Simulation and Modeling Methodologies, Technologies and Applications: 9th International Conference, SIMULTECH 2019 Prague, Czech Republic, July 29-31, 2019, Revised Selected Papers [1st ed.] 9783030558666, 9783030558673

Citation preview

Advances in Intelligent Systems and Computing 1260

Mohammad S. Obaidat Tuncer Ören Helena Szczerbicka Editors

Simulation and Modeling Methodologies, Technologies and Applications 9th International Conference, SIMULTECH 2019, Prague, Czech Republic, July 29–31, 2019, Revised Selected Papers

Advances in Intelligent Systems and Computing Volume 1260

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland Advisory Editors Nikhil R. Pal, Indian Statistical Institute, Kolkata, India Rafael Bello Perez, Faculty of Mathematics, Physics and Computing, Universidad Central de Las Villas, Santa Clara, Cuba Emilio S. Corchado, University of Salamanca, Salamanca, Spain Hani Hagras, School of Computer Science and Electronic Engineering, University of Essex, Colchester, UK László T. Kóczy, Department of Automation, Széchenyi István University, Gyor, Hungary Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX, USA Chin-Teng Lin, Department of Electrical Engineering, National Chiao Tung University, Hsinchu, Taiwan Jie Lu, Faculty of Engineering and Information Technology, University of Technology Sydney, Sydney, NSW, Australia Patricia Melin, Graduate Program of Computer Science, Tijuana Institute of Technology, Tijuana, Mexico Nadia Nedjah, Department of Electronics Engineering, University of Rio de Janeiro, Rio de Janeiro, Brazil Ngoc Thanh Nguyen , Faculty of Computer Science and Management, Wrocław University of Technology, Wrocław, Poland Jun Wang, Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong

The series “Advances in Intelligent Systems and Computing” contains publications on theory, applications, and design methods of Intelligent Systems and Intelligent Computing. Virtually all disciplines such as engineering, natural sciences, computer and information science, ICT, economics, business, e-commerce, environment, healthcare, life science are covered. The list of topics spans all the areas of modern intelligent systems and computing such as: computational intelligence, soft computing including neural networks, fuzzy systems, evolutionary computing and the fusion of these paradigms, social intelligence, ambient intelligence, computational neuroscience, artificial life, virtual worlds and society, cognitive science and systems, Perception and Vision, DNA and immune based systems, self-organizing and adaptive systems, e-Learning and teaching, human-centered and human-centric computing, recommender systems, intelligent control, robotics and mechatronics including human-machine teaming, knowledge-based paradigms, learning paradigms, machine ethics, intelligent data analysis, knowledge management, intelligent agents, intelligent decision making and support, intelligent network security, trust management, interactive entertainment, Web intelligence and multimedia. The publications within “Advances in Intelligent Systems and Computing” are primarily proceedings of important conferences, symposia and congresses. They cover significant recent developments in the field, both of a foundational and applicable character. An important characteristic feature of the series is the short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results. ** Indexing: The books of this series are submitted to ISI Proceedings, EI-Compendex, DBLP, SCOPUS, Google Scholar and Springerlink **

More information about this series at http://www.springer.com/series/11156

Mohammad S. Obaidat Tuncer Ören Helena Szczerbicka •

•

Editors

Simulation and Modeling Methodologies, Technologies and Applications 9th International Conference, SIMULTECH 2019, Prague, Czech Republic, July 29–31, 2019, Revised Selected Papers

123

Editors Mohammad S. Obaidat Fellow of IEEE, Dean of College of Computing and Informatics University of Sharjah Sharjah, United Arab Emirates

Tuncer Ören School of Electrical Engineering and Computer Science University of Ottawa Ottawa, ON, Canada

University of Jordan Amman, Jordan University of Science and Technology Beijing, China Helena Szczerbicka Universität Hannover Hannover, Germany

ISSN 2194-5357 ISSN 2194-5365 (electronic) Advances in Intelligent Systems and Computing ISBN 978-3-030-55866-6 ISBN 978-3-030-55867-3 (eBook) https://doi.org/10.1007/978-3-030-55867-3 © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved 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

The present book includes extended and revised versions of a set of selected papers from the 9th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2019), which was held in Prague, Czech Republic, in the period July 29–31, 2019. SIMULTECH 2019 received 62 paper submissions from 33 countries, of which 12.9% were included in this book. The papers were selected by the event chairs, and their selection is based on a number of criteria that includes the reviews and suggested comments provided by the program committee members, the session chairs’ assessments and also the program chairs' global view of all papers included in the technical program. The authors of selected papers were then invited to submit a revised and extended version of their papers with at least 30% new material. The purpose of the 9th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2019) was to bring together researchers, engineers, scientists and applied mathematicians and practitioners interested in the advances and applications of system simulation. Four simultaneous tracks were held, covering on one side domain-independent methodologies and technologies and on the other side practical work developed in specific application areas. The specific topics listed under each of these tracks highlight the interest of this conference in aspects related to computing, including conceptual modeling, agent-based modeling and simulation, interoperability, ontologies, knowledge-based decision support, petri nets, business process modeling and simulation, among others. The papers selected to be included in this book contribute to the understanding of relevant trends of current research in dynamical systems models and mathematical simulation, simulation in education, e-learning and training, mathematical modeling and simulation of multi-domains systems that consist of boom structure and hydraulic drive systems, simulation models of mangrove forests and environmental modeling and simulation, neural networks modeling and simulation, hint-based configuration of co-simulations with algebraic loops, and dynamic modeling of industrial mixing tanks with a slurry centrifugal pumps and optimal trigger sequence for non-iterative co-simulation with different coupling step size. v

vi

Preface

We would like to thank all the authors for their contributions as well as to the reviewers who have helped ensuring the quality of this publication. We also thank the staff of INSTICC and Springer for their good efforts and kind cooperation. July 2019

Mohammad S. Obaidat Tuncer Ören Helena Szczerbicka

Organization

Conference Chair Mohammad Obaidat

Fellow of IEEE, Dean of College of Computing and Informatics, University of Sharjah, UAE, and with University of Jordan, Jordan, and University of Science and Technology Beijing, China

Program Chair Tuncer Ören (Honorary) Helena Szczerbicka

University of Ottawa, Canada Universität Hannover, Germany

Program Committee Saleh Abdel-Afou Alaliyat Nael Abu-Ghazaleh Lyuba Alboul Mikulas Alexik Ayman Aljarbouh Vera Angelova

Carlos Argáez Gianfranco Balbo Simonetta Balsamo Juan Antonio Barcelo Isaac Barjis

Norwegian University of Science and Technology, Norway University of California, Riverside, USA Sheffield Hallam University, UK University of Zilina, Slovak Republic University of Central Asia, Kyrgyzstan Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Bulgaria University of Iceland, Iceland University of Torino, Italy University of Venezia Ca’ Foscari, Italy Autonoma University of Barcelona, Spain City University of New York, USA

vii

viii

Wolfgang Borutzky Christos Bouras António Brito Agostino Bruzzone Marian Bubak

Larysa Burtseva Francesco Casella Rodrigo Castro Srinivas Chakravarthy Gopinath Chennupati Lawrence Chung Franco Cicirelli Tanja Clees Flavio Correa da Silva Andrea D’Ambrogio Sanjoy Das Guyh Dituba Ngoma Anatoli Djanatliev Atakan Dogan Lorenzo Donatiello Julie Dugdale Dirk Eisenbiegler Sabeur Elkosantini Zuhal Erden Denis Filatov Jason Friedman Claudia Frydman Marco Furini José Manuel Galán Charlotte Gerritsen Anna Golovkinav Alexandra Grancharova Francisco Grimaldo

Organization

Bonn-Rhein-Sieg University of Applied Sciences, Germany University of Patras and CTI&P Diophantus, Greece INESC TEC, Faculdade de Engenharia, Universidade do Porto, Portugal University of Genoa, Italy AGH University of Science and Technology Krakow, Poland / University of Amsterdam, Netherlands Instituto de Ingeniería, Universidad Autónoma de Baja California, Mexico Politecnico di Milano, Italy University of Buenos Aires, Argentina Kettering University, USA Los Alamos National Laboratory, USA The University of Texas at Dallas, USA Università della Calabria, Italy Fraunhofer Institute for Algorithms and Scientific Computing (SCAI), Germany University of Sao Paulo, Brazil Università di Roma “Tor Vergata”, Italy Kansas State University, USA Université du Québec en Abitibi-Témiscamingue, Canada Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany Eskisehir Technical University, Turkey Univ. of Bologna, Italy Laboratoire d’Informatique de Grenoble, France University of Furtwangen, Germany University of Monastir, Tunisia ATILIM University, Turkey Institute of Physics of the Earth, Russian Academy of Sciences, Russian Federation Tel-Aviv University, Israel Aix Marseille University, France Università di Modena e Reggio Emilia, Italy Universidad de Burgos, Spain Vrije Universiteit Amsterdam, Netherlands Saint Petersburg State University, Russian Federation University of Chemical Technology and Metallurgy, Bulgaria Universitat de València, Spain

Organization

Mykola Gusti Ibrahim Hameed Maamar Hamri Cathal Heavey Monika Heiner Martin Hruby Tsan-Sheng Hsu Xiaolin Hu Ralph Huntsinger Giuseppe Iazeolla Nobuaki Ishii Shafagh Jafer Syed Waqar ul Qounain Jaffry Emilio Jimenez Macias Björn Johansson Catholijn Jonker Nina Kargapolova Anniken Karlsen Peter Kemper Etienne Kerre William Knottenbelt Juš Kocijan Petia Koprinkova-Hristova Vladimir Kotev Vladik Kreinovich Claudia Krull Jean Le Fur Willem le Roux Sanghyun Lee Richard Lipka Antonio Lope Maria Celia Lopes Adolfo Lopez-Paredes Ulf Lotzmann Johannes Lüthi José Machado Andrea Marin Carla Martin-Villalba Moreno Marzolla

ix

International Institute for Applied Systems Analysis, Austria NTNU, Norway Laboratoire d’Informatique et Systèmes, France University of Limerick, Ireland Brandenburg University of Technology Cottbus, Germany Brno University of Technology, Czech Republic Institute of Information Science, Academia Sinica, Taiwan, Republic of China Georgia State University, USA California State University, Chico Campus, USA University of Roma TorVergata, Italy Kanagawa University, Japan Embry-Riddle University, USA University of the Punjab, Pakistan University of La Rioja, Spain Chalmers University of Technology, Sweden Delft University of Technology, Netherlands Institute of Computational Mathematics and Mathematical Geophysics, Russian Federation NTNU in Ålesund, Norway College of William and Mary, USA Ghent University, Belgium Imperial College London, UK Jozef Stefan Institute, Slovenia IICT:Bulgarian Academy of Sciences, Bulgaria Bulgarian Academy of Sciences, Bulgaria University of Texas at El Paso, USA Otto-von-Guericke University, Germany IRD (Inst. Res. Development), France CSIR, South Africa University of Michigan, USA University of West Bohemia, Czech Republic University of Porto, Portugal COPPE-UFRJ, Brazil University of Valladolid, Spain University of Koblenz, Germany FH Kufstein Tirol, Austria Institute of Engineering, Polytechnic of Porto, Portugal University of Venice, Italy UNED, Spain University of Bologna, Italy

x

Radek Matušu Nuno Melão Francesco Mercaldo Adel Mhamdi Qi Mi Bozena Mielczarek Federico Milani Gabriele Milani Ramzi Mirshak Vikram Mittal Roberto Montemanni Jairo Montoya-Torres Nobuhiko Mukai Bertie Müller Ivan Mura Navonil Mustafee Nazmun Nahar Àngela Nebot Bao Nguyen Lialia Nikitina Paulo Novais Paulo Oliveira Feng Pan María Parra Santos George Pavlidis Alessandro Pellegrini Sapienza L. Felipe Perrone Alexandre Petrenko Alexandr Petukhov Régis Plateaux Tomas Potuzak Jerzy Respondek M. Riazi Ella Roubtsova Jaroslav Rozman Cristina Ruiz-Martin

Organization

Tomas Bata University in Zlin, Czech Republic Instituto Politécnico de Viseu, Escola Superior de Tecnologia e Gestão de Viseu, Portugal National Research Council of Italy (CNR), Italy RWTH Aachen University, Germany University of Pittsburgh, USA Wroclaw University of Science Technology, Poland CHEM.CO Consultant, Italy Politecnico di Milano, Italy Defence R&D Canada, Canada United States Military Academy, USA University of Modena and Reggio emilia, Italy Universidad de La Sabana, Colombia Tokyo City University, Japan Swansea University, UK Duke Kunshan University, China University of Exeter, UK University of Jyvaskyla, University of Tartu, Finland Universitat Politècnica de Catalunya, Spain Defence R&D Canada and University of Ottawa, Canada Fraunhofer Institute for Algorithms and Scientific Computing (SCAI), Germany Universidade do Minho, Portugal Universidade de Tras os Montes e Alto Douro, Portugal Liaoning Normal University, China University of Valladolid, Spain “Athena” Research Centre, Greece University of Rome, Italy Bucknell University, USA CRIM, Canada Lobachevsky State University of Nizhni Novgorod, Russian Federation SUPMECA, France University of West Bohemia, Czech Republic Silesian University of Technology, Poland Kuwait University, Kuwait Open University of the Netherlands, Netherlands Brno University of Technology, Czech Republic Carleton University, Canada

Organization

Katarzyna Rycerz Paulo Salvador Antonella Santone Jean-François Santucci Frederik Schaff Xiao Song James Spall Mu-Chun Su Peter Summons Antuela Tako Halina Tarasiuk Mamadou Traoré Klaus Troitzsch Zhiying Tu Alfonso Urquia Durk-Jouke van der Zee Svetlana Vasileva-Boyadzhieva Vladimír Veselý Maria Viamonte Manuel Villen-Altamirano Antonio Virdis Friederike Wall Frank Werner Kuan Yew Wong Hui Xiao František Zboril Houxiang Zhang Lin Zhang

xi

Institute of Computer Science, AGH, Krakow, Poland Instituto de Telecomunicações, DETI, University of Aveiro, Portugal University of Molise, Italy SPE UMR CNRS 6134:University of Corsica, France Ruhr-Universität Bochum, Germany Beihang University, China The Johns Hopkins University, USA National Central University, Taiwan, Republic of China University of Newcastle, Australia Loughborough University, UK Warsaw University of Technology, Poland University of Bordeaux, France University of Koblenz-Landau, Koblenz Campus, Germany Harbin Institute of Technology, China Universidad Nacional de Educación a Distancia, Spain University of Groningen, Netherlands Bulgarian Modeling and Simulation Association (BULSIM), Bulgaria Faculty of Information Technology, Brno University of Technology, Czech Republic Instituto Superior de Engenharia do Porto, Portugal Universidad de Malaga, Spain University of Pisa, Italy Alpen-Adria-Universität Klagenfurt, Austria Otto-von-Guericke-Universität Magdeburg, Germany Universiti Teknologi Malaysia, Malaysia Southwestern University of Finance and Economics, China Faculty of Information Technology, Czech Republic Norwegian University of Science and Technology, Norway Beihang University, China

xii

Organization

Additional Reviewers Clarissa Gonzalez Yingze Hou Ninan Theradapuzha Mathew Shihong Wei

Chalmers University of Technology, Sweden JHU, USA Chalmers University of Technology, Sweden JHU, USA

Invited Speakers Hans Vangheluwe Gregory Zacharewicz Tuncer Ören

University of Antwerp, Belgium IMT:Mines Ales, France University of Ottawa, Canada

Contents

Hint-Based Configuration of Co-simulations with Algebraic Loops . . . . Bentley James Oakes, Cláudio Gomes, Franz Rudolf Holzinger, Martin Benedikt, Joachim Denil, and Hans Vangheluwe

1

Shift of Paradigm from Model-Based to Simulation-Based . . . . . . . . . . . Tuncer Ören

29

Assessment of Learning Achievement in an Electronics Program Supported by an Online Simulation Applet . . . . . . . . . . . . . . . . . . . . . . David Valiente, Fernando Rodríguez, Juan Carlos Ferrer, José Luis Alonso, and Susana Fernández de Ávila Application of Artificial Neural Networks for Active Roll Control Based on Actor-Critic Reinforcement Learning . . . . . . . . . . . . . . . . . . . Matthias Bahr, Sebastian Reicherts, Philipp Sieberg, Luca Morss, and Dieter Schramm Optimal Trigger Sequence for Non-iterative Co-simulation with Different Coupling Step Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Franz Rudolf Holzinger, Martin Benedikt, and Daniel Watzenig

46

61

83

Complete Lyapunov Functions: Determination of the ChainRecurrent Set Using the Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Carlos Argáez, Peter Giesl, and Sigurdur Freyr Hafstein A Study on the Dynamic Behavior of a Slurry Mixing and Pumping Process: An Industrial Case . . . . . . . . . . . . . . . . . . . . . . . 122 Ridouane Oulhiq, Khalid Benjelloun, Yassine Kali, Maarouf Saad, and Laurent Deshayes Dynamic Simulation of Two Kinds of Hydraulic Actuated Long Boom Manipulator in Port-Hamiltonian Formulation . . . . . . . . . . . . . . 144 Lingchong Gao, Mei Wang, Haijun Peng, Michael Kleeberger, and Johannes Fottner

xiii

xiv

Contents

Simulating Species Dominance in Mixed Mangrove Forests Considering Species-Specific Responses to Shading, Salinity, and Inundation Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Ian Estacio and Ariel Blanco Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Hint-Based Configuration of Co-simulations with Algebraic Loops Bentley James Oakes1,2(B) , Cl´ audio Gomes1,2 , Franz Rudolf Holzinger3 , 3 Martin Benedikt , Joachim Denil1,2 , and Hans Vangheluwe1,2 1

3

University of Antwerp, Antwerp, Belgium [email protected] 2 Flanders Make vzw, Lommel, Belgium Virtual Vehicle Research GmbH, Graz, Austria

Abstract. Co-simulation is a powerful technique for performing fullsystem simulation. Multiple black-box models and their simulators are combined together to provide the behaviour for a full system. However, the black-box nature of co-simulation and potentially infinite configuration space means that configuration of co-simulations is a challenging problem for today’s practitioners. Our previous work on co-simulation configuration operated on the notion of hints, which allow system engineers to encode their knowledge and insights about the system. These hints, combined with state-of-theart best practices, can then be used to semi-automatically configure the co-simulation. We summarize our previous hint-based configuration work here, and explore the challenging problem of scheduling co-simulations which contain algebraic loops. Solving or “breaking” these loops is required for scheduling, yet this breaking process can induce errors in the cosimulation. This work formalizes this scheduling problem, presents our insights gained about the problem, and details an optimal search algorithm as well as greedy scheduling algorithms. These heuristic algorithms are evaluated on (synthetic) co-simulation scenarios to determine their relative speedup and optimality.

1

Introduction

Cyber-Physical Systems (CPS) marry the complexities of software with the realities of the physical world [24], and are becoming essential systems in today’s world. For example, an airplane or self-driving car relies on safety-critical communication between sensors, controller software, and actuators. A large driver in the complexity of CPS is that their analysis spans multiple domains. Simulating these therefore requires integrating heterogeneous models. For example, the integration of multi-body system models with communication network models. The technique of co-simulation is designed to combine multiple co-simulation units (simulators, each with their own model), in order to compute the behavior of the combined models over time [16,21]. Each unit has its own interface for c Springer Nature Switzerland AG 2021  M. S. Obaidat et al. (Eds.): SIMULTECH 2019, AISC 1260, pp. 1–28, 2021. https://doi.org/10.1007/978-3-030-55867-3_1

2

B. J. Oakes et al.

getting/setting inputs and outputs, and for computing the behaviour of its model over a given interval of time. An example of such an interface is the Functional Mockup Interface (FMI) Standard [7,10]. A master algorithm is then responsible for scheduling the execution and communication of each co-simulation unit. Co-simulation is therefore very promising for representing: a) systems assembled from models in various domains, each with their own most appropriate simulator, and b) black-box models which hide internal details, allowing third-party units to be integrated together, as happens in supplier-integrator relationships in industry. These key benefits have led to multiple modeling and simulation tools allowing the import and export of units implementing the FMI Standard [9], and a wide variety of applications [16]. However, it can be difficult to ensure that the results produced by a cosimulation can be trusted. This is due to not only the black box nature of units, but also to the many ways in which a co-simulation can be computed (i.e., communication frequency between differential-equation-based units, event propagation order, etc. . . ). The correct configuration of a co-simulation also depends on the numerical properties of the participating units. This challenge is aggravated when units themselves may be modified as part of an optimization loop (e.g., design space exploration) and/or impact analysis of sub-model refinements, because the co-simulation user may be unaware of how to account for these modifications in the master algorithm. Prior Work. Our earlier work [18] focused on the core challenge that users do not always know how to configure the co-simulation [25]. To tackle this, we proposed a method, and a tool called HintCO which is summarized in Sect. 2 and available online [13]. In this tool, a user or engineer can write “hints” about the co-simulation and involved units. HintCO then applies state of the art heuristics to configure and run multiple promising co-simulations. This is similar to design-space exploration techniques [28,30]. This approach works well in practice because users usually can tell what properties a correctly configured co-simulation should satisfy. This is demonstrated by an industrial case study in [18], where state of the art co-simulation algorithms failed to produce expected ‘smooth’ (non-oscillatory) results. After specifying a few hints, the top candidate results produced by HintCO were smooth. Motivation. One limitation of HintCO was the assumption that the co-simulation unit couplings do not form algebraic loops, as explained in Sect. 3. However, when differential-algebraic-equation-based units are coupled, algebraic loops can be formed [1]. The ideal technique to solve algebraic loops in co-simulations is fixed point iteration (see, e.g., [15, Section 4] and [26]). However, this technique requires that units support state rollback, which is an optional feature of the FMI standard and is therefore seldom implemented currently. Therefore, the most common technique is to “break” the algebraic loop by employing extrapolations in one (or more) variables in the loop [3,5]. Naturally, variables have different dynamics, hence, care must be taken when choosing the variables to break the loop [19].

Hint-Based Configuration of Co-simulations with Algebraic Loops

3

Our prior work [18] na¨ıvely generates all possible ways in which algebraic loops can be broken, without regard for the dynamics of the variables involved. In the current work, we build on [19] to formalize the problem of breaking algebraic loops in co-simulations, and propose an optimal algorithm to solve it, plus a few cost-effective approximation algorithms. Contributions. Our contributions in this paper are: a) a formalization of the problem of breaking algebraic loops in co-simulations, b) an optimal, but costly, algorithm to solve it, and c) multiple cost-effective heuristic algorithms. 1.1

Paper Layout

The next section (Sect. 2) will provide a brief introduction to the HintCO framework, including the hints and how they shape the search space for finding a correct co-simulation master algorithm. While HintCO has been shown to be effective for an industrial case [18], in Sect. 3 we demonstrate an example with an algebraic loop, where the previous version of the HintCO framework was unable to efficiently schedule this co-simulation. Section 4 formalizes the essential components of our approach. We introduce co-simulation and its involved concepts, as well as the background for our improved approaches to co-simulation scheduling. The concrete problem of scheduling co-simulations in the presence of algebraic loops is formalized in Sect. 5 and a optimal yet costly algorithm is provided. Candidate greedy algorithms for scheduling co-simulations with algebraic loops are presented in Sect. 6, and evaluated on synthetic examples in Sect. 7. Following this, Sect. 8 will discuss related work in the field and compare our approach to past approaches. Finally, we conclude in Sect. 9 with a summary of our research and the steps to extend our framework further.

2

HintCO Framework

This section briefly introduces relevant aspects of the HintCO framework, such that the importance of the contributions made in this paper to the automated configuration of co-simulations can be appreciated. In particular, this section adapts text from [18] to briefly describe the problem statement tackled by the HintCO framework and the elements of the HintCO workflow. This includes exemplifying some of hints the user can state about a co-simulation, and a description of the process for generating candidate configurations for cosimulations. 2.1

The Configuration Problem

As described in [18] and explored in-depth in the thesis of Gomes [14], it can be challenging to configure a co-simulation which satisfies properties to the same degree as the original system. This is due to the inherent approximation of the

4

B. J. Oakes et al.

original system’s behaviour trace by the co-simulation’s behaviour trace, and due to the many possible manners in which a co-simulation can be computed. Assuming that co-simulation units are correctly implemented, the configuration problem can be stated as: given a set of co-simulation units and their interconnections, and a set of properties that the coupled system should satisfy, find the master algorithm that produces co-simulation results satisfying those properties (Problem 1 in [18]).

Fig. 1. The ExecRate and PowerBond hints.

Since the correct co-simulation configuration is not available as our reference oracle, we are forced to rely on the user’s hints as a proxy for the correct set of properties to satisfy. This assumption of hint correctness is quite strong, but allows us to guide the search for a correct co-simulation configuration based on a mapping from these hints to state-of-the-art configuration techniques. 2.2

Workflow

HintCO has four main components, briefly introduced here as part of the HintCO workflow: a) HintCO Hint Language: The user first specifies hints about the system by selecting and configuring built-in hints relevant to the domain of the units involved. b) Generation of Candidate Master Algorithms: HintCO automatically generates candidate co-simulation configurations based on those hints. This is performed by translating those hints into adaptations on the configurations, using state-of-the-art heuristics. c) Scheduling the Co-simulation: A co-simulation schedule is then generated for each candidate configuration using the techniques described in this paper. d) Execution of the Co-simulation: Finally, the co-simulation variants are executed in a ranked order (as determined by the hints) and the results presented to the user for inspection.

Hint-Based Configuration of Co-simulations with Algebraic Loops

2.3

5

Hints

Hints are defined in a small domain-specific language (DSL). DSLs allow experts in the problem space (the system engineers) to describe hints, without having to become experts in the solution space (the co-simulation domain) [29]. As an example, Fig. 1 shows the hints exemplified in [18]. The first hint specifies the frequency of a unit, which is useful to find a communication rate between units, and determines whether a unit represents a time triggered software controller. The second hint defines a power-bond, which declares that energy should not be lost or gained in the communication between two units. Each hint has a number of fields. The description field is for unstructured text, as is commonly seen in industrial requirements. Following this are statements, which can be events or properties. Finally, scopes and patterns specify when the hint is valid. These scopes and patterns have been sourced from [2] and have been utilized for verification of safety-critical automotive requirements in another domain of our work [6]. 2.4

Generating a Configuration Search Space

As will be detailed later in Definition 7, a co-simulation configuration (or master algorithm) has three dimensions in our formalization: – the rate at which co-simulation units execute; – the concrete operations or execution order of those units; – the semantic adaptations (described below) applied to the co-simulation units. It is not feasible to explore all possible configurations, so the hints are used as a way to build a finite ranked list of possible candidate configurations. Semantic adaptations are a technique to create a new co-simulation unit by wrapping the old units [17] (also see Definition 4), thereby changing it’s behavior when inputs are provided, when output are requested, or when time stepping is performed. Some example semantic adaptations are: Extrapolation/Interpolation: Applies the approximation to the input port. Multi-rate: Adapt a co-simulation unit to perform multiple executions per one larger co-simulation step. Power-bond: Whenever two units share a power connection, one of the input ports will correct for the energy dissipated using the technique from [4]. XOR: Is combined with other adaptations to represent alternative configurations. The following exemplifies a search space. Example 1. Figure 2a shows the configuration space used in [18]. Possible adaptations are shown on each co-simulation unit and port: the Load and Plant co-simulation units have a PowerBond adaptation on the v and f ports, and

6

B. J. Oakes et al.

the Environment unit has an XorAdaptation with two alternative multi-rate adaptations. This space represents four alternative master algorithms, because of the two possible rates for the Environment unit. These adaptations are represented in the figure as the execution rate R = {100, 10}, and two possible communication step sizes H = 1 × 10−7 , 1 × 10−6 . The key operation of the HintCO framework is to transform the user’s hints into semantic adaptations, as exemplified in Fig. 2a. This is described in [18, Procedure 1]. Having a configuration space, HintCO then generates a variant diagram, as exemplified in Fig. 2b. This diagram reflects the weighting of the variants as defined by the hints, and each path from root to leave node represents a co-simulation configuration (a variant). HintCO employs a weighted depthfirst search to traverse this tree and generate the variants for scheduling and execution. For example, in Fig. 2b, the search will first select the adaptations of H = 1 × 10−7 for the step-size and then R = 100 for the Environment unit rate, due to the highest weight. The user can opt for generate all variants, or only the top n.

(a) Possible adaptations.

(b) Variant diagram.

Fig. 2. Case study introduced in [18].

2.5

Scheduling and Execution

To properly configure a co-simulation, HintCO must take a variant, and define a concrete operation schedule for how the co-simulation units in the scenario are executed and how values are passed between units. Operation schedules and their creation are further explored in Sect. 4.2. This schedule must comply with the requirements imposed by the adaptations chosen in the variant. These requirements refer to the order in which: inputs can be set; outputs computed; and time advancement operations performed. Thus, the schedule can be different for different variants.

Hint-Based Configuration of Co-simulations with Algebraic Loops

7

Fig. 3. Example operation schedule for the co-simulation scenario in Fig. 2a [18].

Figure 3 demonstrates an example of an operation schedule for the cosimulation scenario in Fig. 2a, as replicated from [18]. The left-hand side of the figure defines the dependencies of function calls, as given by the interaction between the variant’s adaptations and rules defining a valid co-simulation configuration (Definition 9). The right-hand side of the figure depicts a schedule of those operations, as given by a topological ordering of the dependencies. The execution of the variants by HintCO is performed behind-the-scenes by the tool, based on the concrete operation schedule automatically produced. The user receives the co-simulation traces for each variant and can decide whether to continue with the variant tree search or not. In this way, the variant tree generation and the scheduling process are hidden to reduce complexity for the system engineer. An example trace for the Load co-simulation unit from Fig. 2a is shown on the left-hand side of Fig. 4 before applying the hints and adaptation. The smoothed results on the right-hand side of Fig. 4 are the result of the top variant, demonstrating how HintCO can assist in configuring co-simulations.

(a) Signal before adapation

(b) Signal after adapation

Fig. 4. Load signal before and after HintCO adaptations are applied [18].

3

Motivating Example

This example motivates our current work regarding scheduling, as the input/output relationships of the units in the co-simulation form a dependency loop. This is termed as an algebraic loop. The version of HintCO proposed in [18] could produce an operation schedule in the presence of algebraic loops, but HintCO would not consider the dynamics of each connection in deciding which dependency was the least important and

8

B. J. Oakes et al.

could be removed to break the algebraic loop. As discussed in [19] and Sect. 5.3, a lower co-simulation error can be achieved by choosing a more appropriate point to break the loop. 3.1

Example Description

Fig. 5. Example co-simulation network with algebraic loops.

The notion which leads to algebraic loops is feed-through (formalized in Definition 5), where the output of a co-simulation unit algebraically depends on its inputs. That is, when the input value of a unit is modified, the connected output value immediately changes, without the unit executing. Figure 5 shows an co-simulation scenario from [19]. Feed-through is represented in Fig. 5 by the dashed arrows within the co-simulation units. For this motivating example, we consider the case where the user does not provide sufficient hints for HintCO to break the feed-through dependency loops. If a dependency graph (such as Fig. 3) was generated by HintCO, the function calls of these units would produce a graph with a cycle. Then, HintCO would have to make a decision about the best dependency to break. In the previous version of HintCO, this decision was performed without considering the dynamics of the connections. The current work attempts to formalize this problem and present exact and heuristic solutions, such that HintCO can be improved to better schedule these co-simulation scenarios.

4

Formalization of Co-simulation Concepts

This section formalizes the key concepts involved in co-simulation configuration, in a refinement of those presented in our earlier work [18]. In particular, we recall definitions for co-simulation units and co-simulation configuration. These formalizations are required to support Sect. 5 which details the problem of cosimulation scheduling in the presence of algebraic loops.

Hint-Based Configuration of Co-simulations with Algebraic Loops

4.1

9

Co-simulation Units and Relevant Characteristics

The following definitions focus on the concepts of a co-simulation unit and the relevant characteristics which affect co-simulation behaviour. As with our earlier formalization [18], we consider general co-simulation units in our scheduling approach. This notion of co-simulation unit is inspired by and includes the Functional Mockup Units (FMUs). As in our earlier work [18], we follow the notations introduced in [8] and omit the details of initialization. Definition 1 (Co-simulation Unit). [18, Definition 3] A co-simulation unit with identifier c is a structure Sc , Uc , Yc , Rc , setc , getc , doStepc  , where: – – – –

Sc represents the state space; Uc and Yc the set of input and output variables, respectively; Rc : Uc → {true, false} the reactivity of each input (see Definition 3); Dc ⊆ Uc × Yc the set of input/output feed-through dependencies (see Definition 5); – setc : Sc × Uc × V → Sc and getc : Sc × Yc → V are functions to set the inputs and get the outputs, respectively (we abstract the set of values that each input/output variable can take as V); and – doStepc : Sc × R≥0 → Sc is a function that instructs the co-simulation unit to compute its state after a given time duration. When configuring co-simulations, it is crucial to reason about the current time of each co-simulation unit. The following definition defines the state timestamp for each unit, which the FMI Standard leaves implicit. Definition 2 (State Timestamp). [18, Definition 4] Given a communication step size H ∈ R≥0 and H > 0, we say that the state sc ∈ Sc of an co-simulation unit c has timestamp t, denoted as ϕ(sc ) = t when doStepc has been called Ht times with H as parameter. Definition 2 implies that if a co-simulation unit is in state sc at time t, then doStepc (sc , H) will approximate the state at time t + H. If the corresponding model is a continuous one, then an approximation function will be used to estimate the values of the inputs in the interval [t, t + H]. Input Approximation Functions. There are two relevant approximation functions we focus on. There can be an extrapolation on an input, where the value of a input is calculated forward from the last received value. Otherwise there can be an interpolation, where an intermediate value is calculated between the last received value and a value from the sending unit at either the current timestamp, or a (relative) future timestamp. Assumptions can be made about relating these approximation functions to the behaviour of co-simulation units. For example, as mentioned in Sect. 2 the user can provide a hint that an co-simulation unit represents a software controller. As software controllers rely on data from sampled sensors, the software

10

B. J. Oakes et al.

controllers assume that their input readings are not from a future timestamp. Therefore, it can be inferred that a software controller is using an extrapolation approximation function. As interpolations can only be employed when the sending unit can already calculated the value, the choice of scheduling of co-simulation units can also impact which approximation can be used. The interpolation/extrapolation choice also affects the error of the system, as discussed in Sect. 5.3. Extrapolations can induce more error in the co-simulation [19], but can be employed to break algebraic loops as the dependency between units in the same timestep is then removed. In our notation, we choose to leave the approximation function implicit in the doStepc , as reflected in version 2.0 of the FMI Standard. However, we make explicit the requirements of each kind of input approximation in the form of the reactivity Rc . Intuitively, a co-simulation unit c with a reactive input [16] must wait until the co-simulation unit d, which feeds that input to c, executes a step. Then, c may receive that input value. Definition 3 (Reactivity). [18, Definition 5] For a given co-simulation unit c with input u ∈ Uc , Rc (u) = true if the function doStepc makes use of an interpolation of input u. Let t be the timestamp of the state sc prior to a call to doStepc (sc , H), and let d denote the co-simulation unit whose output y ∈ Yd is connected to u. that sc must have been produced from a call Then,  Rc (u) = true means  to setc . . . , u, getd (sd , y) where the state sd of co-simulation unit d satisfies ϕ(sd ) = t + H. Conversely, Rc (u) = false   means that sc must have been produced from a call to setc . . . , u, getd (sd , y) where ϕ(sd ) = t. As the FMI Standard version 2.0 does not include information about reactivity, we make the following assumption for all co-simulation units: Assumption 1. If an co-simulation unit c does not disclose its input approximation scheme for an input u, then we assume that u is approximated with a constant extrapolation. Therefore, Rc (u) = false. We employ the technique of semantic adaptation, introduced in Sect. 2.4 to control the input approximation scheme and reactivity for co-simulation units. Definition 4 (Semantic Adaptation). [18, Definition 2] Semantic adaptation is a technique that allows a new co-simulation unit c to be constructed from a set of co-simulation units, using a custom implementation of the setc ,getc , and doStepc functions [17]. Feed-Through. The concept of feed-through is crucial for the current work. If a co-simulation unit has feed-through, then an output value of a co-simulation unit is a function of the input.

Hint-Based Configuration of Co-simulations with Algebraic Loops

11

Definition 5 (Feed-Through). The input u ∈ Uc of co-simulation unit c feeds-through to output y ∈ Yc , that is, (u, y) ∈ Dc , when there exists two values v1 , v2 ∈ V and some state sc ∈ Sc , such that getc (setc (sc , u, v1 ), y) = getc (setc (sc , u, v2 ), y). This means that the value of y obtained with getc is an algebraic function of the value set for u. Hence, getc should be called to get the value of y only after setc has been called to set the value of u, before doStepc is invoked. Co-simulation units with feed-through will immediately change output values when the corresponding input value changes, even without the co-simulation unit executing a time-step. It is this ‘instant’ change which can form algebraic loops when multiple co-simulation units have feed-through, as in the motivating example in Sect. 3. 4.2

Co-simulation Scenario and Master Algorithms

A co-simulation scenario is a collection of co-simulation units and the input/output connections between them. An example of a co-simulation scenario with four co-simulation units and six connections is shown in Fig. 5 on page 7. Definition 6 (Co-simulation Scenario). [18, Definition 7] A co-simulation scenario is a structure C, L where each co-simulation unit identifier c ∈ C is associated with an co-simulation unit, as defined in Definition 1, means that the output y is connected to input u. Let U =  and L(u) = y  U and Y = c∈C c c∈C Yc , then L : U → Y . Master Algorithms. A master algorithm is the configuration to compute the behaviour of a co-simulation scenario. As stated in Definition 7, a master algorithm combines the co-simulation scenario, the step size, and a scheduling sequence for the scenario. As described previously in Sect. 2.4, our approach is to guide a search through the variation of these parameters, which each induce a different co-simulation behaviour. Definition 7 (Master Algorithm). [18, Definition 10] Given a co-simulation scenario C, L, a step size H, and an (f )i∈N , a master algo  operation schedule rithm is a structure defined as A = C, L, H, (f )i∈N . Co-simulation Step. The execution of a master algorithm A = C, L, H,  (f )i∈N is thus the application of the operation schedule on the co-simulation scenario. One application of this sequence is a co-simulation step. Precisely, executing this schedule in a co-simulation scenario C, L where all co-simulation units c ∈ C have a state sc satisfying ϕ(sc ) = t, will update each co-simulation unit’s state sc to satisfy ϕ(sc ) = t + H, where H is the argument of every call to

12

B. J. Oakes et al.

doStep. Repeated co-simulation steps thus advance the state of the co-simulation scenario, producing a co-simulation behaviour trace. Readers may note that the definition of operation schedule provided in Definition 8 was referred to as a co-simulation step in [18]. This redefinition was performed in order to allow the co-simulation step concept to also refer to the execution of a trigger sequence of co-simulation units, which is further described in Sect. 5.1. This presentation omits the handling of hierarchical co-simulation units, which themselves contain a co-simulation scenario. However, this is accounted for in our treatment, as we consider the sub-scenario to execute whenever the hierarchical co-simulation unit itself executes. Thus, we create the schedule for the top-level elements separately from the schedule for each individual hierarchical co-simulation unit. Operation Schedules. In Definition 7 a master algorithm contains an operation schedule. This operation schedule represents the sequence of function execution for each co-simulation unit in the scenario. That is, the sequence of operations get, set, doStep) which are executed on each port of each co-simulation unit. We formalize this operation schedule concept in Definition 8. Definition 8 (Operation Schedule). [Modified from Definition 8 of [18]] Given a co-simulation scenario C, L, an operation schedule is an ordered sequence of co-simulation unit function calls (f )i∈N with f ∈F =



{setc , getc , doStepc } ,

c∈C

and i denoting the order of the function call. A function call fi comes before a function call fj , written as fi  fj , if i < j, and comes immediately before, written as fi → fj , if i = j − 1. Definition 9 states the restrictions on a well-formed master algorithm, and how the hints provided to HintCO affect this schedule. For example, if an input is reactive (it performs an interpolation approximation - Definition 3), then that input’s get step must occur after the doStep of the preceding co-simulation unit. Definition 9 (Valid Master Algorithm). [Modified from Definition 9 of [18]] A master algorithm is valid with respect to reactivity and the co-simulation scenario couplings if it satisfies the following conditions: 1. A co-simulation step size H > 0. 2. Each function call uses the most recent co-simulation unit state as parameter. For example, if fj = getc (sc , y) then sc must be the result of the most recent call to setc or doStepc , that is, the maximal i such that i < j, and fi = setc (. . .) or fi = doStepc (. . .).

Hint-Based Configuration of Co-simulations with Algebraic Loops

13

3. For every c ∈ C, there exists one, and only one, call to doStepc , and it is done with argument H. 4. Each call to doStepc for c ∈ C must come after every call to setc on the input variables of c. 5. Each call to get is immediately followed by a sequence of calls to set to set the affected input variables. 6. For each c ∈ C and (u, y) ∈ Dc , any call to getc (y, . . .) must be preceded by a call to setc (. . . , u) without any call to doStepc in between. 7. For each c ∈ C and u ∈ Uc satisfying Rc (u) = true, doStepd  getd (L(u), . . .), where L(u) ∈ Yd and d ∈ C. 8. For each c ∈ C and u ∈ Uc satisfying Rc (u) = false, setc (. . . , u)  doStepd , where L(u) ∈ Yd and d ∈ C. Remark 1. Regarding Definition 9: – The most common master algorithms will satisfy conditions 2–4; – Condition 5 is not mandatory but it facilitates the description of Conditions 7 and 8. Furthermore, it makes the implementation simpler. – Conditions 7 and 8 ensure that the reactivity of each input is respected, according to Definition 3. • If Rc (u) = true, then the input approximation is interpolated, and the preceding co-simulation unit must perform doStep before the get call • If Rc (u) = false, then the input approximation is extrapolated, and the set call is performed before the preceding co-simulation unit performs doStep. Generating an Operation Schedule. A particular variant co-simulation configuration (discussed in Sect. 2.4) could define interpolation or extrapolation adaptations on co-simulation units or their input ports. These adaptations interact with the rules defined in Definition 9, which specify the dependencies between the function calls in the operation schedule. This then induces a dependency graph of the function calls in the co-simulation scenario. An example of these ordering dependencies is demonstrated on the left-hand side of Fig. 3 on page 3. In this dependency graph, the operation at the tail of an edge must be executed before the operation at the head of an edge. The dependency graph in Fig. 3 does not contain any cycles, due to the lack of feed-through in the co-simulation scenario (shown in Fig. 2a). A topological sort is therefore sufficient to generate an operation schedule to execute the cosimulation units, as seen on the right-hand side of Fig. 3. This operation schedule approach allows for a great deal of flexibility in the concrete order of operations, as all topological orderings are considered to be behaviourally equivalent. A Prolog-based algorithm for specifying co-simulation scenarios and determining a valid operation schedule is presented in [12]. However, if cycles are present in this dependency graph (as in the motivating example in Sect. 3), the cycle will need to be broken to produce an operation schedule. As this cycle breaking produces errors (due to the extrapolation approximation used), an optimization approach is required to determine the best scheduling, as discussed in the following sections.

