Neural Control of Renewable Electrical Power Systems (Studies in Systems, Decision and Control, 278) 3030474429, 9783030474423

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Studies in Systems, Decision and Control 278

Edgar N. Sánchez Larbi Djilali

Neural Control of Renewable Electrical Power Systems

Studies in Systems, Decision and Control Volume 278

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

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

More information about this series at http://www.springer.com/series/13304

Edgar N. Sánchez Larbi Djilali •

Neural Control of Renewable Electrical Power Systems

123

Edgar N. Sánchez Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional (Cinvestav) Zapopan, Jalisco, Mexico

Larbi Djilali Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional (Cinvestav) Zapopan, Jalisco, Mexico

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-030-47442-3 ISBN 978-3-030-47443-0 (eBook) https://doi.org/10.1007/978-3-030-47443-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Nowadays, the share of power using clean power generators is continuously increasing in all over the world; hence, penetration of distributed energy resources (DERs) into grid is also increasing. On the other hand, in order to reach better exploration of the renewable energy that is generated from different DERS, the microgrid concept is the key. Regarding high penetration of renewable energy generation in medium- and high-voltage networks, modern grid codes enforce the DERs to have Low-Voltage Ride-Through (LVRT) capacity, which is the capability of a specific electric generator to stay connected to the main grid in presence of grid disturbances for a short period. This requirement is one of the most dominant grid connection requirements to be met by DERs. To achieve adequate LVRT capacity, the installed local controllers of each microgrid subsystem should guarantee transient stability to enhance resilience of microgrid operations. Recently, technology advances have forced control engineers to deal with complex systems, which include unknown dynamics, strong interconnection terms, and disturbances. Then, conventional control techniques are unsuccessful to provide an effective solution to control this class of systems. Neural networks are recently a well-established methodology for identification and control of general nonlinear and complex systems. Applying neural networks, control schemes can be developed to be robust in the presence of disturbances, parameter variations, and modeling errors. In this book, robust control schemes based on neural network identification are developed to enhance, firstly, the LVRT capacity of grid-connected Doubly Fed Induction Generator (DFIG)-based Wind Turbine (WT) and, secondly, to improve the LVRT capacity of DERs, which are installed in a grid-connected microgrid. The outline of this work is as follows. In Chap. 1, an introduction, which exposes the state of art and the different research works regarding the wind power system and grid-connected micro grid, is presented. In Chap. 2, mathematical preliminaries, which are used in the development of this book, are introduced. Then, the mathematical model of wind power system is given in Chap. 3 including the mechanical, the DC-link, and the DFIG models. Next, different control schemes based on the proposed neural network identifiers are synthesized in Chap. 4. Those controllers v

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Preface

are the discrete-time neural sliding mode field oriented control, the discrete-time neural sliding mode linearization control, and the discrete-time neural inverse optimal control. All those proposed controllers are synthesized for the grid side converter and the rotor side converter. Simulation results using Matlab are discussed considering the tracking of time-varying reference, robustness to parameter variation, sensitivity to wind speed changing effects, and robustness in presence of grid disturbances. After that, real-time results of the proposed controllers for the DFIG-based WT considering time-varying reference tracking, maximum power extraction, and robustness to grid disturbances are presented in Chap. 5. In Chap. 6, the mathematical models of the solar power system, the battery bank, and their corresponding converters are introduced. In addition, the proposed discrete-time neural sliding mode linearization control is extended for the microgrid components. The real-time simulation results are presented under normal grid conditions regarding time-varying trajectories tracking and in presence of three grid fault type: single-phase-to-ground, two-phase-to-ground, and three-phase-to-ground. Finally, the perspective conclusions and the directions for future researches are presented. This book is intended for researchers and students with a control background and wishing to expand their knowledge of wind power generation and distributed energy resources installed into a grid-connected microgrid. Additionally, this can be useful to the scientist in the automatic control field especially whom looking for innovative control ideas and their applications in renewable energy systems. It will also interest practicing engineers dealing with power generation technologies, who will benefit from the utilization of neural networks in systems modeling and control synthesis. Zapopan, Mexico

Edgar N. Sánchez Larbi Djilali

Acknowledgments

The authors thank CONACyT (from its name in Spanish, which stands for National Council for Science and Technology), Mexico. We thank Cinvestav-IPN (from its name in Spanish, which stands for Advances Studies and Research Center of National Polytechnic Institute), Mexico. We also thank the University of Amar Telidji of Laghouat, Algeria and the Faculty of Electrical Engineering, University Michoacana of San Nicolas de Hidalgo, Morelia, Mexico for facilities provided to accomplish this book publication. Additionally, we thank Dr. Fernando Ornelas-Tellez, University Michoacana of San Nicolas de Hidalgo, Morelia, Mexico, Dr. Alexander G. Loukianov, Cinvestav-Guadalajara, Mexico, Dr. Mohammed Belkheiri, University of Amar Telidji of Laghouat, Algeria, and all of whom have substantially contributed to the development of this book.

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Abstract

Renewable energy generation is one of the most efficient and effective solutions to face global warming. Taking into account high penetration of Distributed Energy Resources (DERs) in medium- and high-voltage networks, modern grid codes enforce the DERs to have Low-Voltage Ride-through (LVRT) enough capacity. In this book, different control schemes based on neural network identification are proposed for a Wind Power System (WPS) based on Doubly Fed Induction Generator (DFIG) connected to the grid and grid-connected microgrid. Firstly, the proposed controllers are used to track the WPS dynamics references which are the DC voltage, the grid power factor, and the stator active and reactive power. The performances of the proposed neural control schemes for the WPS are initially validated via simulations using SimPower ToolBox of Matlab, by means of considering time-varying references tracking, robustness to parameter variations, and wind speed changes. In addition, LVRT capacity enhancement in presence of grid disturbances is tested. Moreover, those controllers are experimentally validated on a 1/4 HP DFIG prototype under both normal and abnormal grid conditions. The experimental results illustrate that the proposed controllers have adequate performances for WPS, even in presence of references variations, and wind turbine speed changes, compared with conventional control schemes. In addition, those controllers are able to operate WPS for extracting the maximum power from the wind under different fault scenarios, enhancing the LVRT capacity and ensuring transient stability. Secondly, a neural sliding mode linearization control scheme is extended to control the generated power from each DER installed into grid-connected microgrid. The proposed microgrid is composed of a WPS, a Solar Power System (SPS), a Battery Bank (BB), and a load demand. In addition, the microgrid under study is interconnected to an IEEE 9-bus system to evaluate its connection performance and response in presence of grid disturbances. The whole system is real-time simulated using an Opal-RT (OP5600) simulator. Results illustrate effectiveness of the proposed control scheme to achieve trajectory tracking for DER active and reactive powers. In addition, the LVRT capability of the proposed control strategy is verified in presence of grid disturbances.

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Contents

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1 5 5 6 6

2 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Discrete-Time Sliding Mode Control . . . . . . . . . . . . . . . . . . . 2.2 Block Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Inverse Optimal Control: Tracking Problem . . . . . . . . . . . . . . 2.4 Neural Networks Identification and Control . . . . . . . . . . . . . . 2.4.1 Discrete-Time Recurrent High Order Neural Networks . 2.4.2 Extended Kalman Filter Training Algorithm . . . . . . . . 2.4.3 Neural Networks for Control Systems . . . . . . . . . . . . . 2.5 Stability Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Wind System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Wind Power Technologies . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Wind Turbine Electrical Generators . . . . . . . . . . 3.1.2 HAWT Components . . . . . . . . . . . . . . . . . . . . . 3.2 Wind Power System Modeling . . . . . . . . . . . . . . . . . . . 3.2.1 Mechanical Part Model . . . . . . . . . . . . . . . . . . . 3.2.2 Electrical Machine and Power Converter Models . 3.3 LVRT and Voltage Dips . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 LVRT Definition . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Voltage Dip Definition . . . . . . . . . . . . . . . . . . . . 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction . . . . . . . . . 1.1 Motivation . . . . . . . 1.2 Objectives . . . . . . . 1.3 Main Contributions . 1.4 Book Outline . . . . .

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5 Experimental Results . . . . . . . . . . . . . . . . . . . . 5.1 Prototype Description . . . . . . . . . . . . . . . . . 5.2 Real-Time SM-FOC . . . . . . . . . . . . . . . . . . 5.2.1 GSC Results . . . . . . . . . . . . . . . . . . 5.2.2 RSC Results . . . . . . . . . . . . . . . . . . 5.2.3 Robustness to Grid Disturbances . . . 5.3 Real-Time N-SMFOC . . . . . . . . . . . . . . . . . 5.3.1 GSC and RSC Identification Results . 5.3.2 GSC Controller Results . . . . . . . . . . 5.3.3 RSC Controller Results . . . . . . . . . . 5.3.4 Robustness to Grid Disturbances . . . 5.4 Real-Time N-SML Control . . . . . . . . . . . . . 5.4.1 Identification . . . . . . . . . . . . . . . . . . 5.4.2 GSC Controller Results . . . . . . . . . . 5.4.3 RSC Controller Results . . . . . . . . . . 5.4.4 Robustness to Grid Disturbances . . . 5.5 Real-Time N-IOC Scheme . . . . . . . . . . . . . 5.5.1 Normal Grid Conditions . . . . . . . . . . 5.5.2 Abnormal Grid Conditions . . . . . . . . 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . .

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4 Neural Control Synthesis . . . . . . . . . . . . . . . . 4.1 Neural Sliding Mode Control . . . . . . . . . . 4.1.1 GSC Controller . . . . . . . . . . . . . . . 4.1.2 RSC Controller . . . . . . . . . . . . . . . 4.2 Simulation Results . . . . . . . . . . . . . . . . . . 4.2.1 GSC Controller . . . . . . . . . . . . . . . 4.2.2 RSC Controller . . . . . . . . . . . . . . . 4.3 Neural Sliding Mode Linearization Control 4.3.1 GSC Controller . . . . . . . . . . . . . . . 4.3.2 RSC Controller . . . . . . . . . . . . . . . 4.4 Simulation Results . . . . . . . . . . . . . . . . . . 4.4.1 GSC Controller . . . . . . . . . . . . . . . 4.4.2 RSC Controller . . . . . . . . . . . . . . . 4.5 Neural Inverse Optimal Control . . . . . . . . . 4.5.1 GSC Controller . . . . . . . . . . . . . . . 4.5.2 RSC Controller . . . . . . . . . . . . . . . 4.6 Simulation Results . . . . . . . . . . . . . . . . . . 4.6.1 GSC Controller . . . . . . . . . . . . . . . 4.6.2 RSC Controller . . . . . . . . . . . . . . . 4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . .

109 109 111 111 113 119 121 122 124 125 128 131 132 134 136 142 144 144 148 153

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6 Microgrid Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Opal-RT Lab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Considered Microgrid . . . . . . . . . . . . . . . . . . . . . . . 6.3 Microgrid Modeling . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Wind Power System . . . . . . . . . . . . . . . . . . 6.3.2 Solar Power System . . . . . . . . . . . . . . . . . . . 6.3.3 Batteries Bank Storage System Modeling . . . 6.4 Microgrid Neural Controllers . . . . . . . . . . . . . . . . . . 6.4.1 Wind Power System Controller . . . . . . . . . . 6.4.2 Solar Power System Controller . . . . . . . . . . . 6.4.3 BB Neural Controller . . . . . . . . . . . . . . . . . . 6.5 Real-Time Simulation Results . . . . . . . . . . . . . . . . . 6.5.1 Normal Grid Conditions . . . . . . . . . . . . . . . . 6.5.2 Abnormal Grid Conditions, First Location . . . 6.5.3 Non-ideal Grid Conditions, Second Location . 6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Appendix A: Neural Controller Stability Analysis . . . . . . . . . . . . . . . . . . 189 Appendix B: The Wind Turbine Modeling . . . . . . . . . . . . . . . . . . . . . . . . 195 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Notations

< k 8 R 2 Zþ k k P ðÞ1 ðÞT ðÞ kk jj X S Dk DV V u0 uk sðxÞ xk vk xi -i yk ek

set of all real numbers sample time for all sliding surface subset belonging to set of positive integers Euclidean norm sum up denotes matrix inverse denotes matrix transpose denotes optimal function the Euclidean norm absolute value compact subset of real numbers subset of real numbers identification error function denotes the Lyapunov difference denotes Lyapunov function candidate the upper control bound the system input the sliding surface Real vector state Estimated vector state Adaptive neural identifier weights Fixed neural identifier weights the output to be controlled tracking error

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Acronyms

BB DFIG EKF FOC GSC HAWT LVRT N-IOC NN N-SM N-SMFOC N-SML RHONN RSC SGUUB SM SM-FOC SM-L SM-VOC SPS VOC VSWT WPS WT

Battery Bank Doubly Fed Induction Generator Extended Kalman Filter Field Oriented Control Grid Side Converter Horizontal Axis Wind Turbine Low-Voltage Ride-Through Neural Inverse Optimal Control Neural Network Neural Sliding Mode Neural Sliding Mode Field Oriented Control Neural Sliding Mode Linearization Recurrent High Order Neural Network Rotor Side Converter Semiglobally Uniformly Ultimately Bounded Sliding Mode Sliding Mode Field Oriented Control Sliding Mode Linearization Sliding Mode Vector Oriented Control Solar Power System Vector Oriented Control Variable Speed Wind Turbine Wind Power System Wind Turbine

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List of Figures

Fig. 2.1 Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

2.2 2.3 2.4 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12

Motion trajectory of continuous-time system with scalar SM control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RHONN scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Identification Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Indirect neural control scheme . . . . . . . . . . . . . . . . . . . . . . . . Wind turbines electrical generators . . . . . . . . . . . . . . . . . . . . . Components of a HAWT . . . . . . . . . . . . . . . . . . . . . . . . . . . . DFIG based WT grid connection . . . . . . . . . . . . . . . . . . . . . . Air flow at a wind turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . Power coefficient Cp according to k and b . . . . . . . . . . . . . . . Schematics of a typical 2MW gearbox design . . . . . . . . . . . . Block diagram of Wind Turbine . . . . . . . . . . . . . . . . . . . . . . . Schematic of HAWT pitch control system . . . . . . . . . . . . . . . Schematic of HAWT pitch control system . . . . . . . . . . . . . . . Pitch control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DFIG based wind turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . DC link electrical configuration . . . . . . . . . . . . . . . . . . . . . . . DFIG electrical configuration . . . . . . . . . . . . . . . . . . . . . . . . . Grid currents and voltages: normal grid condition . . . . . . . . . DC voltage identification and weights . . . . . . . . . . . . . . . . . . Grid d current identification and weights . . . . . . . . . . . . . Grid q current identification and weights . . . . . . . . . . . . . DC voltage and grid power factor with N-SMVOC . . . . . . . . Grid d  q currents with N-SMVOC . . . . . . . . . . . . . . . . . . . Grid active and reactive power with N-SMVOC. . . . . . . . . . . Control d  q signals with N-SMVOC . . . . . . . . . . . . . . . . . . Influence of rg changes on N-SMVOC . . . . . . . . . . . . . . . . . . Influence of lg changes on N-SMVOC . . . . . . . . . . . . . . . . . . Influence of rl changes on N-SMVOC: 10 MX . . . . . . . . . . . . Influence of rl changes on N-SMVOC: 10 X . . . . . . . . . . . . .

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10 17 19 20 24 25 26 26 28 28 29 30 31 31 32 32 35 47 48 48 49 49 50 50 51 51 52 52 53

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Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

List of Figures

4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 4.30 4.31 4.32 4.33 4.34 4.35 4.36 4.37 4.38 4.39 4.40 4.41 4.42 4.43 4.44 4.45 4.46 4.47 4.48 4.49 4.50 4.51 4.52 4.53 4.54 4.55 4.56

Grid voltages and currents . . . . . . . . . . . . . . . . . . . . . . . . . . . Grid faults influences on DC-link . . . . . . . . . . . . . . . . . . . . . . DFIG rotor q currents identification . . . . . . . . . . . . . . . . . DFIG rotor d currents identification . . . . . . . . . . . . . . . . . DFIG rotor d  q currents with N-SMFOC . . . . . . . . . . . . . . N-SMFOC controllers trajectory tracking . . . . . . . . . . . . . . . . N-SMFOC controller trajectory tracking zoom . . . . . . . . . . . . The DFIG control signals for N-MFOC . . . . . . . . . . . . . . . . . Influence of rs change on N-SM-FOC controller . . . . . . . . . . rs change influence zoom . . . . . . . . . . . . . . . . . . . . . . . . . . . . Influence of ls change on N-SMFOC controller . . . . . . . . . . . ls change influence zoom . . . . . . . . . . . . . . . . . . . . . . . . . . . . Influence of lr change on N-SMFOC controller . . . . . . . . . . . lr change influence zoom . . . . . . . . . . . . . . . . . . . . . . . . . . . . Influence of lm change on N-SMFOC controllers . . . . . . . . . . change influence zoom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Speed variation effects on N-SMFOC controller . . . . . . . . . . . Stator powers with N-SMFOC under fault conditions . . . . . . . Stator powers with SM-FOC under fault conditions . . . . . . . . Stator powers with PI-FOC under fault conditions . . . . . . . . . DC voltage identification and weights with N-SML . . . . . . . . Grid d current identification and weights with N-SML . . . Grid q current identification and weights with N-SML . . . DC voltage and grid power factor with N-SML . . . . . . . . . . . Grid d  q currents with N-SML . . . . . . . . . . . . . . . . . . . . . . Grid active and reactive power with N-SML . . . . . . . . . . . . . Control d  q signals with N-SML . . . . . . . . . . . . . . . . . . . . . Influence of rg changes on N-SML . . . . . . . . . . . . . . . . . . . . . Influence of lg changes on N-SML . . . . . . . . . . . . . . . . . . . . . Influence of rl changes on N-SML: 10 MX. . . . . . . . . . . . . . . Influence of rl changes on N-SML: 10 X . . . . . . . . . . . . . . . . Grid faults influences on N-SML . . . . . . . . . . . . . . . . . . . . . . DFIG rotor q currents identification for N-SML . . . . . . . . DFIG rotor d currents identification for N-SML . . . . . . . . DFIG rotor d  q currents with N-SML . . . . . . . . . . . . . . . . . PI-FOC,SM-L, and N-SML controllers trajectory tracking . . . N-SML controllers trajectory tracking zoom . . . . . . . . . . . . . . The DFIG stator d  q currents with N-SML . . . . . . . . . . . . . The DFIG control signals for N-SML . . . . . . . . . . . . . . . . . . . Influence of rs change on N-SML controller . . . . . . . . . . . . . . rs change influence zoom . . . . . . . . . . . . . . . . . . . . . . . . . . . . Influence of ls change on N-SML controller . . . . . . . . . . . . . . ls change influence zoom . . . . . . . . . . . . . . . . . . . . . . . . . . . . Influence of lr change on N-SML controller . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54 54 55 56 56 57 57 58 59 59 60 60 61 61 62 62 64 65 65 66 70 71 72 72 73 73 74 74 75 75 76 76 77 77 78 79 79 80 80 81 81 82 82 83

