 Fs Fb

(5)

Thus, absorption and outflow rates ratio of radionuclide is proportional to the biomass of the biota and the accumulation coefficient in the system “environment”— “biota”. It is defined as parameter Z This means that greater biota biomass and accumulation coefficient of 137 Cs by the biota leads to higher ratio of absorption and outflow of 137 Cs rates, and hence the required substances from the environment into biota biomass. Relationship between radio capacity parameter and absorption and outflow rates is clearly visible here.

3 Modeling of Radionuclides Migration for Some Settlement of Ukraine Modern radioecology lacks methods and models suitable for assessing and forecasting of local ecosystems state for specific settlements of Ukraine. Therefore, specification of existing generalized approaches and models is urgent and important task of modern ecology. It is necessary to have method of operative creation of environmental safety model for some settlements with binding to concrete conditions of any settlement [27, 28]. Such model presence will allow to minimize scope and detail of monitoring and to predict critical situations in ecosystem under study. This makes possible to set limits on environmental capacity to limit excessive anthropogenic pressure in the study area. Chamber models of real ecosystems affected by the Chernobyl accident were developed and analyzed: for the village Haluziia (Manevytskyi district, Volynska oblast), contaminated by 137 Cs, and Kotsiubynchyky (Chortkivskyi district, Ternopilska oblast) contaminated by 90 Sr [29, 30]. Use of chamber models is typical for modeling of radionuclides migration in selected areas. In this case, any element of the ecosystem, agrocenosis (or part of it), where the accumulation of radionuclides can be considered as camera. Agrocenosis is considered as a set of homogeneous chambers, between which there is a transfer of radionuclides characterized by some functions k ji ,qj within this

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approach. These functions describe intensity of radioactive substances flow between the chambers. Apparatus of ordinary differential equations is used in mathematical description of substances transfer in chamber models:  dqi (t)  k ji q j − kil qi − λqi , = dt j=1 i=1 n

n

(6)

where qi (t)—radionuclide content in the chamber i; k ji and k il —transition coefficients between cameras; k ji qj —the amount of substance coming per unit time from chamber j to chamber i; k il qi —the amount of substance leaving per unit time from chamber i to chamber l; λ—constant of radioactive decay. Modified chamber models of typical settlements were created and the process of radionuclide migration (137 Cs and 90 Sr) by trophic chains (chambers) was modeled as a result of research: soil → fodder plants → cow → milk → man. Parameters were established and features of this phenomenon were investigated. Methods and approach for control, forecast and management of radioecological safety for local ecosystems of settlements of Ukraine were obtained. Transition rate of total proportion of radionuclides from chamber to chamber per unit time (year) was used instead of the parameters of radionuclide transition rate from unit weight to chamber to chamber in this modification of chamber models. This method helps to generalize characteristics of ecosystems and obtain integrated characteristics of process of radionuclide transfer in ecosystem. Mathematical model of radioecological processes of typical local ecosystems with estimation of dose loadings formation for population for long term is developed and constructed. This model is suitable for modeling almost any type of local ecosystems specific to territory of Ukraine and can be used to assess ecological status of any agroecosystems. It was shown for the first time that formation of high dose loads in population can be carried out relatively quickly or can have character of slow accumulation, depending on the established parameters of the chamber models according to the calculations on the models. This means fundamentally different dynamics of dose loads formation which can actually occur in different local ecosystems of Ukraine [31]. In radioecology there are two main closely related problems: radionuclides migration and their accumulation in various elements of the ecosystem. There are variety of radiological situations associated with radionuclides incorporation into agriculture. Therefore, radionuclides accumulation by plants from the soil determines initial scale of radionuclides inclusion into trophic chains in system: radioactive emissions—soil—agricultural plants—farm animals—humans. Radionuclides supply to plants depends on number of factors: physicochemical properties of radionuclides, species characteristics of plants, soil properties and its mechanical treatment, climatic factors, reclamation system, fertilizer application, etc. Radionuclides transition from diet to bodies of animals is determined by radionuclides physicochemical properties and as species characteristics and age of animals.

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Numerous studies showed that significant dose loads in humans are formed due to the large values of the transition coefficients in the system “soil – plants” in studied areas. Main reasons for this phenomenon are peat and swamp soils, which dominate in these areas, high degree of moisture and waterlogging of soil, acidic soils, low levels of minerals in territories. This contributes to high cpntamination level of grass and hay, forest products—mushrooms and berries. Use of contaminated forage grasses leads to radionuclides migration in system “grass - farm animals”. In this case, level of radionuclides in milk and meat is increased significantly. The concept of transition coefficient (TC) is used when it comes to the migration of radionuclides by trophic chains. It reflects radionuclides proportion that fall from one element of ecosystem to another. The coefficient shows how many times greater (or less) can be activity of particular radionuclide in elements of ecosystem compared to environment. For the system “soil – plant” TC is ratio of radionuclide activity per 1 kg of air-dry biomass of plant to its content per 1 m2 of soil where these plants are grown. Preliminary estimates showed that there are following main directions for entry of radionuclides to humans for studied areas: through pastures (which are fodder base for dairy and beef cattle); through forest products (mushrooms, wild berries); garden plot (garden). Obtained results on assessment of distribution and redistribution of radionuclides in the agroecosystem showed significant dynamics of dose loads formation in humans. Therefore, it can be argued that radiocapacity parameters can be measure of each element of ecosystem and ecosystem as a whole. Higher reliability of radiocapacity factor and (or) probability of radionuclide retention in each of ecosystem elements lead to higher reliability of constituent elements of ecosystem. It is possible to adequately assess reliability of entire ecosystem through its ability to ensure distribution and redistribution of radionuclide, reflecting its steady state using these parameters of reliability of ecosystem elements and knowing structure of particular ecosystem. Structured block diagram of chamber model is given in Fig. 2. Parameters indicated in the diagram (from a12 to a510 ) mean rates of radionuclide transfer between ecosystem chambers and have dimension: share of radionuclides transferred between the chambers in one year. Methods and model parameters of calculation of transition between cameras are content of specially designed and protected declaration package for utility model [32]. The selected for study settlements are typical. So, the block diagrams of chamber models for them are similar in structure. The model takes into account all main streams of radionuclides 137 Cs and 90 Sr (no other radionuclides were found in these villages). It was possible to find and form a high degree of similarity in the block diagrams of chamber models in the studied areas as a result of research. It is important that these villages are characterized by rather low levels of radionuclide soil contamination from 1–2 Ki/km2 for 137 Cs (Haluziia village) and for 90 Sr—from 1.2 to 1.3 Ki/km2 (Kotsiubynchyky village).

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Fig. 2 Structured block scheme of Haluziia village (Manevytskyi district, Volynska oblast)

Calculated data on radionuclides fluxes by chambers of the studied ecosystems were obtained according to the developed model. Significant dose loads in different categories of population are formed due to constant consumption of milk from cows grazing on contaminated pastures. Formation of dose from use of milk for inhabitants of Haluziia village is up to 40–60% and 70% of the total dose for Kotsiubynchyky village. Average level of daily milk consumption for inhabitants of the studied settlements is from 0 to 3 l (137 Cs milk contamination levels are from 40 to 1000 Bq/l; for Kotsiubynchyky village 90 Sr is from 2 to 30 Bq/l) according to expeditionary research. Current standard for permissible levels of milk contamination is up to 20 Bq/l for 90 Sr, and 100 Bq/l for 137 Cs (permissible levels of 137 Cs and 90 Sr radionuclides in food and drinking water). It is inserted those significant levels of milk contamination are formed in areas not immediately after the accident but increase over time according to simulation data. It explains that significant levels of 137Cs milk contamination were identified in Haluziia village in 1993 and in the Kotsiubynchyky village 90 Sr in 1998. Graphs of expected dose dynamics were constructed for selected social groups— workers, pensioners, children (the division into groups was made because according to expeditionary research in these villages the amount of milk consumption in different population groups differs) based on the simulation data. Figure 3 and 4 show dynamics of dose loads formation in different social groups due to use of the main dose-forming product—milk. The upper curve of Fig. 3 shows radionuclides accumulation and hence dose of milk consumption in Haluziia village for group of workers, the middle—children, and the bottom—pensioners. This course of curves is due to diet, in particular, due

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Fig. 3 Dynamics of dose loads formation of from milk consumption for different social groups of the population Haluziia village: 1—workers; 2—children; 3—pensioners

Fig. 4 Dynamics of dose loads formation of from milk consumption for different social groups of the population Kotsiubynchyky village: 1—workers; 2—children; 3—pensioners

to significant amount of milk consumption by workers and children. The graphs show that the villagers are characterized by a rapid and then slower accumulation of collective dose. Situation for residents of Kotsiubynchyky village (Fig. 4) is characterized by different dynamics of collective dose accumulation. The model demonstrates that no significant doses of 90 Sr should be expected, but over time these doses increase 20 years after the accident. Here you can expect very slow accumulation of dose first and then increase it in all segments of population. At the same time maximum collective doses can be expected for workers; pensioners can be next in dose level, and the lowest doses can be expected for children. It should be noted that in general the dose for Kotsiubynchyky village is almost twice smaller than for the Haluziia village because due to lower contamination levels. The difference between these typical agroecosystems in terms of radioecological processes and collective dose accumulation parameters is clearly related to different radionuclides, climatic factors, and differences in food consumption.

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We believe that these phenomena reflect fundamental features of collective dose formation for Ukraine’s population in relation to agroecosystems with significant contribution of forest component (for example, Haluziia) and for the case of agroecosystems where there is no forest component of collective dose accumulation.

4 Discussion Therefore, modified method of chamber models was developed and applied. The method uses the parameters of radionuclides transition rate between ecosystem chambers not transition rate per unit weight or volume. This approach allows general systematic assessment of state of radionuclide fluxes and predict their dynamics. System type of radiological study of settlements was created. It covers the main links: soils, hay, farm animals, milk, forest products and people. Chamber models of real ecosystems affected by the Chernobyl accident were developed and analyzed. The models take into account all major streams of radionuclides 137 Cs and 90 Sr. It was determined that in settlements of the Haluziia village type significant dose loads were formed not immediately after the accident but only in 1992–1994 according to the simulation results. 30 years after the catastrophe we can observe high collective radiation dose of 137 Cs—from 40 to 80 person\Sv. These areas are characterized by a significant accumulation of the collective dose for the population for 30–40 years after accident, due to the use of 1% of the stock of radionuclides 137Cs in this ecosystem. Insignificant radiation doses are formed in the first decades after the ChNPP accident: 20 years later, the collective dose is 0.3–0.5 people for such settlements as Kotsiubynchyky village where contamination by 90 Sr is dominated. Accumulation of certain dose on 40th year after the accident is expected to be insignificant, no more than 0.1% of the stock of radionuclide 90 Sr in this ecosystem. But over time, we can expect fairly rapid accumulation of the collective dose. As a result of researches the regularity of continuous increase of a collective dose in villages with strontium pollution is revealed. This means that ecosystems of this type can become dangerous over time. It was established and verified that a significant part of the collective dose is not formed locally in these villages but is transfered to other territories through export of milk and meat according to modeling and field research of regional sanitary-epidemiological stations. This phenomenon of collective dose exports outside villages is common characteristic for the whole territory of Ukraine. The results show that environmental safety of area can be achieved only with use of system of protective countermeasures. The proposed modified chamber model is suitable for modeling of almost any type of local ecosystems specific to the territory of Ukraine. It can be used to assess, control and forecast their ecological status for both radionuclide contamination and other pollutants of agroecosystems.

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5 Conclusions Agroecosystem is an important source of radionuclides transition from the environment to humans. Greater the factor of agroecosystem radiocapacity makes it more "reliable" in the sense of radionuclides flow reducing to humans. It is possible to calculate the reliability of this agroecosystem and assess contribution of different components of the agroecosystem due to rates of migration, distribution and redistribution of 137 Cs radionuclides in components of agroecosystem and magnitude of cesium transition to all groups. Depending on the amount of radionuclides falling on the territory it is possible to take countermeasures. Their effectiveness depends on many factors (for example, soil type, humidity, precipitation, etc.), and their benefits can be evaluated. Application of reliability models and theories to investigate ecological processes in different types of ecosystems is useful and heuristic. It allows assessing basic characteristics and fundamental properties of ecosystems by tracking behavior of 137 Cs. Proposed method of reliability assessment can be used to assess level of pollution and transitions of other pollutants in ecosystems of different types.

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28. Zaporozhets, A., Babak, V., Isaienko, V., Babikova, K.: Analysis of the air pollution monitoring system in Ukraine. In: Babak, V., Isaienko, V., Zaporozhets, A. (eds.) Systems, Decision and Control in Energy I. Studies in Systems, Decision and Control, vol. 298, pp. 85–110. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-48583-2_6 29. Matvieieva, I., et al.: Modeling of radioecological processes by the method of chamber models on the example of a village in the Volyn region. Visnyk Natsional noho aviatsiynoho universytetu 3, 173–176 (2005) 30. Matvieieva, I., et al.: Features of radioecological processes in the village of Ternopil region, estimated by the method of chamber models. Visnyk Natsional noho aviatsiynoho universytetu 2, 126–128 (2006) 31. Matvieieva, I.: Radioecological reliability of the local agroecosystem. Naukovi pratsi 221, 66–70 (2014) 32. Babak, V.P., Babak, S.V., Myslovych, M.V., Zaporozhets, A.O., Zvaritch, V.M.: Methods and models for information data analysis. In: Diagnostic Systems For Energy Equipments. Studies in Systems, Decision and Control, vol. 281, pp. 23–70. Springer, Cham (2020). https://doi.org/ 10.1007/978-3-030-44443-3_2

Fossil Fuels

Prospects for the Rational Use of Waste from Uranium Mining Enterprises of Ukraine Maryna Yaroshchuk , Yurii Fomin , Oleksandra Buglak , Oleksandr Vaylo , and Yurii Demikhov

Abstract Data on uranium mining enterprises, types and volumes of accumulated waste, and conditions of their storage are presented. A detailed analysis of the textural-structural, geochemical features of waste, the mineral composition of ore-containing rocks and off-balance ores, mineral, isomorphic sorption forms of uranium, thorium, and rare earth elements is given. The evolution of the composition of waste in the environment, the degree of stability of various minerals, the mobility conditions of uranium, thorium and related elements are considered. An increase in the thorium-uranium ratio in the waste was found. The conclusion is made about the possibility of using waste as a technogenic secondary raw material containing uranium, thorium, rare earths, tungsten, molybdenum, titanium used in nuclear power. The tasks of further research are shown. Data on the material composition of waste will be the necessary basis for the development of rational schemes for ore dressing. Keywords Uranium mining enterprises · Types and composition of waste · Uranium · Thorium · Related elements · Technogenic concentrations of secondary rawmaterials

1 Introduction The relevance of the work is due to the need and the possibility of developing the mineral resource base of nuclear energy in Ukraine. The prospects for the development of the uranium and thorium raw material base due to the exploitation of established, as well as the discovery of new types of deposits, are considered in detail in monographs [1–4]. Technogenic concentrations of uranium, thorium and rare earth M. Yaroshchuk · Y. Fomin · O. Vaylo (B) · Y. Demikhov State Institution “The Institute of Environmental Geochemistry of National Academy of Sciences of Ukraine”, Kyiv, Ukraine e-mail: [email protected] O. Buglak State Ecology Academy of Postgraduate Education and Management, Kyiv, Ukraine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Zaporozhets (ed.), Systems, Decision and Control in Energy III, Studies in Systems, Decision and Control 399, https://doi.org/10.1007/978-3-030-87675-3_10

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elements accumulating in the waste of uranium mining enterprises of Ukraine can be an additional source of raw materials for nuclear energy, which makes their detailed study relevant. The following uranium ore formations have been identified and studied in detail within the Ukrainian Shield: sodium-uranium, potassium-uranium, iron-sodiumuranium and metamorphosed conglomerates [1] (Table 1). Among the known uranium deposits, about 70% are deposits of the sodiumuranium formation, 11% belong to the potassium-uranium formation.

2 Uranium Mining Enterprises The development of the uranium mining industry in Ukraine and, as a result, the accumulation of its waste, began after the discovery and commissioning of the Pervomaisky, Zhovtorichynsk uranium deposits, which have already been worked out, the waste from mining is mainly reclaimed. To date, as a result of the extraction of uranium ores from the Michurynsky and Vatutinsky deposits at the Ingulsky and Smolinsky mining enterprises, industrial waste has accumulated in the dumps (containing rocks, near-ore metasomatites, off-balance ores, products of radiometric sorting of balance ores), the volume of which is millions of cubic meters. Recently came on stream huge Novokostiantynivske deposit, and in the future can be operated several other deposits (Dokuchaevske, Aprelske, Partysanske, Litne, Lisove, Severinivske sodium-uranium formation; Safonovo and the other in the sedimentary cover of the Ukrainian shield—the so-called “Sandstone” type classification IAEA), which will inevitably result in an increase in the number of industrial waste. In Ukraine, enterprises for the extraction and processing of uranium ores are located in the Dnipropetrovsk and Kirovograd regions. In the Kirovograd region, the deposits of uranium ores are localized in two ore nodes—the Kirovograd district proper (around city of Kropyvnytskyi) and the Novokostiantynivsky district (around village of Mala Vyska). Currently, the Shidnyy GZK is engaged in the extraction of uranium ores (the city of Zhovti Vody) and it is carried out by three mines: Ingulskaya (Michurinske deposit), Smolinska (Vatutinske deposit) and Novokostantynivska (Novokostantynivske deposit) (Fig. 1). The Ingulska mine was founded in February 1967; ore deposits are mined at depths from 160 to 420 m; production capacity—470 thousand tons of ore per year; along with the traditional mining technology of uranium extraction, the method of underground block leaching was used at the mine; and also with the use of the Ingul complex since 2011, 3.4 tons of uranium was obtained as a result of processing 25 thousand tons of dump rocks (Fig. 2). Smolinska mine was founded in April 1972; ore deposits are mined at depths from 70 to 640 m; industrial capacity-450 thousand tons of ore per year; since 2011, a heap leaching landfill has been put into operation at the mine; large-scale robots have been carried out for the environmental rehabilitation of the industrial site by radiometric separation of dumps formed during the development of the Vatutinsky

Quartz, albite, aegirine, ribekite, chlorite, phlogopite, carbonate, epidote, microcline

Albite, aegirine, ribekite, arfedsonite, rhodusite, actinolite, tremolite, diopside, carbonates, chlorite

Sodium-uranium

Iron-sodium uranium

Monazite, xenotime, urano-thorianite

3

Thorium

Nedarkevite, Monazite, xenotime uraninite, brannerite, nasturane, coffinite

Uraninite, nasturane, Monazite, xenotime, uranium black, torite coffinite, nenadkevite, brannerite, davidite, uranophane, boltwudite, nangioite

Uraninite, uranium black, uranophane, coffinite, thorogumite

2

1

Quartz, plagioclase, icrocline, biotite, pyroxene, garnet

Uranium

Rock-forming

Minerals

Potassium-uranium

Formations

Sphene, apatite, zircon, ilmenite, anthraxolite, malacone

Sphene, ilmenite, zircon, malacone, apatite, orthite, rutile, graphite

Zircon, apatite, anatase, orthite, fluorite, graphite, cirtolite

4

Accessory

Magnetite, martite, hematite, pyrite, marcasite, pyrrhotite, arsenopyrite, cubanite, bornite, galena, sphalerite

Galena, pyrite, bismuthin, magnetite, hematite, marcasite, sphalerite, chalcopyrite

Cassiterite, molybdenum, galena, pyrrhotite, pyrite, sphalerite, maghemite, chalcopyrite, musiketovite, hematite, arsenopyrite, lellingite

5

Ore

Table 1 Mineralogical and geochemical characteristics of ores of thorium-uranium formations of the Ukrainian shield

(continued)

Ni, Cr, Zn, Cu, V, Mo, Co, Th, Bi, Zr, Ag, Ge, Nb, Ta, Sb, As, P, TR (Y, Sc), Sb

Pb, Bi, Au, TR (Ce, La, Y), Ti, Cu, Be (0,01), Ag, Ge

Ni, Cr, Ti, Cu, Mo, Zr, Pb, V, Zn, Bi, V, Be, TR (V, Sb, Ce, La, Sr), F, P, As, Nb, Sn, Be

6

Companion element

Prospects for the Rational Use of Waste from Uranium … 175

3

Thorium

Uraninite, nasturane, Monazite, xenotime cherni, coffinite, nenadkevite

2

1

Pebbles-quartz, cement-quartz, microcline, sericite, chlorite, biotite, siderite, bitumen (kerite)

Uranium

Rock-forming

Minerals

Note elements used in the nuclear power industry are marked in bold

Metamorphosed conglomerates

Formations

Table 1 (continued)

5

Ore

Zircon, leucoxene, Pyrite, pyrrhotite, sphene, ilmenite, marcasite, tourmaline arsenopyrite, galena, sphalerite, chalcopyrite

4

Accessory Zn, Pb, As, Cu

6

Companion element

176 M. Yaroshchuk et al.

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Fig. 1 The main objects of the Uranium heritage of Ukraine and natural deposits of uranium on the territories of the Uranium heritage of Ukraine [5] with changes and additions

Fig. 2 General view of the Ingulska mine with the main pit head and dumps on the territory

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Fig. 3 General view of the Smolinska mine with the main pit head and dumps on the territory

deposit. For 6 years, more than 5 million tons of rock dumps were removed from the surface, more than 400 tons of metal were extracted additionally (Fig. 3). The Novokostantynivske deposit was discovered in 1975, the mine was put into operation in 2007; according to the operational recalculation, the reserves are about 93,000 tons of uranium-the largest deposit in Europe and one of the ten largest deposits in the world (in July 2011, pilot production of uranium ore was started at the mine; ore deposits are worked at depths from 180 to 300 m; the design capacity is 1500 thousand tons of ore per year [6] (Fig. 4).

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Fig. 4 General view of the Novokostantynivska mine with the main pit head and dumps on the territory

3 Types of Waste During the extraction and processing of uranium ore, solid radioactive waste is formed in the form of: (1) substandard ores (mainly off-balance sheet) and rocks containing them, (2) tailings of radiometric processing plants. Waste from the uranium mining industry interacts with the environment; the processes of this interaction and their environmental consequences are the most important problem [1–7 and others]. This is what determines the relevance of a detailed study of the composition of uranium mining waste and the evolution of this composition in the environment. The waste generated during the extraction of uranium ore is the rocks of quarry overburden, mine and ore dumps, and mine air with dust, which belong to the category of solid and gaseous waste. During the mining cycle waste generation due to the fact that along with the ore of uranium (U = 0.02–0.03%) removed a significant amount of tailings (U—thousandths %) and off-balance ores (U—less than 0.02%); as a result, capital mining, mining-preparatory works and rifled; the wastes are also low-grade ore, rejected the result radiometric sorting and waste rock left over from mechanical enrichment in the processing of low-grade uranium ores (separate uranium from the rock). In the mining process, to obtain 1 ton of conditioned ore, a large amount of waste rock and low-grade (off-balance) ore is separated from the beaten rock mass, so huge masses of waste accumulate in the dumps. Such waste of overburden rocks, waste rocks left over from mechanical enrichment, and substandard uranium raw materials are placed on the surface in dumps.

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Depending on the method of laying waste, the dumps are divided into: conical (waste heaps), ridge and flat. In open-pit mining operations, depending on the storage location, there are dumps: internal, created in the spent space of the quarry; and external, located outside the boundaries of the quarry; as well as combined. Dumps of off-balance ores and waste rocks, the content of radionuclides in which is much higher than clark, occupy many thousands of square meters in mines and quarries and are sources of local pollution of the area. At modern mines, 0.3 tons of waste rock are accounted for per 1 ton of ore extracted by the mine method.

4 Material Composition of Waste As a result of the research, a detailed analysis of the waste from the exploitation of deposits of the sodium-uranium formation, in particular, the content of thorium and rare earth elements in ore-containing rocks, off-balance ores, in which they form technogenic concentrations, was carried out (Table 2). The ore-containing rocks in the deposits of the sodium-uranium formation of Ukraine are granitoids, albitites, various diafluorites, quartered and carbonated their differences. The features of these components of dumps and their changes largely determine the environment in which the leaching and migration of uranium and associated ore elements occurs. In particular, the primary textural and structural features are of great importance: granularity (hypidiomorphic, uniform-grained, leistlike, etc.), banding, massiveness, density, as well as the degree of tectonic processing (cataclysm, mylonitization, brecciation). The mineral composition of ore-containing albitites is determined by the variable ratio of relic minerals of granitoids (plagioclase, quartz, mica), albite of different Table 2 Average uranium content (g/t) in uranium ores and host rocks of albitite deposits of the Ukrainian shield (variations are given in parentheses) Deposit

Host rocks

Albitites ore free

Non-industrial ores

Industrial ores

Severynivske

6.4 (1–22)

13.0 (1–57)

197.7 (101–288) 2409.6 (341–11,128)

Michurinske

10.0 (5–15)

20.3 (7–52)

80.3 (20–180)

835.0 (325–1670)

Pivnichno-Konoplyanske

9.2 (1–27)

15.6 (1–48)

138.8 (57–258)

1060.5 (350–2041)

Yuriivske

14.2 (1–40)

25.1 (7–57)

90.1 (37–167)

1092.0 (355–3506)

Vatutinske

15.9 (3–53)

17.9 (5–65)

142.2 (52–293)

2060.4 (377–5140)

Novokostantynivske

4.8 (0.5–14)

20.8 (0.4–56)

151.0 (51–248)

1449.4 (568–4470)

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generations and secondary dark-colored minerals (hornblende, actinolite, alkaline pyroxenes and amphiboles, chlorite, epidote). Mica minerals (chlorite, sericite, phlogopite) predominate in diafluorites. Of great importance is the content of carbonates, sulfides, organic substances in the host rocks and overburden rocks, the presence of which affects the pH, Eh inside the dumps during leaching and migration of uranium. The content of dispersed phases—iron and manganese hydroxides, which determine the sorption properties of technogenesis products, is also significant. Accessory minerals of near-ore and host rocks are represented by monazite, malacone, xenotim, zircon, thorite, thorianite, ortite, leucoxene, sphene, rutile, apatite, as a rule, containing an isomorphic admixture of uranium, thorium, rare earths. Uranium minerals of the host rocks are most often represented by uraninite, nasturane, coffinite, titanium-containing brannerite, davidite, which form an interspersed crystalline grains, thin veins. Ni, Co, Cr, W, Cu, Pb, Sn, Bi, Mo, which are part of sulfides and oxides, as well as Be, La, are established from the elements accompanying uranium-impurities in the ore-containing rocks of dumps in the high clarke contents. Th, P, rare earths, selenium, rhenium, scandium, which are included in the accessory minerals, and uranium minerals. The mentioned impurity elements and minerals containing them in the course of the evolution of industrial waste may tend to a secondary relative concentration due to the leaching and removal of uranium. Off-balance ores and tailings of the conversion of balance ores differ in a different ratio of relict rock-forming and ore, and, in particular, uranium and uranium-containing minerals. Relict non-metallic minerals in the ores of the developed deposits of the sodium-uranium formation correspond to the composition of minerals of mineralized metasomatites and diafluorites: these are plagioclase, albite of different generations, ferruginous-magnesian alkaline amphiboles, pyroxenes; newly formed-quartz, Fe–Mg and Mg-Ca carbonates, chlorite, phlogopite, epidote, hydrosludes. Accessory minerals of ores are torite, thorianite, apatite (in some places more than 10%), malacone, sphene, arshinovite, zircon, xenotime, as a rule, with an isomorphic admixture of uranium, thorium. These minerals accumulate in the tailings of the enrichment of industrial types of ores. The ores are low-sulfide; sulfides Fe, Pb, Zn, Cu, arsenopyrite (content 1–2%) are established. Among the ore minerals, there are also magnetite, hematite, ilmenite, Fe, Mn hydroxides. The form of distribution of uranium minerals is densely interspersed, veinedinterspersed, fractured, less often lenticular, cataclastic. The forms of separation are crystalline, needle—–like, flow-like, spherulitic, collomorphic, kidney-shaped, filmlike, sorption. It is characterized by the dispersion and zoning of individual secretions, uranium minerals. The uranium minerals in the components of off-balance ores in the dumps presented by the titanates—brannerite, davidicum, a silicate hydrate—coffinite, oxides—uraninite, nenadkevite, the pitchblende, uranium tar with different degrees of oxidation, the hydroxides of uranyl—gummata, becquerelite, scopecam, fourmarierite sladkimi and uranium—phosphate, arsenate, vanadate, sulfates, and

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carbonates in those parts of the dumps, where the products of weathering. In recultivated dumps, ningioite is possible. Microinclusions of radiogenic lead are present in the defects of the crystal lattice of uranium minerals and in the intergranular spaces, small growths of sulfides are characteristic (Tables 3 and 4). A significant amount of the total uranium content is found in dispersed form in quartz, feldspar, mica and in sorption form-in iron hydroxides. Thorium minerals in technogenic products are represented by thorite and thorianite; thorium is isomorphically part of monazite, xenotime; it can be present in the form of an organo-uranium complex-tuholite. The heterogeneity of the physico-mechanical, chemical, and mineralogical properties of industrial products causes the occurrence of various unstable states, meso -, macro -, and microtechnogenic barriers in dumps. The gas component (O2 , CO2 , H2 , hydrogen sulfide, hydrocarbons) is of great importance in the evolution of the composition of industrial products. The destruction of uranium minerals and its transition to a solution will depend on the mineral composition, the size and shape of the secretions, the presence in the form of an isomorphic impurity, or in the form of sorption and dispersed inclusions in rock-forming minerals. Initially, film-shaped, globular uranium will be leached from the near-surface zones of dumps. The crystalline forms of oxides, silicates and titanates are more stable. Table 3 The content of uranium, thorium and rare earths in waste minerals [1]

Mineral

The content of the elements, (%) ThO2

TR

Torit

66–74

0–7

Uranotorite

42–65

2–5

8–17

Macintoshit

45–57

0–2

20–26

Torianite

58–93

6–8

4–39

Uranotorianite

38–59

1–13

36–38

Aldanite

63–64

1–4

15–29

Tuholit

1–48

2–36

6–53

Nenadkevite

~1

~1

59–60

Brannerite

1–8

0–1

40–42

UO2 + UO3 0–1

Davidit

Sl

1–2

7–13

Breggerit

6–15

2–5

65–85

Ortit

–

16–23

–

Fergusonite

1–5

~1

Ningioit

–

3

1–3 39–42

Note Rare earths (according to the study) are represented by cerium, lanthanum, yttrium (tenths of % to 1%), brannerite (2.7– 7.4%); hafnium was found in zircon; strontium was found in apatite; all REE except thulium and promethium in monazite

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Table 4 Results of X-ray spectral analysis of monazite, % [1] Oxide

1

2

3

3a

4

5

La2 O3

12.54

13.98

13.65

10.97

13.10

22.12

Ce2 O3

26.49

28.08

29.60

24.83

30.86

35.75

Pr2 O3

3.13

3.17

3.37

2.39

2.36

2.76

Nd2 O3

10.60

10.09

12.02

10.17

12.98

6, 69

Sm2 O3

2.26

0.92

1.04

0.64

2.05

0.33

Eu2 O3

0.52

0.36

(0,06)

0.38

1.30

0.53

Gd2 O3

1.64

0.47

0.67

1.64

2.05

0.35

Tb2 O3

(0.20)

(0.17)

–

0.23

0.39

(0.19)

Dy2 O3

(0.14)

Not defined

(0.08)

0.83

(0.04)

–

Ho2 O3

(0.14)

Not defined

–

0.28

–

–

Er2 O3

(0.06)

Not defined

–

0.29

(0.11)

–

Tm2 O3

–

Not defined

–

–

–

–

Yb2 O3

(0.03)

Not defined

(0.10)

0.27

0.41

–

Lu2 O3

(0.17)

Not defined

–

0.55

0.37

–

Y2 O2

1.77

1.82

0.68

0.88

(0.11)

0.26

UO2

0.79

Not defined

0.19

1.40

0.31

0.36

ThO2

9.52

10.20

9.14

14.14

5.44

0.82

SiO2

1.77

2.55

–

3.86

–

1.18

P2 O5

28.23

28.19

29.40

26.25

28.12

28.66

Note Concentrations that have not reached the minimum reliably detectable values are shown in parentheses

Secondary, redeposited sorption concentrations of uranium, thorium, radium are possible in clay and mica areas and areas of accumulation of iron hydroxides. A number of works are devoted to the problems of stability and destruction, as well as the formation of minerals of uranium, thorium, associated sulfides, carbonates, and accessory uranium-containing minerals in the zones of hypergenesis and, to a lesser extent, technogenesis [8–12].

5 Evolution of Waste Composition The evolution of industrial waste is the result of a long-term interaction of the material composition of their contents and such factors as the duration of storage of dumps, the degree of compaction, the parameters and shape of dumps and the degree of their disclosure; the position in the relief; hydrochemical regime—the intensity of water exchange and the duration of washing of industrial products, the depth of the groundwater mirror; climate features and its seasonal fluctuations.

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During the long-term interaction of waste with the environment, their composition evolves, which on the one hand leads to the leaching of easily soluble forms of uranium within the dumps and their removal into the environment, and on the other—the redistribution and accumulation of a number of elements resistant to dissolution, which can be considered as possible secondary technogenic concentrations (in particular, thorium, zirconium, rare earths, scandium, molybdenum, tungsten, etc.). The evolution of waste from the extraction and processing of uranium ores is determined by textural-structural and geochemical features, the mineral composition of specific varieties of rocks and ores concentrated in dumps. The most common pattern is the destruction of uranium minerals, its leaching and migration in oxidizing environments, and the deposition and redeposition of uranium under reducing conditions. Acidic environment in dumps can occur as a result of the destruction of sulfides, uranium and all metals are mobile in this environment. In the presence of organic matter in the conditions of its decomposition in an oxidizing environment, acidic and slightly acidic waters are formed as a result of the formation of fulvic acid, carbonic acid. Under these conditions, the migration of uranium and metals takes the form of bicarbonates and complex organometallic compounds. In neutral and slightly alkaline waters, the migration of all metals is difficult, in strongly alkaline waters it does not go at all.

6 Conclusion The relevance of the conducted research is due to the need to develop the mineral resource base of the nuclear power industry of Ukraine. This development is possible due to the rational additional use of waste from uranium mining enterprises. Currently, uranium mining in Ukraine is carried out at the Michurinsky, Novokostantynivsky, Vatutinsky deposits of sodium-uranium (albitite) formation of the Ingulska, Novoknstantynivska and Smolinska mines of Shidnyy GZK. Mine mining of uranium leads to the accumulation of millions of tons of, to varying degrees, solid, radioactive waste, represented mainly by near-ore containing metasomatites, diafluorites, cataclasites, and off-balance poor ores. The dumps of the Ingulska and Smolinska mines amount to approximately 10.6 million tons. At modern mines, up to 0.3 tons of waste is accounted for per 1 ton of ore extracted by the mine method. The waste contains high concentrations of thorium, rare earths, tungsten, molybdenum, titanium, they are represented by uranium, uranium–thorium minerals, isomorphic admixture of rare earth and related elements in rock-forming and accessory minerals, sorption admixture in iron, manganese, aluminum hydroxides. The form of mineral distribution is interspersed, veined-interspersed, fractured, film-like. The evolution of the composition of waste as a result of their interaction with the environment is due to the duration of their storage, climatic conditions, relief, ground water level, the shape of dumps, the degree of their compaction and disclosure. The main importance is the textural and structural geochemical features and

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mineral composition of the dump rocks, the resistance of minerals of oxides, silicates, uranium and thorium titanates, and the dissolution and migration ability of various elements. The content of carbonates, sulfides and organic substances in the rocks of dumps is also of great importance, which determines the Eh and pH during the processes of uranium leaching. The evolution of chemical waste, on the one hand, leads to the leaching of easily soluble forms of uranium and their removal into the environment; on the other hand, to the redistribution and accumulation of thorium, rare earths, and related elements (W, Mo, Ti, Zr) in minerals resistant to dissolution (silicates, titanates, uranium, thorium), which can be considered as possible secondary technogenic concentrations of valuable secondary raw materials. As a result of these processes, an increase in the Th/U ratio is observed in longterm stored and reclaimed waste. The issues of the balance of uranium, thorium, and rare earth elements are complex, and should be studied within the dumps of specific mines. To solve questions about the prospects for the rational use of useful components of waste that form technogenic concentrations (secondary raw materials), it is necessary to solve the following tasks: • in-depth analysis of the material composition of different types of waste using new methods, which will be the necessary basis for the development of schemes for complex enrichment of uranium ores and possible associated extraction of thorium and rare earth elements; • analysis of the stability of mineral phases, the possible share of the accumulation of thorium and rare earths in waste in mineral, isomorphic, sorption forms as a result of the removal of easily soluble uranium. The use of waste as man-made resources is carried out in the uranium-producing countries of Europe, the USA, and Russia.

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6. 7. 8.

9.

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M. Yaroshchuk et al. for the degree of doctor of technical sciences, specialty 21.06.01 – ecological safety. – National Aviation University. Kyiv (2020) State Enterprise Eastern Mining and Processing Plant. https://www.vostgok.com.ua/contacts Verkhovtsev, V.G.: Applied (prospecting and engineering-geological) aspects of studying of platform geostructures of Ukraine. Environ. Ecol. Safety Life 3, 80–92 (2005) Fomin, Yu.A., Demikhov, Y.M., Lazarenko, E.E.: Model of the evolution of the ore-forming system of the Severinovsky uranium deposit (Ukrainian shield). Geochem. Ecol. 8, 169–178 (2003) Fomin, Yu.A., Demihov, Yu.N., Lazarenko, E.E.: Features of the ore-forming fluid of the Novokonstantinovsky deposit (Ukrainian shield). Dopovidi Natl. Acad. Sci. Ukraine 6, 85–91 (2009) Fomin, Yu.A., Demikhov, Y.N., Sushchuk, K.G.: Behavior of uranium in the evolution of albitite deposits of the Kirovograd megablock. Dopovidi Natl. Acad. Sci. Ukraine 1, 126–131 (2010) Fomin, Yu.A.: Thorium in uranium-ore albitites of the Kirovograd megablock. Geokhimiya Ecol. 22, 144–160 (2013) Fomin, Y.O., Demikhov, Y.M., Verkhovtsev, V.G., Dudar, T.V., Borisova, N.N., Kravchuk, Z.N.: Pathfinder elements of uranium mineralization from albitite formation of the Ukrainian shield and their impact on the environment. Environmental Safety and Natural Resources 33(1), 42–58 (2020). https://doi.org/10.32347/2411-4049.2020.1.42-58

Iterative Solution of the Inverse Problem of Resistivity Logging of Oil and Gas Wells: Testing and Examples Mykyta Myrontsov , Oleksiy Karpenko , Oleksandr Trofymchuk , Stanislav Dovgyi , and Yevheniia Anpilova

Abstract Electrometry is one of the main methods of geophysical research of wells, including oil and gas ones. The result of electrometric research is the construction of a geoelectric model of formation evaluation. Building such a model requires solving a mathematical inverse problem, which is currently possible only by software and computing. The paper describes an iterative algorithm for solving the inverse problem of electrical logging. The iterative process for solving the inverse problem at each iteration step contains a procedure for solving the direct problem. To check the correctness of the solution of the direct problem the results of its testing are given. In particular, the results of comparing its results with the results of another program for solving the direct problem of electrical logging are presented. As a model for comparison, bedding with parameters inherent in dense formations and aquifers, oil and gas reservoirs was chosen. The complex of lateral loggingsounding in combination with the focused lateral logging probe is considered as the equipment. The problem of modern use of such a multi-probe electrometry system for determining the conductivity characteristics along the formation with the maximum equal vertical resolution for all probes of the complex is considered. To do this, we used the original method of solving the inverse problem. The method is based on changing the weights of sondes with different vertical resolution. The work resulted in the development and implementation of software and methodological support for quantitative interpretation of electrometry of oil and gas wells, which contains the possibility of solving the inverse problem with high vertical resolution for actual models of sections in the Dnieper-Donetsk basin (DDB) for different types of multiprobe electric logging equipment. The possibility to determine the collectors in three ways is realized: automatically by the selected complex of electrometry, by auxiliary methods, manually. The developed and implemented software and methodological software, which has M. Myrontsov (B) · O. Trofymchuk · S. Dovgyi · Y. Anpilova Institute of Telecommunications and Global Information Space of the National Academy of Sciences of Ukraine, Kyiv, Ukraine e-mail: [email protected] O. Karpenko Taras Shevchenko National University of Kyiv, Kyiv, Ukraine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Zaporozhets (ed.), Systems, Decision and Control in Energy III, Studies in Systems, Decision and Control 399, https://doi.org/10.1007/978-3-030-87675-3_11

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successfully tested on real well material, has been proposed for implementation in production. Keywords Electrometry · The direct problem · The inverse problem · Oil and gas well · Electrical logging

1 Introduction Modeling of mathematical physics problems (solution of a direct problem) is necessary in order to create an iterative method of quantitative interpretation of logging data (solution of the inverse problem), as shown in Fig. 1. The task of creating an effective tool for solving the inverse problem is important from a practical point of view because it is the well logging (WL) that answers the question: where is the useful fluid, how much and at what daily rate it can be extracted? This answer is obtained from a set of methods, one of the main of which is the method of electrometry [1]. From the point of view of electrometry problems, the answer to this question is hidden not in the measured average values of electrical apparent resistivity (AR), but in the values of geoelectric parameters of the section model describing the spatial distribution of spisific resistivity (SR). In this case, other auxiliary methods are added to the electrometry [2]. Such a spatial distribution according to any downhole electrometry can be established only by solving the inverse problem of mathematical physics, the solution of which is a non-trivial scientific and complex technical problem that requires the use of modern computing

Fig. 1 Scheme for solving the inverse problem

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technologies [3–7]. It should be noted that in a number of cases it is possible to significantly simplify the modeling using physically obvious simplifications of the mathematical model [8–11].

2 Electrometry Equipment Two physical principles of electrometric study of penetration layers have been widely used: electric logging (EL) and low-frequency induction logging (IL). We do not consider the technology of high-frequency induction logging isoparametric sounding (VIKIZ) [12] because the actual problems inherent in the conditions of DDB are usually solved with the help of EL and IL [13]. EL is used in wells that are filled with conductive drilling mud (SR of the mud is less than 0.5  m) or weakly conductive solution (SR of the mud is 0.5–5  m). IL is used in wells filled with poorly conductive or non-conductive mud (SR of the mud ismore than 5  m). This classification is quite conditional, but it is used in the domestic WL industry. All three named types of wells (conductive, weakly conductive, non-conductive) are inherent in the conditions of the DDB, so the task of creating new electrometry equipment becomes the task of independent creation of two different EL and IL complexes. The equipment of the basic methods of electrometry, which is practically used in Ukraine, was developed in the middle of the last century and is morally obsolete. Such equipment consists of three types: equipment of lateral logging sounding (BKZ), which is a set of potential and gradient probes of different depths but low spatial resolution; EL equipment consisting of one focused probe—lateral logging equipment (LL); equipment consisting of one focused IL probe—AIK equipment. Only BKZ and LL are used in leading wells; in weakly leading—BKZ, LL and AIK; in non-conductive—only AIK and recently developed equipment for four-probe IL. The use of the complex BKZ + LL + AIK has insurmountable disadvantages: in the wells filled with non-conductive drilling mud, we have only one dimension of IL, which does not allow to establish the fact of changes in conductivity along the formation; in wells filled with conductive solution, we have only one measurement of high-resolution EL, and the change in conductivity along the formation is determining with low spatial resolution; in weakly conducting wells, where both methods (EL and IL) can be used—only two measurements of high resolution, which are also insufficient to determine the three required parameters of thin layer. That is why there is a problem of developing new electrometry equipment that is able to solve current problems in the conditions inherent in DDB [14].