14

5

B. J. Oakes et al.

Scheduling with Algebraic Loops

This section will detail our approach to scheduling co-simulation scenarios with algebraic loops. This approach is based on co-simulation trigger sequences, which are an intuitive scheduling of co-simulation units at a high level. We also present how trigger sequence can be transformed into operation schedules to be executed by HintCO. Third, the cost function for a trigger sequence is defined, determined by the connections within the co-simulation scenario. Finally, we discuss a directed search approach to calculating the optimal trigger sequence. 5.1

Trigger Sequences

As described in Sect. 4.2, execution of a co-simulation scenario requires the production of an operation schedule, which details the precise sequence of function calls within the scenario. However, another (possibly more intuitive and more elegant) approach is to define a trigger sequence for the co-simulation scenario, as is done in [19]. Following this trigger sequence, each co-simulation unit would be visited and executed in turn, with input and output approximation and propagation handled as required. The motivation for defining and utilizing trigger sequences is therefore to consider co-simulation scenarios at an abstract level. The presence of algebraic loops leads to a ‘strong component’ notion, where co-simulation units must be reasoned about as one entity. In particular, the problem statement in Sect. 5.3 deals with co-simulation units, not their individual function calls. Example and Definition. Consider the motivating example co-simulation scenario in Fig. 5, which contains four co-simulation units S1, S2, S3, and S4. There are 24 different permutations of these units, and therefore 24 possible trigger sequences (Definition 10) can be created, such as {S1, S2, S3, S4} or {S2, S4, S3, S1}. Definition 10 (Trigger Sequence). Given a co-simulation scenario C, L, trigger sequence is an ordered sequence of co-simulation unit executions (ci ) with c ∈ C and i denoting the order of the co-simulation unit execution. 5.2

Transformation to Operation Schedule

The definition of a master algorithm in Definition 7 contains a schedule of function calls on the co-simulation units. Therefore, a transformation from a trigger sequence to an operation schedule is required for co-simulation execution. This transformation must respect the constraints defined in Definition 9 for a valid master algorithm. In particular, condition 5 must be followed, in which each call to get an output is immediately followed by a call to set for the associated input. A trigger sequence is thus transformed into an operation schedule by Procedure 1.

Hint-Based Configuration of Co-simulations with Algebraic Loops

15

Procedure 1. Produce an operation schedule from a given trigger sequence: – For each co-simulation unit c in the trigger sequence: • Add the appropriate get and set calls for each input of c to the operation schedule. • Add extrapolation adaptations to co-simulation unit inputs where required (see Sect. 5.3) • Add the doStep call for c to the operation schedule. For example, Example 2 presents the operation schedule for the trigger sequence {S2, S1, S3, S4} for the co-simulation scenario in Fig. 5. Example 2. {getY 11 , setU 21 , doStepS2 , getY 42 , setU 11 , doStepS1 , getY 21 , setU 31 , getY 41 , setU 32 , doStepS3 , getY 31 , setU 41 , getY 12 , setU 42 , doStepS4 } 5.3

Problem Statement

As presented in Sect. 3, our motivating example contains feed-through of inputs and outputs arranged in a cycle. This implies that the dependency graph for operations (as in Fig. 3) would also have a cycle. Section 5.4 will answer the important question of how the above trigger sequence {S2, S1, S3, S4} was created for this co-simulation scenario, despite the cyclic dependency. However, we first focus on what impact the breaking of the algebraic loop has on our co-simulation results. The scheduling of the co-simulation trigger sequence can impact the results of the co-simulation, as seen in related work [19]. This is due to the presence of input approximation algorithms used in the co-simulation, as discussed in Sect. 4.1. Recall that interpolation algorithms may interpolate input values between the previous time step and the next one. This (may) reduce error compared to extrapolations, but interpolations are only available to use on a co-simulation unit when the preceding unit has already executed. That is, any co-simulation unit ci in a trigger sequence can interpolate values from co-simulation units cji . Holzinger and Benedikt [19] take these considerations into account and produce a trigger sequence which minimizes the number of extrapolations performed to reduce error. However, their technique is based on a Travelling Salesman Problem approach, which can be computationally expensive. Instead, Sect. 5.4 presents a directed search algorithm, and Sect. 6 explores heuristic algorithms to perform this scheduling. Therefore, the problem statement considered in the following sections is: given a co-simulation scenario, what is the trigger sequence with a minimum cost, where this cost represents performing extrapolations of co-simulation unit inputs? The following sections investigate trigger sequence creation and define this cost function.

16

5.4

B. J. Oakes et al.

Trigger Sequence Creation

This section will detail how a trigger sequence can be constructed from a cosimulation scenario. First, we describe a scenario graph to intuitively represent the co-simulation units in a scenario and a measure of the dependency strength of their connections. Second, we detail how this graph is used to build a trigger sequence. Finally, a directed search approach for finding an optimal trigger sequence is presented. Scenario Graph. A scenario graph represents the necessary information from a co-simulation scenario to define a trigger sequence. Vertices in this scenario graph represent each co-simulation unit in the scenario. Edges are weighted, directed, and connect vertices which are connected in the original co-simulation scenario. An example co-simulation scenario graph is shown in Fig. 6a. Not shown in this example is that each edge in this scenario graph could represent more than one co-simulation connection in the original scenario. The weight of the edges represent a measure of the interdependence of the units in the original scenario, such as a count of the number of connections. More advanced analysis are possible to represent a more nuanced calculation of sensitivity of the connection, as in [19]. In our formulation of the scenario graph, which follows [19], the weight of each edge represents the cost for performing an extrapolation approximation on the input/output connections in the original co-simulation scenario, which are represented by that edge. This weight can be set through hints from the user, though we are exploring automated determination of costs. Selecting the weight of an edge must also take into account any interpolation or extrapolation information provided by hints on the scenario. That is, the variant generation described in Sect. 2 may determine the approximation for certain co-simulation unit inputs. This sets the weight of the relevant edge to zero, as the hint suggests that the co-simulation unit is constructed to appropriately handle the resulting approximation error. Trigger Sequence Cost Function. Following the definition of the cosimulation graph above, building a trigger sequence involves selection of the co-simulation units to execute, taking into account the weights of the edges between them. For example, consider the situation where the A unit in Fig. 6a is executed first in a trigger sequence. Both the inputs from B and the inputs from C must be extrapolated, for a summed cost of eight. The real complication in determining the optimal trigger sequence arises in that scheduling the execution of a co-simulation sets the costs of outgoing edges of that unit to zero. That is, co-simulation units on outgoing edges will not be forced to extrapolate the output, but instead can rely on interpolation, which is beneficial to the error of co-simulation [19].

Hint-Based Configuration of Co-simulations with Algebraic Loops

(a) Scenario graph.

17

(b) Directed search tree.

Fig. 6. An example scenario graph and a directed search tree for the optimal trigger sequence.

As an example, consider the co-simulation graph in Fig. 6a. Each of the six possible trigger sequences for the co-simulation scenario has a different cost given by this interpretation. If co-simulation unit B is selected for execution in the trigger sequence first, the cost will be four due to the weight of the edge from A. However, the weight on the edge from B to C will then be zero, as C can now interpolate the output value of B. Cost Equation. The following equation defines the cost function for a trigger sequence, in a reformulation of the cost equation found in [19]. Informally, each node is considered and incoming weights from nodes not yet encountered in the trigger sequence are summed. Let G = (V, E) denote the scenario graph, where E ⊆ V × V × R denotes the (positively-)weighted edge set. Given a trigger sequence v0 , v1 , . . . , vn , with distinct v’s, n = |V |, and vi ∈ V for all i = 0, . . . , n, its cost is: n−1 n



w(vj , vi ) c(v0 , v1 , . . . , vn ) = i=0 j=i+1  x if (u, v, x) ∈ E w(u, v) = 0 otherwise.

(1)

An alternative formulation to the above would be to sum up the incoming edge weights for each node, and then subtract the outgoing edge weights for those nodes not yet visited. As an example, consider two trigger sequences {B, C, A} and {A, B, C} for the scenario graph in Fig. 6a. The total cost for these trigger sequences is provided in Example 3.

18

B. J. Oakes et al.

Example 3. Trigger Sequence: {B, C, A} Cost: B = 4, C = 3, A = 0 Total: 7 (optimal)

Trigger Sequence: {A, B, C} Cost: A = 8, B = 0, C = 0 Total: 8

Directed Search. Given Eq. (1) for defining the cost of a trigger sequence, an brute-force algorithm can be easily created: a) all possible trigger sequences are created, b) each is evaluated using Eq. (1), c) the sequence with the lowest cost is the optimal one. However, this brute-force approach is computationally infeasible, as the number of possible trigger sequences is a factorial explosion of the number of co-simulation units. A directed search is instead preferable to find the optimal solution. This directed search builds up a tree of possible sequences, selecting the next branch to expand based on the cost of the branch so far. This search is possible because the cost function is defined for partial trigger sequences, and is consistent as well. That is, adding further nodes to a trigger sequence can only maintain or raise the total cost, so there cannot be a local maxima reached in the search. The directed search begins with the root of the tree as the empty set ∅. Then in an iterative manner the branch with the lowest cost is expanded, by adding as children all those nodes not considered yet in that branch of the tree. For example, Fig. 6b demonstrates the final search tree for the scenario graph in Fig. 6a. The layer just below the root in Fig. 6b considers the execution of each node individually. As partial sequence {B} has the lowest cost, it would be expanded next. Those children ({B, A} and {B, C}) have a higher cost than {C}, so the {C} branch is expanded next into {C, A} and {C, B}. The search returns to {B, C}, which is expanded into {B, C, A} (bolded in Fig. 6b) which is the optimal solution with a cost of seven. This directed search provides the optimal solution, but could be exhaustive and therefore computationally prohibitive. The next section presents the Travelling Salesman and optimal branching approaches, along with heuristic algorithms to find a trigger sequence with near-optimality but at lower computational complexity.

6

Approaches for Constructing Trigger Sequences

This section details approaches for producing a trigger sequence with the lowest cost (as defined by Eq. (1)), while avoiding the computational complexity of the directed search approach described in Sect. 5.4. Each proposed approach will be presented along with a discussion, including counter-examples if known. 6.1

Travelling Salesman Problem

The Travelling Salesman Problem approach to trigger sequence construction is to find a walk (or Hamiltonian cycle) which visits each unit in the scenario graph once. While this approach is intuitive, as one can think of execution as ‘walking’

Hint-Based Configuration of Co-simulations with Algebraic Loops

19

around the scenario graph, this approach is overly restrictive and computationally expensive. First, the trigger sequence to be produced does not have to be a cycle, nor is it required that co-simulation units which are next to each other in the trigger sequence have to be directly connected in the scenario graph. For instance, consider a scenario graph with three nodes A, B, and C, where A and B are connected to each other, and B and C are connected to each other. Clearly there cannot exist a cycle that visits every unit exactly once. Therefore, Hamiltonian cycles are not required for a trigger sequence. A second issue with the Travelling Salesman Problem approach is that the starting unit of the cycle must also be selected, which induces another optimization problem. For example, assume that a Travelling Salesman algorithm gives the optimal solution for the scenario graph in Fig. 6a, which is executing B, C, and then A for a total cost of seven. However, this cannot be treated as a cycle, for while the trigger sequence {B, C, A} has a total cost of seven, the trigger sequence {A, B, C} has a total cost of eight due to the extrapolations required. Based on the above discussion, Travelling Salesman Problem approaches to co-simulation scheduling are certainly intuitive but are not the correct approach.

Fig. 7. Second example scenario graph.

6.2

Optimum Branching

Another approach to trigger sequence construction is to determine the optimum branching for a scenario graph, such as a (minimum) spanning tree. That is, which set of edges spans the entire graph with minimal cost. This approach is also highly intuitive, as the optimal solution must have the minimum weight from incoming edges for each node. However, this approach does not provide the ordering of the nodes which provides that minimal cost. For example, consider again the scenario graph from Fig. 6a. The optimal branching is the edges B → A and A → C, with a cost of six. However, it is unclear how this minimal tree relates to the optimal trigger sequence of {B, C, A}, which has a cost of seven. Therefore, the minimal spanning tree can provide a lower bound on the optimal trigger sequence cost, but (currently) cannot be used to produce this optimal trigger sequence. From this counter-example, it is clear that while the scheduling problem is related to an optimum branching problem, it is not sufficient to apply optimal

20

B. J. Oakes et al.

branching algorithms. This is due to the order of nodes in the trigger sequence affecting the weights of the edges, which is not the case in standard graph problems. 6.3

Greedy Approaches

As discussed in Sect. 5.4, a directed search is able to find the optimal trigger sequence, given enough computational resources. However, it would be beneficial to have algorithms with lower computational complexity for producing trigger sequences. In this section, we present a number of greedy approaches to the trigger sequence construction problem. We note that these algorithms are not optimal and we do not provide formal bounds on the relative error to the optimum. However, our contribution is to provide a selection of algorithms that could be used for scheduling, until future work provides an optimal algorithm (if it exists). These algorithms will be evaluated in Sect. 7, which provides relative error results of the algorithms against the optimum on synthetic scenario graphs. For the below algorithms, we will present examples calculations based on Fig. 6a and Fig. 7. These calculations are not intended to prove optimality, but only to illustrate the operation of the algorithms. As well, differences in implementation tie-breaking may also affect the results. Lowest Incoming. As a simple greedy algorithm, Lowest Incoming selects the node from those remaining which has the lowest incoming weights. The intuition is that the trigger sequence should be built according to choosing the ‘cheapest’ node next. Algorithm: – While not all nodes are in the trigger sequence: • Calculate the node with the lowest sum of incoming edge weights. • Add that node to the trigger sequence. • Remove cost for outgoing edges. Examples Scenario Found Sequence Cost Opt. Sequence Cost Figure 6a {B, C, A} Figure 7 {H, F, E, G}

7 5

{B, C, A} {F, E, H, G}

7 4

Hint-Based Configuration of Co-simulations with Algebraic Loops

6.4

21

Benefit Ratio

The next greedy algorithm we present is Benefit Ratio. In this algorithm, the ratio between the incoming edge weights and the outgoing edge weights for each node is calculated such that the node with the highest ‘benefit’ can be selected next. The ratio is calculated as output weight divided by input weight, and nodes selected from high ratio to low. There are two versions of this algorithm which we evaluate here. The first is static where the benefit ratio is determined at the beginning of the scheduling process. The second version is dynamic, where the benefit ratio is re-calculated after each sequence addition, to take into account that the input weights of other nodes are then reduced. Algorithm 1. Calculate the benefit ratio for each node. 2. While not all nodes are in the trigger sequence. – Add the node with the highest benefit ratio. – If the dynamic version, recalculate the benefit ratios. Examples for the Static Version Scenario Ratios

Found Sequence Cost Opt. Sequence Cost

Figure 6a A = 0.88, B = 1.5, C = 0.83 {B, A, C} {H, F, E, G} Figure 7 E = 0.75, F = 3, G = 0.12, H = 10

9 5

{B, C, A} {F, E, H, G}

7 4

Edge Avoidance. Based on the idea of optimum branching, the Edge Avoidance greedy algorithm has the intention of ensuring that the most expensive edges in the graph have their cost reduced to zero. The most expensive edges are selected such that maximal spanning tree is created and a topological sort gives the trigger sequence. An optimality improvement to this algorithm may be to perform a brute-force search on all possible topological sorts to find the one with the lowest cost. Algorithm 1. Sort the edges in descending order of weight. 2. From each edge from the beginning of that list: – Connect those nodes, unless doing so would create a cycle. 3. Produce the first possible topological sort.

22

B. J. Oakes et al.

Examples Scenario Max. Weight Edges Found Sequence Figure 6a C → A → B Figure 7 H → G, F → G, F→E

7

Cost Opt. Sequence Cost

{C, A, B} 9 Five possibilities, including optimal

{B, C, A} {F, E, H, G}

7 4

Algorithm Evaluation

This section will provide evaluations of the brute-force, directed search, and greedy algorithms presented in Sect. 5.4 and Sect. 6. In particular, measures of relative performance and optimality are discussed to provide insight into their characteristics. 7.1

Set-Up

As we are unaware of a large corpus of co-simulation scenario graphs, synthetic graphs are studied here. One hundred graphs for each sequence length from one to ten were created with cycles, and edges were randomly assigned discrete weights uniformly sampled between zero and nine (inclusive). For each graph, a brute-force search for the optimal trigger sequence is first performed to set a baseline of the performance and optimal cost. Then, the directed search (from Sect. 5.4) is calculated to determine the speedup given. Finally, each greedy algorithm from Sect. 6 is ran to determine the (potential) speedup and cost provided. The calculation effort is given in seconds, as determined by a Python 3.7.3 script running on Xubuntu 19.10 with a Intel i7-8850H CPU at 2.60 GHz. 7.2

Results and Discussion

Figure 8 provides an overview of the algorithm evaluations. Figure 8a presents the average of calculation time (in a log scale) for each algorithm over all graphs of a certain size. Figure 8b presents the relative error of each algorithm versus the optimal given by the brute-force approach. The relative error is calculated by taking the difference between the cost and the optimal cost, then dividing by the optimal cost. A relative error of 0.10 therefore means the cost is 10% worse than the optimal. Figure 9 provides more details by presenting boxplots for each algorithm’s relative error. Concerning computation time, it was expected that the directed search algorithm would be less expensive than the brute-force search. However, the results in Fig. 8a show that the improvement is not as great as expected. The high cost of the Edge Avoidance algorithm was also unanticipated, as it is similar in cost to the directed search, but is not likely to always produce an optimal result.

Hint-Based Configuration of Co-simulations with Algebraic Loops

(a) Calculation effort of each algorithm.

23

(b) Relative error versus the brute-force search.

Fig. 8. Evaluation of the algorithms for both effort and relative error.

However, the other three algorithms show a definite cost improvement over the brute-force and Directed Search approaches. The relative error of the algorithms versus the brute-force search shown in Fig. 8b and Fig. 9 provide interesting insights. The four greedy algorithms show promising results in terms of relative error. For each, the mean relative error is around 10%, the upper quartile falls around 25%, and values are rarely seen above 30%. As reference, a Random Sequence algorithm was also developed which simply makes a random decision which node to take next. For this algorithm, the mean relative error was around 40%, with the lower quartile at around 25%, and upper quartile at around 50%. From the results, two algorithms are clearly superior. First, the Directed Search algorithm should be chosen if the optimal solution is desired and computational resources are sufficient. If a greedy approach is desired, then the Dynamic Benefit Ratio algorithm provides a low relative error at a low performance cost.

8

Related Work

The problem of adequately configuring a co-simulation is not new. We can classify the approaches according to when and which information is used to configure the co-simulation: static and adaptive. An adaptive configuration approach monitors the co-simulation results and adjusts the co-simulation algorithm parameters accordingly. A static configuration approach sets the parameters without running the co-simulation. An overview of the adaptive configuration approaches is given in [18]. In the static configuration category, the following works have the same goal as our work. Rather than starting from a co-simulation scenario, the works in [5,22,27] use a system architecture model to generate a co-simulation scenario and a master algorithm that is consistent with that architecture. The work in [5] uses the input/output feed-through and the kind of model underlying

24

B. J. Oakes et al.

(a) Lowest Incoming.

(b) Edge Avoidance.

(c) Static Benefit Ratio.

(d) Dynamic Benefit Ratio.

Fig. 9. Detailed relative error of each algorithm.

the co-simulation unit (e.g., ODE, DAE, . . . ) to correctly configure the input approximations used. The authors in [27] go even further and use the eCl@ass classification system to automatically link the units. We complement these works by showing other examples of information that are useful to configure the co-simulation, and generating multiple master algorithms, instead of a single one. This is due to the fact that there might not be enough information available to fully specify a single master algorithm. In the domain of scheduling co-simulations, the authors of [11] provide a sequence calculation concept, which is analogous to our trigger sequence. Their work defines an optimization problem minimizing the communication delays between the co-simulation units in the sequence, while taking into account input/output dependencies. The optimization algorithm provides an almost optimal solution over a co-simulation scenario with 14 units. Our work instead focuses on the issue of breaking algebraic loops in the co-simulation scenario, and determining optimal and heuristic algorithms for scheduling. The current work is based off of the approach by Holzinger and Benedikt [19] which examines the Travelling Salesman Problem (TSP) approach to scheduling co-simulations with algebraic loops. We extend that work by presenting further

Hint-Based Configuration of Co-simulations with Algebraic Loops

25

discussion about approaches to the scheduling problem, including a counterexample for the TSP approach. The current work also provides optimal and heuristic-based algorithms with a lower performance cost than the brute-force algorithm found in [19].

9

Conclusion

Configuration of co-simulation scenarios can be complicated due to its blackbox nature, required knowledge of numerical techniques, and lack of a reference solution. In this paper, we have summarized our HintCO technique for (semi-) automatically configuring co-simulations based on user intuitions about the system, as expressed through hints. This work has also presented our advancements in understanding the scheduling problem for co-simulations with algebraic loops. The formalization of the problem suggests that the problem is in or around NP-complexity and that it will be difficult or impossible to arrive at a polynomial-time algorithm. Second, we have determined that typical graph-based algorithms to visit all nodes such as the Travelling Salesman Problem and Minimum Spanning Trees do not sufficiently deal with the issue that node ordering changes the cost function during visitation. As a concrete contribution, we have determined that a directed search (presented in Sect. 5.4) can find an optimal solution, given enough time. We have also presented heuristic algorithms to solve this problem along with results which indicate their performance and a measure of optimality. This work is being integrated as scheduling improvements in our HintCO tool [13]. In particular, hints can now be added to co-simulation scenario connections to indicate their weight in a scenario graph. If an algebraic loop is detected in a scenario then HintCO performs the directed search for the optimal trigger sequence, then transforms it back to a dependency graph to be executed. This search and transformation is performed ‘behind-the-scenes’, so that the user is shielded from the complexity of co-simulation scheduling. 9.1

Future Work

One important direction for our future work is to determine whether there is an algorithm which is less computationally expensive than the directed search approach (Sect. 5.4) but which still guarantees optimality. We suspect that the co-simulation scheduling problem to be in the NP class, but a proof is required. As well, we are investigating integrating additional hints into HintCO to support other scheduling problems. For example, co-simulation can be performed over a network as in [23], where co-simulation units are distributed geographically or within a network. One problem which arises could be the partitioning of cosimulation units to each network node, depending on their dependence on other nodes. This problem has been considered in a slightly different context in [20].

26

B. J. Oakes et al.

Acknowledgments. The authors thank Dr. Guillermo Alberto Perez (University of Antwerp) and Dr. Romain Franceschini (University of Corsica) for illuminating discussions on the trigger sequence cost function and algorithms. This research was partially supported by a PhD fellowship grant from the Research Foundation - Flanders (File Number 1S06316N).

References 1. Arnold, M., Clauß, C., Schierz, T.: Error analysis and error estimates for cosimulation in FMI for model exchange and co-simulation V2.0. In: Progress in Differential-Algebraic Equations, pp. 107–125. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44926-4 6 2. Autili, M., Grunske, L., Lumpe, M., Pelliccione, P., Tang, A.: Aligning qualitative, real-time, and probabilistic property specification patterns using a structured English grammar. IEEE Trans. Softw. Eng. 41(7), 620–638 (2015). https://doi. org/10.1109/TSE.2015.2398877 3. Benedikt, M., Watzenig, D., Zehetner, J., Hofer, A.: Macro-step-size selection and monitoring of the coupling error for weak coupled subsystems in the frequencydomain. In: V International Conference on Computational Methods for Coupled Problems in Science and Engineering, pp. 1–12 (2013) 4. Benedikt, M., Watzenig, D., Zehetner, J., Hofer, A.: NEPCE-a nearly energy preserving coupling element for weak-coupled problems and co-simulation. In: IV International Conference on Computational Methods for Coupled Problems in Science and Engineering, pp. 1–12 (2013) 5. Benedikt, M., Holzinger, F.R.: Automated configuration for non-iterative cosimulation. In: 17th International Conference on Thermal, Mechanical and Multi-Physics Simulation and Experiments in Microelectronics and Microsystems (EuroSimE), pp. 1–7. IEEE (2016). https://doi.org/10.1109/EuroSimE.2016. 7463355 6. Bernaerts, M., Oakes, B., Vanherpen, K., Aelvoet, B., Vangheluwe, H., Denil, J.: Validating industrial requirements with a contract-based approach. In: 2019 ACM/IEEE 22nd International Conference on Model Driven Engineering Languages and Systems Companion (MODELS-C), pp. 18–27. IEEE (2019). https:// doi.org/10.1109/MODELS-C.2019.00010 7. Blockwitz, T., et al.: Functional mockup interface 2.0: the standard for tool independent exchange of simulation models. In: 9th International Modelica Conference, pp. 173–184. Link¨ oping University Electronic Press (2012). https://doi.org/ 10.3384/ecp12076173 8. Broman, D., et al.: Determinate composition of FMUs for co-simulation. In: 11th ACM International Conference on Embedded Software. IEEE Press Piscataway, NJ (2013). Article no. 2 9. FMI: List of tools implementing the FMI standard. https://fmi-standard.org/ tools/ 10. FMI.: Functional mock-up interface for model exchange and co-simulation (2014). https://fmi-standard.org/downloads/ 11. Glumac, S., Kovacic, Z.: Calling sequence calculation for sequential co-simulation master. In: Proceedings of the 2018 ACM SIGSIM Conference on Principles of Advanced Discrete Simulation, pp. 157–160. ACM (2018)

Hint-Based Configuration of Co-simulations with Algebraic Loops

27

12. Gomes, C., L´ ucio, L., Vangheluwe, H.: Semantics of co-simulation algorithms with simulator contracts. In: 2019 ACM/IEEE 22nd International Conference on Model Driven Engineering Languages and Systems Companion (MODELS-C), pp. 784– 789. IEEE (2019) 13. Gomes, C., et al.: HintCO public repository. https://msdl.uantwerpen.be/git/ claudio/HintCO 14. Gomes, C.: Property preservation in co-simulation. Ph.D. thesis, University of Antwerp (2019) 15. Gomes, C., et al.: Co-simulation: state of the art. Technical report, University of Antwerp (2017). http://arxiv.org/abs/1702.00686 16. Gomes, C., et al.: Co-simulation: a survey. ACM Comput. Surv. 51(3) (2018). https://doi.org/10.1145/3179993. Article 49 17. Gomes, C., et al.: Semantic adaptation for FMI co-simulation with hierarchical simulators. Simulation 95(3), 1–29 (2018). https://doi.org/10.1177/0037549718759775 18. Gomes, C., et al.: HintCO – Hint-based configuration of co-simulations. In: Proceedings of the 9th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, SIMULTECH, vol. 1, pp. 57–68. INSTICC. SciTePress (2019). https://doi.org/10.5220/0007830000570068 19. Holzinger, F., Benedikt, M.: Optimal trigger sequence for non-iterative cosimulation. In: Proceedings of the 9th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, pp. 80–87. SCITEPRESS - Science and Technology Publications (2019). https://doi.org/10. 5220/0007833800800087 20. Holzinger, F.R., Benedikt, M.: Hierarchical coupling approach utilizing multiobjective optimization for non-iterative co-simulation. In: Proceedings of the 13th International Modelica Conference, no. 157, pp. 735–740. Link¨ oping University Electronic Press (February 2019). https://doi.org/10.3384/ecp19157735 21. K¨ ubler, R., Schiehlen, W.: Two methods of simulator coupling. Math. Comput. Model. Dyn. Syst. 6(2), 93–113 (2000). https://doi.org/10.1076/13873954(200006)6:2;1-M;FT093 22. Krammer, M., Fritz, J., Karner, M.: Model-based configuration of automotive cosimulation scenarios. In: 48th Annual Simulation Symposium, pp. 155–162. Society for Computer Simulation International (2015) 23. Krammer, M., et al.: The distributed co-simulation protocol for the integration of real-time systems and simulation environments. In: Proceedings of the 50th Computer Simulation Conference, SummerSim 2018, p. 1. Society for Computer Simulation International (July 2018). https://doi.org/10.22360/summersim.2018. scsc.001 24. Lee, E.A.: Cyber physical systems: design challenges. In: 11th IEEE International Symposium on Object Oriented Real-Time Distributed Computing (ISORC), pp. 363–369 (2008). https://doi.org/10.1109/ISORC.2008.25 25. Schweiger, G., et al.: Functional mock-up interface: an empirical survey identifies research challenges and current barriers. In: Proceedings of the American Modelica Conference, pp. 138–146. Link¨ oping University Electronic Press (2018). https:// doi.org/10.3384/ecp18154138 26. Sicklinger, S., et al.: Interface Jacobian-based co-simulation. Int. J. Numer. Methods Eng. 98(6), 418–444 (2014). https://doi.org/10.1002/nme.4637 27. Stecken, J., Lenkenhoff, K., Kuhlenk¨ otter, B.: Classification method for an automated linking of models in the co-simulation of production systems. Procedia CIRP 81, 104–109 (2019). https://doi.org/10.1016/j.procir.2019.03.019

28

B. J. Oakes et al.

28. Van Tendeloo, Y., Vangheluwe, H.: Increasing the performance of a discrete event system specification simulator by means of computational resource usage activity models. Simulation 93(12), 1045–1061 (2017). https://doi.org/10.1177/ 0037549717726595 29. Vangheluwe, H., de Lara, J., Mosterman, P.J.: An introduction to multi-paradigm modelling and simulation. In: Proceedings of AI, Simulation and Planning in High Autonomy Systems, pp. 9–20. SCS (2002) 30. Vanherpen, K., Denil, J., De Meulenaere, P., Vangheluwe, H.: Design-space exploration in model driven engineering: an initial pattern catalogue. In: 1st International Workshop on Combining Modelling with Search and Example-Based Approaches (CMSEBA), vol. 1340, pp. 42–51, CEUR Workshop Proceedings (2014)

Shift of Paradigm from Model-Based to Simulation-Based Tuncer Ören(B) School of Electrical Engineering and Computer Science, University of Ottawa, 800 King Edward Avenue, Ottawa, ON K1N 6N5, Canada [email protected] http://www.site.uottawa.ca/~oren/

Abstract. Simulation-based approach provides a very powerful, rich, and versatile paradigm for many disciplines, methodologies, and studies. Already over 170 disciplines, methodologies, and approaches are simulation-based. The article clarifies the essence and power of simulation-based approach. To reach this aim the following is done: First, as a basis, the essence of simulation which consists of experimentation and experience is covered. The historic rise of the experimentation in early 17th century as a basis of scientific method as well as the additional benefits of simulated experiments are covered. The unique advantages of coupling simulation and the working of real systems are explained. Possibilities offered by simulation to gain three types of experience are clarified. A brief history of modelbased approach as well as the first discipline for which model-based approach was promoted are explained. Richness and versatility of simulation and benefits of its use are detailed. The power gained by first and higher-order synergies of simulation with several other disciplines such as system theories, systems engineering, computers, software engineering, software agents, and reliability are mentioned. The already achieved shift of paradigm from model-based to simulation-based is strongly endorsed. Keywords: Simulation-based discipline · Simulation-based methodology · Simulation-based approach · Shift of paradigm · Model-based approach · Experimentation · Experience

1 Introduction As Appendix A reveals, simulation-based concept is already used in over 170 disciplines, methodologies, and approaches. To explain the versatile use of simulation-based concept and the shift of paradigm from model-based to simulation-based, the following is done: the essence, hence the power of simulation, is clarified, its experimentation, experience, and imitation aspects are explained, a brief-history of model-based approach is outlined, richness and versatility of simulation as well as the benefits of the shift of paradigm to simulation-based paradigm are outlined.

© Springer Nature Switzerland AG 2021 M. S. Obaidat et al. (Eds.): SIMULTECH 2019, AISC 1260, pp. 29–45, 2021. https://doi.org/10.1007/978-3-030-55867-3_2

30

T. Ören

2 Simulation: The Many Aspects 2.1 The Essence of Simulation The term simulation –which based on the concept of similarity– has been used in English since 14th century. The concept of similarity offers a very rich paradigm as a list of over 180 terms of types of similarity attests. The list is divided in 14 categories as follows: Simulation concept, Model, Analogy, Imitation, Behavioral similarity, Functional similarity, Similarity in mathematics, Similarity in linguistics, Similarity in literature, Similarity in art, To be similar, Indistinguishableness, Disguise, similitude under a false appearance, and Non-similarity [1]. At be beginning, the term simulation represented “imitation” to emphasize the positive aspect, and “pretending/feigning” to stress the negative aspect of similarity. When used as a technical term, experimentation and experience are the essence of modeling & simulation (M&S). From the experimentation point of view, simulation is performing goal-directed experiments using a model of a dynamic system. From the experience perspective, simulation is gaining experience by use of a representation of a system. Details of about 100 definitions of simulation and a critical review of are given by Ören [2, 3]. 2.2 Experimentation The traditional system of logic by Aristotle (384 BC–322 BC), “especially his theory of the syllogism (which had) an unparalleled influence on the history of Western thought” concerned chiefly with deductive reasoning” and was expressed in his “Organon” (“Instrument”) [4]. Francis Bacon (1561–1626), as a reaction to Aristotle’s Organon: published New Organon (Novum Organum, 1620) [5] to promote experimentation which is one of the pillars of scientific method [6]. Experimentation can be done using the real system: in vivo, in situ (in place where it occurs), in vitro (under lab. conditions), or in silico (i.e., simulation). Using simulation to perform experiments with dynamic models has several advantages, such as experimentation can be performed for non-existing or unreachable real systems and simulated experiments can be performed under a large number of times including under extreme conditions. As shown in Fig. 1, connectivity of operations of simulation and real system defines two categories of simulation: In stand-alone simulation, operations of the simulation and the system of interest are independent. In integrated simulation (or symbiotic simulation) operations of the simulation and the system of interest are interwoven. In integrated simulation, simulation enriches or supports real-system operations (real-system enriching simulation and real-system support simulation). In enriched (augmented or mixed) reality simulation, real and virtual entities (that can be people or equipment) and the environment can exist at the same time. Hence, operations can take place in a richer augmented reality environment. In real-system support simulation, the system of interest (SOI) and the simulation program operate alternately and provide predictive displays for decision support and on-the-job training.

Not connected

31

Stand-alone simulation

Integrated simulation (or symbiotic simulation) Connected

Operations of real system and simulation are

Shift of Paradigm from Model-Based to Simulation-Based

Role of simulation with respect to the operations of the real system

enriches

Enriched (augmented mixed) reality simulation

or

supports

Simulation provides predictive displays for decision support and on-the-job training.

Fig. 1. Two categories of simulation based on the connectivity of operations of simulation and real system.

2.3 Experience Simulated experiences are used to enhance any one of three types of skills: motor skills (by virtual simulation, or simulators), decision making and communication skills (by constructive simulation, gaming simulation), operational skills (by live simulation); or for entertainment purposes (simulation games). Simulation allows several additional experience possibilities in enriched or augmented realities depending whether the equipment and the operator are real or virtual (Table 1). Table 1. Possibilities offered by simulation for enriched (augmented) reality. Equipment Real Operator

Virtual

Real

-Live simulation -Virtual simulation (a human operator uses real equipment (laser/gun) Simulator Virtual simulator

Virtual

-Automated vehicles* (auto pilot, aircraft without pilot, vehicle without driver)

e.g., an AI aircraft (in dogfight)

* automated vehicles need and can benefit from extensive simulations to be tested under a variety

of operating conditions

2.4 Imitation In visual arts such as painting and sculpture, as well as in literature and in its visual aspects (theater, opera, movies, TV), exiting (real-life based) or imaginary realities are

32

T. Ören

imitated and rendered. John Dewey elaborated on esthetic experience in his book “Art as Experience” [7]. This important possibility is beyond the scope of technical simulation.

3 Model-Based Approach: A Brief History Currently, model-based approach is widely used. Some of the model-based approaches are as follows: • • • • • • • • • • • • • • • • • •

model-based approach model-based clustering model-based consulting model-based decision making model-based definition model-based design model-based development model-based engineering model-based environment model-based exploration model-based feedback model-based image processing model-based interval estimation model-based learning model-based machine learning model-based management model-based method model-based optimization

A brief history of model-based approach may be useful. The first declarative simulation language: “GEST: General Systems Theory implementor” was developed as a doctoral dissertation, at the university of Arizona by Ören [8, 9]. It was based on Wymore’s Theory of Systems Engineering [10]. “Concepts for Advanced Simulation Methodologies” was the first article by Ören and Zeigler [11] where model and experimentation –as well as several components of experimentation– were separated. With this article, simulation has been the starting point for model-based activities. A NATO sponsored Advanced Study Institute resulted with two more publications on model-based activities [12, 13]. The first model-based approach study in an area different than simulation (i.e., in Systems Engineering) was published by Wymore [14].

4 Richness and Versatility of Simulation, and Benefits of Using It As an evidence of the richness and versatility of simulation, in an appendix of a recent article about 750 types of simulation are listed [15]. Details of the evolution of simulation, namely its progress from non-computational simulation to simulation-as-a service, by passing through the levels of computerized simulation, formal simulation,

Shift of Paradigm from Model-Based to Simulation-Based

33

AI-directed simulation, agent-directed simulation (ADS), soft simulation, simulation systems engineering, and simulation-based disciplines, was given by Ören et al. [16]. In the sequel several other aspects of the richness and versatility of simulation are elaborated. 4.1 System Problems and Simulation As shown in Fig. 2, three types of system variables are input, output, and state variables.

Input variable

Output variable

State variable Fig. 2. System variables.

From the systemic point of view, it is well known that there are three types of problems, namely analysis, design, and control problems [17]. As shown in Table 2, in analysis, design and control problems, the values of output, state, or input variables of a system are found, provided that the values of the two other types of variables are known. Table 2. Three types of system problems. Type of problem Given

Find

Analysis

input state

Design

input

Control

output output state

state output input

Simulation can be used all three types of system problems. In analysis and design problems the use of simulation is ubiquitous. In control problems, as shown in Fig. 3, simulation can be used for extensive test of the control strategy before it is applied to the controlled system. 4.2 Types of Inputs The concept of input is ubiquitous in computer applications. However, the possibilities of using both exogenous (generated outside of the system) and endogenous (generated within the system) inputs allow simulation very rich and versatile possibilities. Exogenous inputs include evaluated and filtered external inputs as well as actively perceived inputs of aware systems. Endogenous inputs include inputs generated by self-aware systems. Details of types of exogenous and endogenous inputs are covered in a forthcoming publication; therefore, are not repeated here [18].

34

T. Ören

Controlling system

Controlling system

Simulated system

controlled

Controlled system

Fig. 3. Simulation-based software development for control systems.

4.3 Experimental Condition One of the powers of computerized simulation is to repeat the simulation studies in not only under normal conditions but also under extreme and dangerous conditions. Satell clarifies in his “Why the future of innovation is simulation” article an important aspect of simulation: “When the Thomas Edison was asked about success amidst failure, he said that ‘If I find 10,000 ways something won’t work, I haven’t failed. I am not discouraged, because every wrong attempt discarded is another step forward.’ … ‘It also becomes clear why he regarded success as “1% inspiration and 99% perspiration.’ Failing 10,000 times is a physical and mental undertaking that far exceeds most people’s endurance. Today, however, a new breed of innovators are outsourcing failure to computer simulations and it’s changing everything we thought we knew about business strategy.” [19]. Vozenileck elaborates on the same theme and posits that: “Simulation in health care has massive potential to not just train and educate, but to provide real insight into the latent system flaws. Too often we ‘decide how the world might work’ when we have sophisticated tools to determine exactly how it really works. … One feature of our culture of inquiry within Jump Trading Simulation & Education Center is that we are able and willing to take simulation into the real clinical environment in “in situ” simulations.” [20]. Furthermore, many types of simulation offer variety of possibilities for experimentation and gaining experience.

Shift of Paradigm from Model-Based to Simulation-Based

35

4.4 Synergies of Simulation with Some Disciplines The term synergy –from Greek syn (together) and ergos (working)– means “combined action” and “mutually advantageous conjunction” [21]. “Synergy of disciplines” denotes their mutual contributions or enhancements. Figure 4 represents first order and higher order synergies.

B is enhanced due to contribution of A First order synergy:

A

B

A is enhanced due to contribution of B

Higher order synergies: A

B

A

B

B is enhanced due to the contribution of enhanced A to B

A enhanced by C enhances B …

C

Fig. 4. First order and higher order synergies.

Possible first and higher order synergies of modeling and simulation involve system theories, systems engineering, computers, software engineering, AI (Artificial Intelligence), (software) agents, and reliability (Fig. 5).

36

T. Ören

Simulation System theories

Reliability

Systems engineering

Software agents

Computers

AI

Software engineering

Fig. 5. Possible first and higher order synergies for modeling and simulation with system theories, systems engineering, computers, software engineering, AI, software agents, and reliability.

Synergies of simulation, agents, and systems engineering were elaborated by Yilmaz and Ören [22]. Figure 6 summarizes these very important possibilities. Details and importance of Agent-Directed Simulation are explained by Yilmaz and Ören [23].

Shift of Paradigm from Model-Based to Simulation-Based

37

Systems engineering

Synergy of simulation and SE

Simulationbased SE

ADS-based Systems Engineering

Synergy of ADS and SE

Simulation SE (SE for simulation)

Simulation

Agent simulation

Synergy of SE and agents

Systems Engineering agents Agent-based Systems Engineering

Systems Engineering for ADS

ADS - Synergy of simulation and agents

Software agents

Agent-supported simulation Agent-monitored simulation

Fig. 6. Synergies of simulation, agents, and systems engineering. (Abbreviations: ADS: Agentdirected simulation, SE: Systems engineering)

38

T. Ören

5 Simulation-Based Approach and Its Advantages for Many Disciplines, Methodologies, and Approaches The richness and versatility of simulation covered in Sect. 4 are applicable to many disciplines, methodologies, and approaches. Already the importance of simulation-based engineering science is well established [24]. A first mention of simulation-based approach was in the book of a NATO sponsored Advanced Study Institute [12, 13]. Some other recent references are given by Gianni et al. [25] and Mittal et al. [26] as well as the many chapters of the latter on engineering, systems engineering, software engineering, cyber-physical systems, complex adaptive systems, architectural design, natural sciences, enterprise management, education and training, and military training. The advantages of simulation-based approach were also emphasized by Ören [27] and Ören et al. [28]. Appendix A where over 170 types of simulation-based disciplines, methodologies, and approaches are listed is a testimony to the widespread acceptance of the simulation-based concept. Since all the entries in Appendix A can be searched on internet detailed references are not given.

6 Conclusions Model-based approach has been a sign of maturity for many disciplines and methodologies as they would benefit from model-based approach. Since simulation is modelbased, simulation-based approach is a natural next level of model-based approach where dynamic models can be used for a variety of reasons for experimentation and gaining experience. Therefore, many disciplines, methodologies, and studies can benefit by the adoption of a shift of paradigm from model-based to simulation based. This way, many professionals may have sharpened tools –as Confucius said– in their respective disciplines to perform their work.