List of Figures

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

4.57 4.58 4.59 4.60 4.61 4.62 4.63 4.64 4.65 4.66 4.67 4.68 4.69 4.70 4.71 4.72 4.73 4.74 4.75 4.76 4.77 4.78 4.79 4.80 4.81 4.82 4.83 4.84 4.85 4.86 4.87 4.88 4.89 4.90 4.91 4.92 4.93 5.1 5.2 5.3 5.4 5.5 5.6 5.7

lr change influence zoom . . . . . . . . . . . . . . . . . . . . . . . . . . . . Influence of lm change on N-SML controller . . . . . . . . . . . . . lm change influence zoom . . . . . . . . . . . . . . . . . . . . . . . . . . . . Speed variation effects on N-SML . . . . . . . . . . . . . . . . . . . . . Stator powers with N-SML under fault conditions . . . . . . . . . Grid fault influences on SM-L . . . . . . . . . . . . . . . . . . . . . . . . Stator powers with PI-FOC under fault conditions . . . . . . . . . DC voltage identification and weights with N-IOC . . . . . . . . . Grid d current identification and weights with N-IOC . . . Grid q current identification and weights with N-IOC . . . DC voltage and grid power factor with N-IOC . . . . . . . . . . . . Grid d  q currents with N-IOC . . . . . . . . . . . . . . . . . . . . . . . Grid active and reactive power with N-IOC . . . . . . . . . . . . . . Control d  q signals with N-IOC . . . . . . . . . . . . . . . . . . . . . Influence of rg changes on N-IOC . . . . . . . . . . . . . . . . . . . . . Influence of lg changes on N-IOC. . . . . . . . . . . . . . . . . . . . . . Influence of rl changes on N-IOC: 10 MX . . . . . . . . . . . . . . . Influence of rl changes on N-SML: 100 X . . . . . . . . . . . . . . . Grid faults influences on N-IOC . . . . . . . . . . . . . . . . . . . . . . . DFIG rotor q currents identification for N-IOC . . . . . . . . . DFIG rotor d currents identification for N-IOC. . . . . . . . . DFIG rotor d  q currents with N-SML . . . . . . . . . . . . . . . . . N-IOC controllers trajectory tracking . . . . . . . . . . . . . . . . . . . N-IOC trajectory tracking zoom . . . . . . . . . . . . . . . . . . . . . . . The DFIG stator d  q currents with N-IOC. . . . . . . . . . . . . . The DFIG control signals for N-SML . . . . . . . . . . . . . . . . . . . Influence of rr change on N-IOC . . . . . . . . . . . . . . . . . . . . . . rr changes influence zoom . . . . . . . . . . . . . . . . . . . . . . . . . . . Influence of ls changes on N-IOC . . . . . . . . . . . . . . . . . . . . . . ls changes influence zoom. . . . . . . . . . . . . . . . . . . . . . . . . . . . Influence of lr changes on N-IOC . . . . . . . . . . . . . . . . . . . . . . lr changes influence zoom . . . . . . . . . . . . . . . . . . . . . . . . . . . Influence of lm changes on N-IOC . . . . . . . . . . . . . . . . . . . . . lm changes influence zoom . . . . . . . . . . . . . . . . . . . . . . . . . . . Speed variation effects on N-SML . . . . . . . . . . . . . . . . . . . . . Stator powers with N-IOC under fault conditions . . . . . . . . . . Stator powers with PI-FOC under fault conditions . . . . . . . . . DFIG prototype. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DFIG prototype scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DC-link voltage and grid power factor with SM-VOC . . . . . . Grid active and reactive power . . . . . . . . . . . . . . . . . . . . . . . . Grid currents igd and igq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GSC control signals ugcd and ugcq . . . . . . . . . . . . . . . . . . . . . . Constant wind turbine speed . . . . . . . . . . . . . . . . . . . . . . . . . .

xxi

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 85 85 86 86 87 87 90 91 91 92 93 93 94 94 95 95 96 96 97 98 98 99 99 100 100 102 102 103 103 104 104 105 105 106 107 108 110 110 111 112 112 113 114

xxii

List of Figures

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16

Fig. Fig. Fig. Fig.

5.17 5.18 5.19 5.20

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

5.21 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.30 5.31 5.32 5.33 5.34

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

5.35 5.36 5.37 5.38 5.39 5.40 5.41 5.42 5.43 5.44 5.45 5.46 5.47 5.48

Rotor d  q currents tracking with SM-FOC . . . . . . . . . . . . . Stator active and reactive powers tracking with SM-FOC . . . . Stator currents dynamics with SM-FOC . . . . . . . . . . . . . . . . . RSC control signals urq and urd , first case . . . . . . . . . . . . . . . Variable wind turbine speed . . . . . . . . . . . . . . . . . . . . . . . . . . MPPT rotor current tracking with SM-FOC . . . . . . . . . . . . . . MPPT stator active powers with SM-FOC . . . . . . . . . . . . . . . MPPT stator currents with SM-FOC . . . . . . . . . . . . . . . . . . . . MPPT stator power factor and electromagnetic torque with SM-FOC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RSC control signals urq and urd , fault grid case . . . . . . . . . . . DFIG prototype for fault grid conditions . . . . . . . . . . . . . . . . Grid voltage under fault conditions . . . . . . . . . . . . . . . . . . . . . DC voltage and grid power factor under fault grid conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Active and reactive under fault grid conditions . . . . . . . . . . . . DC voltage identification and weights . . . . . . . . . . . . . . . . . . Grid d current identification and weights . . . . . . . . . . . . . . . . Grid q current identification and weights . . . . . . . . . . . . . . . . Rotor d current identification and weights . . . . . . . . . . . . . . . Rotor q current identification and weights . . . . . . . . . . . . . . . DC voltage and grid power factor tracking with N-SMVOC . Grid powers and currents with N-SMFOC . . . . . . . . . . . . . . . Rotor d  q tracking with N-SMFOC . . . . . . . . . . . . . . . . . . . Stator active and reactive powers with N-SMFOC . . . . . . . . . Stator d  q currents with N-SMFOC . . . . . . . . . . . . . . . . . . . MPPT rotor d  q currents with N-SMFOC . . . . . . . . . . . . . . MPPT stator active and reactive powers with N-SMFOC . . . . MPPT stator power factor and electromagnetic torque with N-SMFOC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grid currents and voltages under fault conditions . . . . . . . . . . DC voltage and grid power factor under fault conditions . . . . Stator powers with N-SMFOC under fault conditions . . . . . . . Stator powers with SM-FOC under fault conditions . . . . . . . . Stator currents with SM-FOC and N-SMFOC. . . . . . . . . . . . . RHONN state identification of DC-link . . . . . . . . . . . . . . . . . Weights vector of DC-link . . . . . . . . . . . . . . . . . . . . . . . . . . . RHONN state identification of DFIG rotor currents . . . . . . . . Weights vector of DFIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DC-link voltage and grid power factor with N-SML . . . . . . . Grid powers and currents with N-SML . . . . . . . . . . . . . . . . . . Grid control signals with N-SML . . . . . . . . . . . . . . . . . . . . . . Rotor currents tracking with N-SML . . . . . . . . . . . . . . . . . . . Stator active and reactive powers tracking with N-SML . . . . .

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114 115 115 116 116 117 117 118

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. . . .

118 119 120 120

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

121 121 122 122 123 123 124 124 125 126 126 127 128 128

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

129 129 130 131 131 132 133 133 134 134 135 135 136 137 138

List of Figures

Fig. Fig. Fig. Fig. Fig. Fig.

5.49 5.50 5.51 5.52 5.53 5.54

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

5.55 5.56 5.57 5.58 5.59 5.60 5.61 5.62 5.63 5.64 5.65 5.66 5.67 5.68 5.69 5.70 5.71

Fig. 5.72 Fig. 5.73 Fig. 5.74 Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

5.75 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14

Stator currents dynamics with N-SML . . . . . . . . . . . . . . . . . . RSC control signals with N-SML, case 1 . . . . . . . . . . . . . . . . Rotor currents tracking with N-SML . . . . . . . . . . . . . . . . . . . Stator active and reactive powers tracking with N-SML . . . . . Stator currents dynamics with N-SML . . . . . . . . . . . . . . . . . . Stator power factor and electromagnetic torque with N-SML . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RSC control signals for N-SML, case 2 . . . . . . . . . . . . . . . . . Stator powers with N-SM-L under grid disturbances . . . . . . . Stator powers with SM-FOC under grid disturbances . . . . . . . DFIG stator currents with SM-FOC under grid disturbances. . Grid currents and voltages: normal grid conditions . . . . . . . . . DC voltage and grid power factor: normal grid conditions . . . Grid powers: normal grid conditions . . . . . . . . . . . . . . . . . . . . DFIG rotor shaft speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DFIG rotor currents: normal grid conditions . . . . . . . . . . . . . . DFIG stator powers: normal grid conditions . . . . . . . . . . . . . . DFIG rotor currents: MPPT, normal grid conditions . . . . . . . . DFIG stator powers: MPPT, normal grid conditions . . . . . . . . Grid currents and voltages: single-phase-to-ground fault . . . . . DC voltage and grid power factor: single-to-ground fault . . . . DFIG stator powers: MPPT, single-phase-to-ground fault . . . . Grid currents and voltages: two-phase-to-ground fault . . . . . . DC voltage and grid power factor: two-phase-to-ground fault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DFIG stator powers: MPPT, two-phase-to-ground fault . . . . . Grid currents and voltages: three-phase to ground fault . . . . . DC voltage and grid power factor: three-phase to ground fault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DFIG stator powers: MPPT, three-phase to ground fault . . . . Opal RT-Lab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microgrid electric structure . . . . . . . . . . . . . . . . . . . . . . . . . . . IEEE 9-bus electric structure. . . . . . . . . . . . . . . . . . . . . . . . . . Wind power system configuration . . . . . . . . . . . . . . . . . . . . . . Solar power system configuration . . . . . . . . . . . . . . . . . . . . . . Storage system configuration . . . . . . . . . . . . . . . . . . . . . . . . . The master and console subsystem in Opal-RT . . . . . . . . . . . Three-phase voltages and current: Ideal grid conditions . . . . . WPS Controlled dynamics under ideal grid conditions . . . . . . DFIG d  q rotor currents under ideal grid conditions . . . . . . WPS controlled dynamics: PI, normal grid conditions . . . . . . THD for the NSML and the PI controllers . . . . . . . . . . . . . . . SPS controlled dynamics under ideal grid conditions . . . . . . . Solar inverter d  q currents: Ideal grid conditions . . . . . . . . .

xxiii

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. . . . .

138 139 139 140 140

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

141 141 142 143 143 144 145 145 146 146 147 147 148 149 149 150 150

. . 151 . . 151 . . 152 . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

152 153 156 157 157 158 158 160 167 169 169 170 171 172 172 173

xxiv

List of Figures

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

6.15 6.16 6.17 6.18 6.19 6.20 6.21 6.22

Fig. Fig. Fig. Fig. Fig.

6.23 6.24 6.25 6.26 6.27

Fig. Fig. Fig. Fig. Fig.

6.28 6.29 6.30 6.31 B.1

BB charging and discharging operating modes . . . . . . . . . . . . BB controlled dynamics: Ideal grid conditions . . . . . . . . . . . . Three-phase voltages and current: Phase-to-ground fault . . . . WPS Controlled dynamics: PI, single-phase-to-ground fault . . WPS Controlled dynamics: Phase-to-ground fault . . . . . . . . . . SPS Controlled dynamics: Phase-to-ground fault . . . . . . . . . . BB Controlled dynamics: Phase-to-ground fault . . . . . . . . . . . Three-phase voltages and current: Two-phase-to-ground fault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . WPS Controlled dynamics: Two-phase-to-ground fault . . . . . . WPS Controlled dynamics: PI, two-phase-to-ground fault . . . SPS Controlled dynamics: Two-phase-to-ground fault . . . . . . BB Controlled dynamics: Two-phase-to-ground fault . . . . . . . Three-phase voltages and current: corresponding to Three-phase-to-ground fault . . . . . . . . . . . . . . . . . . . . . . . . . . WPS Controlled dynamics: Three-phase-to-ground fault . . . . . WPS Controlled dynamics: PI, three-phase-to-ground fault. . . SPS Controlled dynamics: Three-phase-to-ground fault . . . . . BB Controlled dynamics: Three-phase-to-ground fault . . . . . . HAWT simplified scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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. . . . . . .

173 174 175 175 176 176 177

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. . . . .

177 178 178 179 180

. . . . . .

. . . . . .

180 181 181 182 182 198

List of Tables

Table Table Table Table Table Table Table Table Table Table Table Table Table

3.1 3.2 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 6.1 6.2

DFIG parameters . . . . . . . . . . . . . . . . . . . . . . . . . Dip voltage types . . . . . . . . . . . . . . . . . . . . . . . . . MSE and STD values for reference tracking . . . . SE and STD values for parameter variations . . . . MSE and STD values for speed variations . . . . . . MSE and STD values for reference tracking . . . . MSE and STD values for parameter variations . . MSE and STD values for speed variations . . . . . . MSE and STD values for reference tracking . . . . MSE and STD values for parameter variations . . MSE and STD values for speed variations . . . . . . The parameters of the Microgrid subsystems . . . . The IEEE-9-bus power generation . . . . . . . . . . . .

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. . . . . . . . . . . . .

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37 39 58 63 64 78 84 84 101 106 107 167 168

xxv

Chapter 1

Introduction

Renewable energy generation is one of the most efficient and effective solutions to face global warming effects. In order to reach better exploration of the renewable energy that is generated from different Distributed Energy Resources (DERs), the microgrid concept is the key [1]. A microgrid is a low to medium voltage distribution network, which includes alternative sources like Wind Power System (WPS), Solar Power System (SPS), energy storage devices as Battery Bank (BB), local loads, and power electronic devices, among other electrical components. The purpose of using appropriate power converters is to facilitate the DERs connection to the main grid, to guarantee high-generated power quality, and to control power flows [2, 3]. Because of disturbances and voltage fluctuations, inappropriate effects can be produced and might affect the operation of DERs installed into microgrids; those disturbances actually presents big challenges for microgrids control [4]. In the past, the DERs were disconnected during grid disturbances [5]. This discontinuity in power generation might affect stability and reliability of the whole network [6, 7]. Taking into account high penetration of renewable energy generation in medium and high voltage networks, modern grid codes enforce the DERs to have LowVoltage Ride-Through (LVRT) capacity, which is the capability of a specific electric generator to stay connected to the main grid in presence of grid disturbances for a short period; additionally, the installed local controllers should warranty microgrid transient stability, enhance the LVRT capacity of the DERs and guarantee the transient stability to enhance resilience of microgrid operations. Wind electric generation is one of the best alternative electric energy source because its availability, and usefulness for power system in diverse areas. Industry experts predict that if this pace of growth continues as is, by 2050, one third of the world’s electricity needs will be fulfilled using wind power [8]. There are two basic types of wind turbines: Horizontal-axis and Vertical-axis WTs. Variable Speed Wind Turbine (VSWT) generators gained momentum as the preferred choice during © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 E. N. Sánchez and L. Djilali, Neural Control of Renewable Electrical Power Systems, Studies in Systems, Decision and Control 278, https://doi.org/10.1007/978-3-030-47443-0_1

1

2

1 Introduction

the early 1990s [9]. Variable speed requires power electronic converters to operate, to allow WTs for capturing more energy, and to smooth the power fluctuations [9]. The size of WTs varies widely such that the length of the blades is the biggest factor in determining the amount of electricity a WT can be generated [10]. For a VSWT, there are many reasons to use a Double Fed Induction Generator (DFIG), such as noise reduction, effort reductions on the WT shaft, reduced cost inverters, and the possibility for stator active and reactive powers control [11]. The DFIG is an induction generator, which can supply voltage to both the stator and rotor; however, is usually the case that a description of the DFIG includes the induction generator and the back-to-back converter connected to the rotor windings [9]. The most knowing configuration of the DFIG based WT is that where the DFIG rotor is connected to the grid via a bidirectional electronic converter, whereas the stator is directly coupled to the grid. The Grid Side Converter (GSC) can be linked directly to the grid or through a step-up transformer, which is used to control the DC voltage at the output of the DC-link, and also as a voltage source to fed the Rotor Side Converter (RSC), which is directly linked to the DFIG rotor winding terminal [11]. Based on such configuration, the DFIG is very sensitive to the presence of non-ideal grid conditions, due to direct connection of its stator to the grid [12]. Additionally, because of disturbances and voltage fluctuations, inappropriate effects can be produced and might affect the operation of WTs, which lead to tripping of individual turbines and perhaps a whole wind farm, and may introduce poor power quality into the network [9]. These disturbances actually presents big challenges for operation of WTs based on DFIG control. Today, big efforts are being done, by both academia and industry, to avoid grid disturbances effects on the DFIG based WT operation as well as to maintain it connected to the grid and to ensure a permanent power production. This operational condition of the DFIG is well known as Low Voltage Ride-Through (LVRT) capacity [13]. To improve the LVRT capacity of the DFIG, different hardware and software solutions have been already proposed. For hardware components, there are different shunt/series devices included in the DFIG electrical configuration such as protection devices/circuits, which are activated during transients; there are also reactive power injecting devices in order to compensate voltage dips [14]. In [7], combination of a rotor crowbar and DC chopper circuits is the most used and effective solution. The crowbar protects the DFIG rotor, while the DC chopper is utilized to limit the peak value of the DC voltage [15]. This solution is presented in [15] and the obtained results illustrate that the capability of crowbar and DC chopper to reduce the DFIG overcurrent and the DC link overvoltage during grid faults. Another method is proposed in [16], where a DC chopper is combined with a Series Dynamic Resistance (SDR), which are used to operate the DFIG in presence of grid faults. Authors in [15] also discusses the difference between the SDR and the crowbar both combined with DC chopper and the results illustrate that the solution based on the SDR circuits presents better performances compared with the solution that based on the crowbar. These solutions are effective, but do the control scheme more complex; control objective also is changed from tracking to stabilization during grid fault, and increase the total cost and power losses [17]. To overcome hardware solu-