3 Mathematical Modeling As practice shows, the most effective is the use of such methods of solving a direct problem, which can be used in the iterative process of solving the inverse problem (modern methods of solving the inverse problem are iterative processes [1], see

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Fig. 1), which on each step of the iteration requires the solution of a direct problem. That is why the way of development and creation of own software tools for solving direct problems of EL was chosen to achieve this goal. Indeed the method of integrated currents was chosen. Consideration of the effectiveness of algorithms for solving direct problems and methods of their software implementation can not be completed without the most important stage—checking the reliability of the results provided by the developed and implemented algorithms. In other words, it is not enough to have a way to solve a direct problem that produces certain results depending on changes in boundary conditions. In order to say with confidence that we have a way (and its software implementation) to solve a direct problem—we must make sure that the chosen method is correct and all the results are correct (within the permissible error).

4 Testing of the Created Software of Mathematical Modeling The first stage of testing is the study of the potential field, which is calculated when solving a direct problem. Figure 2 shows the spatial distribution of the electric field potential for the case of an unfocused probe (point electrode A and inverted B located on the conductive braid of the cable). Qualitatively and quantitatively analyzing this distribution, the

Fig. 2 Distribution of electric field potential for the case of unfocused probe (point electrode A and inverted B, located on the conductive braid of the cable)

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conclusion was made: the boundary conditions are set correctly (even clearly visible long conductive braiding of the logging cable); there are no extrema in the middle of the solution domain (the result corresponds to the condition of monotonicity for harmonic functions); at the outer boundaries of the region the potential derivative goes to zero. The same analysis was repeated for the boundary conditions corresponding to the focused LL probe (see Fig. 3). The second stage is to check the correspondence of the potential around the axis of the borehole, because in a cylindrical coordinate system (with axial symmetry, such problems are naturally solved in this coordinate system) Laplace equation:   ∂U ∂ 2U 1 ∂ 2U 1 ∂ r + + = 0, r ∂r ∂r ∂r 2 r 2 ∂φ 2

(1)

 when r = 0 we get a feature of the form 1 0. When multiplying Eq. (1) by r 2 :   ∂U ∂ 2U ∂ 2U ∂ r + r2 2 + = 0, r ∂r ∂r ∂r ∂φ 2 when r = 0 we get

∂ 2U ∂φ 2

(2)

= 0 that is also not an informative condition.

Fig. 3 The distribution of the electric field potential for the case of a focused LL probe in the vicinity of the electrodes of the same potential A,Ab,At

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It is customary to eliminate this feature by means of a boundary transition r → 0 (see for example [15]). However, direct use (1) or (2) for problems of unfocused probes where point electrodes are placed on the axis of the well—is impossible. For verification, we used, among other things, the numerical calculation of the values of the coefficients of these probes under the same assumption. The result of such a numerical calculation can be compared with the value obtained by the formula:  K = 4π

M A − BM M A · BM

−

NA− NB NB · NA

−1 .

A similar problem arises in IL problems if we consider the equation for the vector potential in a cylindrical coordinate system. The summing of the current crossing the shell, which contains inside one or more current electrodes of known current strength, was also a test for error of the equations of the discrete model. A very important criterion for the correctness of the choice of algorithm parameters is the sensitivity of the results to changes in the already selected parameters of

Fig. 4 Example of comparing the solution of a direct 2D problem by programs developed and implemented by different authors (SR of drilling mud 2 •m; well radius 0.1 m)

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the discrete model. The study of such sensitivity is painstaking and extensive, but necessary for testing. For example, for the finite difference method, if we grind the grid steps in half and the result changes by 1%, it will be an indication that at least at this point in our method there is most likely no error. Conversely, if the result will change significantly with a slight change in the parameters of the discrete model—we are still far from the correct result. Such checks were made for more complex environmental models. This is necessary because, in the end, one of the tasks of numerical solution of direct problems can be to determine the range of stable operation of a particular equipment or to determine the limits of its sensitivity to a particular parameter of the environment model. Resolving this issue in the analysis of the results is important to understand what we are dealing with when there is a discrepancy between the obtained and expected results: with the actual achievement of the operating range of the equipment or with the error of the chosen method of solving a direct problem. All these checks for the developed software have been performed. Based on this, it was concluded that for the selected parameters of the discrete model, the problem is solved with an error less than the predetermined one. An informative check of the implemented software was the comparison of its results with the results obtained by other software. Various commercial software packages, such as FemLab (University of Florida), were used for this comparison. It should be noted that when using such packages, you need to be very careful, because, for example, FemLabwhen trying to solve a direct EL problem for point electrodes located on the axis of the well, in a cylindrical coordinate system can give a result whose error will not be immediately noticeable. But with the right use of commercial programs, the benefits of testing them for your own programs are hard to overestimate. Another form of testing is to compare the results obtained not with the results obtained by commercial programs, but with the results obtained by programs that have been implemented by other researchers of electrometry problems, for the same models and the same probes. Here is an example of such a comparison, which was performed in Novosibirsk (A. A. Trofimuk Institute of Oil and Gas Geology and Geophysics, SB RAS). A series of calculations was performed (Figs. 4 and 5) developed by the program (“Program 1”) developed by the colleagues and the own program actually developed (“Program 2”). When discussing the results of the comparison, it was found that some differences may arise due to different representations in the modeling of the inverse current electrode (in the actual program, it is not presented as a point electrode, but in the form of a conductive braided cable at a finite distance from direct current electrode). Other discrepancies are the result of the approximate nature of any numerical calculation and in this case objectively reflect the magnitude of its actual error. The main test of the accuracy of the algorithm for solving a direct problem is to assess the performance of the algorithm for solving the inverse problem that uses it

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Fig. 5 Example of comparing the solution of a direct 2D problem by programs developed and implemented by different authors (SR of the drilling mud 2 •m; well radius 0.1 m)

(see Fig. 1). At least the evaluation of the results of the inverse problem in comparison with other data of geological interpretation and lithological dissection of cuts allows to immediately reveal obvious errors in the implementation of the solution of the direct problem. When using the created software to solve direct problems of EL in the iterative process of solving the corresponding inverse problems, no errors were found. The use of the created modeling tools in addition to the described use in the implementation of R & D and the use of software and methodological tools for solving the inverse problem allowed to perform work on the study of the measurement error with the error of solving the inverse problem and investigate the areas of equivalent solutions of the inverse problem.

5 The Method of Solving the Inverse Problem To assess the effectiveness of a method of solving the inverse problem, you can use a criterion based on the division, which was proposed in 1984 [16]. Thus, the efficiency of solving the inverse problem is determined by: the method of calculating

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Fig. 6 Well “North Pokursk” (SR of the drilling mud 1.2–1.3  m). a—VIKIZ diagrams; b—the result of solving the inverse problem BKZ + LL for all K = 1; c—the result of solving the inverse problem BKZ + LL when K A4.0M0.5N = 0; d—diagram of the SP

the measurement data of the probes for the selected parameters of the environment; selection of the parameter “proximity” of the calculated readings of the probe and the real ones; selecting the method of selecting model parameters for the selected parameter “proximity”. These questions can be paraphrased accordingly as questions of choice: • method of solving a direct problem (finite difference, finite elements, integral currents, semi-analytical solution, etc.); • the type of functional that will be minimized when solving the inverse problem; • method of iterative process of solving the inverse problem.

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Enough time has passed since 1984, and we can assume that the choice of iterative process in the modern development of computer technology no longer significantly affects the effectiveness of the inverse problem. The first question will also be considered resolved. Let us focus on the second question, i.e. the question of choosing a functional that is minimized when solving the inverse problem. It turns out that the choice of functional allows to reduce the influence of boundary effects on the solution of the inverse problem. There are several possible approaches to solve the inverse problem. All of them are based on minimizing functional. Here are the main:

n  T

ρi − ρiP 2   T 1 T , F ρ1 , . . . , ρn =  n i=1 ρiT where n is the number of equipment probes; ρiT —calculated values of AR for the model under consideration; ρiP —actually obtained values of AR. There are also some variations of the functional record, which will be minimized in the process of solving the inverse problem. For example, in the form:

n  2   T 1 ρiT − ρiP , F ρ1 , . . . , ρnT =  n i=1 δi ρiT where δi —the relative error of the i probe. Or:

n  T ρ − ρ P 2  1

 T i i T  , F ρ1 , . . . , ρn = n i=1 δi ρiT + χi where χi – the absolute error of the i probe. As a criterion for the proximity of the found solution with the desired true value, we can consider the minimization of the functional [17, 18]:

n

 ρiT − ρiP 2   T T Ki , F ρ1 , . . . , ρn =  ρiT i=1

(3)

where K i —weight coefficients of each probe of the complex, which can be changed by the interpreter. In the presented example n is equal to 8: probes LL, AIK, and 6 probes BKZ (A0.4M0.1 N, A1.0M0.1 N, A2.0M0.5 N, A4.0M0.5 N, N6.0M0.5A, A8. 0M1.0 N, N0.5M2.0A (inverted A2.0M0.5 N).

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Using (3) for a three-layer model of the cut (well + penetration zone of the drilling fluid filtrate (hereinafter—the zone) + part of the formation undamaged by the drilling fluid filtrate (hereinafter—the bed)) we developed an algorithm for solving the inverse problem for BKZ-LL complex, its software implementation is performed (together with the creation of the appropriate friendly graphical interface), its testing on model and well material is performed. To minimize the influence of neighboring formations on the results of solving the inverse problem of EL probes (complexes BKZ, BKZ-LL, or BKZ-LL-AIK) was used to reduce the influence of large probes in determining the parameters of thinbeds (due to the use of (3) we can to change the weights of the influence of an each probe on the results of solving the inverse problem). The obtained results for real well material for BKZ received a positive assessment (confirmed by the relevant acts of implementation) and were recognized as faithful specialists of Ukrspetsgeologiya LLC, Prydniprovska Mining and Chemical Corporation LLC, etc.

6 Examples It should be understood that in carrying out work that depends on the direct participation of commercial structures, technical difficulties such as a ban on the dissemination of information that may contain a trade secret are constantly encountered. Such information includes the names of specific deposits or specific wells. Here are two examples for real wells, which we will tentatively call “NorthPokursk” and ‘Vyngapur” for which the developed software allowed to solve a number of specific practical problems and explore a number of individual issues of interpretation in different terrigenous sections. Figure 6 shows the results of solving the inverse problem BKZ + LL for two different sets of values of weights in (3) (R‘z and R‘b—SR, obtained under the assumption that the whole interval is an interval with penetration, and R“b—obtained under the assumption that the entire interval is impermeable), in comparison with the VIKIZ data and the spontaneous polarization (SP) diagram. When the two largest out-of-focus probes are removed, the formation parameters, in the no penetration, becomes a kind of "middle" between the zone parametersand the formation parameters in the presence of penetration. Therefore, in the presence of large probes and with increasing penetration (which is crucial in this case), the complex BKZ + LL is the most sensitive to beds parameters undamaged by the filtrate of the drilling fluid part of the formation. The change in the penetration zone parameters in this case is the effect of a smaller value. That is, the change in the parameters of the zone is offset by its small size. This is confirmed by VIKIZ data. This fact, generally speaking, complicates the process of interpretation, because without additional material, as follows from the result, we can not unambiguously establish the permeability of the interval.

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In this case, or in the case when it is really impossible to distinguish the intervals “with” and “without” penetration without additional methods, the following should be done. Another approach was used to avoid this ambiguity. We divide the investigated interval by auxiliary methods into intervals with and without penetration. Then on the first intervals we consider as a result of the decision of R‘z and R‘b, and on the second—R“b. This approach is shown in Fig. 7. It is obvious that the parameters of the formation, defined in the assumption of no penetration, differs significantly from the parameters defined in the assumption of penetration of the entire study interval. Indeed, assuming the presence of penetration and its absence, the parametersof the formation almost coincides in the presence of an eight-meter probe in the complex. The use of the created software and methodological tools in addition to the described use in the interpretation of electrometry data allowed to perform work on the study of the relationship between measurement error and the error of solving the inverse problem and to investigate areas of equivalent solutions of the inverse problem. The approaches used to solve not only the problems of electrometry modeling were used in the work. The created methods and the obtained results can be used in solving a wider range of problems [19–25].

7 Conclusions The developed software allowed with the minimum expenses of resources of time by means of modeling: 1.

2.

for a given geometry of the probe part, given initial conditions (supply parameters) and given well conditions, calculate the ranges of the measured values and investigate the characteristics of the probe (vertical resolution, radial resolution, etc.). using the created minimization algorithm to set the optimal value of any probe parameter to reduce the error of setting any geoelectric parameter of the selected formation.

Developed and implemented software and methodological support for quantitative interpretation of electrometry of oil and gas wells, which contains the possibility of solving the inverse problem with high vertical resolution for current models of sections in the conditions of DDB (three-layer models of horizontal formations, vertical wells) for complexes BKZ; BKZ + LL. At the same time, it is possible to determine the collectors in three ways: automatically by the selected complex of electrometry, by auxiliary methods, manually. The developed and implemented software has been successfully tested on model and real well materials.

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Fig. 7 Well “Vyngapur”. The result of solving the inverse problem BKZ + LL using additional information about the nature of penetration in the studied interval

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Mineralogical-Geochemical Properties of Bentonite Clays of the Cherkasy Deposit to Increase the Environmental Safety of Radwaste Disposal at the Vektor Storage Complex Borys Shabalin , Konstiantyn Yaroshenko , Serhii Buhera , Nataliia Mitsiuk , and Oleg Myroshnyk Abstract The lack of comprehensively developed and approved in accordance with the established procedure evidence-based requirements for application of clays as a barrier material poses risks for the safe disposal of radioactive waste in storage facilities at Industrial Complex “Vector” for the period of their operation and closure. The bentonite clay from the Ukrainian largest Cherkasy bentonite and palygorskite clay deposit is considered to be the most acceptable as the main component of the insulating screens in radioactive waste storages. The main properties and features of the composition of the Cherkasy natural bentonite clay (Dashukivka site, II layer) and its variety—alkaline earth bentonite (soda-modified bentonite) that provide the radioactive waste isolation in the storage facilities have been considered. The bentonite clays from the Cherkasy deposit were shown to have good waterproofing and barrier properties, including high sorption capacity of 90Sr and 137Cs—one of the main characteristics ensuring safe disposal of radioactive waste. Alkaline earth bentonite adsorbs 90Sr and 137Cs more efficiently than the natural one. However, 90Sr is adsorbed in greater amounts than 137Cs on both types of bentonite. When the time of interaction with an aqueous solution increases, both types demonstrate redistribution of mobile (exchangeable) and immobile (non-exchangeable) forms of radionuclides. The portion of the immobile form, which does not participate in the migration processes, increases. Keywords Bentonite · Cherkaske deposit · Isolation properties · Caesium · Strontium · Sorption

B. Shabalin (B) · K. Yaroshenko · S. Buhera · N. Mitsiuk State Institution, “The Institute of Environmental Geochemistry of National Academy of Sciences of Ukraine”, Kyiv, Ukraine e-mail: [email protected] O. Myroshnyk National University of Civil Defence of Ukraine, Kharkiv, Ukraine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Zaporozhets (ed.), Systems, Decision and Control in Energy III, Studies in Systems, Decision and Control 399, https://doi.org/10.1007/978-3-030-87675-3_12

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1 Introduction Bentonite clays possess many useful physical, mechanical and chemical properties (plasticity, swelling capacity, high sorption and high radionuclide retardation capacities, etc.). Bentonites are widely used in different countries at nuclear power plants and in almost all types of radioactive waste (RW) storage facilities to provide isolation of radionuclides from the environment [1]. At the same time, in Ukraine, despite extensive designing of the I and II stages of a near-surface radioactive waste storage facility at the Industrial Complex “Vector”, the safety-relevant properties of bentonite clay and methods of its treatment are not fully used. This may lead to inefficient use of bentonite to ensure high geo-ecological safety of the radioactive waste [2]. One of the most promising deposits of bentonite clays which might be used as the main component of the insulating screens in radioactive waste storage facilities is the Cherkaske deposit of bentonite and palygorskite clays (Dashukivka site), which is the largest in Ukraine (about 80% of all balance reserves of bentonites) and one of the largest in Europe [3]. At the same time, the fact that the operator of the facility does not have comprehensively developed and properly approved scientifically justified requirements for clay application as a barrier material threatens the radioactive waste disposal safety for the entire period of the facility operation and closure (which is about 500 years for low-level waste). The purpose of this work is to study mineralogical-geochemical properties of the bentonites from the Cherkaske deposit (Dashukivka site, II horizon) that affect the insulating properties of the buffer materials used in radioactive waste storage facilities.

2 Gneral Information on the Dashukivka Site of Cherkasy Deposit The Cherkaske deposit contains high-quality alkaline earth bentonite clays. The deposit consists of several sites (Dashukivka, Bosivka and Ripky). The deposit was thoroughly explored in 1958–1960. Now its area is about 2.7 km2 . At present, the main extraction is carried out at the Dashukivka site (Fig. 1) by PJSC “Dashukivski Bentonity”. Administratively, the site is located in Cherkasy region near the village of Dashukivka. The Dashukivka site has been thoroughly explored by vertical borehole drilling to the depth of about 43 m. Hydrogeological conditions of the area are favourable for open pit mining. The productive stratum is above the groundwater level. The only water coming in the quarry is rainfall. The Dashukivka site hosts a horizontal stratiform Middle Miocene deposit composed of 5 layers [4] that differ in chemical–mineralogical, physical–mechanical and macroscopic characteristics.

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Fig. 1 a—open-cast mine of the Cherkaske bentonite and palygorskite clay deposit; SEM-images of Dashukivka bentonite (II layer): b—natural; c—alkaline-earth (soda-modified) bentonite (PBA20) [5]

Compared to the other layers, the II layer is of most interest since the content of the smectite (montmorillonite) component is significantly high. The content of the lowdispersed fractions with a size of 0.005–0.001 mm is 86–87.5% (Fig. 2). Bentonite micrographs show dense microaggregates of various configurations, often with clear contours (Fig. 1b, c), as well as the contours of individual fine scaly particles, sometimes of elongated grooved form, which is associated with the crystal chemical properties of montmorillonite. Based on the transmission electron microscopy data, the average size of primary particles does not exceed 30 nm.

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Mass, %

60 50 40 30 20 10 0

0,25-0,1

0,1-0,05

0,05-0,01 0,01-0,005 0,005-0,001 K > Mg > Ca > Al > Fe [6]. This can explain the decrease in the K, Mg, Al and Fe content in the PBA-20 bentonite after soda modification of the natural bentonite. The relative increase in Ca content is likely to result from the additional formation of calcite during the treatment of bentonite with soda. The latter requires more comprehensive studies of the mass balance of elements of natural and soda modified bentonites. The calculated crystal chemical formula of the natural Dashukivka bentonite based on the averaged chemical analyses data is described in the form: (Al1.21 Fe3+ 0.49 Mg0.30 ) [Al0.18 Si3.82 ]O10 (OH)2 + (Ca0.17 Na0.03 ) [7]. Table 2 shows the mineral

0.27 8.89

Soda-modified bentonite (PBA-20), wt%

Na2 O 1.61

2.42

MgO

Oxide (averaged)

Natural bentonite, wt%

Sample

Table 1 Chemical composition of bentonites; oxides (EDS data)

16.51

19.45

Al2 O3 66.17

68.02

SiO2 –

0.18

K2 O 2.27

1.45

CaO

–

1.35

TiO2

4.21

6.86

FeO + Fe2 O3

0.33

–

MnO

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208 Table 2 Mineral composition and cation exchange capacity of the natural Dashukivka bentonite clay (II layer) [7]

B. Shabalin et al. Mineral composition Smectite (montmorillonite)

65–75

Quartz

20–25

Calcite

3–5

Kaolinite

3–5

Mica

5

Feldspar

3

Rutile (anatase)

–

Cation exchange capacity, meq/100 g Na+

2.6

Ca2+

41.3

Mg2+

32.6

K+

1.2

Sum of cations

77.6

composition and the exchange cation content of the natural Dashukivka bentonite clay (II layer) according to [7]. In terms of mineral composition, it is mainly represented by Ca-montmorillonite and finely dispersed quartz. At the bottom of the second layer (7.4 m) Ca-montmorillonite dominates, while the quartz content is quite significant in the upper part of the layer (3.0–5.8 m) and decreases downwards (6.5–7.4 m). The content of carbonates in the bentonite is relatively low (3–5 wt%). Since the sum of calcium and magnesium ions in the exchange complex of the Dashukivka bentonite clays prevails, they belong to alkaline earth calcium ones. Montmorillonite (the main mineral of bentonite clays) is a natural layered aluminosilicate. The basis of a smectite mineral structure is a 2:1 layer, composed of two tetrahedral sheets and one octahedral sheet sandwiched between them (TOT structure) [8]. Two adjacent montmorillonite layers are joined by a weak O–O bond (Van der Waals force), so molecules of water and other polar liquids can freely penetrate between the montmorillonite layers and expand the interlayer distance. The first basal reflection value according to the X-ray diffraction analysis of montmorillonite equals to 1.25 nm and corresponds to one water molecular layer in the interlayer space, while the reflection value 1.55 nm—to two molecular layers. In highly moistened samples, this value is even higher [9]. Isomorphous substitutions in the sheets, mainly in octahedral, produce a net negative layer charge of 0.6–0.9 per formula unit, which is neutralized by the entry of exchange interlayer cations (Na+ , K+ , Ca2+ , Mg2+ , occasionally Fe2+ ) usually in the hydrated form [6]. This facilitates adsorption on the internal and external surfaces of the crystallites. The connection between the layers is weak, so water can enter into the interlayer space causing swelling of the mineral. A wide range of isomorphous substitutions is often found in the octahedral positions of montmorillonite. Depending on the chemical composition of the mineral-forming medium, their structure can contain not only isomorphic Al3+ –Fe3+ or Al3+ –Mg2+ pairs, but also different

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quantity of Al3+ , Fe3+ , Mg2+ , Fe2+ and other ions. In montmorillonite, isomorphous substitutions of Si4+ , Al3+ , Ti4+ , sometimes Fe3+ take place in tetrahedral positions. In montmorillonite, up to 80% of exchange cations are located in the interlayer space, and approximately 20%—on mechanically broken edges of crystals [6], which have the form of thin flat flakes that resemble mica scales. These structural features of montmorillonite determine such specific property of bentonite clays as high adsorption capacity of heavy metals, caesium isotopes and other radionuclides contained in liquid radioactive waste.

4 X-ray Diffraction Analysis The main method used to analyse clay minerals is x-ray diffraction (XRD) [10]. It can be used to identify the mineral composition of clay minerals and to quantify the proportions of different minerals. For approximate quantitative estimation of the montmorillonite content in clays, a method based on the relation between the diffraction peak (060) intensity and the content of the mineral in a sample is used. Also, based on the values of the d001 reflex from oriented air-dry clay samples, it is possible to identify the type of bentonites—alkaline, alkaline-earth or alkaline-alkaline-earth. The mineral composition is identified according to the ASTM standards [11]. Intense reflexes recorded in the diffraction patterns (Shimadzu XRD-6000 diffractometer) of natural bentonite samples (d < 0.005 mm) related to the interplanar spacing d001 of 1.55 nm indicated that they are due to the presence of two molecular layers of water in the interlayer space of montmorillonite. A series of basal reflexes of 0.449, 0.255, 0.170 nm is characteristic of layered aluminosilicates with 2:1 structure (Fig. 3) [5]. Diffraction reflex (060), that is equal to 0.150 nm, indicates that the clay mineral belongs to the dioctahedral series. Narrow and intense lines with the interplanar spacing of 0.335, 0.228, 0.213, 0.198, 0.182, 0.167 and 0.154 nm refer to quartz. The first diffraction reflex of PBA-20 bentonite is shifted to 1.368 nm. In the diffraction pattern of the sample there is a series of basal reflexes of montmorillonite—0.449, 0.246 and 0.146 nm. Based on the averaged data of quantitative (by diffraction reflex intensity) and qualitative X-ray diffraction in accordance with the ASTM standards [11], the main rock-forming mineral of the Dashukivka bentonite (II layer) is montmorillonite (# 00-012-0204, 00-003). Its content here approaches 75 ± 3 wt%. The accompanying mineral is quartz (# 01-089-8937), its content is about 20–25%. The samples also contain calcite (# 01-86-2340)–3–5% and kaolinite (# 01-86-6538)–3–5%.

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Fig. 3 Diffraction patterns of the Dashukivka bentonite clay (II layer): a—natural bentonite, b—soda-modified bentonite (PBA-20), S—smectite, K—kaolinite, Q—quartz, C—calcite [5]

5 Infrared Spectroskopy In the spectrum of the studied bentonites all the most characteristic absorption bands have been identified with VERTEX 70 spectrometer. Infrared spectra of bentonites in the range of 4000–400 cm−1 are typical for dioctahedral Al-smectites. The absorption at 411 and 610 cm–1 in the natural bentonite spectrum (d < 0.005 mm) corresponds to the vibrations of the Si–O–Al (Fe) bonds and the substitution of Si for Al and Fe in the tetrahedral position of montmorillonite (Fig. 4). The intense absorption band with a maximum at 911 cm–1 corresponds to the deformation vibrations of bonds in the SiO4 tetrahedra of the tetrahedral sheets of montmorillonite, and is associated with the vibrations of Al–OH and Fe–OH bonds and a decrease in the tetrahedron symmetry as a result of heterovalent substitutions Al3+ (Fe3+ ) → Si4+ . Intense absorption at 1655 cm–1 is typical for deformation vibrations of water

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Fig. 4 IR spectra of the Dashukivka bentonite (II layer): a—natural, b—soda-modified (PBA-20)

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molecules and corresponds to the presence of divalent cations in the crystal chemical formula of montmorillonite. The intense narrow band vibration at 3619 cm–1 with a maximum at 3391 cm–1 in the wide range 3300–3700 cm–1 is a valence vibration of the O–H bond. It is the result of superimposing of individual absorption bands that correspond to Al–OH– Al and Al–OH–Mg vibrations as well as Mg–OH–Fe, Al–OH–Fe and Fe–OH–Fe vibrations which are associated with isomorphic substitutions in the montmorillonite octahedra. Natural bentonite samples also contain quartz (absorption bands 691, 778 and 796 cm−1 ). The IR spectrum of the soda-modified bentonite (PBA-20) is almost identical to that of the natural sample, there is only a slight shift of the bands, that may be explained by the difference in the structure of the water layer in the interlayer space (the natural bentonite has a double water layer, while the Na-bentonite—single).

6 Differential Thermal Analysis Three endothermic effects are observed on the DTA curves of natural and PBA-20 bentonite (d < 0.005 mm) in the temperature range of 80–950 °C with the maxima at 100–110, 480–510 and 690–720 °C (NETZSCH STA449F1 analyser), which is characteristic of montmorillonite clays (Table 3). The first major endoeffect results from the release of the major amount of the adsorbed water. The amount of this water that is predominantly interlayer water depends both on the nature of the adsorbed ions and on the pre-treatment of the sample (drying conditions, relative humidity, etc.). The dehydration curves are S-shaped and do not have a clear break between the temperature of the complete interlayer water loss and Table 3 Results of the thermal analysis of the Dashukivka bentonite (II layer) Sample

Temperature interval, °C

Maximum, °C

Natural

70–250

100.2

250–720

480 690

−36.97 j/g 7.32 −18.62 j/g

720–980

920

+29.66 j/g 1.63

70–250

107.8 Crossing 176.5

−1.99 j/g

250–760

510 720

−118.8 j/g 7.05 −11.98 j/g

600–960

920

+8.23 j/g

Soda-modified (PBA-20)

Thermal effects*

Mass loss, %

Total mass loss

8.85

17.80

2.17

10.21

0.99

* Indicates thermal effect—the amount of energy during an exothermic reaction, or absorption during an endothermic reaction per mass of substance. Measured in joules per gram

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the temperature of the beginning of the loss of hydroxyl groups. The temperature of 300–320 °C can be taken as a breaking point. At this stage, water evaporated from the air-dry montmorillonite samples makes up to 5–9 wt% of the mineral. The interlayer water loss is accompanied by a cell size reduction along the “c” axis to 0.94 nm, and a change in the “a” and “b” parameters of the crystal lattice. The exact value of the “c” parameter depends on the size of the interlayer ions. Two endothermic effects with maxima at 480–510 and 690–720 °C correspond to elimination of the constitution water and destruction of the clay mineral crystal lattice. The dehydration almost stops at 800 °C. These effects characterize the thermal stability of the mineral and depend on the ratio of Al, Fe and Mg in the octahedral sheets of the lattice. Isomorphous substitution of aluminum with iron helps to decrease the temperature maximum of this effect, while isomorphous substitution of aluminum with magnesium—to increase. In PBA-20 samples, these two maxima have higher values than in the natural sample due to the reduced content of Mg and Fe in the mineral. The hydroxyl water loss neither begins nor ends abruptly. There is almost no dependence of the nature of dehydration on the particle size of the samples. This was to be expected, since the reduction in montmorillonite particle size is, for the most part, a reduction in aggregates rather than primary components, i.e. the separation of scales along the basal planes that are most easily separated, rather than their breaks. The second endothermic effect is due to the loss of hydroxyl groups bonded to the magnesium that is in octahedral coordination, but not the lattice destruction. About 1.5–1.0% of water is still present at the temperature above which there is a loss of the main amount of the hydroxyl water (from 500 to 800 °C). This endothermic effect becomes weak exothermic with a maximum at 910–920 °C, which is caused by formation of new phases—mullite in the form of small needle crystals, the size of which increases with temperature increase, or spinel, cristobalite and feldspar, the formation of which depends on crystallochemical properties of montmorillonites. For the PBA-20 samples, there is a stepwise removal (maximum at 107.8 °C with a slight bend at 176.5 °C) of the adsorbed water, followed by cleavage of the clay mineral hydroxyl groups in the temperature range of 100–200 °C [12]. For PBA-20 samples, the above two maxima have higher values than for natural samples, which indicates its greater thermal stability compared to natural bentonite. The total weight loss of the samples is about 17.8 and 10.31% for natural and PBA-20 bentonites, respectively.

7 Adsorption Characteristics of 90 Sr and 137 Cs from Aqueous Solutions on Bentonite Clays The ability of bentonite clays to adsorb radionuclides is one of the main characteristics to be considered in terms of safe radioactive waste disposal. The adsorption mechanism can be explained by the specific structure and surface inhomogeneity of

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montmorillonite. It is characterized by presence of adsorption centres of different nature—exchange cations, oxygen atoms and hydroxyl groups on the basal planes of clay particles, coordinatively unsaturated ions Mg2+ , Al3+ , Al3+ and hydroxyl groups on the crystal faces and edges. The mechanism of adsorption is explained by the formation of hydrogen bonds between radionuclides and active positively charged complexes of montmorillonite particles, while the selectivity and efficiency in bentonite adsorption is explained by the presence of micro-, meso- and macropores. For bentonite clays, the ion exchange in the interlayer space is the dominant mechanism of 137 Cs and 90 Sr absorption. It accounts for about 80–85% of the total sorption [13]. According to the international classification, the sorption centres located in the interlayer space and the expanded edge regions are marked as FES (Frayed Edge Sites) (Fig. 5) [14]. Based on x-ray diffraction analysis data [15], the mechanism

Fig. 5 Schematic representation of Cs sorption in the interlayer space (a) and on the bentonite surface (b)

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of selective sorption of 137 Cs that can relatively easily lose the hydrate shell is the ion exchange of Cs+ for monovalent cations on frayed edge sites (FES) followed by formation of fixed forms. Access to these centres of large hydrated cations such as Ca2+ , Mg2+ , Sr2+ is sterically limited. It was found that FES are highly selective for Cs+ compared toother single-charge cations. At the same time, nonselective adsorption centres (15–20%) on the basal surface RES (Regular Exchange Sites) play significant role concerning Cs+ ions as a result of the formation of Cs–O bonds in montmorillonite tetrahedral sheets, which lose the desorption capacity even at high concentrations of caesium in the solution (Fig. 5) [16]. The ions with high hydration energy (Ca2+ , Mg2+ , Sr2+ ) are characterised by sorption on RES with the formation of ion exchange forms. PBA-20 bentonite has the highest ion exchange capacity that is directly proportional to montmorillonite intracrystalline swelling ability in the presence of aqueous solutions. That of the natural bentonite is lower. The uptake of 137 Cs and 90 Sr at pH = 3–10 is virtually unchanged or slightly increases by 2–3% with increase in pH, which indirectly indicates the ion exchange mechanism of sorption of these radionuclides in the interlayer space. The sorption of 137 Cs and 90 Sr depends on the presence of macroconcentrations of competing cations in an aqueous solution, and on the ionic strength at all pH values. The sequence of competing cations that effect the sorption of Cs+ and Sr2+ on the soil-absorbing complex is as follows: Al3+ > Fe3+ > Ba2+ > Ca2+ > Mg2+ > K+ > NH4+ > Na+ [17]. The adsorption capacity of bentonite clays towards Cs and Sr is determined not only by the amount of hydrated exchange cations in the voids of the crystal lattice, their size and ratio, and selectivity of exchange centres, but also by macro- and microstructure features [18]. The selectivity of the exchange centres of the aluminosilicate framework (exchange ions Na+ , Ca2+ , Mg2+ ) is determined by steric factors: the size of the hydrated of caesium and strontium cations (Table 4) and the diameter of macro- and micropores. When comparing the sorption/desorption results, the self-dispersing ability of alkaline earth PBA-20 bentonite (decrease in particle size and a significant increase in number of particles per unit volume) should be taken into account. The selfdispersing leads to an increase in the specific surface area and selectivity of the exchange centres. To estimate the ability of bentonite clays to adsorb radionuclides, the physicochemical forms of radionuclide fixation should be determined. They provide important information on the radionuclide adsorption efficiency. They also should be taken into account when assessing the radioactive waste storage safety since the insoluble form remains unchanged and does not participate in the migration processes and the Table 4 Radiuses of hydrated ions of alkali and alkaline earth metals [19] Ion

Na+

K+

Mg2+

Ca2+

Cs+

Sr2+

Radius, nm

0.33

0.27

0.44

0.42

0.23

0.42

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geological environment contamination area is directly proportional to the share of the exchange form. To study the physico-chemical forms of 90 Sr and 137 Cs fixation, we used the sequential leaching procedure. In each filtrate, the degree of desorption of the studied radionuclides was determined by measuring their residual concentration (activity). The intensity of desorption depends on the degree of crystallinity, the disorder in the layers and other defects in the montmorillonite structure. Thus, poor crystallinity and structural defects lead not only to significant desorption of the adsorbed elements (radionuclides) by washing with water, but also to possible destruction of its crystalline structure when saturated with adsorbates. The kinetic adsorption isotherms for the samples of the natural and modified alkaline earth bentonite clays are of complex shapes. The time taken to reach equilibrium and the redistribution of the sorption forms are different for 137 Cs and 90 Sr. For 90 Sr the equilibrium is established in 5–6 days, while for 137 Cs much faster—in about 2 days. The degree of 90 Sr adsorption on modified alkaline earth bentonite clay in aqueous solutions exceeds 85% under equilibrium conditions. For natural samples, this figure is lower by 5–10%. The sorption capacity of the modified form of bentonite clay in relation to 137 Cs is about 75–80% of the initial activity. For the natural bentonite samples, it is 5–10% lower. Desorption of the adsorbed Cs from the natural bentonite in water is in the range of 20–30%, and that of Sr–4–5%. Stronger fixation of Sr and Cs is observed in the Dashukivka modified bentonite (desorption of Sr and Cs is 2.5– 3 and 15%, respectively). This may be explained by the higher dispersion of this clay and the enrichment of the montmorillonite exchange complex with Na ions. With the increase in time of phase interaction, the redistribution of mobile (exchangeable) and immobile (non-exchangeable) forms of radionuclides occurs, and the proportion of the latter increases. This is true predominantly for Cs. In the alkaline earth bentonite, there is a partial substitution of Ca and Mg cations with Na in the interlayer space of natural bentonite, and, as a consequence, a decrease in the particle size and increase in the specific surface area, which leads to increased sorption of radionuclides. As mentioned above, the degree of Sr and Cs sorption is influenced by the competing cations present in a solution and the ionic strength of the solution (mineralization). The increase of the latter causes the decrease in the degree of sorption. For the Chornobyl Exclusion Zone (ChEZ), the groundwater salinity (dry residue) is 0.136 g/dm3 . Therefore, it is quite reasonable to expect an increase in the degree of radionuclide sorption on the Dashukivka bentonites in case of their use in the I and II stages of a near-surface storage facilities at the Industrial Complex “Vector”. To confirm or refute this expectation, the degree of 137 Cs adsorption from model ChEZ groundwater solutions on the Dashukivka natural and modified (PBA-20) bentonite clays (II layer) has been studied in compliance with the acceptance criteria for radioactive waste at the IC “Vector”. In addition to the stable Cs isotopes, a radioactive tracer of 137 Cs isotope was added to the model solution. The activity of the 137 Cs radionuclide was 5,84 × 104 Bq/dm3 (the relative error is 2–3%). The total mineralization of the solution was 0.200 g/dm3 . The solution/bentonite clay ratio

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was 100:1. The dependence of the sorption degree on the initial solution pH values (6.5, 9.0, 11.0) was investigated assuming the increase of the pH values during the dissolution of the cement-concrete barriers in a storage facility. The equilibrium in the sorbent-solution system is established in 12–14 h, the degree of sorption on the modified PBA-20 bentonite is 91–93%, and that on the natural bentonite is 7–10% lower. Increase in the solution pH leads to a slight increase in the degree of sorption (for several %) (Table 5). Increase in the concentration of Ca2+ ions in the model solution by 10 times (from 16 to 160 mg/dm3 ) has virtually no effect in this range on the degree of 137 Cs sorption both on the natural and the soda-modified bentonite (Fig. 6) while 2 times Na+ concentration increase decreases this value by 6–7%, which can be explained by chemical analogy of caesium and competing cations (potassium and sodium). The advantages of the natural Dashukivka bentonite application for engineered barriers construction are determined first of all by the stability of their properties over time and the possibility to predict potential changes in the impervious and anti-migration properties of the barriers with time. [20]. Based on the results of thermodynamic modelling [21], the influence of the bentonite clay properties on the radioactive waste storage barrier stability was evaluated. It was found that the mineral composition and clay properties of the bentonite or sand- bentonite barriers will not change significantly for at least 500 years. New minerals of montmorillonite group will be formed in bentonite, which will increase the sorption properties of the barriers in the near-surface radioactive waste storage facilities. Thus, the experimental studies of the Dashukivka bentonite clay sorption properties (layer II) show that its use as the main component of the underlying screens of radioactive waste storage facilities will significantly reduce or eliminate the migration of the main dose-forming radionuclides to the aeration zone even if the precipitation water comes into the repository. Long-lived radionuclides of Chornobyl origin (plutonium, americium, europium) will also be securely bound by the underlying screen, since their migration behaviour is similar to caesium.

8 Conclusions The research data and analysis of the obtained results showed that: 1.

2.

3.

Bentonite clays of the Cherkaske bentonite and palygorskite clay deposit (Dashukivka site, II layer) have good waterproofing and barrier properties, including significant sorption capacity for 90 Sr and 137 Cs. Alkaline earth bentonite (soda-modified bentonite) absorbs 90 Sr and 137 Cs more efficiently than the natural. However, 90 Sr is adsorbed on both bentonites in greater quantities than 137 Cs. This indicates to the predominant effect of the charge on the binding efficiency, rather than the cation size. With the increase in time of interaction of the both types of bentonite with the aqueous solution, the redistribution of mobile (exchangeable) and immobile

9.0

11.0

2

3

6.5

9.0

11.0

1

2

3

Soda-modified bentonite

6.5

1

Natural bentonite

10.05

8.95

8.9

8.9

7.7

7.5

93.37

92.12

91.30

86.16

83.77

82.95

Degree of 137 Cs sorption, %

3.27

4.31

5.07

5.11

6.34

7.89

Desorption with distilled water

pH of resulting solution

№ of a sample

pH of initial solution

Water-soluble form

Sorption

59.94

55.98

51.17

63.73

61.77

60.25

Desorption with ammoniumacetate buffer

Ion-exchange form

30.16

31.83

35.06

17.32

15.66

14.81

Non- desorbed residue

Immobile form

6.23 ×

4.61 ×

1.408 × 103

1.170 ×

103

4.04 × 102

7.11 ×

9.04 × 102

4.55 × 102

4.08 × 102 102

5.88 × 102

1.25 × 102

0.96 × 102 102

3.80 × 102 1.050 × 103

5.16 ×

0.87 × 102 102

Kdfix ., ml/g

3.53 × 102

Kdion ., ml/g

102

4.86 × 102

Kdgen ., ml/g

Table 5 Results of 137 Cs sorption/desorption on the Dashukivka natural and soda-modified bentonite from model ChEZ groundwater at different pH

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Mineralogical-Geochemical Properties of Bentonite Clays …

Fig. 6 24 h)

4.

219

137 Cs sorption degree dependence on Ca2+ a and Na+ b ions concentration (pH 6,5, exposure

(non-exchangeable) forms of radionuclides occurs, and the proportion of the latter, which does not participate in migration processes, increases. The complex analysis of the Dashukivka bentonite clays, significant reserves and the possibility of improving their technical and economic characteristics allow to recommend the Dashukivka bentonite clays as an additional anti-migration engineered barrier for the construction of the I and II stages of the near-surface radioactive waste storage facility at Industrial Complex “Vector”.