Appendix Over 170 Simulation-based Disciplines, Methodologies, and Approaches – A– simulation-based acquisition simulation-based activity simulation-based after-action review simulation-based analysis simulation-based algorithm simulation-based application simulation-based approach simulation-based architectural design (continued)

Shift of Paradigm from Model-Based to Simulation-Based (continued) simulation-based artificial neural networks simulation-based assessment (of competence) simulation-based assessment (of skills) simulation-based augmented reality –B– simulation-based B-factor analysis simulation-based Bayesian data fusion simulation-based Bayesian filtering simulation-based Bayesian g-formula simulation-based Bayesian inference simulation-based benchmarking simulation-based bias correction simulation-based breeding design simulation-based building energy optimization –C– simulation-based calibration simulation-based case study simulation-based casting simulation-based coaching simulation-based communication workshop simulation-based comparison simulation-based complex adaptive systems simulation-based control simulation-based costing simulation-based creation (of digital twin) simulation-based curriculum simulation-based cyber-physical systems –D– simulation-based data engineering simulation-based decision making simulation-based decision support simulation-based demo simulation-based design simulation-based diagnosis simulation-based discipline simulation-based discovery (continued)

39

40

T. Ören (continued) simulation-based discrete optimization simulation-based drug discovery simulation-based dynamic programming simulation-based dynamic traffic assignment –E– simulation-based e-learning simulation-based e-training simulation-based earthquake scenario simulation-based econometric model simulation-based education (SBE) simulation-based educational workshop simulation-based emergency response planning simulation-based engineering simulation-based environment simulation-based era analysis simulation-based estimation simulation-based estimator simulation-based evaluation simulation-based event-analysis simulation-based experiential learning –F– simulation-based fault detection simulation-based fault injection simulation-based feasibility study –G– simulation-based games simulation-based genetic algorithm –H– simulation-based H-infinity estimator simulation-based hardware verification simulation-based health care simulation-based health care education simulation-based heuristic –I– simulation-based inference simulation-based innovation (continued)

Shift of Paradigm from Model-Based to Simulation-Based (continued) simulation-based instruction simulation-based internal models simulation-based interprofessional education simulation-based intervention simulation-based interview simulation-based invention simulation-based investigation simulation-based irrigation scheduling simulation-based iteration –L– simulation-based learning simulation-based learning methodology –M– simulation-based M-estimator simulation-based M-step value iteration simulation-based mastery learning simulation-based medical education simulation-based medical learning simulation-based medical teaching simulation-based medical training simulation-based method simulation-based methodology simulation-based military training simulation-based monitoring simulation-based movie –N– simulation-based neonatal resuscitation simulation-based neonatal resuscitation curriculum simulation-based network control simulation-based nursing education –O– simulation-based optimal design simulation-based optimization –P– simulation-based paramedic training simulation-based parameter optimization (continued)

41

42

T. Ören (continued) simulation-based patient simulation-based patient safety simulation-based performance analysis simulation-based performance control simulation-based petroleum engineering simulation-based planning simulation-based platform simulation-based post-cardiac arrest care simulation-based proof –Q– simulation-based quality improvement –R– simulation-based reality simulation-based reliability evaluation simulation-based research simulation-based robot simulation-based robot command simulation-based robot control simulation-based robot development simulation-based robot programming simulation-based robot-assisted simulation-based robot-assisted surgical training –S– simulation-based scenario simulation-based scheduling simulation-based science simulation-based search simulation-based security simulation-based simplification simulation-based skill acquisition simulation-based software engineering simulation-based statistical inference simulation-based strategy simulation-based structure optimization simulation-based study simulation-based support (continued)

Shift of Paradigm from Model-Based to Simulation-Based (continued) simulation-based surgery simulation-based surgery planning simulation-based surgical training simulation-based system simulation-based system analysis simulation-based system assessment simulation-based system design simulation-based system development simulation-based system emergency evacuation simulation-based system identification simulation-based system integration simulation-based system reliability simulation-based system specification simulation-based system testing simulation-based system validation simulation-based systems engineering –T– simulation-based t-peel test simulation-based t-test simulation-based t values simulation-based teaching (SBT) simulation-based team training simulation-based testing simulation-based testing platform simulation-based thorascopy training simulation-based tool simulation-based traffic assignment simulation-based training simulation-based tutorial –U– simulation-based UAS swarm selection simulation-based understanding simulation-based usability evaluation –V– simulation-based V&V simulation-based V&V method (continued)

43

44

T. Ören (continued) simulation-based V/F speed control simulation-based validation simulation-based value-at-risk simulation-based verification simulation-based virtual laboratory simulation-based virtual reality simulation-based vocational education simulation-based vocational training simulation-based V-process –W– simulation-based Web course simulation-based workshop –X– simulation-based X-ray system design –Z– simulation-based z-score method

References 1. Ören, T., Mittal, S., Durak, U.: The evolution of simulation and its contributions to many disciplines. Chapter 1. In: Mittal, S., Durak, U., Ören, T. (eds.) Guide to Simulation-Based Disciplines: Advancing our Computational Future, pp. 3–24. Springer, Cham (2017) 2. Ören, T.I.: The many facets of simulation through a collection of about 100 definitions. SCS M&S Magazine 2(2), 82–92 (2011) 3. Ören, T.I.: A critical review of definitions and about 400 types of modeling and simulation. SCS M&S Magazine 2(3), 142–151 (2011) 4. SEP-Aristotle’s Logic: Aristotle’s Logic. In: Stanford Encyclopedia of Philosophy. https:// plato.stanford.edu/entries/aristotle-logic/. Accessed 23 Dec 2019 5. SEP-Francis Bacon. In: Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/ent ries/francis-bacon/. Accessed 25 Dec 2019 6. SEP-Scientific Method. In: Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/ entries/scientific-method/. Accessed 23 Dec 2019 7. Dewey, J.: Art as experience. Minton, Balch and Company. (New York, NY: Perigee (Penguin group) paperback edition, August 2005) (1934) 8. Ören, T.I.: GEST: General System Theory implementor, A combined digital simulation language. Ph.D. Dissertation. Tucson, AZ: University of Arizona (1971) 9. Ören, T.I.: GEST - a modelling and simulation language based on system theoretic concepts. In: Ören, T.I., Zeigler, B.P., Elzas, M.S. (eds.) Simulation and Model-Based Methodologies: An Integrative View, pp. 281–335. Springer, Heidelberg (1984) 10. Wymore, A.W.: A Mathematical Theory of Systems Engineering: The Elements. Krieger, Huntington (1967)

Shift of Paradigm from Model-Based to Simulation-Based

45

11. Ören, T.I., Zeigler, B.P.: Concepts for advanced simulation methodologies. (SCS) Simulation, 32(3), 69–82 (1979) 12. Ören, T.I., Zeigler, B.P., Elzas, M.S. (eds.): Simulation and model-based methodologies: an integrative view. Springer, Heidelberg. NATO ASI Series (1984). https://doi.org/10.1007/9783-642-82144-8_1 13. Ören, T.I.: Model-based activities: A paradigm shift. In: Ören, T.I., Zeigler, B.P., Elzas, M.S. (eds.) Simulation and Model-Based Methodologies: An Integrative View, pp. 3–40. Springer, Heidelberg (1984) 14. Wymore, A.W.: Model-Based Systems Engineering. CRC Press, Boca Raton (1993) 15. Ören, T., Mittal, S., Durak, U.: Modeling and simulation: the essence and increasing importance. In: Niazi, M.A. (ed.) Modeling and Simulation of Complex Communication Networks, pp. 3–26. Stevenage, UK: IET Book Series on Big Data (2019 Invited Chapter) 16. Ören, T., Mittal, S., Durak, U.: The evolution of simulation and its contributions to many disciplines. In: Mittal, S., Durak, U., Ören, T. (eds.) Chapter 1 of Guide to Simulation-Based Disciplines: Advancing Our Computational Future, pp. 3–24. Springer, Cham (2017) 17. Karplus, W.J.: “The spectrum of mathematical modeling and systems simulation. Math. Comput. Simulat. Trans. IMACS 19(1), 3–10 (1977) 18. Ören, T.: Agent-directed simulation and nature-inspired modeling for cyber-physical systems engineering. In: Risco-Martin, J.-L., Mittal, S., Ören, T. (eds.) Simulation for Cyber-Physical Systems Engineering: A Cloud-Based Context. Springer, Cham (2020 – In Preparation) 19. Satell, G.: Why the future of innovation is simulation (2013). Forbes, 15 July 2013. https://www.forbes.com/sites/gregsatell/2013/07/15/why-the-future-of-innovation-issimulation/#7c4533e315e9. Accessed 23 Dec 2019 20. Vozenileck, J.: Why the future of innovation is simulation (2013). https://www.jumpsimul ation.org/research-innovation/our-blog/2013/september/why-the-future-of-innovation-issimulation. Accessed 23 Dec 2019 21. Merriam-Webster. (n.d.): Synergy. In: Merriam-Webster.com Dictionary. 23 Dec 2019. https:// www.merriam-webster.com/dictionary/synergy 22. Yilmaz, L., Ören, T.I. (eds.): Agent-Directed Simulation and Systems Engineering. Wiley Series in Systems Engineering and Management, Berlin (2009) 23. Yilmaz, L., Ören, T.I.: Agent-directed simulation (ADS). In: Yilmaz, L., Ören, T.I. (eds.) Agent-Directed Simulation and Systems Engineering, pp. 111–143. Wiley Series in Systems Engineering and Management, Berlin (2009) 24. NSF (2006). Revolutionizing Engineering Science through Simulation (A Report of the National Science Foundation Blue Ribbon Panel on Simulation-Based Engineering Science. https://www.nsf.gov/pubs/reports/sbes_final_report.pdf. Accessed 24 Dec 2019 25. Gianni, D., D’Ambrogio, A., Tolk, A. (eds.): Modeling and Simulation-Based Systems Engineering Handbook. CRC Press, Boca Raton (2014) 26. Mittal, S., Durak, U., Ören, T. (eds.): Guide to Simulation-Based Disciplines: Advancing Our Computational Future. Springer, Cham (2017) 27. Ören, T.: On the advantages of simulation-based approach in engineering. COJ Electron. Commun. 1(1), 1–3 (2018) 28. Ören, T., Mittal, S., Durak, U.: A shift from model-based to simulation-based paradigm: timeliness and usefulness for many disciplines. International J. Comput. Softw. Eng. 3(1) (2018 – Invited Paper). https://doi.org/10.15344/2456-4451/2018/126

Assessment of Learning Achievement in an Electronics Program Supported by an Online Simulation Applet David Valiente(B) , Fernando Rodr´ıguez, Juan Carlos Ferrer, Jos´e Luis Alonso, ´ and Susana Fern´ andez de Avila Communications Engineering Department, Miguel Hernandez University, Av. Universidad sn, 03202 Elche, Spain {dvaliente,fernando.rodriguezm,jc.ferrer,j.l.alonso,s.fdezavila}@umh.es

Abstract. Nowadays general learning programs in Bachelor’s degrees in engineering have gradually adapted towards the inclusion of more electronics oriented subjects. The prominent growth of technology applications reveals its influence on the design of such education plans at university. Most of them introduce advanced methodologies, which in most cases, are principally sustained by ICT (Information, Communication and Technology) tools. In this work, we assess the effectiveness of a learning program for an electronics course, based on the developed methodology of a previous work, which has been periodically renewed each year, during the six last academic years. In particular, we analyse the achievement of students when using additional digital resources such as circuit simulation, by means of an online Java applet. Moreover, the scope of the analysis has been statistically extended in order to compare with the benefits associated to the use of other digital resources and to extract possible inferences. In this sense, an e-learning platform (M oodle) with auxiliary materials is made available to the students. Besides Java simulation examples and an assignment, video lessons, practical lessons explanations, interactive exercises, and a Problem-Based Learning (PBL) group assignment, are hosted in M oodle. Group testing has been considered for the arrangement of students during the academic year 2018–2019, in order to obtain unbiased achievement results. According to such results, the use of a Java simulation applet (JSA) demonstrates relevant achievement in contrast to the other resources. In addition, the results also comprise the satisfaction and attitude to the whole program, reported by students. Finally, further inferences are extracted from history data of the course, since academic year 2013–2014 to the current academic year, 2018–2019. Keywords: Simulation

1

· Electronics · Education · Applet

Introduction

In the recent days simulation has become a widely extended approach applied to, and integrated in many learning programs, specially within the framework c Springer Nature Switzerland AG 2021  M. S. Obaidat et al. (Eds.): SIMULTECH 2019, AISC 1260, pp. 46–60, 2021. https://doi.org/10.1007/978-3-030-55867-3_3

Assessment of Learning Achievement in an Electronics Program

47

of Bachelor’s degrees in engineering. In many cases, researchers and lecturers have conferred considerable importance to the use of locally installed simulation softwares to cope with a broad variety of problems in engineering [4,8]. There are many examples like virtual labs [5,17,19], web-based courses [6,20], or mobile apps [11,14]. Notwithstanding that computation power of today’s computers is generally sufficient for running many of such softwares, the tendency reveals a rise of online applications, available on the Internet. Besides this, there is also a very important role in the design of the learning programs, and how the use of simulation is addressed within such learning context. Most of the traditional methodologies have been enhanced with different approaches such as blended-based models [1], project-based [2], or problembased [13]. However, the valid attainments of a program mostly depend on the sort of activities designed by lecturers. Therefore the real success highly depends on such design, and its relationships with the stimulation for an active learning of concepts, skills acquisition and motivation [3]. In this context, the aim of this work is to establish goals for the implementation of a program within the previous framework, with particular support of digital resources. According to this, we present a learning program for an engineering course in electronics, mixing traditional theory and laboratory lessons with a blended model sustained by M oodle. Students have available auxiliary digital resources to complement their comprehension and learning. Besides this, we pay special attention to the use of an online circuit simulation applet in Java, from now on JSA [16]. In this manner, lectures can always focus on more exemplifying lessons, supported with the use of simulation in class (either theory, practical or laboratory lessons). This tool does not require any installation, and its use is highly intuitive, as compared to others. This was essential for achieving usability, accessibility and easiness. Such qualities were considered of paramount importance so as to lead students to fully use the tool. Other software like P spice and M atlab/Simulink [9,12] tend to arise issues at the first stages, at least amongst students. Figure 1 presents the main window of the Java applet used in this work, F alstad. The program was initially implemented in three Bachelor’s degrees in engineering [18], since students are enrolled in the same course of electronics, which is taught in the second year of the three degrees: Mechanics; Energy; and Electronics and Industrial Automation. In particular, we aim at assessing the real improvement on the achievement presented by students when using different sort of digital resources, with special attention to the use of JSA. We establish a comparison to extract further inferences and dependencies between the use of different resources and the achievement indicators produced by students. To that end, in this work we extend the previous contribution [18] by introducing group testing during the current academic year, 2018–2019. Students participating in the test group agreed to use and hand in an assignment consisting of JSA exercises, whereas the control group did not take it. Apart from this comparison, some other achievement results will be presented for the five previous academic years. Finally, a custom survey has been designed in order to evaluate in depth

48

D. Valiente et al.

aspects such as concept comprehension, availability and use of resources, and satisfaction and attitude to the current program, as reported by students. The remainder of the paper is: Sect. 2 describes the method and materials for the implementation of the present program; Sect. 3 presents the set of acquired and computed results; Sect. 4 presents the conclusions extracted from the results.

Fig. 1. Main window of the online circuit simulation applet in Java, F alstad [16].

2

Methods and Materials

The present learning program has been sequentially renewed since the academic year 2013–2014, to the current academic year, 2018–2019. The course deals with the fundamentals of electronics and it is taught during the second year of several Bachelor’s degrees in engineering. The initial design pursued the success in general challenges highlighted by the Horizon milestones in higher education [15]. So that, the general goals are: – Promoting active learning methodologies and digital skills. – Facilitating the use of ICT resources to improve digital literacy. And more particularly, the specific objectives are: – Reinforcing the conceptual learning associated to general electronics magnitudes. – Broadening comprehension by support of ICT resources, specially concentrated on the use of JSA.

Assessment of Learning Achievement in an Electronics Program

2.1

49

Course Syllabus

The list of topics comprised in this course is depicted by the following syllabus, presented in Table 1. Synthesizing, four different units are taught during the fifteen weeks the course takes place. It may be observed that the contents cover the main basis of electronics. Students voluntarily attend face to face theory lessons, three hours per week. Besides this, practical lessons and laboratory lessons are also scheduled. Additionally, students are allowed to take advantage of the additional ICT resources available in M oodle. Concretely, they may opt to hand in a PBL group assignment and a dossier with JSA exercise assignment for those participating in the group testing, as it will be described in the following subsection. The contents of such assignments are thought to match real aspects taught during all the lessons and the course, with special incidence on real and typical issues in electronics. This fact usually commits students to enhance their own active knowledge. Furthermore, students can also visualize content related videos; download detailed explanations about practical and laboratory lessons; and test their assimilations through interactive exercises, amongst others. The relevance of the use of JSA is eminent during all the theory lessons, where every conceptual explanation, activity of design or resolution exercise, is not only analytically addressed, but also tested and exemplified with JSA. This also permits comparing theoretical and analytical results with the numerical resolution provided by the simulation tool. In other words, all the design and planned lessons in this program (theory, practical and laboratory) were conceived to be complemented by support of JSA. Thus all the activities, examples and even conceptual explanations made by lecturers, were always validated by exemplification with JSA. As per the perception during the lessons, students tend to assimilate more easily the taught concepts, when they are able to observe the behavior of the resulting waveform, presented by the simulation tool. For instance, the graphical representation of the simultaneous evolution of the voltage, current and power magnitudes in a circuit, aids in the faster assimilation. That is the main reason why all the lessons during the fifteen weeks of the course are always supported with JSA, apart from the use of such tool for the assignments. Finally, Table 2 presents the incorporation of the mentioned resources during each academic year, from 2013–2014 to the current academic year, 2018–2019. The history data associated to the achievement of former academic years will lead to infer whether some resources imply a positive bias on the acquired knowledge, the active learning, and performance of the students. In particular, throughout the six last academic years, the M oodle of the course has incorporated resources such as: i) theory slides for each unit; ii) solved exercises; iii) video lessons: theory, circuit resolution and real applications for laboratory lessons; iv) materials and tools for the PBL group assignment; v) JSA examples. It is worth noticing that the support of JSA is being employed for the past two academic years. The new inclusion during the current academic year has been the video lessons. The inferences on the achievement will be further discussed in the results section.

50

2.2

D. Valiente et al.

Participants

The participants of this study correspond with the total number of students enrolled in the course (266) during the academic year 2018–2019. Figure 2 presents the evolution of the enrollment from the academic year 2013–2014. The total set of students are uniformly distributed into the following Bachelor’s degrees: Mechanics; Energy; and Electronics and Industrial Automation. This is due to the fact that the academic curriculum of these degrees have in common the same electronics course we are considering in this work. Besides this course, students also took a previous course during their first year, in which the basis of circuits resolution (Kirchhoff’s and Ohm’s laws) are addressed. Table 1. Course syllabus and structure: weekly schedule with theory and laboratory lessons. Course syllabus Week

Unit

Content

1

1 - Semiconductors

Electronics principles; semiconductor operation and model

2

2 - Diodes

Physical operation and model

3

2

Rectifiers

4

2

Filters and modulation

5

2

Clipping and clamping circuits

6

2

Regulators

7

3 - Operational Amplifiers (OAs)

OAs operation and examples OAs parameters

8

3

9

3

Applications

10

4 - Transistors

BJT operation, parameters and examples

11

4

BJT AC-configurations

12

4

BJT amplifiers

13

4

Field-effect transistors’ operation

14

4

MOSFET and JFET amplifiers

15

4

Application circuits

Laboratory 8

1

9

1

Review of JSA Practical seminar of JSA

10

2

Applications with diodes: rectifiers

11

2

Applications with diodes: clippling and clamping circuits

12

3

Application with OAs: general amplifiers and rectifiers

13

3

Application with OAs: logical circuits

14

4

Applications with transistors: DC source circuits

15

4

Applications with transistors: AC-amplifier circuits

In order to obtain adequate and unbiased results through formal group testing [7], the arrangement of students into groups was randomly conducted. Two groups (control group and test group) were established by each half of the enrolled students. Students in the test group agreed to work on the JSA assignment, so as to be hand in at the end of the course. The control group followed the standard procedure. Nonetheless, they kept the choice to voluntarily use the rest of available resources during the course.

Assessment of Learning Achievement in an Electronics Program

51

Table 2. Inclusion of digital resources during each academic year. Academic year

2013–2014 2014–2015 2015–2016 2016–2017 2017–2018 2018–2019

M oodle: ICT resources Theory slides

✗











Solved exercises

✗



✗



✗



PBL group assignment ✗

✗









JSA support

✗

✗

✗

✗





Video lessons

✗

✗

✗

✗

✗



Enrolled students per academic year 300

number of students

250

200

150

100

50

0 2013-14 2014-15 2015-16 2016-17 2017-18 2018-19

academic year

Fig. 2. Number of enrolled students during the six last academic year. Table 3. Content blocks of the survey. Survey Questions Aspects 1–5 6–10 11–12

13–14 15–16

Essentials of electronics

Indicators

Do students comprehend and show acquired knowledged on the basis of electronics? Problem resolution Do students transfer their knowledge to circuit resolution? Do they properly apply the studied techniques and laws? Resource availability Do students report knowledge, amongst others, of ICT resources to support their comprehension? And other resources? Use of ICT resources Do students affirm using ICT resources to support comprehension? Attitude and satisfaction Do students appreciate positively the introduction of additional resources throughout the course, in particular JSA? Do they perceive it as a relevant tool for complementing their active learning?

52

2.3

D. Valiente et al.

Assessment

Notwithstanding that one of the main objectives is to assess the final level of achievement associated to the use of the Java simulation applet in the course, a more definite assessment is carried out in terms of: concepts comprehension; skills acquisition; and attitude and satisfaction to the program. Firstly, it is necessary to appraise the achievement perceived by lecturers during the lessons and after correcting the assignments and the final exam (marks).

Fig. 3. Comparison exercise comprised in the JSA assignment. (a): AC-DC full-wave rectifier with diodes. (b) AC-DC full-wave rectifier with operational amplifiers.

However, the design of a specific survey was fundamental in order to analyse in depth other important aspects. It was conceived to assess the comprehension

Assessment of Learning Achievement in an Electronics Program

53

of the basic electronics principles and the ability to apply properly such knowledge to the resolution of real circuits through specific techniques and models. Furthermore, another aspect assessed by the survey was the availability and use of resources made by students in this field. Finally, their attitude and satisfaction to the program. Table 3 summarizes such aspects, assessed in different blocks (1 to 5) of the survey. The answers to this survey were registered for both, the control and test groups in a discrete Likert scale from totally disagree (1) to totally agree (5). Notice that all the blocks of questions presented in Table 3 were validated by five full professors, experts in electronics and with long experience in the teaching of similar subjects in different Bachelor’s degrees. It is also worth mentioning again that the test group agreed to hand in at the end of the course, a specific JSA assignment with a complete variety of exercises related to the taught content during the course. Some examples of such exercises may be observe in Fig. 3, where a comparison of AC-DC converters is depicted. Figure 3(a) represents a full-wave rectifier, implemented with diodes. In contrast, Fig. 3(b) presents a full-wave rectifier, implemented with a operational amplifiers. Students may check the same output provided by different circuits. Hence students in the test group were allowed to notice more explicitly the real behavior of some circuits studied in class, and to compare the results presented in the lessons and the results obtained with JSA. Further comparison outcomes with respect to the control group will be analysed in the results section.

3

Results

This section presents the results obtained in this study. As commented initially, these are basically aimed at verifying the possible implications of using specific ICT resources for the achievement of students, specially regarding the use of simulation by means of JSA. In addition, we also assessed the satisfaction and attitude to the program, as reported by students in the last block of the survey. These results incorporate more historic data of achievement, but also the addition of the current academic year, 2018–2019. 3.1

Achievement

The first outline corresponds to the insight derived from historic data of achievement, registered during the six last academic years, as presented in Table 4. It may be observe that there is a plausible positive trend in the marks distribution during the six last academic years. The six categories in which the marks have been classified are expressed out of a total, and maximum percentage of achievement of 100% (0–10), considering the number of students who took the exam. In contrast, those who did not took the exam are taken into account within the dropout rate. In particular, it may be observed that over the last three academic years, 2016–2017 to 2018-2018, there is a noticeable redistribution to higher marks, with also less percentage of fails, but specially less percentage of

54

D. Valiente et al.

dropout. Such fact is even more prominent during the last two academic years, which indeed are the only two years in which JSA has been introduced in the course (Table 3). Despite this fact, further statistical inferences are presented in the following subsections, so as to extract robust and consistent analysis in this sense. Similarly, Fig. 4 summarizes the previous data with a representation of the mean marks obtained during the six last academic years. Again, it can be confirmed that the introduction of JSA in 2017–2018 and 2018–2019 may have implied the positive increase in the mean marks (greater than 7 out of 10, on the left-side axis). Table 4. Distribution of marks during the last six academic years. Marks

2013–2014

2014–2015

2015–2016

2016–2017

2017–2018

2018–2019

9–10 (A, A+)

2.02%

0.56%

1.13%

1.26%

4.31%

3.55%

8–8.9 (B+, A−)

6.06%

4.44%

5.08%

6.29%

11.21%

10.10%

10.17%

8.81%

10.34%

10.72%

6–6.9 (C−, C, C+)

11.11%

5.05%

7.22%

10.73%

13.84%

12.93%

13.50%

7–7.9 (B−,B)

6.67%

5–5.9 (D)

17.17%

18.89%

20.90%

20.13%

13.79%

16.42%

0.05, etc). This confirms the validation of the hypothesis, although there is not a prominent correlation between the marks and the number of used resources. This would suggest that not all the resources add the same value to the achievement of the students. Table 5. Correlation tests for the marks obtained by students, and the number of used resources during each the course. Marks rp 8–10 6–7.9

0.18

rs

p-value tstudent ttest( α2 ;n−1)

0.15 0.18

−0.11 −0.10 0.15

0.26

3.18

0.12

3.18

5–5.9

0.02

0.03 0.80

−0.25

3.18

χ2test (27.42 > 3.84) and p-value < α (5.70E−7 < 0.05). Table 6. Correlation and contrasts tests between the marks obtained and the specific resources used by students. Resource used

rp

χ2

p-value

χ2test

Video lessons

0.10

2.11 0.096

3.84

Solved exercises

0.25

2.11 0.096

3.84

PBL group assignment

0.33

2.11 0.096

3.84

JSA assignment None

0.79 27.42 5.70E−7 3.84 −0.14

1.16 0.21

3.84

Assessment of Learning Achievement in an Electronics Program

3.4

57

Survey

Finally, Fig. 6 presents the results acquired after the implementation of the custom survey, described in Sect. 2.3 (Table 2), which was finally passed to the control and Control group responses: questions 1-10

Test group responses: questions 1-10 6

5

5

4

4

mean responses

mean responses

6

3

2

1

3

2

1

0

0 1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

questions

questions

(a) Test group responses: questions 11-16 6

5

5

4

4

mean responses

mean responses

Control group responses: questions 11-16 6

3

2

1

3

2

1

0

0 11 12 13 14 15 16 questions

11 12 13 14 15 16 questions

(b)

Fig. 6. Responses to the survey reported by the control group  and the test group , respectively. (a) comprises responses to questions 1–10 and (b) responses to questions 11–16.

58

D. Valiente et al.

test groups, at the end of the current course, 2018–2019. These results are expressed by mean and standard deviation values. The classification of the set of questions follows the same structure described in Table 3, as subdivided into five blocks. More specifically, this set of questions has been presented separately by Fig. 6(a) and Fig. 6(b). The first two blocks, with questions 1–10 regarding to acquisition of concepts, are presented in Fig. 6(a). Whereas the rest of the blocks with questions 11–16 regarding use and availability of resources, and satisfaction and attitude to the program are presented in Fig. 6(b). Once again, there is a substantial gap between the control and the test group. The test group proves a slightly higher level of comprehension, but also higher levels of satisfaction and attitude to the program. Overall, the responses to this survey demonstrate certain benefits achieved by those who took the JSA. Apart from the previous statistical inferences presented in the previous subsection, these outcomes prove the relevant advantage that the use of JSA represent for students. They reported to become more aware of their learning, but also to give more account of self-confidence and motivation towards the next subjects and courses in this field.

4

Conclusions

This approach has implemented an active learning program with support of an online circuit simulation applet. Such program has been assessed in terms of achievement as a case study, during the current academic year, 2018–2019, and compared with respect to the former editions, from 2013–2014. The course deals with basic electronics in three Bachelor’s degrees in engineering, with the aid of several digital resources available via M oodle, but with special reinforcement through simulation and its inclusion in every activity and explanation during the face to face lessons. The main objective was to improve and assess the real achievement of the students, concentrating on the active learning and autonomous comprehension of concepts and skills. In this sense, we produced a first analysis of a set of achievement results, which compared the evolution throughout the six last academic years in terms of marks. It was confirmed that the inclusion of JSA came associated to better achievement results, not only in terms of marks but also in dropout rate. Furthermore, statistical results were also produced through group testing, so as to reveal that the more number of resources used by students did not imply highly dependent improvement of the achievement. As a consequence, we specifically compared the correlations extracted from each single resource and its dependency with the marks. At this point was confirmed that the only highly relevant resource for the achievement of students was JSA. Furthermore, students within the test group who undertook JSA exercises, proved to obtain a better distribution of marks with respect to the control group. It is also worth mentioning that in general, the number of student who pass the course is notably higher in the test group. Finally, the responses to the survey also revealed further comprehension and satisfaction and motivation amongst the students in the test groups.

Assessment of Learning Achievement in an Electronics Program

59

All in all, it can be concluded that the support of an online simulation applet during an electronics course is the most relevant resource for the real achievement and active learning of students. According to the results, the positive outcomes have been statistically validated and this program has been proved as reliable in terms of self-autonomy and long-term learning for students. The historic data comparison and group testing provide robustness to this verification. Acknowledgements. This research has been partially funded by the Spanish Government through the project DPI2016-78361-R (AEI/FEDER, UE); the Valencian Research Council through the project AICO/2017/148; the Valencian Research Council and the European Social Fund through the post-doctoral grant APOSTD/2017/028.

References 1. Aguilar-Pena, J.D., Munoz-Rodriguez, F.J., Rus-Casas, C., Fernandez-Carrasco, J.I.: Blended learning for photovoltaic systems: virtual laboratory with PSPICE. In: 2016 Technologies Applied to Electronics Teaching (TAEE), pp. 1–6, June 2016 2. Amiel, F., Abboud, D., Trocan, M.: A project oriented learning experience for teaching electronics fundamentals. IEEE Commun. Mag. 52(12), 98–100 (2014) 3. Arrosagaray, M., Gonzalez-Peiteado, M., Pino-Juste, M., Rodriguez-Lopez, B.: A comparative study of Spanish adult students attitudes to ICT in classroom, blended and distance language learning modes. Comput. Educ. 134, 31–40 (2019) 4. Dickerson, S.J., Clark, R.M.: A classroom-based simulation-centric approach to microelectronics education. Comput. Appl. Eng. Educ. 26(4), 768–781 (2018) 5. Diwakar, A., Poojary, S., Noronha, S.B.: Virtual labs in engineering education: implementation using free and open source resources. In: 2012 IEEE International Conference on Technology Enhanced Education (ICTEE), pp. 1–4, January 2012 6. Flores, M., Paya, L., Valiente, D., Gallego, J., Reinoso, O.: Deployment of a software to simulate control systems in the state-space. Electronics 8(11), 1205 (2019) 7. Frederiksen, N.: The real test bias: influences of testing on teaching and learning. Am. Psychol. 39(3), 193–202 (1984) 8. Huanyin, Z., Jinsheng, L., Yangjie, W., Hong, X., Min, Q.: Computer simulation for undergraduate engineering education. In: 2009 4th International Conference on Computer Science Education, pp. 1353–1356, July 2009 9. Iyoda, I., Belanger, J.: History of power system simulators to analyze and test of power electronics equipment. In: 2017 IEEE History of Electrotechnology Conference (HISTELCON), pp. 117–120, August 2017 10. Lehmann, E.L.: Testing statistical hypotheses: the story of a book. Stat. Sci. 12(1), 48–52 (1997) 11. Musing, A., Drofenik, U., Kolar, J.W.: New circuit simulation applets for online education in power electronics. In: 2011 5th IEEE International Conference on E-Learning in Industrial Electronics (ICELIE), pp. 70–75, November 2011 12. Peng, L., Bao, L.: Application of matlab/simulink and orcad/pspice software in theory of circuits. In: Wu, Y. (ed.) Software Engineering and Knowledge Engineering: Theory and Practice, pp. 1055–1064. Springer, Heidelberg (2012) 13. Perales, M.A., Barrero, F., Toral, S.L.: Learning achievements using a PBL-based methodology in an introductory electronics course. IEEE Revista Iberoamericana de Tecnologias del Aprendizaje 10(4), 296–301 (2015)

60

D. Valiente et al.

14. Rakhmawati, L., Firdha, A.: The use of mobile learning application to the fundament of digital electronics course. In: IOP Conference Series: Materials Science and Engineering, vol. 296, no. 1, pp. 012–015 (2018) 15. New Media Consortium: New media consortium horizon report. http://www.nmc. org/publication-type/horizon-report/ 16. Falstad, P.: Falstad Simulation Applets (2019). https://www.falstad.com/circuit/ 17. Valiente, D., Berenguer, Y., Pay´ a, L., Peidr´ o, A., Reinoso, O.: Development of a platform to simulate virtual environments for robot localization. In: The 12th Annual International Technology, Education and Development Conference INTED 2018, Valencia, Spain, pp. 1232–1241 (2018) 18. Valiente, D., Paya, L., de Avila, S.F., Ferrer, J.C., Cebollada, S., Reinoso, O.: Active learning program supported by online simulation applet in engineering education. In: 9th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, Simultech, pp. 121–128. Scitepress (2019) 19. Valiente, D., Berenguer, Y., Pay´ a, L., Fonseca Ferreira, N.M., Reinoso, O.: Environment virtualization for visual localization and mapping. In: Merdan, M., Lepuschitz, W., Koppensteiner, G., Balogh, R., Obdrˇza ´lek, D. (eds.) Robotics in Education, pp. 209–221. Springer International Publishing, Cham (2020) 20. Yalcin, N.A., Vatansever, F.: A web-based virtual power electronics laboratory. Comput. Appl. Eng. Educ. 24(1), 71–78 (2016)

Application of Artificial Neural Networks for Active Roll Control Based on Actor-Critic Reinforcement Learning Matthias Bahr1(B) , Sebastian Reicherts2 , Philipp Sieberg2 , Luca Morss3 , and Dieter Schramm2 1 The Hydrogen and Fuel Cell Center ZBT GmbH, Carl-Benz-Straße 201,

47057 Duisburg, Germany [email protected] 2 Chair of Mechatronics, University of Duisburg-Essen, Lotharstraße 1, 47057 Duisburg, Germany 3 Vodafone GmbH, Ferdinand Braun Platz 1, 40549 Dusseldorf, Germany

Abstract. This work shows the application of artificial neural networks for the control task of the roll angle in passenger cars. The training of the artificial neural network is based on the specific actor-critic reinforcement learning training algorithm. It is implemented and trained utilizing the Python API for TensorFlow and set up in a co-simulation with the vehicle simulation realized in IPG CarMaker via MATLAB/Simulink to enable online learning. Subsequently it is validated in different representative driving maneuvers. For showing the practicability of the resulting neural controller it is also validated for different vehicle classes with respect to their corresponding structure, geometries and components. An analytical approach to adjust the resulting controller to various vehicle bodies dependent on physical correlations is presented. Keywords: Artificial neural network · Machine learning · Actor-critic · Reinforcement learning · Active roll control · Vehicle dynamics

1 Introduction Machine Learning and the implementation of Artificial Neural Networks (ANN) is used in many scientific fields. Despite it being more frequently applied in e.g. data science, there are few approaches and publications regarding the usage in terms of controlling. Applying ANNs to control tasks and specifically vehicle dynamics is a rarely explored area. By reason of the relevance of machine learning in neuroinformatics, ANNs are more frequently used in the field of control engineering, whereas using reinforcement learning for the control tasks of vehicle dynamics is a comparative unexplored science. While the active roll control is conventionally realized by PID controlling, Sieberg et al. have shown the effectiveness of a neuro-fuzzy controller [1]. Neuro-fuzzy systems combine the human way of reasoning with learning methods and topologies of neural networks to form a hybrid artificial intelligence able to perform an active roll control. © Springer Nature Switzerland AG 2021 M. S. Obaidat et al. (Eds.): SIMULTECH 2019, AISC 1260, pp. 61–82, 2021. https://doi.org/10.1007/978-3-030-55867-3_4

62

M. Bahr et al.

Boada et al. [2] have shown an approach on the active control using reinforcement learning. An ANN has been trained using simulation to regulate the roll angle of a single unit heavy vehicle model with five degrees of freedom. Even though a simplified model with few degrees of freedom and ideal testing and training conditions are used by Boada et al., the machine learning techniques can be improved to handle control tasks in more realistic scenarios. This work shows the usage for a vehicle simulation made up of a multibody system. Fu et al. [3] have applied a reinforcement learning method for an active suspension control on a quarter-vehicle model with two degrees of freedom. Because of the demand of fast computations regarding the investigated dynamics and vibration control, a critic-only topology has been used instead of a more time-consuming actor-critic model, which consists of two separate ANNs. The flexibility and potential in implementing reinforcement learning has been increased by the vast recent advances in the software and hardware development especially for artificial neural networks. This allows to use more complex and promising structures like the actor-critic model to improve and enlarge the accuracy and practicability of neural controllers to certain tasks. This work will investigate the feasibleness of applying ANNs to control tasks and particularly controlling vehicle dynamics by the example of an active roll control with the specific actor-critic reinforcement learning algorithm. The aim is to show and record that the proposed model proves useful for controlling the roll angle of passenger cars and thus for other control tasks concerning different vehicle dynamics. Therefore, it is not to be aimed to deliver an optimized but a functional controller. The functionality is verified by reducing the roll angle in comparison to a passive stabilization with common spring and damper elements. In [4] the development and validation of an active roll control based on actor-critic reinforcement learning is elaborated. This study extends the validation proposed by the adaption for different vehicles respectively vehicle classes. Furthermore, an analytical approach for this adaption to the corresponding vehicle structures, geometries and components is presented. So, the developed controller is used with respect to the physical boundary conditions. In the following the basics of the considered vehicle dynamics in view of the active roll control in the form of the relevant equations of motion and related influences are presented. Afterwards the applied training algorithm and the topology of the ANN is shown followed by the training process and the associated simulation environment. The developed controller is then validated for different driving conditions and vehicle classes in comparison to a passive roll stabilization. Finally, the whole application is summarized and further researches are proposed.

2 Active Roll Control Driver assistance systems aim to increase either safety or ride comfort or even both. Active roll control fulfills both requirements. The main task of the design of such driver assistance systems is to develop and optimize regulations, that are executed by actuators via a control unit. By doing this, the vehicle components can be arbitrarily influenced within their physical limits.

Application of Artificial Neural Networks for Active Roll Control

63

Vehicle dynamics are characterized by several interactions regarding translations and rotations. The rotational movement and orientation of a vehicle can be described by the Tait-Bryan angles and their derivations in relation to its body-fixed coordinate system. The movement around the vertical axis z is called yaw, the movement around the transverse axis y is called pitch and the movement around the longitudinal axis x is called roll, which is the subject of this study. The visualization of the roll movement and the resulting roll angle ϕ is shown in Fig. 1.

Fig. 1. Visualization of the roll angle in the body-fixed reference system [5].

The roll movement is mainly caused by driving curves or uneven surfaces. Circular drives induce a centrifugal force and a lateral acceleration ay , which tilts the car outside the curve and results in an angular deviation from the reference system. This deviation is quantified with the roll angle ϕ. 2.1 Linear Roll Model To describe the roll movement for fundamental investigations for real time simulations, a simplified linear roll model is proposed. This model is subject to the same assumptions as the single track model, which are listed in [5]. The linear roll model is valid for small roll angles ϕ. Moreover, the front and rear axles are combined to one axle with the roll center W. This roll center is assumed to be at the height of the combined axle. The lateral acceleration ay generates a roll torque, which in turn results in opposite reaction forces of the wheel suspension springs FF and dampers FD and the stabilizers FSt acting on the vehicle chassis. Those forces are geometrically generating a countertorque against the roll movement (see Fig. 2) and thus are stabilizing the vehicle to some extent passively.

64

M. Bahr et al.

Fig. 2. Description of the roll dynamics in a linear roll model [5].

Applying the conservation of momentum with respect to the rolling center leads to the following equations of motion [5]:  θA ϕ¨ = (hS − hW ) cos ϕ · mA ay  + sF,v FF,l,v − FF,r,v  + sF,h FF,l,h − FF,r,h  + sD,v FD,l,v − FD,r,v  + sD,h FD,l,h − FD,r,h  + sSt,v FSt,l,v − FSt,r,v + sSt,h FSt,l,h − FSt,r,h + (hS − hW ) sin ϕ · mA g

(1)

The forces involved by the springs FF , dampers FD and stabiliziers FSt are indexed with v for the front axle and h for the rear axle. Besides that, the index l stands for the left side and r for the right side. The distances s are the specific lever-arms of the chassis forces and hS is the height of the center of mass while hW is the height of the roll center. The moment of inertia of the vehicle chassis is represented by θA . The body mass mA is assumed to equal the total vehicle mass. By involving the equations for the reaction forces and under the assumption of small roll angles ϕ, Eq. (1) leads to [1] −2



1 y + (hS θA [(hS − hW )mA a 2 2 2 d sF,v cF,v + sF,h cF,h ϕ − 2 sD,v v

ϕ¨ =

− hW )mA gϕ  2 d ϕ˙ − M + sD,h St,A ]. h

(2)

Even though this results in a complex dynamical relationship, the numerical correlation between the roll angle, its deviations ϕ, ˙ ϕ¨ and the lateral acceleration ay to the counter roll torque MSt,A is the part that needs to be learned by the ANN. This correlation is dependent on several vehicle and installed component geometries. In addition to the distances s, the vehicle mass mA , the vehicle moment of inertia θA and the height of the center of mass hS and roll center hW are influencing parameters. Furthermore, the damper constants d and the spring constants cF have impact on the roll movement. The total counter roll torque MSt,A is the actuating value and needs to be predicted by the neural controller.

Application of Artificial Neural Networks for Active Roll Control

65

2.2 Stabilizer The stabilizer also known as anti-roll bar is used to reduce the roll angle in addition to the conventional spring and damper components to increase the ride comfort and safety. Moreover, the self-steering of the vehicle can be affected [6]. The working principle is based on connecting the two tires of one axle and thus the resulting reaction forces. The difference between those normal forces is reduced and the vehicle is stabilized regarding the roll movement. Passive Stabilizer A passive stabilizer is made of a torsion bar spring generating a torque against the roll movement. A simplified model of it is shown in Fig. 3.

Fig. 3. Modelling of a passive stabilizer [5].

Dependent on its geometries and the displacement caused by the roll angle ϕ, the stabilizer forces can be approximated by FSt ≈

lSt cSt ϕ b2St

(3)

and resulting in a torque via the lever: MSt = FSt · bSt =

lSt cSt ϕ. bSt

(4)

Due to the fact that the forces FSt are applied on the wheel suspension, the geometrically translated forces FA acting at the point the stabilizer is mounted to the chassis, can be calculated by FA =

lSt FSt 2sSt

(5)

66

M. Bahr et al.

and leading to the torque:   l2 lSt MSt,A = −sSt FA,l + FA,r = − St2 cSt ϕ = − MSt . b bSt St

(6)

Equation (6) shows the geometrical relation of the counter roll torque MSt,A and the stabilizer torque MSt acting on the wheel suspension. Active Stabilizer Active stabilizers are implemented to be able to manipulate the roll motion and vehicle dynamics e.g. the self-steering gradient. To realize this, a passive stabilizer is split into two parts in the middle and an actuator is inserted, which connects its free ends. This actuator can be either hydraulically or electromechanically. In this work an electromechanical actuator is used. The actuator is able to generate a desired torque MSt against the roll movement and thus enables to reduce the roll angle to a large extent or rather influence the vehicle dynamics in a proper way. The working principle is shown in Fig. 4.

Fig. 4. The actuated torque of an active stabilizer [5].

The actuated torque by the electric motor MSt is directly proportional to the counter roll torque MSt,A (see Eq. (6)). The counter roll torque is the value to be set in this used active roll control and thus is the output of the trained ANN. The conversion with the geometrical parameters and an additionally installed transmission to calculate the resulting forces at the wheel suspension is carried out afterwards.