1 Introduction

3

tions drawbacks, different control strategies are adopted to achieve LVRT capacity requirements. The most known controllers used to regulate DFIG dynamics are the conventional Proportional- Integral (PI) and the Proportional-Resonance (PR) [13, 14]. To develop PI and PR control scheme for enhancing the LVRT, a decomposition process is needed, which does control implementation more complex [18]; in addition, the PI and PR controllers are inappropriate for trajectory tracking control, which is a required task for DFIG generated active and reactive power [19]. In the last years, Sliding Mode (SM) control has received much attention due to its effectiveness for nonlinear system and its robustness against disturbances [20]. Different articles have been published for the DFIG control scheme using the SM methodology combined with FOC as in [21], where first order sliding mode controller is proposed for the DFIG stator active and reactive power. In [22], second order sliding mode control is proposed to track the desired values of the DFIG stator reactive power and the electromagnetic torque. In [23], real-time SM based on FOC controller is investigated to extract the maximum power from the wind. However, to reach FOC high-performances, a state transformation is used and precise knowledge of the DFIG parameters is needed. Moreover, to ensure stability and robustness of the SM controller, the disturbances upper bound should be known. For applications this knowledge is not available, which does the selection of the controller gain not easy [24]. In the last decades, Direct Power Control (DPC) is proposed as an alternative approach, which is based on the DFIG nonlinear model. In [25], an SM observer combined with input-output feedback linearization is designed to track the reference value for the DFIG flux and torque. A real-time validation of SM block control algorithm is investigated in [26] to regulate the RSC and the GSC to track DC voltage and active and reactive power references. These control approaches are not robust against parameters variations and disturbances, because they are based on the DFIG mathematical model. All the published works cited above are tested under balanced grid conditions. To operate the DFIG under abnormal grid conditions, various control strategies have been proposed. In [27], a decoupled PI controller is used to operate the RSC and the GSC under unbalanced voltage, where a positive and negative sequences of current and voltage should be known. In [28, 29] Proportional Resonant (PR) and Proportional-Integral Resonant (PI+R) current controllers are presented to control a DFIG under fault grid conditions; this controllers still need negative and positive sequences decomposition of the rotor currents. In addition, decoupled PI, PR and PI+R controllers are highly dependent on the DC-link and DFIG parameters. In [14], an SM controller is used for the electromagnetic torque, the stator reactive power, and the DC-link voltage; the results demonstrate the capabilities of the proposed control scheme to reduce electromagnetic torque and DC voltage ripples under fault grid voltage; but this controller is validated using only simulations and does not take into consideration maximum power extraction and the effects of wind speed changes. In [18], an SM direct power control (SMDPC) strategy for DFIG under unbalanced grid conditions using extended active power is developed; real-time results illustrate that the proposed control scheme can reduce electromagnetic torque ripples in presence of unbalanced grid conditions without needing decomposition process, but it ignores

4

1 Introduction

DC voltage control, which presents a challenge to maintain it constant in presence of fault grid conditions. On the other hand, different methodologies using hardware devices and/or modified control schemes have been developed in order to accomplish LVRT requirements for grid-connected microgrid. In [30], a DC microgrid system and control methods using the PI controller for distributed generation are presented and analyzed under single line-to-ground fault on source side, load side, and DC transmission line. In [1], an LVRT control scheme, which improves power quality during abnormal grid conditions is developed where a cascade voltage and current control loop and a modified droop technique are used to perform primary and secondary level control; the later one controls the reactive power injection. In [31], a hierarchical control that consists of primary and secondary layers is proposed, where the primary layer based on the droop controller is in charge of power regulation, while the secondary layer consists of dynamic consensus algorithm, which is responsible of the LVRT operation of the microgrid. The droop technique is widely used for microgrid control applications [1, 32]; it is usually implemented using voltage and current control loops or power and current control ones in cascade configuration [33]. The droop method can be implemented without communication links between the microgrid subsystems, which improves reliability; its drawbacks are largely discussed in the literature as in [34]. In addition, the droop control based LVRT enhancement for DERs, which are connected to low or medium voltage microgrid, has not been adequately addressed [1], and big efforts are being done to develop an efficient LVRT scheme based control to ensure microgrid stability in presence of grid disturbances. Recently, technology advances have forced control engineers to deal with complex systems, which include unknown dynamics, strong interconnection terms, and disturbances [35]. Then, conventional control techniques are unsuccessful to provide an effective solution to control this class of systems [36]. Neural networks (NNs) have the potential for implementing nonlinear system identification and control due to their capabilities to approximate complex systems and to improve control scheme performances [36]. In [37], the DFIG control based on neural identification is investigated, where an RHONN identifier is used to design an inverse optimal controller; the proposed control scheme in this work is tested in real-time but it does not take into consideration the LVRT in presence of grid disturbances, maximum power extraction, and the effects of wind speed changes on the stator powers control. In [38], a combinational protection of neural network based PI controller for a distribution static synchronous compensator and crowbar protection are proposed to enhance transient stability of DFIG in presence of asymmetrical fault grid; the obtained results in this work illustrate the effectiveness of the proposed technique but addition protection device is till used. In [39], a neural network based integral-derivative controller and the classical one are proposed for DFIG and are compared regarding the impact of each controller on the dynamic behavior of the LVRT; results illustrate that the LVRT behavior have been changed regarding the applied controllers and load types. In [40], an intelligent controller based on the Takagi-Sugeno-Kang-type probabilistic fuzzy neural network with an asymmetric membership function is developed for the reactive and active power control of a three-phase grid connected photovoltaic system

1 Introduction

5

during grid faults; the proposed controller leads to regulate the reactive power to a new desired value in order to complies with the LVRT requirements. In [41], a fault detection and identification based on voltage indicators analysis using neural network have been investigated to develop a convenient control strategy for DFIG based wind turbine and to enhance its LVRT behavior in presence of grid faults. The obtained results in [40, 41] illustrate the effectiveness of the proposed schemes for LVRT enhancement grid tied DERs; however, the proposed schemes do not consider large scale power system with more DER components. A recent work to enhance LVRT capacity using NNs is published in [42], where a neural sliding mode FOC scheme is used to control the DFIG in presence of symmetric and asymmetric voltage dips; the proposed control scheme is validated in real-time and the obtained results confirm its performance; however, chattering still occur.

1.1 Motivation An important feature of wind farms and microgrids are their ability to deal with problems which may arise on the distribution or transmission network to which they are connected. Regarding the causes and effects of voltage unbalance and grid disturbances on the DERs power generation, which might affect their normal operation and can lead to tripping of individual DER and perhaps the whole microgrid from the power system, it is necessity to develop alternative control schemes based on neural networks for each DER. Those controllers should ensure robustness to parameter variations and/or grid disturbances, enhance the LVRT capacity of the microgrid, assure transit stability, and accomplish modern grid requirements.

1.2 Objectives The main objective of this book is • To develop a robust control scheme based on neural network identification for enhancing the LVRT capacity of grid-connected DFIG based WT and grid connected microgrid, assuring transit stability, and accomplishing modern grid requirements. To reach the main purpose of this book, the following particular objectives are considered • Synthesize discrete-time robust controllers based on a neural network identifiers. • Evaluate via simulation the performances of the proposed control schemes for the DFIG controlled dynamics considering time-varying reference tracking, parameter variations and speed changing influences on controlled dynamics. • Test via simulation the LVRT capacity of the DFIG controlled by the proposed controller in presence of grid disturbances.

6

1 Introduction

• Implement in real-time the proposed control schemes for the DFIG. • Extend the proposed control schemes for DERs installed in grid connected microgrid. • Test via real-time simulation using Opal RT lab simulator the extended control schemes.

1.3 Main Contributions The main contributions of this work can be summarized as follows: (1) three control schemes based on neural network identifiers are proposed for the DFIG based WT: discrete-time Neural Sliding Mode Field Oriented Control(N-SMFOC) and discrete-time Neural Sliding Mode Linearization control (N-SML), and discretetime Neural Inverse Optimal Control (N-IOC); all of them are tested via simulation using PowerSim toolBox of Matlab considering with the tracking of time-varying reference, robustness to parameter variation, sensitivity to wind speed changing, and robustness in presence of fault grid conditions. In addition they are compared with conventional controllers such as the Proportional-Integral (PI) one among others, (2) the proposed controllers are based on an Recurrent High Order Neural Network (RHNN) identifier trained online using an Extended Kalman Filter (EKF), which helps to obtain an adequate model of the DFIG under different grid scenarios. Those neural models are used to synthesize the control law, which helps to reject variations caused by parameter changes and/or grid disturbances, (3) the proposed control schemes are real-time implemented and tested for enhancing the LVRT capacity of the DFIG without decomposition of the voltage and currents in positive and negative sequences, and without using any additional device, (4) real-time validation of the proposed controllers is done for controlling the DFIG prototype under both balanced and unbalanced grid conditions, (5) a selected control scheme is extended for each DER installed in a grid-connected microgrid, (6) the proposed microgrid composes of Wind Power System (WPS), Solar Power System (SPS), Battery Bank (BB), and loads; in addition it is connected to an IEEE 9-bus system, (7) real-time simulation using opal RT-lab is done for the whole power system in order to test the proposed control scheme LVRT enhancement in presence of different fault grid scenarios.

1.4 Book Outline In this book, different control schemes are proposed for a DFIG connected to the network and grid-connected microgrid. Those controllers are a Neural Sliding Mode Field Oriented, a Neural Sliding Mode Linearization schemes, and a Neural Inverse Optimal Control. First, performances of the developed controllers are evaluated via simulations, considering time-varying reference tracking, robustness in presence of parameter variations, and speed changing. In addition, all of them are tested in pres-

1.4 Book Outline

7

ence of different fault grid conditions. After that, the proposed control schemes are tested in real-time for tracking time-varying power references and for extracting the maximum power from the wind under ideal and no-ideal grid conditions. Then, the proposed Neural Sliding Mode Linearization control scheme is extended for DERs, which are installed in an AC microgrid and linked to an IEEE 9-bus system. Next, The whole system is real-time simulated using an Opal-RT simulator and the proposed control scheme LVRT enhancement is tested in presence of different grid scenarios: single-phase-to-ground, two-phase-to-ground, and three-phase-to-ground. The outline of this work is as follows • Chapter 2 introduces mathematical preliminaries used in the development of this work, including discrete-time sliding mode control, inverse optimal control, discrete-time high order neural network identifier, extended Kalman filtering, and some stability definitions. • Chapter 3 discusses WT power systems based on the DFIG, including the mathematical models of the mechanical and the electrical parts, and definitions about voltage dips and DFIG LVRT capacity. • Chapter 4 investigates control schemes based on the proposed neural network identifiers. The proposed controllers are the discrete-time neural sliding mode field oriented control, the discrete-time neural sliding mode linearization control, and the discrete-time neural inverse optimal control. All controllers are synthesized for the grid side converter and the rotor side converter. Simulation results using PowerSim toolBox of Matlab are discussed considering the tracking of time-varying reference, robustness to parameter variation, sensitivity to wind speed changing, and robustness in presence of grid disturbances. • Chapter 5 presents real-time results of the proposed controllers for the DFIG based WT considering time-varying reference tracking, maximum power extraction, and robustness to grid disturbances. • Chapter 6 extends the proposed discrete-time neural sliding mode linearization control for the microgrid components. The real-time simulation results are presented under normal grid conditions regarding time-varying trajectories tracking and in presence of three grid fault type: single-phase-to-ground, two-phase-toground, and three-phase-to-ground. • Chapter 7 presents the respective conclusions and future works are stated. Additionally, appendices are included at the end of this book as support. In Appendix A, the stability proof of Theorem 4.1 and Theorem 4.2 are included. The wind turbine aerodynamic model, the DC-link, and the DFIG modeling are presented in Appendix B.

Chapter 2

Mathematical Preliminaries

This chapter briefly describes important mathematical preliminaries required in the development of this work, including discrete-time sliding mode control, the inverse optimal control, discrete-time high order neural network identifier, extended Kalman filter, and stability definitions.

2.1 Discrete-Time Sliding Mode Control In the last years, Sliding Mode (SM) control has received much attention due to its effectiveness for nonlinear system and its robustness against disturbances [43]. Most sliding mode approaches are based on finite-dimensional continuous-time models and lead to discontinuous control action. Once a dynamic system is in sliding mode, its trajectory is confined to a manifold in the state space. Generally, this condition may only be achieved by discontinuous control, switching at theoretically infinite frequency. When challenged with the task of implementing sliding mode control in a practical system are two options [26]: • Direct analogue implementation using a very fast switching device. • Discrete-time implementation. The first method is only suitable for systems with a voltage input allowing the use of analogue switching devices. Most systems use a discrete micro-controller based implementation. However, a discontinuous control designed for a continuous-time system model would lead to chattering when implemented without modifications in discrete-time with a finite sampling rate. This chattering is due to the fact that switching frequency is limited to the sampling rate; however, correct implementation of sliding mode control requires infinite switching frequency. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 E. N. Sánchez and L. Djilali, Neural Control of Renewable Electrical Power Systems, Studies in Systems, Decision and Control 278, https://doi.org/10.1007/978-3-030-47443-0_2

9

10

2 Mathematical Preliminaries

Fig. 2.1 Motion trajectory of continuous-time system with scalar SM control

Before developing the concept of discrete-time sliding mode, let us revisit the principle of sliding mode in continuous-time systems with ideal discontinuous control from an engineering viewpoint. A detailed explication may be found in [43, 44]. Consider a general continuous-time system x˙ = f (x, u, t)

(2.1)

where x ∈ n is the state vector of the system, u ∈ m is the input vector, and f (•) is smooth and bounded vector fields. We assume (2.1) describes the planet dynamics. Using a discontinuous scalar control law u defined as  u=

u 0 if s(x) ≥ 0 −u 0 if s(x) < 0

with s(x) the sliding surface and u 0 the upper control bound. Figure 2.1 illustrates the motion trajectory of continuous-time system with scalar SM control, where the state variable x(t) starts from an initial point x(t = 0), the trajectory reaches the sliding surface s(x) = 0 within finite time tsm and remains on the surface thereafter. The following characteristics of SM systems are noted • The time interval between the initial point t = 0 and the reaching one to the sliding surface s(x) = 0 at tsm is finite, which means that systems controlled with SM exhibit convergence to any manifold consisting of state trajectories. • Once the system is in sliding mode for all t ≥ tsm , its trajectory is confined to the sliding surface s(x) = 0, and the order of the closed-loop system dynamics are less than the order of the original uncontrolled system. • After SM has started at tsm , the system trajectory cannot be backtracked beyond the surface s(x) = 0 like in system without discontinuities. In other words, at any point t0 ≥ tsm , it is not possible to determine the time instance tsm or to reverse calculate the trajectory for t ≤ tsm based on system sate information at t0 . When derived for each sampling point t j = kt, k = 1, 2, . . .. The discrete-time representation of the continuous time system in (2.1) is xk+1 = F(xk ), with

2.1 Discrete-Time Sliding Mode Control

11

Starting from the instance tsm= , the state trajectory belongs to the sliding manifold with s(x(t)), or for a ksm ≥ tsm /t, s(xk ) = 0

∀k ≥ ksm

It seems reasonable to call this motion discrete-time sliding mode. Note that the right-hand side of the system motion equation with discrete-time SM is a continuous state function. Next, we have to derive the discrete-time control law which may generate the SM in a discrete-time system. From (2.1), let us suppose that for any constant control u and any initial condition x(0), its solution can be written in closed form as x(t) = F(x(0), u) (2.2) where the control signal u may be chosen arbitrarily. Using (2.2), and following the procedure below 1. At t = 0, select constant u(x(t = 0), t) for a given time interval t such that s(x(t = t)) = 0. 2. Next, at t = t, select a different constant u(x(t = t), t) such that s(x(t = 2t)) = 0. 3. In general, for each k = 0, 1, . . ., at t = kt select constant u(xk , t) such that s(xk+1 ) = 0. In other words, at each sampling point k, select u k such that this constant control signal will achieve s(xk+1 ) = 0 at the next sampling point (k + 1). During the sampling interval, state x(kt < t < (k + 1)t) may not belong to the manifold, i.e. s(x(t)) = 0 is possible for kt < t < (k + 1)t. However, the discrete-time system. However, the discrete-time system is xk+1 = F(xk , u k ) u k = u(xk )

(2.3) (2.4)

hits the sliding manifold at each sampling point, i.e. s(kk+1 ) = 0, ∀k = 0, 1, . . . is fulfilled. Because F(x(0), u) tends to x(0) as t → 0, the function u(x(0), t) may exceed the available control resources u 0 . As a result, the bounded SM controller steers the state xk to zero only after a finite number of steps ksm . Thus, the manifold is reached after a finite time interval tsm = ksm t, thereafter, the state xk remains on the manifold. Similar to continuous-time systems, this motion may be referred to as discrete-time SM. Note that SM is generated in the discrete-time system with control −u 0 ≤ u ≤ u 0 as a continuous function of the state xk and is piecewise constant during the sampling intervals. Definition 2.1 For the discrete-time dynamic system [26] xk+1 = F(xk , u k )

12

2 Mathematical Preliminaries

discrete-time sliding mode takes place on a subset  of the surface s(x) = 0, s ∈ m , if there exists an open neighborhood  of this subset that for each x ∈  it follows that s(F(xk+1 )) ∈ . In contrast to continuous-time systems, SM may arise in discrete-time systems with a continuous function in the right-hand side of the closed-loop system equation.