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References 1. Sellin, P., Leupin, O.X.: The use of clay as an engineered barrier in radioactive-waste management—a review. Clays Clay Miner. 61(6), 477–498 (2014) 2. Shabalin, B.H.: Perspectives for use of raw bentonite and bentonite-based materials in the nuclear industry in Ukraine to improve environmental safety of radioactive waste management. Ecol. Saf. Environ. Prot. Technol. 1, 60–69. (in Ukrainian) (2019) 3. Mineral resources of Ukraine. Kyiv: state research and production enterprise “State information geological fund of Ukraine”, p. 270 (in Ukrainian) (2018) 4. Ovcharenko, F.D., Kyrychenko, N.G., Ostrovska, A.B., Dovgyi, M.G.: Cherkasy deposit of bentonite and palygorskite clays, p. 124. Naukova Dumka, Kyiv (in Russian) (1996) 5. Yaroshenko, K.K., Shabalin, B.H., Koliabina, I.L., Bondarenko, H.M.: Kinetics of 90Sr and 137Cs sorption by bentonite clays of Cherkasy and Horbky deposits from model solutions of liquid radioactive waste. Collection of scientific articles of the XV International scientific-practical conference “Environmental safety: problems and solutions” (Kharkiv, 9–13 September 2019). Kharkiv, UKRNDIEP, pp. 309–314. (in Ukrainian) (2019) 6. Grim, P.: Mineralogy and Practical use of Clays, p. 512. Mir, Moskov (in Ukrainian) (1957) 7. Shabalin, B.H., Lavrynenko, O.M., Kosorukov, P.O., Buhera, S.P.: Prospects for the use of natural smectite clays in Ukraine to create a geological repository of radioactive waste. Mineral. J. 4(40), 65–78. (in Ukrainian) (2018) 8. Wilson, M.J.: Rock-forming minerals, vol. 3C, Sheet silicates: clay minerals. The Geological Society, p. 736. London (2013) 9. Drits, V.A., Kossovskaya, A.G.: Clay minerals: smectites, mixed-layer formations. Nauka, Moskov, p. 214 (in Russian) (1990) 10. Wilson M.J. (eds).: Handbook of Determinative Methods in Clay Mineralogy, p. 308. BlackieSon Ltd, London (1987) 11. PDF-2. Powder diffraction file. Database. (2003) 12. The Clay Minerals Society Glossary for Clay Science Project, p. 88. (2003) 13. Honcharuk, V., Pshynko, H.: The role of chemical forms of radionuclides in predicting their behaviour in the environment. Herald Natl. Acad. Sci. Ukraine, 10, 3–17. (in Ukrainian) (2011) 14. Essington, M.E.: Soil and Water Chemistry, p. 534. CRC Press, New York (2004) 15. Osipov, V.I., Sokolov, V.N.: Clays and their Structural Forms. Composition, structure and formation of properties, p. 576. GEOS, Moscow (2013) 16. Krupskaya, V.V., Zakusin, S.V., Tyupina, E.A., Chernov, M.S.: Peculiarities of caesium sorption in bentonite barrier systems during disposal of solid radioactive waste. Mining J. 2, 81–87. (in Russian) (2016) 17. Sanzharova, N.I., Sysoeva, A.A., Isamov, N.N., Aleksakhin, R.M., Kuznetsov, V.K., Zhigaeva, T.L.: The role of chemistry in the rehabilitation of agricultural land exposed to radioactive contamination. Russ. Chem. J. 49(3), 26–34. (in Russian) (2005) 18. Tarasevych, Yu.I.: Structure and Surface Chemistry of Layered Silicates, p. 248. Naukova dumka, Kyiv (in Russian) (1988) 19. Erdei-Gruz, T.: Transport phenomena in aqueous solutions, p. 592. Mir, Moskov (in Russian) (1976) 20. Savchenko, V.A.: Development and use of technical soils for near-surface radioactive waste disposal facility. Nucl. Technol. Abroad (7), 3–14. 21. Sabodina, M.N.: Regularities of radionuclide behaviour when creating a technogenic and geochemical clay-based barrier. (Ph.D. Thesis), p. 27. Moscow, (in Russian)

Metal–carbon Nanocomposite for Purification of Natural and Technogenicly Polluted Water from Oil Pollutants Yurii Zabulonov , Vadim Kadoshnikov , Tetyana Melnychenko , Valeriia Kovach , and Liudmyla Sydorchuk Abstract The chapter considers method for obtaining of complex magnetically sensitive nanosorbent with low volatility and positive buoyancy based on thermally expanded graphite and metal–carbon nanosorbent designed to remove non-polar organic liquid including petroleum products from the water surface. The developed sorbent can be used for purification of technogenicly polluted waters from emulsified and dissolved oil products. Combination of thermally expanded graphite and metal–carbon nanosorbent is based on connection of graphite particles with carbon shell of the metal–carbon nanosorbent. It was found that magnetically sensitive core of the metal–carbon nanosorbent with a size of 1.4–1.5 µ and covered with a carbon shell consists of metallic iron alloyed with manganese, cobalt, nickel and iron oxide. Formation of magnetically sensitive oil-containing conglomerates as result of magnetic field influence allows use of magnetic separation to remove them from water medium. The developed sorbent allows to removeup to 95% of oil or oil products from a water surfac, and technogenicly polluted waters. Keywords Thermally expanded graphite · Complex magnetically sensitive nanosorbent · Oil and oil products · Magnetic field · Water area · Technogenicly polluted waters

1 Introduction Pollution of water with oil and oil products is one of the important problems today. Oil production and transportation is often accompanied by spillage which occurs in both regular and emergency situations. Numerous oil storage and transportation companies, agricultural and trucking companies, as well as companies in other industries Y. Zabulonov · V. Kadoshnikov · T. Melnychenko · V. Kovach (B) State Institution “The Institute of Environmental Geochemistry of National Academy of Sciences of Ukraine”, Kyiv, Ukraine V. Kovach · L. Sydorchuk National Aviation University, Kyiv, Ukraine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Zaporozhets (ed.), Systems, Decision and Control in Energy III, Studies in Systems, Decision and Control 399, https://doi.org/10.1007/978-3-030-87675-3_13

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that use oil and oil products are responsible for oil spills. Such companies usually have primitive treatment facilities. Lack of proper treatment system leads to discharge of oil-contaminated sewage into the environment deteriorating sanitary and hygienic condition of soil, air and water bodies. Leakages of oil products related to operation of thermal and nuclear power plants are significant environmental pollution. Oil pollution differs from other anthropogenic impacts. It causes not constant but “volley” load on environment. It is not always possible to make a conclusion about possibility of ecosystem returning to equilibrium [1] assessing consequences of such pollution. There are various ways to remove oil products from the surface of natural reservoirs (mechanical, thermal, biological, oil curing or submersion, sorption). It should be taken in mind that oil removal from water surface mechanically is possible only in case of thick, structured films (tar, fuel oil etc.). This method is unacceptable for removal of thin films. So, sorption method becomes of very important [1, 2]. It is possible to use sorption method for wastewater treatment from oil products, including emulsified and dissolved. Sorbents of various natures—synthetic and natural porous materials—ash, coke, peat, silica gels, aluminogels, natural and modified clays, foam polymers, foam glass and various industrial and agricultural productions became widely used for purification of marine, oceanic and freshwater waters [1–4]. Such characteristics as high hydrophobicity, buoyancy and minimum moisture content are important for oil sorbents. It contributes to effective removal of spent sorbent from the surface of reservoirs. The most important requirements for petroleum sorbents are low toxicity and absence of negative impact on environment. The sorbent “Dulromabsorb” has high sorption properties among plant sorbents. It is a fibrous part of fruits of tropical plant sumuma (seiba, or cotton tree). These trees are widespread in the Republic of Mozambique. Sorbent “Dulromabsorb” has high buoyancy and high oil absorption. High hydrophobicity of the sorbent and low moisture absorption are observed due to presence of thin oil film on the fibers surface [2]. Significant disadvantage of this sorbent is certain technical difficulties that arise during its spraying on polluted surface and subsequent removal. Peat can be used as cheap and accessible oil sorbent. Its sorption activity can be increased after heat treatment (t = 100 – 120 °C) [5]. The method of water surface purification from oil pollution involves contacting water with the peat sorbent until oil contamination is absorbed by the sorbent and then removed from the water surface. Comparative characteristics of petroleum sorbents based on vegetable wastes are shown in Table 1 [2]. Imperfect sorption properties of these sorbents can be seen in the Table 1. However, there are certain economic advantages of using vegetable oil sorbents. The best sorption properties are typical for foamed polymeric materials [2]. They can be used in emergency conditions due to high buoyancy. Significant disadvantage of these oil sorbents is environment pollution with toxic substances formed as a result of polymers biodegradation.

Metal–carbon Nanocomposite for Purification of Natural … Table 1 Sorption characteristics of petroleum sorbents based on vegetable wastes

223

Sorbent

Oil sorption, g/g

Water sorption, g/g

Degree of oil extraction, %

Wheat straw

4.1

4.3

36

Reed

8.2–2.66

4.68

18–30

Wood sawdust

1.72

4.31

10–20

Buckwheat husk

3.05–3.5

2.2

44

Peat

17.71

24.28

74

Mineral oil sorbents do not have this lack. Such sorbents use hydrophobized silica gels, aluminum–silicon porous sorbents of natural origin (zeolite), highly dispersed layered aluminosilicates and synthesized porous materials (foam glass). Application of highly layered aluminosilicates to remove oil contaminants from water is due to the following: absorption of nonpolar hydrocarbons by clays in presence of water is mainly in pores and capillaries located between the basal faces of adjacent crystallites. Water fills micropores in their clay aggregates. coagulation. Amount of absorbed hydrocarbons is determined by the pore volume in microaggregates and depends on the area of crystallites basal faces of layered silicates [6]. Significant disadvantage of clay minerals and zeolites is their high density. It leads to deposition of spent sorbent on the reservoir bottom and to violation of bottom biochemical processes. Foamed aluminosilicates (perlite, expanded clay, foam glass) have high buoyancy with have the necessary hydrophobic-hydrophilic properties. It allows absorbing petroleum products from the aquatic environment and retaining them for a long time. Biomineral sorbents have some interest—biodestructors of petroleum products, immobilized on a porous sorbent, in particular, on foam glass [1]. Separation process of the spent sorbent and cleaned liquid causes certain difficulties. It is possible to facilitate removal of spent sorbent through use of complex magnetically sensitive sorbents with subsequent phase separation by magnetic separation. Use of natural thermoactivated coal is known for oil spills purification. Coal with high sorption capacity has following charactistics: average pore size of 2.2 nm, an average pore volume of 0.14 cm3 /g and a specific surface area of 1336.96 m3 /g. It can be obtained by method of the carbonization of crushed shell walnut in muffle furnace with access to air at a temperature of 700–800 °C for 2 h [7]. High cost of technogenic coal and high density of natural coal (over 1 g/cm2 ) limits possibilities of their use in technologies for purification water areas surface and man-made polluted waters despite the high affinity of coal for non-polar hydrocarbons. Use of coal complicates application of magnetically sensitive sorbents, because hydrophobic surface of coal is incompatible with surface of magnetic media. Properties variability of magnetic nanocomposites based on activated carbon are

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associated with their heterogeneity. It requires finding optimal ways to synthesize nanocomposites by varying conditions, namely ratio of components, temperature, duration of synthesis, pH, etc. However, complex method of introducing magnetite into the activated carbon matrix was developed. Magnetic nanocomposites were synthesized using two different methods: chemical coprecipitation by applying presynthesized Fe3 O4 nanoparticles to the surface of activated carbon and by forming Fe3 O4 nanoparticles in activated carbon medium. Optimal properties were achieved for sorbent obtained by chemical coprecipitation of magnetite nanoparticles on the surface of activated carbon [8, 9]. It is recommended to use thermally expanded graphite (TEG) to increase efficiency of treatment of water and soil [10]. TEG is obtained by thermal shock of carbon raw materials. Product is formed consisting of worm-like graphite particles after heat shock. It so-called “black snow” [11], which is characterized by following parameters: high oil absorption capacity and low water absorption, buoyancy, ability to regenerate sorbent and to return oil to production cycle, low cost and availability [12]. High efficiency of TEG in absorption of petroleum products caused by developed specific surface area 2000 m2 /g and high activity of nanostructured carbon complexes. TEG buoyancy is associated with high hydrophobicity of surface and structure (air contained in TRG pores is not displaced by water) and persists for a long time. Thermoexpanded graphite effectively retains absorbed oil, holding it in pores of the structure. Significant disadvantage of TEG is low bulk density and high sailability. It creates significant difficulties in spraying it on the water surface which is associated with its removal and, consequently, air pollution [12]. High hydrophobicity of TEG provides high affinity for non-polar liquids (oils, oil products) and allows effectively absorption of oil products from water, at the same time creates significant difficulties for combining TEG with magnetically sensitive components. This problem can be solved by obtaining of sorbent based on thermally expanded graphite modified with magnetic phase, represented by metal ferrites with improved performance due to uniform distribution of magnetic phase on the sorbent surface increasing magnetic phase content in TEG and increasing its magnetic properties. The method of obtaining sorbent based on thermally expanded graphite involves impregnation of graphite particles with aqueous solution of salts of ferrous iron, separation of solid and liquid phases followed by drying of solid phase to bulk and heat treatment at 500–1200 °C [13]. The aim of research. Given both the positive and negative properties of thermally expanded graphite, as well as magnetosensitive sorbents, the aim of our work was to develop a method of obtaining a complex magnetosensitive nanosorbent with low volatility and positive buoyancy, designed to remove petroleum products and other nonpolar organic liquids.

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225

2 Materials and Methods Thermally expanded graphite (TEG), metal–carbon nanosorbent (manufactured in Zaporizhiia), crude oil of Nadvirna refinery were used in experimental studies. Following imitators were prepared to study properties of developed sorbent: • ocean waters with sea salt content (DSTU 3853) −35 g/dm 35; • technogenic waters containing 50–60 mg/dm3 of petroleum products, cations of alkali, alkaline earth, transition metals and chlorides, nitrates, borates and phosphates (total mineralization of water does not exceed 5 g/dm3 ); • nuclear water drains with content of cesium and strontium cations 10 mg/dm3 , cobalt 5 mg/dm3 , copper, manganese and iron 2.5 mg/dm3 , amount of organic matter (including complexing agents and surfactants) is 0.64 g/dm3 , inorganic substances −2.89 g/dm3 . Studies of metal–carbon nanosorbent were performed using thermography and scanning electron microscopy (SEM) using a scanning electron microscope JEOL JSM-6490LV (JEOL Ltd., Japan). Plasma-chemical installation was used [14] to increase activity of metal–carbon nanosorbent. High-voltage discharge (10–70 kV) is occurred during the operation. It generates a shock wave in water, causing electrohydraulic effect (electrohydraulic shock), characterized by power 104–105 kg/cm and elastic oscillations of medium (~100 Hz). Indicator of chemical oxygen demand (COD) was determined by dichromate oxidation in accordance with DSTU GOST 31,859: 2018 Water. Determination of chemical oxygen demand (GOST 31,859–2012, IDT; ISO 15705: 2002, NEQ) [15]. The standard corresponds to the international standard ISO 15705: 2002 Water quality—Determination of chemical oxygen demand index (ST-COD)—Small-scale sealed-tube method.

3 The Results and Discussion Application of metal–carbon materials is a promising area for the purification of various types of polluted water. It can be used as an individual sorbent and in combination with other carbon-based sorbents. The main properties of such metal–carbon nanocomposites depend on the chemical composition, structure of the nanocomposite, size and shape of particles, presence and number of electrons in the metal-containing phase and the carbon shell [16]. Metal core interacts with the carbon component in metal–carbon nanocomposite. Metal-containing clusters do not form valence bonds with carbon. Charges are redistributed in the nanostructure and dipole moment occurs due to association of metallic and carbon components. Their association affects electronic and vibrational properties of particles and system as a whole. Symmetry of carbon structure is broken in

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the presence of a metal atom. Behavior of metal–carbon nanocomposites in media and their effect on composite materials is probably due to the effects of shortrange (effects of coordination and electrostatic processes) and long-range (chain mechanism of metal–carbon nanocomposite) [16]. Study of physicochemical properties of metal–carbon nanosorbent (manufactured in Zaporizhia) showed presence of magnetically sensitive core consisting of alloy of metals and metal oxides coated with carbon shell. Carbon content in the initial sample of nanosorbent is 70–75%, iron 9–20%, iron oxides 5 - 21% according to the results of thermographic studies. Results of experiment of metal–carbon nanosorbent by scanning electron microscopy are presented in Fig. 1. It was found that magnetically sensitive core of nanosorbent consists of metallic iron doped with manganese, cobalt, nickel (less than 2% in total) and iron oxide. Analysis of SEM data suggests that size of individual particles (non-aggregated) is 1.5 µm for metal phase and 1.4 µm for oxide form. Sorption properties of the metal–carbon nanosorbent can be enhanced due to its activation by means of electrohydraulic discharge in water medium [14, 17]. Studies showed that after electrohydraulic discharge there is phenomenon of aggregation due to the fusion of individual nanoparticles as a result of electrohydraulic pulse: for metal phase, the particle size increases to 3.3 µm, and for the oxidized form—up to 2.6 µm. However, there is partial destruction of carbon shell and release of carbon in separate phase (Fig. 2).

Fig. 1 Electron microscopic images of original sample surface of metal–carbon nanosorbent (a – d)

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227

Fig. 2 Electron microscopic images of sample surface of metal–carbon nanosorbent after electrohydro discharge treatment (a–b)

Plasma chemical treatment of water dispersion of the sorbent is accompanied by change of the metal phase in chemical composition; by decrease in the metallic iron amount and by increase in oxides amount. Passage of an electric pulse through layer of water is accompanied by electrohydraulic discharge and local high temperature. It leads to redox process resulting in formation of free radicals OH·, H· and oxygen and ozone molecules which leads to partial oxidation of carbon. This phenomenon inactivates surface of the sorbent. In addition, there is a destruction of carbon shell, its separation from the metal core and fusion of individual microparticles of metal into aggregates. From the above it follows that electro-discharge treatment leads to inactivation of surface and to deterioration of the sorption properties of the sorbent. We investigated possibility of combining of properties of TRG and metal–carbon nanosorbent taking into account that thermally expanded graphite has high absorption capacity for oil products. We proposed complex magnetically sensitive nanosorbent [18], consisting of TRG particles. On its surface nanoparticles of metal–carbon sorbent are firmly held due to intermolecular forces. We experimentally established the optimal ratio of TRG–metal–carbon nanosorbent, due to the fact that the metal– carbon nanosorbent contains a metal core and has a negative buoyancy, and the scales of the TRG have a high buoyancy. The ratio is from (2: 1) to (10: 1) that allows to provide also low sailability. Feature of formed composite is its stable buoyancy. It allows nanoparticles to remain on the surface and actively absorb non-polar organic liquids, including oil products. Magnetically sensitive nuclei of the composite are magnetized under the action of external magnetic field. They interact with each other, form magnetically sensitive conglomerates of sorbent saturated with non-polar liquids. It can be easily removed by magnetic separation. Following laboratory studies were performed to assess the ability of the developed sorbent to absorb non-polar liquids from the water surface and from the wateremulsion mixture: 1.

Into the tank with fresh water (tap water was used) was added oil, which is evenly distributed on the surface (Fig. 3), forming a film 140 µm thick, on the surface of which was evenly sprayed 0.8 g of complex magnetically sensitive nanosorbent (Fig. 4), after 30 min the surface dispersion formed was subjected

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Fig. 3 Application of oil film on the water surface during the laboratory experiment (a–water before pollution, b–after pollution)

Fig. 4 Complex magnetically sensitive nanosorbent applied to the surface of the oil film

2.

to a magnetic field, as a result of which magnetically sensitive conglomerates saturated with oil were formed on the surface of the water (Fig. 5), which were removed from the water surface by the magnetic field (Fig. 6). The proposed method allows you to remove up to 95% of oil from the water surface (Fig. 7). 10 cm3 of oil was poured in a tank on the surface of fresh water (tap water) with an area of ≈700 cm2 . It is evenly distributed on the surface, forming a film 140 µm thick. Then it was evenly sprayed 0.8 g of complex magnetosensitive nanosorbent on the surface. After 30 min the resulting dispersion was subjected

Metal–carbon Nanocomposite for Purification of Natural …

229

Fig. 5 The formation under the influence of a magnetic field of magnetically sensitive sorbent conglomerates saturated with oil

3.

4.

to a magnetic field. Then magnetically sensitive oil-saturated conglomerates were formed on the water surface. Then they were removed from the water surface by magnetic separation. The proposed method allows removing up to 95% of oil from the water surface. 15 cm3 of oil was poured in a container on the surface of water solution containing 35 g/dm3 of sea salt (surface area ≈700 cm2 ). It was evenly distributed on the surface forming a layer 210 µm thick. Then it was evenly sprayed 0.9 g complex magnetically sensitive nanosorbent on the surface. After 30 min formed surface dispersion was exposed to a magnetic field. Then saltresistant magnetically sensitive conglomerates saturated with oil on the water surface were formed. They were removed from the water surface by magnetic separation. The proposed method allows removing up to 95% of oil from the surface of solution that mimics ocean water. It was poured 1 dm3 technogenically contaminated water containing 50– 60 mg/dm3 petroleum products, cations of alkali, alkaline earth, transition metals and chlorides, nitrates, borates and phosphates (total water salinity did not exceed 5 g/dm3 ) in a container equipped with mixer. Then 0.2 g of complex magnetically sensitive nanosorbent was added into the container with constant

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Fig. 6 Removal by magnetic field of oil-saturated magnetically sensitive sorbent conglomerates

mixing (for 30 min.). Mixing rate was chosen to ensure uniform distribution of the sorbent in the liquid volume. Formed dispersion was subjected to magnetic field within for 1–2 min to form magnetically sensitive oil-containing conglomerates. Then the dispersion was separated on a magnetic separator (magnetic field strength 12 kA/m). The proposed method allows extraction up to 95% of oil from technogenically polluted water. Our proposed complex magnetically sensitive nanosorbent can be used to purify oil contaminated water from NPP or other non-polar organic liquids. Purification of imitation sewage NPP water containing oil products was performed according to the above given method for purification of technogenically polluted water. The purification efficiency of simulator was evaluated by indicator of chemical oxygen demand (COD). It was found that the COD (mgO2 /dm3 ) decreased from 1300 to 160 after purification.

4 Conclusions 1.

The method for obtaining of complex magnetically sensitive nanosorbent based on thermally expanded graphite and metal–carbon nanosorbent was developed. It allows removing of non-polar organic liquids, including oil and oil products, from the water surface and from technogenically polluted waters.

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Fig. 7 Water after removal of oil pollution (pollution remained on the magnet)

2.

3. 4.

5.

Use of combination of thermally expanded graphite with metal–carbon nanosorbent is based on high affinity of graphite particles to carbon shell of the metal–carbon nanosorbent. Low sailability, combined with positive buoyancy of the developed sorbent allows sorbent application for the purification of contaminated water. Magnetic field treatment of the developed nanosorbent deposited on the surface of oil film on water surface is accompanied by magnetically sensitive conglomerates formation. They can be removed by magnetic separation. Application of the developed nanosorbent for removal of non-polar organic liquids, including oil products, from technogenic polluted waters allows removing of oil products by magnetic separation from water volume rather effectively.

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17. Zabulonov, Y., Popov, O., Burtniak, V., Iatsyshyn, A., Kovach, V., Iatsyshyn, A.: Innovative developments to solve major aspects of environmental and radiation safety of Ukraine. In: Zaporozhets, A., Artemchuk, V. (eds) Systems, decision and control in energy II. Studies in systems, decision and control, vol. 346. pp. 273–292 (2021). https://doi.org/10.1007/978-3030-69189-9_16 18. Zabulonov, Y.L., Kadoshnikov, V.M., Melnychenko, T.I., Pugach, O.V., Lytvynenko, Y.V., Shkapenko, V.V., Odukalets, L.A.: Method removal of non-polar organic liquids from the surface of natural reservoirs and from man-made polluted waters using a complex magnetically sensitive nanosorbent. (2021). https://sis.ukrpatent.org/uk/search/detail/1472524/

Cybersecurity and Computer Science

Method of Improving the Security of 5G Network Architecture Concept for Energy and Other Sectors of the Critical Infrastructure Maksim Iavich , Giorgi Akhalaia , and Sergiy Gnatyuk

Abstract The technology develops very rapidly. Artificial Intelligence is used in the different industries; therefore it has exceeded the limits of being just set of tools for automatization, every day workflow. Technologies are involved in every industry started from IoT to critical infrastructure (for example, energy sector) and processes. Over the last decades remote services and cloud-based technologies have become very important. Because of increased number of connected devices, existing mobile network standards are not able to totally provide reliable services. New critical services have generated the need where latency is more crucial than bandwidth. Hence, engineers have started working on new standard—5th Generation Network. By satisfying 3 key requirements from 3GPP, 5G network will start new era of wireless communications. Which will arise new opportunities for human being. In the other hand, it is obvious that, new technologies, functionalities always form extra threats and attack vectors. As virtualization is core component of 5th generation network, new standard will be more vulnerable to software-based attacks. Even 5G has better encryption and some security improvement, there are still major threats, that should be solved before launching 5G in mass usage. Existing architecture of 5G network has a gap which lets attacker to make a MITM attack. Using MITM testers have performed Battery Draining, MNmap, Downgrading, Sensitive Data Sniffing Attacks and etc. Solution, described in this work represents the concept how should be secure architecture of 5G network. Idea is theoretical and concentrated on the UE attach process with Base Stations. The solution mitigates the MITM attacks. Also the disadvantages of the system are analyzed.

M. Iavich Caucasus University, Tbilisi, Georgia G. Akhalaia Georgian Technical University, Tbilisi, Georgia S. Gnatyuk (B) National Aviation University, Kyiv, Ukraine e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Zaporozhets (ed.), Systems, Decision and Control in Energy III, Studies in Systems, Decision and Control 399, https://doi.org/10.1007/978-3-030-87675-3_14

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Keywords 5G · Security · Secure architecture of 5G · MITM in 5G design · Fake base station · MNmap · Battery draining

1 Introduction Every new technology, services and functionality relates to human’s need. Innovation progress rate depends on urgency of the need and/or on the importance of existing problem solution. Combination of new technologies and artificial intelligence have extremely speed up arising and developing remote 24/7 services, self-managed, automatized processes. High tech robots can parallelize process and work on the same project with other ones. Existing standard cannot reliably handle synchronized processes in real time. Hence, there was a need of improved communication standard that would exceed existing limitations. Modern energy technologies (like other critical infrastructures—see Fig. 1) are becoming progressively more connected to modern digital technologies and ICT networks [1]. This increasing digitalization makes the energy system smarter and enables consumers to better benefit from innovative energy services. Digitalization creates significant risks as an increased exposure to cyberattacks and cyberincidents potentially jeopardizes the security of energy supply and the privacy of consumer data. Therefore, communities have started working on 5G Network, 5th Generation of Telecom type communications. 3GPP has announced 3 key requirements for new standard [2]: 1.

EMBB—Enhanced Mobile Broadband—more than 10 Gb/s;

Fig. 1 Sectors of critical infrastructure

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Fig. 2 Core advantages and target group of 5G Network

2. 3.

URRLC—Ultra Reliable Low Latency—up to 1 ms; MMTC—Massive Machine Type Communication—minimum 1 Million Connections per km2 .

Core advantages and their target groups are shown on the Fig. 2. To achieve desired specification, new standard has to deploy some software and hardware upgrades. First change will be related to the operating spectrum ranges: 1.

2. 3.

Below of 1 GHz, also known as Low-Band. Frequencies from this range are less affected by buildings, so it is used in densely populated areas. However, bandwidth limitation of this band is about 100 Mbps. From 1 to 6 GHz, also known as Mid-Band. This category has more bandwidth (about 1 Gbps), but, also it is more affected by buildings, than Low-band. From 6 to 100 GHz, also known as High-Band or mmWave. This range will have the highest bandwidth and it is about several 10 Gbps.

Operating spectrum ranges are just part of changes made by new standard. Flexible Beamforming is a second key technique in 5G architecture, which promises more energy efficiency and high-quality coverage, service for near and far distances from cell towers. The idea is that, multi-power beams will be used to optimize upload and download rates. Workflow of the Flexible Beam Forming is illustrated on Fig. 3 [3].

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Fig. 3 Illustration of the flexible beam forming

Changes will be done with amount of antennas, which is called Massive Multiple Input Multiple Output—mMIMO. It means, that arrays of antennas will be attached to cell towers, so they will be able to simultaneously serve huge number of clients without interruption. World leading countries have already started deploying 5G Network. At this moment major part of these specifications is theoretical and is going to be developed. Only near the towers can be achieved higher quality of 5G network. Part of technologies used by 5G was imported from 4th Generation Network. Most of them was improved to achieve higher level of security but there are still some issues that should be fixed. It should be noted, that implementing virtualization as a key element, complexity of the 5G architecture has leveled up. Mixing virtualization and AL will save the physical resources, will enrich better power-consumption, better management of resources like network slicing, auto-optimization and etc. [4]. Hence, it makes 5G more software-based then hardware-based. Which means that all software attack vectors should be over checked and considered and marked as high importance.

2 How Does 5G Differ from Previous Generation Networks? Upgraded specifications let different industries to be involved in 5G ecosystem. Thus, will extremely accelerate creation of new products and services like self-operated robots, online critical services like remote surgeries, group VR technologies and so

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Function Analog Voice was delivered was introduced Digital Voice Mobile data has been deployed Mobile Broadband has been started Unified System; URLLC, EMBB, MMTC, Superior Reliability

Fig. 4 Generations of telecom communications with functions

on. Therefor 5G communication will be much more important in future development than telecom standards have ever been. As it is upgraded standard, there are some methodologies and techniques that were transferred from previous generation, 4th generation network. Some of them has vulnerabilities that have not been solved yet. Every new generation of telecom communication has its technical improvement. Development with functionalities are shown on Fig. 4. It should be mentioned that not only technological outcome will have 5G network. It will also positively effect on Global Economy. Qualcomm has mentioned that 13.1 trillion US dollars will be global financial gain and 22.8 m new job places will be created. If we overview articles, related to the 5G, available on the internet we will see that generally most of them is about 5G advantages over 4G Network. There are also lots of argues about security difference. If we summarize major upgrades of 5G, we will get following picture: with more than 100 Mbps average data rate and up to 20 Gbps peak data rate 5G will provide considerable faster internet speed than 4G. Significantly decreased latency empowers 5G network with compatibility of realtime synchronized services/processes which were unable during 4G. By new spectrum design it will natively support shared, unlicensed and licensed spectrum types. Operable spectrum is divided into Low-band, Mid-Band (also known as cmWave) and High-Band (also known as mmWave) [5, 6]. Benefits of 5G should be considered in so called “Helicopter View” and not only per parameter. It will make us look in global scale and catch the sense that 5G is unified, complex, multi-industry network that will bridge nano technologies, AL, humans’ abilities and will trigger new start. Security aspects of new standard are most problematic and arguable subject among cyber security specialists. It is obvious that new services, technologies, functions always arise new threats, attack vectors. However, some methods transferred from previous generation standard also imported existing vulnerabilities. Andy Purdy from Huawei Technologies (USA Office), in Forbes has mentioned that 5G has better encryption algorithm—256 bit, while 4G network used to 128-bit. After all, this does not guarantee total security. There are some cases when data is transferred without encryption. According to the paper new standard also encrypts UE’s location and identity before it will be sent to the BS [7]. Network Infrastructure Providers, Communication Service Providers and Virtual Mob. Network Operators represent major actors in 5G ecosystem. Because of their different privacy and security policies, their incorporation must be done extremely carefully to avoid future misconfiguration or incompatibility-based vulnerabilities.

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3 Security Aspects of 5G Network Smart gadgets like Internet of Things are target of 5th Generation Network. Engineers expect extended battery life for IoT working under 5G. However, there are declared issues related to battery draining. Samsung has officially announced that 5G enabled Samsung smartphones’ battery drain much more faster while operating under 5G than usually. CNET testers also have tested different mobile phones with and without 5G and got the same results—battery draining [8]. Generally, there are 2 reasons, that cause higher battery consumption. The first reason is network switch. That the case tester and smartphone manufacturer are talking about. According to the articles, at this moment 5G is used for data transmission only. So, devices have to operate on several network standards like 4G + 5G, 3G + 5G to uninterruptible hold data transmission and get phone calls as well as messages [9]. Solution is on the way and the problem will be solved after 5G will be capable to support phone calls and messages. Second reason is the result of MITM attack. Using Man in the middle attack, hacker can change device capabilities, disable power saving option and make device to drain battery quicker. MITM represents one of the most powerful and problematic Network Attack. This is a case when intruder stands between UE and network device and sniffs, alters and manipulates traffic, data transferred from UE to network device and vice versa. MITM acts as a medium, so it manages data flow between them. Hence it gives ability for additional attacks like session hijacking, packet/script injection, DNS type attacks and so on. Unfortunately, 5th generation network is not sustainable against MITM attack. Idea of MITM in a form of Fake BS is shown on Fig. 5. Researcher from Germany, Altaf Shaik in his scientific paper has described experimental work. Where he has used man in the middle attack in form of “Fake Base Station” for compromising security of 5G Network. Using MITM he has successfully done MNmap, Downgrading and Battery Draining Attacks [10]. Downgrading attack kicks user to downgrade his/her connection from 5th Generation network to 2G/3G/4G and compromise security using exploits. MNmap is process of identifying and mapping connected devices. It is like fingerprinting active hosts. This gives ability to better determine target group and make a segment-oriented attack. Which will have more probability to be successfully than in blind attack. Using rogue base station, which works as medium between UEBS conversation Altaf Shaik altered device capabilities and disabled Power Saving Mechanism. Which made device modem to continuously measure network signal and doing extra processes. That itself causes battery draining. Modification of attach request and battery draining attack is shown on Fig. 6. The main problem is in “Attach Request”, which is sent without encryption. That’s the attack design for downgrading attack too. So, there should be added protection mechanism to lessen possibility of catching attach request. Also, UE does not monitor, check attacking process. Hence, if someone is trying to sniff or alter the traffic UE will not consider it as malfunctioned activities and will continue attaching process.

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Fig. 5 MITM flow in a form of fake base station in 5G network

Fig. 6 Attach request modification and battery draining

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Generally, Subscriber Identity Module—SIM based mechanism is used for UE authenticating with BS. In some countries, SIMs are used for e-governance. For example: validating users identity on network or on service and sometimes for payment processes [8]. After switching from offline/flight mode and/or while making handover devices have to send International Mobile Subscriber—IMSI and International Mobile Equipment Identity—EMEI to BS. Both are identifying details: IMEI is unique and acts like a fingerprint of device (in level of SIM) and IMSI works for uniquely identifying users connected on the telecom network. As they are sent over the air, they can be sniffed. With a little effort, intruder can determine UE location and identity. Also, it is used to unlock stolen mobile devices. Therefore, it is urgent to avoid or minimize risk of MITM attacks, in this case probability of fake base stations, rogue base stations [11, 12].

4 Novel Architecture Concept Existing architecture, design is vulnerable for MITM type attacks. Which has proven as one of the most powerful network threats. Concept of novel, secure architecture for 5G network is shown on Fig. 7. Process flow is designed to minimize risk of attaching with fake base stations. Scenario according to the new concept should be following: UE before sending unencrypted attach request to desired BS, has to check it’s identity and legitimacy. Core elements of authentication algorithm must be deployed in BS sides. So, all the authorized BSs will have information and confirmation directives about each other, so that they can approve or deny genuine of desired BS. Figure 7 illustrates how attaching process should be managed. BSs with green shields are authorized, genuine ones. Red BS represents fake base station. UE has active connection with legitimate BS (linked with Green Arrow). Before sending attach request to the new BS (red dashed link with Fake BS) it has to check availability details for attaching (blue dashed links) like identity, certificates and etc. Second part is to send requested details with active BS, which itself validates authenticity and authorizes process or denies. The last, third process it to send attach request to the only authorized BS, which has already approved its validity. This architecture does not except existence of fake base stations—MITM, but minimizes risk of trying

Fig. 7 Secure architecture concept for 5G, in terms of UE to BS attaching

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attaching to them. In existing design UE does not monitor capability (CAP) changes. There should be algorithm in UE part, which will monitor CAP changes and if it is done during attach request (when data is transmitted unencrypted, in clear text), user should be alerted and asked to reset changes. Another solution is to make UE CAPs read only during attaching process. It should be mentioned that this concept does not guarantee total security and its efficiency has to be measured in real environment. As extra process, no matter how optimized it is, this flow will take more time. But everything is relative and depends on the need and priorities. There are also cases when UE does not have active connection with BS, so in this situation UE must have list of authorized BSs. Which will be updated every time UE will attach to the network. Important note is that this solution also redirects new attack vectors to BS sides. After this concept, attackers will try to poison records on BS sides to add their fake base station in to the list of authorized BSs. However we could not stop arising new threats as it’s like a chain reaction and every new protection mechanism triggers intruders to find new vulnerabilities and exploits. So, we should try to go on and minimize risk as soon as it will be possible.

5 Conclusions It is obligatory to transfer towards 5G technologies. Securely and successfully implemented standard will cover all the industries and exceed the limits as it is promised. By incorporation general advantages: EMBB, URLLC, MMTC with artificial intelligence, 5th generation network will have scope that had never had telecom communications. Because of its scale, interest of illegally motivated people will extremely arise. So will be about the results of cyber-attack. 5G has some security updated, like better encryption, but cannot avoid one of the most powerful network attack—MITM. Which lets attacker to use different type exploits and successfully make battery drain, MNmap, downgrading attacks, data leakage and so on. This research offers idea, concept how should be done architecture of 5G network, especially attaching process to achieve better security, in terms of minimizing risk of MITM-fake base stations. According to our scenario, UE before sending attach request (clear text) to desired BS (dBS), it must recheck dBS genuine with already connected authorized BS. If it is impossible because of missing active connection, legitimate of dBS must be checked with internal list. Only after successful authentication of dBS should be sent attach request. Second part is to UE must monitor CAP changes and if it is done during attach process, should be asked for resetting changes. It is clear, that every additional process requires extra time. It is obligatory to assess the offered methodic on the real 5G network.

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6 Future Plans For future plan, we are going to test this scenario in micro 5G lab and try to make quantity assessment how effective it will be. How will it effect on productivity and performance for both: network and UE. We are going to set up simulated micro 5G lab and make test environment from 5G enabled micro computers like Raspberry Pi to test ideas, concepts found during scientific studies. Acknowledgements The work was financed by Shota Rustaveli National Science Foundation and was conducted in the frame of the [CARYS-19-121] grant.

References 1. Berdibayev, R., Gnatyuk, S., Yevchenko, Y., Kishchenko, V.: A concept of the architecture and creation for SIEM system in critical infrastructure. Stud. Syst., Decis. Control 346, 221–242 (2021) 2. SK Telecom, in “5G Architecture Design and Implementation Guideline”. (2015) 3. Maheshwari, M.K., Agiwal, M., Saxena, N., Abhishek, R.: “Flexible beamforming in 5G wireless for internet of things”. IETE Tech. Rev. 36(1), 3–16 (2017). https://doi.org/10.1080/ 02564602.2017.1381048 4. Ivezic, M., Ivezic, L.: “5G security & privacy challenges”. In 5G.Security Personal Blog. (2019). https://5g.security/cyber-kinetic/5g-security-privacy-challenges/ 5. Huawei Technologies CO. LTD in “5G Network Architecture—A high Level Perspective”. (2016) 6. Qualcomm Technologies Inc. “What is 5G”, in online article. https://www.qualcomm.com/5g/ what-is-5g 7. Purdy, A.: “Why 5G Can Be More Secure Than 4G”. In Forbes online journal. (2019). https://www.forbes.com/sites/forbestechcouncil/2019/09/23/why-5g-can-be-more-sec ure-than-4g/?sh=2ffcdf1657b2 8. Hanif, M.: “5G Phones Will Drain Your Battery Faster Than You Think”. In online journal. (2020). https://www.rumblerum.com/5g-phones-drain-battery-life/ 9. Samsung in online report “Samsung Phone Battery Drains Quickly on 5G Service”. https:// www.samsung.com/us/support/troubleshooting/TSG01201462/ 10. Shaik, A., Borgaonkar, R., Park, S., Selfert, J.P.: “New vulnerabilities in 4G and 5G cellular access network protocols: exposing device capabilities”, WiSec ’19: Proceedings of the 12th Conference on Security and Privacy in Wireless and Mobile Networks. (2019). https://doi.org/ 10.1145/3317549, ISBN: 9781450367264 11. Yusof, R., Khairuddin, U., Khalid, M.: A new mutation operation for faster convergence in genetic algorithm feature selection. Int. J. Innovative Comput., Inf. Control 18(10), 7363–7380 (2012) 12. Iavich, M., Gnatyuk, S., Odarchenko, R., Bocu, R., Simonov, S.: The novel system of attacks detection in 5G. Lect. Notes Netw. Syst. 226, 580–591 (2021)

The Method of Determining the Elements of Urban Infrastructure Objects Based on Hough Transformation Hennadii Khudov , Vladyslav Khudov , Iryna Yuzova , Yuriy Solomonenko , and Irina Khizhnyak

Abstract The necessity of distinguishing elements of the urban infrastructure in the optoelectronic images of remote sensing of the Earth is substantiated. Known methods for determination of contours and their main disadvantages are analyzed. A method for isolating the contours of urban infrastructure objects on optoelectronic images of remote sensing of the Earth is proposed. The method includes two stages. The first stage implies determining the contours of objects in images. The advanced Canny method was selected as the contour determination method. We considered the main stages of the advanced Canny method for determination of contours of objects on the optoelectronic images. The application of the Hough transformation at the second stage was proposed. The paper reports features in the method for determination of elements of urban infrastructure in the optoelectronic images of remote sensing of the Earth. In contrast to known methods, the method takes into account features of formation of images of remote sensing of the Earth. It highlights color channels and marks out contours and geometric primitives in each color channel; it re-integrated color channels and determines elements of urban infrastructure objects in the space of an output image. The study presents the results of applying the method for determination of elements of urban infrastructure objects in a standard color optoelectronic images of remote sensing of the Earth. We defined elements of urban infrastructure objects, such as roads, houses, streets, building elements and others, as an example. The assessment of the quality of the determination of the contours of urban infrastructure in the image carried out by visual means. Keywords Optoelectronic image · Remote sensing of the earth · Element of urban infrastructure · Determination of elements · The Canny edge detection algorithm · Hough transformation

H. Khudov (B) · I. Yuzova · Y. Solomonenko · I. Khizhnyak Kharkiv National Air Force University, Kharkiv, Ukraine e-mail: [email protected] V. Khudov Kharkiv National University of Radio Electronics, Kharkiv, Ukraine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Zaporozhets (ed.), Systems, Decision and Control in Energy III, Studies in Systems, Decision and Control 399, https://doi.org/10.1007/978-3-030-87675-3_15

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1 Introduction Recently, algorithms for automatic analysis and processing of images, which were obtained from on-board remote sensing systems of the Earth, have become widespread. As a result of: • the availability of images of on-board remote sensing systems of the Earth in the public access; • the penetration of information technologies of Earth remote sensing into different areas of life, for example, agriculture, cartography, monitoring the state of the surface of Earth, navigation, exploration, military affairs, etc. [1–4]. The task of detecting and selecting artificial objects on images of on-board remote sensing systems of the Earth is very important for state registration of structures, buildings, construction in progress, premises, state cadastral registration of objects of real estate, and cartography, etc. [5, 6]. This information will allow: • quickly assess the changes on the terrain; • constantly update data for urban geographic information systems (urban GIS) [3, 5]. Also detection of artificial objects on optical-electronic images which were obtained from on-board remote sensing systems of the Earth is very relevant when translating paper archives of topographic plans of the terrain into their digital map of the terrain. Elements of urban infrastructure, for example, structures, buildings, roads, bridges, and etc., contain many straight lines and are quite contrasting. Thus, if we select the contours with a edge detector in each channel of the color space of the color image representation (such as, RGB). And then we can distinguish geometric primitives (line, circle etc.) using the Hough transformation. If the geometrical primitive is in all three channels of the color space of the RGB at the same time—this is a sign of the artificial occurrence of the object. If the geometrical primitive is only in one channel of the RGB—this is the natural object (such as, a field road, a river etc.). If the geometrical primitive is only in two channels of color space—the classification is difficult. So, this method includes two stages. This is the determination of contours using any edge detection algorithm on first stage. And this is the determination of geometric primitives using the Hough transformation on the second stage. Every day the density of development of large settlements is growing, their architectural appearance is being modified, the length of various communications is increasing, the transport network is becoming more extensive, etc. There are four main models of urban terrain: satellite, network, linear, segment. The main models of street (radial, grid, irregular), main functional areas of urbanized terrain are shown in Fig. 1.