3 Actor-Critic Reinforcement Learning The goal of machine learning is to generate numerical solutions for certain problems using empirical knowledge retrieved from data. The resulting algorithm can be represented by an artificial neural network. Reinforcement learning is one of three paradigms of machine learning besides supervised and unsupervised learning. Its vast majority of approaches can be classified with the actor-only or the critic-only methods [7]. Both the one and the other have advantages and disadvantages with respect to their working principles. Whereas the actor-critic combines those advantages and neglects the disadvantages by training an actor and a critic in two separate yet connected ANNs. The following explains the structure of the employed artificial neural network, the general usage of reinforcement learning and specifically the actor-critic method.

Application of Artificial Neural Networks for Active Roll Control

67

3.1 Artificial Neural Network The term of a neural network arises from the neurosciences and describes the composition of a variety of neurons that resemble a function in a nervous system. For engineering tasks this kind of construct is used for technical problems, where conventional solutions are way to complex or even fail. In this context the term of an artificial neural network is used in order to make clear it is a numerical illustration of a biological construct. In general, an ANN is composed of multiple neurons in a multidimensional configuration. Every single neuron consists of weights, a propagation function and an activation function to generate an output according to multiple inputs In . The weights wn are the parameters to be adjusted during the training process. These are multiplied with their associated inputs and afterwards combined using the propagation function. In this case the propagation function is the sum of the weighted inputs. The propagated value and the activation function y(x) then determine the output of the neuron O, leading to following expression:  n (7) wi · Ii O=y i

The topology of an ANN consists of at least two sequentially layers each with one or more parallel neurons. Those are the input and the output layer. For more complex problems hidden layers are inserted in between to allow high dimensional calculations by a complex matrix system. For each hidden layer and the output layer there also exists a bias B. It is a constant value and is treated as an input for the neurons of the respective layer. The bias is mainly used to include circumstances that are independent of the network inputs e.g. noise. An exemplary architecture of an ANN is shown in Fig. 5.

Fig. 5. Exemplary architecture of an artificial neural network [4].

In this work a fully connected feed-forward network is used, which means that every neuron is connected with each neuron of adjacent layers and there are no recurrent couplings. It is to be mentioned that the proposed actor-critic model requires two separate ANNs. In this case both received the same topology with two hidden layers. The first hidden

68

M. Bahr et al.

layer consists of 100 neurons and the second hidden layer of 20 neurons. The output layer consists of a single neuron. For the actor network this network output corresponds to the counter roll torque MSt,A and for the critic network the network output corresponds to the action-value Q explained in Sect. 3.3. The number of neurons of the input layer is determined by the quantity of dependencies of the output. For the active roll control, the roll angle, its deviations ϕ, ˙ ϕ¨ and the lateral acceleration ay are those dependencies as shown in Sect. 2.1. This leads to four input neurons for each ANN. The activation functions y(x) for each neuron of both networks is the rectified linear unit (ReLU). The only exception is the output neuron of the actor network, the activation of which is calculated by the hyperbolic tangent to enable positive and negative results for the actuated torque. Since the solution set of the hyperbolic tangent is limited to the range of −1 < tanh x < 1 the actor output is normalized and thus multiplied by the maximum counter roll torque MSt,A,max . 3.2 Reinforcement Learning Since the weights of the proposed ANN need to be adjusted to yield the desired outputs dependent on the inputs, a training algorithm is necessary. The scientific study of optimizing those algorithms is called machine learning, which can be subclassified into three categories consisting of supervised, unsupervised learning and reinforcement learning. The key aspect of reinforcement learning is to assign inputs of the artificial neural network to certain outputs, which accomplish a maximum reward [8]. For that the training algorithm and the ANN is interacting with the system environment to be controlled directly instead of learning based on stored data sets as it is the case with supervised and unsupervised learning. In terms of reinforcement learning the used ANN(s) combined with the training algorithm are called agent. The set of the inputs of the ANNs is called state S and the set of the outputs is called action A. The aim of reinforcement learning is therefore to let the agent find the best action it can take depending on the actual state. This is done without using any information about which is a suitable or correct action to take in which state or whether the last performed action was right or wrong. Instead the agent needs to explore and take several actions to find those, that lead to the highest reward R. This reward is specified by the reward function, which in this study is given by R = 1 − ϕ2.

(8)

The smaller the amount of the roll angle ϕ, the higher the reward R. That means that the agent ideally reduces the roll angle to ϕ = 0 and returns a reward of R = 1 regardless of the physical limits. The reward represents the feedback of the system environment. As the result the agent can justify the last executed action relatively to the amount of the reward. Since the chosen action of the agent needs to be performed and evaluated by the consequential state, respectively the roll angle, the state and reward are delayed. The resulting flow chart is shown in Fig. 6.

Application of Artificial Neural Networks for Active Roll Control

69

Fig. 6. Flow chart of reinforcement learning [8].

In every time step t the agent sends an action At depending on the current level of training to the system environment getting back the resulting state St+1 and reward Rt+1 . The agent’s policy π summarizes the assignment of the actions to the respective states: At = π (St )

(9)

The goal of reinforcement learning is to optimize this policy. This is achieved by two processes called exploitation and exploration which cannot be followed simultaneously. On the one hand, the agent needs to find the action that maximize the reward by exploration instead of taking the same action for specific states repeatedly dependent on the actual policy. In this study this is guaranteed by adding a random variation in form of a Gaussian distribution to the action chosen by the agent. Simultaneously, on the other hand, the agent needs to follow its actual policy exploiting its hitherto successful behavior and get closer to its goal. In order to make this possible, the standard deviation of the Gaussian distribution used for exploration is reduced exponentially over the time. The time delayed reward and the exploration are the two most important characteristics of reinforcement learning and distinguish it from other learning methods [8]. Reinforcement learning methods can deal with unknown, dynamic and interactive system environments. In contrast to supervised and unsupervised learning this method does not utilize existing data sets but rather in the process of training, produces data in the form of follow-up states induced by the choice of the previous action. The space of possible states is therefore only restricted by the system limits and all relevant states and actions involved by the training. 3.3 Actor-Critic The goal of the agent is to adjust its weights until it reaches the optimal policy π mentioned in Eq. (9). This is achieved by adjusting those with a weight specific gradient ∇J (w) and the learning rate α leading to wt+1 = wt + α · ∇J (w).

(10)

70

M. Bahr et al.

The learning rate α is a decisively parameter for the speed of the training and determines the step size with which the weights are adjusted. The agent in reinforcement learning can be an actor-only, a critic-only or a combined actor-critic model, which differ in their weight adjusting algorithms, respectively calculating the weights gradients. While the approximation of the action-value function is used for the critic, the actor uses a policy gradient approximation [9]. The approximation of the action-value Q is called Q-learning and is a variant of temporal difference learning. In this case, the critic evaluates the actual policy dependent on the states and the corresponding actions that were taken. It quantifies the expected return Gt , that can be achieved by taking the action At starting from the state St based on the current pursued policy π with Qπ,t (s, a) = Eπ [ Gt |St = s, At = a]. The return Gt matches the sum of cumulative rewards given by tb Gt = γRt · Rt t=0

(11)

(12)

and is affected by the discount factor γR . It quantifies the relative influence of successive rewards within a batch size of tb . This proceeding complies to a Markov Decision Process (MDP) and is called a discounted reward MDP for γR < 1 and an undiscounted reward MDP for γR = 1 [10]. For the weight adjustment the critic calculates the loss E, which is the mean squared error of the reward Rt and the difference of two successive action-values: E=

2 1 Rt + γR · Qπ,t+1 − Qπ,t 2

(13)

This loss E is used to calculate the weight gradient ∇J (w) via backpropagation by   ∂E ∂E ∂yj ∂xj ∇J wi,j = = · · ∂wi,j ∂yj ∂xi ∂wi,j

(14)

with xj being the summed up and weighted inputs and yj the output of the neuron j. This resulting gradient can be used to change the weight wi,j connecting the neuron i and neuron j. For the learning method of policy gradient approximation used by the actor network the policy π is parameterized with the weights w. Thus, a policy is trained, which assigns the action directly to the state by Eq. (9). Generally, the actor tries to maximize the return Gt by applying a gradient method with backpropagation. In the proposed actor-critic model the actor obtains the action-value Q from the critic and change its weights to get the maximum possible action-value. This leads to the weight gradient of   ∇w π (St , At , wt ) . (15) ∇J (wt ) = Eπ Qπ,t π (St , At , wt ) For a detailed explanation and derivation of the method, the literature of Sutton et al. is recommended [8]. The resulting partitioned agent with separate ANNs and learning methods for the actor and the critic is shown in Fig. 7.

Application of Artificial Neural Networks for Active Roll Control

71

Fig. 7. Flow chart of the used actor-critic model [10].

While the critic minimizes the loss E with a gradient descent method, the actor maximizes the action-value Q with a gradient ascent method. Since both are working with backpropagation, the gradients of the critic backpropagated to its inputs, which are the states S and the action A, are forwarded to the actor. This allows the actor to propagate the gradients regarding the action value from the output of the critic network back to every single weight of the actor network. By reason of using a gradient ascent for the actor, the learning rate αA gets a change of sign in comparison to the gradient descent of the critic: αA = −αC = −α

(16)

The learning rates could also be chosen with different absolute values, but it is highly recommended to equalize them to prevent an asynchronous training process. In this work the learning rates are adjusted individually by the optimization algorithm Adam anyway [11]. When the training is finished, the actor network forms the neural controller, while the critic network can be regarded as a learning aid during the training process.

4 Training Setup The training is performed simulative including three software packages. The necessary data used for the online training is generated by IPG CarMaker. It allows to simulate whole vehicle systems under realistic conditions. In its virtual environment the road, vehicle and several vehicle parts can be selected and customized, while the detailed numerical model is inaccessible for the user. So IPG CarMaker is a kind of black box. It is directly integrated into MATLAB/Simulink via a defined interface and supplies the dynamical driving and vehicle states used for the training. Via a TCP/IP interface, MATLAB/Simulink forwards the data to the ANN and its training algorithm. These are implemented by TensorFlow libraries inside a python script. Based on its training progress and the delivered driving conditions it calculates the counter roll torque MSt,A

72

M. Bahr et al.

for the active roll control. This in turn is then transmitted back to MATLAB/Simulink via TCP/IP. At this point, the given torque MSt is used for the steady state space model of the active stabilizer calculating the resulting forces acting on the vehicle. Afterwards, those forces are transmitted to IPG CarMaker, which affect the sequential driving conditions and close the simulation loop. The whole simulation is carried out with a simulation time step of one millisecond and is visualized by a flow chart in Fig. 8.

Fig. 8. Flow chart of the simulation environment [4, 12–14].

For training a road with even surfaces is used. The chosen vehicle is a Sport Utility Vehicle (SUV). Its relevant vehicle and component parameters regarding the roll movement used are summarized in Table 1. Table 1. Relevant vehicle parameter for the SUV. Symbol Meaning

Value

θA

Vehicle inertia

945.339 kgm2

mA

Vehicle mass

2072 kg

hS

Height of center of mass

0.682 m

hW

Height of roll center

0.326 m

sF,v

Lever-arm of front springs

0.784 m

cF,v

Stiffness of front springs

25,000 N/m

sF,h

Lever-arm of rear springs

0.769 m

cF,h

Stiffness of rear springs

25,000 N/m

sD,v

Lever-arm of front dampers 0.784 m

dv

Damping of front dampers

2500 Ns/m

sD,h

Lever-arm of rear dampers

0.769 m

dh

Damping of rear dampers

3000 Ns/m (continued)

Application of Artificial Neural Networks for Active Roll Control

73

Table 1. (continued) Symbol Meaning

Value

lSt,v

Length of front stabilizer

1.58 m

bSt,v

Width of front stabilizer

0.2 m

cSt,v

Stiffness of front stabilizer

8251.4 N/m

lSt,h

Length of rear stabilizer

1.58 m

bSt,h

Width of rear stabilizer

0.2 m

cSt,h

Stiffness of rear stabilizer

6833.5 N/m

The methodology of machine learning is to provide the ANN with enough training data to let it develop and represent an algorithm that ensures satisfactory results. However, instead of covering up the entire investigated state space during the training, it is intended to inter- and extrapolate the knowledge given by the training data over the entire or most of the state space. As mentioned in Sect. 2 the lateral acceleration is the main cause for the roll movement, so driving maneuvers are chosen for training that induce such lateral acceleration. These consist of stationary circular drives according to ISO 4138 in both directions with a curvature radius of rc = 100 m and a slalom around pylons with a constant distance of dP = 36 m.

Fig. 9. Lateral acceleration of the training maneuvers [4].

The training maneuvers are repeated and randomly arranged for the whole training duration. The resulting lateral accelerations for the non-controlled reference vehicle are shown representatively in Fig. 9. Since the lateral acceleration, the roll angle and its deviations have dynamic interactions, the lateral accelerations caused by the driving maneuvers change throughout the training progress and thus the state space is increased automatically. This is one advantage of using reinforcement learning and its online training capability. Furthermore, the state space is increased by adding random variance

74

M. Bahr et al.

to enable the exploration (see Sect. 3.2). Due to that and the randomly initialized weights, the success and progress of the training is highly dependent on randomness. The lateral acceleration ay , the roll angle ϕ, the roll velocity ϕ˙ and the roll acceleration ϕ¨ happen to be the state vector S and thus the input of the agent. Since the electromechanical actuator has a time delayed transfer behavior and thus acts as a firstorder delay element (PT1-element), its response time must also be learned from the agent. For that reason the state is supplemented by the actuated counter roll torque of the last time step MSt,A,t−1 leading to

(17) S T = ay ϕ ϕ˙ ϕ¨ MSt,A,t−1 . The output of the agent determines the counter roll torque to be set by the actuator MSt,A . To be able to adapt the resulting controller to different vehicles and components, the action A is normalized by the hyperbolic tangent (see Sect. 3.1). Consequently it is multiplied by the maximum counter roll torque MSt,A,max : MSt,A = A · MSt,A,max .

(18)

For the SUV the maximum counter roll torque is defined to be MSt,A,max = 1000 Nm. Since there is an active stabilizer model for both the front and the rear axle, the given torque is used for both actuators equally. The results presented in Sect. 5 required 173 training sessions. Each of them differs either from the used hyperparameters, activation functions, topologies of the ANNs or optimization algorithms. The training session leading to the presented controller took about 20 million iteration steps according to about 37 h of simulation time. The final used hyperparameters are summarized in Table 2. Table 2. Final hyperparameters of the agent. Symbol Meaning

Value

α

Learning rate

0.0001

nH

Number of hidden layers

2

nH ,1

Number of neurons in first hidden layer

100

nH ,2

Number of neurons in second hidden layer 20

σ

Variation standard deviation

2

γσ

Variation standard deviation decay

0.9999

γR

Reward decay

0.9

5 Validation The validation is executed simulative as well as the training inside the same framework and is split into two parts. The first part shows the variation and extrapolation for different

Application of Artificial Neural Networks for Active Roll Control

75

driving maneuvers. The second part deals with the adaption of the resulting controller to vehicles of different vehicle classes. For the validation the agent only consists of the actor and the critic is not used. Since the training is finished the actor becomes the neural controller. 5.1 Driving Maneuvers Four different driving maneuvers are used to validate the neural controller. These consist of the training maneuvers with varied radii, pylon distances and speeds and are supplemented by the double lane change (ISO 3888-1) and braking during stationary cornering. By adding further driving maneuvers and varying them, the interpolation and extrapolation of the developed controller can be evaluated. In each case it is compared with the passive stabilization. To assess the roll behavior, the roll angle is compared. For each maneuver one representative case is selected. The validated stationary circular drive (ISO 4138) is driven at the vehicle speed of v = 50 km/h and a curvature radius of rC = 40 m. The roll angle of the vehicle with passive stabilization and the neural controller are shown in Fig. 10.

Fig. 10. Stationary circular drive at v = 50 km/h and rC = 40 m [4].

Even though the roll angle caused by the neural controller is reduced in comparison to a passive stabilization, it shows different behavior for positive and negative lateral accelerations. This may be caused by the random distribution of the training maneuvers mentioned in Sect. 4. These training maneuvers were frequently duplicated to fit for the training duration and mixed randomly afterwards. This random arrangement combined with the decaying exploration could have possibly led to the observed difference. Applying the neural controller reduces the roll angle in the left-handed curve from ϕpassive = 4°

76

M. Bahr et al.

to ϕnc = 1.47°. This is equivalent to a 63.25% reduction. In the right-handed curve the roll angle is decreased to ϕnc = 0.28°, resulting in a 93% reduction. The validation of the slalom drive is shown exemplary at v = 50 km/h and a pylon distance of dP = 18 m in Fig. 11. The result of the slalom ride reflects the previous findings regarding the discriminative controlling for negative and positive lateral accelerations. Nevertheless, it can be stated that the neuronal controller reduces the roll angle in comparison to a passive stabilization at any time and thus is inter- and extrapolatable to different vehicle speeds, curvature radii and pylon distances.

Fig. 11. Slalom drive at v = 50 km/h and dP = 18 m [4].

To extend the validation with driving maneuvers that are unseen by the neural controller during the training, the double lane change (ISO 3888-1) is driven at v = 60 km/h. The resulting roll angle can be seen in Fig. 12. The driving dynamic of a double lane change is comparable to that of a slalom drive. Therefore, the results are similar. Due to the fact, that the roll angle of the neural controller is reduced at any time, it is proven that it can also be extrapolated to other maneuvers and driving situations. Braking during stationary cornering is a comparatively critical driving maneuver. To test the neural controller under such conditions, the circular drive at v = 70 km/h and rC = 100 m with a braking process 10 s after the curve entrance for both directions are driven. Those results are shown in Fig. 13. Even in critical driving situations the neural controller is able to reduce the roll angle compared to the passive stabilizer. Although the small fluctuations at t ≈ 90 s and t ≈ 110 s must be noted. The oscillating roll angle can be perceived as uncomfortable by the driver and thus the neural controller is not fulfilling the function of an active roll control reasonably. Despite this, the neural controller works in proper way.

Application of Artificial Neural Networks for Active Roll Control

77

Fig. 12. Double lane change at v = 60 km/h [4].

Fig. 13. Braking during stationary cornering at v = 70 km/h and rC = 100 m.

5.2 Vehicle Classes For the validation of the neural controller on different vehicle classes, three additional passenger cars are simulated, each representing a specific vehicle class. The classification for vehicle classes used in this work are defined by the European Commission [15].

78

M. Bahr et al.

The SUV validated in Sect. 5.1 represents the class J. Further the multi purpose car representing the class M, the small car representing the class B and sport coupé representing the class S are validated with the developed active roll control. For simplification of the comparison all vehicles are performed with an active stabilizer for both axles. Since the various used passenger cars are different constructions with diverse dynamic behaviors, the maximum counter roll torque MSt,A,max must be adjusted to their specific properties regarding the roll movement. Therefore, the normalization of the controller output is beneficial (see Sect. 3.1). The vehicle-specific inertia θA is a significant parameter for the adaption of the developed controller to other passenger cars. So, the maximum counter roll torque MSt,A,max was adjusted proportional to it. The torque and inertia combinations for the addressed vehicles are summarized in Table 3. Table 3. Proportional adjustment of the counter roll torque over the vehicle inertia. Vehicle

Vehicle class [15] Inertia θA [kg m2 ] Max. counter roll torque MSt,A,max [Nm]

SUV

J

945.339

1000

Multi purpose car M

654.502

692.346

Small car

B

433.106

458.149

Sport coupé

S

568.612

601.490

This approach is a simplification for adapting the neural controller to several vehicle constructions and dynamics. There are a lot of further parameters affecting the roll movement and thus the active roll control e.g. the vehicle mass. For more accurate investigations those parameters can be involved for the adaption. By reason of conciseness the other differing parameters are not listed, as these are not considered. Each of the three additional vehicles are validated by the stationary circular drive (ISO 4138) at the vehicle speed of v = 70 km/h and a curvature radius of rC = 100 m and compared with their passive stabilization. The multi purpose car is a compact van and is validated with a maximum counter roll torque of MSt,A,max = 692.346 Nm. The resulting roll angle is visualized in Fig. 14 The neural controller has succeeded in reducing the roll angle at any time. The roll angle in the left-handed curve is reduced from ϕpassive = 2.14° to ϕnc = 1.08, resulting in a 49.53% reduction. In the right-handed curve a roll angle of ϕnc = 0.55° and thus a 74.30% reduction is achieved. The percentage reduction is smaller than for the neural controller of the SUV. An interesting result is the proportion of the asymmetrical control behavior. The ratio of the reduction in the right-handed curve to the reduction in the left-handed curve is 1.5. This ratio for the SUV driving a stationary circular drive at the vehicle speed of v = 70 km/h and a curvature radius of rC = 100 m is 1.47.

Application of Artificial Neural Networks for Active Roll Control

79

Fig. 14. Multi purpose car stationary circular drive at v = 70 km/h and rC = 100 m.

The small car represents the class B and is validated with a maximum counter roll torque of MSt,A,max = 458.149 Nm. Figure 15 shows the resulting roll angle. The roll angle in the left-handed curve is reduced from ϕpassive = 1.63° to ϕnc = 0.91 with a 44.17% reduction. In the right-handed curve a roll angle of ϕnc = 0.55° and thus a 66.26% reduction is achieved. The reduction ratio is 1.5.

Fig. 15. Small car stationary circular drive at v = 70 km/h and rC = 100 m.

80

M. Bahr et al.

The class S is represented by the sport coupé with a maximum counter roll torque of MSt,A,max = 601.49 Nm. In Fig. 16 the resulting roll angle of its active roll control is shown. The roll angle in the left-handed curve is reduced from ϕpassive = 1.65° to ϕnc = 0.71 with a 56.97% reduction. In the right-handed curve a roll angle of ϕnc = 0.23° and thus an 86.06% reduction is achieved. The resulting reduction ratio is 1.51.

Fig. 16. Sport coupé stationary circular drive at v = 70 km/h and rC = 100 m.

Looking at the ratio of the reduction in the right-handed curve to the reduction in the left-handed curve, there is a reproducibility of the neural controller. For all four considered vehicles this ratio is nearly 1.5. It seems this ratio caused by the neural controller is nearly independent of the used vehicle properties except for the inertia θA . It can be interpreted, that if the proportion of the inertia to the maximum counter roll torque MSt,A,max remains constant, the ratio also remains nearly the same. This confirms the assumption that the inertia θA is the main cause of the roll movement, even if the absolute and relative reduction differs. Since the ratio for the SUV is always around 1.47 for every tested stationary circular drive with varied vehicle speed and curvature radii, it also seems to be independent of the vehicle speed and curvature radius.

6 Conclusion In this work, an active roll control with an artificial neural network based on an actor critic reinforcement learning method has been successfully developed. The neural controller was realized with the TensorFlow open-source software library and combined with the simulation of an entire vehicle in IPG CarMaker and the active roll stabilization in MATLAB/Simulink communicating via a TCP/IP interface. A real-time control

Application of Artificial Neural Networks for Active Roll Control

81

is created by a guaranteed calculation of the counter roll torque in a fixed time interval. The developed neural controller is able to reduce the roll angle in comparison to a passive stabilization at any time. Thus, the functionality of the controller is given. The results show that the neural controller behaves asymmetric for positive and negative lateral accelerations. However, it has been proven that artificial neural network trained by a reinforcement actor-critic algorithm is able to form an active roll control and can potentially replace conventional control strategies. Moreover, an analytical approach for an adaption of the developed neural controller to several different vehicle classes has been made and successfully implemented. According to the presented results the conclusion can be drawn that there is a correlation between ratio of the maximum counter roll torque to the vehicle inertia and the ratio reduction of negative roll angles and lateral accelerations to the reduction of positive ones. The neural controller is applicable for several different driving conditions and constructions of passenger cars. Nevertheless, the training algorithm can be enhanced and improved. For example, a regularization on the weight adjustment to ensure the minimal optimal weights and symmetric behavior for positive and negative lateral accelerations can be applied. Besides that, the training could be executed with reward functions involving the roll velocity and the roll acceleration to possibly achieve even better results regarding emerging fluctuations at braking in a stationary circular drive that are uncomfortable for the driver. Normalizing the state vector elements could also lead to an improvement and speed up the training duration. After optimizing the training algorithm and the used artificial neural network structure, it is suggested to implement a roll angle function into the reward function that calculates a desired roll angle dependent on the actual driving state instead of reducing it as much as possible. The resulting roll movement would be perceived less rigid and thus more comfortable.

References 1. Sieberg, P., Schmid, M., Reicherts, S., Schramm, D.: Computation time optimization of a model-based predictive roll stabilization by neuro-fuzzy systems. In: Bargende, M., Reuss, H.C., Wagner, A., Wiedemann, J. (eds.) 19. Internationales Stuttgarter Symposium, pp. 499–512. Springer, Wiesbaden (2019) 2. Boada, M., Boada, B., Gauchia Babe, A., Calvo, J., Diaz, V.: Active roll control using reinforcement learning for a single unit heavy vehicle. Int. J. Heavy Veh. Syst. 16(4), 412–430 (2009) 3. Fu, Z.-J., Li, B., Ning, X.-B., Xie, W.-D.: Online Adaptive optimal control of vehicle active suspension systems using single-network approximate dynamic programming. Math. Probl. Eng. 2017 (2017). 9 p. 4. Bahr, M., Reicherts, S., Sieberg, P., Morss, L., Schramm, D.: Development and validation of active roll control based on actor-critic neural network reinforcement learning. In: Proceedings of the 9th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, pp. 36–46. SCITEPRESS, Prague (2019) 5. Schramm, D., Hiller, M., Bardini, R.: Vehicle Dynamics – Modeling and Simulation, 2nd edn. Springer, Heidelberg (2018) 6. Ammon, D.: Modellbildung und Systementwicklung in der Fahrzeugdynamik. Vieweg+Teubner, Stuttgart (2013)

82

M. Bahr et al.

7. Konda, V., Tsitsiklis, J.: Actor-critic algorithms. SIAM J. Control Optim. 42(4), 1143–1166 (2003) 8. Sutton, R., Barto, A.: Reinforcement Learning: An Introduction, 2nd edn. The MIT Press, London (2018) 9. Sutton, R., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. In: NIPS 1999 Proceedings of the 12th International Conference on Neural Information Processing Systems, pp. 1057–1063. The MIT Press, Cambridge (1999) 10. Szepesvári, C.: Algorithms for Reinforcement Learning, 1st edn. Morgan & Claypool Publishers, New York (2010) 11. Kingma, D.P., Ba, J.: Adam: a method for stochastic optimization. In: Proceedings of the 3rd International Conference on Learning Representations, p. 13. arXiv.org, Ithaca New York (2015) 12. IPG Automotive’s Official Website. https://ipg-automotive.com/. Accessed 09 Feb 2019 13. MathWorks’ Official Website. https://de.mathworks.com/. Accessed 09 Feb 2019 14. TensorFlow’s Official Website. https://www.tensorflow.org/. Accessed 09 Feb 2019 15. REGULATION (EC) No 139/2004. https://ec.europa.eu/competition/mergers/cases/decisi ons/m5518_20090724_20310_en.pdf. Accessed 08 Dec 2019

Optimal Trigger Sequence for Non-iterative Co-simulation with Different Coupling Step Sizes Franz Rudolf Holzinger1(B) , Martin Benedikt1 , and Daniel Watzenig1,2 1

2

Virtual Vehicle Research GmbH, Inffeldgasse 21/A/II, Graz, Austria {franz.holzinger,martin.benedikt,daniel.watzenig}@v2c2.at Institute of Automation and Control, Graz University of Technology, Graz, Austria [email protected]

Abstract. The definition of a suitable trigger sequence is challenging during the configuration of non-iterative co-simulation. Therefore, a trigger sequence approach is presented for interacting subsystem in a sequential co-simulation framework. For this purpose, the dependencies between the subsystems are used to describe a co-simulation graph. According to the co-simulation graph an optimization approach for the optimal trigger sequence is derived. Furthermore, the subsystem execution behaviour is discussed with respect to different coupling step sizes. Therefore, the impact of the underlying scheduling algorithm is analysed. A transformation of the co-simulation graph is introduced in order to consider the scheduling behaviour. This enables the usage of solving algorithms designed for equal coupling time steps. In addition to that, an extension of the co-simulation graph is done by weighting of the coupling signals. The weighting of coupling signals allows the prioritization of the subsystems. This affects the trigger sequence and consequently the simulation results. In this context, different weighting approaches are discussed and compared by an example. Keywords: Trigger sequence optimization Different coupling time steps

1

· Sequential co-simulation ·

Introduction

In the last decades, the demands on simulation especially in the automotive industry have increased dramatically, with multi-disciplinary task on the one hand and domain-specific simulation environments on the other hand. As a consequent, techniques such as co-simulation have developed to handle this challenge. Co-simulation as an integration platform allows the combination of different subsystems and ensures the correct data exchange between the subsystems. From the technical point of view in addition to data exchange, the integration of subsystems is a challenge. The several subsystems are often modelled in specific simulation environments from different vendors in their separate modelling c Springer Nature Switzerland AG 2021  M. S. Obaidat et al. (Eds.): SIMULTECH 2019, AISC 1260, pp. 83–103, 2021. https://doi.org/10.1007/978-3-030-55867-3_5

84

F. R. Holzinger et al.

languages. In addition to that, the subsystems are solved with special solving algorithms and different step sizes, which may be variable. In this context FMI has established as a standard for the interface to ease the integration challenge, for details see [1]. Besides the problem of integrating the subsystems into the co-simulation framework, the challenge is to parametrize or configure the co-simulation. There are three major types of co-simulation configuration parameters: 1. coupling step size of the subsystems; 2. extrapolation filter of the coupling signals; 3. coupling mechanism and trigger sequence. These three parameter types have a significant impact to the quality of the co-simulation results. The challenge to configure the co-simulation and an automated configuration approach have been discussed by the authors in the past, for more detail see [4]. An additional approach for the configuration of co-simulation based on user-hints is described in [13]. The coupling step size defines the time steps of the data exchange between the subsystems. It seems apparent that small step sizes and thus high data exchange frequency lead to better simulation results. However, a small coupling step size expenses the performance and increases the simulation duration. Therefore, it is essentially to define appropriate coupling step size, with a threshold between simulation accuracy and simulation performance. There are several approaches to get proper coupling step sizes. The definition of coupling time steps, depending on the instantaneous frequency of the coupling signals is discussed in [4,7]. Adaptive time steps based on a measure of the coupling error is shown in [8,16,23,25] and an approach for coupling time steps regarding the coupling signal derivatives is described in [14]. The extrapolation of the coupling signals is necessary to solve the causality problem to due the coupling between interconnected subsystems. The most common extrapolation technique is zero-order-hold (ZOH), where the last known value is used to calculate the next coupling step. There are further coupling signal based extrapolation approaches, which compensate for instance the extrapolation error in the sense of energy preserving, see [3]. An extended approach for the coupling of stiff subsystems is described in [6]. A model-based coupling predictor approach is shown in [15] and [9] introduces a model-based pre-step stabilization method. Beside to the coupling step size and the extrapolation, the coupling mechanism and the trigger sequence have an impact to the result of the co-simulation. The coupling mechanism can be classified into two major categories: parallel and sequential. In the parallel case, all subsystems are calculated at the same time. This allows a high simulation performance due to the parallel execution, but leads to the fact, that all coupling signals of the subsystems must be extrapolated in each time step. Sequential execution of the subsystems increases the simulation duration, because the subsystems have to wait on the results of the other subsystems, but

Optimal Trigger Sequence for Non-iterative Co-simulation

85

due to the sequential execution not every coupling signal has to be extrapolated and this decreases the overall coupling error. Which signals are extrapolated depends mainly on the defined trigger sequence. The trigger sequence describes the execution or calculation order of the subsystems. A trigger sequence approach based on the minimization of the communication delay between the subsystems is described in [12]. In [18] the authors discuss a graph representation of the co-simulation with an appropriate optimization formalism. Several heuristic approaches to solve this optimization problem are analysed in [24]. A hierarchical approach, which allows a trade-off between the simulation duration and simulation accuracy is discussed in [21]. These trigger sequence approaches assume that all subsystems Si have the same coupling step size ΔTi . The assumption of equal coupling step sizes allows to negligence the underlying scheduling algorithm. With the usage of different coupling time steps this assumption is no longer permissible. This work discusses the problem of the trigger sequence with different coupling step sizes. Furthermore, a method is shown, which allows to use the developed solving approach for equal step sizes even for different coupling time steps. The work is outlined as follows. Section 2 discusses the topic trigger sequence for non-iterative co-simulation. Therefore, the co-simulation framework is interpreted as an optimization problem in the form of a co-simulation graph to get a suitable trigger sequence. The impact of different coupling time steps of the subsystems on the trigger sequence is discussed in Sect. 3 and it is shown, that the graph representation of the co-simulation is still possible. Section 4 discusses different approaches to weight the edges of the co-simulation graph to affects the results of the trigger sequence based on subsystem and coupling signal properties. All methods and approaches are illustrated and discussed on an example in Sect. 5. The co-simulation example is shown in Fig. 1 and consists of four subsystems SA , SB , SC and SD .

2

Co-simulation Trigger Sequence

Sequential co-simulation executes one subsystem after the other, i.e., in each time step only one system is calculating. Apart from an increased simulation duration compared to a parallel execution scheme, the results of the already calculated subsystems can used directly. This reduces the number of extrapolations in total, which typically reduces the overall coupling error and increases the simulation accuracy. Nevertheless, the definition of a proper trigger sequence is challenging, due to factorial relation between the number of included subsystems and the possible combinations of the trigger sequence. The execution behaviour of four subsystems SA , SB , SC and SD is shown in Fig. 2. In a first step it is assumed, that all subsystems depends on each other, i.e., all subsystems are connected. In addition to that, all subsystems have the same coupling step size ΔTA = ΔTB = ΔTC = ΔTD = 2 s. The execution of the subsystems is depicted over the simulation time ts and the wall-clock time tw . For reasons of illustration, it is assumed that all subsystems calculate in

86

F. R. Holzinger et al. uba uad

ybc ucb

SB

ucd

yab

SA

SC

ycd

yad ydc yda

SD

uda udc

Fig. 1. Co-simulation topology [18].

real-time, i.e., one second in simulation ts is one second in wall-clock time tw . This assumption is for illustrative purposes only and has no effect to the execution and simulation behaviour of the subsystems. Each block represents the execution (i.e., simulation) of one coupling step of a subsystem. The letters inside the blocks show the required extrapolation of the depending subsystems. For an exemplary defined trigger sequence S := {ABCD}, where subsystem SA is calculated first, that means, the subsystem SA has to extrapolate all inputs of the connected subsystems, illustrated as b, c, d. The following calculating subsystem SB can now use the results of SA , i.e., only the coupling signals of SC and SD (c, d) have to be extrapolated. For the last executed subsystem SD all required incoming coupling signals are available, that means, no signal has to be extrapolated. After each subsystem has done a calculation step, subsystem SA starts to calculate the next step. The execution of the subsystem follows the defined trigger sequence S. SA :

b,c,d

SB :

b,c,d c,d

b,c,d c,d

SC :

c,d

d

d

d

SD : 0 0

2 4

8

4 12

16

20

6 ts 24 tw

Fig. 2. Sequential coupling mechanism with equal coupling step sizes.

For the supposedly simple example with m = 4, subsystems there exists m! = 24 possible variations of the trigger sequences. Depending on the defined trigger sequence, the induced coupling error of the entire co-simulation may differ, which leads to different simulation results. It is hardly possible for co-simulation users and application engineers to define appropriate trigger sequences to reduce the coupling effects, especially for an increasing number of subsystems. The following section describes an approach to get an optimized trigger sequence based on a graph representation for equal coupling time steps. This approach was already introduced in [18].

Optimal Trigger Sequence for Non-iterative Co-simulation

2.1

87

Co-simulation Graph

To define an appropriate trigger sequence for co-simulation an optimization problem is derived. Therefore, the co-simulation structure is interpreted as an asymmetric, directed graph, the so called co-simulation graph. In this context, each node represents a subsystem and the edges show the dependencies from one subsystem to the other. The weight of the edges is a measure for the strength of the dependency. The co-simulation graph from the example in Fig. 1 is shown in Fig. 3. cbc

SB

cab

cda

SA

SC ccd

cdc

cad

SD Fig. 3. Co-simulation Graph [18].

The connections between the subsystems of the co-simulation, i.e., linkage of an output yi to an input uj , can be described with the linking matrix: u = L · y,

(1)

whereby, y = [y1 , y2 , . . . , yn ]T is a vector of all outputs yi and the vector u = [u1 , u2 , . . . , un ] includes all inputs ui . Both vectors have a length of n, where n is the number of connections within the co-simulation network. The linking matrix L with the size n × n is an orthogonal matrix, this assumes, that each output yi is linked to one single input ui . In the case of a multiply connected outputs, the outputs can be duplicated. The linking matrix L of the co-simulation example is as follows: ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ yab 0 0 0 0 0 1 uad ⎢ uba ⎥ ⎢ 1 0 0 0 0 0 ⎥ ⎢yad ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ucb ⎥ ⎢ 0 0 1 0 0 0 ⎥ ⎢ ybc ⎥ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ (2) ⎢ ucd ⎥ ⎢ 0 0 0 0 1 0 ⎥ · ⎢ ycd ⎥ , ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎣ udc ⎦ ⎣ 0 1 0 0 0 0 ⎦ ⎣yda ⎦ 0 0 0 1 0 0 uda ydc The linking matrix L describes the connections between inputs ui and outputs yi . Nevertheless, there is no allocation to the subsystems Sj . Therefore, the linking matrix L is extended to describe the dependency matrix A as follows: T  (3) A = TT · L · S .

88

F. R. Holzinger et al.

The dependency matrix A has the dimension m×m and describes the dependencies between the subsystems Sj based on the linking matrix L. The two matrices S and T have the dimension n×m and transform the connection dependencies L into the subsystem dependencies A. In relation to the example the two matrices are as follows: ⎤ ⎡ 1 0 0 0 ⎡ ⎤ ⎢1 0 0 0⎥ 1 0 0 0 0 0 ⎥ ⎢ ⎢0 1 0 0⎥ ⎢0 1 0 0 0 0⎥ ⎥ ⎥ and TT = ⎢ S=⎢ (4) ⎢0 0 1 1⎥ ⎣0 0 1 1 0 0⎦ . ⎥ ⎢ ⎣0 0 0 1⎦ 0 0 0 0 1 1 0 0 0 1 The transformation matrix S describes the relation between the outputs yi to the subsystems Sj and the matrix T describes the relation between the inputs ui and the subsystems Sj . Both matrices, such like the linking matrix L depend on the co-simulation structure only. According to the example from Fig. 1 the dependency matrix A is given by (4) as follows: ⎡ ⎤ 0 1 0 1 ⎢0 0 1 0⎥ ⎥ A=⎢ (5) ⎣0 0 0 1⎦ . 1 0 1 0 The resulting dependency matrix or adjacency matrix A in (5) is a special representation of the co-simulation graph from Fig. 3, where each edge has a weight of cij = 1. The representation of the co-simulation as graph allows the usage of solving strategies for graphs to derive the trigger sequence. 2.2

Minimum Sum of Edges

The weights of the edges of co-simulation graph are a measure for the sensitivity of the extrapolated inputs to the subsystems results. The higher the weight of an edge, the higher is the impact to the coupling error. The trigger sequence defines the execution order of the subsystems and consequently the extrapolation of the coupling signals. A suitable trigger sequence prevents the extrapolation of critical and sensitive coupling signals. According to the co-simulation graph, that means to avoid high weighted edges. An optimization problem can be derived by minimizing the sum of the edges based on the trigger sequence, as follows:

ek , (6) min S

k∈S

where S represents the subsystem indexes of trigger sequence, e.g. S = [3, 1, 2, 4] for a trigger sequence {CABD}. According to the co-simulation graph the summand ek for the considered node k is determined as the sum of the incoming

Optimal Trigger Sequence for Non-iterative Co-simulation

edges for all n subsystems: ek =

n

Ai,k .

89

(7)

i=1

This step corresponds to the calculation of the subsystem Sk and therefore all incoming coupling signals has to be extrapolated. After the simulation the results of the subsystem Sk are available for all other subsystems, i.e. there is no extrapolation needed for these coupling signals. Regarding the co-simulation graph that means, these edges can be neglected and so in a second step the outgoing edges of the node k are erased: Ak,i = 0

∀i = 1..n

(8)

The determination of the trigger sequence can be interpreted as an extended travelling-sales-man problem (TSP), where the shortest distance between the nodes is requested. In contrast to the classical TSP problem, the edges of already visited nodes are removed, see (8). In this work a brute-force algorithm, i.e., evaluation of all possible combinations, is used to solve the optimization problem. The number of possible combinations of the trigger sequence increases factorially. The identification of the optimal trigger sequence by the brute-force algorithm can lead to a high calculation effort. Nevertheless, for a small number of subsystems the calculation effort of the brute-force algorithm can be neglected. However, in order to reduce the calculation effort alternative approach should be used. Therefore, different heuristic optimization approaches are discussed in [24].

3

Different Coupling Step Sizes

In co-simulation, it is not mandatory for subsystems to use the same step size. Conversely, if the dynamics of the subsystems have huge differences, it does not make sense to use the same coupling step sizes. Nevertheless, if different coupling step sizes are used, the execution order of the subsystems depends not only on the defined trigger sequence but also on the underlying scheduling, e.g. latestfirst algorithm. The latest-first scheduling algorithm defines, which subsystem calculates next with respect to the simulation progress of the several subsystems. The subsystem with the lowest simulation progress, i.e., smallest simulation time, calculates next. The defined trigger sequence only matters, if the subsystems are at the same coupling time step. For instance, Fig. 4 shows the behaviour of the execution order of four subsystems SA , SB , SC and SD with different coupling step sizes ΔTA = 2 s, ΔTB = 3 s, ΔTC = 1 s and ΔTD = 2 s based on the latest-first scheduling algorithm. The trigger sequence for the considered example is defined as follows: S := {ABCD}. This sequence can be seen in the first execution of each subsystem in Fig. 4. The subsystem SA calculates first with ΔTA = 2 s, after that subsystem SB starts to simulate its step ΔTB . In the first two seconds SB uses the already calculated results from SA , in the third second of the coupling step, the inputs

90

F. R. Holzinger et al.

SA :

b,c,d

SB :

c,d a,c d

c,d

SC :

c,d c

d

SD :

c d,b a,c,d

d

d

c

c

b,c

0 0

4

8

12

16

20

6 ts 24 tw

Fig. 4. Sequential coupling mechanism with different coupling step sizes.

from subsystem SA have to be extrapolated too. Subsystem SC does one step and has to extrapolate the subsystem SD and the subsystem SD has the first second of the coupling step all required inputs available, for the second half of the step it has to extrapolate the inputs coming from SC . Contrary to expectations, the calculation does not start again with subsystem SA but with subsystem SC , due to the fact, that the simulation progress of subsystem SC is the latest. After this simulation step, the three subsystems SA , SC and SD are at the same coupling time ts = 2 s and so the trigger sequence defines the next calculating subsystem SA . This is continued accordingly. Finally, referring to this example and the chosen coupling step sizes the execution sequence repeats every ts = 6 s, i.e., the least-common-multiple step size ΔTlcm . The co-simulation graph does not describe such a behaviour, i.e., the trigger sequence with respect to the minimum sum of the edges described in Sect. 2.2 of the co-simulation graph is no longer valid, due to the underlying scheduling mechanism. Considering the interaction between two subsystems Si and Sj , the extrapolated fractions related to the greatest-common-divisor step size ΔTgcd and the least-common-multiple step size ΔTlcm remain constant ΔTlcm /ΔTgcd , independent which subsystem calculates first. For instance, Fig. 5 a shows the execution of two subsystems Si and Sj , where ΔTi = 3 s and ΔTj = 1 s, with ΔTgcd = 1 s and ΔTgcd = 3 s. The subsystem Si calculates first and has to extrapolate the inputs of subsystem Sj along the coupling time step ΔTi = 3 · ΔTgcd . The subsystem Sj calculates afterwards three times in a row, without extrapolation. Regarding the ratio of the coupling step sizes, the inputs of subsystem Si has to be extrapolated all the time. In contrary, if the step size ΔTi = 1 s and subsystem Sj has the step size ΔTj = 3 s, subsystem Si extrapolates one step ΔTi = ΔTgcd , see Fig. 5 c. Due to the latest first algorithm Sj calculates next with ΔTj = 3 s = 3 · ΔTgcd , whereby the first ΔTgcd the subsystem calculates with the data from Si but the rest has to be extrapolated. This means, for a step size ratio ΔTi /ΔTj = 1/3 the subsystem Si extrapolates 33.3% of all extrapolations and Sj extrapolates 66.6%. This consideration is also given if the step sizes are not multiples of each other. Figure 5 b shows two subsystems Si and Sj with a step size ΔTi = 3 s and ΔTj = 2 s, i.e., ΔTgcd = 1 s and ΔTlcm = 6 s. Consequently, for the ratio ΔTi /ΔTj = 3/2 the subsystem Si extrapolates 5 · ΔTgcd of ΔTlcm , i.e. 83.3%.