2.2 Block Control Consider the following perturbed discrete-time nonlinear system [45] xk+1 = f (xk , k) + B(xk )u(xk , k) + d(xk ) yk = h(xk )

(2.5) (2.6)

where k ∈ Z+ ∪ {0} denotes the discrete-time, xk ∈ n is the state vector of the system, u k ∈ m is the input vector, yk ∈  p is the output vector to be controlled, the vector f (•), B(•), h(•), and d(•) are smooth and bounded vector fields. We assume that the nonlinear non autonomous system (2.5) and (2.6) describes the complete plan dynamics. Moreover, d(•) is a disturbances term. System (2.5), (2.6) can be rewritten in the block controllable form as follows xi,k+1 = f (i, x¯i,k , k) + B(i, x¯i,k )xi+1,k + di,k xr,k+1 = f (r, xk , k) + B(r, xi,k )u k + dr,k yk = x1,k , i = 1, . . . , r − 1

(2.7)

T T T    where xk = x1,k · · · xi,k · · · xr,k , x¯i,k = x1,k · · · xi,k , dk = d1, k · · · di,k · · · dr,k , i = 1, . . . , r − 1, and the numbers set (n 1 , . . . , n r ) define the structure of (2.7), satisfying n 1 ≤ n 2 · · · ≤ n r ≤ m. Let us define the following transformation z 1,k = x1,k − x1,r e f,k z 2,k = x2,k − x2,r e f,k  −1   = x2,k − B(1, xi,k ) K 1 z 1,k − f (1, x1,k , k) − d1,k .. . zr,k = xr,k − xr,r e f,k

(2.8)

where yd,k = x1,r e f,k is the desired trajectory, xi,r e f,k is the desired value for xi,k with i = 1, 2, . . . , r , and K i is a Schur matrix. Using (2.8), (2.7) can be expressed as

2.2 Block Control

13

z 1,k+1 = K 1 z 1,k + B1 z 2,k .. . zr −1,k+1 = K r −1 zr −1,k + Br −1 zr,k zr,k+1 = f (r, xk , k) + B(r, xi,k )u k + dr,k − xr,r e f,k+1

(2.9)

To design the control law, the sliding mode block control technique is used. The manifold can be derived from the block control procedure as sk = zr,k . Thus, (2.9) is rewritten using the new variables as z 1,k+1 = K 1 z 1,k + B1 z 2,k .. . zr −1,k+1 = K r −1 zr −1,k + Br −1 sk sk+1 = f (r, xk , k) + B(r, xi,k )u k + dr,k − xr,r e f,k+1

(2.10)

The control objective is to force the output x1,k to track a reference signal x1,r e f,k and rejecting the effects of disturbances d(xk ). Then, the system (2.10) can be used to synthesize the control input u(xk , k) as state in Theorem 2.1. Theorem 2.1 ([26]) For system (2.5), the control law defined as u(xk , k) = u eq (xk , k) + u s (xk , k) ensures trajectory tracking, with u eq (xk , k) is determined using discrete-time sliding modes [43] and u s (xk , k) is a stabilizing term. The corresponding proof is detailed in [36]. From (2.10), we obtain the equivalent control law is calculated as [20]   u eq (xk , k) = −B(r, xk )−1 f (r, xk , k) − d(xk ) + xr,r e f,k+1

(2.11)

Applying u(xk , k) = u eq (xk , k) to (2.5), the state of the closed loop system reaches the sliding manifold in one sample time. However, it is appropriate to add a stabilizing term u s (xk , k), in order to reach the sliding manifold asymptotically u s (xk , k) = B(r, xk )−1 (kc sk )

(2.12)

where kc is Schur matrix [46]. Applying u(xk , k) = u eq (xk , k) + u s (xk , k) to (2.10), the sliding manifold is (2.13) sk+1 = (kc sk ) Then, the sliding manifold sk = 0 is reached asymptotically. Taking into account the boundedness of the control signal u(xk , k) < u 0 , u 0 > 0, the following control law is selected [43]

14

2 Mathematical Preliminaries

u(xk , k) =

⎧ ⎪ ⎨

u(xk , k)

⎪ ⎩ u0

u eq (xk ,k)

u eq (xk ,k)

  if u eq (xk , k) ≤ u 0   if u eq (xk , k) > u 0

(2.14)

where • stands for the Euclidean norm. The stability proof using (2.14) is presented in [20]. In order to implement this control law, the precise knowledge of the system parameters and the upper perturbation bound are needed. This fact does the selection of the gain controller not easy; for the general case, a high gain controller is selected, which produces chattering and may leads to instability of the controlled system [24, 43]. For overcome this problem, an RHONN identifier trained on-line with an EKF is proposed. This method helps to obtain an adequate model of the controlled system. Based on such neural identifier, the controller will be synthesized.

2.3 Inverse Optimal Control: Tracking Problem Consider perturbed discrete-time nonlinear system (2.5), (2.6). Taking the output variable to be controlled yk contains the full state vector (case of the system under study), the objective is to force the controlled dynamics to track selected trajectories, then the tracking error is as follows ek = xk − xr e f,k

(2.15)

with xr e f,k the desired trajectory. The error dynamics at k + 1 is expressed by ek+1 = f (xk , k) + B(xk )u(xk , k) + dk − xr e f,k+1

(2.16)

The respective cost function is minimized by solving the Hamilton–Jacobi–Bellman (HJB) Partial Differential Equation (PDE). However, this class of equations solution is difficult to obtain [47]. For tracking trajectory, the cost function of system (2.16) is selected as ∞   l(ek ) + u kT Ru k (2.17) J (ek ) = k=0

where J : n → + is a performance measure, l : n → + is a positive semidefine function, and R : n → nxm is a positive real symmetric matrix. When the cost function J is optimal, it is noted as J ∗ and it will be defined as Lyapunov function. From the optimality principle of Bellman, for the infinite horizon optimization case, the selected Lyapunov function V (ek ) is time-invariant and should satisfy the discrete-time Bellman equation defined as follows V (ek ) = min l(ek ) + u kT R(ek )u k + V (ek+1 ) uk

(2.18)

2.3 Inverse Optimal Control: Tracking Problem

15

Hence, the discrete-time Hamiltonian equation is expressed as follows H (ek , u k ) = l(ek ) + u kT R(ek )u k + V (ek+1 ) − V (ek )

(2.19)

The optimal control law is obtained from the following equality H (ek , u k ) = 0

(2.20)

and the gradient of (2.19) right-hand side is calculated with respect to u k [45], then ∂ V (ek+1 ) 1 u ∗k = − R(ek )−1 B(xk )T 2 ∂ek+1

(2.21)

and V (0) = 0 is the boundary condition of V (ek ) which should be satisfied, u ∗k is the optimal control law. Using (2.21) in (2.18), the discrete-time HJB equation is V (ek ) =

∂ V (ek+1 ) 1 ∂ V T (ek+1 ) R(ek )−1 B(xk )T + l(ek ) + V (ek+1 ) 4 ∂ek+1 ∂ek+1

(2.22)

Obtaining the solution of the HJB-PDE (2.22) for V (ek ) is not simple. To do that, The discrete-time IOC technique and a Lyapunov function are used to synthesize the respective control law [37, 47]. To state the above problem as an IOC one, the following definition is established. Definition 1: ([45]) For system (2.5), the control law in (2.21) is considered to be IOC (globally) stabilizing if: • (i) it ensures that (2.21) is (global) asymptotic stability for ek = 0, and • (ii) it minimizes the cost function (2.17), for which the V (ek ) is positive definite function such that V := V (ek+1 ) − V (ek ) + u ∗k B(xk )u ∗k ≤ 0

(2.23)

Thus, the IOC synthesizing is based on the V (ek ) previous definition. Then, Definition 2: ([45]) Let select V (xk ), which is established to be radially bounded positive definite function such that for each xk there exist u k , and V (ek , u k ) < 0

(2.24)

where V (ek ) is a discrete-time Control Lyapunov Function (CLF), which should be defined to satisfy conditions in i) and ii) of the definition 1. To do that, the CLF is selected as follows 1 (2.25) V (ek ) = ekT Pek 2 with P ∈ nxn , and P = P T > 0. By selecting an appropriate matrix P, the control signal (2.21) guarantees the equilibrium point ek = 0 of (2.16) stability. Additionally,

16

2 Mathematical Preliminaries

the control law (2.21) with (2.25), which is considered as an inverse optimal control law for (2.5) optimizes the meaningful cost function in (2.17). Additionally, by using (2.21) in (2.25), the IOC law is established as follows u ∗k

1 =− 2

1 R + B(xk )T P B(xk ) 2

−1

  B(xk )T P f (xk ) − xr e f,k+1

(2.26)

where P and R are positive definite matrices. To implement the discrete-time IOC scheme, a precise mathematical model of the controlled system is needed. In addition, this control algorithm is not robust in presence of parameter variation, unknown dynamics, and the external disturbances, which are always exist in real applications. To avoid this defects and improve the robustness of this control algorithm, an RHONN identifier trained on-line with an EKF is proposed. By using the neural identifier an adequate mathematical model of the controlled system is obtained, which helps to synthesize to IOC, and improves its robustness.

2.4 Neural Networks Identification and Control 2.4.1 Discrete-Time Recurrent High Order Neural Networks Recentness crucial achievement on Neural Networks (NN) techniques, do them a good alternative for nonlinear control. An RHONN identification consists of adjusting the parameters of an appropriate selected model according to an adaptive law [35, 36]. Using a series-parallel configuration, the estimated state variable of the nonlinear system in (2.1) by RHONN identifier with n neurons and m inputs is given by [48] (2.27) χi,k+1 = wiT φi (xk , u k , k) where χi,k is the  state of the ith neuron which identifies the ith component of xk , xk =  x1,k , . . . , xn,k is the state vector, wi,k ∈  L i are the adjustable synaptic weights of NN, u ∈ m is the input vector of NN and φi (xk , u k , k) ∈  L i is defined as ⎡  di j (1) ⎤ j∈I1 ζi j  di j (2) ⎥ ⎥ ⎢ ⎢ ⎥ j∈I2 ζi j ⎥ ⎢ ⎢ ⎢ ⎥ φi (xk , u k , k) = ⎢ ⎥=⎢ . ⎥ . ⎦ ⎣ ⎣ ⎦ .  d (L ) ij i φi L i ,k j∈I L ζi j ⎡

φi1 φi1,k .. .

⎤

(2.28)

i

where di j,k are non-negative integers. L i is the connections number. I1 , I2 , . . . , I L i is a non-ordered subset collection of 1, 2, . . . , n + m, n is the state dimension, m is the inputs dimension, and ζi is defined as

2.4 Neural Networks Identification and Control

17

Fig. 2.2 RHONN scheme

⎡ ⎤ ⎤ S(x1 ) ζi1 ⎢ . ⎥ ⎢ .. ⎥ ⎢ .. ⎥ ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ S(xn ) ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ζ ζi = ⎢ in ⎥ = ⎢ ⎥ ⎢ . ⎥ ⎢ u 1,k ⎥ ⎣ .. ⎦ ⎢ .. ⎥ ⎣ . ⎦ ζin+m u m,k ⎡

(2.29)

where u = [u 1,k , u 2,k , . . . , u m,k ]T is the input vector to the network. The function S(.) is a hyperbolic tangent function defined as S(xk ) = αi tanh(βi xk )

(2.30)

where xk is a state variable; α and β are positive constants. To approximate the nonlinear model (2.1), the discrete-time RHONN in (2.27) is modified as [35] χi,k+1 = wiT φi (xk ) + iT ϕi (xk , u k )

(2.31)

where wi,k represents the adjustable weights, and i,k is fixed weights, ϕi is a linear function of the state vector or vector input u k depending to the system structure or external inputs to the RHONN model. An ith RHONN scheme is depicted in Fig. 2.2.

2.4.2 Extended Kalman Filter Training Algorithm The best well-known training approach for recurrent networks is the back propagation through time learning. However, it is a first order gradient descent method and hence its learning speed could be very slow [36, 48]. Recently, EKF based algorithms have been introduced to train NNs [49]. The training objective is to minimize the error

18

2 Mathematical Preliminaries

between the real and the estimated state for each i-th neuron. The EKF training algorithm is written as follows [49] K i,k = Pi,k Hi,k Mi,k wi,k+1 = wi,k + ηi K i,k ei,k T Pi,k+1 = Pi,k − K i,k Hi,k Pi,k + Q i,k −1  T Mi,k = Ri,k + Hi,k Pi,k Hi,k ei,k = xi,k − χi,k

(2.32)

where ei ∈ R is the respective identification error, Pi ∈ R L i ×L i is the prediction error associated covariance matrix, wi ∈ R li is the weights vector, L i is the number of NN weights, ηi is a design parameter, K i,k ∈ R L i ×m is the Kalman gain matrix, Q i,k ∈ R L i ×L i and Ri,k ∈ R m×m are the state and the measurement noise associated covariance matrices, Hi ∈ R L i ×m is a matrix, which is the derivative of each NN state xi to respect to the weights wi j which is calculated as 

Hi j,k

∂χi,k = ∂wi j,k

T (2.33) wi,k =wi,k+1

where {i = 1, . . . , n} and { j = 1, . . . , L i } and k is the iteration number. Pi , Q i and Ri are diagonal matrices.It is important to note that Hi,k , K i,k and Pi,k for the EKF are bounded [50]. Therefore, there exist constants H¯ i > 0, K¯ i > 0 and P¯i > 0 such that:    Hi,k  ≤    K i,k  ≤    Pi,k  ≤

H¯ i K¯ i

(2.34)

P¯i

The training is performed on-line, using a series-parallel configuration as displayed in Fig. 2.3.

2.4.3 Neural Networks for Control Systems Neural networks have become a well-established methodology as exemplified by their applications to identification and control of general nonlinear complex systems; the use of high order NNs for modeling and learning has recently increased [35]. Using NNs, control algorithms can be developed to be robust to uncertainties and modeling errors. The most used NN structures are Feed-forward networks and Recurrent ones. The latter type offers a better suited tool to model and control of nonlinear systems [36, 48]. There exist different training algorithms for neural networks,

2.4 Neural Networks Identification and Control Fig. 2.3 Identification Scheme

19

Unknown System EKF

Neural Identifier

which, however, normally encounter technical problems such as local minimum, slow learning, and high sensitivity to initial conditions, among others. As a viable alternative, new training algorithms, for example, those based on Kalman filtering, have been proposed [36, 48, 49]. Artificial NNs have been used to formulate a variety of control strategies [35]. There are two basic approaches [35]: • Direct control system design: It means that the controller is a neural network. A neural network controller is often advantageous when the real-time platform available prohibits complicated solutions. The implementation is simple while the design and tuning are difficult. • Indirect control system design: It is the selected control configuration for this report. The approach is comprised of an identification model, whose parameters are updated on-line in such a way that the error between the neural identifier output and the system model output is very small. The controller receives information from the neural identifier and outputs signals, which forces the plant to perform a prespecified task. The controller type which are used in this configuration are conventional controllers such as sliding mode, feedback linearization, block control, and . . . etc. The model is typically trained in advanced, but the controller is designed on-line. As will appear, the indirect design is very flexible; thus it is the most appropriated for most of the common control problems. A block diagram of the indirect adaptive control architecture is pictured in Fig. 2.4 The increasing use of NN to modeling and control is in great part due to the following features that makes them particularly attractive [51]: • NN are universal approximators. It has been proven that any continuous nonlinear function can be approximated arbitrarily well over a compact set by a multilayer NN, which consist of one or more hidden layers [52]. • Learning and adaptation. The intelligence of NNs comes from their generalization ability with respect to unknown data. Online adaptation of the weights is possible.

20

2 Mathematical Preliminaries

Fig. 2.4 Indirect neural control scheme

Neural Identifier Controller Unknown System

2.5 Stability Definitions Definition 2.2 ([36, 53]) The system (2.3) is said to be forced or to have an input. In contrast, the system described by an equation without explicit presence of an input u k , that is, (2.35) xk+1 = F(xk ) is said to be unforced or autonomous. It can be obtained after selecting the input u k as a feedback function of the state u k = ξ(xk )

(2.36)

xk+1 = F(xk , ξ(xk ))

(2.37)

Such substitution eliminates u k

and yields unforced system (2.37). Definition 2.3 A subset S ∈ n is bounded if there exists r > 0 such that xk ≤ r for all x ∈ S, [53]. Definition 2.4 ([36]) The solution of (2.3) and (2.36) is semiglobally uniformly ultimately bounded (SGUUB), if for any , a compact subset of n and all xk0 ∈ , there exists an  > 0 and a number N (, xk0 ) such that xk <  for all k ≥ k0 + N . In other words, the solution of (2.3) is said to be SGUUB if, for any a priori given (arbitrarily large) bounded set  any a priori given (arbitrarily small) set 0 , which contains (0, 0) as an interior point, there exists a control law (2.36) such that every trajectory of the closed loop system starting from  enters the set 0 = {xk | xk < } in a finite time and remains in it thereafter [36].

2.5 Stability Definitions

21

Theorem 2.2 ([36]) Let a Lyapunov function for a discrete-time system (2.3), which satisfies the following properties γ1 ( xk ) ≤ V (xk ) ≤ γ2 ( xk ) V (xk ) = V (xk+1 ) − V (xk ) ≤ −γ3 ( xk ) + γ3 ()

where  is a positive constant, γ1 (•) and γ2 (•) are strictly increasing functions, and γ3 (•) is continuous, nondecreasing function. Thus if V (xk ) < 0 for xk > , then xk is uniformly ultimately bounded, i.e., there is a time instant k T , such that

xk < , ∀k < k T . Theorem 2.3 ((Separation Principle), [54]) The asymptotic stabilization problem of the system (2.3)–(2.36), via estimated state feedback u(k) = ξ(χk ) χk+1 = F(χx , u k , yk )

(2.38)

is solvable if and only if the system (2.3)–(2.36) is asymptotically stabilizability and exponentially detectable.

Chapter 3

Wind System Modeling

Wind electric generation is one of the best alternative electric energy sources because of its huge availability and usefulness for power systems in diverse areas. For a VSWT, there are many reasons to use the DFIG, such as noise reduction, effort reductions on the WT shaft, reduced cost inverters, and the possibility for stator active and reactive powers control. In this chapter, the HAWT mechanical and electrical parts models are presented. In addition, the grid voltage dips types, the grid disturbances influences on the DFIG operation, and the LVRT capacity requirements are introduced.

3.1 Wind Power Technologies 3.1.1 Wind Turbine Electrical Generators Before a discussion of WT modeling, a general overview of wind power technologies is presented to justify where the DFIG fits within different types of technologies used for wind power generation, as illustrated in Fig. 3.1. In terms of the turbine speed, there are two main classifications of WT generator systems, constant speed and variable speed, which requires power electronic converters to operate with generators, allowing wind turbines to capture more energy, and smoothing the power fluctuations by essentially using the rotor as a flywheel. Power electronics offers the most effective method for providing variable speed operation and it can supply reactive power to the utility grid [9]. The DFIG is currently the most popular electric machine for HAWT power applications. The majority of large capacity WTs of more than 1 MW available from manufacturers such as Vestas, General Electric-Wind, etc. use DFIGs [55]. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 E. N. Sánchez and L. Djilali, Neural Control of Renewable Electrical Power Systems, Studies in Systems, Decision and Control 278, https://doi.org/10.1007/978-3-030-47443-0_3

23

24

3 Wind System Modeling Mechanical Energy Source (Wind)

Direct Driven

Multi-polar Synchrounous Machine

Large Power Electronic Converter

Conventional Synchrounous Machine

Permanent Magnet Machine Large Power Electronic Converter

Induction Machine

Cage Rotor Machine

Wound Rotor Machine

Large Power Electronic Converter

Small Power Electronic Converter

DFIG

Wound Rotor Machine

Gear box

Electrical Grid

Fig. 3.1 Wind turbines electrical generators

3.1.2 HAWT Components HAWTs have the main rotor shaft and electrical generator at the top of a tower, and may be pointed into or out of the wind. Most have gearboxes, which turn the slow rotation of the blades into a quicker rotation which is more suitable to drive electrical generators. The major components of modern HAWTs are presented in Fig. 3.2, [55, 56] 1. Rotor Blades: The turbine blades are made of high density wood, glass fiber reinforced polyester. Modern WTs have 2 or 3 blades [56]. 2. Gear box: Two types of gear box are used in wind turbines [56]. • Parallel shaft: It is used in small turbines, its design is simple, its maintenance is easy, high mass material and low power. • Planetary: It is used in large turbines, it requires complex design and skilled person for its maintenance.

3.1 Wind Power Technologies

25

Pitch Control Low-speed shaft Gearbox

Aerogenerator

Generatro

Wind Direction

Brake

Anemometer

Controller

Yaw drive Yaw motor Blade

Tower

High-speed shaft

Nacelle

Wind vane

Fig. 3.2 Components of a HAWT

3. Electric generator: Conversion of mechanical power into electrical one can be accomplished by one of the following types of the electrical Machine: Synchronous, Induction, and Direct Current.