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Fig. 1 The main models of street on urban terrain: a—radial; b—grid; c—irregular

2 Methods for Determining the Contours of Objects in Images There are three main groups of methods for determining the contours of objects in images [1, 2]: • method of high-frequency filtering; • method of functional approximation; • method of spatial differentiation. Common to all methods is the desire to consider the contours as a area of sharp difference in the brightness function of the image f (x, y). These methods are distinguished by the mathematical model of the concept of “edge” and the algorithm for finding edge points [1, 2].

2.1 Method of High-Frequency Filtering The method is based on the fact that the information about the contour of the object is contained in the high-frequency components of the image spectrum. The method of determining of contour is high-frequency image filtering, which is performed using fast convolution and threshold processing. Filtration is described by expression (1) [1]:   G(x, y) = F −1 f i (u, v), H i (u, v) ,

(1)

where f i (u, v) is the Fourier transform of the function f (x, y); H i (u, v) is transfer function of the high-frequency filter; F −1 (. . .) is Fourier inverse transform operator. The transition from G(x, y) to the contour image L(x, y) is carried out by comparison with the threshold value T , namely (expression (2)):

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 L(x, y) =

1, f or G(x, y) ≥ T ; 0, f or G(x, y) < T.

(2)

It is necessary to select the type and parameters of the transfer function of the high-frequency filter for implement the method and achieve an acceptable result. It is a difficult task.

2.2 Method of Functional Approximation The method is based on the fact that each point of the image with coordinates (x, y) is surrounded by some neighborhood ρ with center at this point. A step function of the expression (3) is determined in the neighborhood [1]: ⎧ f or (ax + by ≥ t), (x, y) ∈ ρ; ⎨ c, ˆ S(x, y, a, b, t, c, d) = (c + d), f or (ax + by < t), (x, y) ∈ ρ; ⎩ 0, f or (x, y) ∈ / ρ,

(3)

where a, b, t, c, d is the numerical parameters [1]. The quality of the function approximation S(x, y) by a function ˆ S(x, y, a, b, t, c, d) in the neighborhood is determined by the metric (4) [1]: ˆ = d(S, S)

¨ 

2 ˆ S(x, y) − S(x, y, a, b, t, c, d) d xd y.

(4)

ρ

The point lies on the contour of the image if there are parameters of the approxˆ imating function S(x, y, a, b, t, c, d) that provide a given quality of approximation ˆ (value) d(S, S). Otherwise, it is assumed that the point does not belong to the contour. ˆ The method of determining the parameters of a function S(x, y, a, b, t, c, d) is known as the Hückel operator and is based on a Fourier series decomposition [1]. The method of functional approximation is also quite complex. The solution of the optimization task of selecting the parameters of the function must be performed for each point of the image. It is requires a significant amount of calculations.

2.3 The Method of Differentiation The method of differentiation is based on the fact that at the points of the contour of the object the modulus of the gradient of the function f (x, y) takes the maximum values. The functional diagram of the method of spatial differentiation is shown in Fig. 2 [2]. In this case, at each point with coordinates (x, y) is calculated (expression (5)):

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differential (gradient) operator

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threshold operator the gradient image G ( x, y )

the binary image b( x, y )

Fig. 2 Functional diagram of the method of spatial differentiation [2]

G(x, y) = |∇ f (x, y)| =



G 2x + G 2y ,

(5)

where G x = ∂ f ∂(x,y) , G y = ∂ f (x,y) are the partial derivatives. x ∂y Next, the given threshold value T is determined by the function (expression (6)):  b(x, y) =

1, i f |∇ f (x, y)| ≥ T ; 0, i f |∇ f (x, y)| < T.

(6)

In practice instead of function f (x, y) the function of discrete argument F(x, y) is set. Then the implementation of the method of differentiation involves the replacement of partial derivatives G x = ∂ f ∂(x,y) and G y = ∂ f (x,y) discrete estimates of the x ∂y corresponding derivatives. In practice, methods of numerical differentiation give acceptable results only if the image is pre-processed in order to suppress noise, increase contrast, emphasize the contours [1, 2]. Since the image is discrete, it is impossible to directly calculate the partial derivatives and the modulus of the gradient by expression (5) in computer memory [1]. It is necessary to use the gradient method [1, 2] that to use differentiation to determine the brightness difference at any point of the image. The gradient method is based on the procedures of spatial differential operators. Usually the brightness difference operator is represented in the form of a mask linear filter [1, 2]. The mask (which is a matrix of coefficients) slides across the image field, occupying alternately all possible positions in the process of processing. In each position, the mask plays the role of a window. Through this window the pixels of the image are selected and element by element multiplication is performed on the corresponding element of the mask, followed by the summation of all products. The resulting number is considered as a countdown of the original image at a point corresponding to the center of symmetry of the window. The choice of differentiation operator is the main difference between different methods. The main operators of differentiation are known: Roberts, Prewitt, Sobel, Kirsch and others [1, 2]. In [1] are proposed methods of digital image processing which based on the use of a two-dimensional differential scalar Laplace operator. The main disadvantages are the impossibility of determining the direction of the contour, not the detecting, but only the emphasis on the brightness difference. In [2] gradient methods are proposed to detect the contours. In these methods is calculated the full vector of the image gradient. The main disadvantages are the

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complexity of solving the Bayes problem, the need for a priori knowledge of the conditional probabilities of the gradient values. In [2] heterogeneous methods of spatial differentiation (Sobel operator, Prewitt operator, Roberts cross, sequential masking, etc.) are used to detect the contours in the image. The main disadvantages of the methods are the presence of gaps, points and strokes that form a background, the need for knowledge of the initial approach to the desired contour and significant computational costs. The disadvantage of methods of sequential masking is the reduction of image contrast, image blur. In [3] methods of binding (stitching) of contours, comparison with use of a template are proposed. The disadvantages of the methods are the complexity of practical implementation, high computational costs, the need to use a priori information about the original image. In [4] it is noted that the methods of determining objects that are technologically organized using artificial neural networks have a number of advantages. The main advantage is the speed of implementation of operations. This is due to the properties of neural networks in terms of adaptive learning, self-organization, generalization, the ability to perform calculations in time close to real etc. In [4] proposed a model for detecting small objects on images from unmanned aerial vehicles, which is based on the architecture of the family of two-pass models R-CNN. In [5] the multitasking convolutional neural network RoadNet (Wuhan University, China) is used to determine road networks on images from air monitoring systems. However, using the RoadNet network allows you to define only road networks. There are some difficulties in identifying other elements of urban infrastructure: the network needs training, exact initial values etc. In [6] proposed the use of neural networks for mapping and land cadastre using images from the WorldView-2 system (DigitalGlobe, USA). The methods proposed in [6] solve problems in rural terrain. The application of methods [6] to determine the elements of urban infrastructure is complicated. In [7] it was proposed to use the ant method to determine the contours of the images from the on-board system. The main disadvantage of the method is the presence on the resulting image of a large number of contours of small objects (“garbage” objects). In [8], the application of multi-scale image processing from on-board systems based on the ant algorithm is proposed. The main disadvantage of the method is the selection not of contours, but of areas where there may be objects of interest. In [9] was proposed the method of artificial bee colony for thematic segmentation of optical-electronic images. The disadvantage of the method is a significant computing resource. In [10] are developed the results [9]. The optimization problem of image segmentation from on-board air monitoring systems by the method of artificial bee colony is formulated. The disadvantage of this method is not defining the contours of the objects of interest, but the areas that are potential objects of interest.

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3 The Method of Determining the Elements of Urban Infrastructure Objects Based on Hough Transformation We use the advanced Canny method for the first stage [11]. The first step of the method of determining the elements of urban infrastructure objects in the optoelectronic images of remote sensing of the Earth includes the following blocks (Fig. 3). Identification of elements of urban infrastructure objects in the image (7): f (X, Col),

(7)

where X = (x, y) is the vector of pixel coordinates of the image; Col is the color space of the image representation provides definition of Q(X, P) contours with P(p1 , p2 , …, pN ) parameter vector. We can see from Q(X, P) contour notation that the function, which describes the contour, depends on two parameters. They are X pixel coordinates characterized by brightness and geometric parameters of P contour. As we noted above, elements of urban infrastructure objects are contrast and consist of simple geometric primitives.

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Fig. 3 General diagram of the method of determining the elements of urban infrastructure objects based on Hough transformation in the optoelectronic images of remote sensing of the earth

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Therefore, we consider the method for determination of elements of urban infrastructure objects as a two-step method. The first stage analyzes the decrypting feature— brightness. The second stage analyzes the decrypting feature—the geometric shape and parameters of P element of the urban infrastructure object. Figure 1 shows a general diagram of the method for determination of elements of urban infrastructure objects in the optoelectronic images of remote sensing of the Earth. The diagram represents the sequence of actions of the proposed method. f (X, Col) function transforms into f (X) function, where f (X) is the brightness of the tone image, which changes in the interval of [0…255], in the first stage of the analysis of the brightness and the definition of contours in the tone image. There is a contour point at X point when the absolute value of the discrete gradient exceeds a certain threshold level m > 0 (expression (8)): | f (X)| = | f (X + I)− f (X)| ≥ m → {X ∈ Q(X, P)},

(8)

where I is the unit matrix. We use the advanced Canny method for the first stage [11]. The first step of the method for determination of elements of urban infrastructure objects in the optoelectronic images of remote sensing of the Earth includes the following blocks (Fig. 3). Block 1 on Fig. 3. Smoothing. It is necessary to reduce the effect of noise on boundary determination. The Gaussian filter is used for it (expression (9)): f (x, y) =

1 e 2π σ



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where σ is the blur parameter. One should choose the value of the blur parameter to provide the highest noise suppression. We use the larger value of the parameter to highlight large contours, and the less one to highlight thin contours and small details. Block 2 on Fig. 3. Search for a gradient. One uses the Sobel operator to determine the gradient in the image after the Gaussian filter (expression (9)). The basis of Sobel’s transformation is the assumption that the brightness difference function becomes much larger on borders. One can conclude from this assumption that it is sufficient to differentiate f (x, y) brightness functions (10) and (11) to find the borders: f (x + ∂ x, y) − f (x, y) ∂ f (x, y) = x = , ∂x ∂x

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∂ f (x, y) f (x, y + ∂ y) − f (x, y) = y = . ∂y ∂y

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One can measure ∂ x and ∂ y between two points in pixels by expressions (12) and (13) in discrete images:

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x = f (i + 1, j)− f (i, j),

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y = f (i, j + 1)− f (i, j).

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The expression (14) determines the value of G gradient: G=



(x)2 + (y)2 ,

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and we determine θ direction from expression (15):  θ = arctan

 x . y

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We evaluate the gradient using masks in expressions (14) and (15): 

 −1 1 , 0 0   −1 0 y = . 1 0

x =

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The major disadvantage of using (16) and (17) masks in the classic Canny method [11] is a large number of errors due to noise. In addition, the use of masks of the pair order in the classic Canny method [11] does not make it possible to carry out evaluation based on a pixel centered on the mask. Unlike the classic Canny method [11], we use the Sobel operator with masks (18) and (19) for evaluation of the gradient: ⎛

⎞ −1 0 1 K G x = ⎝ −2 0 2 ⎠, −1 0 1 ⎛ ⎞ 1 2 1 K G y = ⎝ 0 0 0 ⎠. −1 −2 −1

(18)

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There is the coefficient 2 used for the average elements in expressions (18) and (19) (in comparison to (16) and (17)). In contrast to the classic Canny method [11], we reduce the smoothing effect. Values and the direction of a value of G gradient take the forms (20) and (21), respectively: G=



G 2x + G 2y ,

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 |G x | θ = arctan   . G y  

(21)

As a result of using the Sobel operator with masks (18) and (19), the intensity of each pixel of the output image is equal to the gradient of the brightness vector. Block 3 on Fig. 3. Suppression of false maxima. The objective of this stage is to turn “blurred” boundaries into “clear” ones. We achieve this by maintenance of local maxima and removal of everything else. There are the following steps performed for each pixel: – the rounding of direction of the gradient to the nearest value, which is a multiple of 45° (Fig. 4a); Fig. 4 Search for local maxima: a—p and r maxima are interpolated (deleted); b—the principle of suppression of false maxima

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– if there is the local maximum reached at the current point in the direction of the gradient, then it is part of a boundary; – otherwise the point is deleted (Fig. 4b). Figure 4b illustrates the suppression principle. All pixels have an upward orientation in Fig. 4b, so we compare the gradient value at these points to the pixels below and above. The pixels outlined by white color in Fig. 4b remain in the output image, the other ones will be suppressed. Block 4 on Fig. 3. Figure 5 illustrates the double threshold filtration. Each pixel, which exceeds the upper threshold, is a “strong” pixel. Each pixel, which falls between two thresholds, is a “weak” one. The brightness of “weak” pixels assumes a fixed average value and will be refined in the next stage. Pixels, which are smaller than the lower threshold, are deleted. The using of the double threshold gives possibility to reduce an effect of noise (due to the upper threshold) and not to lose “tails” (due to the lower threshold). Block 5 on Fig. 3. Unlike in the classic Canny method [11], we trace the uncertainty area in the block 5 on Fig. 3. The task is to isolate groups of pixels, which got an intermediate value in the previous stage, and to assign them to a boundary (if they are connected to one of the boundaries) or to suppress them (otherwise). This block makes it possible to reduce missing of pixels in definition of object contours. Thus, we determine contours of objects at the first stage of the method for determination of elements of urban infrastructure objects in images made by air monitoring systems. The main decrypting feature is the brightness of pixels in the image. We choose the advanced Canny method. Unlike the known method, it uses the Sobel operator with masks of (3 × 3) to evaluate the gradient and it traces the area of uncertainty additionally, which reduces the smoothing effect and missing of pixels in determination of object contours. The second stage analyzes the decryption feature—the geometric shape and P = (p1 , p2 , …, pN ) parameters of elements of the urban infrastructure object. We use the

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Hough transform for straight lines [12] to determine (p1 , p2 , …, pN ) parameters at the second stage. One calls P = (p1 , p2 , …, pN ) space parametric. For each (x, y) contour point, we perform the procedure of increasing of the value of all cells of P = (p1 , p2 , …, pN ) parametric space with (p1 , p2 , …, pN ) coordinates, which satisfy Eq. (22), by one: P( p1 , p2 , . . . , p N ) = P ∗ ( p1 , p2 , . . . , p N ) + 1( p1 , p2 , . . . , p N ) : Q(X, p1 , p2 , . . . , p N ) = 0,

(21)

where P(p1 , p2 , …, pN ) is the new value of the cell of the parametric space; P* (p1 , p2 , …, pN ) is the previous value of a cell of the parametric space. The coordinates of cells in the parametric space will generally correspond to the figures in the image after recalculation for all points of the contour. The elements of urban infrastructure objects generally have the shape of a straight line. Therefore, one can represent the equation of a line, which passes through a point with (x, y) coordinates, as (22) (Fig. 6): x cos(ϕ) + y sin(ϕ) = ρ,

(22)

where ρ is the distance from the origin of coordinates to the straight line (a beam); ϕ is the angle between the abscissa and the beam.. One can represent the parametric space as (23): P(ρ, ϕ)

(23)

each its cell has (ρ, ϕ) coordinates. Thus, for each (x, y) contour point, we perform the procedure of increasing of the value of all cells of the parametric space (23) with coordinates by Eq. (24), (x, y): P(ρ, ϕ) = P ∗ (ρ, ϕ) + 1(ρ, ϕ) : Q(X, ρ, ϕ) = 0, Fig. 6 Parameters of the direct element of urban infrastructure objects

(24)

y

ρ φ

x

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where P(ρ, ϕ) is the new value of a cell of the parametric space; P* (ρ, ϕ) is the previous value of a cell of the parametric space. Thus, we consider the method for determination of elements of urban infrastructure objects as a two-step method. The first stage analyzes the decryption feature— brightness. The advanced Canny method was chosen for the analysis. Unlike the known method, it uses the Sobel operator masks of (3 × 3) sizes to evaluate the gradient and it traces an uncertainty area, which reduces the smoothing effect and missing of pixels in determination of contours of objects. The second stage analyzes the decryption feature—the geometric shape of an element of an urban infrastructure object. At this point, the Hough method was chosen for the analysis. One should note that this method is applicable for tone images. Therefore, for further research, let us look at features of the method for determination of elements of urban infrastructure objects in color images of on-board remote sensing systems of the Earth. We have color images of on-board remote sensing systems of the Earth presented in the Red–Green–Blue (RGB) color space. This fact is taken into account when determining elements of urban infrastructure objects in color images of on-board remote sensing systems of the Earth. In contrast to known methods [9–11] of contour determination, there is recommendation not to proceed to a tone image immediately and then to the binarized image in processing of color images, due to the fact that elements of urban infrastructure objects are in all three RGB color channels at the same time. If object elements are in one color channel only, an element may be of natural origin (for example, river); if objects are in two color channels at the same time, element classification is difficult (this may be, for example, field road, etc.). The above is an additional decryption feature of elements of urban infrastructure objects in color images made by air monitoring systems. In view of the above, Fig. 7 shows the sequence of actions of the method. The method involves: – an input of the output color image: f(X) (X = (x, y)); – definition of color channels in the output f (X) color image: f R (X), f G (X), f B (X) (where f R (X), f G (X), f B (X) are the images by Red, Green, and Blue color channels, respectively); – definition of the brightness channel in each color channel of the output image: f R (X), f G (X), f B (X); – determination of elements of urban infrastructure objects in images made by air monitoring systems in each color channel by the method (Fig. 1) and obtaining of images by each color channel: fsR (X), fsG (X), fsB (X) (where fsR (X), fsG (X), fsB (X) are the images with defined elements of urban infrastructure in Red, Green, and Blue color channels, respectively); – back transfer to RGB color model (integration of color channels); – obtaining of the processed fs(X) image. Therefore, there are the following features of the method for determination of elements of urban infrastructure objects in color images made by air monitoring

260 Fig. 7 The method for determination of elements of urban infrastructure objects in color images made by air monitoring systems

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Definition of color channels in f(X) output image

Definition of a brightness channel in each color channel in f(X) output image

Identification of elements of urban infrastructure in each color channel by the method (Fig. 1)

Back transfer to RGB color model (integration of color channels)

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systems. Unlike the known ones, the method takes into account features of formation of images made by air monitoring systems. It highlights color channels and marks out contours and geometric primitives in each color channel. It re-integrates color channels and identifies elements of urban infrastructure objects in the space of an output image. Figure 8 shows the image acquired from Ikonos spacecraft air system (DigitalGlobe, United States) [13]. The image is in RGB color space. The image size is (3000 × 4000) pixels. The image (Fig. 8) is a typical image of a city with elements of urban infrastructure objects. Therefore, there is only one typical image, which takes into account all features of city images made by air monitoring systems, in the experimental studies. Figure 9 shows the implementation of the first stage of the method for determination of elements of urban infrastructure objects according to the results of air monitoring. The first stage analyzed the decryption feature—brightness—and used the advanced Canny method.

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Fig. 8 The image acquired from Ikonos spacecraft air system (DigitalGlobe, United States) [13]

Figure 10 shows the image after the second stage of the method for determination of elements of urban infrastructure objects according to the results of air monitoring. Figures 9 and 10 show images in pseudo-colors after integration of color channels for visual demonstration of the method. The rules for integration of color channels are as follows: – if a pixel of the image relates to the boundary in all three channels of the Red– Green–Blue color space at the same time, this pixel becomes white; – if a pixel of the image relates to the boundary in the Red channel and there is no determination of a boundary in the Green and Blue channels, this pixel becomes red; – if a pixel of the image relates to the boundary in the Blue channel and there is no determination of a boundary in the Green and Red channels, the pixel becomes blue; – if a pixel of the image relates to the boundary in the Green channel and there is no determination of a boundary in Red and Blue channels, it becomes green; – if a pixel of the image relates to the boundary in the Green and Blue channels at the same time and there is no determination of a boundary in the Red channel, this pixel becomes blue;

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Fig. 9 The image after the first stage of the method for determination of elements of urban infrastructure objects according to the results of air monitoring

– if a pixel of the image relates to the boundary in the Red and Blue channels at the same time and there is no determination of a boundary in the Green channel, the pixel becomes purple; – if a pixel of the image relates to the boundary in the Red and Green channels at the same time and there is determination of a boundary in the Blue channel, the pixel becomes yellow; – if there are no boundaries defined in all three Red–Green–Blue channels at the same time, the pixel becomes black. Figure 11 shows the processed image for demonstration of the use of the additional decryption feature for determination of elements of urban infrastructure objects in color images. We obtained colors in Fig. 11 according to the following rules: – if there is a geometric primitive in all three channels of the color space of the RGB image representation (Red–Green–Blue) at the same time, the pixel is blue; – if there is a geometric primitive in one of the three channels of the color space of the RGB image representation (Red–Green–Blue), the pixel is green; – if there is a geometric primitive in two channels at the same time and it is absent in another channel in the color space of the RGB image representation (Red–Green– Blue), the pixel is green.

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Fig. 10 The image after the second stage of the method for determination of elements of urban infrastructure objects according to the results of air monitoring

Fig. 11 The image with identified elements of urban infrastructure objects taking into account their presence in processing channels

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Fig. 12 The identified elements of urban infrastructure objects

Figure 12 shows only the identified elements of urban infrastructure objects, which are of artificial origin. Such elements are blue in Fig. 11. Decryption of the defined elements of urban infrastructure objects, their identification, thematic classification and more are the subject of further research and remain beyond the scope of this study.

4 Conclusions Thus, it is proposed the two-stage method as a method for selecting the contours of urban infrastructure objects on images of Earth remote sensing. The first stage is the selection of contours by the Canny edge detection algorithm. The second stage is the selection of geometric primitives using the Hough transform. The assessment of the quality of the selection of the contours of urban infrastructure on the image carried out by visual means. To improve the performance of this method, it is promising to use a multi-scale image processing method.

References 1. Gonzalez, R.C., Woods, R.E.: Digital Image Processing, 4th edn. p. 1192. Prentice Hall, Upper Saddle Rever, (2017)

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2. Gupta, V., Singh, D., Sharma, P.: Image segmentation using various edge detection operators: a comparative study. Int. J. Innovative Res. Comput. Commun. Eng. 4(8), 14819–14824 (2016) 3. Saito, S., Aoki, Y.: Building and road detection from large aerial imagery [Electronic resource]. Proceedings of SPIE—The International Society for Optical Engineering, vol. 9405. (2015). Peim doctypy. https://www.researchgate.net/publication/273453681_Bui lding_and_road_detection_from_large_aerial_imagery 4. Manzanera, A., Nguyen, T., Xu, X.: Line and circle detection using dense one-to-one Hough transforms on greyscale images. EURASIP J. Image Video Process. 34, 1773–2000. Springer, (2016) 5. Liu, Y., Yao, J., Lu, X., Xia, M., Wang, X.: RoadNet: learning to comprehensively analyze road networks in complex urban scenes from high-resolution remotely sensed images. IEEE Trans. Geosci. Remote Sens. 57(4), 2043–2056 (2019) 6. Nyandwi, E., Koeva, M., Kohli, D., Bennett, R.: Comparing human versus machinedriven 3 cadastral boundary feature extraction [Electronic resource]. (2019). Peim doctypy: https://www.researchgate.net/publication/333454602_Comparing_Human_versus_ Machine-Driven_Cadastral_Boundary_Feature_Extraction 7. Ruban, I., Khudov, H., Khudov, V., Khizhnyak, I., Makoveichuk, O.: Segmentation of the images obtained from on-board optoelectronic surveillance systems by the evolutionary method. Eastern-Eur. J. Enterp. Technol. 5/9(89), 49–57 (2017) 8. Ruban, I., Khudov, V., Khudov, H., Khizhnyak, I.: An improved method for segmentation of a multiscale sequence of optoelectronic images. Problems of Infocommunications. Science and Technology (PIC S&T’2017): 4 International Scientific-Practice Conference, October, 10–13, 2017. Thesis of reports.—Kharkiv, pp. 212–213. (2017) 9. Ruban, I., Khudov, V., Makoveichuk, O., Khudov, H., Khizhnyak, I.: A swarm method for segmentation of images obtained from on-board optoelectronic surveillance systems. Problems of Infocommunications. Science and Technology (PIC S&T’2018): 5 International ScientificPractice Conference, October, 9–12, 2018: Thesis of Reports.—Kharkiv, pp. 613–618 (2018) 10. Ruban, I., Khudov, H., Makoveichuk, O., Khizhnyak, I., Khudov, V., Podlipaiev, V., Shumeiko, V., Atrasevych, O., Nikitin, A., Khudov, R.: Segmentation of opticalelectronic images from onboard systems of remote sensing of the earth by the artificial bee colony method. Eastern-Eur. J. Enterp. Technol. 2/9(98), 37–45 (2019) 11. Canny, J.F.: A computational approach to edge detection. IEEE Trans. Pattern Anal. Mach. Intell. 8, 679–698 (1986) 12. Ramlau, R., Scherzer, O.: The Radon Transform, p. 321. Berlin, Boston, Walter de Gruyter GmbH, (2019) 13. Ikonos Satellite Image Gallery, http://www.satimagingcorp.com/gallery/ikonos/ (accessed 12 Nov 2019)

Application of Discriminant Analysis in the Interpretation of Well-Logging Data Oleksiy Karpenko , Mykyta Myrontsov , and Yevheniia Anpilova

Abstract During the geophysical data interpretation of oil and gas wells, situations arise when, for various reasons, there is a lack of necessary information. Information—for carrying out traditional complex geological interpretation. The main typical tasks are the identification of reservoir rocks in the sections of wells, the determination of their characteristics, and the determination of the nature of the rock saturation. The lack of information can be compensated for by using pattern recognition methods. Here, even several curves of different logging methods, addition the combined data, allow solving the tasks of oil and gas geology with a certain probability. Methods, using artificial neural networks and, much less often, methods of discriminant analysis are quite popular at the present stage of geophysical research. The discriminant analysis method is quite easy to use and understandable from a mathematical standpoint. It is undeservedly little used in geophysical research. Below we will consider examples of using this statistical method to solve the tasks of assessing the probability of the presence of gas-saturated and water-saturated rocks in the productive sections of the wells of the Precarpathian trough gas field. The lithological features of the rocks of gas fields in the study area are the frequent alternation of clays, sandstones and siltstones in the sections of wells. As a result, the curves of geophysical parameters are monotonic, poorly differentiated lines. This significantly complicates the quantitative and qualitative geological interpretation of well logging data. Pattern recognition methods can be of great help in solving tasks associated with determining the intervals represented by gas-saturated reservoir rocks in such cases. Keywords Well logging · Gas saturation · Rocks · Reservoir rock · Discriminant analysis · Interpretation

O. Karpenko (B) Taras Shevchenko National University of Kyiv, Kyiv, Ukraine M. Myrontsov · Y. Anpilova Institute of Telecommunications and Global Information Space of the National Academy of Sciences of Ukraine, Kyiv, Ukraine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Zaporozhets (ed.), Systems, Decision and Control in Energy III, Studies in Systems, Decision and Control 399, https://doi.org/10.1007/978-3-030-87675-3_16

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1 Introduction The conditions for using the discriminant analysis is the conformity of the probability distribution of input quantities to the normal theoretical law of distribution of random variables. Therefore, a mandatory stage of data preparation is the study of the peculiarities of their distribution [1, 2]. When checking for the best correspondence to the theoretical distribution law, the nonparametric Kolmogorov–Smirnov λ criterion and the criterion χ 2 are expedient in this situation [1, 2]. It is established that for almost all reference groups the distributions of quantities: electrical resistivity, αsp , obey the normal law, and the values of standard deviations of imaginary resistivity—lognormal [3]. Therefore, the values of the natural logarithms of these standard deviations were used in further calculations. The procedure of linear discriminant analysis, its use in the interpretation of well-logging geophysical data is described in detail in [1–5]. The physical meaning of the linear discriminant function is that it is an equation of a hyperplane in a multidimensional feature space, and this plane is drawn so that on one side of it is the maximum number of objects of one group, and on the other side—the maximum number of an alternative group [3, 4]. Discriminant analysis allows not only to solve classification problems, but also to determine the informativeness of individual classification features, helps to select a rational set of geophysical parameters or research methods.

2 Purpose and Objectives of the Study Let us consider the possibilities of discriminant analysis of well-logging information in order to assess the nature of the saturation of layers and strata in well sections on the example of the Niklovitske gas field. The deposit is located in the outer zone of the Precarpathian Depression. Productive deposits are represented by a rhythmic alternation of sand-siltstone and clay rocks, which create a typical thin-layered stratum of the Dashav Suite of Neogene age. Diagnosis of such strata according to geological (field) and geophysical studies is very difficult due to the high clay content of rocks, the thin layer structure of sediments. It should be noted that another important aspect of the Outer Zone of the Precarpathian Depression is characteristic, first of all, of Neogene deposits. This is a different degree of consolidation of rocks at different depths. Gas-saturated deposits are located here at depths of several hundred meters to 3 or more kilometers. Numerous studies of rock material—core and sludge—show that rocks in the upper parts of well sections—up to 1 and more km are usually in a weakly compacted state, represented by weakly consolidated clays, sandstones and siltstones [6]. Their reservoir and filtration properties vary widely, which makes it difficult to carry out a qualitative interpretation of geophysical materials, especially when using traditional methods based on percolation evaluation of the properties of individual lithological groups of thin-layer section.

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Figure 1 shows the distribution of the main statistical characteristics of geophysical parameters of standard logging methods for the intervals of sections of wells of the Niklovitske field, where numerical tests of formations were carried out. The research was carried out within the strata ND-3–ND-10 of the Lower Dashava Subsuite. The results of field tests are divided into four groups according to the nature of saturation and values of flow rates: “gas”, “gas + water”, “water”, “dry”. From Fig. 1 it is seen, that the distributions of parameters for individual groups of layers overlap significantly. This feature of the deposits of the Dashava Suite complicates the use of certain geophysical indicators for the diagnosis of section rocks by the nature of saturation and provides a basis for the use of discriminant analysis of multidimensional data and other methods of pattern recognition [7–11]. The main task of linear discriminant analysis in the study of specific sediments is to increase the informativeness of geophysical research in the detection of additional gas-saturated layers in the section, which were not revealed by perforation, as well as to assess the effectiveness of this method in a limited amount of geophysical information digitized curves of standard logging. The data of standard logging include: curves of electrical resistivity of 2.25 m gradient probe, 0.5 m potential probe, as well as the curve of the SP method. Digitization was performed with a quantization step of 0.2 m.

Fig. 1 Statistical characteristics of geophysical parameters of rocks within the ND-3–ND-10 horizons of the Niklovitske gas field depending on the nature of saturation according to the formation tests

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To carry out the above studies, wells № 8, 10, 12, 25 of the Niklovitske gas field were selected for analysis, for which, in addition to the absolute values of the resistivity of electric probes and the magnitude of the natural potentials SP, the following values were used: relative potentials, the ratio of the gradient and potential probes data, the values of the standard deviation of the resistivity of the gradient and potential probes, as well as the ratio of the variances of the resistivity of the two probes. Previous studies have shown that in thin-layered clay deposits on the curves of large gradient probes (2.25 and 4.25 m length) in the gas-saturated intervals, the value of the variance of the electrical resistivity increases significantly. The values of variances were determined using the method of a moving strip with a window width (strip) of 1.8 m. Thus, the number of indicators, different in nature and geometry of measurements, increased from 3 to 8. After the procedure, the transformations of geophysical information were the averaging of the curves was performed with the increase of the sampling step to 0.6 m in order to reduce the influence on the results of the interpretation of random displacements of the curves with each other in separate sections [12, 13].

3 Research Methods One of the problems in choosing informative features is to determine the minimum parameters that would satisfy the required reliability of the analysis results [11, 12, 14, 15]. For a large number of ways to assess the informativeness of individual input parameters, there is a general algorithm. Assume that a set of data is used to solve the task, and the fraction of correct conclusions using the recognition algorithm is equal to p. If the efficiency is not changed when the parameter is added to the data set ai when the task is repeated, it means that the new parameter does not contain any additional information. If the efficiency of solving the task has changed ( pi > p), then the parameter contains new information, and the difference between pi − p is a measure of the informativeness of the parameter. For our tasks we will use the λ Wilks’ (lambda) and partial λ (Partial lambda) criteria [4, 5, 15], which are implemented in the popular statistical application packages Statistica 12, NCSS, SPSS and other. The value of F-statistics is calculated as: F = ((n − q − p)/(q − 1)) · ((1 − Par tial_lambda)/Par tial_lambda), (1) where: n—number of observations; q—number of reference groups; p—number of variables; Par tial_lambda—partial λ.

Application of Discriminant Analysis in the Interpretation … Table 1 Parameters for minimizing the number of input features for discriminant analysis

271 λ Wilks’ Partial λ F-statistics

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0.937

10.5

R gradient-probe (1)

0.254

0.906

16.1

R potential-probe (2)

0.265

0.870

23.1

(1)/(2)

0.237

0.972

4.5

Ln(σ(R gradient-probe)) (3)

0.262

0.880

21.2

Ln(σ(R potential-probe)) (4) 0.244

0.946

8.8

(3)/(4)

0.927

12.2

0.249

4 Research Results The maximum values of F-statistics correspond to the input parameters-variables, which create the maximum differences of the discriminant function between the reference groups. The Mahalanobis distance (criterion) D 2 is used as a linear discriminant function [16–18]. Tables 1, and 2 shows the results of assessing the informativeness of the input features and the division of reference observations into reference groups based on the analysis of the calculated values of discriminant functions. According to the results of calculations of F-statistics revealed (see Table 1), that significant in relation to the informativeness in the separation of rocks by the nature of saturation, were the values of the electrical resistivity of standard probes— gradient and potential probes, as well as the standard deviation of the R gradient probe in the window sliding along the wellbore. Confirmation of this fact is the Table 2 The efficiency of classification of rocks by the nature of saturation in the reference groups using discriminant analysis (Niklovitske deposit) Test results

The result of the classification (distribution of observations by groups) Gas

Water

Gas + water

“Dry”

Efficiency separation, %

16

84.5

All input geophysical parameters Gas

201

12

9

Water

1

52

24

2

65.8

Gas + water

0

23

41

2

62.1

“Dry”

2

5

24

61

66.3

204

92

98

81

74.7

Overall result

Three parameters: R gradient-probe, R potential-probe, Ln(σ(R gradient-probe,)) Gas

203

6

6

23

85.3

Water

0

45

28

6

57.0

Gas + water

0

19

44

3

66.7

“Dry” Overall result

1

6

24

61

66.3

204

76

102

93

74.3

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calculated values of the efficiency of rock extraction in the reference samples of wells of the Niklovitske gas field by the nature of saturation (Table 2). As you can see, the overall efficiency of the division of rocks into four classes (groups) by the nature of saturation is 74.7% (all input data) and 74.3% (data of the three most informative features—the electrical resistivity of the gradient and potential probes, and also—the logarithm of the standard deviation of the electrical resistivity of the gradient probe—Ln (σ (R gradient-probe)). The reference observations are separated in gas-saturated intervals, where the efficiency of their isolation is 85%, which is a rather high figure with a limited amount of information obtained by the results of research by three geophysical methods. Figure 2 shows the results of the assessment of the nature of rock saturation in the clay-sandy sections of the Dashava Suite of wells of the Niklovitske gas field using the method of discriminant analysis. The right column is a graphical representation of the probability of distribution of rocks by the nature of saturation (recall that at the initial stage of data preparation was averaged geophysical parameters with a step of 0.6 m depth). As can be seen from Fig. 2, in the intervals of depths of 1020–1157 m of the well 10, not opened by perforation, the section is represented mainly by gas-saturated rock thickness. By comparison with the perforation intervals below, in which significant gas inflows were obtained, it can be concluded that the upper specified rock strata are of commercial interest. The distribution of the probability of assigning rocks to the appropriate groups by the nature of saturation is formal, so in some intervals there are quite high values. The probability of the presence of gas-saturated rocks at the level of 0.98–0.99 indicates that the geophysical features of such rocks are quite close to the statistical characteristics of the center of the corresponding reference group. At some intervals, with the predominant probabilities of one group, signs of another appear, for example, in the gas-bearing stratum there are layers and strata with a low probability of gassaturated rocks. This fact is a manifestation of several factors, including—limited input geophysical information, inaccuracies in the digitization of logging curves, a simplified linear model of the discriminant function, a very complex structure of the real physical-geological model of the section, etc. In this case, only the main component of the probability of the nature of the saturation of the rocks should be taken into account [19]. Within the perforation intervals, even with unambiguous test results, there are different groups of rocks with different reservoir properties [20]. Therefore, the samples in the reference groups cannot be considered homogeneous. This leads to some reduction in the efficiency of classification in discriminant analysis. It can be considered that with a small number of methods of well-logging, in the absence of the necessary information for qualitative interpretation by standard methods, the use of an alternative method—discriminant analysis as a method of recognizing images of gas and water saturated rocks is appropriate.

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Fig. 2 Estimation of rock saturation in the wells sections of the lower Dashava suite of Niklovitske gas field based on the results of discriminant analysis

5 Discussion and Conclusions Thus, the main purpose of this study was considered and found practical confirmation—to assess the effectiveness and feasibility of discriminant analysis in geological conditions of thin-layer section of sandy-clay sediments with a limited set curves of well-logging methods. The method can be effectively applied in the old hydrocarbon deposits/fields in the search for previously undetected productive formations and intervals of well sections. Its use requires the presence of two prerequisites—the

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presence of logging data recorded in digital form, and a sufficient number of test intervals with the received inflows of fluids to create reference groups. However, as we have found, in many cases, especially with the predominant content of clay rocks in the section, the use of linear discriminant analysis does not give high values of the efficiency of separation of rocks by the nature of saturation. In such cases, other, more progressive methods of pattern recognition should be used [11, 14, 21].

References 1. Kazhdan, A.B., Guskov, O.I.: Mathematical Methods in Geology. Moskow, Nedra, 1–251 (1990). https://ua1lib.org/book/2834088/6a77e4?id=2834088&secret=6a77e4 2. Dmitriev, V.I., (ed.): Computational Mathematics and Engineering in Exploration Geophysics: A Geophysics Handbook. Moskow, Nedra, 1–498 (1990). https://www.ozon.ru/product/vychis litelnye-matematika-i-tehnika-v-1-razvedochnoy-geofizike-8688009/ 3. Dech, V.N., Knoring, L.D.: Unconventional Methods of Complex Processing and Interpretation of Geological and Geophysical Observations in the Sections of Wells. Leningrad, Nedra, 1–192 (1978). https://www.twirpx.com/file/3095326/ 4. Shatakhtsyan, A.R.: Formal clustering method application to data on large and super-large ore deposits. Transition Zone Geosyst. 2(1), 33–41 (2018). https://doi.org/10.30730/2541-8912. 2018.2.1.033-041 5. Rao, C.R.: An asymptotic expansion of the distribution of Wilks’ criterion. Bull. Int. Stat. Inst. 33, 177–181 (1951). http://repository.ias.ac.in/71616/ 6. Loktev, A.V.: Reasons for the omission of productive horizons in the clay layer of the Neogene of the outer zone of the precarpathian depression and measures to prevent them. Explor. Develop. Oil Gas Fields. Ivano-Frankivsk, 8(3), 123–126 (2003). http://elar.nung.edu.ua/bitstream/123 456789/5440/1/30p.pdf 7. Baldwin, J.L., Bateman, R.M., Wheatley, C.L.: Application of a neural network to the problem of mineral identification from well logs. Log Anal. 3, 279– 293 (1990). https://onepetro.org/petrophysics/article-abstract/170779/Application-Of-A-Neu ral-Network-To-The-Problem-Of?redirectedFrom=PDF 8. Benaouda, B., Wadge, G., Whitmarh, R.B., Rothwell, R.G., MacLeod, C.: Inferring the lithology of borehole rocks by applying neural network classifiers to downhole logs—an example from the ocean drilling program. Geophys. J. Int. 136, 477–491 (1999). https://doi. org/10.1046/j.1365-246X.1999.00746.x 9. Saggaf, M.M., Nebrija, Ed.L.: Estimation of missing logs by regularized neural networks. AAPG Bull. 87(8), 1377–1389 (2003). https://doi.org/10.1306/03110301030 10. Saggaf, M.M., Nebrija, Ed.L.: A fuzzy logic approach for the estimation of facies from wire-line logs. AAPG Bull. 87(7), 1223–1240 (2003). https://doi.org/10.1306/02260301019 11. Karpenko, O. Neural networks technologies in oil and gas well logging. 17th Int. Conf. Geoinformatics—Theoret. Appl. Aspects (2018). https://doi.org/10.3997/2214-4609.201801798 12. Karpenko, O., Myrontsov, M., Karpenko, I., Sobol, V. Detection conditions of gas-saturated layers by the result of complex interpretation of non-electrical well logging data. Monitoring 2020 Conference «Monitoring of Geological Processes and Ecological Condition of the Environment», Extended Abstracts, Kyiv (2020). https://doi.org/10.3997/2214-4609.202 056034 13. Myrontsov, M.L., Karpenko, O.M., Trofymchuk, O.M., Okharie, V.O.: Examples of determination of spatial and geoelectric parameters of productive beds of deposits of the DniproDonetsk depth. XIV International Scientific Conference «Monitoring of Geological Processes

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Noise Immunity of Devices of Automated Systems for Technological Control of Energy Facilities in the Almaty Region B. R. Kangozhin , O. A. Baimuratov , M. S. Zharmagambetova , S. S. Dautov , and D. B. Kangozhin

Abstract The article presents the results of a study of the noise immunity of devices and elements of automated systems for technological control of energy facilities, associated with the use of relay cables. The analysis of the data obtained in this study is based on theoretical research methods, and the computational and experimental method were applied to determine the level of pulsed electromagnetic interference at 220 kV Taldykurgan, Stroitelnaya (Construction), Sary-Ozek, Zavodskaya and SS220 “Shu” substations. All energy facilities considered in this article are located in the Almaty region. The influence of shielding cables on the level of electromagnetic influences on devices and elements of the automation system is revealed. It has been experimentally established that impulse noise in some relay circuits exceeds the permissible values during short-circuit and switching in the electrical network. The actual screening factors of control cables in operational conditions are determined. Keywords Electromagnetic compatibility · Electromagnetic interference · Aggregated interference · Radiated pulse interference · Electromagnetic environment · Grounding device · Relay protection and automation · Electrical substation · Automated process of control systems

B. R. Kangozhin (B) · M. S. Zharmagambetova Al-Farabi Kazakh National University, Almaty 050040, Kazakhstan e-mail: [email protected] O. A. Baimuratov · M. S. Zharmagambetova Suleyman Demirel University, Kaskelen 040900, Kazakhstan M. S. Zharmagambetova · S. S. Dautov Kazakh Academy of Transport and Communications Named After M. Tynyshpayev, Almaty 050012, Kazakhstan D. B. Kangozhin JSC “NC” Kazakhstan Garysh Sapary”, Nur-Sultan, Kazakhstan © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Zaporozhets (ed.), Systems, Decision and Control in Energy III, Studies in Systems, Decision and Control 399, https://doi.org/10.1007/978-3-030-87675-3_17

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1 Introduction Electromagnetic Compatibility today has a broad spectrum in various technical areas, representing a cluster of specific requirements and recommendations. This condition is manifested in work [1], when developing the design of a highly efficient tactical radar system, where careful attention is paid to electromagnetic compatibility (EMC) in order to ensure high performance and accuracy in an adverse electromagnetic environment (EME). In [2, 3] a wide range of issues related to the study of electromagnetic compatibility in railway systems is considered. It also discusses in detail the various types of interference phenomena, namely conducted interference, capacitive interference phenomena, induced interference phenomena and radiated interference phenomena. In [3], issues of railway EMC and safety are considered: • from a Railway EMC perspective it is fundamental that equipment should operate safely; • in this context it is vital that traction and rolling stock should not interfere with signalling and communication systems. Unstable power supplies pose a serious threat to the reliability and longevity of data, making a UPS a necessity. The UPS shall be immune to power noises/disturbances and shall not generate such noises as well. EMI/EMC evaluation of UPS thus becomes very essential to ensure that the system works satisfactorily in the intended environment. The paper [4] discusses various available specification for EMC evaluation of UPS and analyzes the working environment of UPS. It also highlights probable EMI sources and suppression and immunity improvement techniques for compliance of UPS to national and international norms. The authors in [5] provided the research results of the electromagnetic compatibility of intelligent devices of automated process control systems at a high-voltage electrical substation. The noise immunity of relay protection and automation (RPA) devices associated with the use of electric control cables is investigated. In [6], highpower microwave radiation (UHF) with a frequency of >1 GHz is considered, which can penetrate into electronic systems through the front or rear openings and propagate like a real signal. Sometimes parasitic resonance can amplify the HPM signal and cause havoc in these systems. As noted in [6], microwave can also cause breakdown in low-pressure devices such as hydrogen thyratrons and cause parasitic alarms in the control circuits that use them. Corona generated pulsed currents on high voltage transmission line conductors radiate electromagnetic fields, which, in turn, can interfere with communication systems, and radio as well as television receivers operating nearby [7]. In work [7], the corona generated electromagnetic interference (EMI) field in dB at an observation point near the ground has been computed assuming the ground as a perfect conductor. It has been observed that the interference field starts abruptly at the corona inception voltage and it increases with the voltage stress on the conductor.