Optimal Trigger Sequence for Non-iterative Co-simulation Si :

Si :

j

j

j

Sj :

Sj : 0 0

3

i 0

ts 6 tw 3

0

4

(a) ΔTi /ΔTj = 3/1 Si :

Si :

0

j

j

Sj :

i 0

i

i

0

ts 6 tw 3

3

6 ts 12 tw

8

(b) ΔTi /ΔTj = 3/2

j

Sj :

91

0

4

6 ts 12 tw

8

(d) ΔTi /ΔTj = 2/3

(c) ΔTi /ΔTj = 1/3

Fig. 5. Extrapolation behaviour regarding ratio of the coupling step sizes: (a) ΔTi /ΔTj = 3/1; (b) ΔTi /ΔTj = 3/2 ; (c) ΔTi /ΔTj = 1/3; (d) ΔTi /ΔTj = 2/3.

A step size ratio ΔTi /ΔTj = 2/3 is illustrated in Fig. 5 d and leads to an extrapolation ratio of 50% for Si . The ratios can be determined for any coupling step size pairs. Table 1 shows the extrapolation ratios Qij with respect to the coupling step sizes ΔT˜i and ΔT˜j . For instance, the ratio of the coupling step sizes of the subsystems Si and Sj is 5/3 (represented by Q(5, 3)), i.e., the coupling signals from subsystem Sj to the subsystem Si are extrapolated 80% of the simulation time and the coupling signals from subsystem Si to Sj are extrapolated the rest of the time, i.e., 20%. The absolute value of the coupling step size ΔTj is not relevant coefficients Qij . Therefore, the step sizes are scaled by the great-common-divisor ΔT˜i = ΔTi /ΔTgcd . Table 1. Weights for latest first scheduling algorithm regarding the coupling step size ratio of the subsystems.   Q ΔT˜i , ΔT˜j 1

2

3

4

5

6

7

8

9

10

1 2 3 4 5 6 7 8 9 10

0.5000 1.0000 0.8333 1.0000 0.9000 1.0000 0.9286 1.0000 0.9444 1.0000

0.3333 0.5000 1.0000 0.7500 0.8000 1.0000 0.8571 0.8750 1.0000 0.9000

0.2500 0.5000 0.5000 1.0000 0.7000 0.8333 0.7857 1.0000 0.8333 0.9000

0.2000 0.3000 0.4000 0.5000 1.0000 0.6667 0.7143 0.7500 0.7778 1.0000

0.1667 0.3333 0.5000 0.5000 0.5000 1.0000 0.6429 0.7500 0.8333 0.8000

0.1429 0.2143 0.2857 0.3571 0.4286 0.5000 1.0000 0.6250 0.6667 0.7000

0.1250 0.2500 0.2500 0.5000 0.3750 0.5000 0.5000 1.0000 0.6111 0.7000

0.1111 0.1667 0.3333 0.2778 0.3333 0.5000 0.4444 0.5000 1.0000 0.6000

0.1000 0.2000 0.2000 0.3000 0.5000 0.4000 0.4000 0.5000 0.5000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

Based on the extrapolation behaviour of subsystem Si in Fig. 5 and the step size ratio ΔTi /ΔTj , the dependency matrix can be scaled as follows: (1)

Aij = Wij · Aij ,

(9)

92

F. R. Holzinger et al.

where, the weights Wij = Q(ΔT˜i , ΔT˜j ) result from the ratio of the step sizes and can be taken directly from Table 1. The scale of the dependency matrix in (9) considers the reduced extrapolation of subsystem Si . The additional extrapolation effort of Sj caused by the different coupling step sizes can be determined as follows: (2) (10) Aij = (1 − Wij ) · Aji , whereby the weighting of Aji , i.e., the edge from subsystem Sj to Si , is mapped to the weight Aij scaled by the 1 − Wij . Finally, the transformed dependency ˜ is given as sum of (9) and (10), as follows: matrix A ˜ ij = Wij · Aij + (1 − Wij ) · Aji . A

(11)

˜ Applied to the dependency matrix in (5) the transformed dependency matrix A with exemplary step sizes ΔTA = 2 s, ΔTB = 3 s, ΔTC = 1 s and ΔTD = 2 s is given as: ⎡ ⎤ 0 0.5000 0 1.0000 ⎢ ⎥ ˜ = ⎢ 0.1667 0 1.0000 0 ⎥ A (12) ⎣ 0 0.6667 0 1.0000 ⎦ 1.0000 0 1.0000 0 ˜ has In contrary to the dependency matrix in (5) the transformed matrix A 8 entries. The additional entries, i.e., extra connections and edges in the cosimulation graph, are based on the bi-directional extrapolation behaviour with different coupling step sizes. For instance, if there is no dependency from subsystem SB to SA and the subsystem SA calculates first, parts of SB has also to be extrapolate, due to the different step sizes. The dependency transformation in (11) changes not only the weighting of the edges but also the connections of the graph. Subsystems that are not directly dependent in any direction have no dependencies after the transformation. The transformed dependency matrix in (11) has the same properties regarding the edge behaviour than a substitution graph with equal step sizes. Due to this transformation, the underlying scheduling algorithm need not to be considered explicitly. Thus, the complexity to find a trigger sequence for different coupling step sizes decreases. In addition to that, the graph transformation allows to use heuristic solving algorithms to identify suitable trigger sequences, which are designed for subsystems with constant step sizes. Various representatives are described and compared in [24].

4

Weighting of Connections

The previous analysis of the trigger sequence approach assume that every connection and coupling signal have the same impact to the co-simulation results and to the entire coupling error. Generally, this is not the case. The coupling signals affect the co-simulation differently, i.e., some coupling signals have a bigger impact to the simulation results and the impact of other coupling signals

Optimal Trigger Sequence for Non-iterative Co-simulation

93

is negligible. Therefore, the calculation of the dependency matrix A from (3) is extended as follows: T  (13) A = TT · C · L · S , where C is a diagonal matrix with the weights ci of the coupling signals. Regarding to the co-simulation example in Fig. 1 the extended dependency matrix as well as the transformed matrix according to (11) are given as follows: ⎡ ⎡ ⎤ ⎤ 0 c˜ab 0 c˜ad 0 cab 0 cad ⎢ ⎢ 0 0 cbc 0 ⎥ ⎥ ⎥ ˜ = ⎢ c˜ba 0 c˜bc 0 ⎥ and A (14) A=⎢ ⎣ 0 c˜cb 0 c˜cd ⎦ ⎣ 0 0 0 ccd ⎦ cda 0 cdc 0 c˜da 0 c˜dc 0 The weights ci allow to prioritize individual coupling signals. This has an effect on the dependency matrix A and consequently on the resulting trigger sequence. In the following different approaches to the choice of the weighting coefficient ci are discussed. 4.1

Number of Extrapolations

An obvious approach for an appropriate trigger sequence is to minimize the number of extrapolated inputs. It is assumed, that all coupling signals have the same significance to the co-simulation results. In the case of different coupling time steps ΔTj , the coupling signal extrapolation of subsystems with higher step sizes has a bigger impact to the results, than coupling signals with smaller time steps. Therefore, the weight of the incoming coupling signals ci of subsystem Sj has to be scaled with the coupling step size ΔTj , as follows: ci =

ΔTj . ΔTgcd

(15)

To determine the trigger sequence, only the co-simulation structure, i.e., the dependency matrix, the linking matrix, and the coupling step sizes of the subsystems are required. This allows the usage of this approach in an early simulation phase, where no additional information, e.g. simulation results, of the subsystems is given. 4.2

Direct Feed-Through

Additional information about the subsystems allows a more dedicated choice of the coupling signal weight ci . Even the availability of meta-information, e.g. the knowledge of the direct feed-through, could have a significant impact to the prioritization of the coupling signals and to the resulting trigger sequence. If an input signal of a subsystem is marked with a direct feed-through to an output, the weight of this coupling signal is set to ci = 1, otherwise ci = 0. The resulting trigger sequence is given by the minimum number of extrapolation of inputs with the property of direct feed-through. If the co-simulation network includes

94

F. R. Holzinger et al.

different step sizes, the signals with direct feed-through behaviour are scaled by the subsystem step size ΔTj , as in (15). Like the scaling of the coupling signals in Sect. 4.1, this approach is based on meta-information only and does not require detailed subsystem or coupling signal information, i.e., it can be used without additional subsystem analysis if available. The meta-information of the direct feed-through is supported for instance by the FMI-standard, for details see [1]. 4.3

Mean Extrapolation Error

The coupling error of the co-simulation results among others from the local extrapolation error of the coupling signals. Typically, the reduction of the extrapolation error leads to a smaller coupling error and more accurate results. It is obvious to find a trigger sequence, which minimize the extrapolation error. Therefore, the weight of the coupling signals based on the local extrapolation error is given as root-mean-squared error as follows:  N

1 2

1 (ei [k]) , (16) ci = ypp,i N k=1

where ei [k] = yi [k] − ui [k] and N is the number of coupling steps. The coefficients ci are scaled by the range of the coupling signals ypp,i = max(yi )−min(yi ). To identify the local extrapolation error ei of the coupling signals the simulation results, i.e., output signals yi and input signals ui , are required. Therefore, at least one simulation run has to be done. For this run it is assumed, that all coupling signals are extrapolated, i.e., a parallel coupling mechanism is set. Otherwise, the extrapolation error and thus the weighting is distorted, due to the fact, that not every coupling signal is extrapolated. ˆi This can be avoided, if instead of the input ui an approximated input u is used, which is an estimation based on the output yi and the set extrapolation technique. For instance, due to ZOH coupling, the estimated output is given as u ˆi [k] = yi [k − 1]. This calculation can be done in a post-process and is independent to the set coupling mechanism. 4.4

Signal and Subsystem Properties

The previous approaches to define the weights of the coupling signals ci in Sect. 4.1–4.3 consider local coupling behaviour of the one hand or a subset of subsystem properties on the other hand. Nevertheless, the coupling signal behaviour and subsystem properties, e.g. signal frequency and system dynamics, are widely neglected. The impact of the extrapolation of an input signal depends on both the behaviour of the coupling signal and the dynamic of the subsystem. For instance, the extrapolation of a coupling signal with a high dynamic is not critical if the subsystem has a damping behaviour. Vice versa, the input extrapolation of subsystems with high dynamic can be neglected, if the coupling signals

Optimal Trigger Sequence for Non-iterative Co-simulation

95

have a barley excitation. However, if the signal and the subsystem have high dynamic, this leads to a critical coupling signal and the extrapolation of this signal can cause significant coupling errors. Therefore, a definition of a proper trigger sequence requires the knowledge of the subsystem behaviour and the coupling signal behaviour. This topic is described in [4] and an approach can be found in [18]. In addition, an application for the configuration of real-time co-simulation based on the same idea is discussed in [20]. In general, the properties of the coupling signals and the subsystems are not available. Therefore, it is required to determine the properties in a post-process after a simulation run. In the following some signal and subsystem characteristics are discussed for the weighting approach. Signal Classification. The weight of the coupling signal is given by the dynamic behaviour of the one hand and the suitability of the coupling signal for extrapolation on the other. Therefore, the coupling signal can be classified in three major types: continuous, discontinuous and discrete. In general, the extrapolation of continuous coupling signals is less critical than the extrapolation of discontinuity or discrete signals, due to the fact, that it is easier to estimate the upcoming behaviour of a continuous signal. Typically, the modelbased extrapolation methods are based on dynamic subsystems and continuous coupling signals. A detail description regarding the classification can be found in [19]. Coupling Signal Frequency. In general, the extrapolation of rapidly changing coupling signals, i.e., coupling signals with high frequency components, w.r.t. the coupling step size cause higher extrapolation errors than signal coupling with a low frequency. Especially in combination with the dynamical behaviour of the subsystem, the coupling signal frequency has a significant impact to the cosimulation results. Therefore, it is an important quantity for the determination of the weight ci of the coupling signal. An approximation of the instantaneous coupling signal frequency based on a block-wise empirical-mode-decomposition method is discussed in [17]. An additional approach which determines the dynamic of the coupling signals based on an appropriate step size analysis is shown in [10]. Subsystem Dynamic. Uncertainties caused by extrapolation at the subsystem input can be increased or reduced depending on the subsystem dynamic. For instance, a subsystem with a low-pass characteristic has damping behaviour and reduces high frequency components at the subsystem inputs. Subsystems with a high dynamic gains the input uncertainties. The dynamic of the subsystem is a significant indicator for the criticality of coupling signals. In addition to classical system identification methods, e.g. RLS (see [22]), a suitable approach to estimate the dynamic behaviour based on the ratio of the appropriate step sizes between the inputs and output signals of the subsystem

96

F. R. Holzinger et al.

is used. This is sufficient, due to the fact, that only the rough behaviour of the subsystem and not the dynamic in detail is needed. The identification of the appropriate step sizes is discussed in [10] and therefore, the selection of suitable time frames is analysed in [11]. The resulting quantities from the coupling signal and subsystem analysis are combined by using a fuzzy approach. Therefore, the quantities are transformed into the weights of the coupling signals by expert knowledge in form of simple rules.

5

Simulation Example

The different approaches to define the coupling signal weights ci and the resulting trigger sequences are compared on a co-simulation example. The topology of the example with the input and output connections is shown in Fig. 1. It consists of four subsystems SA , SB , SC and SD . A detailed description of the several subsystems can be found in [18], whereby parts of the subsystems are based on an example in [6]. In a first analysis all subsystems have the same coupling step size and afterwards different coupling step sizes for the subsystems are used.

5.1

Equal Coupling Step Size

All subsystems have the same coupling step size ΔTA = ΔTB = ΔTC = ΔTD = 1 ms. Due to the equal coupling step sizes, the execution order of the subsystems behaves as shown in Fig. 2, i.e., the scheduling algorithm is not considered. The extended dependency matrix A with the weighted connections is given in (14). The resulting weights of the edges for the approaches described in Sect. 4.1–4.4 is shown in Table 2. Table 2. Connection and edge weights of the co-simulation graph regarding different weighting approaches for equal coupling step sizes. Connection/edge weights

cab

cad

cbc

ccd

cda

cdc

Number of extrapolation

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

Direct feed-through

1.0000 0.0000 0.0000 1.0000 1.0000 0.0000

Mean extrapolation error

0.1347 0.2181 1.0000 0.1875 0.1333 0.2278

Signal and subsystem properties 1.0000 0.3500 0.2500 0.7200 0.6700 0.2500

All coefficients of the method according to the number of extrapolations are cij = 1, i.e., all inputs have the same impact to the results. In the case of direct feed-through analysis, only the weights cab , ccd and cda is set. These inputs of the several subsystems have a direct feed-through behaviour to an output. The coefficients based on the mean extrapolation error and the signal and subsystem

Optimal Trigger Sequence for Non-iterative Co-simulation

97

properties are determined in a post-process. In the case of the mean extrapolation error, the weights are calculated according to (16). The weights based on the signal and subsystem properties, results from a dynamic and frequency analysis, more details are described in [18]. Due to better comparable coefficients and results, the weighting of the mean extrapolation error and the signal and subsystem characteristics are scaled with their maximum weight. This has no effect on the resulting trigger sequence. The results of the objective function (7) for all possible combinations of the trigger sequence regarding the different weighting approaches are illustrated in Fig. 6. The evaluation of the objective function with a weighting according the number of extrapolations in Fig. 6 a variates between 2 and 4. The results correspond to the actual number of extrapolated inputs for a coupling step of each subsystem. In the best case, i.e., for the sequences {ABCD}, {ABDC}, {ADCB} and {DABC}, only 2 inputs have to be extrapolated in each coupling step. For comparison, if parallel coupling mechanism is used, all 6 inputs have to be extrapolated. The optimum trigger sequence is inconclusive, all 4 solutions equivalent regarding the weighting approach. Figure 6 b shows the results of the objective function with the weights of the edges based on the direct feed-through. The results are the actual number of extrapolated inputs with a direct feed-through behaviour. The example includes three subsystem with a direct feed-through. The trigger sequence {CDAB} delivers the best solution. In this case no input signal with a direct feed-through to an output is extrapolated. The results of the trigger sequence permutation based on the mean extrapolation error is shown in Fig. 6 c. The results correspond to the scaled mean extrapolation error of the subsystems. The best extrapolation error is given by the trigger sequences {ABDC} and {ADBC}. The two sequences differ in the trigger order of subsystems SD and SB . The two subsystems have no direct dependency, i.e., they are not connected. Due to this fact, there is no difference if subsystem SB of subsystem SD calculates first. The evaluation according the signal and subsystem properties is illustrated in Fig. 6 c. The trigger sequence {CDAB} shows the best results. Comparing the different evaluations and results, similarities can be seen between the weighting approaches. The weighting according the number of extrapolations in Fig. 6 a and weighting based on the mean extrapolation error in Fig. 6 c show a similar behaviour. The optimal solution of the mean extrapolation error approach can also be found in the best solutions of the number of extrapolations. The direct feed-through in Fig. 6 b and the signal and subsystem properties in Fig. 6 d have the same optimal solution with the trigger sequence {CDAB}. The approaches provide individually optimal solutions of the trigger sequence regarding the different weightings. Figure 7 shows the simulation results of the two coupling signals yab and yad according the co-simulation example, whereby all possible trigger sequences are illustrated in gray. The monolithic simulation, i.e., where all subsystems

98

F. R. Holzinger et al.

(a) Number of extrapolations

(b) Direct feed-through

(c) Mean extrapolation error

(d) Signal and subsystem properties

Fig. 6. Evaluation of the objective function regarding different weights of the cosimulation graph with equal step sizes: (a) number of extrapolation; (b) direct feedthrough; (c) mean extrapolation error; (d) signal and subsystem properties.

are included in one model and solved with a single solver, is depicted with a dashed line. The optimal trigger sequence, regarding the simulation accuracy, i.e., minimum distance to the monolithic simulation, is illustrated with black solid lines. The results of the different trigger sequences show partly huge deviations and discontinuities in the coupling signals, see Fig. 7 b. These discontinuities are caused by a direct feed-through from the input of the subsystem SA to the output ya b. The trigger sequence of the optimal solution based on the simulation accuracy has the execution order {CDAB}. The optimal solutions according the direct feed-through weighting and the optimal solution based on the signal and subsystem properties provide the same trigger sequence. These both approaches use partly information about the system behaviour. 5.2

Different Coupling Step Sizes

The weights and the structure of the co-simulation graph changes, due to the usage of different coupling step sizes. The following example assumes the coupling

Optimal Trigger Sequence for Non-iterative Co-simulation

99

step sizes ΔTA = 2 ms, ΔTB = 3 ms, ΔTC = 1 ms and ΔTD = 2 ms. The execution of the subsystems is equivalent to the behaviour shown in Fig. 4, i.e., in addition to the subsystem dependencies the scheduling of the subsystems has to be considered. ˜ from (14) for the different Table 3 shows the weights of the transformed graph A weighting approaches. In addition to the graph transformation (11), the weights of the coupling signals are scaled by the step sizes ΔTi according to (15). The evaluation of the trigger sequences regarding the different weighting approaches for various coupling step sizes is shown in Fig. 8. The weighting 12.5 12 11.5 11 10.5 10 9.5 9 0

0.05

0.1

0.15

0.1

0.15

(a) Coupling signal SA , yab 60 40 20 0 -20

0

0.05

(b) Coupling signal SA , yad

Fig. 7. Simulation results of the permutation, monolithic simulation and the optimal trigger sequence {CDAB} of the coupling signals: (a) SA , yab ; (b) SA , yad [18]. Table 3. Connection and edge weights of the co-simulation graph regarding different weighting approaches and different coupling step sizes. Connection/edge weights

c˜ab

c˜ad

c˜ba

c˜bc

c˜cb

c˜cd

c˜da

c˜dc

Number of extrapolation

1.500 2.000 0.500 1.000 0.667 1.500 2.000 1.000

Direct feed-through

1.500 0.000 0.500 0.000 0.000 1.000 2.000 0.000

Mean extrapolation error

0.202 0.436 0.067 1.000 0.667 0.301 0.267 0.228

Signal and subsystem properties 1.500 0.700 0.500 0.250 0.167 0.845 1.340 0.250

100

F. R. Holzinger et al.

based on the number of extrapolated inputs is illustrated in Fig. 8 a and the connection weights according the direct feed-through behaviour is shown in Fig. 8 b. The mean extrapolation error weighting approach is shown in Fig. 8 c and Fig. 8 d depicts the evaluation of the trigger sequences based on analysed coupling signal and subsystem properties. As in the example with equal coupling step sizes in Fig. 6, similarities can be identified between the individual weighting methods in Fig. 8. The evaluation of the objective function based on the number of extrapolations in Fig. 8 a and mean extrapolation error in Fig. 8 c provide a single optimum trigger sequence {ABCD}. The solutions based on the direct feed-through weighting approach in Fig. 8 b and the signal and subsystem weightings in Fig. 8 d are similar. Both methods have an optimal trigger sequence {CDAB}. For this example, the optimal solution corresponds to the optimal trigger sequence according equal step sizes, see Fig. 6 c and 6 d.

(a) Number of extrapolations

(b) Direct feed-through

(c) Mean extrapolation error

(d) Signal and subsystem properties

Fig. 8. Evaluation of the objective function regarding different weights of the cosimulation graph with different step sizes: (a) number of extrapolation; (b) direct feed-through; (c) mean extrapolation error; (d) signal and subsystem properties.

Optimal Trigger Sequence for Non-iterative Co-simulation

101

Nevertheless, the differences of the evaluated trigger sequences within the several weighting approaches are smaller compared to the evaluation with equal coupling step sizes in Fig. 6. Consequently, the effect of the trigger sequence on the simulation results decreases due to different coupling step sizes. Single highly weighted edges dominate the optimal trigger sequence in the case of equal coupling step sizes. If different coupling step sizes are used, these high weights are scaled on the one hand and they are mapped to opposite edges on the other hand, see (11). These reduces the weights of dominant edges and thus the difference between the several trigger sequences. However, despite different step sizes, there are unique optimal solutions for the trigger sequence according the different weighting approaches.

6

Conclusions and Further Work

The work shows an optimization approach of the trigger sequence for noniterative co-simulation. Therefore, a co-simulation graph is derived from the co-simulation network. Based on the co-simulation graph an optimization problem for the trigger sequence is formalized. Furthermore, the work addresses the trigger sequence for sequential cosimulation with different coupling step sizes. Therefore, the execution of the subsystems and the underlying scheduling algorithm are analysed. A transformation of the co-simulation graph is proposed to rescale the dependency matrix. The transformed graph can be interpreted as a substitution graph with equal step sizes, which has the same properties regarding the extrapolation behaviour. Consequently, optimization algorithms, which are designed for subsystems with equal coupling step sizes, can still be used. In addition to the trigger sequence approach for different coupling step sizes, several weighting approaches for the coupling signals are proposed and analysed. These enable the prioritization of the coupling signals according to subsystem meta-information, e.g. direct feed-through, coupling error analysis or coupling signals and subsystem behaviour. The weighting of the coupling signals affects the co-simulation graph and the resulting trigger sequence. The different approaches are demonstrated on an co-simulation example. In further works other scheduling approaches in addition to the latest-first scheduling algorithm will be analysed and their effect to the co-simulation graph will be derived. In a further step, the approach will be extended for hierarchical co-simulation, where subsets of sequentially calculating subsystems are executed in parallel. This allows a trade-off between simulation accuracy and simulation duration. Remark Patent. The presented work describes a part of an automatic configuration approach for non-iterative co-simulation. This is supported by the co-simulation platform Model.CONNECTTM [2] from AVL. The automated configuration approach is protected by a pending European patent [5].

102

F. R. Holzinger et al.

Acknowledgements. The publication was written at Virtual Vehicle Research GmbH in Graz, Austria. The authors would like to acknowledge the financial support within the COMET K2 Competence Centers for Excellent Technologies from the Austrian Federal Ministry for Climate Action (BMK), the Austrian Federal Ministry for Digital and Economic Affairs (BMDW), the Province of Styria (Dept. 12) and the Styrian Business Promotion Agency (SFG). The Austrian Research Promotion Agency (FFG) has been authorised for the programme management.

References 1. Functional mock-up interface for model exchange and co-simulation (2014) 2. AVL: Model.CONNECTtm , the neutral model integration and co-simulation platform connecting virtual and real components (2020). (https://www.avl.com/-/ model-connect). Accessed 31 Jan 2020 3. Benedikt, M., Hofer, A.: Guidelines for the application of a coupling method for non-iterative co-simulation. In: 2013 8th EUROSIM Congress on Modelling and Simulation (EUROSIM), pp. 244–249, September 2013 4. Benedikt, M., Holzinger, F.R.: Automated configuration for non-iterative cosimulation. In: 2016 17th International Conference on Thermal, Mechanical and Multi-Physics Simulation and Experiments in Microelectronics and Microsystems (EuroSimE), pp. 1–7, April 2016. https://doi.org/10.1109/EuroSimE.2016.7463355 5. Benedikt, M., Bernasch, J., Holzinger, F., Watzenig, D.: Method for configuration a co-simulation for a total system. EP3188053 (A1)—05-07-2017 (2016). http:// www.google.it/patents/US4741207 6. Benedikt, M., Drenth, E.: Relaxing stiff system integration by smoothing techniques for non-iterative co-simulation. In: IUTAM Symposium on Co-Simulation and Solver-Coupling (2018) 7. Benedikt, M., Watzenig, D., Zehetner, J., Hofer, A.: Macro-step-size selection and monitoring of the coupling error for weak coupled subsystems in the frequencydomain. In: Proceedings, pp. 1–12 (2013) 8. Busch, M., Schweizer, B.: An explicit approach for controlling the macro-step size of co-simulation methods. In: Bernadini, D. (ed.) Proceedings of the 7th European Nonlinear Dynamics Conference (ENOC 2011), 24 - 29 July 2011, Rome, Italy, pp. 1–6 (2011). http://tubiblio.ulb.tu-darmstadt.de/77923/ 9. Genser, S., Benedikt, M. (eds.) A pre-step stabilization method for non-iterative co-simulation and effects of interface-jacobians identification. In: Simulation and Modeling Methodologies, Technologies and Applications - Advances in Intelligent Systems and Computing, Springer (2019) 10. Genser, S., Holzinger, F., Stettinger, G.: Computation of the ideal step-size - a constant and adaptive approach. In: ICCMSE - 15th International Conference of Computational Methods in Sciences and Engineering (2019) 11. Genser, S., Stettinger, G., Holzinger, F.: Identification of informative time frames for system identification purposes. In: ICCMSE - 15th International Conference of Computational Methods in Sciences and Engineering (2019) 12. Glumac, S., Kovacic, Z.: Calling sequence calculation for sequential co-simulation master. In: Proceedings of the 2018 ACM SIGSIM Conference on Principles of Advanced Discrete Simulation, SIGSIM-PADS 2018, pp. 157–160. ACM, New York (2018). https://doi.org/10.1145/3200921.3200924, http://doi.acm.org/ 10.1145/3200921.3200924

Optimal Trigger Sequence for Non-iterative Co-simulation

103

13. Gomes, C., Oakes, B.J., Moradi, M., G´ amiz, A.T., Mendo, J.C., Dutr´e, S., Denil, J., Vangheluwe, H.: Hintco – hint-based configuration of co-simulations. In: Proceedings of the 9th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - vol. 1: SIMULTECH, pp. 57–68. INSTICC, SciTePress (2019). https://doi.org/10.5220/0007830000570068 14. G¨ unther, F., Mallebrein, G., Ulbrich, H.: A modular technique for automotive system simulation (2012) 15. Haid, T., Stettinger, G., Watzenig, D., Benedikt, M.: A model-based corrector approach for explicit co-simulation using subspace identification. In: The 5th Joint International Conference on Multibody System Dynamics (2018) 16. Hoepfer, M.: Towards a comprehensive framework for co-simulation of dynamic models with an emphasis on time stepping (2011) 17. Holzinger, F.R., Benedikt, M.: Online instantaneous frequency estimation utilizing empirical mode decomposition and hermite splines. In: 2014 22nd European Signal Processing Conference (EUSIPCO), pp. 446–450 (2014) 18. Holzinger, F., Benedikt, M.: Optimal trigger sequence for non-iterative cosimulation. In: Proceedings of the 9th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, pp. 80–87. SCITEPRESS - Science and Technology Publications. https://doi.org/10.5220/0007833800800087 19. Holzinger, F., Genser, S., Stettinger, G.: Identification and classification of signal types and critical loops to support configuration of co-simulation. In: ICCMSE 15th International Conference of Computational Methods in Sciences and Engineering (2019) 20. Holzinger, F., Haid, T., Genser, S., Weiß, G.B., Stettinger, G., Benedikt, M., Watzenig, D. (eds.) configuration of co-simulation for cyber-physical systems: an automated approach. In: SummerSim 2020, Society for Computer Simulation International (2020) 21. Holzinger, F.R., Benedikt, M.: Hierarchical coupling approach utilizing multiobjective optimization for non-iterative co-simulation 1348, 735–740 (2019). http://dx.doi.org/10.3384/ecp19157735 22. Ljung, L.: System Identification: Theory for the User. PTR Prentice Hall Information and System Sciences Series, 2nd edn. Prentice Hall, New Jersey (1999) 23. Meyer, T., Schweizer, B.: Error estimation approach for controlling the comunication step size for semi-implicit co-simulation methods. PAMM 15(1), 63–64 (2015). https://doi.org/10.1002/pamm.201510022. https://onlinelibrary. wiley.com/doi/abs/10.1002/pamm.201510022 24. Oakes, B.J., Gomes, C., Holzinger, F.R., Benedikt, M., Denil, J., Vangheluwe, H. (eds.) Hint-based configuration of co-simulations with algebraic loops. Simulation and Modeling Methodologies, Technologies and Applications - Advances in Intelligent Systems and Computing, Springer (2020 - submitted) 25. Sadjina, S., Kyllingstad, L.T., Skjong, S., Pedersen, E.: Energy conservation and power bonds in co-simulations: mon-iterative adaptive step size control and error estimation. Eng. Comput. 33(3), 607–620 (2017)

Complete Lyapunov Functions: Determination of the Chain-Recurrent Set Using the Gradient Carlos Arg´aez1(B) , Peter Giesl2 , and Sigurdur Freyr Hafstein1 1

Science Institute, University of Iceland, Dunhagi 5, 107, Reykjav´ık, Iceland {carlos,shafstein}@hi.is 2 University of Sussex, Falmer BN1 9QH, UK [email protected]

Abstract. Complete Lyapunov functions (CLF) are scalar-valued functions, which are non-increasing along solutions of a given autonomous ordinary differential equation. They separate the phase-space into the chain-recurrent set, where the CLF is constant along solutions, and the set where the flow is gradient-like and the CLF is strictly decreases along solutions. Moreover, one can deduce the stability of connected components of the chain-recurrent set from the CLF. While the existence of CLFs was shown about 50 years ago, in recent years algorithms to construct CLFs have been designed to determine the chainrecurrent set using the orbital derivative. These algorithms require iterative methods that constructed better and better approximations to a CLF, based on previous iterations. A drawback of these methods is the overestimation of the chainrecurrent set, which has been addressed by different methods. In this paper, we construct a CLF using the previous method, but in contrast to previous work we will use the norm of the gradient of the computed CLF, rather than its orbital derivative, to determine the chain-recurrent set. We will show in this paper that this new approach determines the chain-recurrent set very well without the need of iterations or further methods to reduce the overestimation. Keywords: Complete Lyapunov functions · Chain-recurrent set · Dynamical systems

1 Introduction In this paper we study the dynamics of a general time-autonomous system of differential equations, given by (1), x˙ = f(x), (1) where x ∈ Rn and n ∈ N. We assume that f : Rn → Rn is a continuously differentiable vector field and x˙ denotes the derivative with respect to time. A solution x(t) to (1) with initial value ξ is a continuously differentiable function that satisfies the ODE (1) and such that x(0) = ξ; if the solution is defined for all t ∈ R, then this defines a dynamical system through St ξ = x(t). The first author in this paper is supported by the Icelandic Research Fund (Rann´ıs), Iceland grant number 1163074-052, Complete Lyapunov functions: Efficient numerical computation. c Springer Nature Switzerland AG 2021  M. S. Obaidat et al. (Eds.): SIMULTECH 2019, AISC 1260, pp. 104–121, 2021. https://doi.org/10.1007/978-3-030-55867-3_6

Complete Lyapunov Functions

105

Since analytical solutions to initial value problems are usually not obtainable, one could attempt to use numerical methods for a large collection of initial conditions for (1). This is computationally demanding, would only represent particular solutions, and might not reveal the general behaviour of the dynamical system over the whole phase space. Special solutions of (1) are equilibria, i.e. points x0 with f(x0 ) = 0, implying that the solution starting at the equilibrium will not change in time, but is a constant solution. An equilibrium point is called stable if solutions starting at all adjacent points remain close for all future times. It is called attractive if all adjacent solutions will converge to it as time grows. That means that for an attractive equilibrium x0 one can find an open ball Bδ (x0 ) centered at x0 and with radius δ > 0, such that St x − x0  → 0 as t → ∞ for all x ∈ Bδ (x0 ). In this case, we can define the basin of attraction of x0 by A(x0 ) = {x ∈ Rn | St x → x0 as t → ∞}. The attractivity and stability are two different concepts and do not imply one another, however, when both occur, then we talk about an asymptotically stable equilibrium. An asymptotically stable equilibrium is the simplest example of an attractor. An (local) attractor is a compact, invariant set, that attracts a neighborhood of itself. Further examples include asymptotically stable periodic orbits. One method to analyse dynamical systems is to find the boundaries of the attractors’ basins of attraction. A collection of methods exist to that aim. Among them, one can point out computing the invariant manifolds which form the boundaries of the attractors’ basins of attraction [23]. Another well-known methodologies are set oriented methods [16] or the cell mapping approach [19]. All these methods require large computational effort. An attractor and its basin of attraction can be characterised by a Lyapunov function, [24], originally introduced by Aleksandr Mikhailovich Lyapunov in 1893. A Lyapunov function is a scalar-valued function that attains its minimum on the attractor. Furthermore, its domain is the basin of attraction and it is strictly decreasing along all solutions apart from those on the attractor, where it is constant. An example for a Lyapunov function is the energy in a dissipative physical system. The advantage of the Lyapunov function is that it describes the behaviour of the system without computing its explicit solutions. However, this function is only defined in the basin of attraction of one attractor and describes this subset of the phase space. Even if the attractor is an equilibrium, it is hard to obtain a Lyapunov function. If the system under analysis is linear, then it is possible to obtain a quadratic Lyapunov function relatively easily. However, if the system is not linear then, in general, one requires numerical algorithms to construct a Lyapunov function, see [18]. A complete Lyapunov function is an extension of the classical Lyapunov function for one attractor. It is defined on the whole state space and was introduced in [14, 15, 20, 21]. A complete Lyapunov function characterises the complete qualitative behaviour of the dynamical system on the whole phase space and not just in a neighbourhood of one particular attractor. Therefore, it allows to describe the different basins of attraction for all attractors of the dynamical system. Furthermore, it divides the state-space into two disjoint areas: The gradient-like flow, where the system’s trajectories flow through, and

106

C. Arg´aez et al.

the chain-recurrent set, where infinitesimal perturbations can make the system recurrent. These two areas describe fundamentally different behaviours. The regions in which the system is recurrent or almost recurrent, in the sense that εtrajectories that are arbitrarily close to true system’s solutions are recurrent, are usually referred to as the chain-recurrent set. This set can be shown to equal the intersection of all attractors and all corresponding repellers; for a precise definition see, e.g. [14]. The dynamics outside of the chain-recurrent set are similar to a gradient system, i.e. a system (1) where the right-hand side f(x) is given by the gradient ∇U(x) of a function U : Rn → R. This set includes the transient behaviour of the system. The authors have developed an algorithm to construct CLFs (for further reading, please refer to [2–6, 8, 10–12]). Thus, we are able to identify the chain-recurrent set and the gradient-like flow. The algorithm constructs a CLF by approximating solutions of a linear PDE, fixing values of the orbital derivative. It starts by fixing the values of the orbital derivative to −1, however, this problem does not have a solution on the chainrecurrent set, where the orbital derivative must be zero. Hence, we iteratively adjust the values of the orbital derivative, using the information of the previous iteration. The chain-recurrent set is then characterised as the set of points, where the orbital derivative of the approximating function is zero or close to zero. The algorithm often overestimates the chain-recurrent set, i.e. the area where the approximating function has orbital derivative close to zero is larger than the actual chain-recurrent set. We have recently proposed a general algorithm to reduce the overestimation using geometric properties [7, 9]. In this paper, however, we use a new method to determine the chain-recurrent set, using the norm of the gradient of the computed CLF. It turns out that this gives a much more accurate indication of the chain-recurrent set without the need of further iterations or further algorithms to reduce the overestimation, and is thus preferable to the previous method. However, it requires a dense evaluation grid, where the norm of the gradient needs to be evaluated. Let us give an idea of why the norm of the gradient is a good indicator of the chainrecurrent set. Consider the simple ODE x˙ = −x. In this example, the chain-recurrent set consists of the equilibrium at the origin, which is an attractor. When computing the solution to the PDE V  (x) = −1, where V  (x) = ∇V (x) f (x) denotes the orbital derivative, we find the solution  ln x + c+ if x > 0 V (x) = ln |x| + c− if x < 0 with arbitrary constants c+ , c− ∈ R; note that V is not defined at the equilibrium. Its gradient is given by ∇V (x) =

1 if x = 0. |x|

The mesh-free collocation method, which we use to approximate V , however, produces a smooth function v(x), which fulfills the PDE in all given collocation points (which cannot includes the equilibrium) and minimises the norm in a certain Hilbert space. Moreover, if the fill distance of the collocation points converges to 0, i.e. they become

Complete Lyapunov Functions

107

denser and denser, then v converges to V , and the same holds for its gradient. This means that ∇v(x) becomes very large for x is close to 0. However, since v is a smooth 1 function and ∇v(x) ≈ −1 −x = x changes sign at x = 0, we expect that ∇v(0) = 0. Hence, the chain-recurrent set is characterised by the points x fulfilling the condition ∇v(x) ≈ 0, while close to it we have ∇v(x) 1. Note that the same behaviour occurs for a repeller; consider e.g. the system x˙ = x. The argument fails, however, for a non-hyperbolic equilibrium. Let us give an overview of the contents of the paper: Sect. 2 contains a description of the construction of the Complete Lyapunov function and the computation of its gradient, as well as a description and pseudo-code on how the implementation is done. In Sect. 3 we apply the method to several examples and compare our results to the previous method. In Sect. 4 we discuss the results and give our conclusions.

2 Construction of Complete Lyapunov Functions 2.1 Mesh-Free Collocation Mesh-free collocation, in particular using Radial Basis Functions (RBFs), is a powerful method to solve (generalised) interpolation problems, e.g. linear PDEs [13, 26]. In particular, they can be used to construct CLFs when posed as a generalized interpolation problem. RBFs are real-valued functions whose evaluation depends only on the norm of a point in Rn . Common examples of RBFs are Gaussians and multiquadrics. In this paper, we use Wendland functions as RBF, which are compactly supported and positive definite functions [25]. They have the advantage of being expressed as algebraic polynomials on their compact support. Further, the corresponding Reproducing Kernel Hilbert Space H is norm-equivalent to a Sobolev space. 2.2 Wendland Functions The general form of a Wendland function [25] is ψ(x) := ψl,k (cx), where c > 0 determines the size of the compact support and k ∈ N is a smoothness parameter. For our application the parameter l is fixed as l = n2  + k + 1. The Reproducing Kernel Hilbert Space corresponding to ψl,k contains the same functions as the Sobolev space k+(n+1)/2 (Rn ) and the spaces are norm equivalent. W2 The functions ψl,k are defined by the recursion: For l ∈ N and k ∈ N0 , we define ψl,0 (r) = (1 − r)l+ , ψl,k+1 (r) =

1 r

(2) tψl,k (t)dt

l l for r ∈ R+ 0 , where x+ = x for x ≥ 0 and x+ = 0 for x < 0. Note that x+ := (x+ ) .

108

C. Arg´aez et al.

2.3

Collocation Points

To construct a CLF for system (1), we use mesh-free collocation with RBFs. The method finds the norm-minimal function in the Reproducing Kernel Hilbert space, that satisfies the PDE V  (x) = −1 at all collocation points x; this is a special type of a generalised interpolation problem. It turns out that the solution is a linear combination of the Riesz-representatives, which can easily be calculated in a Reproducing Kernel Hilbert space. The coefficients of the linear combination are determined by solving a system of linear equations, given by the so-called collocation matrix. As collocation points, we use a subset, X = {x1 , . . . , xN } ⊂ Rn , of a hexagonal grid with fineness-parameter αHexa-basis ∈ R+ constructed according to the equation: {αHexa-basis ∑nk=1 ik ωk : ik ∈ Z} , ωk = ∑k−1 e j + (k + 1)εk ek j=1 ε j 1 . and εk = 2k(k + 1)

(3)

Here e j is the usual jth unit vector. These basis vectors are shown in red colour in Fig. 1 in R2 , while the canonical vectors are shown in black. The hexagonal grid has been shown to minimize the condition numbers of the collocation matrices for a fixed fill distance, i.e. a measure of the density of the collocation grid [22]. The collocation points must not include any equilibrium, i.e. any point x with f(x) = 0. In fact, including an equilibrium in the set of collocation points X renders the collocation matrix singular.

Fig. 1. Black: Canonical basis. Red: Hexagonal basis set. Image taken from [7].

Practically, we compute the solution v of the generalised interpolation problem V  (x) = −1 by solving a system of N linear equations, where N is the number of collocation points. v(x) = ∑Nk=1 βk xk − x, f(xk )ψ1 (x − xk ), v (x) = ∑Nk=1 βk − ψ1 (x − xk )f(x), f(xk ) + ψ2 (x − xk )x − xk , f(x)  ·xk − x, f(xk ) ,

(4)

Complete Lyapunov Functions dψ

109

(r)

where ψ1 , ψ2 are given by ψ j (r) = 1r j−1 for r > 0 and j = 1, 2 and ψ0 (r) = ψl,k is dr a Wendland function. Moreover, ·, · denotes the standard scalar product and  ·  the Euclidean norm in Rn , β ∈ RN is the solution to Aβ = r, rk = r(xk ) and A is the N × N matrix with entries ai j = ψ2 (xi − x j )xi − x j , f(xi )x j − xi , f(x j ) −ψ1 (xi − x j )f(xi ), f(x j ) for i = j and

(5)

aii = −ψ1 (0)f(xi )2 .