3.2 Wind Power System Modeling A typically configuration of a DFIG based WT is presented in Fig. 3.3. The proposed VSWT model is composed of the following systems • Aerodynamic model where the turbine torque is valuated as a function of the wind speed and the low speed turbine shaft. • Gearbox model which is used to transmit the captured power from the low speed turbine shaft to the high speed turbine one. • High speed turbine shaft model, which presents the mechanical equation of the electrical machine. • Pitch model, evaluates the pitch angle dynamics as a function of pitch reference angle. • Electrical machine and power converters transform the generator power into a grid current as a function of voltage grid.

26

3 Wind System Modeling Grid DFIG Wind

Back-to-Back Converter

Pref

MPPT Algorithm

Proposed Control Algorithm

Qref

Pitch Control Fig. 3.3 DFIG based WT grid connection

3.2.1 Mechanical Part Model For HAWT, the air flow is presented in Fig. 3.4, with v1 is the upstream wind speed, v2 is the downstream wind speed, v is the speed of the wind passing through the

Downstream Upstream

v1

v S

S1 Wind

Fig. 3.4 Air flow at a wind turbine

v2 S2

3.2 Wind Power System Modeling

27

aerogenerator, S1 and S2 are respecting the upstream and downstream sections of the air tube, S is the surface swept by the blade. The available wind power is given by Pw =

1 ρ Sv13 2

(3.1)

The ratio between the extracted power from the wind and the available total one is   2   2   v2  v2   Paer o 1  Cp = = 1 −  1 +  Pw 2 v1   v1 

(3.2)

Let define the relative speed λ as λ=

v2 t r = v1 v

(3.3)

with ρ is the air density, v is the wind speed, t is the rotational speed of the turbine, and r is the radius of a blade. So the expression of the extracted power for the wind is defined as a function of the power coefficient C p as follows Paer o =

1 C p ρ Sv 3 2

(3.4)

The power coefficient C p represents the yield aerodynamics of the WT, which depends on the turbine characteristics defined by the following empirical formula for variable speed WT [57]     −c6 1 x C p (λ, β) = c1 c2 − c3 β − c4 − c5 exp  

(3.5)

1 with c1 = 0.5; c2 = 116; c3 = 0.4; c4 = 0; c5 = 5; c6 = 21 and 1 = λ+0.08β − 0.035 , where λ is the relative speed, β is the blade angle. Figure 3.5 displays the 1+β 3 curves of the power factor λ for different β values. This factor presents a theoretical limit, called the Betz limit, which is equal to 0.593 [58]. The gearbox is a mechanical device capable of transferring torque loads from a primary mover to a rotary output, typically with a different relation of angular velocity and torque. In the case of WTs the gearbox connects the low-speed shaft and the generator; therefore its gear ratio generally is dictated by the requirement of the generator and the angular velocity of the rotor [59]. In the case of electrical power production with an asynchronous generator, the output of the gearbox (which is connected to the generator) usually operates in the ranges from 50 Hz to 60 Hz or 1, 500 rpm to 1, 800 rpm. This depends on the frequency of the grid to which the generator is connected, and on the number of poles of the generator [60]. In many of today’s modern machines the generator is able to operate at a greater range of

28

3 Wind System Modeling 0.4 X: 9.7 Y: 0.3556

Power Coefficient C

p

0.3 0.2 0.1 0 -0.1

=2

0

=4

0

= 60

-0.2

=8

0

= 10

-0.3 0

5

0

10

15

Relative Speed Fig. 3.5 Power coefficient C p according to λ and β

two Parallel stages

Output shaft

Planetary stage

Input shaft

Fig. 3.6 Schematics of a typical 2 MW gearbox design

speeds. Regardless of this advantage a speed increment still is required, although it might not be as significant as that needed for the asynchronous generator [59]. There are two main types of gearboxes, parallel shaft gearboxes and planetary gearboxes as seen in Fig. 3.6. • Parallel Shaft Gearbox: Parallel shaft gearboxes are a collection of simple gear stages. Each gear stage is composed of two shafts, a gear, and a pinion. For a gearbox that is designed to increase the angular velocity, the gear is on the input shaft and the pinion is on the output shaft [59]. • Planetary Gearbox: The epicyclic or planetary gearbox offers several advantages compared to the parallel-shaft configuration, including a higher gear ratio in a smaller “package”. This compact configuration has the advantage of reducing the

3.2 Wind Power System Modeling

29

overall mass of the gearbox, which is an important requirement for wind turbine gearboxes, because the head mass of the turbine is kept to a minimum. Additionally, the planetary configuration has the capability of handling greater torque loads [59]. A gearbox is typically used in a WT to increase rotational speed from a low-speed rotor to a high-speed electrical generator. It is supposed to be rigid and modeled by a simple gain. The elasticity and the friction of the multiplier are neglected. Energy losses in the multiplier are considered zero. The mechanical torque of the WT is composed of the multiplier ratio to get the mechanical torque on the generator shaft as follows Tmec = G1 Tt (3.6) mec = Gt where Tmec is the mechanical torque (N − m), Tt is the aero-generator torque (N − m), mec is the mechanical generator speed (rad/s), t is the turbine speed (rad/s), G is gear ratio gain. The fundamental equation of the High Speed Shaft is written as follows [21] J

dmec + f mec = Tmec − Tem dt

(3.7)

where Tmec is the mechanical torque (N − m), Tt is the aero-generator torque (N − m), mec is the mechanical generator speed (rad/s), J is the total rotating parts inertia (K g.m 2 ), f is the friction coefficient, G is gear ratio gain which connects the turbine and the generator. Figure 3.7 presents a HAWT block diagram. Nowadays, pitch angle control systems are commonly employed in medium to large WTs to keep the captured wind power close to the rated value above the rated wind

−

= −

(

)

−

+ =

+

− =

− Fig. 3.7 Block diagram of Wind Turbine

−

30

3 Wind System Modeling

Fig. 3.8 Schematic of HAWT pitch control system

speed, [21, 57], by adjusting the blades angle. Figure 3.8 illustrates an example of pitch control system installed on HAWT. In general, pitch modeling is done in three steps 1. Generating the reference angle βr e f . 2. Regulation of the orientation angle. 3. Regulation of the speed of variation of the angle. Figure 3.9 gives the pitch control system modeling steps The blade angle regulation is achieved by generating a reference rate of angle change, which is obtained by saturated Proportional-Integral (PI); the error of instantaneous electrical power and its reference is considered to be controlled. to adjust the pitch angle, a servo-mechanism system is considered where its constant time is directly related to the power of the wind system. Then, the pitch angle is obtained by integrating the angle variations. Figure 3.10 shows the pitch control system block diagram.

3.2.2 Electrical Machine and Power Converter Models Today, most of the installed WTs are variable speed based on DFIGs [55]. The DFIG rotor is coupled to the grid via a power electronic converter, while the DFIG stator is linked directly to the grid as presented in Fig. 3.11; this configuration allows to decouple the network frequency and the DFIG rotation speed [55]. In fact, a GSC is linked to the grid via the step-up transformer terminals, which provides a stable DC voltage at the output. The DC-link voltage is used for feeding the RSC, which is directly coupled to the rotor winding terminals of the DFIG.

3.2 Wind Power System Modeling

31

Fig. 3.9 Schematic of HAWT pitch control system

Pref

+ −

Pelec

β max βref + PI β min

. β max angle controller

. β . β min

1 s

β

Fig. 3.10 Pitch control

By using the control system, the bidirectional converter assures energy generation at nominal grid frequency and nominal grid voltage independently of the rotor speed. The converter main aim is to compensate for the difference between the speed of the rotor and the synchronous speed [22, 55]. The main characteristics may be summarized as follows • • • • •

Limited operating speed range (−30% to +20%) Small scale power electronic converter (reduced power losses and price) Complete control of active power and reactive power exchanged with the grid Need for slip-rings Need for gearbox

The following sub-sections present the DC-link and the DFIG models.

32

3 Wind System Modeling

Greabox

DFIG

Grid

Ps , Q S

Back To Back Converter DC AC

AC DC

RSC V dc GSC

Pg , Q g

Transformer

Wind Turbine

Fig. 3.11 DFIG based wind turbine

3.2.2.1

DC-Link Model

The RSC is connected via the DC-link to the GSC, which is in turn connected to the stator terminals directly or through a step-up transformer. The circuit of the GSC can be consider as a STATCOM, as shown in Fig. 3.12. Amusing balanced conditions, the AC-side circuit equations can be described as lg

abc di g,t

dt

abc abc = −r g i ga,t − u abc g,t + u gc,t

(3.8)

abc are the three-phase grid currents (A), u abc where i g,t g,t are the three phase grid voltages abc (V ), u gc,t are the three phase GSC voltages (V ), which are present the control input of the DC-link voltage, r g is the grid line resistance () and l g is the grid inductance (H ).

Fig. 3.12 DC link electrical configuration

3.2 Wind Power System Modeling

33

In order to understand the dynamic behavior of the GSC, the space vector representation is used. By means of this representation tool, it is possible to use the differential equations defining the behavior of the GSC variables such as the currents and voltages [55]. In addition, by applying the space vector notation to the GSC three-phase modeling (3.8) it is possible to represent the electrical equations in d − q reference frame. There are normally three reference frames; stationary reference frame (usually referred to the applied voltage of the electrical system), synchronous reference frame (usually aligned with internal flux, stator, air-gap or rotor flux), and a rotor reference frame (aligned with the rotating shaft). One of the advantage of the d − q reference frame is in the ease in which control schemes can be implemented [61]. The mathematical model of the GSC in the d − q reference frame can be obtained by applying the Park transformation to (3.8), as follows [12] rg di gd,t 1 1 = − i gd,t + ωg i gq,t − u gd,t + u gdc,t dt lg lg lg di gq,t rg 1 1 = −ωg i gd,t − i gq,t − u gq,t + u gqc,t dt lg lg lg

(3.9) (3.10)

where i gd,t , i gq,t are the d − q component of the grid current, u gd,t , u gq,t are the grid voltage d − q component, and u gdc,t − u gqc,t are the GSC voltage d − q component. Neglecting the harmonics distortion due by the electronic switches, the GSC and electrical transformer losses, and the power balance between the ac and dc sides of the GSC, the DC voltage at the output of the DC link can be determined as [12]   3 dUdc,t = i gd,t u gd,t + i gq,t u gq,t dt 2cg Udc,t

(3.11)

Using the Euler method, The discrete-time mathematic model of DC-link in the d − q reference frame is defined as [26]   3 u gd,k i gd,k + ϒ1,k Udc,k+1 = Udc,k + ts 2CUdc,k   Rg 1 i gd,k+1 = i gd,k − ts i gd,k + u gcd,k − ϒ2,k Lg Lg   Rg 1 i gq,k+1 = i gq,k − ts i gq,k + u gcq,k − ϒ3,k Lg Lg with Udc,k = 0, where ϒ1,k = u gq,k i gq,k , ϒ2,k = −ws i gq,k −

1 u , ϒ3,k L g gd,k

(3.12) (3.13) (3.14) = ws i gd,k

− are the DC-link interconnection terms, which includes the step-up transformer voltages. The grid active and reactive power flows between the grid and the GSC are [12, 26] 1 u L g gq,k

34

3 Wind System Modeling

Pg,k = i gd,k u gd,k + i gq,k u gq,k

(3.15)

Q g,k = i gd,k u gq,k − i gq,k u gd,k

(3.16)

and the grid power factor is obtained by [12, 26] f g,k = 

Pg,k 2 Pg,k + Q 2g,k

(3.17)

with ωg is the synchrony frequency (rad/s), i gd,k , i gq,k are the grid d − q currents, u gdc,k , u gqc,k , u gd,k and u gq,k are the d − q voltages in the GSC and step-up transformer respectively, r g is the grid line resistances (), l g is the grid line inductances of (H ), cg is the DC-link capacitance (F), ts is the sample time. For more details about the DC-link modeling see Appendix B.2.

3.2.2.2

DFIG Model

The most used electric machines for HAWT is the DFIG. There are many reasons to use a DFIG, such as noise reduction, effort reductions on the WT shaft, reduced cost inverters and the possibility for stator active and reactive powers control [11]. The DFIG rotor is coupled to the grid via power electronic converters (rectifier, filter, and inverter), while the DFIG stator directly is linked to the grid; this configuration allows a decoupling between the grid frequency and the DFIG rotation speed [22]. The DFIG stator and rotor is presented in Fig. 3.13 where θ is the displacement between the stator and rotor windings. The three phase stator and rotor mathematical model of the DFIG is described as • The DFIG stator equations

abc dφs,t abc = u abc s,t − r s i s,t dt abc abc abc φs,t = L ss i s,t + Msr ir,t

(3.18) (3.19)

• The DFIG rotor equations

abc dφr,t abc abc = u r,t − rr ir,t dt abc abc abc = L rr ir,t + Mr s i s,t φr,t

(3.20) (3.21)

abc with u abc s,t are the three-phase stator voltages (V ), i s,t are the three-phase stator curabc abc are the three-phase stator flux (W b), u r,t are the three-phase rotor rents (A),φs,t

3.2 Wind Power System Modeling

35

Fig. 3.13 DFIG electrical configuration

stator

rotor

abc abc voltages (V ), ir,t are the three-phase rotor currents (A), φr,t are the three-phase rotor flux (W b), rs is the stator winding per phase leakage resistance (), rr is the rotor winding per phase leakage resistance (), L ls is the stator winding per phase leakage inductance, L ss is the stator self inductance, L lr is the rotor winding per phase leakage inductance, L rr is the rotor self inductance, Msr is the stator to rotor mutual inductance, Mrr is the rotor to stator mutual inductance, Ms is the stator mutual inductance, and Mr is the rotor mutual inductance. The ideal machine can be described by a set of six first-order differential equations, one for each winding. These differential equations are coupled to each other by the mutual inductances between the windings. The stator-to-rotor coupling terms are a function of the rotor position, therefore when the rotor rotates the coupling terms change with time. This problem is solved when the induction machine equations are transferred to the quadrature d − q rotating reference frame [9]. By using the Park transformation [26], the DFIG stator components in the d − q stator reference frame are obtained as

dφsd,t = u sd,t − rs i sd,t + ωs φsq,t dt dφsq,t = u sq,t − rs i sq,t − ωs φsd,t dt φsd,t = ls i sd,t + lm ir d,t φsq,t = ls i sq,t + lm irq,t

(3.22) (3.23) (3.24) (3.25)

36

3 Wind System Modeling

and the DFIG rotor components in the d − q reference frame are given as dφr d,t dt dφrq,t dt φr d,t φrq,t

= u r d,t − rr ir d,t + (ωs − ωr )φrq,t

(3.26)

= u rq,t − rr irq,t − (ωs − ωr )φr d,t

(3.27)

= lr ir d,t + lm i sd,t = lr irq,t + lm i sq,t

(3.28) (3.29)

with lm = 23 Ms is the stator-rotor mutual inductance (H ), ls = L ls + lm is the stator indcutance (H ), and lr = L lr + lm is the rotor inductance (H ), i sd,t , i sq,t , ir d,t , and irq,t are the stator and rotor d − q currents (A), φsd,t , φsq,t , φr d,t , and φrq,t are the stator and rotor d − q flux (W b), u sd,t , u sq,t , u r d,t , and u rq,t are the stator and rotor voltages (V ), ωs and ωr are the stator and rotor angular velocities (rad/s). The detail of the DFIG modeling is explained in Appendix B.3. To obtain the state representation of the DFIG in the d − q reference frame, a combination of (3.22)–(3.29) is needed. The selection of the DFIG state vector is variated depending the control objective. In this study, and taking into account the control objective, which is the control of stator active and reactive power generated by the DFIG through the control of the rotor quantities, and considering the real-time implementation of the designed controller, the DFIG state vector is selected as the stator and rotor d − q currents. Using (3.24), (3.25) in (3.22), (3.23) and (3.28), (3.29) in (3.26), (3.27) respectively, then di sd,t dt di sq,t ls dt dir d,t lr dt dirq,t lr dt ls

dir d,t dt dirq,t + lm dt di sd,t + lm dt di sq,t + lm dt

+ lm

= u sd,t − rs i sd,t + ls ωs i sq,t + lm ωs irq,t

(3.30)

= u sq,t − rs i sq,t − ls ωs i sd,t − lm ωs ir d,t

(3.31)

= u r d,t − rr ir d,t + lr (ωs − ωr )irq,t + lm (ωs − ωr )i sq,t

(3.32)

= u rq,t − rr irq,t − lr (ωs − ωr )ir d,t − lm (ωs − ωr )i sd,t

(3.33)

di

Substituting didtr d,t and dtrq,t obtained from (3.32), (3.33) in (3.30), (3.31) and the terms di sd,t di sq,t , dt calculated from (3.30), (3.31) in (3.32), (3.33), the state representation of dt the DFIG is determined as di sd,t dt di sq,t dt dir d,t dt dirq,t dt

= a1 i sd,t + (ωs + a2 ωr ) i sq,t + a3 ir d,t + a4 irq,t + d1 u sd,t − b1 u r d,t

(3.34)

= − (ωs + a2 ωr ) i sd,t + a1 i sq,t − a4 ir d,t + a3 irq,t + d1 u sq,t − b1 u rq,t

(3.35)

= a5 i sd,t − a6 ωr i sq,t − a7 ir d,t + (ωs − a8 ωr ) irq,t − d2 u sd,t + b2 u r d,t

(3.36)

= a6 ωr i sd,t + a5 i sq,t − (ωs − a8 ωr ) ir d,t − a7 irq,t − d2 u sq,t + b2 u rq,t

(3.37)

3.2 Wind Power System Modeling

37

In order to present the DFIG d − q state model in the discrete-time domain, the Euler method is used, then   i sd,k+1 = i sd,k + ts a1 i sd,k + (ωs + a2 ωr ) i sq,k + ϒ4,k − b1 u r d,k   i sq,k+1 = i sq,k + ts − (ωs + a2 ωr ) i sd,k + a1 i sq,k + ϒ5,k − b1 u rq,k   ir d,k+1 = ir d,k+1 + ts a7 ir d,k + (ωs − a8 ωr ) irq,k + ϒ6,k + b2 u r d,k   irq,k+1 = irq,k + ts − (ωs − a8 ωr ) ir d,k − a7 irq,k + ϒ7,k + b2 u rq,k l2 −rs , a2 = σ lms lr , a3 = σlmlsrlrr , a4 = σlmlr ωr , d1 = σ1ls , b1 = σllms lr , σ ls  l2 lm , a7 = σrlrr , a8 = σ1 , d2 = σllms lr , b2 = σ1lr , and σ = 1 − lsmlr , ϒ4,k = σ lr

with a1 =

(3.38) (3.39) (3.40) (3.41)

a5 = σlmlsrlsr ,

a6 = a3 ir d,k + a4 irq,k + d1 u sd,k and ϒ5,k = −a4 ir d,k + a3 irq,k + d1 u sq,k , are the DFIG stator interconnecting terms, which include the rotor currents and the stator voltages. ϒ6,k = a5 i sd,k − a6 ωr i sq,k − d2 u sd,k , and ϒ7,k = a6 ωr i sd,k + a5 i sq,k − d2 u sq,k are the DFIG rotor interconnecting terms, which include the stator currents and voltages. For more detail about the DFIG state representation model see [26]. The DFIG stator active and reactive power are defined as

Table 3.1 DFIG parameters Meanings Base power (Pb ) Base voltage (Ub ) Grid reactance (X l ) Grid resistance (Rg ) DC link capacitance (C Base angular frequency (wb ) Angular moment of inertia Stator resistance (Rs ) Rotor resistance (Rr ) Stator reactance Rotor reactance Magnetizing reactance (X m ) GSC switching frequency RSC switching frequency

Ps,k = u sd,k i sd,k + u sq,k i sq,k

(3.42)

Q s,k = u sq,k i sd,k − u sd,k i sq,k

(3.43)

Units 185 VA 179.63 V 0.0045 pu 0.0014 pu 0.185 pu 379.99 rad/s 0.23 s 0.1609 pu 0.0502 pu 2.4308 pu 2.4308 pu 2.3175 pu 4860 Hz 1620 Hz

38

3 Wind System Modeling

The electromagnetic torque is given by   Tem,k = plm i sq,k ir d,k − i sd,k irq,k

(3.44)

with p is the number of pole pairs. The DFIG prototype are given in Table 3.1.