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In paper [8] reviews, the electromagnetic field pollution generated by digital computers with their benchmark results. It reviews the theoretical analysis and prediction for the generated pollution of PCs. Further, it reviews the practical measurements carried out for a typical Compaq personal computer, in a shielded chamber, to find the field strength at different frequency bands for the spectrum of the emission. It also covers LAN emission measurements and how this analysis and data is useful to ameliorate the compatibility of the equipment for various sensitive receivers working around personal computers [8]. In [9], specific issues are presented that accompany measurements of conducted electromagnetic interference in multi-converter systems. A well-known example of this problem is the influence of EMI generated by power electronic converters, implemented, for example, in drive systems or dimmers, as well as on the transmission reliability of a power transmission line (PLC) [10–12], usually used in advanced measuring infrastructure (AMI) to read smart power meters [13–17]. The analysis of works [1–17] determines the relevance of studies of electromagnetic compatibility (EMC) in an automated process control system (APCS), which has a significant impact on noise immunity [13]. As a result of the analysis of works [18–21], it was determined that the influence of the degree of shielding of cables on the accident rate of the APCS has not been sufficiently studied. Failures and malfunctions of relay protection and automation devices lead to the shutdown of the main high-voltage equipment at electrical substations and to power supply interruptions. Through short-circuit currents in electrical networks lead to the appearance of hidden defects in the electrical insulation of highvoltage power equipment [22, 23]. Thus, the reduction in the accident rate of APCS devices is associated with the noise immunity of SMART devices [24] when exposed to radiated pulse interference (RPI), which is the subject of the sought article. Devices of automated technological control systems (ATCS) should be tested for noise immunity in accordance with [25, 26]. At the energy facility, such an electromagnetic environment (EME) must be provided so that, under any operating modes of the electrical network, electromagnetic interference (EMI) does not exceed the levels acceptable for SMART equipment. The transition from the relay-contact element base of the ASTU to SMART devices (SU) revealed the problem of noise immunity of the latter. All relay protection and automation devices providing reliable, safe distribution and transmission of electrical energy are complex technical devices. They have many geographically dispersed functional modules and have a very large number of various states. The interconnection of relay protection and automation elements is provided by a variety of control cables (Fig. 1) [11, 12]. In particular, Fig. 1 shows the connection of the relay protection and automation equipment with high-voltage equipment at the considered substations PS-220 kV in Almaty region [27]. In general, the assessment of the SU noise immunity from the RPI is carried out in a unified complex of works to ensure the EMC of the ACS, including the examination of the EMI, the standardization of parameters for the severity of the tests of the SU, the development of measures to ensure EMC [28].

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Fig. 1 Communication of relay protection and automation equipment with high-voltage equipment substation-220 kV of Almaty region

Sources of RPI are lightning strikes, switching and short circuits on equipment of open switchgears (OSG). Electromagnetic communication occurs in the presence of simultaneous electrical and magnetic influences between two or more electric long lines: overhead lines, buses and cables of the ASTU [29]. Short-circuit currents (SC), flowing through the wires of the SS busbar, cause a magnetic flux, which, according to the law of electromagnetic induction, excites the EMF of self-induction [29] in the circuits of the ACS, the value of which determines the level of the influencing RPI. In accordance with [30], the transfer of the RPI from the source to the receiver is carried out through an electromagnetic field by radiation. The type of cables used, the presence of a shield, have a particular effect on the noise immunity of the SU. The shielding factor of a cable is the ability of its shield to one way or another to reduce the level of the electromagnetic PI and is defined as the ratio of the residual noise to the value of external noise penetrating the RPI from the external environment [31]. In practice, the Kshld factor is a general shielding factor, it shows how many times, compared to a single wire, the radiated noise is attenuated by adjacent cores in the cable, adjacent cables in a cable duct or tray, metal structures of the cable duct, screens and cable sheaths.

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The shielding factor depends on the material, thickness and density of the shield. The curves of the dependence of the screening coefficient on the interference frequency were considered in [31]. The electromagnetic environment at the substation in terms of the RPI is determined by the characteristic sources of electromagnetic influences that can affect the SU equipment, including: – Transient processes in high voltage circuits during commutation by power switches and line disconnectors at outdoor switchgear; – Transients in high voltage circuits during short circuits, operation of arresters or surge suppressors. In this case, as a result of the high-frequency transient process of the discharge of the equipment and busbars to the ground, the radiation of the RPI occurs. The amplitude-frequency characteristics of impulse noise arising in cables vary in a wide range and depend on the route and length of the cables, the load at the ends of the cables. The frequency spectrum varies from tens of kilohertz to several megahertz. The amplitude of impulse noise can range from tens of volts to tens of kilovolts. The most important are: voltages and currents of transient processes arising during commutations, the mutual position of wires, buses and cables of the ASTU. The RPI levels at the SU ATCS are normalized depending on the severity of tests [13, 25] and depend on many parameters.

2 Methods To determine the levels of RPI, computational and experimental methods are used with the use of computer programs. All models describing communication through an electromagnetic field are built on the basis of the well-known Maxwell equations. The most commonly used form for numerical solutions is the theory of long lines [32, 33]. Using line theory, you can quickly and accurately solve problems associated with the electromagnetic interaction of cables and lines. In electrically long lines, voltages and currents cannot be considered independently of each other. They are connected to each other through the characteristic line impedance. In the time domain, a line is considered electrically long if the rise time of the pulses transmitted along it is of the same order of magnitude as the propagation time of the pulse along the line. In the frequency domain, a line is considered electrically long if the complex amplitudes of the voltage and current pulses depend on the location on the line. This effect occurs if the wavelength is of the same order of magnitude as the line length or less. To calculate the induced overvoltage in cable lines, the computer program “Interference” was used. The physical and mathematical model is described in [34–36]. The mechanism of induction overvoltage generation is described in [37]. For sections of the cable line located underground, the attenuation coefficient of the electromagnetic field can be used [38].

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To calculate RPI in cable lines, you need to know the shielding factors. In [37], a method is given for calculating the screening coefficients based on generalizations of the research results presented in [29]. The shielding property of cable screens, trays, channels and other elements is explained by the fact that the noise source induces a current in the screen, the electromagnetic field of which compensates for the electromagnetic interference field. To obtain a high screening coefficient, it is necessary to provide the lowest possible resistance of the loop through which the screening current flows [31]. At frequencies up to 10 MHz, the shielding current flows over the shield and closes in the ground through the shield grounding points. Only at frequencies above 10 MHz can the shielding current be closed through the capacitance between the shield and ground, as well as in the shield itself [31]. The frequency of interference on the territory of the substation, as a rule, is below 10 MHz, therefore, to ensure their effective shielding, it is necessary to ensure the minimum resistance of the loop through which the shielding current flows. This is ensured by reliable galvanic connection of the screen on both sides to the grounding device (GD) with conductors with minimum inductance. The method for calculating the screening coefficients is described in [30]. The frequency of the RPI during switching, breakdown of electrical insulation, operation of arresters in the primary high voltage circuits of the ORU-110, 220 kV is 50–1000 kHz. The equivalent circuit for calculating the screening coefficient is given in [13, 28]. Existing calculation methods give approximate estimates of the shielding coefficients, since it is difficult to take into account the set of various parameters inherent in cable lines: the cross-section of the conductors, the number of cores, the design of the screens, and the laying methods. Only the results of full-scale and simulation experiments in combination with numerical calculations of the shielding coefficient give a picture of the noise immunity of the SU RPA [28, 29]. To calculate the induced overvoltage in cable lines, the “Interference” program was used. In the program, the calculation of induced over voltages is performed using field theory and the theory of long lines [33–37].

3 Experiment Results To determine the values of the RPI on the ASTU devices, simulation tests were carried out at the 220 kV substation of the Almaty MES branch of KEGOC [11]. To simulate field impulse radiated noise, the routes for laying relay cables from the electrical equipment of 110 kV outdoor switchgear, 220 kV outdoor switchgear to the switchgear of the indicated substations were determined. For measurements, 1–2 samples were selected from each group of cables: cables with a screen, cables without a screen, cables in a sheath. On the selected section of the substation, parallel to the cable route, an insulated wire (1) was suspended at a height of 1.5 m relative to the ground level, simulating to the HV network (Fig. 2a). The control wire (3) was laid on the surface of the earth along the route of relay cables from the equipment of the outdoor switchgear-220 kV,

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Fig. 2 Scheme for simulating radiated pulse interference: 1—wire simulating the HV bus, 2—relay protection and automation cables, 3—control wire

the outdoor switchgear-110 kV, the AT-1 and AT-2 autotransformers to the building of the control room, where the switchgear 220/110 kV, the main control board and the DC board are located. Control wires were laid along the surface of the ground next to the route of laying the relay cables so that the length of the wire was approximately equal to the length of the cable (Fig. 2). Oscilloscope “FLUKE 199” and pulse voltmeter BI-5 M are alternately connected to the laid control wire and to the relay protection and automation circuits. A high-frequency pulse generator GBQI–4P is connected to wire 1. On the control wire and on the selected cables, with the generator turned off, the background noise values were measured with a BI-5 M pulse voltmeter or a FLUKE 199 oscilloscope. Then, at a fixed amplitude and frequency of oscillations of the current pulse GBQI–4P, measurements of the induced noise were carried out: on the control wire (Uwire ) and on the selected cables (Uloop ). Oscilloscope “FLUKE 199” and pulse voltmeter BI-5 M are alternately connected to the laid control wire (Fig. 2). Figures 3 and 4 show the results of measuring the shielding coefficient for existing cables of the relay protection and automation circuits. The measured coefficients of shielding of cables coming to the substation control house with the outdoor switchgear-220/110 kV vary from 1.1 to 8.2 [27].

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Fig. 3 The results of measuring the shielding coefficient at the 220 kV “Taldykurgan” SS current transformer cable L-159 (Kekr = 2.8)

Fig. 4 Results of measurement of the shielding coefficient at the 220 kV “Shu” substation. Control and signaling cable V-220 kV L-2733 (Kekr = 2.6)

The results of measurements and calculations of the values of the shielding coefficient (Kshld) and radiated pulse interference (RPI) at the substations “Taldykurgan”, “Stroitelnaya” (Construction), “Sary-Ozek”, “Zavodskaya” and PS-220 “Shu” of the Almaty region are shown in Table 1 (Figs. 5 and 6). Impulse radiated interference in existing relay protection and automation circuits during short-circuit and switching on outdoor switchgear-110/220 kV may exceed permissible levels. To ensure the required shielding factor and noise immunity of SMART ASTU, it is necessary to replace control cables with shielded ones and use cables with higher Kshld .

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Table 1 The results of measurements and calculations of values at substations in the Almaty region №

SS name

Kshld

RPI, kV

Min

Max

Max. when switching

Max. at short circuit

1

SS 220/110/10 kV Taldykorgan

1.3

8.2

2.55

1.92

2

SS-220/110/10/6 kV “Shu”

1.02

6.5

1.33

2.72

3

SS 220/35/6 kV “Construction (No. 120)”

1.1

2.8

0.4

3.954

4

SS-220/110/35/10 kV “Sary-Ozek (No. 126)”

1.0

6.7

0.6

5.07

5

SS-220/10 kV 1.1 “Zavodskaya (No. 149)”

2.8

0.3

3.086

Fig. 5 Values of the measured shielding coefficients Kshld

Fig. 6 Radiated pulse interference values

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4 Discussions The development of power grids and the technological energy efficiency of power supply systems is one of the factors of sustainable economic development [39–43]. The design processes and operation of power supply facilities should also contain a number of solutions and tasks to ensure and improve technological energy efficiency, which is a centered aspect on which the reliability of the power supply system depends [39, 42, 43]. The main attention in this work is paid to the electromagnetic compatibility of equipment and system elements, since it is also an important aspect in the issue of noise immunity of devices of automated systems for technological control of energy facilities [41, 44–46]. Electromagnetic compatibility in digital communications provides advantages such as increased efficiency, better transmission accuracy, better noise immunity, etc. [47, 48]. EMC assessment is important not only from a technical, but also from a legal point of view, since harmonized standards should contain detailed guidelines to allow the EMC assessment for typical application of devices in systems. The presented theoretical and experimental results [1–49], as well as the analysis results, can serve as recommendations for practitioners involved in interference measurements, and the proposed approach and method can be used as a basis for the development of reliable standards for assessing electromagnetic compatibility within the investigated frequency range interference. The overall shielding caused by adjacent conductors in a cable, adjacent cables in a cable duct or tray, metal structures of the cable duct and cable sheaths has a low shielding coefficient [27]. Computer calculations of RPI in secondary circuits during switching in primary circuits were carried out in the INTERFERENCES program “Simulation of impulse pickups and over voltages in branched cable lines”. The results of calculations of the maximum RPI indicate that the existing Kshld does not allow providing EMC of the SU RPA. Studies of electromagnetic influences in the devices of the ASTU SS of the Almaty region have shown that impulse noise in existing relay cables during short-circuit and commutations exceeds the permissible values. From the calculations carried out in the Interference program, it follows that the use of shielded cables allows reducing the levels of RPI in relay circuits to acceptable values and are necessary measures. Shielding factor is a characteristic of the ASTU cable and is used in calculations in the INTERFERENCES program. On real objects, the oscillation frequency of impulse noise can vary from tens of kilohertz to tens of megahertz. Measurements by the high-frequency pulse generator are performed at three frequencies. The results allow us to establish the dependence of the PI level on frequency. The measurement results when simulating the PI are reduced to the real value of the frequency of the high-frequency component of the short-circuit current. Researches allow to establish that for relay cables: at low frequencies in the range (50 ÷ 500) Hz, grounding of reserve wires of cables is more effective than grounding of screens; at high frequencies (0.5 ÷ 1) MHz and with aperiodic impulses (1.2 / 50; 8/20) µs—grounding of

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cable shields is more efficient; at frequencies (5 × 10−5 ÷ 10) MHz it is necessary to ground cable screens and reserve conductors on both sides [31]. From the works performed by a group of researchers [18–21, 29, 41, 44–46], it is possible to determine the potential of the topic: Risk-oriented provision of electromagnetic compatibility, scientific research in the field of electromagnetic compatibility. For the Republic of Kazakhstan, from the conducted research and modern solutions, we recommend considering the issue of the implementation/embedding of a risk-based approach in ensuring the reliability of microprocessor-based hardware SMART systems and technologies in an unfavorable electromagnetic environment at power facilities.

5 Conclusions Practical implementation of methods, models and recommendations used in the world makes it possible to provide a guaranteed level of noise immunity of devices in automated systems for technological control of energy facilities, which is a hypothesis and an uncertain parameter. The presented theoretical and experimental results in the direction of this article and research can be expanded and applied to carry out research work in power facilities on the territory of the Republic of Kazakhstan, in order to reduce emergency situations and prevent them. This paper presents the results of the analysis of the electromagnetic compatibility of devices at energy facilities of the Almaty region. As a result of the conducted research and diagnostics, it was found that the RPI in relay circuits during short-circuit and switching at the switchgear of the HV substation can exceed the permissible values for SMART relay protection and automation devices, which indicates their low noise immunity. It is also shown that an increase in the shielding factor of cables can reduce the levels of RPI on SMART devices to acceptable values, which will increase their noise immunity. Taking into account all the recommendations, the use of shielded cables is a necessary measure to ensure the EMC of SMART devices of SS ASTU. The results of experiments at the energy facilities “Taldykurgan”, “Construction”, “Sary-Ozek”, “Zavodskaya” and PS-220 “Shu” are summarized in Table 1, which shows the parameters of the screening factors of the existing control cables and the values of the pulsed radiated disturbances. Pulse electromagnetic interference affecting the operability of devices and elements of automated process control systems was determined in the 220 kV Taldykurgan, Stroitelnaya, Sary-Ozek, Zavodskaya substations and in the Shu substation 220 kV. It has been experimentally established that impulse noise in some relay circuits exceeds the permissible values during short-circuit and switching in the electrical network; solutions for their elimination have also been proposed. The actual screening factors of control cables under operating conditions are determined.

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Environmental Safety

Ecological Aspects of the Assessment of Safety Limits of the Near Surface of Radioactive Wastes in the Chornobyl Exclusion Zone Yuriy Olkhovyk , Sergey Mikhalovsky , and Andrew B. Cundy

Abstract The rationale for using a geochemical factor as an alternative to radiation exposure dose criteria for the assessment of the safety of the near surface disposal of radioactive wastes in the Chornobyl Exclusion Zone (CEZ) has been discussed. The absence of human habitation in the vicinity of the burial site (likely for thousands of years) eliminates the use of conservative reference exposure scenarios. Taking into account the impracticality of using conventional exposure dose criteria, and that the only factor that could potentially enable large scale migration of radionuclides beyond the boundaries of the disposal site is groundwater transport, we suggest to apply the geochemical safety criterion using a fixed value of the specific activity of an individual radionuclide in the groundwater present in the Quaternary subsurface deposits. This allows avoiding problems associated with large-scale uncertainties in the interpretation of the exposure dose and risk assessment. Keywords Radioactive wastes · Near surface · Radiation safety · Exposure dose · Risk uncertainty

1 Introduction In 1998, at the site of the industrial complex “Vector” located within the Chornobyl Exclusion Zone (CEZ), a radioactive waste storage facility was built. The purpose of this facility is the subsurface disposal of short-lived low and intermediate radioactive Y. Olkhovyk (B) State Institution «The Institute of Environmental Geochemistry of National Academy of Sciences of Ukraine», Kyiv, Ukraine e-mail: [email protected] S. Mikhalovsky ANAMAD Ltd, Brighton, UK Chuiko Institute of Surface Chemistry, National Academy of Sciences of Ukraine, Kyiv, Ukraine A. B. Cundy University of Southampton, Southampton, UK © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Zaporozhets (ed.), Systems, Decision and Control in Energy III, Studies in Systems, Decision and Control 399, https://doi.org/10.1007/978-3-030-87675-3_18

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wastes generated by the explosion of the 4th reactor of the Chornobyl nuclear power plant (NPP), and by the continued operation of the remaining reactors until their decommissioning. Analysis of the data available for the projected Vector storage capacity and of National Standards of radioactive waste classification allows the total waste radioactivity that could be buried in the Vector subsurface storage facilities to be estimated at 2 × 1016 Bq [1]. This figure does not account for the activity of ionising sources to be stored at the Centralised spent fuel storage facility, CSFSF, for long-term storage of spent nuclear fuel built at the Vector site with the support of the UK Department of Energy and Climate Change. It exceeds the activity of radioactive wastes accumulated through the operation of all NPPs in Ukraine to date. At present, the total activity of liquid and solid radioactive wastes stored at operational NPPs is ca. 2 × 1014 Bq, and the total activity of radioactive wastes stored at the Chornobyl NPP is ca. 4 × 1014 Bq, with a total volume of ca 22,500 m3 [2]. A specific feature of the operational radioactive wastes is the absence of long-lived radionuclides. Their activity is determined mainly by 137 Cs. Although the content of another radioisotope capable of migration, 90 Sr has not been determined, it can be estimated from the nature of the parent nuclei (137 Xe for 137 Cs and 90 Rb for 90 Sr) that the radiostrontium activity accounts for 10–20% of radiocaesium activity.

2 Literature Analysis and Problem Statement According to the International Atomic Energy Agency (IAEA)’s fundamental principles, the main goal of ensuring nuclear safety is human and environmental protection against harmful effects of ionising radiation [3]. It means that the primary aim of radioactive waste storage is its isolation from the biosphere with the help of natural and engineered barriers for the whole period during which this waste can remain dangerous for humans and the environment (including after the storage closure). During this period radionuclides could migrate from the storage site, dissipate in the environment and eventually increase dose to the surrounding population. The safety requirements for storage facilities after their closure are defined in the IAEA safety standards [4], and are based on the potential radiation dose and/or risk estimates to human health of the radionuclides migrating from the disposal facility in the environment as a result of possible natural processes or accidents. Safe disposal of radioactive wastes means a reasonable assurance that the longterm doses and risks to a representative member of public do not exceed the limits or constraints that were applied in the projected criteria. The safety criteria for the protection of humans and the environment after the storage facility closure are based on the calculated doses and/or risks for the population that could be exposed to ionising radiation in the future due to possible natural processes affecting the disposal facility [4].

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Current approaches and calculations to determine the radiological criteria for radioactive waste acceptance for storage are based on the anthropocentric methodology of safety assessment aimed at ensuring the protection of human health and the environment. The exposure dose is the traditional indicator of radiation safety in radioactive waste handling and particularly in estimating disposal safety. The latter in particular requires consideration of varying long-term scenarios. It is known however that the exposure dose is closely related to the habits of individual persons and the lifestyle of the population as a whole, which can only be foreseen for a limited period of time. Hence the doses and risks calculated for a distant future should be treated as abstract safety indicators rather than taken at face value and used for comparison with current quantitative parameters. The International Commission on Radiological Protection acknowledged that “…forecasts of health detriment over periods longer than several hundred years should be examined critically” due to uncertainties in estimating individual and collective doses over long periods of time, and these uncertainties increase with time [5].

3 The Purpose and Objectives of the Study The safety assessment of a real near surface storage facility requires establishing the limits for the radionuclide content in individual package units, as well as in the whole facility. In this paper a new approach for assessing the activity limits of the radioactive wastes disposed of in the near surface storage sites at the Vector Complex, using geochemical criteria, is described. The new approach does not contradict the methodology for deriving radioactivity safety limits described in the IAEA recommendations for assessing the near surface disposal facilities [6]. In addition to the use of the dose criteria, the ISAM (Improvement of Safety Assessment Methodologies for Near Surface Disposal Facilities) methodology recommends using environmental contamination levels such as groundwater radionuclide concentration and respectively, activity. The geochemical approach incorporates consideration of degradation processes in engineered barriers under the influence of environmental factors, and the protective (natural barriering or attenuation) properties of the natural geological environment which determine the radionuclide migration. Regarding the uncertainties associated with the safety assessment of the Vector disposal sites, it should be noted that the main (and arguably insurmountable) uncertainty arises from the unique nature and scale of radioactive contamination in the CEZ, where these near surface facilities are located. Never before the humankind faced with such a large scale contamination of residential areas by long living plutonium isotopes, which made the territories located within the CEZ uninhabitable for thousands of years. Within the framework of the Global Environmental Fund project in Ukraine [7], the State Agency of Ukraine on Exclusion Zone Management (SAUEZM) developed the draft Strategy of the conversion of the exclusion zone to the territory useful to the society for the period of 2021–2030. This strategy envisages the creation of a site for special industrial applications which aims at

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decommissioning of the Chornobyl NPP and transforming the object Shelter into an ecologically safe system. It has also elaborated the rules for handling and long-term storage and disposal of radioactive wastes. This site will occupy approximately 40% of the CEZ territory which is highly contaminated with 239 + 240 Pu alpha emitters. Due to their long half-lives, this territory will remain unsuitable for living practically forever. If the decision on assigning the status of territories that are eternally uninhabitable by humans to some contaminated sites in the CEZ will be reached, it will require creating or choosing an adequate methodology for assessing the safety of radioactive waste disposal on such territories. At present the existence of such territories has been acknowledged but their boundaries, and the criteria for taking such a decision have not been specified [8]. It should be emphasised that the unfitness for human habitation of the territories adjacent to the radioactive waste disposal site for at least a thousand of years is a consequence of the residual contamination caused by the Chornobyl NPP accident. Considering the current level of contamination of the so-called inner zone (within 5 km around Chornobyl NPP), where the near surface storage facilities are located, this situation will remain practically unchanged for hundreds of years (Figs. 1 and 2) [9]. Such levels of contamination eliminate any possibility for human dwelling in the inner CEZ for hundreds to thousands of years, which significantly exceeds the period required for lowering the radioactive waste activity in the storage facilities (which are dominated in dose terms by the medium lived (ca. 30 years half-life) 137 Cs and 90 Sr) to a safe threshold. These circumstances exclude the use of conservative reference exposure scenarios, which stipulate permanent residence and activities of the population on the contaminated territory mentioned in [10], in particular: – permanent residence on the territory within the 10-km inner zone; – various activities such as road construction, house building, use of water resources, drilling of wells through the storage space, etc. Taking into account the likely absence of permanent residents on the territories adjacent to the decommissioned near-surface storage facilities on the Vector site for at least a thousand years, the only likely cause of potential population exposure to the radioactivity present in the wastes is radionuclide dissipation in the environment, and migration with groundwater. An alternative route to the environment from discharge to surface water from the deteriorated near surface storage sites located at the Chystogaliv moraine ridge, where the CSFSF is being built, is unlikely to contribute long lived radionuclides such as plutonium isotopes [11]. In general, due to the combination of landscape features in the CEZ, the tendency for 137 Cs and 90 Sr migration with the surface waters is much lower than their migration through the vadose zone, which reaches 14 m in depth [11, 12]. Of the three main radionuclides contributing to the exposure dose at present, 137 Cs, 90 Sr and 239 Pu, the first two have half-lives T½ of 30 and 29.1 years, respectively and their long-term contribution will be continuously decreasing. 239 Pu with T½ = 24,065 years along with other plutonium isotopes will be the main contributor to dose

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Fig. 1 The level of contamination with Pu-(238 + 239 + 240), kBq/m2 , in CEZ in year 2015 (a) and year 2515 (b, forecast) (adapted from [8])

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Fig. 2 The level of average annual effective dose for adult population, mSv, in CEZ in year 2515 (forecast) (adapted from [8]). The 10-km (radius) zone is the territory around the NPP where the main work on the liquidation of the Chornobyl [9]

over the longer term. It has been established that the soil of the Chystogaliv moraine ridge creates a natural barrier preventing plutonium dissipation in the environment [11, 13]. Although the sands of the Chystogaliv moraine ridge have moderate or weak sorption capacity, they are characterised by a large content of finely dispersed clays such as montmorillonite and bentonite which have significant cation exchange and sorption properties. They retain plutonium mostly existing in the cationic form in the contaminated water via the cation exchange mechanism, with Kd > 4000 in sandy loam [13]. The potential migration of radioactive aerosols that can form as a result of a hypothetical fire on the site, even in the conditions of the uppermost isolation layer degradation, cannot make a significant contribution to the dose formation. The modelling of the secondary transfer of radioactive aerosols formed as a result of the largest most recent fire on 26–29 April 2015 beyond the CEZ showed that it was negligible, and the increase of the volumetric activity of 137 Cs and the radiation dose in the air outside the CEZ was not statistically significant. The additional expected effective dose of internal radiation of adults caused by the inhalation of radioactive aerosols formed by the fire in April 2015 in the CEZ did not exceed effective doses formed in normal conditions from the natural background radiation [14]. For the so-called “inner zone” which is located around the Vector Complex site, a hydrological model was developed in [15] (Fig. 3). According to this model, the radionuclides which flow from the Vector storage facilities underneath into the Quaternary aquifer will migrate to the north where they will discharge in the river Sakhan, atributary of the river Prypyat’, at a distance of 8000 m from the Vector

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Fig. 3 The groundwater levels and flow patterns in the Quaternary aquifer of the “inner zone” around the Vector Complex site (adapted from 14]). The groundwater levels above sea level are shown with contour lines and flow directions in the upper aquifer are shown with brown arrows

site. The groundwater travel rate in the Quaternary aquifer in the proposed model is < 40 m/year, and the travel time of migration from the Vector site to the point of discharge at the river Sakhan is between 210 and 340 years. The forecast of the average annual effective dose for adult population in the CEZ (Fig. 2) means that the possible discharge points of the contaminated water from the Quaternary horizon into surface waters of the river Sakhan or the Rodvyno watercourse would remain unsuitable for human habitation for more than 500 years. This estimate is based on calculating the dose intake with the standard amount of food and water consumption. The estimated travel time of water migration to the discharge points, 210– 340 years, does not take into account the ability of the soil to retain radionuclides by adsorption [15]. This mechanism will significantly slow down the radionuclide migration. For example, for 90 Sr it will take 4600 years and for 137 Cs—tens of thousands of years to reach the points of discharge, due to interaction with clay and other minerals in the subsurface, which allows to effectively eliminate them from longterm forecasts due to their relatively short half-lives [15, 16]. Further, the retention of plutonium isotopes and 241 Am by soil is much stronger; it has been estimated that it will take hundreds of thousands to millions of years for these isotopes to reach the

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surface waters [15]. In addition, water exchange between the Quaternary horizon and underlying, lower Eocene, strata is expected during contaminated water migration due to faults in the Kyiv marlstone, which further reduces the specific radioactivity of Quaternary horizon water, although the quantitative estimates of this factor are not possible due to the lack of data. Even if the contaminated water from the near surface storage facilities of the Vector Complex site reach the points of discharge in the environment, it will be so diluted with surface waters that no noticeable increase of radioactivity in the river Prypyat’ will occur. The average intra-annual runoff of the Sakhan river is 0.6 m3 /s, whereas it is 400 m3 /s for the river Prypyat’. This river is the main route of the radionuclides to the biosphere outside the CEZ. Thus, a formal approach to using dose criteria on territories without human habitation in the CEZ, which will practically remain unsuitable for living for hundreds to thousands of years could lead to paradoxical results, which set up activity limits for near surface storage facilities exceeding any realistic radionuclide activity accumulated in the radioactive wastes.

4 Research Methods and Results The proposed approach described here focuses on the potential impact of the ionising radiation on the environment rather than human beings, who will not be living on the affected territories [17]. The International Commission on Radiological Protection (ICRP) has developed a concept of Reference Animals and Plants (RAPs), which are representative species of flora and fauna living in different environments [18]. For near surface storage facilities such reference organisms could be certain invertebrates whose habitat coincides with the area of the facility such as snails or earthworms. In some countries, for example, Russia, the recommended threshold of acceptable ionising radiation dose for invertebrates (earthworms, bees, snails and other molluscs) has been established at 10 mGy per day [19]. However, if the radioactive wastewater leaves the storage site at depths exceeding the habitat of these representative species which is, for example, the case for the Radioactive Waste Disposal Site (RWDS) “Buriyakivka”, this approach is invalidated. The cases mentioned above lead to a conclusion that on the territories uninhabitable for humans for over a thousand years the main protective function against radioactive contamination will be served by natural barriers, mostly soils and underlying strata comprising horizons saturated with water. The role of engineered barriers will be limited to protecting personnel operating the storage facilities. This conclusion raises a question regarding the selection of criteria that could be used to define the safety of the near surface radioactive waste burial sites in the CEZ from the point of view of future generations. The forecasts of technogenic impacts on site personnel and the general population, and safety norms, should be meaningful and use physically correct models for

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the appropriate timescale. The ethical principles of our responsibility to future generations demand envisaging at least the same level of their protection against radioactivity as is provided today, and over the whole period of time when the radioactive wastes are considered dangerous. The experts of the Nuclear Energy Agency (NEA) recommend using alternative indicators, among which the most relevant to the scenario discussed in this paper, seems to be the concentration of radionuclides in the water in the biosphere [20]. Its calculation requires using models that take into account surface and subsurface conditions, and the degree of dilution of any contaminated waste plumes in the near surface aquifers. As the use of the population exposure dose criteria is irrelevant due to the predicted absence of human habitation and assuming that the only cause of large scale radionuclide migration beyond the disposal site is the groundwaters, it is rational to suggest geochemical criteria such as the fixed value of the specific activity of an individual radionuclide in the Quaternary horizon aquifer, which is measured in a well located down the groundwater flow gradient, or apply the concept of the radiation capacity to describe the geotechnical system (GTS) “disposal site—aeration zone—aquifer”. The radiation capacity is a measure of the equilibrium accumulation of the radioactive contamination in the GTS elements at the defined intake level of radionuclides. Even if their concentration in the water discharge does not exceed the permitted levels, the radiation accumulation could reach dangerous levels [21]. This determines the limits of the environment to remaining safe to its biosphere. Once a dynamic equilibrium between the source of radionuclide contamination, which is the degraded storage site in the GTS, and the absorbing media, which are the vadose zone and the aquifer, is reached, a constant level of the bulk activity in groundwater at some distance from the storage body could be established. In our opinion, the total activity of radionuclides at equilibrium in all GTS elements represents the ecological capacity of the GTS in question at respective migration parameters. Thus, the radioecological capacity for each individual radionuclide could be calculated providing the necessary input for defining the quantitative criteria of radioactive waste storage acceptance. It could be used as a basis for calculating the activity limits of radionuclides for site disposal. The dynamic equilibrium could change over time, due to prolonged changes in rainfall dynamics (from global heating), leading to changes in boundary conditions for the groundwater over time. The proposed approach to assessing the ecological capacity of the near surface burial site offers the most comprehensive description of the processes involved in the interaction between the contaminant and the environment. However the mathematical calculations of the ecological capacity of the storage site pose a significant challenge because of the lithological diversity of the territory where the storage facilities are located and due to the limited data on the sorption processes and mass exchange in the GTS elements. It may be necessary to use simpler models recommended by IAEA, which are compatible and proportionate with the available data. The insufficient quantitative data on the radionuclide migration in the GTS components may increase the uncertainty of forecasts rather than improve their accuracy if too complex models are applied.

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In any case, the safety assessment is carried out by comparing the calculated indicator data with the criteria defined in the regulatory documents. An alternative (and integrative) geochemical indicator allows avoiding obstacles faced by alternative approaches to assessing safety, the dose interpretation and risks in the conditions of unlimited and insurmountable uncertainties.

5 Discussion and Conclusions A risk assessment using different criteria is always associated with uncertainties that could cast doubt on the credibility of its conclusions. The majority of such uncertainties originating from insufficient knowledge of the characteristics of the storage site could be overcome by carrying out more detailed research aimed at getting real values of the required parameters such as subsurface porosity and permeability, groundwater flow rates and their variability, radionuclide concentrations in groundwater and seasonal variability. Reducing the impact of uncertainties on the accuracy of the safety assessment could be achieved by improving and developing the radioecological monitoring system of the storage facilities, in particular those located at the Vector Complex territory. It will ensure choosing the right criteria and justifying the safety assessment forecast for radioactive waste burial. The work presented here allows drawing the following key conclusions. 1.

2.

3.

The presence of a significant number of diverse radionuclides in the radioactive wastes planned for disposal at the near surface sites of the Vector Complex (within the CEZ) requires determining scientifically proven activity limits for their safe storage at its territory. Taking into account large variations of the parameters currently used to assess the protective properties of the main components of the engineered barriers, and the absence of quantitative assessment of their long-term degradation resistance after decommissioning of the radioactive waste storage facilities, it would be appropriate to use alternative geochemical criteria which do not rely on the exposure dose to the population. Considering the guaranteed absence of human habitation for thousands of years on the contaminated territory this will allow avoiding challenges associated with the interpretation of the doses and risks in the conditions of unlimited and insurmountable uncertainties. The most appropriate criterion for the safety and risks assessment appears to be the concentration of radionuclides in the water in the biosphere. It can be calculated using models which account for the conditions and degree of dilution and sorption of the contaminated water flows in the Quaternary aquifer present at the site.

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Environmental Hazards of the Donbas Hydrosphere at the Final Stage of the Coal Mines Flooding Yevheniia Anpilova , Yevhenii Yakovliev , Oleksandr Trofymchuk , Mykyta Myrontsov , and Oleksiy Karpenko

Abstract A military conflict takes place in the largest coal mining overloaded region in the world from the technogenic point of view, which is one of the largest and most dangerous natural technogenic geosystems (NTGS) with a high density of potentially hazardous objects (PHO). New natural and man-made hazards, unparalleled in world history, are emerging from the uncontrolled flooding of dozens of mines and dangerous irreversible changes in environmental parameters. An increase in the uncontrolled flooding of the coal mines in the Siversky Donets River Basin leads to an increase in the polluted mine water flow into the soil aquifer and rivers as local drains, which forms irreversible pollution of surface and underground sources in the boundaries of the Donbas coal mining area. Usage of water from the local water supply resources (dug wells, individual boreholes etc.), the health and sanitary state of which is often unknown or dangerous, increases in such periods. Besides, the top-down movement of rocks above the flooded workings causes the development of hydro-geomechanical shocks in the form of technogenic earthquakes. Such a military and technogenic vulnerability of the environmental state of the drinking and water supply sources (DWS) forms a high risk of emergencies of the water-ecological nature, including health epidemics. The increasing impact of water-environmental risks due to additional pollution of both surface and groundwater sources of drinking and domestic water, caused by flooding and inundation of geochemically contaminated catchment landscapes, must be taken into account. Taking into account the above-mentioned conditions in the Donbas NTGS, an indicative scheme of express investigations, that could allow evaluating the most vulnerable chain critical for the health and safety provision (HSP) in this region, was created.

Y. Anpilova (B) · Y. Yakovliev · O. Trofymchuk · M. Myrontsov Institute of Telecommunications and Global Information Space of the National Academy of Sciences of Ukraine, Kyiv, Ukraine e-mail: [email protected] O. Karpenko Taras Shevchenko National University of Kyiv, Kyiv, Ukraine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Zaporozhets (ed.), Systems, Decision and Control in Energy III, Studies in Systems, Decision and Control 399, https://doi.org/10.1007/978-3-030-87675-3_19

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Keywords Mining area · Geological environment · Post-mining · Flooding · Regional aquicludes · Environmental protection · Surface subsidence · GIS · Vulnerability of the environmental state · Pollutions

1 Introduction Protection of the water infrastructure is crucial for the life of millions of people. The UNICEF report states that in the countries affected by military conflicts, children under 15 years old have a three times higher risk of dying from diseases related to the poor quality of water and sanitary conditions than violence. For children under five years old, this risk is 20 times higher. Thus, according to the UNICEF data [1], since the beginning of this year, four attacks on the water infrastructure have occurred in the East of Ukraine. In total, starting from 2017 year, and 380 similar attacks have occurred and resulted in the death of 14 people during the maintenance works. At the beginning of May of this year, crucial water supply objects suffered from attacks. In the evening on May 5, four shells hit the territory and building of the pumping station, where the purified water is kept, near the borderline close to Donetsk. It caused disconnection of the water supply. Two more serious incidents occurred at the pumping station near Shumy, which provides water to about 3.1 mln inhabitants living on both sides of the confrontation line. Traditionally, back to the Soviet time, Donbas was one of the regions with the highest technogenic impact on the environment—emissions of harmful substances into the air, contaminated wastewater emissions into the natural water objects, and many waste disposal polygons. It was connected to natural-resource factors of the material production industries development (coal and chemical industries, mining and smelting complex, mechanical engineering), in which the region led, as well as historical peculiarities of the industrial complexes development. After all, in Donbas, the coal was intensively mined on a comparatively small area of 15 thousand km2 during 200 years. During this period, 20 bln tons of rocks were withdrawn from the subsoil, including 15 bln tons of coal, with the regional activation of the surface water flows into the aquifers depressing holes and into the coal workings of active and decommissioned mines. Simultaneous withdrawal of large volumes of coal and rocks has led to the regional deformations (mainly subsidence) of the ground surface. Nowadays, subsidence of ground surface at 1.5–2 m on average is observed on the territory of 8 thousand km2 . In total, 600 km3 have experienced geospatial deformation [2, 3]. The current level of the technogenic hazard of Donbas significantly resulted from the presence of potentially hazardous objects (PHO) on its territory. In 2009, 157 coal mines, 108 hydro-technical objects, 537 gas stations, 12 quarries functioned in the Donetsk region. There were also 11 railway stations, 115 bridges and overpasses, 1 overland tunnel, 13 main pipelines and pipeline connections in the Donetsk region. There were 69 coal mines, 66 hydro-technical objects, 247 gas stations, 3 quarries, 2

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railway stations, 13 bridges, 5 main pipelines and pipeline connections, 4 oil deposits in the Luhansk region. At the beginning of 2013, there were 3020 PHO in the Donetsk region (about 13% of their total amount in Ukraine or 114 objects per 1000 km2 of the territory), 1220 objects in the Luhansk region (5% of the total amount or 46 objects per 1000 km2 of the territory). In the Donetsk region, 1443 objects were explosive, 17—radiation hazardous, 522—flammable, 111—hydrodynamically dangerous, 22—biologically dangerous, 17 objects had the first level of chemical hazard, 63—the second level, 91—the third level, 69—the fourth level. In the Luhansk region, 717 objects were explosive, 7—radiation hazardous, 798—flammable, 65—hydrodynamically dangerous, 12—biologically dangerous, 6 objects had the first level of chemical hazard, 29—the second level, 43—the third level, 6—the fourth level. Natural and technogenic-disturbed conditions in the coal mining regions of Donbas at the current stage of functioning of the mining regions, urban-industrial agglomerations, objects of critical infrastructure, including surface and underground DWS systems, have specific peculiarities that are mainly developed under the influence of the numerous mines flooding factors. The following conditions can be attributed to the Donbas territory natural conditions peculiarities: 1.

Geological-structural: • diversity of forms and sizes of crease structures of the basin; • mines being a part of certain geomorphological elements with a high degree of the relief dismemberment in the areas of water collection development above the minefields; • lithological and filtration anisotropy of coal deposits in time and space.

2.

Hydrogeological: • dependency of underground drinking and low-mineralized water areas on the extension of rocks and layering of high and low permeable rocks that constitute aquifers of coal deposits; • reduction of their areas and territories as a result of activation of the interconnection of surface and deep groundwater at the beginning of flooding of unprofitable mines and quarries.