More detailed explanations on this construction are given in [17, Chapter 3]. If no collocation point x j is an equilibrium for the system, i.e. f(x j ) = 0 for all j, then the matrix A is positive definite and the system of equations Aβ = r has a unique solution. The last assertion will hold true independently of whether the underlying discretized PDE has a solution or not, while error estimates are obviously only available if the PDE has a solution. As it can be seen in Eq. (4), we use two set of points: x j that represent the collocation points and x that represent the evaluation points. If we evaluate the approximation v at a collocation point x j we have v (x j ) = −1 by construction. If the PDE has a solution, then error estimates ensure that V  (x) and v (x) are close if the collocation points are sufficiently dense. However, as the PDE has no solution at points of the chainrecurrent set, these error estimates are not applicable. In previous work the values of v (x) were used for subsequent iterations of the method and thus each evaluation point was associated with an appropriate collocation point [2, 4, 5, 8]. In particular, the first approximation v to a CLF is given by a function which satisfies the PDE v (x) = −1 at all collocation points and is norm minimal in the corresponding Reproducing Kernel Hilbert space H. In later iterations we solve v (x) = r j for different r j ≤ 0, determined by previous approximations. However, in this paper we do not need to reiterate on previous approximations. It will be shown that to have a good approximation to the chain-recurrent set, it is sufficient to find the function that satisfies the PDE v (x) = −1 at all collocation points. In this paper, we use a dense Cartesian evaluation grid, namely a finite subset of hZn with small h > 0. The cardinality of the evaluation grid is denoted by Ξ. As introduced and explained in [4] it is advantageous to use an “almost” normalized approach, i.e., replace the original dynamical system (1) by x˙ = ˆf(x), where

ˆf(x) = 

f(x) δ2 + f(x)2

(6)

with a small parameter δ > 0. This new system has the same trajectories as the original one, but the speed with which trajectories are passed through is more uniform, i.e. ˆf(x) ≈ 1 if x is not an equilibrium. This normalization already reduces significantly the overestimation of the chain-recurrent set.

110

2.4

C. Arg´aez et al.

Gradient of v

The gradient of a function v(x1 , . . . , xn ), is the vector field defined as   ∂v ∂v ∇v(x1 , . . . , xn ) = (x1 , . . . , xn ), . . . , (x1 , . . . , xn ) . ∂x1 ∂xn As explained in the introduction, in this paper we will use the criterion ∇v(x1 , . . . , xn ) ≈ 0 to determine the points in the chain-recurrent set. In practice, we will choose a small parameter γ2 and determine the points x = (x1 , . . . , xn ) with ∇v(x) ≤ γ2 . Therefore, we calculate the first derivative of the function v, see (4), as: N ∂ ∂v (x) = ∑ βk ((xk − x)T f(xk )ψ1 (x − xk )) ∂xi ∂xi k=1

=

N

∑ βk



− fi (xk )ψ1 (x − xk ) + (x − xk )i (xk − x)T f(xk )ψ2 (x − xk )

k=1

Code. In this section, we explain the algorithm used to compute the gradient. Our code is a continuation of the published, free-distributed code, LyapXool [11]. for j=1:Ξ for k=1:N for i=1:n ∇v( j, i)+ = β(k) ∗ (− fˆi (xk ) ∗ ψ1 (y j − xk ) + (y j − xk )i ∗ (xk − y j )T ˆf(xk ) ∗ ψ2 (y j − xk ) end end end Here ∇v( j, i) is the ith component of the gradient of v at the evaluation point y j and we use the collocation points xk , k = 1, 2, . . . , N. As it can be seen, the computation of the gradient vector is a factor n more workload than just computing the CLF and its orbital derivative, which previously was analysed in [8].

3 Results We approximate the solution of V  (x) = −1 by v, using the collocation points X. Previously, when we analysed the orbital derivative, we defined a tolerance parameter −1 < γ1 ≤ 0, and marked a collocation point x j to be poorly approximated, i.e., an element of our approximation of the chain-recurrent set (x j ∈ X 0 ), if there is at least one point x associated to the point x j , i.e. near to x j , such that v(x) > γ. The well approximated points, i.e., for which the condition v (x) < γ holds for all x near x j belong to our approximation of the area of the gradient-like flow (x ∈ X − ). Now we look for points in the evaluation grid such that ∇v(x) ≈ 0, so we define 0 < γ2 and a collocation point x j such that ∇v(x) ≤ γ for some evaluation point x associated to x j is considered to belong to X 0 . In the following, we compare the results of our new method with those of the previous one for several examples.

Complete Lyapunov Functions

111

3.1 Two Orbits Consider the system (1) with   −x(x2 + y2 − 1/4)(x2 + y2 − 1) − y f(x, y) = . −y(x2 + y2 − 1/4)(x2 + y2 − 1) + x

(7)

This system has an asymptotically stable equilibrium at the origin and two periodic circular orbits: an asymptotically stable periodic orbit at Ω1 = {(x, y) ∈ R2 | x2 +y2 = 1} and a repelling periodic orbit at Ω2 = {(x, y) ∈ R2 | x2 + y2 = 1/4}. To compute the CLF we used the Wendland function ψ4,2 . The collocation points were set in a region [−1.5, 1.5]×[−1.5, 1.5] ⊂ R2 and we used a hexagonal grid (3) with αHexa−basis = 0.0131. We computed this example with the almost-normalized method x˙ = ˆf(x) with δ2 = 10−8 . Figure 2 shows the function v, displaying a maximum at the repelling periodic orbit and minima at the attractive periodic orbit as well as the asymptotically stable equilibrium at the origin. Figure 3 shows the orbital derivative v , which is −1 by construction apart from the points in the chain-recurrent set. Using the previous method, namely the orbital derivative, the chain-recurrent set is obtained by choosing the points satisfying v (x) ≥ γ1 with the critical parameter γ1 = −0.25, see Fig. 4 where the failing points of the evaluation grid are plotted. Using our new method, we determine the chain-recurrent set as points satisfying ∇v(x) ≤ γ2 with γ2 = 0.9. Figures 5 and 6 display the norm of the gradient and Fig. 7 shows the points satisfying ∇v(x) ≤ γ2 with γ2 = 0.9. The evaluation grid was computed with the Cartesian grid with a distance parameter h = 0.0007. For this example, N = 60, 456 and Ξ = 18, 369, 796.

Fig. 2. Complete Lyapunov function for system (7). It clearly shows two orbits and one attractor at the origin.

The CLF, Fig. 2, approximated by our method for system (4) shows a system whose behaviour has two circular period orbits with r1 = 1/2 and r2 = 1. Furthermore, the presence of an attractor at the origin is clear. Figure 3 shows that the condition −1 is satisfied for all points in the gradient-like flow while the values of the orbital derivative

112

C. Arg´aez et al.

Fig. 3. Orbital derivative for system (7). As it is seen, all points in the gradient-like flow satisfy the condition −1.

over the chain-recurrent set are clearly different. Points with an orbital derivative larger than γ = −0.25 give us our first approximation to the chain-recurrent set, Fig. 4. All components of the chain-recurrent set, the equilibrium and the two periodic orbit, are present, but the periodic orbits are over-estimated. That overestimation was reduced using geometrical properties in [7].

Fig. 4. Chain-recurrent set for system (7) constructed by filtering the orbital derivative. Two orbits are seen at r = 1 and r = 1/2. The attractive origin is also found. The critical value to filter the orbital derivative was γ1 = −0.25.

Let us now discuss the norm of the gradient of the CLF, see Fig. 5. As discussed in the introduction, the norm is very large close to the chain-recurrent set, and close to 0 directly on the chain-recurrent set, cf. Fig. 6

Complete Lyapunov Functions

113

Figure 7 shows the points with norm ∇v(x) ≤ γ2 and displays precisely the same two orbits and the critical point. In this case, however, the norm of the gradient vector of v was used instead of the orbital derivative. Compared to the previous method, see Fig. 4, the new method shows the chain-recurrent set much sharper and with hardly any overestimation. A direct comparison between the two sets is shown in Fig. 8. Figure 8 reveals how much the old method overestimates the chain-recurrent set in comparison to the new method.

Fig. 5. Norm of the gradient of v for system (7).

Fig. 6. Norm of ∇v for system (7). The figure has been cut to show the behaviour close the chain-recurrent set.

114

3.2

C. Arg´aez et al.

Van der Pol Oscillator

The Van der Pol oscillator is a classical example in dynamical systems. It is represented by Eq. (8).     y x˙ . (8) = f(x, y) = y˙ (1 − x2 )y − x For computing the CLF associated to system (8), we set αHexa-basis = 0.027 over the area defined by [−3, 3]2 ⊂ R2 . As before we use the almost normalized method with δ2 = 10−8 . The Wendland function used is ψ4,2 and we have used the critical values γ1 = −0.25 for the orbital derivative and γ2 = 0.9 for the norm of the gradient. The evaluation grid was computed with the Cartesian grid with a distance parameter h = 0.0015. For this example, N = 61, 446 and Ξ = 16, 008, 001.

Fig. 7. Chain-recurrent set for system (7) constructed by filtering the norm of the gradient of the Lyapunov Function. Two orbits are seen at r = 1 and r = 1/2. The attractive origin is also found. The critical value to filter the orbital derivative was γ2 = 0.9.

Fig. 8. Chain-recurrent set for system (7). Black: using the orbital derivative (previous method). Red: using the gradient of the complete Lyapunov function (new method).

Complete Lyapunov Functions

115

Fig. 9. Complete Lyapunov function for system (8). The repeller at the origin and the attractive periodic orbit are clearly seen.

Fig. 10. Orbital derivative for system (8).

Figures 9 and 10 show the computed CLF and its orbital derivative associated to the system (8). Again, we can see that the behaviour of this system is represented clearly by the CLF – it has a minimum at the periodic orbit and a maximum at the unstable equilibrium at the origin. As well, the orbital derivative is −1 at all point in the gradientlike flow, while being larger on the chain-recurrent set. The chain-recurrent set obtained by plotting the points where the orbital derivative satisfies v (x) ≥ γ1 is shown in Fig. 11. Again the periodic orbit for this system is clearly overestimated.

116

C. Arg´aez et al.

Fig. 11. Chain-recurrent set for system (8) obtained by the orbital derivative.

Figure 12 displays the norm of the gradient of the CLF. Near the periodic orbit the norm is again very large, while it is close to zero on the periodic orbit. The chainrecurrent set obtained by plotting the points with ∇v(x) ≤ γ2 is shown in Fig. 13. As is clearly seen in Fig. 13, the chain-recurrent set obtained with the norm of the gradient of v is sharper around the periodic orbit; it is not fully closed, but the shape is clearly visible. However, there is overestimation around the equilibrium, which will be addressed in Sect. 4.

Fig. 12. Norm of the gradient of v for system (8).

Complete Lyapunov Functions

117

Fig. 13. Chain-recurrent set for system (8) obtained by the norm of the gradient.

3.3 Homoclinic Orbit In dynamical systems, a homoclinic orbit is a trajectory which connects an equilibrium to itself. In this paper, we consider the following system with a homoclinic orbit,     x˙ x(1 − x2 − y2 ) − y((x − 1)2 + (x2 + y2 − 1)2 ) = f(x, y) = . (9) y˙ y(1 − x2 − y2 ) + x((x − 1)2 + (x2 + y2 − 1)2 ) For this system, the origin is an unstable focus. The system has an asymptotically stable homoclinic orbit at a circle centred at the origin and with radius 1, connecting the equilibrium (1, 0) with itself. We used the Wendland function ψ4,2 for our computations. The collocation points were set in a region [−1.5, 1.5] × [−1.5, 1.5] ⊂ R2 and we used a hexagonal grid (3) with αHexa−basis = 0.0131. We computed this example with the almost-normalized method x˙ = ˆf(x) with δ2 = 10−8 . We used the following critical values: γ1 = −0.25 for the orbital derivative and γ2 = 0.9 for the norm of the gradient. The evaluation grid was computed with the Cartesian grid with a distance parameter h = 0.0007. For this example, N = 60, 456 and Ξ = 18, 369, 796. Figures 14 and 15 show a CLF and its orbital derivative associated to system (9). Again, the system’s behaviour is represented clearly by the CLF. The function satisfies the condition v (x) ≈ −1 at all point in the gradient-like flow, while this fails over the chain-recurrent set. The chain-recurrent set obtained by the orbital derivative is shown in Fig. 16 for failing points of the evaluation grid. Unlike the previous examples, this system is rather complicated. Plotting the points with orbital derivative v (x) ≤ γ1 = −0.25 does not sufficiently classify the orbit; some parts are overestimated and some are missing. Using a γ1 closer to −1, as done in [2, 4, 5, 8], helps to fully determine the homoclinic orbit. However, that will also bring more overestimation. Figure 17 displays norm of the gradient of the CLF, while Fig. 18 shows the chainrecurrent set obtained through the gradient of the CLF.

118

C. Arg´aez et al.

Fig. 14. Complete Lyapunov function for system (9).

Fig. 15. Orbital derivative of the complete Lyapunov function for system (9). It clearly shows how the approximation to −1 succeeds over the gradient-like flow while it fails over the chainrecurrent set.

Fig. 16. Chain-recurrent set for system (9) obtained by the orbital derivative.

Complete Lyapunov Functions

119

Fig. 17. Norm of the gradient of v for system (9).

Fig. 18. Chain-recurrent set for system (9) obtained by the gradient.

As for the system (8), this orbit is not closed either. However, it is sufficiently welldefined. The overestimation is minor on the orbit and even if it is considerable around the equilibrium, that is not an issue as discussed in Sect. 4.

4 Discussion The new method, using the norm of the gradient to classify the chain-recurrent set, gives sharper image of the chain-recurrent set, i.e. it has less overestimation. Further, it misses less parts of it. These results have been obtained without iterations. The determination of equilibria and higher-dimensional sets in the chain-recurrent set requires two different critical values; this was observed for the previous method in [12], where we have discussed the problem of finding appropriate critical values for the chain-recurrent set when using the orbital derivative. As is explained in [12], we should consider two different critical values, one for orbits and another for equilibrium points. Indeed, for a

120

C. Arg´aez et al.

system (1) the equilibrium points can be computed by solving f(x) = 0, often analytically. For these reasons, in this paper we have focused on classifying the orbits. The new method requires higher computational effort when compared to one iteration of the previous one: first, we require a dense evaluating – this is related to the fact that the norm of the gradient is only close to zero on a very small set. Indeed, the norm of the gradient becomes large close to the area where it is small! However, this also delivers a sharp resolution of the chain-recurrent set. In future work we intend to first use a course evaluation grid and then refine it locally in areas of interest. Second, the computation of the gradient, as explained in Sect. 2.4, is computationally more costly by factor n.

5 Conclusions We have introduced a new method to determine the chain-recurrent set of dynamical systems using approximations to complete Lyapunov functions. While previous methods determined the chain-recurrent set by finding points where the orbital derivative of the complete Lyapunov functions is close to zero, i.e. v (x) ≈ 0, the new method uses the norm of its gradient and determines points where ∇v(x) ≈ 0. The new method is able to determine the chain-recurrent set better, both by detecting all areas and by reducing its overestimation. The method will be extended in the future in order to automatically determine stable and unstable components of the chain-recurrent set, as well as stable and unstable directions by determining minima and maxima of the function v.

References 1. Anderson, J., Papachristodoulou, A.: Advances in computational Lyapunov analysis using sum-of-squares programming. Discrete Contin. Dyn. Syst. Ser. B 20(8), 2361–2381 (2015) 2. Arg´aez, C., Giesl, P., Hafstein, S.: Analysing dynamical systems towards computing complete Lyapunov functions. In: Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, SIMULTECH 2017, Madrid, Spain (2017) 3. Arg´aez, C., Giesl, P., Hafstein, S.: Computation of complete Lyapunov functions for threedimensional systems. In: Proceedings of the 57rd IEEE Conference on Decision and Control (CDC), Miami Beach, FL, USA, 2018, pp. 4059-4064 (2018) 4. Arg´aez, C., Giesl, P., Hafstein, S.: Computational approach for complete Lyapunov functions. In: Dynamical Systems in Theoretical Perspective, Springer Proceedings in Mathematics and Statistics, vol. 248, pp. 1–11. Springer (2018) 5. Arg´aez, C., Giesl, P., Hafstein, S.: Iterative construction of complete Lyapunov functions. In: Proceedings of 8th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2018), pp. 211–222 (2018). https://doi.org/ 10.5220/0006835402110222. ISBN 978-989-758-323-0 6. Arg´aez, C., Giesl, P., Hafstein, S.: Construction of a complete Lyapunov function using quadratic programming. In: Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics (ICINCO), Porto, Portugal, 2018, pp. 560–568 (2018) 7. Arg´aez, C., Giesl, P., Hafstein, S.: Middle point reduction of the chain-recurrent set. In: Proceedings of the 9th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH), Prague, Czech Republic, 2019, pp. 141–152 (2019)

Complete Lyapunov Functions

121

8. Arg´aez, C., Giesl, P., Hafstein, S.: Complete Lyapunov functions: computation and applications. In: Simulation and Modeling Methodologies, Technologies and Applications Series: Advances in Intelligent Systems and Computing, vol. 873, pp. 200–221. Springer (2019) 9. Arg´aez, C., Giesl, P., Hafstein, S.: Clustering algorithm for generalized recurrences using complete Lyapunov functions. In: Proceedings of the 16th International Conference on Informatics in Control, Automation and Robotics (ICINCO), Prague, Czech Republic, 2019, pp. 138–146 (2019) 10. Arg´aez, C., Giesl, P., Hafstein, S.: Improved estimation of the chain-recurrent set. In: Proceedings of the 18th European Control Conference (ECC), Napoli, Italy, 2019, pp. 1622– 1627 (2019) 11. Arg´aez, C., Berthet, J.-C., Bj¨ornsson, H., Giesl, P., Hafstein, S.: LyapXool - a program to compute complete Lyapunov functions. SoftwareX 10, 100325 (2019). ISSN 2352-7110 12. Arg´aez, C., Giesl, P., Hafstein, S.: Critical tolerance evolution: classification of the chainrecurrent set. In: Proceedings of the 15th International Conference on Dynamical Systems: Theory and Applications (DSTA), Volume: Mathematical and Numerical Aspects of Dynamical System Analysis, Lodz, Poland, pp. 21–32 (2019) 13. Buhmann, M.: Radial Basis Functions: Theory and Implementations. Cambridge University Press, Cambridge (2003) 14. Conley, C.: Isolated invariant sets and the morse index. In: CBMS Regional Conference Series no. 38. American Mathematical Society (1978) 15. Conley, C.: The gradient structure of a flow I. Ergodic Theory Dynam. Syst. 8, 11–26 (1988) 16. Dellnitz, M., Junge, O.: Set oriented numerical methods for dynamical systems. Handbook of dynamical systems, vol. 2, pp. 221–264. North-Holland, Amsterdam (2002) 17. Giesl, P.: Construction of Global Lyapunov Functions Using Radial Basis Functions. Lecture Notes in Math, vol. 1904. Springer, Heidelberg (2007) 18. Giesl, P., Hafstein, S.: Review on computational methods for Lyapunov functions. Discrete Contin. Dyn. Syst. Ser. B 20(8), 2291–2331 (2015) 19. Hsu, C.S.: Cell-to-Cell Mapping. Applied Mathematical Sciences, vol. 64. Springer, New York (1987) 20. Hurley, M.: Chain recurrence, semiflows, and gradients. J. Dyn. Diff. Equat. 7(3), 437–456 (1995) 21. Hurley, M.: Lyapunov functions and attractors in arbitrary metric spaces. Proc. Amer. Math. Soc. 126, 245–256 (1998) 22. Iske, A.: Perfect centre placement for radial basis function methods. Technical report TUMM9809, TU Munich, Germany (1998) 23. Krauskopf, B., Osinga, H., Doedel, E.J., Henderson, M., Guckenheimer, J., Vladimirsky, A., Dellnitz, M., Junge, O.: A survey of methods for computing (un)stable manifolds of vector fields. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 15(3), 763–791 (2005) 24. Lyapunov, A.M.: The general problem of the stability of motion. Internat. J. Control 55(3), ´ 521–790 (1992). Translated by A. T. Fuller from Edouard Davaux’s French translation (1907) of the 1892 Russian original 25. Wendland, H.: Error estimates for interpolation by compactly supported Radial Basis Functions of minimal degree. J. Approx. Theory 93, 258–272 (1998) 26. Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)

A Study on the Dynamic Behavior of a Slurry Mixing and Pumping Process: An Industrial Case Ridouane Oulhiq1,2(B) , Khalid Benjelloun1,2 , Yassine Kali3 , Maarouf Saad3 , and Laurent Deshayes2 1

Ecole Mohammadia d’Ing´enieurs, Mohammed V University, Rabat, Morocco [email protected] 2 Mohammed VI Polytechnic University, Benguerir, Morocco 3 ´ Ecole de Technologie Sup´erieure, Montreal, QC H3C 1K3, Canada

Abstract. Flow behavior of solid-liquid flow systems depends on the solids’ properties, the carrier fluid and the interactions between the two phases. The solids’ properties of slurry pumping systems such as granulometry and density change as a function of time and location. The above-mentioned changes impact directly the slurry pumping flow rate. In this paper, a dynamic model of an industrial mixing tank with a slurry centrifugal pump is proposed. The dynamic model takes into account the different parts of the pump; the hydraulic part, the induction motor and the system head. A graphical method is then used to estimate the hydraulic part and the induction motor parameters. Regarding the system head, the model is generalized for both homogeneous and heterogeneous flows. The overall model is simulated using MATLAB/Simulink software and validated through a comparison between the results obtained and the real industrial data of a slurry mixing and pumping unit. Finally, the effects of density and level variations on the flow dynamic behavior are studied.

Keywords: Slurry pumping Heterogeneous slurry

1

· Solid-liquid flows · Centrifugal pump ·

Introduction

Slurry pumping systems are extensively used in several chemical processes to pump solid-liquid mixtures between the different units of a process. In general, the behavior of solid-liquid flows are quite complex and hard to understand in addition to several identified features that might make them more complicated. This imperfect knowledge of the process causes uncertainties that might degrade and deteriorate the desired performances in industrial flow controllers. Taking into account the necessity of stable and accurate flows to guarantee the quality of the final product, an accurate dynamic model is needed, for a better understanding, simulation and control of the process. c Springer Nature Switzerland AG 2021  M. S. Obaidat et al. (Eds.): SIMULTECH 2019, AISC 1260, pp. 122–143, 2021. https://doi.org/10.1007/978-3-030-55867-3_7

A Study on the Dynamic Behavior of a Slurry Mixing and Pumping Process

123

In slurry pumping processes, the centrifugal pump is generally used [27]. The dynamic behavior of the flow of a centrifugal pump is related to the pump motor, the hydraulic part of the pump and the hydraulic application or the system head. In [18] a dynamic model for the system pump/pipeline is proposed and simulated. The system head is discussed for water flow and heterogeneous flow. However, the model assumes that the pump drive behaves like a constant torque system and that the centrifugal pump ensures the affinity laws. In [12], the different parts of the pump are discussed. The parameters of the hydraulic part are calculated based on the physical properties of the pump. However, the system head is not detailed and the variation of flow is not discussed. In [9], the hydraulic part of the pump is described by the pump head and the load torque without considering the density of the pump fluid. The induction motor is modeled using the dqo coordination including a variable frequency drive. In [26], the centrifugal pump is modeled based on the hydraulic system without considering the pump’s motor and without discussing the system head of the pump. In [19], the centrifugal pump is modeled based on the model of the hydraulic application, the model of the hydraulic part and the induction motor model. The hydraulic application model assumes an homogeneous slurry while the hydraulic part’s parameters are estimated graphically based on the pump characteristic curves. Finally, for the pump’s motor, the dq model is used [15] assuming that a set of parameters are well-known. Based on the literature revue below where it can be noticed that few works tackled this topic. This work proposes a dynamic model of the slurry centrifugal pump based on the interaction of all the different sub-systems, namely, the hydraulic part, the induction motor and the system head. The hydraulic part and the induction motor parameters are estimated using a graphical method eliminating the need of a prior knowledge of the system’s physical properties. A variable frequency drive is added to the system to provide the possibility to implement a controller. Otherwise, regarding the system head, a generalized approach is proposed for both homogeneous and heterogeneous flows while taking into account the effect of the flow rate variation on the friction factor. The centrifugal pump is then coupled with a mixing tank that is modeled based on mass balance equations [5,19]. The detailed models are simulated and evaluated through a comparison with real industrial data of a slurry mixing and pumping process. The outline of this chapter is summarized as follows. Section 2 presents the dynamic analysis of the slurry centrifugal pump and the mixing tank. Section 3 exposes the simulation and the performance results compared to the industrial data, with a study on the effects of density and level variations on the flow dynamic behavior. Section 4 concludes the chapter.

2

Process Modeling

In the studied industrial process, the slurry feed is delivered to a cylindrical mixing tank in which it is kept in agitation in order to avoid the sedimentation

124

R. Oulhiq et al.

of the solid. The mixed slurry is then pumped, using a centrifugal pump driven by an induction motor, to downstream processes (refer to Fig. 1). 2.1

Centrifugal Pump

The centrifugal pump is modeled based on its different parts, namely, the hydraulic part, the induction motor and the system head:

Fig. 1. Pump hydraulic application parameters.

Hydraulic Part. The pump head Hp (m) is a function of flow Fs2 (m3 /h) and shaft speed ωp (rad/s). The equation describing this fact is [12,13] given by: 2 Hp = ah Fs2 + bh Fs2 .ωp + ch ωp2

(1)

where ah , bh and ch are constant parameters fixed from the physical properties of the pump. In this chapter, these parameters are determined using the H-Q curves shown in Fig. 2 of the used pump [19]. Otherwise, the pump torque also known under the name of load torque Tp (N.m) is described by the following equation [12,13]: 2 Tp = −ds2 at Fs2 + ds2 bt Fs2 ωp + ct ωp2

(2)

where ds2 denotes the pump slurry density while at , bt and ct are constants fixed from the physical properties of the pump. Once again, the above three constants will be fixed in this work from the H-Q curves of the used pump (refer to Fig. 2). Induction Motor. The induction motor is made up of three parts, namely, the mechanical part, the electromechanical part and the variable frequency drive that are described as follows:

A Study on the Dynamic Behavior of a Slurry Mixing and Pumping Process

125

Fig. 2. The pump H-Q curve.

Mechanical Part: This part of the pump is described [4,25] by the following equation: 1 dωp Cf = (Te − Tp ) − ωp (3) dt Jm + Jp Jm + Jp Where: Jm : Jp : Cf : Te :

is is is is

the the the the

moment of inertia of the pump mechanical parts (Kg m2 ); moment of inertia of the fluid inside the pump impeller (Kg m2 ); friction losses coefficient of pump induction motor (Kg m2 /s); torque produced by the pump induction motor (N m).

The moment of inertia Jp depends on the fluid density [14]:  Jp = 2πhds2

R2

R1

r3 dr = CJ ds2

(4)

126

R. Oulhiq et al.

Where: CJ : r: h: R1 : R2 :

a constant (m5 ); the radius at a given slice of the disc forming the impeller (m); the height of the impeller (m); the inside diameter of the impeller (m); the outside diameter of the impeller (m).

is is is is is

Electromechanical Part. The torque Te is calculated based on a model describing the speed-torque curve characteristic [1]. With a change in the equation’s denominator to take account of the re-generative breaking part of the induction motor: t0 V 2 s Te = 2 (5) s + t1 |s| + t2 ⎧ ws − wp ⎪ s= ⎪ ⎪ ⎨ ws

with:

⎪ ⎪ 120f ⎪ ⎩ws = p

(6)

Where: s: ws : f: p: V:

is is is is is

the the the the the

slip; synchronous speed (rpm); supply frequency (Hz); pole number; amplitude of the terminal voltage (V).

Variable Frequency Drive (VFD). The induction motor is supplied through a three phase VFD. The used VFD is based on the frequency control method, assuming that the voltage-frequency ratio is stable:

with:

V = cte f

(7)

 V = M V (Vmax − Vmin ) + Vmin f = M V (fmax − fmin ) + fmin

(8)

Where: Vmin : Vmax : fmin : fmax : MV :

is is is is is

the the the the the

minimum VFD voltage; maximum VFD voltage; minimum VFD frequency; maximum VFD frequency; setpoint of the VFD (%).

A Study on the Dynamic Behavior of a Slurry Mixing and Pumping Process

127

System Head. The system head is composed of four parts, namely, the static head (Hs ) known also under the name of elevation, the pressure head (Hp ), the total head losses (Hl ), that are frictions and other resistances in the piping system and the velocity head (Hv ). The system head [17] between point 1 and point 2 as depicted in Fig. 1) is described by: Hsys = Hs + Hpr + Hl + Hv

(9)

Where: Hsys : Hs : Hpr : Hl : Hv :

is is is is is

the the the the the

system head between point 1 and point 2 (m); static head of the system (m); pressure head (m); total head losses in the system (m); velocity head (m);

The static head Hs of the system measures the elevation difference between the suction point (point 1) and the discharge point (point 2): Hs = z2 − z1 = z2 − Lm

(10)

Where: z1 : is the elevation at point 1 (m); z2 : is the elevation at point 2 (m); Lm : is the mixing tank level (m); The pressure head Hpr is the difference between the suction and the discharge pressures. It is described by the equation: Hpr =

P2 − P1 ds2 g

(11)

Where: P1 : is the pressure at point 1 (Pa); P2 : is the pressure at point 2 (Pa); ds2 : is the density of slurry (kg/m3 ). Since P1 = P2 = Patm , where Patm is the atmospheric pressure, then: Hpr = 0

(12)

Regarding the velocity head Hv , it is the energy lost into the system due to the velocity of the liquid moving through the system. The Hv formula is given as follows: v 2 − v12 Hv = 2 (13) 2g

128

R. Oulhiq et al.

Where: v1 : is the slurry speed at point 1 (m/s); v2 : is the slurry speed at point 2 (m/s); Otherwise, considering that: 4Fs2 πDp2

v2 =

and assuming that v1 ≈ 0, Eq. (13) becomes: Hv =

2Fs2 gπDp2

(14)

Regarding the total head losses Hl , it is the equivalent head in meters necessary to overcome the resistance developed by the friction caused by the fluid flow through a pipe and other resistances in the piping system, such as: Hl = Hlf + Hll

(15)

Where: Hlf : is the friction losses (m); Hll : is the local losses (m); Furthermore, for The friction losses Hlf , a generalized expression for homogeneous and heterogeneous flows is proposed based on the following formula [17,22]: Hlf = (1 + Cv φ)

f v22 Lmt 2gDp

(16)

Where: φ: is a dimensionless parameter (for homogeneous flows we take φ = 0); Cv : is the volume concentration of solids; f : is the Darcy friction factor of the slurry for homogeneous flows while is the Darcy friction factor of the carrier fluid for heterogeneous flows; Dp : is the pipe diameter (m). Moreover, substituting Cv and v2 with their respective expressions in Eq. (16), the following equation is deduced:  Hlf =

 1+

ds2 − df dp − df

 2 f Lmt 4Fs φ 2gDp πDp2

(17)

where dp and df represent the dry product and the carrier fluid densities, respectively. The Darcy friction factor f and the parameter φ depend on the pipe properties and on the nature of the pumped fluid. Table 1 summarizes the friction factor formulas used for different natures of fluids [7,17,22,24].

A Study on the Dynamic Behavior of a Slurry Mixing and Pumping Process

129

Table 1. The friction factor formulas. Slurry nature

Flow regime

Newtonian

Laminar Re < Rec

Turbulent Re > Rec

Non-newtonian Bingham plastic

Laminar Re < Rec

Friction factor (f ) 64 v2 Dp ds2 f = ; Re = Re μ Re is the Reynolds number; Rec = 2100; μ is the absolute viscosity of slurry (Pa.s)   e 1 2.51 + = −2log10 ; Re = √ √ f 3.7Dp Re f v2 Dp ds2 μ e is the absolute pipe roughness (m)   1 64He4 He f = 64 − + ; 2 3 8 Re 6Re 3f Re Re =

v2 Dp ds2

; He =

2 ds2 τ 0 Dp

η η2 He is the Hedstrom number; Rec is determined graphically based on He; η is the plastic viscosity; τ0 is the yield stress Turbulent Re > Rec Pseudo-plastic

Laminar Re < Remc

A graphical method is used. n  64 n f = ; Rem = 8 Re⎛ 6n + 2 m ⎞ (2−n) n ⎝ v2 ⎠ ds2 Dp K

Rem is the modified Reynolds number n is the power law exponent; K is the power law coefficient  (2+n)/(1+n) 1 (1 + 3n)2 Critical Re = Remc f = 101n 2+n (2 + n)(2+n)/(1+n) Remc = 6464n (1 + 3n)2 1 Turbulent Re > Remc √ = f   1−n/2  f 2 0.4 log10 Rem − 0.75 n 4 n1.2 64 Yield pseudo-plastic Laminar Re < Remc f = ; Rem ⎞ ⎛ n  (2−n) n n ⎝ v2 ⎠ Rem = 8 ds2 Dp 6n + 2 K 64 Critical Re = Remc f = ; Remc = Remc (2 + n)(2+n)/(1+n) 6464n Φ (1 + 3n)2  (2−n) 2x(1 − x) x2 (1−x)2 + + 1+3n 1 + 2n 1+n Ψ = (1 − x)n 2 τ0  τ0 2/(n−2) ds2 Dp He = K2 K x is calculated from: 3232 He = n (2 +   2−n  n 2+n n x 1 n) 1+n 1+n (1 − x) 1−x Turbulent Re > Remc For smooth pipes:: 2 2.69 4.53 − 2.95 + log10 (1 − x) √ = f n n

 0.68 4.53 − log10 Rem f 2−n + n n For rough pipes: 2 Dp 2.65 + 6.0 − √ = 4.07log10 f 2e n

130

R. Oulhiq et al.

Considering the local losses Hll that are associated with the piping components, such as valves and fittings, pipe diameter enlargement and reduction and pipe entrance and exit. The local losses are included in computations either using the equivalent length of the valve or fitting or by using a resistance factor (or K factor) multiplied by the velocity head. Notice that this method is only applicable to turbulent flow through pipe fittings and valves [17]. Hence, the local losses are described by: Hll =

n v22

Ki 2g i

(18)

where Ki are the local losses resistance factors. Simplifying Eq. (9), the relation of the system head is obtained as follows: Hsys = α + (β + γf )Fs2

(19)

Where: α = z2 − Lm 2 n 4 1 β= (1 + i Ki ) 2 2g πDp 2   1 4 Lmt ds2 − df γ= φ 2g π.Dp2 Dp dp − df Given that the friction factor depends on the fluid’s velocity and properties, it must be updated in each iteration of the dynamic model simulation. Flow Dynamics. In addition to understanding the effects of fluid in steady flow, the interest is also in started flows and transient flows. For such flows, the generalized form of the Navier-Stokes equation that can be found in [20] is used: ds

Dv = −∇P + ds Fext + Fvis Dt

(20)

Where: ds : v :  Fext : Fvis : D Dt :

is is is is is

3

the slurry density (Kg/m ); the flow speed vector (m/s); the external forces vector (N/Kg); the viscous forces vector (Pa/m); substantial derivative.

Developing the above equation gives: ds

∂v  + ds (v ∇)v = −∇P + ds Fext + Fviscous ∂t

(21)

A Study on the Dynamic Behavior of a Slurry Mixing and Pumping Process

131

The Eq. (21) is then expanded in the cartesian coordinates x, y and z as follows: ⎧ ∂vx ∂Px x ds ∂t + ds vx ∂v ⎪ ∂x = − ∂x + ds Fextx + Fvisx ⎪ ⎪ ⎪ ⎨ ∂P ∂v ∂v (22) ds ∂ty + ds vy ∂yy = − ∂yy + ds Fexty + Fvisy ⎪ ⎪ ⎪ ⎪ ⎩ ∂vz ∂Pz z ds ∂t + ds vz ∂v ∂z = − ∂z + ds Fextz + Fvisz Computing the integral of equation (22) from 1 to 2 for k ∈ [x, y, z] gives:  j  j  j  j  j ∂vk ∂vk ∂Pk dk + dk = − dk + ds d s vk ds Fextk dk + Fvisk dk ∂t ∂k ∂k i i i i i (23) Moreover, assuming that vy ≈ 0 and that the mixing tank is perfectly mixed ds ≈ ds2 , Eq. 23 becomes: ⎧ 2 2 1 2 ∂vx 2 ⎪ ⎪ ⎨ 1 ds2 ∂t dx + 2 ds2 v2 = −P2 + 1 ds2 Fextx dx − 1 Fvisx dx (24) ⎪ 2 ⎪ ⎩ 2 d ∂vz dz − 1 d v 2 = P + 2 d F dz − 1 Fvisz dz s2 1 1 1 s2 ∂t 1 s2 extz 2 Otherwise, the addition of the equations given in 24 result in:  2  2 ∂vx ∂vz dx + dz = − ds2 gHpr − ds2 gHs + ds2 gHp − ds2 gHl ds2 ds2 ∂t ∂t 1 1 (25) − ds2 gHv = ds2 g(Hp − Hsys ) where: Hsys = Hpr + Hs + Hv + Hl

(26)

Equation (25) is then simplified by breaking the terms in the left hand side into several terms (refer to Fig. 1):  a  b  c  2  2 ∂vx ∂vx ∂vx ∂vx ∂vx dx = dx + dx + dx + dx ds2 ds2 ds2 ds2 ds2 ∂t ∂t ∂t ∂t ∂t 1 1 a b c  2  b ∂vx ∂vx dx + dx ≈ ds2 ds2 ∂t ∂t a c ∂v2 ≈ ds2 (La−b + Lc−2 ) ∂t (27)  a  b  c  2  2 ∂vz ∂vz ∂vz ∂vz ∂vz dz = dz + dz + dz + dz ds2 ds2 ds2 ds2 ds2 ∂t ∂t ∂t ∂t ∂t 1 1 a b c  c ∂vz dz ≈ ds2 ∂t b ∂v2 ≈ ds2 Lb−c ∂t (28)

132

R. Oulhiq et al.

Where: La−b : is the length of pipe between a and b (m); Lb−c : is the length of pipe between b and c (m); Lc−2 : is the length of pipe between c and 2 (m). Hence, combining Eqs. (25), (27) and (28) will result in: g dv2 = (Hp − Hsys ) dt La−2

(29)

Where: La−2 : is the length of pipe between a and 2 (m). Then, the equation describing the variation of flow in function of the pump head Hp and the system head Hsys is obtained as follows: dFs2 Ap g = (Hp − Hsys ) dt La−2

(30)

Where: La−2 : is the length of pipe between a and 2 (m); Ap : is the pipe section (m2 ). 2.2

Mixing Tank

Based on mass balance equations, the dynamic behavior of the mixing tank is described by the following equations [10,23]: d(Mpm + Mwm ) = dw Fw + ds1 Fs1 − ds2 Fs2 dt

(31)

dLm 4 = (Fs1 + Fw − Fs2 ) 2 dt πDm

(32)

Where: Mpm : ds1 : Fs1 : ds2 : Fs2 : Mwm : dw : Fw : Lm : Dm :

is the total mass of dry product in the mixing tank (Kg); 3 is the density of slurry at the inlet of the mixing tank (Kg/m ); is the volumetric flow rate of slurry at the inlet of the mixing tank (m3 /s); 3 is the density of slurry at the outlet of the mixing tank (Kg/m ); is the volumetric flow rate of slurry at the outlet of the mixing tank (m3 /s); is the total mass of water in the mixing tank (Kg); 3 is the density of water (Kg/m ); is volumetric flow rate of water at the inlet of the mixing tank (m3 /s); is the level of slurry in the mixing tank (m); is the diameter of the mixing tank (m).

A Study on the Dynamic Behavior of a Slurry Mixing and Pumping Process

133

It is assumed that the mixing tank is perfectly mixed. The perfect mixing assumption is valid for low-viscosity liquids that receive an adequate degree of agitation [21]. Then, the outlet density is given by: ds2 =

4 2 L πDm m

(Mpm + Mwm )

(33)

From Eqs. (31), (32) and (33), the formula describing the density variation is derived as follows:

 d(ds2 ) 4 (dw − ds2 )Fw + (ds1 − ds2 )Fs1 = 2 dt πDm Lm

2.3

(34)

Global Model

The global model of the system shown in Fig. 3 is deduced from Eqs. (32), (34), (30) and (3) as follows: dLm 4 = (Fs1 + Fw − Fs2 ) 2 dt πDm

 d(ds2 ) 4 = (dw − ds2 )Fw + (ds1 − ds2 )Fs1 2 dt πDm Lm

 dFs2 2 = CF (ah − β − γf )Fs2 + bh ωp Fs2 + ch ωp2 − α dt dωp 1 Cf 2 = (Te + ds2 at Fs2 − ds2 bt Fs2 ωp − ct ωp2 ) − ωp dt Jm + Jp Jm + Jp

(35)

(36) (37) (38)

The model takes into consideration the density and the friction factor variations and assume that the mixing tank is perfectly mixed. The model inputs and outputs shown in Fig. 3 are as follows: the model manipulated inputs are

Fig. 3. The mixing unit block diagram.

134

R. Oulhiq et al.

the slurry flow rate (Fs1 ) and the water flow rate (Fw ); The model disturbance input is the slurry density (ds1 ); The model outputs are the mixing tank level (Lm ), the slurry density (ds2 ), the slurry flow rate (Fs2 ) and the induction motor speed (wp ).

3

Model Simulation and Validation

In this part, numerical simulation is conducted using Matlab/Simulink environment. The simulation blocks are created based on Level-2 Matlab S-functions. The simulation parameters are obtained from a real industrial unit. To validate the model, historical data of the same unit is used. Additionally, the simulation results are exposed. 3.1

Simulation Parameters

The used parameters in this simulation are listed in Table 2. These parameters are the real parameters of the studied industrial unit. The unknown parameters ah , bh and ch in (1)) are determined using the pump characteristic curve shown in Fig. 2. In this part, three different operating points are chosen in order to get three equations with the three variables [19]. Figures 4 and 5 show the estimated H-Q curve of the pump compared to the real H-Q curve. It is clear that the estimated H-Q curve matches the real one. The parameters at , bt and ct of the load torque Tp in Eq. (2) are determined using the Eq. (42) [11] and the efficiency characteristic curve depicted in Fig. 5. Otherwise, the efficiency of the pump is defined by the equation: η=

Ph Pp

(39)

where Pp is the brake horse power and Ph is the hydraulic horse power. The hydraulic horse power Ph is the energy delivered to the fluid per time unit: Ph = ds2 gFs2 Hp

(40)

The brake horse power Pp is the actual power delivered to the pump shaft and it is related to the load torque and the shaft speed by the equation: Pp = ωp T p

(41)

Moreover, the load torque is derived using Eqs. (39), (40) and (41) as follows: Tp =

ds2 gFs2 Hp ηωp

(42)

Furthermore, using Eq. (42)) and the efficiency characteristic curve shown in Fig. 5, three different operating points are chosen in order to get three equations with the three variables [19].

A Study on the Dynamic Behavior of a Slurry Mixing and Pumping Process H-Q curve of the system

110

n = 300 n = 400 n = 500 n = 600 n = 700 n = 800 n = 900 n = 1000 n = 1100 Hsys

100 90 80 70

Head (m)

135

60 50 40 30 20 10 0

0

200

400

600 Flow (m3/h)

800

1000

1200

Fig. 4. The estimated H-Q curve of the pump.