3.3 LVRT and Voltage Dips 3.3.1 LVRT Definition Due to direct grid connection of the DFIG to the utility grid, grid outages affect the DFIG behavior. In addition, modern wind farms must be operated as conventional power plants due to agreements for clean energy following the Kyoto Protocol. Due to grid disturbances and grid voltage fluctuations, inappropriate effects can be produced, which might affect the operation of distributed generators such as wind power systems; those disturbances actually present big challenges for DFIG operation [1]. In the past, Distributed Energy Resources (DERs) were preferred to be disconnected during grid disturbances [5]. This discontinuity in power generation might affect stability and reliability of the whole network [6, 7] taking into account high penetration of renewable energy generation in medium and high voltage networks. Considering these facts, modern grid codes enforce the DERs to have Low-Voltage Ride-through (LVRT) capacity, which is the capability of the electrical generator to stay connected to utility grid in presence of low voltage for a short period; local controllers should guarantee DFIG transient stability as established in IEEE 519—1992 [62] and IEEE 1547—2003 [63]. The control strategy must allow the wind turbine [55]: • To remain connected to the power system without consuming active power during faults. • To be provided with reactive power during faults to allow voltage recovery. • To return to normal operation conditions after faults.

3.3.2 Voltage Dip Definition Voltage dips are the most common power disturbance; it is defined as brief reductions in voltage, typically lasting from a cycle to a second or tens of milliseconds to hundreds of milliseconds. Longer periods of low or high voltage are referred to as “undervoltage” or “overvoltage” [55]. They normally occur as a result of network disturbances, with the degree of disturbance mainly determined by amplitude reduction/increase and time duration. Voltage dips are normally due to events within the network, e.g. the occurrence and termination of a short-circuit fault or other extreme increase in current due to motor starting, transformer switching etc. In networks with

3.3 LVRT and Voltage Dips Table 3.2 Dip voltage types Fault type Voltage dip (phase and neutral) Three-phase Single-phase Phase-to-phase Three-phase-toground

A B C E

39

Voltage dip Voltage dip (phase and after a phases y or y transformer

Voltage dip phases after after a y y transformer

A C D F

A C D G

A D C F

WTs a single-phase, two-phase or three-phase voltage dip in the network can result in voltage unbalance at the point of common coupling with the generators [9]. In three-phase grids, dips can be divided into two broad categories [55]: 1. Symmetric dips, when voltages of the three phases fall in the same proportion. 2. Asymmetric dips, where all three phase drops are not equal and the voltage is unbalanced. Bollen and co-workers propose a more intuitive approach to the characterization of three-phase voltage dips [61]. The ABC classification method distinguishes between seven dip types (A to G) by analyzing possible types of short circuits and dip propagation through transformers. The behavior of the doubly fed induction machine under voltage dips is well detailed in Chap. 6 of [55]. The results of this chapter is resumed as (Table 3.2): • In case of voltage dip, the stator flux may be separated into three components: Positive, negative, and natural ones. • The positive component is always present in normal or abnormal operation conditions, this positive component rotates at synchronous speed. It induces an electromagnetic force in the rotor proportional to the machine slip. • The negative component only appears in cases of asymmetric voltage dips since it is generated by negative sequence of the grid voltage. It also rotates at synchronous speed but in the opposite direction. • The natural component is a transient one which appears due to voltage variations. In case of total symmetric voltage dips, the natural flux is constant, does not rotate and its amplitude decays exponentially from its initial value to zero. In case of partial symmetric voltage dips the stator flux is divided on two component: (a) the forced flux, which its amplitude is constant proportional to the grid voltage and it is rotating at grid frequency. (b) natural flux, which is constant and its amplitude decays exponentially from its initial value to zero. In case of asymmetric voltage dips, the natural flux is a function of voltage dip as well it depends on the fault type (phase-to ground, phase-to-phase) and on the time of appearance.

40

3 Wind System Modeling

• As the natural and the negative fluxes rotate at relatively high speed with respect to the rotor, they induce voltages in the rotor that are significantly greater than those appearing under normal operation. • If these overvoltages exceed the limits of the rotor converter, control of the current is lost momentarily or even permanently. In this situation, overcurrents appear which can damage the converter.

3.4 Conclusion In this chapter, the variable speed wind turbine both mechanical and electrical parts are discussed. For the mechanical part, the aerodynamic, gearbox, high speed, and pitch models are introduced and for the electrical part, the DC-link and DFIG models are presented. In addition, an introduction to voltage dips and LVRT capacity of the DFIG generator is stated. Presence of grid disturbances, which can produce a high ripples in the generated active and reactive power, and in the DC-link voltage present a big challenge for the DFIG control. To avoid that, several methodologies using hardware devices and/ or modified control schemes have been developed in order to accomplish LVRT requirements. For hardware components, there are different shunt/series devices are included in the DFIG electrical configuration such as protection devices/circuit, which are activated during transient; there are also reactive power injecting devices in order to compensate voltage dips. However, these solutions are effective, but they do the control scheme more complex, increase the total cost and power losses. To overcome the hardware solutions drawback, different control strategies are adopted to achieve the LVRT capacity requirements. In the next chapter, different control algorithms are developed for the DFIG prototype. The proposed control schemes are evaluated under normal and abnormal grid conditions.

Chapter 4

Neural Control Synthesis

In this chapter, the Neural Sliding Mode Field Oriented Control (N-SMFOC), the Neural Sliding Mode Linearization (N-SML), and the Neural Inverse Optimal Control (N-IOC) schemes are proposed for DFIG prototype. By using neural identifiers, an adequate model of the DFIG and of the DC-link are obtained for different grid scenarios, which helps the controllers to reject disturbances caused by fault grid conditions and/or parameter variations. The proposed controllers performances are evaluated for time-varying tracking reference, parameter variations, and win d speed changing effects. In addition, both of them are tested in presence of three different grid fault conditions: single-phase-to-ground, two-phase-to-ground, and three-phase-toground.

4.1 Neural Sliding Mode Control This DFIG control scheme includes two parts: (a) The GSC controller, which allows to regulate the DC-link voltage and the electric power factor at the step-up transformer independently. (b) The RSC controller, which permits to control independently the DFIG stator active and reactive powers. To develop control schemes for the GSC and the RSC, it is usually assumed that all parameters are known. In reality, this assumption is not fulfilled. Hence, in order to improve robustness in presence of parameter variations and disturbances, an RHONN is used for the DC-link and the DFIG identifications. In order to select the RHONN identifier, the following considerations are taken into account • The neural identifier structure is mainly based on the plant mathematical model.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 E. N. Sánchez and L. Djilali, Neural Control of Renewable Electrical Power Systems, Studies in Systems, Decision and Control 278, https://doi.org/10.1007/978-3-030-47443-0_4

41

42

4 Neural Control Synthesis

• The selected neural identifier should be implemented using the minimum possible number of sensors that helps to reduce the control scheme cost. • Taking into account the neural identification properties such as its capability to absorb the perturbation [36], the interconnection terms are considered as disturbances. • The proposed neural identifier should reduce the coupling in the mathematical model and simplify the control scheme. Taking into consideration neural identification properties such as the capability to absorb disturbances [36], the proposed neural model (4.1) and (4.2) for system (2.5), (2.6) is rewritten as ˆ n (xk ) χk+1 = fˆ(xk ) + Bu ˆ yˆk = h(xk )

(4.1) (4.2)

where χk ∈ n is the estimated state vector, xk ∈ n is the state vector of the system, u k ∈ m is the input vector, yˆk ∈  p is the estimated output vector to be controlled, Bˆ is a square matrix presents the control matrix composed by the fixed weights, ˆ and the vector fˆ(•), and h(•) are smooth and bounded vector fields. To design the neural controllers, the discrete-time separation principle is applied as established in Theorem 2.3, which allows to determine the NN identifier without taking into account the controller and vice versa [36, 54]. Taking the output variable to be controlled yˆk = xk contains the full state vector and the relative degree of the system equal to one (case of the system under study), the objective is to force the controlled dynamics to track selected trajectories, so let define the tracking error as follows sk = xk − xr e f,k

(4.3)

Using (2.7)–(2.10), we obtain the sliding surface at k + 1 is expressed as follows ˆ n (xk ) − xr e f,k+1 sk+1 = fˆ(xk ) + Bu

(4.4)

The N-SM controller is synthesized as stated in Theorem 4.1. Theorem 4.1 For system (4.1) and (4.2), the neural sliding mode control law u n (xk ) = u eqn (xk , k) + u sn (xk , k)

(4.5)

ensures that the respective tracking error (4.3) is ultimately bounded, achieving that the system outputs converge to a neighborhood of their respective desired trajectory. The corresponding proof is included in Appendix A.1. From (4.4), the control law u eqn (xk ) is calculated from the sliding variable sk+1 = 0 [20], as follows   (4.6) u eqn (xk , k) = − Bˆ −1 fˆ(xk ) − xr e f,k+1

4.1 Neural Sliding Mode Control

43

Applying u n (xk , k) = u eqn (xk , k) to (4.4), the state of the closed loop system reaches the sliding manifold in one sample time. However, it is appropriate to add a stabilizing term u sn (xk , k), in order to reach the sliding manifold asymptotically u sn (xk , k) = Bˆ −1 (kc sk )

(4.7)

where kc is Schur matrix [46]. Applying u n (xk , k) = u eqn (xk , k) + u sn (xk , k) to (4.4), the sliding manifold is (4.8) sk+1 = (kc sk ) Then, the sliding manifold sk = 0 is reached asymptotically. Taking into account the boundedness of the control signal  u n (xk , k) < u 0 , u 0 > 0, the following control law is selected [43] ⎧   ⎪ if u eqn (xk , k) ≤ u 0 ⎨ u n (xk , k) u n (xk , k) = . (4.9)   ⎪ u eqn (xk , k) > u 0 ⎩ u 0 u eqn (xk ,k) if u (x ,k)   eqn

k

4.1.1 GSC Controller To force the capacitor voltage u dc to track the desired values u dcr e f , and to maintain the grid power factor f g at the nominal value, this N-SMVOC scheme is proposed. In order to simplify the DC-link voltage control, the Voltage Oriented Control (VOC) approach is applied. This method consists of aligning the grid voltage u gd,k on d axis, forces u gq,k equal to zero [26], which ensures direct control of the DC voltage Udc,k by means of the i dg,k current and independent control for the powers flowing between the GSC and the grid [64]. The discrete-time voltage oriented model of the DC-link can be obtained from (3.12) and (3.12) as follows

3 Udc,k+1 = Udc,k + ts u gd,k i gd,k 2CUdc,k

Rg 1 1 i gd,k+1 = i gd,k − ts i gd,k + u gcd,k + ws i gq,k + u gd,k Lg Lg Lg

Rg 1 i gq,k+1 = i gq,k − ts i gq,k + u gcq,k − ws i gd,k Lg Lg

(4.10) (4.11) (4.12)

The grid active and reactive power at the step-up transformer are established as [64] Pg,k = u gd,k i gd,k Q g,k = −u gd,k i gq,k

(4.13)

44

4 Neural Control Synthesis

Based on the above consideration, the proposed RHONN identifier for the DC-link is selected as Uˆ dc,k+1 = w11 S(Udc,k ) + w12 S(Udc,k )S(i gq,k ) + 1 i gd,k iˆgd,k+1 = w21 S(i gd,k ) + w22 S(i gq,k ) + w23 S(Udc,k ) + 2 u gcd,k iˆgq,k+1 = w31 S(i gq,k ) + w32 S(i gd,k ) + 3 u gcq,k

(4.14) (4.15) (4.16)

where Uˆ dc,k , iˆgd,k and iˆgq,k are the estimate dynamics of Udc,k , i gd,k and i gq,k respectively, wi j are the adaptive NN weights, and i are the fixed NN weights with i = 1, . . . , 3 and j = 1, . . . , 3. From (4.14), the N-SM control for the DC voltage eq is determined using Theorem 4.1 such that the equivalent control i gd is calculated form s1n,k+1 as eq

i gd =

1 (Udcr e f − w11 S(Udc,k ) − w12 S(Udc,k )S(i gq,k )) 1

(4.17)

s is defined as and the stabilizing term i gd,k s = k1c s1n,k i gd,k

(4.18)

where s1n,k = u dcr e f − uˆ dc is the DC voltage sliding surface. The i dgr e f,k is equal to the control law of the DC voltage. The grid −q− current reference i gqr e f,k is given by  i gqr e f,k = −i gd,k

1 − f gr2 e f f gr e f

(4.19)

Let define xˆ g,k = [iˆgd,k , iˆgq,k ]T , whose expression at k + 1 written as xˆ g,k+1 = fˆg (xk ) + Bˆ g Ugc,k

(4.20)



w21 S(i gd,k ) + w22 S(i gq,k ) + w23 S(u dc,k ) i 2 0 , xk = gd,k , , Bˆ g = fˆg (xk ) = i gq,k w31 S(i gq,k ) + w32 S(i gd,k ) 0 3



u gcd,k eq . Using Theorem 4.1, the equivalent control vector Ugc,k = u gcq,k eq eq [u gcd,k , u gcg,k ]T is given by and Ugc,k =

eq Ugc,k = Bˆ g−1 (xr e f,k − fˆg (xk )) s = [u sgcd,k , u sgcq,k ]T is defined as The stabilizing term vector Ugc,k

(4.21)

4.1 Neural Sliding Mode Control

45

s Ugc,k =

k2 0 s 0 k3 2n,k

(4.22)

where s2n,k is the sliding surface of the grid d − q current defined as s2n,k = xˆ g,k − xr e f,k .

4.1.2 RSC Controller The control objective for the RSC is to force the rotor currents ir d and irq to track the desired references trajectories ir dr e f and irqr e f defined form specified stator active Psr e f and reactive Q sr e f powers dynamics. From (3.42), (3.43), and (3.43), there exists a strong coupling between flux, voltages, and currents, which causes the DFIG control to be particularly difficult. In order to simplify the control algorithm, we apply FOC approach [11, 64]. It is assumed φsq,t = 0 u sd,t = 0

and and

φsd,t = φs u sq,t = ωs φs = u s

(4.23) (4.24)

Taking into account simplifications (4.23) and (4.24) and substituting in (3.24) and (3.25), we obtain the stator currents as us Lm − ir d,t ws L s Ls Lm =− irq,t Ls

i sd,t =

(4.25)

i sq,t

(4.26)

and the expression for the stator powers are [11, 12] Lm irq,t Ls Lm u 2s − us ir d,t Qs = L s ωs Ls

Ps,t = −u s

(4.27) (4.28)

In order to obtain the Field Oriented Model of the DFIG, we need to establish the relationship between the rotor currents and voltages as follows [11, 64] 1 Rr dir d,t = u r d,t − ir,td + gωs irq,t dt Lr σ Lr σ dirq,t 1 Rr L m us = u rq,t − irq,t − gωs ir d,t + g dt Lr σ Lr σ Ls Lr σ with g = (ws − wr )/ws is the slip.

(4.29) (4.30)

46

4 Neural Control Synthesis

Using the Euler method, the discrete-time Field Oriented Model of DFIG is written as

Rr 1 (4.31) ir d,k + gωs irq,k + u r d,k Lr σ Lr σ

Rr L m us 1 − ts irq,k − gωs ir d,k + g + u rq,k (4.32) Lr σ Ls Lr σ Lr σ

ir d,k+1 = ir d,k − ts irq,k+1 = irq,k

with ts is sample time. Considering the DFIG field oriented model [12], the RHONN identifier for the DFIG is proposed as iˆr d,k+1 = w11,k S(ir d,k ) + w12,k S(ir d,k )S(irq,k ) + w13,k S(irq,k ) + 1 u r d,k(4.33) iˆrq,k+1 = w21,k S(irq,k ) + w22,k S(ir d,k )S(irq,k ) + w23,k S(ir d,k ) + 2 u r d,k(4.34) eq,k

Using Theorem 4.1, the equivalent control u r Ureq =

eq

ur d eq u rq

is given by

= −Br−1



fˆd fˆq

(4.35)

with Br = [1 0; 0 2 ], fˆd = w11,k S(ir d,k ) + w12,k S(ir d,k )S(irq,k ) + w13,k S(irq,k ), and fˆq = w21,k S(irq,k ) + w22,k S(ir d,k )S(irq,k ) + w23,k S(ir d,k ). s is defined as The stabilizing term u r,k

s Ur,k

u rs d,k = s u rq,k

= Br−1 K r s3n,k

(4.36)

kc4 0 is included in order to reach the sliding surface asymp0 kc5 totically such that K r is a Schur matrix, s3,k is the DFIG rotor d − q currents T  sliding surface defined as s3n,k = xˆr,k − xrr e f,k , xˆr,k = iˆr d,k , iˆrq,k and xrr e f,k =  T ir dr e f,k , irqr e f,k . The rotor d − q currents reference is defined from the desired stator active and reactive power as follows

where K r =

Psr e f,k ls u s lm Q sr e f,k ls us =− + . u s lm ωs lm

irqr e f,k = −

(4.37)

ir dr e f,k

(4.38)

Grid Currents (A)

4.2 Simulation Results

47

(a)

2

iga

0

i i

-2 5

5.02

5.04

5.06

5.08

5.1

5.12

5.14

5.16

5.18

gb gc

5.2

Grid Voltages (V)

t [Sec]

(b) 200

u

0

u

ga gb

ugc

-200 5

5.02

5.04

5.06

5.08

5.1

5.12

5.14

5.16

5.18

5.2

t [Sec] Fig. 4.1 Grid currents and voltages: normal grid condition

4.2 Simulation Results These proposed controllers as applied to the DC-link and the DFIG respectively have been simulated using SimPower System tools of Matlab.1 The DFIG prototype is connected to an infinite bus through three-phase transmission lines.