The following conditions can be attributed to the Donbas territory technogenic disturbed conditions peculiarities: 1.

Ways and system of coal deposits works: • presence of a very complicated system of mine workings in a wide range of depths with a large amount of closed and flooded mines; • extension of the scope of work in the plan; • long term (decades) operation of mines, predominant mining of coal layers without stowing of mining workings and active development of technogenic cracking.

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Hydro-geodynamic: • spatial and unstable nature of the underground water movement in the boundaries of active and closed mines; • presence of regional depressing holes that extend far beyond the boundaries of minefields; • presence of local depressing holes, that move together with the cleansing work of mines and areas of technogenic cracking above them; • enhancement of filtration feed with atmospheric sediments and surface water above the minefields due to cracking development after the layer collapse above the used treatment facilities; • increase of permeability of rocks in the areas of collapse above the mine workings layer; • absorption of surface water and its ingress into the mine workings in the areas of disturbed mountain massifs under the watercourses and water bodies with partial or total capture of the surface run-off; • formation of a new natural-technogenic structure of the underground flow within the boundaries of the “watermaker—river valley” filtration field after the total flooding of the mines and renewal of the depressing hole.

3.

Engineering-geological and engineering and seismic geological: • formation of troughs of the ground surface subsidence above the minefields (the sediment length sometimes reaches 3–4 m and more), in which flooded and waterlogged areas were formed; • consolidation (sometimes swelling) of rocks after a certain period of their collapse from the mine workings layer; • hydrostatic and dehydration compression of soil during depressing holes formation; • dense industrial and residential pressure above the majority of coal mining complexes with the additional technogenic feed of groundwater and formation of technogenic ground aquifers (so-called “water tables”); • development of areas of hydro-geomechanical pressure and technogenic earthquakes during accelerated subsidence of rocks above the flooded workings.

2 Literature Analysis and Problem Statement This article is a continuation of previous research by the authors [4–8] in the area of improving the efficiency of management decisions, environmental safety issues, development of appropriate software etc. Some ecological, informational and other aspects of monitoring system models are presented in [9–11].

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Land rehabilitation aimed at returning mined-out land to its former suitability for the adoption of new land uses and other aspects of post-mining have been considered by the authors [12–24]. Regional disturbance of the geomechanical balance of geological structures as a result of the withdrawal of large volumes of coal masses and disturbance of interconnection of surface and underground water in the area of active and slow water exchange has harmed the quality of surface water and ecological resource mode of the surface watercourses. The total resource of the surface water of Donbas is formed by the basins of the rivers Dnipro, Siversky Donets and small rivers of Pryazovia. Under natural conditions, the main feed of the Donbas river is received due to precipitation and, first of all, due to the spring melting of snow which provides 40– 80% of their run-off. Feed from the underground water is significant only within the boundaries of the Donets Hills where due to the deep embedding of the river valleys, aquifers are actively drained in the coal and layered deposits. Drainage (mine) water is crucial for the river run-off feed. The total amount of the water emissions into the surface water resources as of 1995 was about 25 m3 /s, and as of nowadays (according to the authors’ calculations) it was 24.2 m3 /s or 87.0 thousand m3 /h (Table 1). It must be noted that industrial objects located in Donbas dump about 70 m3 /s (252 m3 /h) of water into the rivers. At that, about 39 m3 /s (140 thousand m3 /h) of water is taken from the rivers for residential and industrial usage. Determination of the impact of coal mining companies on the formation of the surface run-off was based on the comparison of natural indicators of the rivers mode Table 1 Scale of the mining works at the main river basins of Donbas (as of 2012/2013) [7] Donbas river basins of the I order

Area of the impact of the minefields, thousands of km2

Volumes of the mine water emissions thousands of m3 /h

Coal mining (in 2013) mln of tons per year

Total number of mines with water drainage, active/closed

Module of the mine water emission, m3 /h per km2

Basin of the river Dnipro-Samara

1.59

14.70

18.17

31/4

0.8

Basin of the 2.29 Siversky Donets river

30.18

18.09

65/29

1.1

Small rivers of Pryazovia (the Kalmius river and the Mius river)

2.14

42.09

15.71

87/27

3.5

In total in Donbas

6.02

86.97

51.96

183/61

1.3

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in the non-disturbed state with such indices under the conditions of technogenic impact. However, solving this task was complicated by the following conditions: • Development of coal deposits in Donbas, which was accompanied by drainage of the mountain massif and emission of drainage water into the surface watercourses, started more than 130 years ago. • Yet, systematic study of the hydrological mode of the Donbas rivers was founded only in the second half of the 1940s of the last century, that is to say, that at that time, the technogenic constituent had existed and no attempts to single it out and evaluate it were made. As of today, there is no long-term systematic data on the volumes of the mine water emissions into the rivers. For the rivers of Donbas, an extremely wide range of values of losses is typical at different periods of the year. At this range of losses, the comparison issue is quite difficult, as the impact of the emission of mine water can be minor during the flood period and significant during the baseflow period. That is why only average long-term losses of rivers or losses of the 50% availability can be used for comparison. There is a large number of enterprises in Donbas using water-intensive technologies and providing significant volumes of emissions, which exceeds three times the emissions of mine water (up to 70 m3 /s or 252 thousand m3 /h is dumped from urban-industrial agglomerations). The canal Siversky Donets—Donbas has delivered up to 35 m3 /s of water since 1958, and the canal Dnipro—Donbas has delivered 45 m3 /s of water starting from the 1980s to cover residential needs. After using this water and irreversible technological losses a significant amount is dumped into rivers, totally ruining the water structure of the loss part of the river run-off. Unfortunately, all existing hydrological directories do not take into account these circumstances. They include only results of processing perennial materials without separating a technogenic constituent of feed. The stated circumstances somewhat reduce the reliability of the taken evaluations, however even based on their development the impact of enterprises of the coal mining complex on the formation of the surface run-off is significant in many cases.

3 Research Results In terms of the region, conditions of the accumulation of the resources and development of the underground water quality in the area of the Donbas coal mining complexes impact are complex and diverse. It is sufficient to compare the equivalent underground run-off under the natural and technogenic conditions in the boundaries of the mine drainage regional depression. At the average perennial value of the underground run-off module of the Donbas rivers under the low-disturbed conditions (middle of the XX century) 80 m3 /24 h km2 its value in the boundaries of the peak development of the regional depression reached 250 m3 /24 h km2 (beginning of the XXI century) or exceeded three times natural values.

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The analysis of the changes of the mine water inflows water balance structure and underground constituent of the river run-off in the boundaries of the regional depression demonstrates an increase in the rocks permeability and infiltration feed of the underground water as well as the not one-time flow of the mine water during the capture of the river run-off by mine workings. Under these conditions, auto- rehabilitation flooding of the mine workings and renewal of depressions will lead to the accelerated approaching of polluted mine water to the ground surface and an increase in its flow into the riverbeds relative to natural (background) values. The above said specifics of the ecological technogenic recovery of hydrogeological conditions of different areas of the Donetsk and Luhansk regions at the current stage of formation results from a complex interaction of geological (structure of water supplying rocks, their solubility, etc.), physical-geographical (amount of sediments, development of the river network, climate, etc.) and, in the last decades, technogenic (disturbance of the geomechanical balance of geological structures and coal massif, drainage by the mines, quarries, water intakes of the surface and underground constituents of the hydrosphere, infiltration of technogenic pollutions, etc.) factors. At the same time, it is worth noticing separate regional patterns in spreading the underground water, its resources and its quality. Thus, in the areas of shallow and open depositions of coal supplying rocks (“open Donbas—central and eastern parts of the Donetsk region and south-west part of the Luhansk region) development of fracturedporous rocks with sufficient washing of water supplying rocks to 100–200 m. of depth is characteristic. Northern regions of the Donetsk and Luhansk regions have a hydrogeological structure of the artesian basin with the surface development of aquifers in the loose sedimentary rocks. At that, the lower aquifers (3rd and 4th from the surface) contain mineralized (saline) water that determines the increased salinity of the mine water and its pollution impact on the rivers, springs, and wells and first from the surface ground aquifer. The aggregate data of the coal mining pressure in the hydrogeological regions and basins are presented in Table 2. Table 2 The aggregate data of the coal mining pressure in the hydrogeological regions and basins (1 January 2014) [7] Hydrogeological provinces, artesian basins (AB)

Area, km2 Region

Minefields

Towns, urban settlements

Towns, urban settlements over mines

Donetsk fold region

22,963

4219

2272

963.9 14.3

Dnipro AB

649

114.9

Dnipro-Don AB

311

–

–

–

Donetsk-Don AB

4343

28

281.5

21.9

Hydrogeological Province of the Ukrainian Shield

3931

–

48.5

–

789

–

12.2

–

4896

2729

1000.1

Black Sea AB Total

8743

41,080

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The main natural-technogenic factors of drinking underground water resources formation in the boundaries of the underground water basins of Donbas and its interaction with the surface run-off is the specifics of structural-tectonics construction, relief dismemberment for the areas of active water exchange taking into consideration water intake areas and boundaries of geological structures for the areas of slow water exchange and impact of the mine water drainage on the formation of depressing sinkholes and emissions into the river run-off of the Siversky Donets, Luhan, Kalmius rivers, etc. The critical complication of the water-ecological constituent of the civil defence and limited reliability of the exploitation of the DWS systems is connected to the fact that 18% of the minefields are built up—more than 1 thousand km2 of the territory of 63 cities and 91 small towns of Donbas (25 and 51% of their area is explored accordingly) are located above the minefields. Today, under the conditions of the fragmented nature of the water sources ecological monitoring system in the area of the Joint Forces Operation and lack of funding, the threat of polluting impact of toxic leachate emissions from the industrial and domestic waste polygons, mainly located in the lowered forms of relief with the low natural protection of groundwater from pollution, increases. The Table 3 presents the critical parameters for monitoring the Donbas environment in the post-mining and military conflict phases. GIS and Remote Sensing methods have been used by the authors for anthropogenic impact assessment, environmental monitoring and management (Fig. 1). Mining activities, in old mining areas, are often carried out simultaneously by several mines at different depths in the same area, with a growing number of mines that have flooded uncontrollably since the outbreak of armed conflict. These circumstances significantly increase the occurrence of the following risks: contamination of surface systems of drinking and supply water sources, engineeringgeomechanical and seismic-geological deformations, destruction of surface technological complexes. Nowadays, complex changes occur in the system of the water run-off of the river basins and local basins of the underground water. They are conditioned by the increase of dissipated polluted water run-off from the flooded mines. Determination of the forecast impact of the coal mining enterprises on the formation of the surface run-off, primarily of the Siversky Donets river basin, which is going to be the main source of the DWS for long, is complicated because of the reduction of the regional water-ecological monitoring level, including: – no long-term systematic data on the volumes of the mine water emissions into the rivers; – significant unsteadiness of the Donbas rivers losses in different seasons of the year, which increases the relative part of inflows of the mine water and industrial run-offs and raises the level of pollution of the surface water resources as DWS sources; – the natural wide range of dissolved salts content is characteristic for the rivers of Donbas, especially within water catch area of Pryazovia. Mineralization of water in them is changed from 0.2–0.3 g/l during the flood period to 3.5–5.0 g/l

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Table 3 List of critical parameters of the Donbas environment monitoring at the stage of postmining (flooding of mines) and military conflict The main technogenic sources of the environmental impact

Factors of the environmental impact

Source of environmental impact data

Subsidence, elevation of the ground surface, cracking, disturbance of foundations, loss of Mine depressing of the endurance of rocks, increase of underground water geochemical and gas Areas of the flooding pollution, seismic of the lands quakes Areas of geochemical and gas-geochemical surface anomalies

Regional, territorial, object ecological monitoring of the natural environment Environmental survey (investigation) Analysis of remote materials

Large technogenic objects

Factor forming elements of technogenic objects

Mines

Deformation of the ground surface, technogenic earthquakes

Urban-industrial agglomerations

Air emissions, leakage Flooding of residential of water and heat from and industrial massifs, the networks, etc. chemical pollution of landscapes

Potentially hazardous objects, hazardous chemical wastes, waste pits

Air emissions, water emissions, accumulation of toxic waste

Chemical pollution of landscapes, lower layers of atmosphere, surface run-off

Nature preserve fund (biodiversity)

Loss of biodiversity

Limited potential of the region recovery

Ecological investigations Analysis of remote materials Materials of cadasters of natural resources

during the baseflow period. During the mine water mineralization 2.0–4.0 g/l its dumping into the surface water flows can have a prevailing negative effect taking into consideration reduced surface run-off during the summer-autumn and winter baseflows; – even under the current conditions, there is a big amount of enterprises with water-intensive technologies and significant volumes of emissions in Donbas, which, according to the preliminary estimates, can exceed the volumes of the mine water emissions thus forming significant risks of emergencies of the water-environmental origin. According to the existing schemes of the water usage, after its use, mine, industrial and residential wastewater and its irreversible technological losses flow into the rivers, totally changing the resource and hydrochemical indicators of the river runoff. Unfortunately, as a consequence of the surface and underground water monitoring system destruction, the existing current hydrogeological data do not allow taking into account the impact of these factors with certainty.

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Fig. 1 Main factors of the technogenic impacts on the environmental components of the Donbas region

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4 Conclusions Thus in the authors’ opinion, the supreme measures of improving the system of the environmental monitoring of the surface and underground hydrospheres and engineering and seismic geophysical conditions of the coal massif provided the mines flooding, and continuation of the armed conflict are: 1.

2.

3.

4.

5.

6.

Urgent creation of the international scientific-methodical expert committee on the development and reconstruction of the environmental monitoring of the hydrosphere, taking into account the experience of Ruhr (Germany) and Wales (Great Britain). Development and implementation of the new complex of methodical and directive documents on the monitoring of the surface and underground hydrosphere providing the final flooding of mines and continuation of the armed conflict. Preparation of the informational bulletin on the evaluation of the current environmental state of the surface and underground water of Donbas and areas of improvement of the hydrosphere monitoring. Complex ecological investigation of the surface and underground water of Donbas and evaluation of threats to the civil defence of the water-ecological origin. Scientific substantiation of the maximum permissible changes and technogenic pressure on the surface and underground hydrospheres of the coal mining area of Donbas. Transition of the drinking and water supply systems to the prevailing usage of the water resources protected from the surface pollution of the underground water of drinking quality.

References 1. New UNICEF report highlights scale and impact of attacks on water and sanitation facilities on children in conflict-affected countries. https://www.unicef.org/ukraine/press-releases/newunicef-report-highlights-scale-and-impact-attacks-water-and-sanitation. 2. Udalov, I.V.: Transformation of the geological environment under the influence of man-made processes (in the conditions of the north-eastern Donbass), Kharkiv, KhNU. Karazina, 176 p. (In Russian) (2016) 3. Liuta, N.H.: Ekolohichnyi stan dovkillia ta yevropeiski perspektyvy Ukrainy. Ecological state of the environment and European prospects of Ukraine. Mineral resources of Ukraine, 1, 6–11 (In Ukrainian) (2011) 4. Myrontsov, M., Karpenko, O., Trofymchuk, O., Okhariev, V., Anpilova, Y.: Increasing vertical resolution in electrometry of oil and gas wells. Systems, Decision and Control in Energy II. Studies in Systems, Decision and Control, vol. 346, pp. 101–117 (2021). https://doi.org/10. 1007/978-3-030-69189-9_6 5. Trofymchuk, O., Yakovliev, Y., Anpilova, Y., Myrontsov, M., Okhariev, V. Ecological situation of post-mining regions in Ukraine. In: Zaporozhets A., Artemchuk, V. (eds.) Systems, Decision and Control in Energy II. Studies in Systems, Decision and Control, vol. 346, pp. 293–306 (2021). https://doi.org/10.1007/978-3-030-69189-9_17

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Complex Oxygen Regimes of Water Objects Under the Anthropogenic Loading Inna Skurativska , Sergii Skurativskyi , Oleksandr Popov , Deineka Viktoriia , Eduard Mykhliuk , and Maksym Dement

Abstract Under conditions of intense anthropogenic loading in combination with climate change, the significant degradation of water resources of the planet, especially surface waters, is observed. The surface waters are the environment where a vast part of natural ecosystems dwell, and provide the major source of drinking water. They are also an extremely important element of technological processes in industry, agriculture, and etc. Therefore, the assessment, control, and forecasting of surface water quality are in a focus of many scientific investigations. This report deals with the oxygen regime of water objects (basins of self-purification) which is studied by the methods of mathematical modeling. The oxygen regime description consists of the balance equations governing the dynamics of oxygen, i.e. biochemical oxygen demand (BOD) and dissolved oxygen (DO), phosphorus, and phytoplankton. The resulting nonlinear dynamical system is studied by the numerical and qualitative analysis methods. It is shown that the model possesses the steady solutions in a vicinity of which the nonlinear periodic regimes can occur. When the parameters of nonlinearity vary, the periodic regimes lose their stability and multiperiodic regimes appear. Among complex system’s solutions, there are also chaotic regimes. Furthermore, we developed the model for the river system which consists of coupled dynamical systems describing the bio-chemical processes in two connecting self-purificating basins. This model possesses the nonlinear periodic regimes as well.

I. Skurativska (B) · S. Skurativskyi Subbotin Institute of Geophysics of NAS of Ukraine, Kyiv, Ukraine S. Skurativskyi · O. Popov State Institution, The Institute of Environmental Geochemistry of National Academy of Sciences of Ukraine, Kyiv, Ukraine O. Popov Pukhov Institute for Modelling in Energy Engineering of NAS of Ukraine, Kyiv, Ukraine Interregional Academy of Personnel Management, Kyiv, Ukraine D. Viktoriia · E. Mykhliuk · M. Dement National University of Civil Defence of Ukraine, Kharkiv, Ukraine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Zaporozhets (ed.), Systems, Decision and Control in Energy III, Studies in Systems, Decision and Control 399, https://doi.org/10.1007/978-3-030-87675-3_20

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Keywords Modified Streeter–Phelps equation · Water pollution · Basin of self-purification · Nonlinear dynamical systems · Bifurcations · Chaos

1 Introduction Water is a good solvent and can transport substances which adversely affect biological systems, including human health, over long distances through river systems. Due to its geographical (geological, relief) location, water objects are the endpoint of the waste water accumulation of the most of man-made substances formed in a catchment area. At the same time, natural waters are in constant circulation and actively interact with the main components of the biosphere, atmosphere, lithosphere, aquatic, and terrestrial ecosystems [1–3]. Therefore, hydro-ecological features of surface water quality serve the natural indicator of ecological welfare of both aquatic ecosystems and the entire catchment area [4]. Since surface runoff is formed in the catchment area, where different pollution sources (agricultural complexes, industrial facilities, mining facilities [5]) are located, it becomes obvious that to assess water quality [6–8] we should quantify the impact of these facilities, find out the nature of pollution and ways to overcome the negative impact of polluting components (involving natural factors) on the ecosystem’s elements. Determining the oxygen regime of surface waters is an important component of assessing the state of aquatic ecosystems and catchment areas. Dissolved oxygen ensures the viability of living organisms and the self-purificating function of the aquatic ecosystem. The participation of oxygen in the processes of biological, chemical and physical–mechanical self-purification of water objects indicates that the assessment of the concentration of dissolved oxygen is of great practical and general scientific importance [9, 10]. Taking into account the complexity of conducting a series of ecological experiments, the synergy of ecosystem components [11, 12], the lack of comprehensive information about their interaction, the tasks of estimating, predicting, and controlling the oxygen regime using mathematical modeling remain insufficiently studied [13–18]. Especially this concerns the phenomena of self-organization inherent in ecosystems. To take into account the nonlinear and cooperative effects [19, 20], we generalize our previous investigations [13–18] of the processes of oxygen regime formation. Therefore, the purpose of these studies is to classify the nonlinear oxygen regimes that can be observed in the target system and to establish their bifurcations.

2 Mathematical Model for the Oxygen Regime Description The surface water quality [21] of a water object is characterized by the oxygen dynamics, which in turn essentially depends on different biogenic elements (here we consider the phosphorus dynamics) and the phytoplankton behavior [4, 19]. In

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particular, the oxygen transformations are characterized by the biochemical oxygen demand (BOD) representing the indicator of the total organic content reacting with oxygen [22, 23] and dissolved oxygen (DO), i.e. amount of oxygen containing in water [24]. For modeling BOD and DO dynamics, it is known the Streeter and Phelps model [6, 7] and its modifications [6, 22, 25–27]. These models are mostly linear with large number of parameters and do not take into account accompanied processes affecting the oxygen regime. Therefore, to develop the mathematical model describing the dynamics of the quantities BOD, DO, phytoplankton, and phosphorus, we construct the balance equations for the corresponding concentrations C B O D , C D O , C PhT and C P . At first, consider the processes governing the incoming and outcoming oxygen fluxes. Sources of oxygen supply primarily include: re-aeration (invasion), i.e. saturation (enrichment) of water with oxygen; oxygen supply with external water flows from a catchment area; oxygen replenishment due to photosynthesis of algae and higher aquatic plants. Oxygen consumption items that cannot be neglected in the analysis of the oxygen regime of aquatic environment include evasion, i.e. the process of oxygen transfer from water to air and is based on the same physical principles as the invasion process; destruction of organic compounds (respiration of microorganisms); oxidation of inorganic compounds with the formation of oxides, such as the process of nitrification, the spending on respiration of aquatic organisms. Closely related to the dynamics of BOD and DO is the phenomenon of reservoir eutrophication representing the water enrichment with nutrients (mainly phosphorus and nitrogen), which cause the growth of primary organic matter production due to intensification of metabolic processes in algae and higher aquatic vegetation. The main indications of reservoir eutrophication are the predominance of production processes over destructive, which is accompanied by a significant increase in nutrients. This phenomenon, in turn, provokes the growth of biomass of phytoplankton, phytobenthos, filamentous algae to the level of water “blooming”, reducing the concentration of dissolved oxygen. On the one hand, the growth of phytoplankton biomass is a positive phenomenon because the forage base for aquatic organisms and oxygen level increase. However, there comes a moment when the balance between the growth of algal biomass, the formation of organic matter, oxygen and destructive processes is disturbed. Ultimately, the physicochemical properties of the environment change: the level of nutrients and organic matter increases, the level of dissolved oxygen decreases, anaerobic zones appear and expand, turbidity increases and water transparency decreases, and so on. As a result of the activity of algae and their decomposition, the oxygen content in the water decreases and the concentration of toxic substances increases, which causes mass death of fish and invertebrates [4]. However, the process of reservoir eutrophication is determined not only by the accumulation of nutrients, but also by the degree of water exchange, the reservoir depth and volume, and the oxygen saturation degree. Therefore, the process of eutrophication is considered as a result of the interaction of biotic and abiotic factors, which gives it an ecosystem character,

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and its study requires a systematic approach and the use of methods of mathematical modeling. Thus, the resulting mathematical model for the oxygen regime dynamics reads as follows: dC B O D dt dC D O dt dC PhT dt dC P dt

QBOD  q  − k1 (C D O ) + C B O D + m 1 C PhT − λC B O D C D O W W  Q DO q  = + k2 C ∗ + αC PhT − k2 + C D O − k1 (C D O )C B O D W W Q PhT = + [k3 (C D O , C P ) − m]C PhT , W QP q = − n 1 C PhT + n 2 C B O D − C P , W W =

(1)

where Q B O D ,Q D O ,Q PhT , and Q P denote rates of addition of corresponding component from the outside; W is the volume of water object; q stands for the incoming flow; C ∗ is the oxygen concentration at its saturation; the function k1 (C D O ) is the rate for oxygen consumption by BOD; k2 is the aeration coefficient; the function k3 (C D O , C P ) and m are the rate of phytoplankton growth and death, respectively; m 1 = γ m, γ stands for the fraction of dead phytoplankton for oxidation;αC PhT is the rate of oxygen adding during photosynthesis;λ is the coefficient of non-conservativity at the nonlinear interaction of organic substance and oxygen; n 1 is the rate of phosphorus depletion due to phytoplankton; n 2 is the rate of phosphorus production from organic substance (detritus). To identify the functions k1 (C D O ) and k3 (C D O , C P ), the Michaelis–Menten kinetics [6, 28] is used. In particular, k1 (C D O ) = k1

Cn CDO CP and k3 (C D O , C P ) = k3 n D O n · . K DO + CDO Kν + C DO Kμ + C P

Here the parameters k1 and k3 are the constants of saturation defining the behavior of functions at infinity, n is an empirical paramet88er. The parameters K D O , K ν , and K μ stand for the half saturation constants. It should be noted that system (1) can be regarded as a generalization of the wellknown Streeter–Phelps equations for the assessment of water quality [6, 7, 27, 29]. Next, let us write system (1) in dimensionless form applying the scale transformation {C B O D ; C D O ; C PhT ; C P } = C D O {x; y; z; u}, where C D O  is a charactristic value of DO concentration; x, y, z, u are the dimensionless quantities. Thus, utilizing the notatins

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  Q DO Q PhT QBOD 1 QP + k2 C ∗ = B, = A, = F, =C C D O  W W C D O  W C D O  W C D O  q K DO Kν Kμ = S, λC D O  = λ, = K DO , = K ν, = K μ, C D O  C D O  C D O  W and dropping the bar over the symbols, the final form of dynamical system is as follows:   dx k1 y = A− + S x + m 1 z − λx y, dt KBOD + y dy k1 y = B + αz − [k2 + S]y − x, dt KBOD + y   dz yn u = F + k3 n − m z, · dt Kν + yn Kμ + u du = C − n 1 z + n 2 x − Su. (2) dt Now we are going to consider the solutions of autonomous system (2) using the qualitative analysis method [30]. According to the method, at first, the fixed points of system (2) should be derived. Then the stability of these points is considered in the linear approximation. In particular, we are interested in the conditions of lose of stability for the fixed points. These conditions provide the constraints for the model’s parameters at which the development of nonlinear periodic solutions (limit cycle) can be possible. To validate the limit cycle appearance, the direct numerical integration of system (2) is used.

3 Stationary Solutions of the Mathematical Model Describing the Oxygen Regime Thus, the fixed points of system (2) representing the steady oxygen regimes in water objects satisfy the nonlinear algebraic system  k1 y + S x + m 1 z − λx y = 0, KBOD + y k1 y x = 0, B + αz − [k2 + S]y − KBOD + y   yn u F + k3 n − m z = 0, · Kν + yn Kμ + u 

A−

C − n 1 z + n 2 x − Su = 0.

(3)

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To get solutions of this nonlinear system, the iterational numerical Newton’s process is used. In the matrix form, the numerical procedure is described by the sequence:   X p+1 = X p − J −1 X p G X p , p = 1, 2, . . . , where J −1 is the inverse matrix for the matrix of linearization [31] 

 ⎛ ⎞ k1 x − S + λy + K BkO1Dy+y − λx + (KK B O D+y) m1 0 2 B O D 

⎜ ⎟ ⎜ k1 x − K BkO1Dy+y α 0 ⎟ − S + k2 + (KK B O D+y) 2 ⎜ ⎟ BOD J =⎜ ⎟ nk3 K νn uy n−1 z ⎜ ⎟ 0 J J 2 33 34 ⎝ ⎠ ( K μ +u )( K νn +y n ) n2

0

(4)

−n 1 −S

Here J33 = −m + 

k3 uy n K μ k3 zy n  , J34 =  . 2  K μ + u K νn + y n K μ + u K νn + y n

Using Newton’s procedure, let us consider the dependence of coordinates of fixed points on the model’s parameter variation. Taking into account that the nonlinearity is related with the variable y while other variables are incorporated into system (2) linearly, the steady solutions is useful to study via the method of parameter mapping [30]. To do this, from the first and the last equations of system (3) we get the following expressions: x = a1 + a2 z, u = b1 + b2 z, where  Kμ + y A Kμ + y m1 , a2 = a1 = K μ (S + λy) + y(k1 + S + λy) K μ (S + λy) + y(k1 + S + λy) C + n 2 a1 n 2 a2 − n 1 , b2 = . b1 = S S 

When these expressions are inserted into the third equation of system (3), it is obtained the quadratic equation with respect to z:      F K νn + y n K μ + u + k3 y n u − m K νn + y n K μ + u z = 0, or H1 z 2 + H2 z + H3 = 0,

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where       H1 = b2 k3 y n − m K νn + y n , H2 = b2 F − m K μ K νn + y n + b1 k3 y n − m K νn + y n ,   n H3 = F b1 + K μ K ν + y n .

Its roots are as follows:   −H2 − H22 − 4H1 H3 −H2 + H22 − 4H1 H3 z− = , z+ = . 2H1 2H1 Note that the right parts of the relations derived depend on the variable y only. Thus, inserting into the second equation of system (3) the expressions for x and z, we lead to the relation with respect to the variable y and the model’s parameters. Solving the resulting equation for α, it is easy to get the following expression α = (k2 + S)

B yx y + k1 − , z z (K B O D + y)z

(5)

where z should be replaced by z + or z − and x is given above. In what follows, we fix the parameter values: A = 0.933, B = 4.974, C = 2, F = 3, k1 = 1.5, k2 = 0.7, W = 1, q = 0.2, n = 0.44, K B O D = 4.3, K μ = 1.14, K ν = 0.296, λ = 0.7, k3 = 4.65, n 1 = 0.65, n 2 = 0.9, m = 2.3, m 1 = 0.45.

(6)

Using these parameters, one can depict the function α(y) in Fig. 1, where the profile of α(y) at z = z − is shown by the solid curve and α(y) at z = z + —by the dashed curve. From Fig. 1 it follows that, depending on α, we can derive from one to four roots of system (3). In particular, at α = α0 = 0.05 (Fig. 1) there are three points lying in the curve derived at z = z − and one root belonging to the curve plotted at z = z + . The coordinates of three fixed points now can be evaluated by Newton’s method with the accuracy 10−5 and then the eigenvalues of the linearized matrix J can be obtained as well. Thus, the coordinates of Point I: (123.30026; 0.27971; 131.47888; 137.54482) and corresponding eigenvalues ξ = (−39.42; 0.031 ± 0.33i; −0.17) of the matrix J at this point. Since there is the complex-valued eigenvalues with positive real parts, the fixed point is an unstable focus. Point II: (0.54449; 5.22832; 3.59295; 0.77313), the eigenvalues ξ = (−4.69; −0.56 ± 1.48i; −0.84). Since all real parts of ξ j are negative, then the fixed point is a stable focus.

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Fig. 1 The graph of function α(y) defined by relation (5). Here y1 , y2 , and y3 are the y-coordinates of the fixed points I, II, and III, respectively

Point III: (68.77574; 0.42813; 95.05617; 10.55825), the eigenvalues ξ = (−21.43; −0.16 ± 1.19i; 0.16). Since there is a positive eigenvalue and pair complex-valued ones, then the fixed point is a saddle-focus. It is worth noting that the dependence α(y) allows one to consider the influence of the parameters of nonlinearity λ and n on the position of the roots of system (3), i.e. the shapes of curves in Fig. 1. In particular, when the parameter n increases, i.e. n = 0.8 > 0.44, the curves (Fig. 2a) corresponding to z = z + are stretched along Oα. This causes the growth of interval of α, where there is more than one root of system (3). When the parameter λ decreases from 0.7 to 0.2, the curve corresponding to z = z − displaces along vertical axes (Fig. 2b). To get an idea of the complex structure of the phase space, the analysis of the convergence of Newton’s method to the solutions of system (3) is carried out. We choose the initial conditions for running iterations of Newton’s method from the domain  = {x, u|0 ≤ x ≤ 300, 0 ≤ u ≤ 250}. Let us restrict the case with z = z + . Then, as a result of finding the solution of system (3), we obtain one of the three possible roots of the system. Thus, each starting point for Newton’s method is uniquely associated with a certain root of system (3), which is denoted by a shade of black (Fig. 3a). The structure of the set of points and enlargement of the selected domain (Fig. 3b) tell us about the fractal nature of the resulting structure. After derivation of the coordinates of stationary points, the type of fixed point can be found out by studying the eigenvalues of the matrix of the linearized system. The most interesting cases are related to the changes of the point’s types. These changes in turn define the bifurcations of the system’s phase space. In particular,

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Fig. 2 The positions of roots of system (3) depending on the parameter n (a) and the parameter λ (b)

we are going to consider the most intriguing case when the real parts of the pair of eigenvalues change their sign from minus to plus. This case is called the AndronovHopf bifurcation and corresponds to the birth of a periodic regime in a vicinity of a fixed point. To identify the Andronov-Hopf bifurcation, we fix the parameter values as above except for the parameter λ. When λ varies, all eigenvalues ρ of the matrix J are evaluated simultaneously with the coordinate of the fixed point I. The resulting dependence of Re(ρ) on λ is depicted in Fig. 4 (solid curve). From the analysis of

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Fig. 3 Diagram of the results of convergence of Newton’s method iterations. Shades of black indicate the initial conditions under which Newton’s method coincides with one of the three solutions of system (3)

Fig. 4 The dependence of the real part of complex-valued eigenvalues of the matrix J on the parameter λ evaluated for the fixed point I

this figure it follows that at increasing λ the eigenvalue’s real part intersects horizontal axes at about 0.34. It is important to estimate the position of the Andronov-Hopf bifurcation when other parameters vary. In particular, when the parameter of nonlinearity n changes from 0.44 to 0.46, the zero of the function Re(ρ) moves to the left, i.e. the moment of the periodic regime displaces toward the lower values of λ (dashed curve in Fig. 4). Thus, if we take λ a little bit larger than 0.34, we can expect to observe the stable limit cycle appearance.

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To make sure that periodic oscillations appear, when λ grows after the AndronovHopf bifurcation, we perform the numerical integration of system (2) in a vicinity of the fixed point I. It turned out that, indeed, it is observed the oscillations producing the limit cycle in the system’s phase space. Moreover, their amplitude increases when λ grows.

4 Periodic and Chaotic Oxygen Regimes Now we consider the limit cycle development in more detail at the variation of the parameters n and λ. To evaluate the limit cycle, we choose the initial data for the numerical integration of system (2) close to the fixed point I at n = 0.44. After omitting the transient processes, the system approaches the periodic regime, the phase portrait’s projection of which is depicted in Fig. 5a. When n = 0.48, the double period bifurcation occurs and double limit cycle (Fig. 5b) exists. The result of another period doubling bifurcation is shown in Fig. 6a representing the quadruple limit cycle at n = 0.50. Finally, at n = 0.56 there is a chaotic attractor (Fig. 6b) in the phase space. To find out the geometric structure of the chaotic attractor, the Poincare section technique can be used. To apply it, we choose the section plane : z = 150 and evaluate the points of intersection of the plane and a trajectory. In particular, at n = 0.54 in this plane the set of points forming the localized islands (Fig. 7a) is observed. When n = 0.56, the points of intersections form the strip (Fig. 7b) of fractal structure. Such changes in Poincare section structure indicate the occurring of bifurcations in the chaotic attractor during the parameter n variation. The fact that we observed the “shuffled” narrow strip means that the geometric dimension of the chaotic attractor is measured not by an integer but by a fractional number [30]. Using the same technique, it is useful to analyze the bifurcation diagram describing the development of the stable attractors at increasing n. For diagram construction,

Fig. 5 The projections of the phase portraits on the plane (x; z) of the limit cycles at n = 0.44 (a) and n = 0.48 (b)

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Fig. 6 The phase portraits of the limit cycles at n = 0.50 (a) and n = 0.56 (b)

Fig. 7 The Poincare sections of the chaotic attractors existing at n = 0.54 (a) and n = 0.56 (b) (here λ = 0.7)

we put the parameter n along horizontal axes, while the values of x-coordinates of intersection points along vertical one. Thus, we construct the Poincare diagram (Fig. 8a), when λ = 0.7 and n grows from 0.44, when the limit cycle exists and produces the leftmost point in Fig. 8a, to 0.64 corresponding to another limit cycle existence. This diagram shows the moments of period doubling bifurcations and chaotic attractor development. There is also the separated “window” of solutions with quadruple period. Note that this “window” disappears, when λ increases to 0.72 (Fig. 8b). Instead of the developed chaotic zone, we observe the structure which is inherent to multiperiodic regimes obeying the period-doubling scenario. In the similar spirit, the Poincare diagrams are constructed at lower λ, i.e. λ = 0.68 (Fig. 9a) and λ = 0.66 (Fig. 9b). Analyzing these diagrams we see that the decreasing λ causes the growth of attractor’s size and “window” size too. Moreover, the chaotic zones become more wide and intensive. There is the coexisting chaotic attractor producing the points in the diagrams beyond the main set of points.

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Fig. 8 The Poincare bifurcation diagrams at λ = 0.70 (a) and λ = 0.72 (b), when the parameter n grows

Fig. 9 The Poincare bifurcation diagrams at λ = 0.68 (a) and λ = 0.66 (b) when the parameter n grows

The results of qualitative analysis concerning the periodic regimes can be supplemented by the application of shooting method [30, 32]. According to the method, the initial value problem transforms to the boundary value problem, which is considered over a period of the limit cycle. This method also allows one to evaluate a point lying on the limit cycle with high accuracy, its Floquet multipliers characterizing the limit cycle’s stability, and to derive the unstable periodic solutions which are important for understanding the scenarios of bifurcations. In particular, consider the limit cycle existing at n = 0.44 and λ = 0.7 (Fig. 5a). To realize the shooting method algorithm, the procedures and functions from the “Mathematica” can be used. Thus, starting from the initial data (112.6084; 0.4004; 194.8777; 53.3995) which is close to the limit cycle approximate value of period T = 17.6, after application of shooting method we get the refined point’s coordinates (112.6084; 0.4003; 194.7685; 53.4728) and limit cycle period T = 17.5019. Simultaneously, the Floquet multipliers ρi =

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(−0.8327; 0.9999; −0.1398; 0) are evaluated. Since |ρi | < 1, we can conclude that the limit cycle considered is stable. From the Poincare diagram of Fig. 8a it follows that this periodic solution at increasing n is destroyed and instead another periodic regime appears. To find out what happened with the previous regime, we fix n = 0.45 and apply the shooting method to the data derived at n = 0.44. Then we get the coordinates of cycle’s point (112.6084; 0.4037; 197.0695; 51.7384), its period T = 17.1364, and the Floquet multipliers ρ = (−1.0971; 0.9999; −0.1221; 0). Since |ρi | > 1, it means that the derived periodic regime is unstable (repeller). If we take the initial data on the double limit cycle when Ts = 34.4, then we obtain the cycle’s point (136.2579; 0.2075; 92.5914; 200.5915), its refined period Ts = 34.2307, and Floquet multipliers ρ = (0.9999; 0.6831; 0.0260; 0). From the inequality |ρi | < 1 it follows the stability of the double limit cycle (attractor).  Comparing the periods of stable and unstable limit cycles, we get Ts Tu = 1.9558 ≈ 2. From this it follows that we indeed deal with the period doubling bifurcation after which the pair of periodic trajectories coexists. Thus, we shown that dynamical system (2) possesses the periodic (both stable and unstable) solutions, which undergo the several periodic doubling bifurcations, and chaotic solutions.

5 The Construction of the Mathematical Model for the River System Using the Nonlinear Dynamical Model of the Basin of Self-Purification One of the possible applications of model (1) is the assessment of oxygen regime for drainage systems. To do this, we assume that the drainage system can be represented by the pair of basins of self-purification. Let these basins join successively. Assume also that the first basin possesses N incoming streams with a rate q j , j = 1, ..., N each. Then the stream  outcoming from the first basin and incoming into the second basin has the rate Nj=1 q j . The same stream leaves the second basin to provide the mass conservation. Then, using model (1), the mathematical model for these joined basins reads as follows: dC B1 O D dt dC D1 O dt dC 1PhT dt dC 1P dt

   = A1 − k11 C D1 O + S1 C B1 O D + m 11 C 1PhT − λ1 C B1 O D C D1 O ,    = B1 + α 1 C 1PhT − k21 + S1 C D1 O − k11 C D1 O C B1 O D ,    = F1 + k31 C D1 O , C 1P − m 1 C 1PhT , = C1 − n 1 C 1PhT + n 2 C B1 O D − S1 C 1P ,

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dC B2 O D dt dC D2 O dt dC 2PhT dt dC 2P dt where A1 = S1,2

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   = C B1 O D S2 − k12 C D2 O + S2 C B2 O D + m 21 C 2PhT − λ2 C B2 O D C D2 O ,    = C D1 O S2 + k22 C ∗,2 + α 2 C 2PhT − k22 + S2 C D2 O − k12 C D2 O C B2 O D ,    = C 1PhT S2 + k32 C D2 O , C 2P − m 2 C 2PhT , = C 1P S2 − n 21 C 2PhT + n 22 C B2 O D − S2 C 2P ,

N

j

q j cB O D , W1

B1 = N  qj i = Wj=11,2 , k1i C D O = j=1

N

j

q j cD O W1

(7)

N

qjc

j

N

+ k21 C ∗,1 , F1 = j=1W1 PhT , C1 = i  i k1i C iD O (C iD O )n i i i C = k , k , C i i · 3 3 DO P K Di O +C iD O ( K νi )n +(C iD O )n j=1

j

q j cP W1

j=1

C iP

K μi +C iP

, ,

i = 1, 2. The system obtained is highly dimensional and nonlinear. Therefore, let’s study it by numerical methods. We can assume that all quantities of system (7) are dimensionless (due to application of the procedure mentioned above). Thus, the parameters for the first basin are chosen close to set (6), namely A1 = 0.933, B1 = 4.974, F1 = 3, C1 = 2, S1 = 0.2, k11 = 1.5, k21 = 0.7, n 11 = 0.65, n 22 = 0.9, k31 = 4.65, K D O = 4.3, K ν = 0.296, K μ = 1.14, λ1 = 0.7, α 1 = 0.05, m 11 = 0.45, m 1 = 2.3, and the empirical coefficient n I = 0.6. For the second basin we prescribed the following parameter values λ2 = 2.3, n I I = 0.5, while the rest of parameters are the same as for the first basin. The parameter C ∗,2 characterizing the processes of aeration in the basin is chosen as a control parameter. We start from C ∗,2 = 1.42. Omitting the transient processes, system (7) approaches the double periodic regime the phase portrait projection of which is depicted in Fig. 10a. When C ∗,2 = 1.57 and C ∗,2 = 3.4, the corresponding phase portrait projections are shown in Fig. 10b and c, respectively. Comparing the amplitude values of the functions from Fig. 10, we can conclude that the increase in oxygen supply in the second basin causes the decrease in the concentration of the nutrient element phosphorus in water. It should be noted that the phosphorus concentration changes quite sharply when the parameter C ∗,2 changes. From the analysis of the phase portraits it also follows that the dynamics of oxygen regime becomes more complex when C ∗,2 increases.