Fig. 5. The real H-Q curve of the pump.

Electromechanical torque as a function of motor speed

8

f = 30 m f = 40 m f = 50 m f = 60 m f = 70 m

7

Torque (N.m)

6 5 4 3 2 1 0

0

200

400

600

800 Speed (rpm)

1000

1200

1400

Fig. 6. The induction motor speed-Torque curve.

The induction motor parameters t0 , t1 and t2 given in Eq. (5) are determined graphically based on the motor speed-torque curve. Three operating points (speed, Torque) are chosen to get three equations with the three unknowns. The equations are solved and the parameters are determined. Figure 6 shows the characteristic curve of the simulated induction motor.

136

R. Oulhiq et al.

Table 2. Simulation parameters.

3.2

Parameter Value

Unit

Parameter Value

Unit

z2

11

m

Dp

0.150

m

z2 

11

m

Dp

0.150

m

1.5

–

g

9.81

m/s2

Jm

5.24

Kg m2

p

6

–

Ki

2

CJ

0.7

Kg m

Cf

0.062

Kg m2 /s

Vmin

165

V

Vmax

660

V

fmin

12.5

Hz

fmax

50

Hz

La−2

29

m

φ

8.1

–

dp

2900

Kg/m3

df

1000

Kg/m3

Dm

14.5

m

ah

−0.000021 s2 /m5

bh

−0.0000016 s /m

ch

0.000075

s2 /m

at

−25.80

N s2 /(kg m2 ) bt

−0.0118

N s2 /kg

2

3

ct

0.1299

N s /m

t0

0.1875

N m/V2

t1

1.4375

–

t2

0.0182

–

2

Simulation and Validation

As mentioned earlier, the simulation is performed using the Matlab/Simulink software. The simulation blocks are created based on Level-2 Matlab s-functions as shown in Fig. 7. S-functions define how a block works during different parts of simulation, such as initialization, update, derivatives, outputs and termination [16]. Besides the characteristic of readability and felexibility [6], s-function blocks can be used for continuous, discrete or hybrid systems [3]. In addition, the Matlab control system toolbox which is abundant and comprehensive afford the capability to connect the simulated models to control systems [8].

Fig. 7. The mixing and pumping unit simulation using Matlab/Simulink.

A Study on the Dynamic Behavior of a Slurry Mixing and Pumping Process

137

The s-function blocks are based on a set of inputs and outputs. The developed model inputs are the slurry flow rate (Fs1 ), the water flow rate (Fw ), the slurry density (ds1 ) and the VFD setpoint (M V ). The outputs are the mixing tank level (Lm ), the slurry density (ds2 ), the slurry flow rate (Fs2 ) and the induction motor speed (wp ). Mixing tank density

1550

Real Model

1540 1530

Density (Kg/m3)

1520 1510 1500 1490 1480 1470 1460 1450

0

0.5

1

1.5

2

2.5

3

Time (s)

104

Fig. 8. Comparison between real density and model density. Mixing tank level

12

Real Model

11.5

Level (m)

11

10.5

10

9.5

9

0

0.5

1

1.5

2

2.5

Time (s)

Fig. 9. Comparison between real level and model level.

3 104

138

R. Oulhiq et al.

Considering the model validation, industrial data of a mixing and pumping unit is used with a sampling time of 1 s. To evaluate the model the real inputs are injected into the simulation and the outputs are compared to the real outputs. Induction Motor Speed

450

370

Real Model

360 350 340 4800

4900

5000

6000

Speed (rpm)

400

350

300 1000

2000

3000

4000

7000

8000

9000

10000

Time (s)

Fig. 10. Comparison between real motor speed and model motor speed.

Figure 8 shows the result of the estimated slurry density compared to the real density. The trend between the simulation and experiment results is almost the same with an acceptable error. This error is due to the simplification in the mixing tank dynamics model by considering a perfect mixing. In terms of the mixing tank level and the induction motor speed, as shown in Figs. 9 and 10 respectively, the simulation results follow the experiment results quite closely. For the outlet flow rate shown in Fig. 11, the simulation results has similar trend with the experimental results with a slight difference in the magnitude of the seasonal component. This is explained by the existence of model uncertainties due to unmodeled and unmeasured disturbances. 3.3

Simulation Results

As seen in the modeling part, density and level may be considered as measured disturbances affecting the slurry flow rate. In this part, the effect of density and level variations in the flow dynamic behavior is studied. This is being done by means of two different tests, one for density and the other one for level. As a result, their effects on the system response gain and time constant are evaluated.

A Study on the Dynamic Behavior of a Slurry Mixing and Pumping Process

139

Pump Flow

170

Real Model

160 150

Flow (m3/h)

140 130 120 110

140 130

100

120 110

90

2000 2100 2200 2300

80 1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Time (s)

Fig. 11. Comparison between real pump flow and model pump flow.

Density Effect on the Flow Dynamic Behavior. In this case, the slurry density is varied keeping the other inputs constants. Figure 12 shows the effect of the slurry density on the system outputs. In the manipulation, the slurry densi3 3 ties from 1800 Kg/m to 2000 kg/m are tested. This increase in the outlet slurry density increases the system head. The rise in the system head decreases the outlet slurry flow rate. Then, as the flow decreases the load torque decreases and the induction motor speed increases. Thus, a change in slurry density impacts inversely the outlet slurry flow rate. Which is consistent with fact that the higher is the density, the greater is the resistance to flow [2]. Considering the flow response as a first order system, Fig. 13 shows the effect of the density change on the gain and the time constant. The gain of the system decreases as the density increases whereas the time constant increases by the increase of the density. This is explained by the inertia of the slurry inside the impeller. As the density of the slurry increases the inertia of the slurry increases. Level Effect on the Flow Dynamic Behavior. To study the effect of the mixing tank level on the system outputs, the initial level is varied. Figure 14 shows the effect of varying the mixing tank level on the system outputs. The mixing tank level is increased from 3 m to 10 m. This results in an increase in the outlet slurry flow rate that is explained by a change in the static head of the system. This increase of the flow increases the load torque of the pump which in return decreases the speed of the pump. Figure 15 shows the effect of varying level on the gain and the time constant of the flow response. As seen with the

R. Oulhiq et al.

Density

2000

1500

1000 150

200

250

Head (m)

Torque (N.m)

50

200

600 500 400 250

20

200

Time (s) Speed

650

300

25

15 150

300

250

Time (s) System head

30

700

200

60

40 150

300

Time (s) Load torque

150

MV

70

MV (%)

Density (Kg/m3)

140

250

300

Time (s) Flow

350

Flow (m3/h)

300

550

250

200

ds2 = 1200 kg/m3

500

ds2 = 1500 kg/m3

150 450 150

200

250

ds2 = 1800 kg/m3

300

200

Time (s)

250

300

Time (s)

Fig. 12. Density effect on the flow dynamic behavior.

Gain as a function of density

8.5

18

16 Time constant (s)

7.5 7 6.5 6

15 14 13 12

5.5 5 1200

Time constant as a function of density

17

8

Gain (m)

Speed (rpm)

600

11 1300

1400

1500

1600 3

1700

1800

10 1200

1300

1400

1500

1600 3

Fig. 13. Density effect on the flow gain and time constant.

1700

1800

Level

14 12 10 8 6 4 2 150

200

250

60 50 40 150

300

200

700 600 500 400 180

200

220

240

250

260

280

300

25 20 15 150

200

250

Time (s) Speed

300

Time (s) Flow

340

600

320

580

300 Flow (m3/h)

560 540 520

280 260 240 220 Lm = 4 m Lm = 7 m Lm = 10 m

200

500

180 480 180

200

220

240

260

160 150

280

200

250

Time (s)

300

Time (s)

Fig. 14. Level effect on the flow dynamic behavior.

Gain as a function of level

Time constant as a function of level

14.6 14.4

7

14.2 Time constant (s)

Gain (m)

Speed (rpm)

300

Time (s) System head Head (m)

torque (N.m)

Time (s) Load torque

160

141

MV

70

MV (%)

Level (m)

A Study on the Dynamic Behavior of a Slurry Mixing and Pumping Process

6.5

6

14 13.8 13.6 13.4 13.2 13

5.5

12.8 3

4

5

6

7

level (m)

8

9

10

3

4

5

6

7

level (m)

Fig. 15. Level effect on the flow gain and time constant.

8

9

10

142

R. Oulhiq et al.

density effect, the gain of the system is inversely impacted by the variation of the mixing tank level whereas the time constant increases as the level increases.

4

Conclusion

In this work, a dynamic model of a mixing tank with a slurry centrifugal pump is proposed. The centrifugal is modeled based on its different parts; the hydraulic application, the induction motor and the system head. The hydraulic part is modeled taking into account the density of the pumped slurry. Concerning the induction motor, a graphical method is used for modeling its dynamic behavior. The system head is then detailed taking into account the type of the pumped slurry either homogeneous or heterogeneous. The simulation of the process is conducted using level 2 Matlab s-functions and Simulink environment. To validate the model, industrial data of a mixing and pumping process is used. Additionally, the effects of density and level variations on the flow dynamic behavior are studied. According to the simulation results, besides the induction motor speed effect, the slurry flow rate is also affected by the slurry density and the suction level. The change in slurry density and level impacts indirectly the slurry flow rate by affecting the load torque and/or the system head. For the system response, the gain decreases as density and level increase whereas the time constant increases with the increase of density and level. The used approach in developing the dynamic simulation makes full use of the advantages of Matlab software. This approach not only increases the developing efficiency and flexibility greatly but also offers the possibility to connect control algorithms easily to the simulated models. Eventually, this model can be used as a digital twin of the real process either to test new control strategies or to train operators in skills they will need in their daily tasks. For future work, further research will be pursued to propose new methods to control slurry pumping flow rate in presence of model uncertainties and fluctuating disturbances.

References 1. Aree, P.: Analytical approach to determine speed-torque curve of induction motor from manufacturer data. Procedia Comput. Sci. 86, 293–296 (2016) 2. Blevins, T.L., Nixon, M.: Control Loop Foundation: Batch and Continuous Processes. ISA, Pittsburgh (2010) 3. Chakravarty, S.: Technology and Engineering Applications of Simulink. InTech, London (2012) 4. Chan, T.F., Shi, K.: Applied Intelligent Control of Induction Motordrives. Wiley, Hoboken (2011) 5. Deng, S.Y.: Nonlinear & linear MIMO control of an industrial mixing process. Master’s thesis, McGill University (2002) 6. Du, W.: Informatics and Management Science. Springer, London (2013) 7. El-Emam, N., Kamel, A., El-Shafei, M., El-Batrawy, A.: New equation calculates friction factor for turbulent flow on non-newtonian fluids. Oil Gas J. 101(36), 74 (2003)

A Study on the Dynamic Behavior of a Slurry Mixing and Pumping Process

143

8. Gan, D.S.H.Y., Gromiha, P.G.M.M.: Advanced Intelligent Computing Theories and Applications. Springer, Heidelberg (2010) 9. Ghafouri, J., Khayatzadeh, H., Khayatzadeh, A.: Dynamic modeling of variable speed centrifugal pump utilizing matlab/simulink. Int. J. Sci. Eng. Invest. 1(5), 1–7 (2012) 10. Hougen, O.A., Watson, K.M., Ragatz, R.A.: Chemical Process Principles: Material and Energy Balances, vol. 1. Wiley, Hoboken (1954) 11. Isermann, R.: Mechatronic Systems: Fundamentals. Springer, Heidelberg (2007) 12. Kallesøe, C.S., Cocquempot, V., Izadi-Zamanabadi, R.: Model based fault detection in a centrifugal pump application. IEEE Trans. Control Syst. Technol. 14(2), 204–215 (2006) 13. Kallesøe, C.S., Izaili-Zamanabadi, R., Rasmussen, H., Cocquempot, V.: Model based fault diagnosis in a centrifugal pump application using structural analysis. In: Proceedings of the 2004 IEEE International Conference on Control Applications, 2004. vol. 2, pp. 1229–1235, September 2004 14. Kallesøe, C.S.: Fault detection and isolation in centrifugal pumps. Ph.D. thesis, Aalborg University, Institute of Electric Systems (2005) 15. Manekar, G., Bodkhe, S.B.: Modeling methods of three phase induction motor. In: National Conference on Innovative Paradigms in Engineering & Technology, pp. 16–20 (2013) 16. MATLAB: What is an S-function? The MathWorks Inc. (2020) 17. Menon, S.: Piping Calculations Manual. McGraw Hill Professional, New York (2004) 18. Miedema, S.: Modeling and simulation of the dynamic behavior of a pump/pipeline system. In: 17th Annual Meeting & Technical Conference of the Western Dredging Association. New Orleans (1996) 19. Oulhiq, R., Benjelloun., K., Saad., M., Kali., Y., Deshayes., L.: Dynamic modeling and simulation of a slurry mixing and pumping process: an industrial case. In: Proceedings of the 9th International Conference on Simulation and Modeling Methodologies, Technologies and Applications SIMULTECH, vol. 1, pp. 27–35, July 2019 20. Peube, J.L.: Fundamentals of Fluid Mechanics and Transport Phenomena. ISTE, London (2009) 21. Seborg, D.E., Mellichamp, D.A., Edgar, T.F., Doyle III, F.J.: Process Dynamics and Control. Wiley, Hoboken (2010) 22. Shamlou, P.A.: Processing of Solid-Liquid Suspensions. Elsevier, Amsterdam (2016) 23. Skogestad, S.: Chemical and Energy Process Engineering. CRC Press, Cambridge (2008) 24. Southard, M.Z., Green, D.W.: Perry’s Chemical Engineers’ Handbook. McGrawHill Education, New York (2018) 25. Trzynadlowski, A.M.: Control of Induction Motors. Elsevier, Amsterdam (2000) 26. Valtr, J.: Mass flow estimation and control in pump driven hydronic systems. Master’s thesis, Czech Technical University, Czech Republic (2017) 27. Wilson, K.C., Addie, G.R., Sellgren, A., Clift, R.: Slurry Transport Using Centrifugal Pumps. Springer, Boston (2006)

Dynamic Simulation of Two Kinds of Hydraulic Actuated Long Boom Manipulator in Port-Hamiltonian Formulation Lingchong Gao1(B) , Mei Wang2 , Haijun Peng3 , Michael Kleeberger1 , and Johannes Fottner1 1 Chair of Materiels Handling, Material Flow, Logistics, Technical University of Munich, Garching 85748, Germany [email protected] 2 Chair of Automatic Control, Technical University of Munich, Garching 85718, Germany 3 Department of Engineering Mechanics, Dalian University of Technology, Dalian 116023, People’s Republic of China

Abstract. The boom systems of mobile cranes and aerial platform vehicles can be described as hydraulic actuated long boom manipulators. The purpose of this paper is to develop a complete mathematical model for such a boom system which is a multi-domains system consisting of the boom structure and hydraulic drive system. The hydraulic system and the boom structure are described in the port-Hamiltonian formulation. The port-Hamiltonian systems can be easily interconnected through energy exchanges, thus allowing the description of a complex system as a composition of subsystems. The structure of the long boom manipulator is specified as two main types, telescopic boom, and folding boom. These two boom types are correspondingly simplified as rotational non-homogeneous Timoshenko beam and double rotational Timoshenko beams. A structure-preserving discretization for the Timoshenko beam model is applied to transfer the boom model from infinite into finite. Then the interconnections between the hydraulic model and discretized boom structure model are illustrated and simulations of two types of long boom manipulators are accomplished in MATLAB/Simulink. Keywords: Port-Hamiltonian system · Structure-preserving discretization · Hydraulic cylinder · Telescopic boom · Folding boom

Supported by Deutsche Forschungsgemeinschaftand (DFG) and National Science Foundation of China (NSFC). c Springer Nature Switzerland AG 2021  M. S. Obaidat et al. (Eds.): SIMULTECH 2019, AISC 1260, pp. 144–166, 2021. https://doi.org/10.1007/978-3-030-55867-3_8

Dynamic Simulation

1 1.1

145

Introduction Technical Background

In order to lift heavy loads with large radius and heights for material handling tasks, or to reach high altitude operation site, the boom systems of mobile cranes and aerial platform vehicles are normally designed as, long boom manipulators. The boom structure is usually designed as light as possible with telescopic or folding mechanism due to the concern of self-weight reduction and the grantee for mobility and transportability. Such a boom structure with limited stiffness always performs a strong flexible behavior, especially in the cases with heavy loads or high acceleration. The hydraulic system consists of hydraulic actuators along with an electro-hydraulic servo system that is equipped to control the movements of the boom structure. Because of the elastic hydraulic oil and flow characteristics of hydraulic components, the dynamic response of the hydraulic system during start-up and braking stages of the operations could affect the boom structure significantly. This kind of hydraulic actuated long boom manipulator can lead to a hybrid system with complex dynamic behavior. The flexibility of the boom structure can cause intense vibration responses when there are load or motion changes for the whole system. These unstable dynamic responses could result in strong oscillation at the boom tip and compromise the location accuracy or personal safety. There are some research achievements about the dynamic analysis of the boom system of mobile cranes and fire-rescue turntable ladder. Some researchers focus on the dynamic analysis of the boom structure, some others have investigated the analysis of the whole boom system including hydraulic system and use hydraulic actuators to control the structure vibration. Sun added the equations of a hydraulic drive system including essential hydraulic components into the formulation of the finite element model of boom structure of the mobile crane and obtained a complete model to describe the dynamic behavior of the hydraulic actuated boom system [1]. This method has been tested on the calculation of slewing, lifting and luffing operations of lattice boom cranes [2,3], and the structure and hydraulic system were described in the complete model with details. Focusing on the long fire-rescue turntable ladder, Sawodny modeled its boom system (lattice ladder) as a flexible multi-body system [4] and the hydraulic drive system was included in the mathematical model as a set of equations [5]. In the work of Pertsch [6], a Euler-Bernoulli beam model in low dimensional space for the fire-rescue turntable ladder was derived. In their recent work [7], they considered the rotational motion and developed a model of an aerial ladder for the coupled bending-torsional vibration. An active vibration damping control has been developed and validated on real equipment. Nguyen developed a multi-body dynamic model using a chain of rigid bodies connected with rotational springs and dampers to reflect the flexibility of the telescopic ladder, he included a pre-tensioned rope and investigated its effect to the vibration at the tip of the ladder [8].

146

L. Gao et al.

Fig. 1. Hydraulic actuated long boom manipulators.

Our current research force is on the telescopic boom systems used by mobile cranes and aerial platform vehicles. They can be classified as two main kinds of long boom manipulators as shown in Fig. 1. The left one is a telescopic boom with a platform attached at the boom’s tip, which is a typical boom system for smaller aerial platform vehicles. The middle one is for mobile cranes, there are sets of pulleys with rope passing though to act as a hoisting mechanism. The number of the sections of the telescopic boom can be up to eight for large mobile cranes. The larger aerial platform vehicles which are designed for the tasks of fire fighting or rescue mission for high building, usually equip the folding telescope boom system as the right one in the illustration. According to the fact that the length dimension of a telescopic boom is larger than the dimension of its cross-section and the assumption of small linear deflection dynamic respond, the objects can be summarized as the dynamic problem of rotation beam. For the single boom system, the differences of the boom section are considered as the different parameters of the cross-section. The boom model is considered as continuum beam with the ignorance of the overlap between each pair of the boom sections. For the double boom system, the boom model can be derived as double rotation beams with the same assumption as a single boom system. All the rotation motions are driven by the individual luffing mechanism which is controlled by the output of the hydraulic cylinder. The hydraulic actuated long boom manipulators can be regarded as a multiphysical system consisting of the mechanical structure, hydraulic drives, and electrical control system. Such a hybrid system consisting of subsystems of different domains is more complex than a single domain system due to the different dynamic characteristics of different domains, and it also arises the difficulty of control design. Many industry applications have strong demands for the improvement of performance and accuracy, these lead to a requirement of a better controller for the hydraulic actuated system. For example, in the steel industry, the rolling mills require a very strict thickness tolerance. Kugi resented a control strategy for the active compensation of the roll eccentricity based on the

Dynamic Simulation

147

system model including the hydraulic cylinder and some modern control theories such as the factorization approach and the projection theorem were applied [9]. There is a kind of modeling method for the multi-domain system called the bond-graph method [10] which is also used for hydraulic actuated systems, such as hydraulic motion simulator [11]. The bond-graph is a graphical approach to illustrate the energy transformation through the energy ports with the bonds of power-conjugate variables, efforts and flows, and the energy conversion between components and different physical systems. The mathematical representation of a physical system described as a bond-graph can lead to a formulation called port-Hamiltonian (PH) system. 1.2

Methodology: Port-Hamiltonian System

The reason why the port-Hamiltonian framework is a proper modeling method for multi-domain systems comes from the important property of Dirac structure. A crucial property of the Dirac structure is that the standard interconnection of Dirac structures is again a Dirac structure [12]. On the basis of this principle, it is allowed to model different domains or subsystems separately, and then assemble into a complete system model with standard interconnection. The space of the power variables is F ∗ E and it can be defined by a linear space F (space of generalized velocities or flows) with its dual denotes as E = F ∗ (space of generalized forces or efforts) [12]. According to the definition given by Duidam [12], there exists a canonically defined symmetric bilinear form on F ∗ E  (f1 , e1 ), (f2 , e2 ) := e1 | f2  + e2 | f1  ,

(1)

where fi ⊂ F , ei ⊂ E and  ., .  denotes the duality product between F and its dual space E . A constant Dirac structure on F is a linear subspace D ⊂ F × E with the property D = D ⊥ [12]. A multi-domain system which can be formulated as the subsystems described as Dirac structure connecting by ports, is still a Dirac structure. A suitable space of power variables and the Dirac structure defined on it are the foundation to definite a port-Hamiltonian system. The system energy function Hamiltonian H (x ) is used to build the mathematical model of the system. The state space variables x reflect the system flow variables by the definition of x˙ = f and the system effort variables e are given by the coenergy variables ∂H (x )/∂x . All the subsystems can be described in the form of ∂H (x)) + G(x)u x˙ = (J (x) − R(x))( ∂x (2) ∂H (x)) y = G (x)( ∂x which is a useful formulation because the matrix J (x) is a skew-symmetric matrix and the components of J (x) are smooth functions of the state variables. The matrix R has to be symmetric and positive semi-definite R = R ≥ 0. A dynamic system with such formation is called a port-Hamiltonian system. The port-Hamiltonian formulation has already been applied to describe the dynamic behavior of the flexible beam and to design the corresponding control

148

L. Gao et al.

strategies. In the work of Maccheli [13], the transitional and rotational deflections and momenta of a Timoshenko beam were chosen as state variables to build the framework of the port-Hamiltonian system. He also extended the portHamiltonian approach with the power conserving interconnection to the multibody system [14]. A structure-preserving discretization method is essential to transfer the infinite-dimensional model in the port-Hamiltonian formulation to a solvable finite-dimensional model and the property of Dirac structure is preserved in the discretized model [15,16]. Wang discretized the port-Hamiltonian formulation of a planar Timoshenko beam into a finite-dimensional model with a geometric pseudo-spectral discretization method and designed a feed-forward motion control strategy based on the discretized model [17]. Bo considered a double flexible manipulator as two connected Euler-Bernoulli beams and designed an energy-based controller bases on the port-Hamiltonian formulation with generalized canonical transformations [18]. Some researchers also investigated the application of port-Hamiltonian formulation on the modeling and controller design of the hydraulic system. Kugi designed a nonlinear controller for a classical hydraulic piston actuator system based on the port-Hamiltonian model [19]. In his doctor thesis [20], Stadlmayr presented a port-Hamiltonian representation of a hydraulic actuated manipulator with a flexible boom structure and designed a MIMO-controller with feedforward and feedback to accomplish path following and vibration suppression [21]. Based on the current research achievements and our previous work, we investigate the dynamic behavior of the planar motion of two kinds of long boom manipulators with a hydraulic drive system in the port-Hamiltonian formulation. The boom structure is considered as planar Timoshenko beams particularly, and original Timoshenko beam equations are derived to fit the purpose of large rigid body displacement.

2 2.1

System Description Port-Hamiltonian Model of 2-D Timoshenko Beam

A Rotational Homogeneous Timoshenko Beam with Displacement. During the luffing operation, the boom structure can be simplified as an ideal beam model in a plane. The axial loads can be neglected, such as the inertial force and pressure force along the axis, due to the stiffness on these directions are rather larger and the corresponding deflections can be also ignored. With the purpose to obtain a suitable port-Hamiltonian representation of the boom structure, we start with the formulation of a rotational homogeneous Timoshenko beam with small displacement to describe the dynamic behavior of the boom structure. In Fig. 2, t is the time variable and z is the spatial coordinate along the equilibrium position of the beam, and we (z, t) is the lateral deflection of the beam from the equilibrium position and Ψe (z, t) is the rotation of the beam’s cross section.

Dynamic Simulation

149

Fig. 2. A rotational homogeneous Timoshenko beam with displacement at bottom joint.

The boundary condition of the boom structure can be considered as a free tip with a fixed end, and the fixed end rotates around a fixed axis with the angle displacement θ(t) and has a small displacement along axis Y . Then the original Timoshenko beam model is transferred to a rotating Timoshenko beam with small displacement. We use the trajectory length of the point on the beam and the rotation angle in the global coordinate as new system variables with the definition as w(z, t) = we (z, t) + z · θ(t) + d(t) · cos θ(t) (3) ψ(z, t) = ψe (z, t) + θ(t) which still fulfill the original PDEs in a new form 2

2

∂ψ ∂ w ρ ∂∂tw 2 − K ∂z 2 + K ∂z = 0   2 2 Iρ ∂∂tψ2 − EI ∂∂zψ2 + K ψ − ∂w ∂z = 0

(4)

In (4), the coefficients ρ, Iρ , E, I are the mass per unit length, the mass moment of inertia of the cross section, Young’s modulus and the moment of the inertia of the cross section respectively. And in the coefficient K = kGA, G is the modulus of elasticity in shear, A is the area of cross section and k is a constant depending on the shape of the cross section. Port-Hamiltionian Model of Timoshenko Beam. For a homogeneous Timoshenko beam, the coefficients are constant and its mechanical energy is given as following form:  HB (t) =

Hdz 0

H=

L

 1 2 2 2 ρw˙ + Iρ ψ˙ 2 + K (ψ − ∂z w) + EI (∂z ψ) . 2

(5)

According to the mechanical energy formulation (5) the potential elastic energy is the function of the shear and bending deformations, which can be

150

L. Gao et al.

written as:

εt (z, t) = ∂z w(z, t) − ψ(z, t) εr (z, t) = ∂z ψ(z, t).

(6)

The associated co-energy variables are shear force Kεt (z, t) and bending momentum EIεr (z, t). The kinetic energy is the function of the translational and rotational momenta which are given as ˙ t), pt (z, t) = ρw(z, ˙ t), pr (z, t) = Iρ ψ(z,

(7)

and the co-energy variables are translational velocity (pt (z, t))/ρ and rotational velocity pr (z, t)/Iρ . According to the definition of new state variables, the original PDEs can be rewritten in a form ⎡ ⎤ ⎡ ⎤⎡ ⎤ 0 0 ∂z 0 p˙t δpt H ⎢ p˙r ⎥ ⎢ 0 0 1 ∂z ⎥ ⎢ δpr H ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎥ (8) ⎣ ε˙t ⎦ ⎣ ∂z −1 0 0 ⎦ ⎣ δεt H ⎦ . ε˙r 0 ∂z 0 0 δεr H We denote e ∈ E , f ∈ F as the effort and flow variables separately. They are related to the time derivative of state variables f = −x˙ and the associated co-energy variables as ⎡ ⎤ ⎡ p ⎤ ⎡ ⎡ p⎤ ⎤ p˙t e t δpt H f t ⎢ ⎥ ⎢ pr ⎥ ⎢ ⎢ f pr ⎥ ⎥ ⎥ = −⎢ p˙r⎥, e = ⎢ eε ⎥ = ⎢ δpr H ⎥ . f =⎢ (9) ε ⎣ ε˙t ⎦ ⎣ e t ⎦ ⎣ δεt H ⎦ ⎣ f t⎦ εr εr f ε˙r e δεr H The total energy (we neglect the gravity potential energy and just focus on the quadratic energy functions) becomes the following formulation:  H˙ B = 0

L



L

∂x H · x˙ dz = −

e  · f dz.

(10)

0

Applying integration by parts on Eq. (10), one obtains L H˙ B = (ept eεt + epr eεr )|0 .

Defining the boundary flow and effort variables as ⎡ t⎤ ⎡ p ⎤ e t |∂Z f∂ ⎢ f r ⎥ ⎢ epr | ⎥ ⎢ t∂ ⎥ = ⎢ ε ∂Z ⎥ , ⎣ e∂ ⎦ ⎣ e t |∂Z ⎦ er∂ eεr |∂Z

(11)

(12)

where e|∂Z denotes the restriction on the border of the domain Z = [0, L]. Comparing the right hand sides of (10) and (11), it is clear that the increase of the

Dynamic Simulation

151

total energy is equal to the power through the borders. And the power continuity and conservation equation is fulfilled as    e  · f dz + f∂t · et∂ + f∂r · er∂ ∂Z = 0. (13) Z

Finally, Eq. (8) can be rewrote shortly as − f = J (z)∂x H = J (z)e,

(14)

where J (z) is a skew-symmetric differential operator as that in Eq.(2). Using the definition of flow, effort variables (9) and the system state equations (14), one can define a bilinear geometric structure, a Dirac structure D: D = {(f , f0t , f∂r , e, et∂ , er∂ ) ∈ F × E |

(15)

− f = J (z)e and Eq.(11) holds }.

Structure-Preserving Spatial Discretization. In order to solve the distributed parameter model of a Timoshenko beam, we need to transfer its portHamiltonian model into a lumped-parameter model. The skew-symmetric differential operator has to be retained in the new discretized model, which means that the discretization method should preserve such property. The geometric discretization of a Timoshenko beam has proven to provide a good approximation of system properties in [15] and [17], such as the spectrum of the differential operators. Some essential steps in [17] and [23] are reviewed in this section. Based on the formulation (8), we rewrote the Dirac structure (15) as ⎡ ⎡ ⎤ ⎤ 0 0 1 0 ⎢ ⎢ 1 0⎥ 0 1⎥ ⎥ e∗ ⎥ ∂z e + ⎢ (16) −f=⎢ ⎣ 0 −1 ⎣1 0 ⎦ ⎦ 0 0 0 1 so we can classify the effort vectors as e and e∗ , depending on whether it is subject to differentiation or not.According to the pseudo-spectral method proposed for canonical systems of two conservation laws, we define different approximation bases for the flows f v ∈ {f pt , f pr , f εt , f εr } and the efforts ev ∈ {ept , epr , eεt , eεr }. f v (t, z) ≈

N −1  k=0

fkv ϕk (z), ev∗ (t, z) ≈

N −1 

ev∗,k ϕk (z), ev (t, z) ≈

N 

k=0

evi φi (z).

i=0

(17)

The time dependent coefficients are collected in the vectors f v , ev∗ ∈ RN and e v ∈ RN +1 , v ∈ {pt , pr , εt , εr }

(18)

ϕk (z) and φi (z) are the basis functions for flows and efforts, and satisfying the exact differentiation or compatibility condition [16]: E = span {φ0 , . . . , φN } F = span {ϕ0 , . . . , ϕN −1 } ∂x E = F .

(19)

152

L. Gao et al.

Then we chose the interpolation Lagrange polynomials of degree N and N −1 as suitable bases functions for the efforts and flows: ϕk (z) =

N −1  j=0,j=k

z − zj , φi (z) = zk − zj

N  j=0,j=i

z − ζj . ζi − ζj

(20)

zk ∈ (0, L), k = 0, . . . , N − 1 and ζi ∈ (0, L), i = 0, . . . , N are the collocation points for ϕk (z) and φi (z) respectively.  Denote φ = [φ0 , . . . , φN ] the vector of effort basis functions and   φ(0) Φ= . (21) φ(L)     t/r t/r t/r t/r t/r t/r and e∂ = e0 , eL be the vectors of boundary Let f∂ = f0 , fL flows and efforts corresponding to translational or rotational motion. Inserting (17) into (16) and (12), one obtains the linear system of equations ⎡ p ⎤ ⎡ ⎤⎡ p ⎤ ⎡ ⎤ ⎡ pt ⎤ D 0 00 f t e t e∗ ⎢ f pr ⎥ ⎢ ⎥ ⎢ epr ⎥ ⎢ ⎥ ⎢ ep∗r ⎥ 0 D I 0 ⎥ ⎢ ⎥⎢ ε ⎥ + ⎢ ⎥⎢ ε ⎥ −⎢ ⎣ f εt ⎦ = ⎣ D 0 ⎦ ⎣ e t ⎦ ⎣ 0 −I ⎦ ⎣ e∗t ⎦ εr εr f e eε∗r 0 D 0 0 ⎡ t⎤ ⎡ ⎤⎡ p ⎤ (22) Φ f∂ e t ⎢fr ⎥ ⎢ Φ ⎥ ⎢ epr ⎥ ⎢ t∂ ⎥ = ⎢ ⎥⎢ ⎥. ⎣ e∂ ⎦ ⎣ Φ ⎦ ⎣ eεt ⎦ Φ et∂ eεr The elements of the derivative matrix D ∈ RN ×(N +1) are obtained from the spatial derivative of the effort bases functions at the flow collocation points: Dk+1,i+1 = ∂x φi (zk ) ,

(23)

with i = 0, . . . , N, k = 0, . . . , N −1. In accordance with the distributed parameter model, an additional coupling term with identity matrices I = IN appears on the right hand side of (23). Replacing the approximations of flows and efforts in the energy balance, we obtain   H˙ ≈ (ev ) M f v (24) v∈{pt ,pr ,εt ,εr }

with the elements of the non-square matrix M ∈ R(N +1)×N defined as  Mi+1,k+1 =

L

φi (z)ϕk (z)dz.

(25)

0

The right hand side of (24) consists of degenerated bilinear forms between the vectors of discrete flows and efforts. Due to the degeneracy (the kernel of

Dynamic Simulation

153

M is non-empty), this bilinear form does not qualify to define a Dirac structure on the finite-dimensional bond space of reduced flows and efforts Fr × Er with (ept , epr , eεt , eεr , et∂ , er∂ ) ∈ Er ∈ R4N +8 and (f pt , f pr , f εt , f εr , f∂t , f∂r ) ∈ Fr ∈ R4N +4 . It can, however, be shown that the power continuity equation (13) is approximated via     (ev ) M f v + (eu∂ ) M f∂u = 0. (26) v∈{pt ,pr ,εi ,εj }

u∈{t,r}

To obtain a non-degenerate power pairing, vectors of reduced effort variables e˜v ∈ RN are defined: e˜v = M T ev . (27) 

These shall be, we discretize the constitutive equation e = (δx H) , derived from a discrete energy. Note that x˙ = −f holds, i.e. states and flows are discretized with respect to the same basis. We can define x˙ v = −f v and replace the approximation in H = Z H dz with Hamiltonian density. We obtain   1 1 1 v v  v v , , Ks , Kb , c (x ) Sx , c ∈ (28) H≈ 2 v ρ Iρ where the elements of S ∈ RN ×N are given by 

L

Si+1,j+1 =

ϕi (z)ϕj (z)dz.

(29)

0

The required discretized constitutive relation is simply v

e˜ =



∂H ∂xv



= cv Sxv , ∀v.

(30)

On the other hand, the relation for the discretized effort vectors ev∗ in the flow (or state) bases becomes ev∗ = cv xv = S −1 ev = S −1 M  ev .

(31)

According to [15], the discretised Timoshenko beam can be formulated into an input-/output (I/O) representation:   0 J s1 f= (32) ·e J s2 0

154

with

L. Gao et al.

  t r , f εr , f∂L f = f pt , −et∂0 , f pr , −er∂0 , f εt , f∂L   t r e = e˜pr , f∂0 , e˜pr , f∂0 , e˜εt , et∂L , eεr , er∂L    −1 ⎡  ⎤ −D M 0 ⎢ ⎥ −φ φ 0 L ⎥ J s1 = ⎢ ⎣ −S −1 M    M  −1  −D   M  −1 ⎦ 0 −φ φ φ 0 L L    −1 ⎤    −1  ⎡ −1  −D M M −S M ⎥ ⎢ −φ φ0 0 φ L ⎥ ⎢ Js2 = ⎣    L−1 ⎦ −D M 0   −φL φ0

Indeed, Js1 = −Js2 , i.e. the interconnection matrix J ∈ R(4N +4)×(4N +4) is skew-symmetric. We obtain the explicit state space model in linear port-Hamiltonian form X˙ = J4N ×4N Q4N ×4N X + G4N ×4 U Y = G 4N ×4 Q4N ×4N X + D4×4 U .

(33)

The vector xv ∈ RN is merged in the overall state vector X ∈ R4N , Q4N ×4N = blockdiag {S/ρ, S/Iρ , Kb S, Ks S} is the overall energy (Hessian) matrix, and U ∈ R4 , Y ∈ R4 are the vectors of boundary inputs and collocated, power conjugated outputs. They are composed of the elements of the boundary flow an effort vectors. In the terms of the physical boundary variables we have ⎡ ⎤ ⎡ ⎡ ⎤ ⎡ ⎤ ⎤ v(0) Y1 −Q(0) U1 ⎢ Y2 ⎥ ⎢ −M (0) ⎥ ⎢ U2 ⎥ ⎢ ω(0) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ U (t) = ⎢ ⎣ U3 ⎦ = ⎣ Q(L) ⎦ , Y (t) = ⎣ Y3 ⎦ = ⎣ v(L) ⎦ . M (L) ω(L) U4 Y4 2.2

Port-Hamiltonian Model of Hydraulic System

State-Space Representation. The planar motion (luffing operation) of the hydraulic actuated manipulator is driven by the luffing cylinder(s). The luffing cylinder can be simply considered as the hydraulic cylinder actuator of Fig. 1. A 4/3-way proportional directional valve controls the movement of the asymmetric cylinder, which actuates the luffing movement of the boom system. The directional valve is connected with a pressure pump and a tank. The supply pressure pS is determined by the relief valve, and the tank pressure is pT . The volumetric flows through the directional valve, Q1 and Q2 , are simply given by Qi = kv



Δpi xv , i = 1, 2

(34)

Dynamic Simulation

155

Fig. 3. Diagram of the hydraulic system of a hydraulic cylinder [23]

with the position of the valve core xv , the pressure difference Δpi and the valve coefficient kv , which can be considered as constant when the properties of the flow through the valve ports are constant and the type of valve core is slide valve. Then, we assume that there is no internal or external leakage flows and the temperature remains constant. The continuity equations of the cylinder chambers can be described as d d (As) = Q1 , (αA(l − s)) = Q2 . dt dt

(35)

Using the linearized constitutive law of the constant (isothermal) bulk modulus Eoil ∂p (36) Eoil = ρ , ∂ρ we can rewrite Eq. (34) as a well-known formation   Eoil P p˙1 = −A + Q1 V01 + As m   Eoil P p˙2 = αA − Q2 V02 + αA(l − s) m

(37)

with the piston’s momentum P and its mass m. And the motion equations of the piston are considered as the following form P S˙ = m P˙ = (p1 A − p2 αA − F ) .

(38)

Eq. (38) and (37) constitute a state model of the valve-controlled hydraulic cylinder of Fig. 3 with the state vector x = [s, P, p1 , p2 ] .

156

L. Gao et al.

Energy Description of Isentropic Fluid. The fluid inside the chambers of the cylinder can be seen as isentropic with the assumption that no heat transfer exists between the environment and the considering system, the change of the energy only performs as the work of expansion. The system energy can be simplified as dU = −pdV. (39) Following the derivation procedures from [9] and with the definition of isothermal bulk modules, we get the specific energy of the fluid obeying the constitutive law (36) in the form: 

p0 +Eoil +u0 ρ0 −(p+Eoil )e((p0 −p)/Eoil )

u(p) = with u0 = u (p0 ).



ρ0

(40)

For the sake of convenience we will subsequently choose p0 = 0 and u0 = 0. Hence the internal energy results in M

      U (p) = V ρ(p) u(p) = V E oil e(p/E oil ) − 1 − p .

(41)

Using the above formulation, we can acquire the energy function of the fluid inside the hydraulic cylinder, which is essential to building the Hamiltonian function of the system. Port-Hamiltonian Representation of Hydraulic Cylinder. The specific internal energy u(p) of the fluid in the two chambers of the hydraulic cylinder can be determined by (40), then the energy stored in the hydraulic cylinder of Fig. 1 is given by    pi UHydr = V1 ρ1 u (p1 ) + V2 ρ2 u (p2 ) = Eoil Vi e(pi /Eoli ) − −1 Eoil (42) i=1,2 V1 = V01 + As, V2 = V02 + αA(l − s), the subscripts refer to the corresponding quantities of the chamber 1 and chamber 2 (rob side). Assuming that the kinetic energy of fluid mass can be neglected compared to the kinetic energy of the piston, the total energy Ec of the hydraulic cylinder is P2 . (43) Ec = UHyd + 2m We can use the total energy Ec as the Hamiltonian function Hc , then the port-Hamiltonian model of the hydraulic cylinder takes the form as (2) with the input vector u = [F, Q1 , Q2 ] , the the state vector x = [s, P, p1 , p2 ] and! hydraulic cylinder can be described as 0  x˙ = J (x)∂x Hc + g(x)u (44) Σ0 : y = g(x) ∂x Hc

Dynamic Simulation

The matrices J (x) and g(x) are ⎡ ⎤ 0 1 0 0 ⎢ −1 0 Eoil A/V1 −Eoil αA/V2⎥ ⎥ J (x) = ⎢ ⎣ 0 −Eoil A/V1 ⎦ 0 0 0 0 0 Eoil αA/V2 ⎡ ⎤ 0 0 0 ⎢ −1 ⎥ 0 0 ⎥ g(x) = ⎢ ⎣ 0 Eoil /V1 ⎦ 0 0 0 −Eoil /V2

157

(45)

(46)

In order to illustrate the system output y, more details of the model is given by

⎡

⎤ αA (Γ2 ) − A (Γ2 ) ⎢ ⎥ ∂Hc ⎢  (pP/m  ⎥, 1 /Eoil ) ∂x = ⎣ V − 1 ⎦ 1 e (p2 /Eoil ) −1 V2 e with Γi = pi − E oil e(pi /Eoil ) + Eoil , i = 1, 2. Then using the definition (41), the system output is ⎡ ⎤ 0 −1 0 0 ∂H c ⎦ ∂Hc 0 = ⎣ 0 0 Eoil /V1 y = g(x) ∂x ∂x 0 0 0 −Eoil /V2 ⎤ ⎡ ⎤ ⎡ −v  −p/m  = ⎣ Eoil e(p1 /Eoil ) − 1  ⎦ = ⎣ ρ1 h1 ⎦ −ρ2 h2 −Eoil e(p2 /Eoil ) − 1

(47)

(48)

The variables h1 and h2 are the mass specific enthalpy h = H/M = u + pv in the both chambers of the cylinder. The change of the system energy can be obtained by the product of input u and output y as dEc = y  u = h1 ρ1 Q1 −h2 ρ2 Q2 −v · F     dt ˙1 M

(49)

˙2 M

In Eq. (49), the first two parts represent the energy changed by the fluid mass flows to or from the chambers respectively, M˙ 1 or M˙ 2 , associated with the corresponding enthalpy in the two chambers h1 or h2 , and the third part represents the work transferred to the boom structure by cylinder force F . This reflects the energy conservation of isentropic fluid by the energy transfer between the mechanical energy and the energy stored in the fluid.