4.2.1 GSC Controller To evaluate the N-SMVOC performances for tracking of the DC voltage desired value and to keep at the unit value the grid power factor, simulation results for the GSC are presented considering time-varying reference trajectory tracking, robustness to DClink parameter variations, load resistances, and grid disturbances. Figure 4.1 presents the three-phase grid currents (a) and voltages (b) as applied to the GSC considering balanced grid conditions. The grid voltage has a peak-to-peak value of 220 V applied to the GSC corresponding amplitude currents of 1.8 A peak-to-peak value. • Time-Varying Reference Tracking The objective of this experiment is to test the proposed controller capabilities for tracking the DC voltage time-varying reference and to keep constant at the unit value the grid power factor independently. To do that, the DC-link parameter are maintained at their nominal values, the load resistance is selected to be constant, and normal grid conditions are considered, (Fig. 4.1). Figures 4.2, 4.3 and 4.4 present 1 Matlab,

Simulink. de 1994–2018 ©The Math Works, Inc.

48

4 Neural Control Synthesis

DC Voltage (V)

(a)

100

Udc

real

Uestimate

0 0

10

20

30

40

50

60

70

80

90

DC Voltage Weights

t [Sec]

(b)

40

1,1 1,2

20

1,3 1

0 0

10

20

30

40

50

60

70

80

90

t [Sec]

Grid -d- Current (A)

Fig. 4.2 DC voltage identification and weights

(a)

3 2

igd

1 0 0

iigd estimate

10

20

30

40

50

60

70

80

90

Grid -d- weights

t [Sec]

(b)

2

1,1 1,2

1

1,3

0 0

1

20

40

60

80

100

120

140

160

180

200

t [Sec] Fig. 4.3 Grid −d− current identification and weights

the DC link neural identification according to the RHONN identifier presented in (4.14)–(4.16) and the behavior of the neural weights. The identification results are included as follows: Fig. 4.2 displays the DC voltage estimation (a) and the respective weights behavior (b). Figure 4.3 displays the grid −d− current estimation (a) and the RHONN weights behavior (b). The grid −q− current estimation and its RHONN weights are presented in Fig. 4.4a and b respectively. The proposed RHONN identifier for each DC-link variables is composed of adjustable weighs and fixed ones. From

Grid -d- Current (A)

4.2 Simulation Results

49

(a)

2

igd

1

i

gq estimate

0

-1 0

10

20

30

40

50

60

70

80

90

Grid -q- weights

t [Sec]

(b)

2 1

1,1 1,2

0

1

0

20

40

60

80

100

120

140

160

180

200

t [Sec] Fig. 4.4 Grid −q− current identification and weights

DC-link Voltage (V)

(a)

100 U

0 0

dc

Udc ref

10

20

30

40

50

60

70

80

90

Grid Power Factor

t [Sec]

(b) 1

0.9 0

fg f

10

20

30

40

50

60

70

80

g ref

90

t [Sec] Fig. 4.5 DC voltage and grid power factor with N-SMVOC

the obtained result of the DC-link identification, it is clear to see that the proposed neural identifier provides a good identification for DC-link real state vector and also the RHONN identifier dynamics and also the neural weights are bounded. Figure 4.5 presents the DC-link voltage (a) and the grid power factor (b) as controlled by the N-SMVOC used for the GSC. From the obtained results, it is clear to see that the proposed N-SMVOC controller forces the dynamics to reach adequately their desired values which are time-varying for the DC-link voltage, its amplitude is changed from 150 to 90 V; and the grid power factor desired value is selected to be constant at the

4 Neural Control Synthesis

Grid -d- Current (A)

50

(a)

3 2 1

i

0 0

10

20

30

40

50

60

70

gd

80

90

Grid -q- Current (A)

t [Sec]

(b)

5 0

igq

-5 0

10

20

30

40

50

60

70

80

90

t [Sec]

Active Power (W)

Fig. 4.6 Grid d − q currents with N-SMVOC

(a)

150

Reactive Power (VAr)

100 50 0 0

i

10

20

30

40

50

60

70

gd

80

90

t [Sec]

(b)

100 0

-100

i

0

10

20

30

40

50

60

70

80

gq

90

t [Sec]

Fig. 4.7 Grid active and reactive power with N-SMVOC

unit value. Figure 4.6 displays the grid d − q currents flow between the grid and the DFIG rotor. The grid active and reactive power are illustrated in Fig. 4.7a and b respectively. The control u gcd and u gcq signals as applied to the GSC are presented in Fig. 4.8a and b respectively. From the obtained results, it is clear to observe that the grid current on axis −d− is related to the DC voltage behavior; the grid current on axis −q− depends on the grid power factor f g dynamics. In addition, the GSC controller forces the DC-link voltage to track a desired time-varying trajectory and keeps at the unit value the grid power factor. Moreover, the controlled dynamics and

Control -d- Signal (V)

4.2 Simulation Results

(a)

50 0

-50 0

Control -q- Signal (V)

51

ugdc

10

20

30

40

50

60

70

80

90

t [Sec]

(b)

50

ugqc

0

-50 0

10

20

30

40

50

60

70

80

90

t [Sec]

Fig. 4.8 Control d − q signals with N-SMVOC

the control signals u gcd and u gcq are bounded. We can conclude that the proposed controller ensures good tracking performances for the DC voltage and the grid power factor independently, even in presence of time-varying references. • Robustness to Parameter Variations To examine the robustness of the proposed controller N-SMVOC in presence of parameter variations, the DC-link parameters and the load resistance values are changed. The obtained results are presented as follows: Fig. 4.9 illustrates the dynamDC-link Voltage (V)

(a)

100

Udc U

0 0

10

20

30

40

50

60

70

dc ref

80

90

Grid Power Factor

t [Sec]

(b) 1

0.8 0

f f

10

20

30

40

50

t [Sec] Fig. 4.9 Influence of r g changes on N-SMVOC

60

70

80

g g ref

90

4 Neural Control Synthesis

DC-link Voltage (V)

52

(a)

100

U U

0 0

10

20

30

40

50

60

70

dc dc ref

80

90

Grid Power Factor

t [Sec] (b) 1 f

0.8 0

g

fg ref

10

20

30

40

50

60

70

80

90

t [Sec] Fig. 4.10 Influence of l g changes on N-SMVOC

ics of the DC-link (a) and the grid power factor (b) controlled by the N-SMVOC when the grid resistance r g is increased by 200% of its nominal value. Figure 4.10 presents the dynamics of the DC voltage and the grid power factor (a) and (b) respectively as controlled by the N-SMVOC when the grid inductance l g is augmented by 100% of its nominal value. The influences of the load resistance rl on the DC voltage (a) and the grid power factor (b) are presented in Figs. 4.11 and 4.12 respectively. The resistance load rl is changed from 10  to 10 M. From the obtained results, it is DC-link Voltage (V)

(a)

100

Udc U

0 0

10

20

30

40

50

60

70

dc ref

80

90

Grid Power Factor

t [Sec]

(b) 1

0.8 0

fg f

10

20

30

40

50

t [Sec] Fig. 4.11 Influence of rl changes on N-SMVOC: 10 M

60

70

80

g ref

90

4.2 Simulation Results

53

DC-link Voltage (V)

(a)

100

0 0

Udc Udc ref

10

20

30

40

50

60

70

80

90

Grid Power Factor

t [Sec]

(b) 1

0.8 0

f

g

fg ref

10

20

30

40

50

60

70

80

90

t [Sec] Fig. 4.12 Influence of rl changes on N-SMVOC: 10 

very clear to observe that the parameter variations and the resistance load changes have no impact on the DC voltage and the grid power factor tracking trajectories controlled by N-SMVOC. In addition, the proposed controller ensures decoupling between the d − q axes. • Robustness to Grid Disturbances The objective of this test is to test the performance of the proposed controller in presence fault grid conditions. In order to do so, the DC voltage is forced to track a constant value equals to 150 V and the following grid disturbances are simulated, (Fig. 4.13). This fault grid is defined as momentary voltage dips. • From t = 5 s to 7 s, a single-phase-to-ground fault is applied, leading to increase the magnitude of two-phase 220 V to 450 V and to decrease the magnitude of the other phase. • From t = 10 s to 12 s, an asymmetric (two-phases) voltage dips is introduced by provoking a phase-to-phase fault grid. This fault leads to increase the voltage amplitude of two phases and increases the voltage amplitude of other phase. • From t = 15 s to 17 s, symmetric voltage dip is applied by introducing a threephase-to-ground fault, the three-phase voltage amplitude is decreased. • Normal conditions are considered for the remaining time. Note that each voltage dip is applied for 2 s which is more than 30 cycle for grid of 60 Hz and less than 3 s. According to I E E E1159−1995, Fig. 4.14 presents the dynamics of the DC voltage (a) and the grid power factor (b) as controlled by the

54

4 Neural Control Synthesis

DC-link Voltage (V)

Fig. 4.13 Grid voltages and currents

(a)

200 150

Udc Udc ref

100

Grid Power Factor

Remaining time

t = 5s to t = 7s

t = 10s to t = 12s

t = 15s to t = 17s

(b)

1.1 1

f

0.9 0.8

g

fg ref Balanced grid voltages

Phase-to-ground Fault

Phase-to-phase Fault

3-Phase-to-ground Fault

Fig. 4.14 Grid faults influences on DC-link

N-SMVOC. From these results, it is clearly to observe grid faults have no significant impact on the DC voltage and the grid power factor as controlled by the N-SMVOC, thanks to the proposed neural identifier, which helps to approximate the DC-link dynamics controlled under different grid scenarios. In addition, based on the neural identifier, an exact equivalent part of the SM control is obtained, which helps the control law to remove the effects of the grid disturbances and ensuring stability.

4.2 Simulation Results

55

4.2.2 RSC Controller In this part, a comparison is investigated between three control schemes which are: PI-FOC, SM-FOC and N-SMOFC for the DFIG stator active and reactive powers regulation, considering a tracking test, robustness to DFIG parameters variations, sensitivity to speed variation, and robustness in presence of grid faults. Two of the control schemes are based on existing control strategies which are the decoupled PIFOC and SM-FOC scheme as reported in [21] and [57] respectively, while the third one presents a novel control strategy which is the contribution of the present paper named the N-SMFOC. These control schemes as applied to the DFIG prototype connected to a 220/60 Hz grid have been simulated using SimPower toolbox of Matlab. • Time-Varying Reference Tracking

Rotor -q- Current (A)

The purpose of this test is to compare trajectory tracking of the three control schemes above motioned to track a time-varying reference, while the DFIG parameters are maintained at their nominal values and the DFIG speed is kept at its synchronous speed, which is equal to 189 rad/s. Figure 4.15 presents identification results for the DFIG rotor −q− currents and its corresponding neural identifier weights. Figure 4.16 presents identification results for the DFIG rotor −d− currents and its corresponding neural identifier weights. From the identification results, it is very clear to observe that the proposed RHONN identifier for the DFIG performs adequately; in addition all the neural identifier weights are bounded. Figure 4.17 illustrates simulation results of the DFIG rotor −q− current (a) and the DFIG rotor −d− current (b) as controlled

(a)

0

-0.5 i

-1

i

Current -q- Weights

-1.5 0

10

20

30

40

50

rq rq estimate

60

70

80

90

60

70

80

90

t [Sec]

(b)

2 2,1

1 0 0

2,2 2,3 2

10

20

30

40

50

t [Sec] Fig. 4.15 DFIG rotor −q− currents identification

4 Neural Control Synthesis

Grid -d- Current (A)

56

(a)

1

0.5

ird i

0 0

rd estimate

10

20

30

40

50

60

70

80

90

60

70

80

90

t [Sec] Grid -q- Current (A)

(b)

6 4 2 0 -2 0

1,1 1,2 1,3 1

10

20

30

40

50

t [Sec]

Rotor -q- current (A)

Fig. 4.16 DFIG rotor −d− currents identification

(a)

0

-1

i

-2 0

Rotor -d- current (A)

irq

10

20

30

40

50

60

70

rq ref

80

90

t [Sec]

(b)

1

0.5 0 0

i

rd

ird ref

10

20

30

40

50

60

70

80

90

t [Sec] Fig. 4.17 DFIG rotor d − q currents with N-SMFOC

by the N-SMFOC. The rotor currents are defined from a desired stator active and reactive power using (4.37) and (4.38) respectively, where the rotor −q− currents reference is selected to be a varying-time trajectory, its amplitude changes from −1 to −0.2 A while, the rotor −d− current reference is fixed at constant value equals to 0.6 A. Figure 4.18 displays the stator active and reactive power trajectory tracking response as controlled by PI-FOC (blue), SM-FOC (green), and N-SMFOC (red) controllers. Zoom of the obtained results is presented in Fig. 4.19. The control

57

(a)

200 100

Ps with N−SM

19.9

20

20.1

Ps with SM Ps with PI

0

Psref

0

Stator Reactive Power [VAr]

Stator Active Power [W]

4.2 Simulation Results

10

20

30

40

50 t [Sec]

60

70

80

90

(b)

30 20

Qs with N−SM Qs with SM

10

Qs with PI 19.9

0 0

10

20

20

Qsref

20.1

30

40

50 t [Sec]

60

70

80

90

Active Power [W]

Fig. 4.18 N-SMFOC controllers trajectory tracking

(a)

100 P with N-SM s

50

Ps with SM P with PI s

Psref

0 39.8

39.85

39.9

39.95

40

40.05

40.1

40.15

40.2

40.05

40.1

40.15

40.2

Reactive Power [VAr]

t [Sec] (b)

40 Q with N-SM s

20

Q with SM s

Qs with PI Q

0 39.8

s ref

39.85

39.9

39.95

40

t [Sec]

Fig. 4.19 N-SMFOC controller trajectory tracking zoom

signals u rq and u rq as applied to the RSC are plotted in Fig. 4.20. From these results, it is possible to see that all controllers successfully force the dynamics to reach the desired trajectories with sluggish responses time for the PI-FOC and the SM-FOC. In addition, we can observe that the N-SMFOC and the SM-FOC controllers ensure a perfect decoupling between the control axes; nevertheless a strong coupling is presented for PI-FOC trajectory tracking. As results of this test, we can conclude that the proposed control has better tracking performance in presence of trajectory

4 Neural Control Synthesis

Control -q- Signal (V)

58

0

-50 0

Control -d- Signal (V)

(a)

50

urq

10

20

30

40

50

60

70

80

90

t [Sec]

(b)

50 0

-50 0

u

10

20

30

40

50

60

70

80

rd

90

t [Sec]

Fig. 4.20 The DFIG control signals for N-MFOC Table 4.1 MSE and STD values for reference tracking Case Tracking Active power with PI-FOC

MSE STD Reactive power with PI-FOC MSE STD Active power with SM-FOC MSE STD Reactive power with SM-FOC MSE STD Active power with N-SMFOC MSE STD Reactive power with MSE N-SMFOC STD

0.3732 0.6063 0.052 0.2263 0.1575 0.3835 0.0139 0.0450 1.28e − 4 0.0156 2.1e − 5 0.0283

variations with perfect decoupling on the control axes. Table 4.1 presents statistical characteristics of Mean Square Error (MSE) and Standard Deviation (STD) for this tracking test. • Robustness to Parameter Variations To test the robustness in presence of parameters variations of the proposed control scheme, and to compare its characteristic with the PI-FOC and SM-FOC controllers, the DFIG parameters are changed, and the DFIG mechanical speed is maintained at

59

(a)

200 100

Ps with N−SM PS with SM

0

19.9

20

20.1

Ps with PI Psref

10

0

20

50

40

30

60

70

90

80

t [Sec]

Stator Reactive Power [VAr]

Stator Active Power [W]

4.2 Simulation Results

(b)

30 20

Qs with N−SM

with N−SMC s Qs Qwith SM

10

Q with SMC

s Qs Qwith PI with PI s

19.9

0 0

10

20

Qsref Qsref

20.1

20

30

40

50

60

70

80

90

t [Sec]

Active Power [W]

Fig. 4.21 Influence of rs change on N-SM-FOC controller

(a)

100 P with N-SM s

50

P with SM s

P with PI s

0 39.8

P

s ref

39.85

39.9

39.95

40

40.05

40.1

40.15

40.2

40.05

40.1

40.15

40.2

Reactive Power [VAr]

t [Sec]

(b)

40 Qs with N-SM

20 0 39.8

Qs with SM Qs with PI Qs ref

39.85

39.9

39.95

40

t [Sec]

Fig. 4.22 rs change influence zoom

its synchronous speed (189 rad/s). Figure 4.21 illustrates the dynamics of the stator active power (a) and the stator reactive power (b) for PI-FOC, SM-FOC and NSMFOC controllers, when the stator resistance rs is increased by 200% of its nominal value. For the rotor resistance rr similar results are obtained. Figure 4.23 presents the dynamics of stator active power (a) and stator reactive power (b) for PI-FOC, SM-FOC and N-SMFOC controllers; when the stator inductance ls is augmented by 90% of its nominal value. Figure 4.25 displays the dynamics of stator active power

4 Neural Control Synthesis

(a)

200 Ps with N−SMC

100

Ps with SMC

20

19.9

20.1

Ps with PI

0

Psref

10

0

Stator Reactive Power [VAr]

Stator Active Power [W]

60

20

40

30

t [Sec]

60

50

70

90

80

(b)

30 20

Qs with N−SMC Qs with SMC

10

Qs with PI 19.9

0 0

10

20

20

Qsref

20.1

40

30

t [Sec]

60

50

70

80

90

Active Power [W]

Fig. 4.23 Influence of ls change on N-SMFOC controller

(a)

100 Ps with N-SM

50

P with SM s

Ps with PI Psref

0 39.8

39.85

39.9

39.95

40

40.05

40.1

40.15

40.2

40.25

40.3

40.1

40.15

40.2

40.25

40.3

Reactive Power [VAr]

t [Sec] (b)

30 Qs with N-SM

25

Qs with SM Qs with PI Q

20 39.8

s ref

39.85

39.9

39.95

40

40.05

t [Sec]

Fig. 4.24 ls change influence zoom

(a) and stator reactive power (b) for PI-FOC, SM-FOC and N-SMFOC controllers; when the rotor inductance lr is changed by 90% of its nominal value. Figure 4.27 displays the dynamics of stator active power (a) and stator reactive power (b) for PI-FOC, SM-FOC and N-SM-FOC controllers; when the mutual inductance lm is decreased by 90% of its nominal value. Zooms of the active and reactive power response for parameter variations are included as follows: Fig. 4.22 corresponds to stator resistance rs ones, Fig. 4.24 is related to stator inductance ls ones, Fig. 4.26 displays the effects of rotor inductance lr ones, and finally Fig. 4.28 corresponds to mutual inductance lm ones.