6 Conclusions Summarizing, we developed the nonlinear mathematical model describing the oxygen regime in water objects. This model is based on the well-known Streeter and Phelps model dealing with the dynamics of BOD and OD. To generalize the Streeter and Phelps model, we incorporate additional factors affecting the oxygen

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Fig. 10 The phase portraits of system (7) at the following parameters C ∗,2 = 1.42 (a), C ∗,2 = 1.57 (b), C ∗,2 = 3.4 (c)

regime, namely the dynamics of phosphorus and phytoplankton. It should be noted that the study of resulting dynamical system is a complex challenge due to nonlinearity and the high dimension of the system’s phase space. Therefore, we apply the qualitative and numerical analysis methods, which provide the information on the model’s dynamics without construction of exact system’s solutions. We thus shown that model can possess several steady solutions of different stability. The variation of the parameters of nonlinearity causes the change of their stability type, i.e. the bifurcation occurs. In particular, we derived the conditions when the Andronov-Hopf bifurcation takes place. It is shown that due to this bifurcation the nonlinear periodic solution develops. Applying Poincare section technique, we revealed the chain of period doubling bifurcations and the formation of the chaotic attractors. It is worth to note that the revealed attractors represent the limiting states of the system’s evolution under the certain set of initial data for the system. Their properties (period, frequency, statistical characteristics, and etc.) are defined by the structure of the system only. This allows one to understand the role of each system’s component in the formation of complex dynamical patterns of system’s evolution. In other words, these studies shed light on the phenomena of self-organization which emerge in the ecological systems [8, 12, 19, 20].

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Mathematical Software for Estimation of the Air Pollution Level During Emergency Flowing of Gas Well for Education and Advanced Training of Specialists in the Oil and Gas Industry Oleksandr Popov , Teodoziia Yatsyshyn , Anna Iatsyshyn , Yulia Mykhailiuk , Yevhen Romanenko , and Valentyna Kovalenko Abstract Operation of oil and gas wells and unforeseen emergencies are accompanied by disturbances in natural state of the atmosphere, soils, reservoirs. It threatens personnel and surrounding areas population. Therefore, oil and gas industry are always subject to increased environmental hazards. International studies confirm that about 80% of accidents and technogenic disasters are related to the human factor. After all, quite common causes of errors are design deficiencies in the workplace equipment or errors in training or instruction of personnel. Therefore, oil and gas workers need constant training and acquisition of competencies to prevent emergencies. Currently, modeling is only one tool for research and solution of current problems of environmental safety of gas condensate fields. This is especially true for those questions that cannot be answered in practice, namely study of causes and accidents forecasting with low probability of occurrence, but with great destructive consequences. For this aim, it is proposed to use number of mathematical models developed by the authors of this publication, namely: model of gas mixture leakage during non-burning flowing of gas well; model of steady flow of gas mixture from the well; model of volley leakage of gases mixture from well; model of pollutants distribution in atmospheric air during flowing of gas well. Process of future specialists training for oil and gas industry for specialties 101 “Ecology”, 103 “Earth Sciences”, 183 “Environmental Protection Technologies”, 184 “Mining”, 185 “Oil and Gas Engineering and Technologies” should be based on use of powerful scientific and O. Popov (B) · A. Iatsyshyn · V. Kovalenko State Institution The Institute of Environmental Geochemistry of National Academy of Sciences of Ukraine, Kyiv, Ukraine O. Popov G.E Pukhov Institute for Modelling in Energy Engineering of NAS of Ukraine, Kyiv , Ukraine Interregional Academy of Personnel Management, Kyiv, Ukraine T. Yatsyshyn · Y. Mykhailiuk Ivano-Frankivsk National Technical University of Oil and Gas, Ivano-Frankivsk, Ukraine Y. Romanenko National Aviation University, Kyiv, Ukraine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Zaporozhets (ed.), Systems, Decision and Control in Energy III, Studies in Systems, Decision and Control 399, https://doi.org/10.1007/978-3-030-87675-3_21

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methodological training base with application of modern scientific achievements and innovative developments. Therefore, we consider it appropriate to supplement curricula for the students and graduate students training in the outlined specialties by studying issues of: development of mathematical models and software to solve problems of emergency prevention during spread of hazardous substances in flowing wells; use of specialized software for solving problems of emergency prevention. Keywords Mathematical modeling · Oil and gas complex · Gas flowing · Air pollution · Environmental safety · Training of future specialists

1 Introduction Today oil and gas industry as a whole is a complex object of increased environmental danger. Many of oil and gas facilities at all stages of the life cycle are a source of increased environmental hazard. Regulated activities and unforeseen emergencies of these facilities are accompanied by violation of natural state of the atmosphere, soils, reservoirs, reservoirs. Currently, many countries have significant number of oil and gas wells. At the same time, oil and gas wells pose threat to the environment both during normal technological processes and during emergencies [1]. In modern conditions of technological era there is a constant improvement of oil and gas equipment, tools and systems of emergency diagnostics and protection [2, 3]. But there is always possibility of uncontrolled or poorly controlled phenomena and processes. Emergencies and accidents pose special danger to the biosphere and humans. One of the possible accidents that pose a special threat is open flowing of wells. After all, toxic substances enter the atmospheric air in large quantities under conditions of both ignited and non-burning flowing of gas well. Subsequently, these gases under action of wind and due to turbulent diffusion spread through the drilling site and beyond (Fig. 1), posing threat to the environment, health of personnel and residents of surrounding areas [4]. Performing of the following tasks can make personnel ready to prevent emergencies in case of air pollution by gas emissions from flowing well. First, it is necessary to perform an operational forecast on the accident fact, and secondly—early calculation of impurities spread into the atmosphere under fixed conditions [5–7]. Studies [8, 9] emphasize that air pollution is characterized by spatio-temporal heterogeneity, non-stationary emission intensities and changes in weather conditions. Operation of the oil and gas complex often disturbs balance of natural geological systems—causes an increase in seismic activity, which can lead to disasters. Extraction of hydrocarbons from subsoil can provoke local earthquakes of significant destructive force. Well-known historical precedents include powerful earthquakes near oil and gas wells in the Rockies (USA), gas fields in Uzbekistan and local 4-magnitude earthquake occurred in March 1986 at the Khrestyshchenske gas condensate field in Ukraine. Another factor that relates to both technogenic and natural risk is the growth of seismic activity in regions where such events were not

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Fig. 1 Accidents at oil and gas wells

observed before and increase in seismic activity in seismically active areas, Fig. 2 [10]. Different models are used to predict possible emergency impact on the environment. Oil and gas workers should have skills to develop and apply mathematical models, to model and forecast level of air pollution in wells in various critical situations. It is important for further professional activities.

Fig. 2 Seismic activity for 30.06.2021 [10]

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The latest requirements for environmental impact assessment provide for need to predict air pollution in emergency situations, but do not contain any specific recommendations for this procedure. Thus, the urgent tasks are: (1) for scientists and practitioners—to increase safety on drilling site and beyond by building adequate mathematical models of spatial distribution of hazardous substances in flowing gas well; (2) problem solving of assessing and forecasting level of air pollution in areas of wells in different critical situations; (3) for scientific and pedagogical workers, students and graduate students—recommendations development and updating of educational-professional and educational-scientific programs of training of future specialists in the specialties: 101 “Ecology”, 103 “Earth Sciences”, 183 “Environmental protection technologies” 184 “Mining”, 185 “Oil and gas engineering and technologies” concerning problems of mathematical models development of spatial distribution of dangerous substances at flowing gas well.

2 Analysis of Literature Sources Analysis of scientific literature determined following: 1.

2.

3. 4.

5.

Publications of Adamenko Ya. O., Semchuk Ya. M., Orfanova M. M., Podavalova Yu. A., Vladymyrov V. A., Makarevych V. N., Rybalov O. V., Obykhod H. O., Storchak S. S., Tetelmyn V. V., Pichtel J., Di Toro D. M., Thomas J. L. and other scientists studied some problematic issues: risk assessment for population from oil and gas production, impact of petroleum products and other fluids on soil, water resources, air, public health, conditions of pollutants, ways to overcome existing pollution in oil and gas facilities complex, etc. New technological solutions for effective overcoming of gas well flowing are offered in number of works [11–15]: method of gas formation flooding is described; technology of using underground directed explosions use; developed mathematical models of dangerous substances distribution from planar and linear source are presented; GIS analysis of measuring results of methane, ethane, propane, carbon dioxide and hydrogen sulfide concentrations in areas of gas wells during emergency non-flaming flowing is presented; calculation algorithm of parameters of emissions and gross emissions of harmful substances from flare installations of hydrocarbon mixes burning is resulted. Theoretical studies consideration of atmospheric turbulence and spread of industrial emissions is devoted to publications [16, 17, 18]. Use of engineering methods for air pollution calculating developed in The Federal State Budgetary Institution “Voeikov Main Geophysical Observatory” (USSR, 1986) is devoted to the work [19]. Use of the EPA USA methodology to assess risks to public health from emissions of toxic substances by large industrial facilities in the air is described in the publication [20].

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6.

7.

8. 9.

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Use of one-dimensional model to determine maximum concentration and distance where it occurs as result of chemically hazardous object release under given meteorological conditions is described in the study [21]. Application features of model of non-stationary emission regime for risk study to public health in the case of explosion at potentially dangerous object is presented in the publication [22]. Model description of the International Atomic Energy Agency to find emission source power according to monitoring data is given in [23]. Peculiarities of future specialists training in specialties 103 “Earth Sciences” and 183 “Environmental Protection Technologies” are considered in publications [24–33]. We conclude following based on the above analysis and generalization:

– Analyzed above scientific works do not raise issue of mathematical models development for modeling and determining distribution in space and time of pollutants concentration emitted during accident at a gas well. – Performed analysis showed that leakage modeling issues of gases mixture from non-burning flowing gas well at different emission regimes need to be developed and applied for education and advanced training of future specialists in oil and gas industry. – Existing mathematical models and techniques on description of pollutants distribution in the air from emissions of technogenic sources of pollution currently possess shortcomings and limitations. – There is a need to develop mathematical tools that will adequately describe both movement of gas mixture along well and pollutants migration in the atmosphere under different emission regimes and meteorological scenarios. – There is a need to form methodology for managing environmental safety of oil and gas facilities taking into account multifactorial impacts. – There is a need to develop recommendations and update educational-professional and educational-scientific training programs for future specialists in the following specialties: 101 “Ecology”, 103 “Earth Sciences”, 183 “Environmental Protection Technologies”, 184 “Mining”, 185 “Oil and Gas engineering and technology” on application of mathematical models of spatial distribution of hazardous substances in flowing gas well. The aim of the research is to substantiate feasibility and studying of prospects for mathematical software application to assess air pollution level during emergency flowing of gas well in training of specialists in the oil and gas industry.

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3 Features of Atmospheric Air Pollution Modeling at Non-Burning Flowing of Gas Well 3.1 Flowing of Gas Well and Physical Features of Air Pollution First of all, let’s consider features of level and distribution forecasting of air pollution during open flowing of gas well, which includes following stages: (1) determination (calculation) of gas emissions, their parameters and composition; (2) calculation of pollutants migration in space and time in the surface layer of the atmosphere. Studies [1, 14] emphasize that emergence of fountains is facilitated by several factors: – long stops and violations of drilling cycle; – incompetent application of accident elimination methods; – formations opening with sharply different lithological and physical characteristics and presence of abnormally high formation pressures; – other factors. Scattering in the atmosphere of accidental natural gas emissions depends on many interrelated causes and emissions conditions. Nature of impurities movement in the air is determined by their own physical properties. Propagation and scattering of accidental emissions in the air occurs as result of its transfer by wind and turbulent diffusion due to the presence in the atmosphere of chaotic vortices that interact in complex way with each other and with the earth’s surface [4, 34, 35]. Scattering of impurities slows down and its cloud with a significant concentration can be transported over long distances at steady state of the atmosphere. Elementary volume of air, out of equilibrium, tends to continue its movement in unstable state of the atmosphere. Under these conditions, the cloud of impurities quickly erodes [8]. The concept of effective source height is introduced: He f = Hs + H,

(1)

where H s —height of emission source (flare pipe), m; H—the magnitude of impurities rise above the source, m. Following empirical formula is used to calculate the lift height H (m) [31]: H =

  3, 3g Rs T 1, 5W0 Rs 2, 5 + , u Ta u 2

(2)

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where W 0 —average velocity of pollutants out of gas well, m/s; Rs —radius of the wellhead, m; u—wind speed at the height of the weather vane 10 m, m/s; T — overheating of gases; T a —ambient air temperature in absolute scale; g = 9,8 m/s2 — acceleration of gravity. Flowing lasts several tens of seconds before the combustible gas ignites or ignites in the case of an instantaneous release. The wind carries away gas cloud formed at height as a whole, and its expansion is caused by turbulent diffusion both in wind direction and in directions perpendicular to it. Time of ejection to moment of ignition of the jet will be 10–30 min in the case of emergency flowing of well (continuous source of constant intensity). Gas ejected from the well is distributed in the form of jet as in stationary leak. Period of sharp change in flow is small in the case of discharge from the well and soon after the start of flowing it is stabilized by value determined by size of shaft and formation characteristics. Level of surface concentration and distribution of impurities in the atmosphere is affected by fog, precipitation and solar radiation [14]. In previous studies, the authors of this publication [1, 4] have already described in detail the following models: – mathematical model of gas mixture leakage during non-burning flowing of gas well; – mathematical model of steady flow of gases mixture from well; – mathematical model of volley leakage of gases mixture from well.

3.2 Mathematical Model of Pollutant Distribution in Atmospheric Air During Flowing of Gas Well In order to build mathematical model of pollutants distribution in the air during emission from gas well it was necessary to accept following statements: (1) environment is continuous and incompressible; (2) atmospheric turbulence is isotropic and inhomogeneous; (3) during transfer of impurities there is only turbulent diffusion of gradient type, molecular diffusion is neglected; (4) pollutants emitted into the atmosphere are considered as “passive impurities”, i.e. those that do not change aerodynamics of the air flow into which they fall; (5) air movement is stationary; (6) gases mixture enters atmosphere; its deposition rate is quite low, i.e. it is neglected [1, 36]. Taking into account the above statements and assumptions, we found function q(t, x, y, z), which determines pollutants concentration in space and time under conditions of volley emission at time t = 0 impurities mass M point source which is located at the point (0, 0, H ef ). The empirical equation of turbulent diffusion with the corresponding initial and boundary conditions needed to be solved [37]: ∂q ∂q ∂q ∂q +u +v +w + λq ∂t ∂x ∂y ∂z

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∂ 2q ∂ 2q ∂ 2q + K + K y z ∂x2 ∂ y2 ∂z 2 + Mδ(t)δ(x)δ(y)δ(z − Hef ),

= Kx

(3)

initial conditions: uq = Mδ(x)δ(y)δ(z − Hef ) at t = 0; + wq + βq = 0 at z = Limit conditions: q → 0 at x 2 + y 2 + z 2 → ∞ and K z ∂q ∂z z0 , – u, v, w—coordinates of the wind speed vector, m/s; – K x , K y , K z —turbulent diffusion coefficients, m2 /s; – λ—parameter that takes into account change in pollutants concentration due to such processes as chemical transformation, leaching by precipitation, absorption by the underlying surface, etc., c–1 ; – z0 —parameter that determines roughness of underlying surface, m; – β—parameter that characterizes pollutant interaction with underlying surface (reflection or absorption), m/s; – t—time of distribution of pollutants, p; – δ—Dirac delta function. Idea of Eq. (3) solving was to reduce number of calculations by splitting of three-dimensional problem into sequence of one-dimensional problems so that both structure of solution and its basic properties are preserved [38]. In this case, due to homogeneous boundary conditions, fundamental solution of the spatial equation was presented as combination (gluing) of fundamental solutions of the corresponding problems. ∂ f (t, xi ) ∂t

= K xi (t) ∂

2

f (t, xi ) ∂ xi2

−u i (t) ∂ f ∂(t,xi xi ) + δ(t − t0 )δ(xi − x0 ),

f (t0 , xi ) = 0.

(4)

For each spatial coordinate i = 1, 2, 3. The item λ(t)·q(t) of the equation characterizes the loss of an impurity in the process of its diffusion. It was excluded from the solution by replacing function q(t, x, y, z) = e

t − λ(t) dt t0

· f (t, x).

(5)

The coordinate system was changed to solve Eq. (4): t ξ=

t K x (t) dt = η(t), ϑ = x −

t0

u(t) dt,

(6)

t0

∂f ∂2 f ∂f ∂f ∂f ∂ f ∂2 f , = · K x (t) − u(t); = ; 2 = ∂t ∂ξ ∂ϑ ∂x ∂ϑ ∂ x ∂ϑ 2

(7)

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following nonlinear differential equation in partial derivatives was obtained performing the substitution taking into account the function substitution   −1  η−1 (ξ ) ∂2 f  ∂f  e 0 η (ξ )λ(t) dt  −1    δ η (ξ ) δ ϑ − x0 + = + u(t) dt . ∂ξ ∂ϑ 2 K x η−1 (ξ ) t0

(8)

We used the method of Fourier transform [33] on the variable ϑ and performed an inverse substitution for the function and coordinates to solve it. The obtained solutions of equations [38] were joined together (4) to finally obtain solution of Eq. (3). Convolution procedure [38] of the equation solution (3) on t was performed to find the concentration function q(t, x, y, z) under the conditions of short-term pollutant emission of duration t 1 . To t 1 was first replaced by t to find function that describes spatial distribution of concentration of continuous emissions of intensity M from a point source, in the model for the source of short-term action Then boundary was taken from right-hand side of obtained equality, directing t to infinity. Axis Ox was located along the wind direction during solving Eq. (3). It is not always convenient. When solving practical problems, the coordinate system is usually positioned so that the abscissa axis is directed to the east, i.e. main coordinate system Oxyz is fixed. Under such conditions the direction of the wind may differ from direction of axis Ox and form with it certain angle α. It is necessary to perform modeling in the coordinate system Ox 1 y1 z to use models under the following conditions, where axis Ox 1 is directed along the wind direction and then transfer results to the coordinate system Oxyz using the transition formulas:

x1 = x cos α + y sin α, y1 = −x sin α + y cos α.

(9)

Analytical dependences forming the mathematical model of air pollution under non-stationary and stationary conditions of emissions from flowing of gas well took following form after all mathematical transformations: for instantaneous emission source: 

Me−

2 x cos α+y sin α−u H t +(−x sin α+y cos α)2 ef 4K t

 8π π K 2 K z t 3  (z−H )2 (z+He f −2z0 )2 − λ(x cosu α+y sin α) − 4Kez ft − He f 4K t z × e +e ; ·e

q(t, x, y, z) =

for source of short-term emission duration t 1 :

(10)

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− λ(x cosu α+y sin α) He f

t1

2 x cos α+y sin α−u H (t−τ ) +(−x sin α+y cos α)2 ef 4K (t−τ )

e−   · 8π π K 2 K z (t − τ )3 0  (z−He f )2 (z+He f −2z0 )2 − 4K z (t−τ − 4K z (t−τ ) ) +e dτ ; × e

q(t, x, y, z) =

Me

(11)

for continuous emission source: ⎛

 − 21

u H (x cos α+y sin α) ef

2K Me q(x, y, z) = √ 4π K K z

x 2 +y 2 K

 u 2H z−He )2 ( ef + ·

⎜e ⎜  + ⎜ 2 ⎜ z−He f ) ( x 2 +y 2 ⎜ + Kz K ⎜ ·⎜   2 ⎜ 2 2 (z+He f −2z0 )2 · u He f ⎜ − 21 x +y + K Kz K ⎜ e ⎜+  ⎝ 2 z+H −2z x 2 +y 2 + ( ef 0) Kz

K

K

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ − λ(x cos α+y sin α) ⎟ uH ef , ⎟·e ⎟ ⎟ ⎟ ⎟ ⎠

Kz

(12) where u He f —is a wind speed at effective height of emission source, m/s; wg is rate of gravitational deposition for heavy impurities, m/s. When building the model it was assumed that K x = K y = K. M means intensity of emissions in g/s in the analytical dependence (12). The wind speed at the effective height of emission source is determined by the formula [32]: u He f = u(10) ·

Heεf − z 0ε 10ε − z 0ε

,

(13)

where u(10)—wind speed at height of 10 m, m/s; ε is dimensionless parameter that depends on category of atmospheric stability. The publication [4] presents results of modeling and measurement data comparison for methane and hydrogen sulfide. Under such flowing conditions of the investigated well in the surface layer of the atmosphere, the maximum levels of concentrations of all substances exceeded the corresponding maximum allowable concentrations by a maximum of 4–5 times. It possesses significant risk to health of well personnel and surrounding area during inhaling air at such emergencies. Thus, the developed mathematical tools are effective mean for solving problems of emergency prevention in non-flammable flowing of gas well. It is necessary to apply mathematical model of steady flow of gas mixture from the well in order to determine the emission capacities for each component of the gas mixture in the stationary mode

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of emergency flowing. The model of volley leakage of a mixture of gases should be used in order to determine the mass of the emission in the non-stationary mode of release of toxic gases from the wellhead. These models take into account all main physical factors influencing movement of gas along the wellbore. Description of the models is presented in the form of set of corresponding algebraic and differential equations that describe movement of gas through the well. It is necessary to have data on gas-dynamic and geometric characteristics of reservoir and well use them. Such data are indicated in working design and passport of each well [1]. Mathematical models of pollutants distribution in the air allow to determine distribution of impurities in space and time in adjacent to the well area under conditions of volley flowing. The input data of these models are mass (non-stationary gushing mode) or intensity (stationary gushing mode) of emission, parameters of source and emission conditions, parameters of pollutants and meteorological conditions. The developed mathematical models have significant advantages over existing analogues—models and methodological recommendations based on the Gaussian distribution. Thus, they take into account peculiarity of underlying surface and impurities interaction with environment (precipitation leaching, absorption by underlying surface, chemical transformation), vertical structure of boundary layer. They can also be used to determine concentration of any gaseous pollutants with different densities. Also, they allow to determine distribution of pollutants concentration in any emission regimes and meteorological scenarios. This allows to solve wider class of problems related to control of atmospheric air in areas of gas wells [1, 4]. Verification of developed models adequacy according to actual emergency flowing of one of the gas wells (in Poltava region, Ukraine) showed their high accuracy: modeling error did not exceed 15% for all components of the gas mixture. Thus, the developed models should be used to solve problem of operational forecasting of accident and early calculation of impurities distribution in the atmosphere under fixed conditions. This approach can be useful in integrated design of new wells and in order to plan safety measures on existing ones, in particular to assess maximum possible concentrations for planned emissions.

4 Features of Education and Advanced Training of Environmentally Conscious Specialists for Oil and Gas Industry (Frontline Performers of Oil and Gas Companies) Practical experience confirms that about 80% of accidents and technogenic disasters are related to human factor. Principle of “lifelong education” should be prior in oil and gas companies and should be applied to both management and production staff [1]. Often errors causes are design deficiencies in the workplace equipment or errors in training. It is impossible to completely eliminate human errors, so improvement

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of structures, training, education of workers to gain more experience are extremely important in professional activities of oil and gas workers. Information support in this case is important resource for staff at various levels and for population. Pollution Prevention Information Clearinghouse (PPIC) is an example of such information system. It aims to reduce and eliminate industrial pollution through education and public awareness [39]. This information service helps companies reduce industrial waste by providing advice and inquiries. In Ukraine, there are some organizations that work to support adoption of environmentally efficient decisions, but still unfinished network and imperfect specifics of certain industries [1]. For example, Dragon Oil (UAE) is guided by principles based on ISO 9001, ISO 14,001 and OHSAS 18,001 standards. It defines following:—compliance with HSE legislation;—consequences reduction of all types of incidents/accidents with potentially harmful impact on personnel, assets and/or the environment;—application of generally accepted levels of good environmental practice at oil field; create transparent HSE performance criteria. These goals can by achieved by the next actions: the company develops and maintains an appropriate HSE management system; it holds HSE accountable at all levels of management, staff and contractors within their areas of responsibility; it develops a positive culture of environmental protection in the company through effective communication and training; the company conducts HSE reviews and audits to identify deficiencies in organization and ensure compliance with local, international and corporate standards and norms; ensures that contractors demonstrate full compliance with this policy; develops and implements risk-based approach to doing business [40]. Sociogenic elements of management provide implementation of the “lifelong education” principle and promotion of environmental awareness of oil and gas workers at all levels., where There is a need in the oil and gas sector to attract modern technologies due to obsolete, inefficient, environmentally dangerous, resourceintensive technologies. It does not meet standards of society based on vector of sustainable development. The principle of environmental “lifelong education” is extremely important for all hierarchies of oil and gas institutions. It is necessary to conduct systematic training of personnel since more than 70% of emergencies at oil and gas fountains occur due to technological regime violations of drilling, improper installation and operation of preventors, i.e. incompetence of service personnel. Theoretical provisions of “lifelong education” formed basis for reforming national education systems in the world (Germany, Great Britain, USA, Japan, Canada, Eastern Europe and the “third world” [41]. There is a need to promote environmental approaches in various fields according to the concept of greening in Ukraine and the main points of the Paris Agreement. Environmental education at different levels of human development is one of keys to environmental safety. Three-level environmental education should be basis for knowledge and skills in seeing environmental issues [42, 43] in particular workplace. Any employee meets real problem situations and competent response can prevent or eliminate possible negative consequences. Therefore, it is necessary to create conditions for implementation of training courses, seminars and other tools that provide environmental knowledge in relevant areas [27,

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44–47]. Also, modern innovative developments should be introduced in training of future specialists. Improving of education quality is one of important issues in any society development. The modern world is evolving and changing rapidly, information technologies are updated and improved. Therefore the domestic higher education system does not have time to adapt curricula and plans to requirements of the market and society. This problem is relevant in field of training in the following specialties: 183 “Environmental Protection Technologies” and 103 “Earth Sciences” [27]. Therefore, we believe that it is important in training of future specialists in the outlined specialties to add topics for modeling processes of gas distribution in the air at emergency flowing wells. Choice of mathematical and software tools used in the educational process should be based on need to form professional skills in students and graduate students, development of systematic thinking, ability to select optimal tool for solving particular application problem [27]. This will greatly enrich their experience and make it possible to understand peculiarities of such events modeling related to spread of gas in the air during emergency well flowing. It is extremely important in emergencies preventing. For example, the “Standard of Higher Education of Ukraine for the Master’s Level”, branch of knowledge 18—“Production and Technology” in specialty 183—“Environmental Protection Technologies” determined that main purpose of future professionals training is following: formation of professional competencies necessary for innovative research and production activities for development and implementation of modern technologies for environmental protection. Thus, the developed mathematical models for gas distribution in the air during emergency flowing of well was introduced in educational process of Ivano-Frankivsk National Technical University of Oil and Gas of the Ministry of Education and Science of Ukraine, as educational elements of disciplines: “Ecology Of Oil And Gas Complex”, “Product Life Cycle Assessment and Environmental Management”, “Fundamentals of Ecology in Oil and Gas Industry”, “Environmental Protection Technologies”. Also, new discipline “Assessment of Product Life Cycle Characteristics and Environmental Management” was developed; recommendations and updations of educational-professional and educational-scientific training programs for future specialists in the following specialties were prepared: 101 “Ecology”, 103 “Earth Sciences”, 183 “Environmental Protection Technologies”, 184 “Mining”, 185 “Oil and Gas Engineering and Technology” on application of mathematical models of spatial distribution of hazardous substances in gas well flowing. It is also planned to implement these models for training of graduate students in the specialty 183 “Environmental Protection Technologies” in the State Institution “The Institute of Environmental Geochemistry of National Academy of Sciences of Ukraine”. Therefore, the following measures should be taken in order to increase efficiency of education and advanced training of specialists in oil and gas industry on issues of risk reduction during spatial distribution of hazardous substances in gas well flowing:

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– supplemention of curricula for training of students and graduate students to ensure acquisition of competencies on risks reduction during spatial spread of hazardous substances at gas well flowing; introducing of issues study: on development of mathematical models and software tools for solving problems of emergency prevention during the spread of hazardous substances during the gushing of a gas well; application of specialized software to solve problems of emergency prevention; – expanding topics of bachelor’s, master’s, dissertation works of students and scientific degrees with problems on various aspects of mathematical models and software development to solve problems of emergency prevention during spread of hazardous substances in flowing gas well; – organization and conduction of seminars for advanced training of specialists to solve problems of emergency prevention in spread of hazardous substances during flowing gas well.

5 Conclusions Oil and gas industry are always subject of increased environmental hazard. Operation of oil and gas wells and unforeseen emergencies are accompanied by disturbances in natural state of the atmosphere, soils, reservoirs and threatens personnel and surrounding areas population. International experience confirms that about 80% of accidents and technogenic disasters are related to the human factor. After all, quite common causes of errors are structural defects in workplace equipment or errors in training. Therefore, oil and gas workers need constant training and acquisition of competencies to prevent emergencies. The “lifelong education” principle should be prior in oil and gas companies and should be applied to both management and production staff. It is proposed to use the developed mathematical models of steady and volley leakage of gases mixture from well. These models take into account all main factors influencing intensity of gas mixture leakage during emergency flowing gas well. Main parameters that determine distribution of pollutants concentration in space and time include following: emission parameters; meteorological characteristics; roughness of underlying surface; parameters of impurities interaction with environment; propagation time and deposition rate of impurities. Also, it is proposed to use the developed mathematical model of pollutants scattering in the air, which takes into account all the main factors influencing this process in contrast to existing ones. This model allows to effectively solve air monitoring problems in areas of gas wells and preventive forecasting of emergencies related to emergency flowing. Adequacy checking of the developed mathematical models proved that modeling error did not exceed 15% for all components of gas mixture. This indicates high adequacy of the developed models and their use prospects to solve urgent problems of air protection in areas of gas wells.

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Process of future specialists training for oil and gas industry, in particular in the following specialties: 101 “Ecology”, 103 “Earth Sciences”, 183 “Environmental Protection Technologies”, 184 “Mining”, 185 “Oil and Gas Engineering and Technologies” should be based on use of powerful scientific and methodological training base with use of modern scientific achievements and innovative developments. Therefore, we consider it appropriate to supplement curricula for students and graduate students training of the outlined specialties by studying issues of: development of mathematical models and software to solve problems of emergency prevention during spread of hazardous substances in flowing wells; use of specialized software for solving problems of emergency prevention. For this purpose, it is proposed to use number of mathematical models developed by the authors of this publication, namely: model of gas mixture leakage during non-burning flowing of gas well; model of steady flow of gas mixture from the well; model of volley leakage of gases mixture from the well; model of pollutants distribution in the atmospheric air during gas well flowing. The developed models were tested and implemented in the training of future specialists at the Ivano-Frankivsk National Technical University of Oil and Gas of the Ministry of Education and Science of Ukraine. Positive feedback and approval was received from students and faculty. Therefore, these models are planned to be implemented in other universities at institutions where future specialists are trained in the following specialties: 101, 103, 183, 184, 185. Further research will focus on development and improvement of specialized software and modeling complex, which is an effective tool to support decision-making in management of environmental safety of ambient air in the areas of gas wells. This software and modeling complex includes functional modules. Their use will allow constructing of electronic maps of possible pollution spread during the emergency flowing of gas well and assessing of risks to health of personnel and surrounding areas population.

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Environmental Assessment of Recreational Territories of Ukraine Nataliia Ridei , Tetiana Khitrenko , Valeriia Kovach , Oleg Karagodin , Hrushchynska Natalia , and Oleksii Mykhalchenko

Abstract Conceptual-categorical apparatus of recreation in Ukraine according to normative-legal, reference and scientific-methodical literary sources is analyzed. Peculiarities of structural organization of recreational sphere, state and prospects of its development are established. Structural and organizational scheme of conceptual and categorical apparatus of recreation is developed. Existing approaches and classification features on basis of developed classifications are specified and the author’s approaches, signs and kinds are offered. The paper analysis current methods of environmental assessment of soils, lands, surface waters of rural areas, social and natural systems in the agrosphere, including anthropogenic-industrial, social-domestic and recreational load for social, recreational-tourist and environmental support of the organization and development of recreational activities on territories of the agrosphere. Indicators and parameters of scientific and methodological support necessary for assessing the ecological condition of recreational areas of the agrosphere are highlighted. Structural and logical scheme of organization of environmantal and social assessment of recreational areas of agrosphere is developed, which allows adaptation of current scientific and methodological recommendations of leading scientific institutions of Ukraine to provide initial, current and forecast information data for substantiation and construction of environmental monitoring of recreational areas of the agrosphere. Qualitative assessment of lands, agroecological assessment of soils N. Ridei (B) National Pedagogical Dragomanov University, Kyiv, Ukraine T. Khitrenko Institute of Agroecology and Environmental Management of National Academy of Agrarian Sciences of Ukraine, Kyiv, Ukraine V. Kovach State Institution “The Institute of Environmental Geochemistry of National Academy of Sciences of Ukraine”, Kyiv, Ukraine V. Kovach · H. Natalia · O. Mykhalchenko National Aviation University, Kyiv, Ukraine O. Karagodin Interregional Academy of Personnel Management, Kyiv, Ukraine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Zaporozhets (ed.), Systems, Decision and Control in Energy III, Studies in Systems, Decision and Control 399, https://doi.org/10.1007/978-3-030-87675-3_22

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for growing grain, technical, fodder and vegetable crops and assessment of suitability were carried out. It was found that soils of Southern Pryluchchyna possess unfavorable conditions for growing of studied groups of crops due to excess of mobile forms of lead relative to maximum allowable concentrations. Accordingly, these soils are unsuitable for growing products for baby and dietary nutrition. Studied soils of Vinnytske Pobuzhzhia possess optimal and acceptable conditions for growing of above groups of crops, are suitable for growing products for baby and dietary food. Forecast for obtaining environmentally safe and biologically complete products on these soils is made. Keywords Recreation · Recreational areas of agro-sphere · Environmental control · Normative-legal regulation · Agroecological assessment of soils and assessment of suitabilit

1 Introduction Ukraine possesses great potential for development of recreational activities. However, scientific and legislative regulation does not develop mechanisms for environmental monitoring [1–3] of recreational areas of agro-sphere. Different approaches, methods, mathematical and software tools for assessing of the territories state determining various indicators and parameters for them are studied in publications [3–21]. The works of N. Fomenko, P. Masliak, O. Beidykand others are devoted to study main content and tasks of recreation, classification of recreational resources, recreational zoning. Developers of methods for assessing of agricultural land quality are prominent agro-ecologists: O. Sozinov, V. Medvediev, A. Siryi, M. Kozlov, B. Prister, O. Furdychko, O. Tarariko, O. Rakoid, N. Makarenko, M. Lisovyi. The chapter aim is to determine functional purpose of recreational areas of the agrosphere. Tasks: • Analysis of conceptual and categorical apparatus of recreation in Ukraine according to normative-legal, scientific, scientific-methodical and reference literary sources; • Theoretical analysis of scientific approaches to classification systems of recreational resources in domestic and foreign scientific sources, identification of classification features, distinguishing of approaches, building of a systematization scheme of existing classifications; • Analysis of approaches and methods for assessing of environmantal and social condition of territories; identification and recommendation of objective indicators and parameters for assessing of state recreational areas of the agrosphere to develop their own methodological approaches;

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• Development of structural and logical scheme of organization of environmental and social assessment of recreational areas of agrosphere; conduction of agroecological assessment and assessment of soils suitability for growing baby and diet food; • Recommendations for studied areas to conduct optimal recreational activities.

2 Methods and Techniques The study used method of qualitative assessment and certification of land using agroecological method (A. I. Siryi, M. V. Kozlov, 2002) [22], DSTU 4288: 2004 “Soil quality. Soil passport” [23], “Assessment of Ukraine’s agricultural lands suitability for creation of environmentally friendly raw material zones and farms for the production baby food and dietary food” (O. G. Tarariko, 1998) [24] and “Assessment of agricultural land suitability for special raw materials zones” (O. I. Furdychka, 2006) [25], “Classification of soil quality to obtain environmentally friendly and biologically complete food” [26, 27] and others. The research was conducted within the following topics: ‘Development of program for sustainable nature management of local agroecosystems and scientific justification of their environmental safety” (DRC № 0109U000955), “Substantiation of scientific concept and development of measures to form professional and practical competence of environmental management specialists” № 0112U001684) and “Scientific substantiation of principles and practical recommendations for systematic analysis of sustainable development of rural areas” (DRC № 0115U003406).

3 Results There is following definion according to the Regulations on recreational activities within territories and objects of nature reserve fund of Ukraine, approved by the Ministry of Environment of Ukraine: recreation—is restoration outside of permanent place of residence in the places of nature reserves and objects of mental, spiritual and physical strength, carried out by general health, cultural and cognitive recreation, tourism, rehabilitation, etc. The concept of “recreation” in dictionaries-reference books is defined as a simple recovery, reproduction of physical and spiritual forces expended by man in the process of work, study and household activities; in narrow sense—various types of human activity in free time, which are aimed at restoring strength and meeting wide range of personal and social needs to increase socio-labor and cultural potential of society, formation of new personality traits and qualities, and so on. There are three forms of time spending in recreation: tourism, treatment and recreation. Ye. Prystupa, O. Zhdanova, M. Lynets and other scientists note that “recreation” is all forms of human leisure.

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The concept of “recreational sphere” is not clearly defined at the legislative level. Concepts of “recreation and tourism business” and “tourism and resorts” are used in the Law of Ukraine “On Resorts” (2000), the President Decree “On main directions of tourism development in Ukraine until 2010” (1999), the order of the Cabinet of Ministers “On approval of the Strategy for development of tourism and resorts” (2008) [28, 29]. Recreational area at the legislative level is defined as natural complex that possesses necessary prerequisites for its use for recreational purposes. Thus, recreational area of the agrosphere is a part of agro-landscape with favorable environmental conditions, which can be used for organization of recreational activities [30, 31]. Objects, phenomena and processes of natural and anthropogenic origin that are used or can be used for development of recreation, according to P. Masliak belong to recreational resources. M. Pokolodna distinguishes two approaches to classification of recreational resources: genetic (by the origin of recreational resources) and situational (depending on use of recreational resources). According to origin there are natural, historical, cultural and socio-economic, and according to the use—tourist, medical and recreational resources. In our opinion, situational approach to classification does not have classification feature—conduction of recreational activities. In addition following types should be included—garden, cognitive and memorial types of recreational resources [32, 33]. Genetic approach is distinguished by” nativeness”. It should include environmental recreational resources based on natural. It should their use in harmony with nature and ensure environmentally balanced development [34]. Among natural recreational resources are medical, climatic, beach, landscape, water, biological, relief resources and objects of nature reserves, which in turn are divided into subspecies. Separately, we propose to distinguish: egional ethnographic cultural and historical types of recreational resources and refer them to historical and cultural subspecies. S. Boholiubova proposes to divide natural resources into: exhaustive (nonreproducible, reproducible and relatively reproducible), inexhaustible (space, climatic, water), replaceable and irreplaceable. The author uses material approach. However, in our opinion, water recreational resources cannot belong to inexhaustible resources, because healing properties of water after use are virtually irreversible. The scientist also proposes the division of recreational resources into: natural resources, conditions and phenomena (landscape, water resources, vegetation, etc.), natural and anthropogenic (hunting grounds, artificial ponds, resources under special protection), anthropogenic (water parks, museums) and others), socio-historical (cultural sites, monuments, historical sites) and infrastructure (sanatoriums, boarding houses, recreation centers, service infrastructure and others). Recreational resources are allocated by use—intensively, extensively and unused. According to the geosphere approach of Y. Hiletskyi recreational resources are divided into natural-geographical and socio-geographical with subspecies. This approach is carried out, in our opinion n”by genesis”. We offer to expand these types. For example, pedospheric recreational resources need to be included in lithosphere subspecies, as Ukraine possesses the most fertile soils in the world. An important

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type of atmospheric recreational resources is ecoclimatic, which should be environmentally friendly for the biosphere. We propose to include healing resources in the hydrosphere subspecies, as it is environmentally friendly water that has healing properties. In the biosphere subspecies we distinguish genetic recreational resources as a separate species. Our vision of unified scheme of modern approaches to recreational resources classification with the author’s addition is presented in Fig. 1. We analyzed current methods of environmental assessment of soils, lands, surface waters of rural areas, social and natural systems in agrosphere, including anthropogenic-industrial, social-domestic and recreational load for social,

Fig. 1 Scheme of modern approaches to the classification of recreational resources (* author’s development)

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recreational-tourist and environmental support of organization and development of recreational activities in the agrosphere of Ukraine. Indicators and parameters of scientific and methodological support necessary for assessing the ecological condition of recreational areas of the agrosphere are highlighted. In our opinion, the methodology for assessing environmental condition of recreational areas of agrosphere should be comprehensive. It should include indicators of environemntal condition of lands, surface waters, biodiversity of the studied ecosystem, climate, sociosphere and levels of anthropogenic and recreational impacts. In our opinion it is expedient to assess environmental condition of lands on the basis of following parameters: provision by agricultural lands, environmental and agrochemical assessment of soils (quality class), environmental and toxicological pollution, environmental stability of the territory. Surface waters, their safety and suitability for recreational activities and vacationers are best assessed by parameters in indicators: biological, hydrological, hydromorphological, physicochemical, chemical, water resources and water quality class. In our opinion, trere are follofing main indicators allowing effecient assess of natural resources availability: species composition of flora and fauna, namely forest resources, valuable natural resources, including NRF, mineral waters, healing springs, medical natural resources in the studied areas. Presence of anthropogenic impact, its intensity can be assessed by analyzing of the following indicators: building density, number of landfills (including unorganized), sewage, mineral fertilizers and pesticides, number of business entities, agricultural machinery, refueling points and etc. Group of climatic indicators (sum of effective temperatures, precipitation, favorable weather, sunshine, radiation balance, average wind speed) is relevant for assessment of recreational areas. It can be used to assess comfort of these areas in terms of optimal weather conditions. In our opinion it is necessary to give a separate assessment of the environmental condition for each group of these indicators and parameters to effectively, fully and objectively assess environmental conditions of recreational areas of the agrosphere. It is necessary to deduce the integrated estimation of an ecological condition of territories at general estimation of environmental condition of recreational territories. Therefore, we recommend using a point approach (scale within 1–100 points) for more accurate and effective assessment of groups of indicators. Groups of indicators for parametric assessment of ecological and social condition are presented in Fig. 2 during structural and functional analysis of the content and structure of scientific and methodological support of ecological and social assessment: areas of the agrosphere, as a whole, soil cover (and its differentiation by quality categories), land agricultural purposes (their stability, stability by type of land, degradation, ratio of arable land to ecologically stable lands, and quality classes), recreational areas of the agrosphere and their potential for development of recreational industry (infrastructure, aesthetic, medical, tourist, natural and agricultural) potentials), surface waters (by classes and categories of their quality, summarized by radiation, geochemical, environmental and sanitary criteria, levels of toxicity to living

Fig. 2 Structural logical scheme of environemntal and social assessment organization of recreational areas of agrosphere