158

3 3.1

L. Gao et al.

Port-Hamiltonian Model of the Hydraulic Actuated Long Boom Manipulators Modeling of the Telescopic Boom Structure

One distinguishing feature of telescopic structure is the coefficients for each boom section ρ, Iρ , I of are not constant. It means that the homogeneous assumption is no longer valid. However, based on the assumption of small linear displacement, each section of telescopic boom can be model as a rotational Timoshenko beam with small displacement and to be connected end to end. The telescopic boom structure can be modeled as a N -stepwise beam with different coefficients for each boom section, a non-homogeneous Timoshenko beam. The boundary conditions between each two sections should be specified as fixed according to the continuous conditions. The overlapping parts of the boom sections and the telescopic mechanism are neglected in the purpose of modeling simplification.

Fig. 4. Simplified model of telescopic boom: the rotational non-homogeneous Timoshenko beam.

Figure 4 presents a rotational non-homogeneous beam with two sections as an example to illustrate the interconnection between both beam section. The local coordinate Ze1 Oe1 Ye1 of the first beam section rotates around the X axis at point O(Oe1 ). The local coordinate Ze2 Oe2 Ye2 attached on the point Oe1 which is the tip of the first beam section. According to the small displacement assumption, we consider the transnational velocity v1 (transnational output of the first beam) is vertical to the tangent direction of the deflection curve of the first beam at point Oe2 . Due to the geometric shapes of the long boom manipulators are rather slender and long, we use the rotational displacement of the cross section as the direction angle of the tangent line of the deflection curve at point Oe2 . The then coordinate of the second beam can be defined as: the origin is placed at Oe2 , the tangent of the deflection curve of the first beam at point Oe2 is defined as Oe2 Ze2 axis. So that the coordinate Ze1 Oe1 Ye1 is moving

Dynamic Simulation

159

with a rotational velocity ω1 and a transnational velocity v1 which is always vertical to the axis Oe2 Ze2 . Now we can expand the model from two beam sections to the boom with multiple sections. According to the definition of power variables (6) and (7), we can define the new power variables for the k th section of the N -stepwise beam in the domain of Dk := [Lk−1 , Lk ] as pt,k , pr,k , εt,k , εr,k , k = 1, . . . , n. For each part of the stepwise Timoshenko beam, a Dirac structure Dk can be acquired by using the corresponding power variables. According to the port-Hamiltonian form (32), the I/O represent of the k th ! boom is represented as system k ⎧   ⎪ ∂H k ⎪ k k ˙ ⎪ X = J + Gk U k ⎪ k ⎪ ∂x ⎪ ⎪ ⎪   ⎪ ⎨  k  ∂H k  k Y = G + Dk U k Σk : (50) k ∂x ⎪ ⎪ ⎪   k ⎪ ⎪ k ⎪ U k = vk−1 ωk−1 Qkk Mkk ⎪ ⎪ ⎪  ⎩ k  k Y = −Qkk−1 − Mk−1 vkk ωkk the superscripts indicate that the variables are belong to the k th beam and the subscripts represent the connection points. The telescopic boom structure can be considered as the combination of multiple subsystem and each of them is a homogeneous Timoshenko beam. Based on the geometrical and mechanical continuous conditions, the flows and efforts of the k th and (k + 1) th beam sections through the connection point Lk have the following relations f∂t,k (Lk , t) = f∂t,k+1 (Lk , t) f∂r,k (Lk , t) = f∂r,k+1 (Lk , t) t,k+1 (Ln , t) et,k ∂ (Lk , t) = e∂

(51)

r,k+1 (Lk , t) er,k ∂ (Lk , t) = e∂

which mean that the inputs and outputs of the adjacent two beams have similar relations at the connection point Lk Ykk = Ukk+1 , Ykk+1 = −Ukk . 3.2

(52)

Modeling of the Folding Boom Structure

The folding boom structure can be modeled as one rotational beam attached on the tip of another rotational beam. Figure 5 presents a model of double rotational Timoshenko beams. The definition of the coordinates and assumptions are as same as the model in Fig. 4, the difference is that the coordinate Ze2 Oe2 Ye2 rotates around the axis Ze2 at point Oe2 with an angle displacement θ2 . For the joint connection, one big difference compared to the fixed connection is the angle displacements of both beams are no longer continues at the connection joint. One additional input of angle changes the relative angle displacement

160

L. Gao et al.

between both sides of the beams, therefore the force and velocity are not continues nether. Because only the components along the axis Ye1 offer efforts for the lateral bending vibration on the first beam 1. We also consider the effort of the inertial force of beam 2 along the axis Ze2 which applies at the tip of the first boom as a part of input force.

Fig. 5. Simplified model of folding boom: double rotational Timoshenko.

The connection condition in the double rotational Timoshenko model has the following relations between the beams at both sides of the connect joint Lk : f∂t,k (Lk , t) cos θk (t) = f∂t,k+1 (Lk , t) f r,k (Lk , t) + θ˙k (t) = f r,k+1 (Lk , t) ∂

∂

t,k+1 (Ln , t) cos θ(t) + mk+1 f˙∂t,k (Lk , t) sin2 θk (t) et,k ∂ (Lk , t) = e∂

(53)

r,k+1 er,k (Lk , t) ∂ (Lk , t) = e∂

3.3

Interconnection Between Hydraulic and Structure Systems

Now we focus on the interconnection between the hydraulic system and the structure system. The luffing mechanism performs as the interconnection and the hydraulic cylinder in it acts as the actuator. The output of the hydraulic system, the velocity of the piston, can be transferred to the input angle velocity of the first beam at its boundary (z = 0) for the telescopic boom, or the input angle velocities of both boom at their own boundary (z = 0). For the sake of simplicity, we neglect the mass and inertial moment of the luffing mechanism, including the piston and the additional links. Based on the dimensions of luffing mechanism (the length of the cylinder and the positions of the joints), the rotation of the boom can be expressed as θ(t) = ∂s θ · s(t), the actuate force and moment have the relation F = ∂s · θM .

Dynamic Simulation

161

Therefor we have the I/O transfer relation between the hydraulic system and the boom structure as U0r,1 = ω01 = ∂s θ · s˙ = ∂s θ · yv uF = F (t) = ∂s θ · M01 = ∂s θ · Y0r,1

(54)

which illustrates the interconnection between the two systems in different domains. Finally, we have the complete model of the hydraulic actuated telescopic boom system. The load at the boom’s tip and the hydraulic flows though the cylinder are the system inputs, the velocity responds at the boom’s tip and the enthalpy change of the hydraulic system are the system outputs.

4

Simulation and Results

Firstm, an evaluation of the approximation quality of the finite-dimensional model of non-homogeneous Timoshenko beam based on the geometric discretization is needed. We define a stepped beam model with three sets of parameters as Table 1. The model of each part of the beam are built as a port-Hamiltonian system, discretized and implemented in MATLAB/Simulink individually. Then connect these three I/O representations as a series system. A finite element model using the given parameters (Table 1) is also built in NODYA, a dynamic finite element analysis programme developed by our institute. An eigenvalue analysis is applied firstly to check the frequencies, and the next step is to check the respond behaviors by the dynamic analysis. Table 1. System parameters [23]. Parameter

Beam 1 Beam 2 Beam 3

Length

0.3 m

Width

0.02 m

Depth

0.005 m 0.004 m 0.003 m

Density

7850 kg

0.015 m 0.01 m

Yong’s modulus 210 GPa

4.1

Poisson’s ratio

0.33

Shear factor

5/6

Simulation of the Telescopic Boom System

The results of eigenvalue analysis is listed in Table 2, the deviations between the two models are less than 4% [23]. It means that the port-Hamiltonian system representation and the corresponding geometric discretization are still suitable for the non-homogeneous Timoshenko beam.

162

L. Gao et al. Table 2. Eigenvalue analysis results [23]. Mode num. FE-

PH

Deviation

1

7.37 Hz

7.54 Hz 2%

2

27.5 Hz

28.2 Hz 3%

3

65.9 Hz

67.9 Hz 3%

4

136.4 Hz 141 Hz 3%

Figure 6 (b) represents the transient responds of the FE-model and the portHamiltonian model, both have the definition of input ω01 as Fig. 6 (a) shows. The amplitudes of the responding angle velocity at the boom’s tip are very close. And the difference of the eigenvalue is also reflected. Next, we can evaluate the model of complete telescopic boom system using the verified port-Hamiltonian model of boom structure. The model of the hydraulic system with the parameter set as in Table 3 is also implemented in Matlab/Simulink. Then the model can be easily connected to the model of boom structure by the specified input and output ports.

Fig. 6. Desired input and the transient responds of FE-model and port-Hamiltonian model [23]

Dynamic Simulation

163

Table 3. Parameters of hydraulic system [23]. Parameters

Value

Supply pressure

10 bar

Tank pressure

0 bar

Piston Area and area ratio 0.0001 m2 , 0.75 Bulk modulus

1.2 GPa

Piston stoke length

0.25 m

Fig. 7. Transient respond of the complete model of the telescopic boom, including hydraulic system [23].

In order to avoid the strong nonlinear characteristic of luffing mechanism, we set the limit of the input angle displacement to π/3. Figure 7 shows the transient responds of the boom structure model and the complete model including the hydraulic system. In this case, the influence of the hydraulic system is reflected by the respond time delay. 4.2

Simulation of the Folding Boom System

Based on the model of telescopic boom system, we change the connection condition between beam 1 and beam 2 from fixed to rotational as joint 2, the rotation joint between beam 1 and the ground is joint 1, but the connection condition

Fig. 8. Input angle velocities at joint 1 and joint 2.

164

L. Gao et al.

between beam 2 and beam 3 is still fixed. The above modification derives a double rotational beam with non-homogeneous beam section, which also reflects the feature of some large long boom manipulators due to the fact that they are designed with telescopic boom structure for the folding boom. We add the motion of joint 2 with the same curve of input angle velocity as joint 1, as Fig. 6(a) in the simulation of telescopic boom model. The motion sequence is designed as Fig. 8, that the joint 1 rotates π/3 beginning at 1s, and the joint 2 rotates π/3 beginning at 3 s.

Fig. 9. The output angle velocities at the tips of the first boom and the second boom.

We choose the angle velocity at the tips of the first and second boom as the system outputs. Figure 9 shows that the second boom rotates with the first boom since 1s because the joint 2 has 0 input and the whole boom system performs like the previous telescopic boom system. But after 3s, there is relative angle velocity at joint 2 and the second boom starts to rotates independently. Figure 10 is the angle displacement at the tips of the first boom and the second boom, the results are calculated by the integration of the curves in Fig. 9.

Fig. 10. Desired input and the transient responds of FE-model and port-Hamiltonian model.

Dynamic Simulation

5

165

Conclusion and Outlook

In this paper, we present the port-Hamiltonian formulation of two kinds of long boom manipulators, telescopic boom system and folding boom system. The hydraulic cylinder and the boom structure were modeled in the port-Hamiltonian formulation respectively. A port-Hamiltonian model of the Timoshenko beam with large displacement and rotation is proposed, to describe the telescopic boom structure and the folding boom structure with different connection conditions. And a structure-preserving discretization method is applied to transfer PDEs of the Timoshenko beam model into a solvable finite-dimensional model. The discretized port-Hamiltonian model has been proven by the comparison with the finite element model. The individual port-Hamiltonian models are integrated by interconnection ports into complete models to investigate the interaction between the hydraulic system and the boom structure in the dynamic simulation. However, due to the assumptions, we have made, the current model can only be effective for small linear deflection. The motion of the long boom manipulator is limited as planar motion and only the bending deflection is considered. To develop a more universal model, we have the plan to du as following: (1) A spatial beam model and its port-Hamiltonian formulation will be developed to describe the dynamic behavior of the long boom manipulation in 3-D space. The geometrically exact Euler-Bernoulli beam element could be a suitable option. (2) Besides the luffing operation, the long boom manipulators can also be influenced by other operations, such as hoisting of mobile cranes. The change of the load applying at the tip of the boom can stimulate remarkable vibration. The dynamic simulation of more operations for the long boom manipulators will be included. (3) An energy-based controller will be developed for the complete long boom manipulator system, to suppress and control the vibration during the motion or stimulated by the input forces.

References 1. Sun, G., Kleeberger, M.: Dynamic responses of hydraulic mobile crane with consideration of the drive system. Mech. Mach. Theory 38(12), 1489–1508 (2003) 2. Sun, G., Kleeberger, M., Liu, J.: Complete dynamic calculation of lattice mobile crane during hoisting motion. Mech. Mach. Theory 40(4), 447–466 (2005) 3. Sun, G., Liu, J.: Dynamic responses of hydraulic crane during luffing motion. Mech. Mach. Theory 41(11), 1273–1288 (2006) 4. Zuyev, A., Sawodny, O.: Stabilization of a flexible manipulator model with passive joints. IFAC Proc. Vol. 38(1), 784–789 (2005) 5. Sawodny, O., Aschemann, H., Bulach, A.: Mechatronical designed control of firerescue turntable-ladders as flexible link robots. IFAC Proc. Vol. 35(1), 509–514 (2002) 6. Pertsch, A., Zimmert, N., Sawodny, O. (eds.): Modeling a fire-rescue turntable ladder as piecewise Euler-Bernoulli beam with a tip mass. IEEE (2009)

166

L. Gao et al.

7. Pertsch, A., Sawodny, O.: Modelling and control of coupled bending and torsional vibrations of an articulated aerial ladder. Mechatronics 33, 34–48 (2016) 8. Nguyen, V.T., Schmidt, T., Leonhardt, T.: Effect of pre-tensioned loads to vibration at the ladder tip in raising and lowering processes on a turntable ladder. J. Mech. Sci. Technol. 33(5), 2003–2010 (2019) 9. Kugi, A., Haas, W., Schlacher, K., Aistleitner, K., Frank, H.M., Rigler, G.W.: Active compensation of roll eccentricity in rolling mills. IEEE Trans. Ind. Appl. 36(2), 625–632 (2000) 10. Gawthrop, P.J., Bevan, G.P.: Bond-graph modeling. IEEE Control Syst. Mag. 27(2), 24–45 (2007) 11. Zhao, Q., Gao, F.: Bond graph modelling of hydraulic six-degree-of-freedom motion simulator. Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci. 226(12), 2887–2901 (2012) 12. Duindam, V., Macchelli, A., Stramigioli, S., Bruyninckx, H.: Modeling and Control of Complex Physical Systems: The port-Hamiltonian Approach. Springer, Heidelberg (2009) 13. Macchelli, A., Melchiorri, C.: Modeling and control of the timoshenko beam. The distributed port hamiltonian approach. SIAM J. Control Optimiz. 43(2), 743–767 (2004) 14. Macchelli, A., Melchiorri, C., Stramigioli, S.: Port-based modeling and simulation of mechanical systems with rigid and flexible links. IEEE Trans. Rob. 25(5), 1016– 1029 (2009) 15. Moulla, R., Lefevre, L., Maschke, B.: Pseudo-spectral methods for the spatial symplectic reduction of open systems of conservation laws. J. Comput. Phys. 231(4), 1272–1292 (2012) 16. Vu, N.M.T., Lefevre, L., Nouailletas, R., Br´emond, S.: Geometric discretization for a plasma control model. IFAC Proc. Vol. 46(2), 755–760 (2013) 17. Wang, M., Bestler, A., Kotyczka, P.: Modeling, discretization and motion control of a flexible beam in the port-hamiltonian framework. IFAC-PapersOnLine 50(1), 6799–6806 (2017) 18. Bo, X., Fujimoto, K., Hayakawa, Y.: Control of two-link flexible manipulators via generalized canonical transformation. In: IEEE Conference on Robotics, Automation and Mechatronics, vol. 1, pp. 107–112. IEEE (2004) 19. Kugi, A., Kemmetm¨ uller, W.: New energy-based nonlinear controller for hydraulic piston actuators. Eur. J. Control 10(2), 163–173 (2004) 20. Stadlmayr, R.: On a combination of feedforward and feedback control for mechatronic systems. Shaker (2009) 21. Stadlmayr, R., Schlacher, K. (eds.): Modelling and Control of a Hydraulic Actuated Large Scale Manipulator, vol. 1. Wiley, New York (2004) 22. Cardoso-Ribeiro, F.L., Matignon, D., Pommier-Budinger, V.: A power-preserving discretization using weak formulation of piezoelectric beam with distributed control ports. IFAC-PapersOnLine 49(8), 290–297 (2016) 23. Gao, L., Mei, W., Kleeberger, M., Peng, H., Fottner, J.: Modeling and discretization of hydraulic actuated telescopic boom system in port-Bamiltonian formulation. In: Proceedings of the 9th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, pp. 69–79. SCITEPRESS-Science and Technology Publications, Lda. (2019)

Simulating Species Dominance in Mixed Mangrove Forests Considering Species-Specific Responses to Shading, Salinity, and Inundation Frequency Ian Estacio(B)

and Ariel Blanco

University of the Philippines Diliman, Quezon City, Philippines {icestacio,acblanco}@up.edu.ph

Abstract. To ensure that mangrove forest conservation efforts are successful, simulation models of mangrove forests are developed to forecast outcomes of different environmental scenarios. This paper presents MaDS (Mangrove species Dominance Simulator), an agent-based model that simulates the structures of mixed mangrove forest stands by considering specific responses of mangrove species to shading, salinity, and inundation frequencies. By simulating certain inundation and salinity conditions in a 50 m × 50 m plot, the model forecasts the resulting dominance of different mangrove species. The model uses different species-specific parameters for differentiating responses to environmental factors. The model was validated by conducting an experiment where different test sites in Katunggan It-Ibajay (KII) Eco-park, Aklan, Philippines were simulated given actual site salinity and inundation frequency values to see if simulated species dominance matched actual site species dominance. The validation experiment showed simulated dominant species matched with dominant species in the sites given that different groups of species are tested. An experiment on species abundance at different combinations of salinity and inundation frequency values was also conducted. Results of the simulations imply that observed zonation in mangrove forests is not only caused by the environmental conditions in the site but also by the number and characteristics of species in the site. Keyword: Agent-based modeling · Geographic information science · Environmental modeling · Forest stand

1 Introduction Mangroves, along with seagrasses and wetlands, are classified by the International Group of Experts on Blue Carbon (IGEBC) as a blue carbon ecosystem. These blue carbon ecosystems are far more effective in storing carbon in the atmosphere than terrestrial ecosystems, even storing up to 100 times faster and more permanently [1]. In the current times where anthropogenic global warming is increasing due to anthropogenic causes [2], blue carbon ecosystems need to be conserved in a strategic and science-based method. © Springer Nature Switzerland AG 2021 M. S. Obaidat et al. (Eds.): SIMULTECH 2019, AISC 1260, pp. 167–183, 2021. https://doi.org/10.1007/978-3-030-55867-3_9

168

I. Estacio and A. Blanco

Fig. 1. Sample of a mangrove zonation pattern observed in mangrove forests. The figure is from Smith [3].

A problem encountered in conservation of mangrove forests is the lack of knowledge in species zonation, where species tend to group themselves in different parts of the mangrove forest (see Fig. 1). Species zonation patterns occur at different geographic scale such as the estuarine location and intertidal position [3]. Zonation patterns are mainly explained by the spatially-varying conditions throughout the forest, hence species observed in a particular site reflect the conditions in the site. In general, the main environmental drivers that cause observed zonation patterns in mangrove forests are salinity and inundation frequency [4]. Because of the complexity of distribution patterns of species observed in mangrove forests, the causes of these patterns are difficult to infer. Agent-based models, which treat mangrove trees as agents and describe the processes occurring in it, may be helpful in understanding the resulting bottom-up zonation patterns. With this in mind, this paper developed an agent-based model called MaDS (Mangrove Dominance Simulator), which simulates the resulting mixed mangrove forest stand and dominance of each mangrove species given salinity, inundation frequency, and shading conditions. This model is adapted and modified from the model by Estacio et al. [5] to take into account Inundation Frequency conditions. The model was validated by comparing simulations of the different test sites in a study area to actual site data. As an experiment, the abundances of different mangroves species given different environmental conditions were simulated. The MaDS model can be used be used for assessing mangrove conservation strategies. The model can also be used in explaining the species zonation in a mangrove forest. Given the characteristics of species in the site and the site conditions, the model can give relationships between these inputs to the resulting patterns of species dominance.

Simulating Species Dominance in Mixed Mangrove Forests

169

2 Agent-Based Modeling for Mangrove Forests Agent-based modelling (ABM) or Individual-based modelling (IBM) has been used in modeling mangrove forests for different applications [5–9]. The MaDS model developed in this paper differentiates itself from the other models by considering multiple species (nine in this paper) and the inundation frequency, an environmental factor that dictates intertidal positions of mangrove species. In agent-based modelling of mangrove forest stands, individual mangrove trees are treated as agents. Being agents, mangrove trees are described routinely through their DBH (diameter at breast height), height, and other variables. In all models, individual mangrove trees follow three main processes: recruitment, growth, and mortality. Individual trees also sense the conditions of their environment which may affect their individual growth. The trees also compete with other trees; the method of describing the competition differ with each model. Through the mangrove agents’ behavior, the processes they undergo, their sensing of their environment conditions, and their interaction with one another, a unique mangrove forest stand will appear. Differing the environment, the behavior of the mangrove agents, and the initial conditions of the mangrove population will lead to different mangrove forest stand structures. The properties of the resulting mangrove forest stands are correlated with the set simulation parameters to answer different ecological questions.

Fig. 2. Orthophoto (a), salinity raster file (b), and inundation frequency raster file (c) of KII Mangrove Eco-park.

170

I. Estacio and A. Blanco

3 Development of the MaDS Model 3.1 Setting of the Environment Through Remote Sensing and GIS Data The mangrove forest chosen as the environment for the MaDS model is the Katunggan It Ibajay (KII) Mangrove Eco-park in Aklan, Philippines (see Fig. 2a). KII Eco-park is known for having the most number of mangrove species in any site in the Philippines, with a total of 28 species [10]. Raster files of salinity (see Fig. 2b) and inundation frequency (see Fig. 2c) were acquired and processed through remote sensing and GIS techniques. These raster files were used as environment variables that are sensed by the mangrove trees. 3.2 Overview, Design Concepts, and Details (ODD) of the Model The agent-based mixed mangrove forest stand model was given the name MaDS (Mangrove species Dominance Simulator) and was designed using the ODD (Overview, Design concepts, and Details) protocol [11]. The description of the ODD is stated in the following subsections. The overview of MaDS is shown in Fig. 3. The AnyLogic Simulation software was used for developing the model.

Fig. 3. Overview of the MaDS model, which includes the entities, scales, and processes.

Purpose. The purpose of the MaDS model is to simulate the resulting structure of a mangrove forest composed of different mangrove species of different species-specific characteristics and environmental responses, given salinity and inundation frequency conditions in a plot. By specifying the environmental conditions and the species present in the plot, the resulting mangrove forest stand can be forecasted. The results of the simulation may explain observed zonation patterns in a given mangrove forest.

Simulating Species Dominance in Mixed Mangrove Forests

171

Entities, State Variables, and Scales. MaDS has two entities: the mangrove agents and the environment spatial units which states the environmental conditions. A mangrove agent are defined in space by the x and y position of the center of its trunk. It is characterized by its species, which gives the species-specific parameters such as growth rate, allometry, wood density, and responses to environmental conditions (see Table 1) [5]. The state of a mangrove agent is defined by its DBH (Diameter at breast height), height, crown radius, above-ground biomass, and maturity state. A mangrove agent also has a memory of its growth for the last five years, which affects its mortality. Environment spatial units are grid cells that contain the salinity, inundation frequency, and above- and below-ground intensity of competition. The salinity and inundation frequency cells both have 5 m lengths while the above- and below-ground intensity of competition cells both have 0.2 m lengths. The time step of the simulation is 1 year and the extent time of the simulation depends on user input, which can range up to more than 1000 years. Simulation is implemented in a square plot of 50 m × 50 m size. Process Overview and Scheduling. Each year, the following processes are scheduled as follows: 1. 2. 3. 4. 5. 6. 7. 8.

Maturity transition (Mangrove), Recruitment (Mangrove), Salinity Response (Mangrove), Inundation Frequency Response (Mangrove), Competition Response (Mangrove), Growth (Mangrove), Mortality (Mangrove), and Updating of competition intensity (Environment).

Each process involving the mangrove agent (processes 1 – 8) are implemented by all agents synchronously. Mangrove agents update their state variables by the end of the growth process. After these processes, the Environment updates the new competition intensity grids from the new forest structure. The processes stated above are described in the Submodels section. Design Concepts. The subsections below gives details on the design concepts of the MaDS model. Emergence. The structure of a mixed mangrove forest stand emerges from the several years of recruitment, growth, and mortality of different species of mangroves. The structure of the forest stands is represented by the tree count and total above-ground biomass per species. The dominating species are inferred from these indicators. Sensing. A mangrove agent senses the environment conditions in its location which are salinity, inundation frequency, and intensity of competition. Interaction. Mangrove agents interact indirectly with one another through their competition for light and below-ground resources. A mangrove tree experiences reduced

3000

1000

1000

1500

Tree

Nypa fruticans Palm

Camptostemon Tree philippinense

Tree

Tree

Shrub

Tree

Avicennia rumphiana

Sonneratia alba

Xylocarpus granatum

Ceriops decandra

Bruguiera cylindrica

45

500

2000

1800

Tree

Avicennia officinalis

1000

Tree

45

20

80

70

60

45

300

100

40

200

150

200

200

200

100

200

200

200 83.34 0.6500

47.98 0.7316

60.58 0.6731

36.30 0.9075

46.57 0.2911

38.94 0.2782

28.77 0.2397

38.36 0.4262

68.47 0.8100

34.39 0.7250

90.71 0.6721

70.01 0.6443

49.31 0.4867

96.74 0.6167

19.09 0.0318 135.65 0.7367

33.26 0.1663

43.15 0.5394

G

High

Low

Tolerant

Tolerant

Tolerant

Low

Low

Low

Intolerant Mid

Tolerant

Tolerant

Intolerant Mid

Intolerant High

Intolerant High

Shading tolerance

33

67

34

44

75

33

90

63

85

Salt Smax (ppt) tolerance

b3

ρ (wood density)

Hmax (cm) Dmax (cm) Agemax (year) b2

Form

Mid

Mid

Mid

Low

High

Low

High

Low

Low

Lowest intertidal zone

Salinity/Inundation Inundation frequency response frequency [5] response [4]

Salinity response [5]

Biomass Shading estimation response [5] [5]

Growth [5]

Avicennia marina

Species

Table 1. Species-specific parameters for each of the nine species in the model.

172 I. Estacio and A. Blanco

Simulating Species Dominance in Mixed Mangrove Forests

173

growth when there are other trees in its neighboring area. The magnitude of reduction is dependent on the species and size of the tree and the number, proximity, and size of neighboring trees. Stochastic. The location of establishment of mangrove saplings at initialization are chosen such that saplings cover the whole plot in a grid-like manner but the nearest distances between saplings are still variable. The location of the establishment of produced seedlings within the range of dispersal from the parent tree are also chosen randomly. A seedling being established on a location also depends on a probability. There is a higher chance the seedling will be established if the intensity of competition in the location is lower. Therefore, when a tree dies and canopy gap arises, the species of seedlings that will be established in the area is stochastic.

Fig. 4. A 100-year mangrove forest stand dominated by Avicennia rumphiana, visualized in (a) 2D view of POV from the sky and (b) 3D view.

174

I. Estacio and A. Blanco

Observation. The structure of a simulated mixed mangrove forest stand is visualized in both 2D view and 3D view (see Fig. 4). Each tree is represented by two concentric cylinders where the first cylinder represents the trunk and the other represents the crown. Each tree is also color-coded according to species so that each tree can be identified by species. Each year, the model also records the individual tree count and total above-ground biomass per species so that trends of dominance through time can be observed per species. Two charts are displayed real-time which track the total above-ground biomass per species and the total above-ground biomass of the whole forest stand. Initialization. At the start of the simulation, an initial number of saplings, N, of DBH = 1.27 cm are placed around the plot with the idea that saplings are spaced as much as possible. The plot√is first divided into grids where the number of columns and rows is the rounded-down N . For each grid, a sapling of random species is placed at a random location within the grid. Once all grids already contain a sapling, the remaining number of saplings that still haven’t been placed are placed randomly around the plot. All species of mangroves in the plot almost have the same number of saplings with only a maximum difference of one sapling between two species. The initial number of saplings can range from 50 to 200. Depending on the experiment, the user can just specify the salinity and inundation frequency values in the plot or use actual site data. Data Input. The model does not use external data inputs for the time-varying processes in the model. The specified salinity and inundation frequency conditions specified at initialization do not change through the extent of the simulation. Submodels. The MaDS model is composed of eight submodels which represent the processes implemented in the model (as stated in the Process Overview and Scheduling section). All the submodels, except for the Inundation Frequency Response and Updating of Competition Intensity, were adapted from Estacio et al. [5]. The following subsections describe the stated two submodels. Inundation Frequency Response. The inundation frequency response, Ir , is computed based on the model by Yang et al. [12]. The model states that the limit inundation frequency (IFS, limit ) tolerated by a tree is a function of the salinity condition (S) it experiences, the species-specific maximum loading condition (MLC), and the speciesspecific maximum inundation frequency (IFmax ). It is given by the equation: S = MLC IF max − IF S,limit

(1)

The IFmax was assigned based on the intertidal position of the species, which is the observed position of the mangrove species due to inundation (see Table 2). The growth of Kandelia Candel is reduced at around 50 ppt [13], so it was assumed that the maximum salinity of Kandelia Obovata is around this value. The MLC of Kandelia Obovata is at 99 [12], so it was assumed that the MLC is twice the Smax .

Simulating Species Dominance in Mixed Mangrove Forests

175

Table 2. Maximum inundation frequency for each intertidal zone. Intertidal zone IFmax Low

1

Mid

0.75

High

0.5

Manipulating the equation to derive the inundation frequency limit given a salinity value, the equation was derived to be: IF S,limit = IF max −

S 2Smax

(2)

It is assumed that if a mangrove tree is in a position where the inundation frequency is beyond its limit (IF ≥ IFS, limit ), the tree won’t grow. The inundation frequency response of a mangrove is computed as: ⎧ 0 ≤ IF < IFS,limit − C ⎨ 1; (3) Ir = e(−|ln(0.01)|∗(IF−IFS,limit +C )/C ) ; IFS,limit − C ≤ IF < IFS,limit ⎩ 0; IFS,limit ≤ IFIFS,limit ≤ IF where C is assigned a value of 0.05, meaning at 0.05 frequency away from the IFS, limit , the growth of a mangrove starts to reduce due to exposure to inundation. The inundation frequency response is affected by both the inundation frequency and salinity (see Fig. 5). Inundation Frequency Response is favorable for a certain low range of both inundation frequency and salinity.

Fig. 5. Inundation frequency response of Xylocarpus granatum through varying salinity and inundation frequency gradients. Xylocarpus granatum’s IFmax is 0.75, hence as salinity values approach 0, the edge of optimum response approaches 0.75.

176

I. Estacio and A. Blanco

Updating of Competition Intensity. The grid storing the spatial pattern of intensity of competition in the whole plot is updated after all processes involving the mangroves have occurred. For every point in every center of every grid (of size 0.2 m), the intensity of competition at point (x,y), F(x,y), is computed by adding all the FONs that overlap with the point as shown below [7]:  FON n (x, y) (4) F(x, y) = N

The value of this intensity of competition will be used by the recruitment model to compute for the probability of establishment of a sapling given the intensity of competition at the point of establishment.

4 Validation of the Model To validate the model, simulation results of species dominance given actual site salinity and inundation frequency data were compared with the actual species dominance in the site. A raster map of the mangrove species in the area was acquired from the Mangrove Remote Sensing (MaRS) project of the IAMBlueCECAM (Integrated Assessment and Modelling of Blue Carbon Ecosystems for Conservation and Adaptive Management) program (see Fig. 6). Six test sites were chosen in KII Eco-park and marked in the raster species map using ArcGIS software. The dominating species and other abundant species for each site were noted through visual inspection (see Table 3). 4.1 Validation Settings For this validation experiment, 15 saplings per species and with DBH of 1.27 cm were placed around the plot at the start of each simulation run. 10 replications of 300-year simulation runs were executed per test site. 300 years was used as this is the forest stand age where the second generation of trees are already dominating [14]. The annual median total above-ground biomass (AGB) of each species for the 10 replications were acquired. In case for a given site, the simulated most dominant species by the 300th year did not match the actual dominant species, another set of simulation experiment was conducted in the site but this time, the (one or more) simulated species more dominant than the actual site dominant species was removed – leaving a group of fewer species in the simulations. 4.2 Validation Results Results of this validation experiment are shown as dominance curves of species in relation to the forest stand age (see Fig. 7). For test sites dominated by Avicennia rumphiana (test sites 1, 2, and 6), the simulated dominant species by the 300th year is also Avicennia rumphiana. For all years also, Avicennia rumphiana was the most dominating species,

Simulating Species Dominance in Mixed Mangrove Forests

177

Fig. 6. The six test sites in KII chosen as validation plots overlaid on top of the species map.

Table 3. Dominant species and other abundant species per test site. Test site

Dominant species

Other abundant species

1

Avicennia rumphiana

2

Avicennia rumphiana

3

Nypa fruticans

Ceriops decandra, Xylocarpus granatum

4

Nypa fruticans

Avicennia marina, Sonneratia alba

5

Nypa fruticans

Xylocarpus granatum, Sonneratia alba

6

Avicennia rumphiana

hence the resulting dominant species of the simulation matches the dominant species in these sites. For test sites dominated by Nypa fruticans (test sites 3, 4, and 5), simulation results of species dominance by the 300th year given all nine species did not show Nypa fruticans. For test site 3, the simulated dominant species is Avicennia officinalis, followed by Sonneratia alba, then Nypa fruticans. For test sites 4 and 5, Avicennia rumphiana is the simulated dominant species then followed by Nypa fruticans. Simulations were again conducted for test sites 3, 4, and 5 with species more dominating than Nypa fruticans removed (see Fig. 8). Removing the one or two most dominant

178

I. Estacio and A. Blanco

Fig. 7. Mangrove species dominance given nine species over a 300 year-period for the different test sites in KII Eco-park.

Simulating Species Dominance in Mixed Mangrove Forests

179

species in the simulations, the results showed that Nypa fruticans already dominated the test sites by the 300th year. In test site 5, Xylocarpus granatum is also abundant, which was also observed for the site.

Fig. 8 Mangrove species dominance given different group of species over a 300 year-period for test sites 3, 4, and 5 in KII Eco-park.

Results of these simulations imply that observed zonation in mangrove forests is not only caused by the environmental conditions in the site but also by the number and characteristics of species in the site. Nypa fruticans may be the dominating species in test sites 3, 4, and 5 as not only are the environmental conditions favorable, but also because other species that may outcompete it was not recruited in the area. This brings a limitation to the MaDS model that only models spatial processes within a 50 m square plot. To further explain zonation in mangrove forests, accurate designing of the recruitment process in a wider scale may be the key.

180

I. Estacio and A. Blanco

5 Simulation Experiment of Species Dominance at Different Environment Conditions An experiment was conducted where model simulations were used to observe at what combinations of salinity and inundation frequency value will mangrove species survive and thrive. Different simulation runs covering different pairs of salinity and inundation frequency were executed. 5.1 Experiment Settings For salinity, values were varied from 1 to 37 ppt with an interval of 3 ppt. 1 ppt was used as the minimum salinity value as mangroves generally dominate in saline areas and they are outcompeted by terrestrial trees in freshwater areas. 37 ppt was used as the maximum salinity value as 35 ppt is the average salinity value of seawater and a little leeway was given for values exceeding the average. For inundation frequency, values were varied from 0 to 1 with an interval of 0.05. Per pair of salinity and inundation frequency value, 10 replications of 300-year simulation runs were executed. In each simulation, the subject salinity and inundation frequency value was placed constant throughout the whole plot and 15 saplings per species and with DBH of 1.27 cm were placed around the plot. After the 300th year of every simulation, the abundance of each species represented by their total AGB was examined. Since multitude of simulation runs were need to be executed (2400 runs), a Monte Carlo experiment was executed. A Monte Carlo experiment executes multiple simulation runs where certain parameters are varied for each run. In this experiment, the salinity and inundation frequency values were varied according to the chosen range and interval above. To handle the data from several simulation runs, the PostgreSQL open-source database management system was used. In PostgreSQL, the median 300th year total AGB of 10 simulation replications was derived for every combination of species, salinity, and inundation frequency. 5.2 Experiment Results From the simulation runs of different salinity and inundation frequency conditions, the abundance surfaces of the nine mangrove species with respect to salinity and inundation frequency were derived (see Fig. 9). For different pair combinations of salinity and inundation frequency values, only a number of mangrove species thrive. It can be observed from the derived abundance surfaces that the first factor that dictates the abundance of a species is its environmental tolerances, which is mostly dictated by the species’ SUOG in the salinity axis and the IFmax in the inundation frequency axis. As an example, for Nypa fruticans¸ its abundance along the salinity axis borders about its SUOG value of 29. For Avicennia rumphiana, its abundance along the inundation frequency axis borders about its IFmax value of 0.5.

Simulating Species Dominance in Mixed Mangrove Forests

181

Fig. 9. Abundance of nine species in KII Eco-park given different combinations of salinity and inundation frequency conditions.

The same is also true for Sonneratia alba and Bruguiera cylindrica. They do not thrive at environments of low-inundation frequency as it is outcompeted by the much

182

I. Estacio and A. Blanco

larger Avicennia rumphiana. At inundation frequencies where Avicennia rumphiana cannot survive, Sonneratia alba and Bruguiera cylindrica already thrive. It can also be observed that species with high growth rates (parameter G) has a higher chance of being abundant (e.g., Avicennia rumphiana and Nypa fruticans) as it can easily outgrow other species. Contrary to expectations, shading tolerance don’t seem to play a significant role in the abundance of a species, at least in the simulations. To summarize, the abundance of a mangrove species in an environment condition is dictated by its tolerance to environment conditions (which is controlled by the SUOG and IFmax ) and its ability to outcompete other species (controlled by the growth rate G and shading tolerance). The results of this experiment support the results of the validation experiment that zonation is not only a result of gradients of environment conditions but also of the competition between species. Hence, the presence of species that establishes in the site may be a huge determinant for resulting mangrove zonation.

6 Conclusions This paper developed a model named MaDS (Mangrove species Dominance Simulator), an agent-based mangrove forest stand model which simulates the development of mixed mangrove forests on a 50 m × 50 m plot given the different specific properties of mangrove species and the environmental conditions in the plot. To develop the model, salinity and inundation frequency data in the form of raster files, derived from remote sensing and GIS techniques, were used to describe the spatial environment of the model. The agent-based model was designed using the ODD (Overview, Design, and Details) protocol and was developed using the AnyLogic software. The model was validated by conducting an experiment where simulation results of species dominance given actual site salinity and inundation frequency data were compared with the actual species dominance in the site. Results of the validation experiment simulating the environmental conditions of six test sites in KII Eco-park showed simulated dominant species matched with dominant species in the sites given that different groups of species are tested. Using the MaDS model, an experiment was conducted where simulations at different combinations of salinity and inundation frequency values were used to observe which mangrove species will survive and thrive. Results from the experiment showed that zonation is a result of both the environment conditions in the site and the presence of other mangrove species.

References 1. Fortes, M., Salmo, S.: Status of mangrove research and management in the Philippines. In: Challenges and Opportunities. State of the Mangrove Summit: Southern Luzon Proceedings, pp. 50–60 (2015) 2. Powell, J.L.: Climate scientists virtually unanimous: anthropogenic global warming is true. Bull. Sci. Technol. Soc. 35(5–6), 121–124 (2015)

Simulating Species Dominance in Mixed Mangrove Forests

183

3. Smith, T.: Forest structure. In: Robertson, A.I., Alongi, D.M. (eds.) Tropical Mangrove Ecosystems, pp. 101–136. American Geophysical Union, Washington, D.C. (1992) 4. Duke, N., Ball, M., Ellison, J.: Factors influencing biodiversity and distributional gradients in mangroves. Glob. Ecol. Biogeogr. 7, 27–47 (1998) 5. Estacio, I., Quinton, K., Macatulad, E., Salmo, S.: A species-specific individual-based simulation model of mixed mangrove forest stands. In: Proceedings of the 9th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, pp. 153–164 (2019) 6. Chen, R., Twilley, R.R.: A gap dynamic model of mangrove forest development along gradients of soil salinity and nutrient resources. J. Ecol. 86, 37–51 (1998) 7. Berger, U., Hildenbrandt, H.: A new approach to spatially explicit modelling of forest dynamics: spacing, ageing and neighbourhood competition of mangrove trees. Ecol. Model. 132, 287–302 (2000) 8. Jiang, J., DeAngelis, D., Smith, T., III., Teh, S., Koh, H.L.: Spatial pattern formation of coastal vegetation in response to external gradients and positive feedbacks affecting soil porewater salinity: a model study. Landsc. Ecol. 27(1), 109–119 (2012) 9. Grueters, U., Seltmann, T., Schmidt, H., Horn, H., Pranchai, A., Vovides, A.G., Peters, R., Vogt, J., Dahdouh-Guebas, F., Berger, U.: The mangrove forest dynamics model mesoFON. Ecol. Model 291, 28–41 (2014) 10. Barrientos, K., Apolonio, J.: Species diversity and soil carbon sequestration potential of mangrove species at Katunggan It Ibajay (KII) Eco-Park in Aklan, Philippines. Prism 22(1) (2017). 11. Grimm, V., Berger, U., DeAngelis, D., Polhill, G., Giske, J., Railsback, S.: The ODD protocol: a review and first update. Ecol. Model. 221(23), 2760–2768 (2010) 12. Yang, S.C., Shih, S.S., Hwang, G.W., Adams, J., Lee, H.Y., Chen, C.P.: The salinity gradient influences on the inundation tolerance thresholds of mangrove forests. Ecol. Eng 51, 59–65 (2013) 13. Qiu, D., Lin, P., Guo, S.Z.: Effects of salinity on leaf characteristics and CO2/H2O exchange of Kandelia candel (L.) Druce seedlings. JFS 53, 13–19 (2007) 14. Bormann, F.H., Likens, G.E.: Pattern and Process in a Forested Ecosystem. Springer, New York (1979)

Author Index

A Alonso, José Luis, 46 Argáez, Carlos, 104 B Bahr, Matthias, 61 Benedikt, Martin, 1, 83 Benjelloun, Khalid, 122 Blanco, Ariel, 167 D Denil, Joachim, 1 Deshayes, Laurent, 122 E Estacio, Ian, 167 F Fernández de Ávila, Susana, 46 Ferrer, Juan Carlos, 46 Fottner, Johannes, 144

K Kali, Yassine, 122 Kleeberger, Michael, 144 M Morss, Luca, 61 O Oakes, Bentley James, 1 Ören, Tuncer, 29 Oulhiq, Ridouane, 122 P Peng, Haijun, 144 R Reicherts, Sebastian, 61 Rodríguez, Fernando, 46 S Saad, Maarouf, 122 Schramm, Dieter, 61 Sieberg, Philipp, 61

G Gao, Lingchong, 144 Giesl, Peter, 104 Gomes, Cláudio, 1

V Valiente, David, 46 Vangheluwe, Hans, 1

H Hafstein, Sigurdur Freyr, 104 Holzinger, Franz Rudolf, 1, 83

W Wang, Mei, 144 Watzenig, Daniel, 83

© Springer Nature Switzerland AG 2021 M. S. Obaidat et al. (Eds.): SIMULTECH 2019, AISC 1260, p. 185, 2021. https://doi.org/10.1007/978-3-030-55867-3