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organisms), natural and social systems (in the state and development of human potential, and economic parameters ohmic, educational, demographic in nature, conjunctural in terms of employment, infrastructural and cultural organization, environmental safety and security of life, compliance with quality of life standards; state of geosphere transformation (land, water, atmosphere, subsoil, use of flora and fauna), including biodiversity). Structural and logical scheme of organization of ecological and social assessment of recreational areas of the agrosphere is developed. It allows adaptatiton of current scientific and methodological recommendations of leading scientific institutions of Ukraine to provide initial, current and forecast information data for substantiation and construction of ecological monitoring of recreational areas of the agrosphere. In future it will contribute in planning of recreational activities types in the agrosphere, environmental, social measures of ecological optimization of the state and development of recreational areas of the agrosphere on the sustainability basis. The team of authors carried out an ecological assessment of the lands of the studied areas based on the proposed approaches. Qualitative land evaluation was performed using listed above methods. In the territory of the Southern Pobuzhzhia high quality (good) lands are occupied by typical chernozems, which belong to the IV class of soil suitability [26, 27] (growth class 61.87). They are close to the first group, but possess slightly lower productivity. These soils are well supplied by nutrirents. They have favorable physicochemical and agrophysical properties, reduce the quality of land, weakly expressed negative properties of soils. Soils of medium quality (satisfactory) are occupied by podzolic chernozems, which belong to the V class of soil suitability (growth class 58.83), and dark gray podzolic (growth class 51). These soils have an average supply of nutrients and productive moisture. Quality of lands is reduced by more pronounced negative properties of soils (low and medium acidity, etc.) and technological properties of land plots (dismemberment by a network of beams, erosion, etc.). Soils types are established and their classes of suitability for agricultural production of Pryluky district are determined according to conducted environmental and agrochemical assessment. Vast majority of soil cover of studied areas (total area 1665.4 ha) are of average quality (satisfactory lands of V and VI class [26, 27]). These lands have restrictions for use in agriculture, in particular due to intensive erosion, located on slopes of 7–10° or more, poor drainage, high stonyness, low water holding capacity, frequent flooding, adverse climatic conditions. It can be used for growing perennial grasses or for haymaking. Lands of high quality (good lands of class IV) include soil cover of 1 plot (№ 22) with an area of 102.8 hectares, they can be used for agriculture, pastures, forests, reserves and for natural food. Low quality lands (VII class) were found on 5 studied plots (№ 3, 5, 6, 8, 10) with a total area of 407.1 ha. From the ecological point of view, cultivation of unpretentious crops on these soils is possible under condition of intensive agrotechnical measures to improve soil cover condition. It should be noted that low soil quality of these areas is observed due to significant excess of maximum allowable concentration of lead. Therefore, we consider it appropriate to recommend growing in these areas of industrial crops, such as oilseed rape, soybeans, flax (for technical needs), or energy

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crops—sorghum, bitter gourd, Jerusalem artichoke, millet, some species of willow and poplar, hybrids of tobacco and sorrel used as fuel. Agroecological assessment of lands was carried out in order to establish their capacity for growing of grain, technical, fodder and vegetable crops. Soil cover of the studied lands of the southern part of Pryluky district and Vinnytsia Pobuzhzhia is characterized by different conditions for growing crops. Typology of Pryluchchyna soils is represented by typical chernozems: leached coarse-grained light loam (68.5%), leached coarse-grained light-loam lightly washed (19.25%), deep lowhumus coarse-grained light loam (12.25%), and in Vinnytsia district—chernozems podzolic medium loam (45%), dark gray podzolic medium loam (40%), gray podzolic medium loam and chernozems leached medium loam (15%). Indicators of soils ecological stability of the southern part of Pryluchchyna: reaction of the soil environment of these soils (pH = 4.8–6.2), humus content (2.12– 3.75%), thickness of a humus layer (over 70 cm). All these parameters formed admissible conditions for growing of different groups of crops. The indicators of Vinnytsia Pobuzhzhia are similar. During 2011–2016 in the studies areas average values of active temperatures sum over 10 C were—2760–28,600 C, Selyaninov hydrothermal coefficient (SHC)—was 1.1–1.2. It created optimal conditions for all types of crops. Content of easily hydrolyzed nitrogen (according to Cornfield) in the soils of southern Pryluchchyna is low and very low (70–137 mg/kg). It is an acceptable condition for growing fodder crops in the fields № 1, 2, 4, 5, 9, 11–15, 17, 27–29. Rest of fields possess unacceptable conditions for growing of abovelisted groups of crops. Content of easily hydrolyzed Nitrogen in the studied fields of Vinnytsia Pobuzhzhia is within the range of 108–160 mg/kg. So, it is reffered to low (100– 150 mg/kg) and medium (in the field № 9 it is above 150 mg/kg) levels, which is inadmissible and admissible for cultivation of grain, technical and vegetable crops. Fodder crops are less demanding to nitrogen content. So, safety level over 100 mg/kg is acceptable and optimal for their cultivation. It was found that content of mobile phosphorus compounds (according to Chirikov’s method) in the soils of Southern Pryluchchyna is high and increased (83–167 mg/kg). It creates optimal and acceptable conditions for growing of the studied groups of crops. The soils of Vinnytsia Pobuzhzhia possess average content of mobile phosphorus compounds (80–99 mg/kg), which creates acceptable conditions for crops growing. The soils of both studied areas possess high and increased content of mobile forms of potassium (69–110 mg/kg and 83–167 mg/kg by the Chirikov’s method). They are characterized by suitable and limited conditions for formation of high quality environmentally friendly crops. Content of mobile forms of microelements (in particular zinc and copper) creates optimal and acceptable conditions for crops growing. The soils of Southern Priluchchina possess copper content 3.4–4.83 mg/kg, which is optimal condition, and the rest creates acceptable conditions. Sanitary and hygienic condition of soils was assessed by content of: mobile forms of heavy metals, including cadmium and lead, pesticide residues (DDT and its metabolites) and density of radionuclide contamination—cesium 137. Lead content

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in the soils of Southern Pryluchchyna exceeds level for safe agricultural products. Remaining indicators do not exceed the permissible concentrations and create favorable conditions for crops growing. Assessment of land suitability and data of environmental and agrochemical assessment of the studied soils are presented in Table 1. It should be noted that soils of Southern Pryluchchyna are unsuitable for growing agricultural products for baby and dietary nutrition. In these areas, we can recommend such recreational activities as: local history, landscape, history, cognitive, pilgrimage and other types not related to use of agricultural products grown on these soils for childrens nutrition. The soils of Vinnytsia Pobuzhzhia are suitable and conditionally suitable for growing agricultural products for baby and dietary nutrition. We make the first forecasts of suitability of the studied areas for specific types of recreational activities based on analysis of soil cover and land. These territories are respectively suitable for: health, medical, gastro-, apitourism, various gastro festivals and other types with possible use of grown agricultural products on these soils. Forecasts for biologically complete ecologically safe raw materials and agricultural products were made based on the obtained data. It is established that ecologically safe products can be obtained on the studied soils of Vinnytsia Pobuzhzhia, as indicators of content of mobile forms of heavy metals (cadmium, lead), pesticide residues and density of radionuclide contamination (cesium-137, strontium-90) do not exceed the MPC. Biologically complete products can be obtained at the field №9. It is predicted to obtain biologically defective raw materials on the rest of the studied soils due to low content of mobile forms of easily hydrolyzed nitrogen (108– 142 mg/kg). The soils of fields 1, 3, 4, 10 have insufficient content of mobile forms of zinc (1.7–1.9 mg/kg). The field №10 (pH −4.6) is reffered to moderately acidic soils. On the soils of Southern Priluchchina we forecast obtaining of biologically defective and ecologically dangerous products due to following reasons: excess of lead (3.94–7.09 mg/kg), strontium-90 (0.03–0.07 Ki/km2 ) relative to the MPC and insufficient content of mobile forms of nitrogen (70–137 mg/kg) in all studied areas. Soils of fields № 1–3, 9–12, 14, 19–20, 29–30 are partially moderately acidic soils. According to the classification of environmentally safe and biologically complete products and raw materials, according to gradation of soils by classes soils of Vinnytsia Pobuzhzhia belong to class II ecologically safe and biologically inferior— products and raw materials meet only environmental safety standards and partially (field #9) to the I class—ecologically safe and biologically complete agroproducts and raw materials—products and raw materials that meet the established requirements of biological completeness and norms of ecological safety. Soils of southern Pryluchchyna belong to class IV—ecologically dangerous and biologically defective—products and raw materials do not meet requirements of biological integrity and environmental safety standards. Existing recreational infrastructure of the studied areas is analyzed. Share of potential-recreational areas of Pryluchchyna, according to recreational-tourist zoning is 30–35% in the land structure. According to territorial organization of tourism, the north-eastern and central parts have more developed recreational sphere. District

Soil quality class

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Mobile phosphorus content, mg/kg

Easily hydrolyzed nitrogen content, mg/kg

Humus content in the arable layer %

Reaction of soil solution, pH H2O

The amount of absorbed bases, mg-eq/100 g of soil

Agrochemical indicators

Indicators of ecological stability of soils

Factors

Bohdanivka village, Pryluky district

passport

№

98

97

65

72

79

67

86

104

81

55

66

96

119

120

78

90

80

±

±

±

±

±

±

±

±

±

±

±

±

±

±

±

±

±

The content of metabolic potassium, mg/kg

3.85

3.81

3.5

3.98

4.35

4.36

4.33

3.94

4.83

3.81

3.73

3.81

3.8

4.61

3.73

3.93

4.38

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

Content of mobile forms of microelements (mg/kg): copper

±

±

±

A

±

±

±

±

A

±

±

±

±

A

±

±

A

(continued)

5.01

5.3

5.8

6.21

5.86

5.82

5.66

5.11

6.78

4.76

4.33

4.73

5.49

6.33

4.33

5.46

6.42

Zinc

Table 1 Date of of land suitability assessment according to indicators of ecological stability, fertility and parameters of sanitary and hygienic condition of soils

Environmental Assessment of Recreational Territories of Ukraine 363

VI

VI

VI

VI

IV

VI

V

V

VI

V

V

V

VI

18.

19.

20.

21.

5.5 22.

23.

24.

25.

26.

27.

28.

29.

30.

5

5

5.6

5.5

5.1

5.4

5.3

6.2

5.3

5

5

5.6

15

22

±

±

13

±

V

V

V

1.

2.

3.

6.2

6.3

6.2

A

A

A

19.1

19.2

19.1

15

19

±

16

12

±

±

17

±

A

19

15

±

±

35

18

±

A

19

A

2.78 2.79 2.78

± ±

3.04

±

±

3.75

3.39

± ±

2.94

2.41

3.03

2.76

3.06

2.9

2.81

2.83

2.9

2.88

±

±

±

±

±

A

A

±

±

±

±

±

±

±

±

±

±

±

±

±

110

113

110

94

132

101

101

88

99

90

95

98

±

99

±

92

98

70

±

±

±

±

±

±

±

±

±

±

±

±

±

±

±

±

±

±

±

±

71

73

71

99

158

111

144

92

130

142

146

167

111

90

120

95

±

±

±

±

±

±

±

±

±

±

±

±

±

±

±

±

Mobile phosphorus content, mg/kg

Easily hydrolyzed nitrogen content, mg/kg

Humus content in the arable layer %

Reaction of soil solution, pH H2O

The amount of absorbed bases, mg-eq/100 g of soil

Agrochemical indicators

Indicators of ecological stability of soils

Factors

Parpurivtsi village, Vinnytsa region

Soil quality class

passport

№

Table 1 (continued)

84

88

84

97

151

126

89

67

99

110

146

141

83

92

118

107

±

±

±

±

±

±

±

±

±

±

±

±

±

±

±

±

The content of metabolic potassium, mg/kg

1.70

1.79

1.70

3.44

4.05

3.73

3.62

3.2

4.2

3.68

4.44

4.28

3.98

3.67

4.44

3.2

±

±

±

A

A

A

A

A

A

A

A

A

A

A

A

A

Content of mobile forms of microelements (mg/kg): copper

±

±

±

±

A

±

±

±

±

±

±

A

A

±

±

±

(continued)

1.9

2.1

1.9

5.55

6.75

5.2

5.86

5.2

5.82

5.3

5.97

6.72

6.44

5.4

5.6

3.94

Zinc

364 N. Ridei et al.

V

V

V

V

V

IV

VI

4.

5.

6.

7.

8.

9.

10.

23.7

10.1

±

21.5

21.5

21.6

21.5

19.1

A

A

A

A

A

A

Soil quality class

4.6

6.5

6.4

6.4

6.5

6.4

6.2

VI

VI

VII

V

VII

1.

2.

3.

4.

5.

2.48

3.51

3.45

3.45

3.46

3.45

2.77 142 140 140 160

± ± ± ± 108

140

±

±

108

±

±

A

±

±

±

±

±

69

110

98

98

99

98

69

0.18

0.2

0.14

0.08

0.18

A

A

A

A

A

Content of mobile forms of heavy metals (mg/kg): cadmium

5.24

6.24

6.08

3.94

5.74

Lead

N

N

N

N

N

Parameters of sanitary and hygienic condition

Factors

±

A

A

A

A

A

±

80

99

96

96

98

96

80

±

±

±

±

±

±

±

The content of metabolic potassium, mg/kg

0

0

0

0

0

A

A

A

A

A

Pesticide residues (DDT and its metabolites)

±

±

±

±

±

±

±

Mobile phosphorus content, mg/kg

Easily hydrolyzed nitrogen content, mg/kg

Humus content in the arable layer %

Reaction of soil solution, pH H2O

The amount of absorbed bases, mg-eq/100 g of soil

Agrochemical indicators

Indicators of ecological stability of soils

Factors

Bohdanivka village, Pryluky district

№ passport

Soil quality class

passport

№

Table 1 (continued)

±

±

±

±

±

±

±

1.7

2.1

2.5

2.5

2.9

2.5

1.8

Zinc

0.13

0.06

0.06

0.06

0.06

A

A

A

A

A

±

±

±

±

±

±

±

(continued)

Density of radionuclide contamination (Ki/km2 ): cesium-137

1.65

2.0

1.70

1.70

1.90

1.70

1.68

Content of mobile forms of microelements (mg/kg): copper

Environmental Assessment of Recreational Territories of Ukraine 365

Soil quality class

VII

VI

VII

VI

VII

VI

VI

VI

VI

VI

VI

VI

VI

VI

VI

VI

IV

VI

V

№ passport

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

Table 1 (continued)

0.15

0.18

0.26

0.16

0.12

0.14

0.13

0.14

0.1

0.1

0.12

0.18

0.09

0.1

0.1

0.13

0.16

0.14

0.14

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

Content of mobile forms of heavy metals (mg/kg): cadmium

4.96

6.53

6.17

6.37

5.68

5.91

5.98

6.31

4.92

4.59

6.45

6.35

5.85

4.55

7.09

7.08

6.31

6.08

6.16

Lead

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

Parameters of sanitary and hygienic condition

Factors

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

Pesticide residues (DDT and its metabolites)

0.06

0.13

0.06

0.06

0.06

0.13

0.13

0.13

0.06

0.13

0.13

0.13

0.06

0.06

0.06

0.06

0.06

0.06

0.06

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

(continued)

Density of radionuclide contamination (Ki/km2 ): cesium-137

366 N. Ridei et al.

V

VI

V

V

V

VI

25.

26.

27.

28.

29.

30.

V

V

V

V

V

V

V

IV

VI

2.

3.

4.

5.

6.

7.

8.

9.

10.

0.0013

0.0014

0.0016

0.0016

0.0017

0.0016

0.0015

0.0015

0.0016

0.0015

0.09

0.14

0.16

0.1

0.13

0.08

1.25 1.21

A

1.29

1.29

1.3

1.29

1.27

1.29

1.31

1.29

5.13

5.68

6.09

6.17

4.98

4.98

Lead

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

Content of mobile forms of heavy metals (mg/kg): cadmium

A

A

A

A

A

A

A

A

A

A

N

N

N

N

N

N

Parameters of sanitary and hygienic condition

Factors

A—appliable, ±—limited appliable, N—non appliable

V

1.

Parpurivtsi village, Vinnytsa region

Soil quality class

№ passport

Table 1 (continued)

0.0056

0.0062

0.0071

0.0063

0.0064

0.0071

0.0057

0.0061

0.0064

0.0061

0

0

0

0

0

0

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

Pesticide residues (DDT and its metabolites)

0.1

0.1

0.09

0.06

0.13

0.12

0.08

0.06

0.1

0.06

0.13

0.13

0.13

0.13

0.13

0.06

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

Density of radionuclide contamination (Ki/km2 ): cesium-137

Environmental Assessment of Recreational Territories of Ukraine 367

368

N. Ridei et al.

center is listed as settlement with the number of tourism objects from 21 to 30, the main type of recreational activity is ecological. Pryluky district has all the necessary opportunities and resources for development of recreational activities and tourism business, especially—rural, green and agritourism. On the territory of the district there are Gustinsky Holy Trinity Convent known to general public, large number of churches. It promotes development of religious tourism, pilgrimage. It is established that share of potential recreational areas of Vinnytsia Pobuzhzhia is over 20% in the land structure. According to territorial organization of tourism the district has developed recreational sphere. Typical types of tourism are: green, rural, cultural and cognitive. The recreational and tourist sphere of the district has significant number of monuments, in particular: nature—2, history—75, architecture—8, archeology—141, monumental art—4. Following types of recreational facilities are distinguished according to recreational purpose and direction of recreational activities of Vinnytsia Pobuzhzhia: religious-theological, cultural-historical, landscape-architectural, garden-park, traditional-household, social-rehabilitation-recreational, agro-recreational tourist and ecological-recreational. There is a great potential for development of gastronomic tourism in rural areas with development of gourmet routes, as the local national cuisine is represented by fairly wide range of dishes. Polyvariance of recreational and tourist routes allows provision of preferences and needs of various target groups of vacationers. For example, visiting historical and cultural, garden, architectural monuments, it is possible to service vacationers on the territory of farmsteads, rehabilitation and recovery in the agrosphere, agritourism, apitourism, various types of amateur tourism (hunting, fishing, berry picking, learning traditional crafts—pottery, blacksmithing, embroidery, weaving, weaving, etc.).

4 Conclusions Development in the field of recreation and tourism requires approving of the macrotourist zoning of Ukraine at the state level. It will promote development of recreation in regions with high recreational potential. Theoretical analysis of scientific approaches to the classification systems of recreational resources in scientific sources allowed to identify classification features of recreational resources and to build systematization scheme for existing classifications with additions. Systematization of approaches and diagnosis of recreational resources signs contributes to improvement of existing classifications of recreational resources of the agrosphere to establish their purpose. It will qualitatively assess and optimize use of recreational potential. The developed structural and logical organization scheme of ecological and social assessment of recreational areas of the agrosphere takes into account the current state of scientific and methodological support of assessment process, includes parametric

Environmental Assessment of Recreational Territories of Ukraine

369

characteristics of ecological condition of soils, lands, surface waters, social, ecosystems and potential opportunities for recreation. It allows developing a technique of ecological monitoring of various types of recreational territories of agrosphere on an example of the studied regions in the following researches. The agroecological assessment of soils for growing grain, fodder, technical and vegetable crops and assessment of suitability for obtaining ecologically safe products suitable for children’s, dietary nutrition were carried out. It is established that the soils of southern Pryluchchyna possess mobile forms of lead, which exceeds the maximum allowable concentrations by 2–3.5 times. They are unsuitable for growing agricultural products for baby and dietary nutrition. The lands of Vinnytsia Pobuzhzhia have optimal and acceptable conditions for grain, technical, fodder and vegetable crops. They are suitable and limitedly suitable for growing agricultural products that will be used for baby and dietary food. Accordingly, agricultural products are safe for local population and vacationers. The forecast for obtaining biologically complete ecologically safe raw materials and agricultural products was carried out. It is possible to obtain ecologically safe products and, in part, biologically complete on the studied soils of Vinnytsia Pobuzhye. We predict production of biologically defective and ecologically dangerous products on the soils of southern Pryluchchyna due to contamination with heavy metal (lead), insufficient content of mobile forms of nitrogen, some soils are moderately acidic. The territories of southern Pryluchchyna are recommended for local lore, landscape, historical, cognitive, pilgrimage and other types of recreational activities and tourism. Vinnytsia Pobuzhzhia has favorable conditions, first of all, for: health, medical, gastro-, apitourism, holding various gastro-festivals and other types of recreational activities.

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Legal Aspects of the Use of Renewable Energy Sources and the Implementation of the Concept of “Green Economy” in Ukraine in the Context of Sustainable Development Strategy Volodymyr Yermolenko , Olena Hafurova , Maryna Deineha , Tamara Novak , and Yuliia Shovkun Abstract The article examines the features of legal support for the use of renewable energy sources and the implementation of the concept of “green economy” in Ukraine in the context of sustainable development strategy. The essence of the concept of “green economy”, basic principles, directions of realization are clarified. The current state of renewable energy in Ukraine, directions of stimulation to its development are determined. The international standards and provisions of the current domestic legislation on the certain problems are analyzed. It is established that a number of important steps have been taken in Ukraine to improve the legal framework both in the energy sector in general and in the field of renewable energy. However, despite the significant potential of almost all types of renewable energy sources in Ukraine, the large number of adopted regulations, the share of renewable energy in the energy balance of the country remains insignificant. Ukraine’s renewable energy sector operates in a generally unfavorable climate, which is affected by political risks, economic instability, military threats, and insufficient investment protection. The necessity of development of renewable energy in Ukraine by introduction of principles of sustainability in economic, social and ecological spheres is substantiated. Keywords Renewable energy · Renewable energy sources · “Green” economy · “Green” development · Sustainable consumption and production · Sustainable development

1 Introduction The leading trend of modern social development is the spread of globalization processes, in which the level of depletion of natural resources is growing. The needs V. Yermolenko · O. Hafurova · M. Deineha (B) · T. Novak · Y. Shovkun National University of Life and Environmental Sciences of Ukraine, Kyiv, Ukraine e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Zaporozhets (ed.), Systems, Decision and Control in Energy III, Studies in Systems, Decision and Control 399, https://doi.org/10.1007/978-3-030-87675-3_23

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for natural resources far exceed the volume and speed of their reproduction. As a result, inevitably there is a depletion of natural resources, which leads to their shortage [1–3]. In the second half of the twentieth century, the question arose about the prospects for the preservation of humanity on the planet in connection with the global environmental crisis. The need to find an acceptable balance in relations between society and the environment requires the consolidation of actions of the world community, the development and adoption of international legal instruments aimed at ensuring a favorable environment at an increasing rate of exploitation of natural resources, and strengthening of technogenic impact on the environment [4–6]. A necessary factor in changing approaches to the use of natural resources is the transition to a strategy of sustainable development and one of the means to ensure it—a “green” economy, the essence of which is to improve human well-being and strengthen social justice while reducing risks to the environment and scarcity of natural resources [7–10]. The use of renewable energy sources in Ukraine is one of the means of practical implementation of the concept of “green” economy, the most important area of energy policy of our country, aimed at saving traditional fuel and energy resources and improving the environment. Increasing the use of renewable energy sources in the energy balance of Ukraine will increase the level of diversification of energy sources, which will strengthen the energy independence of the state. According to the International Renewable Energy Agency (IRENA), Ukraine has the largest technical potential among the countries of Southeast Europe for the use of renewable energy sources, the greatest is the technical feasibility of using wind and solar power plants. Figures 1 and 2, according to IRENA, show the current state of the distribution of energy produced from different types of renewable energy sources, respectively, in the world and in Ukraine. At the same time, the State Agency on Energy Efficiency and Energy Saving of Ukraine estimates the technically feasible potential for energy production from renewable energy sources and alternative fuels at more than 98.0 million tons of conventional fuel per year (Table 1). This actually allows to replace about 50% of the total energy consumption of our state. In recent years, Ukraine has seen a gradual increase in the use of renewable energy sources, but the difficult economic situation in the country does not contribute to their development.

2 Literature Analysis and Problem Statement Such scientists in the field of economists have devoted their research to identifying the features of the use of renewable energy sources and the implementation of the concept of “green economy” in Ukraine in terms of sustainable development as: O. Borisov, I. Doronina, M. Drapak, N. Zakharkevych, S. Ignatieva, Ja. Kvach, S.

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Fig. 1 Distribution of energy produced from different types of renewable energy sources in the world. (Source https://www.irena.org/Statistics/View-Data-by-Topic/Capacity-and-Generation/Tec hnologies)

Kudrya, O. Lavrinenko, N. Orlova, A. Okhotina, O. Rybalkin, K. Firsova, O. Chmyr and others. However, a significant part of the issues related to the legal provision of certain issues remains out of the attention of scientists.

3 Purpose and Objectives of the Study Research of the current state of legal support for the use of renewable energy sources in Ukraine in the context of the implementation of the concept of “green” economy as a component of sustainable development.

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Fig. 2 Distribution of energy produced from different types of renewable energy sources in Ukraine. (Source https://www.irena.org/Statistics/View-Data-by-Topic/Capacity-and-Generation/ Technologies) Table 1 Technically achievable potential for energy production from renewable energy sources and alternative fuels (according to the State Agency on Energy Efficiency and Energy Saving of Ukraine, source: https://saee.gov.ua/uk/activity/vidnovlyuvana-enerhetyka/potentsial) Directions of development of renewable energy sources

Annual technically achievable energy potential, million tons of conventional fuel

Wind energy

28.0

Solar energy

6.0

Small hydropower

3.0

Bioenergy

31.0

Geothermal thermal energy

12.0

Environmental energy (heat pumps)

18.0

The total amount of replacement of traditional fuel and energy resources

98.0

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4 Research Results The instability of development trends inherent in the modern world, the turbulence of the world economy, the imbalance of economic, social and environmental trends necessitate the formation of new economic models within the paradigm of sustainable development, which has become central to humanity in the XXI century. The need to move towards sustainable development in the world was largely due to the realization of a critical exacerbation of the disparity between economic development and environmental degradation. Over the past 30 years, there has been a significant increase in world GDP—more than four times, which has increased the living standards of hundreds of millions of people [7, 8]. However, this growth has been largely due to the global depletion of natural capital and the degradation of ecosystems. At the beginning of the XXI century environmental problems were growing rapidly in the world: growing shortage of fresh water and food, climate change, biodiversity loss and forests, desertification and many others. Here are just some of these problems: (1) in the world 40% of land degrades due to reduced soil fertility, erosion and depletion; land productivity decreases, which in pessimistic scenarios can lead to a loss of 50% of potential yield; (2) lands where a third of the world’s population lives are threatened by desertification; (3) almost 1 billion people lack clean drinking water; 2.6 billion people don‘t have access to adequate sanitation; 1.4 million children under the age of five die each year due to lack of clean water and lack of access to essential sanitation; In the future, water scarcity will only increase, and in 20 years its reserves will meet only 60% of world needs; 4) the disappearance of forests on the planet continue; in 2000–2010 the area of forests decreased annually by 5.2 million hectares; (5) climate change may affect about 2 billion people living in coastal areas, etc. [11]. If modern “anti-steel” trends continue, the use of natural resources and pollution in the next half century will increase several times. Preservation of negative environmental trends can lead to extremely dangerous consequences for all mankind and individual countries. In 2015, UN member states unanimously adopted a new Agenda—a bold global agenda for a sustainable future until 2030. To replace the Millennium Development Goals (2000–2015), 17 new principles of Sustainable Development Goals (2016– 2030) were developed to ensure the balance of all three components: economic, social and environmental. The UN Agenda declares its readiness to save the planet from degradation, primarily through the introduction of rational patterns of consumption and production, rational use of natural resources and taking urgent measures to combat the environmental crisis. Sustainable development goals are also designed to facilitate the transition to new economic models. Therefore, in the context of the strategy of sustainability in the world, new models of the economy related to environmental factors have become widespread both in theory and in practice (Table 2).

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Table 2 Economic models related to environmental factors Economic model

Specifics

Green economy

It is economy aims at reducing environmental risks and ecological scarcities, and that aims for sustainable development without degrading the environment

Green growth

It is economy based on the understanding that as long as economic growth remains a predominant goal, a decoupling of economic growth from resource use and adverse environmental impacts is required

Low-carbon economy

Is an economy based on low-carbon power sources that therefore has a minimal output of greenhouse gas emissions into the atmosphere, specifically carbon dioxide

Bioeconomy

Is an economy encompassing the sustainable production of renewable resources from land, fisheries and aquaculture environments and their conversion into food, feed, fiber bio-based products and bio-energy as well as the related public goods

Blue economy

Is an emerging concept which encourages better stewardship of our ocean or ‘blue’ resources

The foundations for the formation of the concept of “green” economy were laid within the concept of sustainable development in the late 80’s of last century. The new vision of the economy is especially clear in the concept documents of international organizations. UN documents, including UNEP (United Nations Environment Program), UNDESA (United Nations Department of Economic and Social Affairs), UNCTAD (United Nations Conference on Trade and Development), UNCSD (United Nations Conference on Sustainable Development), along with the Sustainable Development Strategy and the concept of “green” economy, its basic principles, advantages, risks and generalized international experience in this field. Thus, official UNEP documents state that the “green” is an economy that leads to increased human well-being and social justice while significantly reducing risks to the environment and the scarcity of environmental resources. The emphasis is on the efficient use of natural resources, taking into account existing social factors. In a “green” economy, UNEP believes, income and employment growth should be driven by public and private investment that reduces pollution, increases resource efficiency, prevents biodiversity loss, and expands ecosystem services [12]. UNDESA’s definition of a green economy is similar. The “green” economy is a favorable component of the overall goal of sustainable development. It should be interpreted, focusing primarily on the sign of ensuring economic and social progress on the basis of environmentally friendly activities. Important features of such an economy are: efficient use of natural resources; preservation and increase of natural capital; pollution reduction; low carbon emissions; prevention of loss of ecosystem services and biodiversity; income growth and employment [13]. The new model of the economy is reflected in the priorities of practical activities of many states and private business. Thus, the European Community has adopted a Program for the development of green economy, circular economy, bioeconomy for 2030–2050, The Paris Climate Agreement of 2015 directs all states to transition

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to a low-carbon economy. In the non-financial, social and environmental reporting of companies, the consolidation of sustainable development goals is becoming increasingly important. In December 2014, the UN adopted a program for the development of a “green” economy “EaP GREEN” (“Greening the economy of the Eastern Partnership of the European Union”)—a large regional program implemented by UNECE (UN Economic Commission for Europe), OECD (Organization for Economic Cooperation and Development), UNEP (United Nations Environment Program) and UNIDO (United Nations Industrial Development Organization), aimed at assisting the six Eastern Partnership countries of the European Union: Armenia, Azerbaijan, Belarus, Georgia, the Republic of Moldova and Ukraine– to the transition to a “green” economy. The goal of the program is the transition of the Eastern Partnership countries to a “green” model of business development and business by distinguishing between economic growth and environmental degradation and depletion of resources, in particular: integration of sustainable consumption and production into national development plans, legislation and regulations in order to create a sound legal basis for the development of “green” growth policy in accordance with the approaches of the European Union; encouraging the use of strategic environmental assessment and environmental impact assessment as important tools for economic development planning based on the principles of environmental sustainability; ensuring the transition to a “green” model of business development and adaptation by adapting and demonstrating the benefits of applying practices and methods of sustainable consumption and production in certain sectors of the economy. In International Documents (Declaration of the United Nations Conference on the Environment (Stockholm, 1972), Report of the United Nations Conference on Environment and Development (Rio de Janeiro, 1992), Earth Charter (The Hague, 2000), Johannesburg Declaration of Sustainable Development (Johannesburg, September 4, 2002) and others) The UN defines the principles of sustainable development and “green” economy (Table 3). In pursuance of the “Agenda for the XXI century.” the UN has developed strategies, programs, and plans for the transition to sustainable development in almost all countries of the world. Ukraine has also adopted a number of regulations in this area. The Decree of the President of Ukraine of September 30, 2019 № 722 stipulates that the Sustainable Development Goals of Ukraine for the period up to 2030 are guidelines for the development of draft forecast and program documents, draft regulations to ensure the balance of economic, social and environmental dimensions of sustainable development of Ukraine. The adopted Strategy of the state ecological policy of Ukraine for the period till 2030 in many respects corresponds to the purposes of transition to sustainable development and “green” economy: raising the level of public environmental awareness; improving the environmental situation and the level of environmental safety; achieving a safe environment for human health; integration of environmental policy and improvement of the system of integrated environmental management; cessation of losses of biological and landscape diversity and formation of ecological network; ensuring ecologically balanced nature management; improving regional environmental policy. Strategic priorities of our state are

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Table 3 Principles of sustainable development and “green” economy The principle of sustainable development and the “green economy”

The content of the principle

Fair distribution of wealth

Achieving social and economic justice, fair distribution of world natural resources

Economic equality and justice

Providing financial and technological assistance to less developed countries in order to minimize the economic gap and maintain environmental sustainability

Equality of generation

Managing environmental resources and ecosystems in a way that restores and preserves the value of natural resources for future generations

Preventive approach

Early detection of environmental risk

The right to development is determined

Human development in harmony with the natural environment is fundamental to achieving sustainable development

Taking into account external factor

Market prices must reflect real social and environmental costs and benefits

International cooperation

Regular purposeful and coordinated joint activities of participants in international relations, carried out on the basis of generally accepted principles and norms of international law and aimed at reconciling their interests to achieve common goals of sustainable development

International responsibility

Determination by the norms of international law of the responsibility of subjects arising for the commission of an international legal tort in the field of sustainable development

Awareness, participation and responsibility

Free access to information about the state of the environment, participation in environmental decision-making processes, responsibility for these decisions

Sustainable production and consumption of natural resources

Use of goods and services that meet basic needs and improve the quality of life with minimal use of natural resources and with the least harm to the environment

Strategic, coordinated and comprehensive planning

Will accelerate the achievement of socio-economic and environmental sustainability

The transition of the traditional economy to “green” to sustainable development is substantiated

Based on the developments of branch sciences and practice of their use (continued)

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Table 3 (continued) The principle of sustainable development and the “green economy”

The content of the principle

Revaluation of welfare

Human well-being and quality of life, the state of the environment should become the main development goals, which should receive adequate indicators of evaluation

Gender equality

Ensuring gender equality, women play an extremely important role in implementing change in nature management and development

Conservation of biodiversity and prevention of environmental pollution

An integral part of human well-being

Management focused on protecting the resilience of ecosystems

Preventing their irreversible damage

fixed in other regulations, in particular: Energy Strategy of Ukraine for the period up to 2030, strategic programs of the Cabinet of Ministers of Ukraine, etc. At the same time, an effective model of legal regulation that would ensure the integration of efforts for economic growth, the pursuit of social justice and environmental management, hasn’t yet been created in Ukraine. The vast majority of measures proposed to ensure sustainable development are of a general recommendatory nature. One of such measures is the introduction and development of renewable energy in Ukraine. Today in our country there is a steady trend towards the development of renewable energy sources, as at the present stage of development the problem of interaction between energy and the environment acquires new features, negatively affecting the environment in general and individual natural resources in particular. The greater the scale of development of energy supply and consumption, the more intense the impact on all components of the environment. A developed energy sector inevitably leads to environmental pollution. In 1994, the Verkhovna Rada of Ukraine adopted the Law of Ukraine “On Energy Conservation” of July 1, 1994, which defined the legal, economic, social and environmental bases of energy conservation for all businesses, as well as for citizens, and provided for the use of renewable sources. energy to ensure the energy security of the state. Further legal regulation of public relations in the field of renewable energy sources and promotion of their expansion in the energy sector was acquired with the adoption of the Law of Ukraine “On Alternative Energy Sources” of February 20, 2003, which provides that renewable energy sources are sources that are constantly exist or periodically appear in the natural environment, namely: solar energy, wind, aerothermal, geothermal, hydrothermal, energy of waves and tides, hydropower, biomass energy, gas from organic waste, gas sewage treatment plants, biogas. In September 2010, the Protocol on Ukraine’s Accession to the Energy Community Treaty was signed, ratified by the Law of Ukraine “On Ratification of the Protocol on Ukraine’s Accession to the Energy Community Treaty” of December 15, 2010. According to this Law, on February 1, 2011 Ukraine became a full member of

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the Energy Community and undertook to implement the main acts of EU energy legislation into national legislation. Ukraine’s accession to the Energy Community has provided opportunities and tools for structural reform in the energy sector. The introduction of European norms and standards of the acquis communautaire in the energy sector, as well as in the field of environmental protection allows our country to gradually restructure the economy and embark on the path of sustainable development. In October 2012, the Council of Ministers of the Energy Community adopted Decision D/2012/04/MC-EnC “On the implementation of Directive 2009/28/EC and amending Article 20 of the Treaty establishing the Energy Community”, according to which each party to the Treaty should bring into force the laws, regulations and administrative provisions necessary to comply with Directive 2009/28/EC of the European Parliament and of the Council of 23 April 2009 on the promotion of the use of energy from renewable sources. According to decision D/2012/04/MC-EnC, Ukraine has committed itself to reach 11% of energy produced from renewable energy sources in the overall structure of energy consumption by 2020. In fulfillment of the undertaken commitments, in 2013 the Cabinet of Ministers of Ukraine by the order of July 24, 2013 № 1071 approved the updated Energy Strategy of Ukraine for the period up to 2030, which defines that the development of renewable energy sources is an important factor in improving energy security and reducing the anthropogenic impact of energy on the environment. Large-scale use of the potential of renewable energy sources in Ukraine has not only domestic but also significant international significance as an important factor in combating global climate change in general, improving the overall state of energy security in Europe. The Association Agreement between Ukraine, of the one part, and the European Union, the European Atomic Energy Community and their Member States, of the other part, of 27 June 2014 (ratified on 16 September 2014) defines mutual cooperation in the field of development and support for renewable energy, taking into account the principles of economic feasibility and environmental protection (Article 338). In pursuance of the provisions of the Association Agreement, on October 1, 2014, the Cabinet of Ministers of Ukraine adopted the National Renewable Energy Action Plan for the period up to 2020, which stipulates that the development of renewable energy should be the result of consistent and balanced public policy. security, industrial development and diversification of energy sources. And in 2017, the Cabinet of Ministers of Ukraine adopted an order “On approval of the Energy Strategy of Ukraine for the period up to 2035” Security, energy efficiency, competitiveness”, which, defining the main strategic goals of the state in energy, is a benchmark in establishing state priorities in energy policy. Ukraine’s energy strategy for the period up to 2035 “Security, energy efficiency, competitiveness” sets quite ambitious goals in terms of the development of alternative energy. To stimulate the development of renewable energy, the Law of Ukraine “On Amendments to Certain Laws of Ukraine on Ensuring Competitive Conditions for Electricity Production from Alternative Energy Sources” of April 25, 2019 was adopted, according to which: the “green” tariff is attached to the euro; “green” tariff

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for electricity from biomass and biogas increased by 10% to 12.39 e ct/kWh per hour; abolished the requirements for the local component and introduced a surcharge to the “green” tariff of 5% and 10% for the use of Ukrainian-made equipment at the level of 30% and 50%; introduced “green” tariff for geothermal electrical installations, for solar and wind power plants of private households with a capacity of up to 30 kW. It should be noted that today the level of “green” tariffs in Ukraine is one of the highest in Europe [14]. According to the Law of Ukraine “On the Electricity Market” № 2019-VIII of April 13, 2017, it is possible to enter into long-term contracts for the purchase of electricity produced at a “green” tariff until 2030. The Tax and Customs Codes of Ukraine contain provisions that provide: reduction of land tax for renewable energy companies; tax exemption: profits from the main activities of enterprises in the field of energy, producing electricity from renewable sources; profits of biofuel producers received from the sale of biofuels; exemption from value added tax on operations on import into the customs territory of Ukraine of equipment operating on renewable energy sources, equipment and materials for the production of alternative fuels or for the production of energy from renewable energy sources, as well as exemption from import duties on such equipment, equipment and materials, etc. Thus, in particular, the Tax Code of Ukraine stipulates that until December 31, 2022, transactions on import of the following goods into the customs territory of Ukraine are exempt from value added tax: wind power generating sets, solar photovoltaic panels, inverters and transformers of appropriate capacities. The Law of Ukraine “On Energy Lands and the Legal Regime of Special Zones of Energy Facilities” of July 9, 2010 stipulates that on land (industry, transport, communications, energy, defense and other purposes) alternative accommodation facilities may be located. energy companies that use renewable energy sources, regardless of the purpose of such land. Thanks to these legislative incentives, the total capacity of renewable electricity facilities in Ukraine, which has a “green” tariff, has increased by 68% [15]. According to the Report of the International Renewable Energy Agency from 46 European countries, Ukrainian renewable energy is the twenty-second largest, excluding large hydropower plants [16]. However, today the rapid growth of the issuance of technical conditions and connections to the energy system of Ukraine of renewable energy facilities (namely wind power plants and solar power plants) leads to a certain destabilization of its sustainable operation. As of 2018, the level achieved in our country is 9% (including large hydropower plants) of its total production in Ukraine, and excluding hydropower plants, this share does not exceed 2% [17], while having a fairly high cost according to the Institute of Renewable of Energy of Ukraine [18]. The efficiency of the “green” economy, including the state of development of renewable energy, in the world is determined on the basis of expert assessment according to the Global Green Economy Index (GGEI), prepared by the rating agency Dual Citizen. The GGEI index uses quantitative and qualitative indicators to determine the performance of each state on four indicators: management and climate change (media coverage, international forums, climate change results); efficiency of industries (construction, transport, energy, tourism, etc.); markets and investments

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(renewable energy, eco-technologies, stimulation of ecological investments, corporate stability); environment (agriculture, air and water quality, biological and natural environment, forests). In 2018, according to GGEI Ukraine took 120th place [19]. Another important indicator is the Environmental Performance Index (ERI)—a comprehensive indicator of the assessment of environmental policy of the state and its individual actors. The index is determined by 24 performance indicators covering health, the environment and the viability of the ecosystem. These indicators make it possible to determine the extent to which countries have achieved the set goals of environmental policy. In 2018, Ukraine took 109th place among 132 countries with a rate of 52.87% [19]. Ukraine’s low indicators indicate the need to strengthen the state’s efforts to ensure sustainable development for a number of factors, including in the field of renewable energy development.

5 Discussion and Conclusions The world is now facing the global challenges of a rapidly growing population and the increasing pressure on the environment related to it that should be prevented. The survival and development of humanity requires the transition to a “green” economy [20]. The significant role of renewable energy in our quest for a just transition to reach the Sustainable Development Goals is unquestionable. In particular, the global challenge of reaching net zero carbon emissions by 2050 will require the most technically feasible, costeffective, and socially acceptable interventions [21, 22], with immediate and continual actions from all stakeholders—governments, businesses, investors, and citizens—if we are to continue on a sustainable pathway as outlined in a recent report of the International Energy Agency [23]. In recent years, Ukraine has taken a number of important steps to improve the legal framework both in the energy sector in general and in the field of renewable energy. However, this wasn’t enough. The commitment to reach 11% of renewable energy in the country’s overall energy consumption by 2020 has failed to reach. Despite the significant potential of almost all types of renewable energy sources in Ukraine, the large number of adopted regulations, the share of renewable energy in the energy balance [24] of the country remains insignificant—less than 9%. The total share of renewable energy sources in the energy balance of Ukraine is shown in Fig. 3. Ukraine’s renewable energy sector operates in a generally unfavorable climate, which is affected by political risks, economic instability, military threats, and insufficient investment protection. In addition, the commitment to transpose the main pieces of EU energy legislation into national law is fragmentary and somewhat formalistic [25]. In addition, EU legislation in all areas is a dynamic rather than a sustainable process. European integration should be seen not only as a foreign policy, but above all as a domestic one, which includes issues of public policy in one area or another,

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The total share of renewable energy sources in the energy balance of Ukraine, % 2020 2019 2018 2017 2016 2015 2014 0

2

4

6 Fact

8

10

12

Plan

Fig. 3 The total share of renewable energy sources in the energy balance of Ukraine (according to the State Agency on Energy Efficiency and Energy Conservation of Ukraine, source: https://saee. gov.ua/sites/default/files/RE_SAEE_2019.pdf)

as well as the approximation of legislation and its proper implementation and application. Considering that changes in the energy sector can only begin through internal reforms.

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