61

(a)

200 100

Ps with N−SM

20

19.9

Ps with SM

20.1

Ps with PI

0

Psref

10

0

Stator Reactive Power [VAr]

Stator Active Power [W]

4.2 Simulation Results

20

40

30

t [Sec]

50

60

70

90

80

(b)

30 20

Q Qss with with N−SMC N−SM Q Qss with with SMC SM

10

Q Qss with with PI PI 19.9

0 0

10

20

20

Q Qsref sref

20.1

30

40

t [Sec]

50

60

70

80

90

Active Power [W]

Fig. 4.25 Influence of lr change on N-SMFOC controller

(a)

100 Ps with N-SM

50

Ps with SM Ps with PI Ps ref

0 39.8

39.85

39.9

39.95

40

40.05

40.1

40.15

40.2

40.25

40.3

40.1

40.15

40.2

40.25

40.3

Reactive Power [VAr]

t [Sec]

(b)

30 Qs with N-SM

25

Qs with SM Qs with PI Qs ref

20 39.8

39.85

39.9

39.95

40

40.05

t [Sec]

Fig. 4.26 lr change influence zoom

From simulation results, we can observe that parameter variations have an important impact on the stator powers controlled by the PI-FOC controller where a high coupling between the control axes and a sluggish response time are presented. Whereas, the SM-FOC shows better performance compared with the PI-FOC by reducing coupling between the control axes; however, there is a sluggish response time still appears in the controlled dynamics in presence of parameter variations. Furthermore, the proposed controller (N-SMFOC) has an excellent performance in presence of

4 Neural Control Synthesis

(a)

200 100

Ps with N−SM

20

19.9

20.1

Ps with SM

0

Ps with PI Psref

40

30

20

10

0

Stator Reactive Power [VAr]

Stator Active Power [W]

62

t [Sec]

60

50

70

90

80

(b)

30 20

Q Qss with with N−SMC N−SM Q Qss with with SMC SM

10

Q Qss with with PI PI 19.9

0 0

20

Q Qsref sref

40

30

20

10

20.1

t [Sec]

60

50

70

80

90

Active Power [W]

Fig. 4.27 Influence of lm change on N-SMFOC controllers

(a)

100 Ps with N-SM

50

Ps with SM P with PI s

Psref

0 39.8

39.85

39.9

39.95

40

40.05

40.1

40.15

40.2

40.25

40.3

40.1

40.15

40.2

40.25

40.3

Reactive Power [VAr]

t [Sec] 30

(b) Q with N-SM s

20

Qs with SM

10

Q with PI s

Qsref

0 39.8

39.85

39.9

39.95

40

40.05

t [Sec] Fig. 4.28 change influence zoom

parameter variations; decoupling is ensured, and the response time is improved compared with the two other controllers. Table 4.2 presents statistical characteristics for this robustness test. • Sensitivity to Wind Speed Variations In order to analyze the effect of wind speed changes on the stator powers, we force the controlled system to track constant references, while the DFIG parameters are

4.2 Simulation Results

63

Table 4.2 SE and STD values for parameter variations Case Tracking – Rs Ls Active power with PI-FOC

Lr

Lm

MSE

0.3387

0.1159

1.5371

0.2637

STD MSE

0.5784 0.0417

0.3406 0.1378

1.2394 0.5927

0.5142 0.2684

STD Active power MSE with SM-FOC STD Reactive MSE power with SM-FOC STD Active power MSE with N-SMFOC STD Reactive MSE power with N-SMFOC STD

0.205 0.1717

0.0552 0.0709

0.7691 0.1076

0.5161 0.1554

0.4420 0.0145

0.2610 0.0515

0.2430 0.0603

0.3572 0.0169

0.1248 1.03e − 4

0.2513 2.8e − 5

0.3113 2.6e − 5

0.1350 1.65e − 4

0.0509 0.0011

0.0274 7.5e − 5

0.0267 6e − 5

0.0487 4.46e − 5

0.0334

0.0210

0.0191

0.0165

Reactive power with PI-FOC

maintained at their nominal values and we vary the mechanical speed at instant 9.6 s from sub-synchronous to super-synchronous regime (170 rad/s → 200 rad/s), and reversing it at instant 10 s. Figure 4.29 illustrates dynamics of stator reactive power for PI-FOC, SM-FOC and N-SMFOC when the mechanical speed is changed. From this test, we conclude that speed changes has an important effect on the stator reactive power controlled by the standard PI-FOC controller, whereas this impact is reduced by the SM-FOC controller, while the speed variation has less effect on the controlled system using N-SMFOC controller. Table 4.3 presents statistical characteristics for this speed variations test. • Robustness to Grid Disturbances In order to examine the proposed controller performance in presence of grid disturbance and to evaluate its capabilities to improve the LVRT capacity of the DFIG, the same grid disturbances as presented in Fig. 4.13 are applied. Figure 4.30 displays the DFIG stator active and reactive power dynamics as controlled by N-SMFOC in presence of grid disturbances. Figure 4.31 displays the DFIG stator active and reactive power dynamics as controlled by SM-FOC in presence of grid disturbances. The behavior of the DFIG stator active and reactive power as controlled by the PI-FOC

64

4 Neural Control Synthesis

(a)

25

Qs with N-SM

20

Reactive Power [VAr]

Q

15 9

9.2

9.4

9.6

9.8

25

10

10.2

(b)

10.4

10.6

11

s

Q

9.2

9.4

9.6

9.8

25

10

10.2

(c)

10.4

10.6

sref

10.8

11

Q with PI

20 15 9

10.8

Q with SM

20 15 9

sref

s

Qsref

9.2

9.4

9.6

9.8

10

10.2

10.4

10.6

10.8

11

Fig. 4.29 Speed variation effects on N-SMFOC controller Table 4.3 MSE and STD values for speed variations Case Speed variation Active power with PI-FOC

MSE STD Reactive power with PI-FOC MSE STD Active power with SM-FOC MSE STD Reactive power with SM-FOC MSE STD Active power with N-SMFOC MSE STD Reactive power with MSE N-SMFOC STD

0.4077 0.6400 0.4147 0.6367 0.3180 0.4671 0.0126 0.0970 3.43e − 4 0.0707 7.52e − 4 0.0327

in presence of grid disturbances is displayed Fig. 4.32. From these results, it is very clear to see that grid disturbances have no significant impact on the stator active and reactive power behavior controlled by N-SMFOC controller compared with the PI-FOC and the SM-FOC controllers thanks to the proposed neural identifier, which helps to approximate the DFIG rotor currents dynamics under different grid scenarios and under any rotor currents frequency. In addition, based on the neural identifier, an exact equivalent part of the SM control is obtained, which helps the control law to remove the effects of grid disturbances.

155

65 (a)

160

P with N-SM s

150

Reactive Power [Var]

Active Power [W]

4.3 Neural Sliding Mode Linearization Control

Psref t = 4s to t = 9s

23

t = 9s to t = 14s (b)

t = 14s to t = 20s

22 21 20

Qs with N-SM Qsref Ground to Phase Fault

Phase to Phase Fault Three-Phase to Ground Fault

Fig. 4.30 Stator powers with N-SMFOC under fault conditions

Fig. 4.31 Stator powers with SM-FOC under fault conditions

4.3 Neural Sliding Mode Linearization Control By applying the separation principle as established in Theorem 2.3, which allows to determine the NN identifier without taking into account the controller and vice versa [36, 54]. Considering (4.1) and (4.2), and taking the output variable to be controlled yˆk = xk contains the full state vector and the relative degree of the system equal to one (case of the system under study), the objective is to force the controlled dynamics to track selected trajectories; the N-SML controller is synthesized by means of Theorem 4.2.

66

4 Neural Control Synthesis

Fig. 4.32 Stator powers with PI-FOC under fault conditions

Theorem 4.2 For system (4.1) and (4.2), the neural sliding mode linearization control law   (4.39) u n (xk ) = − Bˆ −1 fˆ(xk ) + vn (xk ) allows to use the discrete-time SM methodology to define vn (xk ) in order to ensure that the respective tracking error (4.3) is ultimately bounded, achieving that the system outputs converge towards a neighborhood of their respective desired trajectory The corresponding proof is included in Appendix A.2. The proposed control law (4.39) is composed by two parts: the neural linearization control part − Bˆ −1 fˆ(xk ) and the sliding mode part vn (xk ), which is determined by a discrete-time SM as below. By evaluating (4.2) at k + 1 we obtain ˆ n (xk ) yˆk+1 = χk+1 = fˆ(xk ) + Bu

(4.40)

Matrix Bˆ is invertible because it is composed by the fixed weights. Substituting (4.39) in (4.40), the decoupled linear system is obtained as yˆk+1 = vn (xk )

(4.41)

with vn (xk ) determined by means of the discrete-time SM methodology [26, 43] as  vn (xk ) =

  vn (xk ) if veqn (xk ) ≤ u 0   v (x ) u 0 veqn (xk ) if veqn (xk ) > u 0 eqn

k

(4.42)

4.3 Neural Sliding Mode Linearization Control

67

with  veqn (xk ) < u 0 , u 0 > 0 is the bound of the control signal. The equivalent control given by veqn (xk ) is (4.43) veqn (xk ) = xr e f,k+1 with xr e f,k+1 the reference signal to be tracked. The equivalent control is calculated by evaluating the sliding manifold sk+1 = 0, with sk as sk = χk − xr e f,k

(4.44)

The stabilizing term vsn (xk ) is calculated as vsn (xk ) = kc sn,k

(4.45)

4.3.1 GSC Controller To force the capacitor voltage Udc,k to track the desired values Udcr e f,k , and to maintain the grid power factor f g at the nominal value, the N-SML control scheme is proposed. The DC-link NN model (4.14)–(4.16) can be rewritten as χ1,k+1 = fˆ1 (x1,k ) + Bˆ 1 u 1,k yˆ1,k = hˆ 1 (x1,k )

(4.46)

where x1,k = [Udc,k i gd,k i gq,k ]T is the state vector, χ1,k = [Uˆ dc,k iˆgd,k iˆgq,k ]T is the estimate vector, u 1,k = [u gcd,k u gcq,k ]T is the control vector, yˆ1 = [hˆ 11 (x1,k ) hˆ 12 (x1,k ) hˆ 13 (x1,k )]T = [Uˆ dc,k iˆgd,k iˆgq,k ]T is the variable vector to be controlled, Bˆ 1 = diag [2 , 3 ], and fˆ1 (x1,k ) = [ fˆ11 (x1,k ) fˆ12 (x1,k ) fˆ13 (x1,k )] such as fˆ11 (x1,k ) = w11 S(Udc,k ) + w12 S(Udc,k )S(i gq,k ) + 1 i gd,k fˆ12 (x1,k ) = w21 S(i gd,k ) + w22 S(i gq,k ) + w23 S(Udc,k ) fˆ13 (x1,k ) = w31 S(i gq,k ) + w32 S(i gd,k ) To control the DC-link voltage a cascade controller is used. From (4.46), the linearizing control law for the DC voltage is controlled by the grid −d− current. Using Theorem 4.2, the DC voltage neural linearization control part is calculated as i gd,k =

 1  ˆ − f 11 (xk ) + v1,k 1

(4.47)

with 1 a fixed weight and v1,k the DC voltage decoupled control part such that a discrete-time SM is used to define it as established in Theorem 4.2, where the eq equivalent control u 1n,k is calculated form s1n,k+1 as

68

4 Neural Control Synthesis eq

u 1n,k = Udcr e f,k+1

(4.48)

with Udcr e f,k+1 the DC voltage desired value and the stabilizing term u s1n,k is defined as (4.49) u 1n,k = kc1 s1n,k with −1 < kc1 < 1 and s1n,k = Udc,k − Udcr e f,k . The i dg,k reference is equal to the control law (4.47)  1  ˆ − f 11 (xk ) + v1,k (4.50) i gdr e f,k = 1 The desired value of the grid −q− current is calculated using (4.19). Using Theorem 4.2, the grid currents neural linearization control law is calculated as

u gcd,k u gcq,k

=

B1−1



− fˆ12 (xk ) + v2,k − fˆ13 (xk ) + v3,k

(4.51)

Discrete-time SM control is used to obtain v2,k , v3,k . The sliding manifold is defined

− i gdr e f,k iˆ eq eq The equivalent control vector [u 2n,k u 3n,k ]T is given by as s2n,k = ˆgd,k i gq,k − i gqr e f,k

eq

u 2n,k eq u 3n,k



i = gdr e f,k+1 i gqr e f,k+1

(4.52)

The stabilizing term vector [u s2n,k u s3n,k ]T is defined as

u s2n,k u s3n,k

=

kc2 0 s . 0 kc3 2n,k

(4.53)

4.3.2 RSC Controller The objective of this discrete-time N-SM-L controller is to force the rotor currents ir d,k and irq,k to track a desired references and to provide robustness in presence of parameter variations and external disturbances. The DFIG neural identifier is proposed as follows fˆ2 (x2,k ) + Bˆ 2 u r,k χ2,k+1 = (4.54) yˆ2,k = hˆ 2 (x2,k ) where x2,k = [i sd,k i sq,k ir d,k irq,k ]T is the state vector, χ2,k = [iˆsd,k iˆsq,k iˆr d,k iˆrq,k ]T is the estimate vector, yˆ2,k = [hˆ 21 (x2,k ) hˆ 22 (x2,k )]T = [iˆr d,k iˆrq,k ]T is the variable vector to be controlled, and fˆ2 (x2,k ) = [ fˆ21 (xk ) fˆ22 (xk fˆ23 (xk ) fˆ24 (xk )] such as

4.3 Neural Sliding Mode Linearization Control

69

fˆ21 (x2,k ) = w11,k S(i sd,k ) + w12,k S(i sq,k ) + w13,k S(i sd,k )S(i sq,k ) fˆ22 (x2,k ) = w21,k S(i sd,k ) + w22,k S(i sq,k ) + w23,k S(i sq,k )S(i sd,k ) fˆ23 (x2,k ) = w31,k S(ir d,k ) + w32,k S(irq,k ) + w33,k S(ir d,k )S(irq,k ) fˆ24 (x2,k ) = w41,k S(irq,k ) + w42,k S(ir d,k ) + w43,k S(ir d,k )S(irq,k ) and



Bˆ 21 Bˆ 22



1 0 3 0 = 0 2 0 4

T

The DFIG rotor currents control is obtained by using Theorem 4.2 as follows

u r d,k u rq,k

−1 = B22



− fˆ23 (xk ) + v4,k − fˆ24 (xk ) + v5,k

(4.55)

with v4,k , v5,k decoupled control law defined by using discrete-time SM controller as established in Theorem 4.2 such that is the sliding manifold is given as s3n,k =

iˆr d (k) − ir dqr e f,k eq eq . The equivalent control [u 4n,k u 5n,k ]T is calculated by iˆrq (k) − irqr e f,k

eq

u 4n,k eq u 5n,k



i = r dr e f,k+1 irqr e f,k+1

(4.56)

The stabilizing term vector [u s4n,k u s5n,k ]T is defined as

u s4n,k u s5n,k



k−4c 0 = s . 0 k5c 3n,k

(4.57)

4.4 Simulation Results The proposed N-SML controller, which is used to control the DC-link voltage and the grid power factor through the control of the GSC, and the stator active and reactive power controller by means of the DFIG rotor currents through the RSC are simulated using SimPower Toolbox of Matlab. The DFIG prototype is connected to an infinite bus through three-phase transmission lines.

70

4 Neural Control Synthesis

4.4.1 GSC Controller • Time-Varying Reference Tracking

DC voltage (V)

The capabilities of the proposed N-SML controllers to track a time-varying desired trajectory for the DC voltage and at the same time the unit value for the grid power factor are tested in this section. The DC-link parameters are maintained at their nominal values, the load resistance is taken constant, and normal grid conditions are selected, see Fig. 4.1. Neural identification results for the DC-link are presented as follows: Fig. 4.33a illustrates the DC voltage real (blue) and estimated (red) dynamics; the corresponding neural identifier weights are presented in Fig. 4.33b. The grid d − q currents real, estimated dynamics, and their corresponding neural identifier weights are presented in Figs. 4.34 and 4.35 respectively. We note that the proposed neural identifier weight vector for each state of the DC-link is composed of adaptive weights and single fixed ones. From the obtained result of the DC-link identification, it possible to observe that the proposed neural identifier provides a good identification for the DC-link real states and the respective RHONN identifier weights are bounded. Figure 4.36 presents the DC-link voltage (a) and the grid power factor (b) as controlled by the N-SML controller. From the obtained results, it is clear to see that the proposed N-SML controller forces the controlled dynamics to achieve their desired values. A time-varying desired trajectory for the DC-link voltage is selected, where its amplitude changing from 150 to 90 V and the unit value is chosen for the grid power factor. The grid d − q currents and the gird active and reactive power dynamics flow between the grid and the DFIG rotor winding terminals are displayed in Figs. 4.37 and 4.38 respectively. Figure 4.39 presents the control u gdc and u gqc signals as applied to the GSC. From the obtained results, it is clear to observe that

(a)

100

Udcreal U

0 0

10

20

30

40

50

60

70

80

estimate

90

DC Voltage Weights

t [Sec]

(b)

100

1,1 1,2

50 0 0

1,3 1

10

20

30

40

50

t [Sec]

Fig. 4.33 DC voltage identification and weights with N-SML

60

70

80

90

Grid -d- Current (A)

4.4 Simulation Results

71

(a)

3 2

igd

1 0 0

i

10

20

30

40

50

60

70

gd estimate

80

90

Current -d- Weights

t [Sec] (b)

3

2,1

2

2,2

1

2,3

0

-1 0

2

10

20

30

40

50

60

70

80

90

t [Sec] Fig. 4.34 Grid −d− current identification and weights with N-SML

the grid active power (Pg ) and the grid current on axis −d− are related to the DC voltage behavior; the grid reactive power (Q g ) and the grid current on axis −q− depend on the grid power factor f g dynamics. In addition, the proposed N-SML controller forces the DC-link voltage to track a desired time-varying trajectory and keeps at unit value the grid power factor. Moreover, the controlled dynamics and the control signals u gcd and u gcq are bounded. We can conclude that the proposed controller ensures excellent tracking performances for the DC voltage and the grid power factor independently even in presence of reference trajectory variations. • Robustness to Parameter Variations The objective of this experiment is to examine the proposed N-SML controller performances in presence of the DC-link parameter variations and load resistance changes. The obtained results are exposed as follows: Fig. 4.40 presents the DC-link voltage (a) and the grid power factor (b) dynamics as controlled by the N-SML when the grid resistance r g is increased by the 200% of its nominal value. Figure 4.41 displays the dynamics of the voltage at the output of the DC-link (a) and the grid power factor (b) as controlled by the N-SML when the grid inductance l g is augmented by the 100% of its nominal value. Figures 4.42 and 4.43 offers the dynamics of the DC voltage (a) and the grid power factor (b) as controlled by the N-SML when the load resistance rl is variated form 10 M to 10  respectively. From the obtained results, it is very clear to observe that parameter variations and load resistance changes have no effects on the DC voltage and the grid power factor controlled by N-SML. In addition, the proposed controller ensures decoupling between the d − q axes. Note that the proposed N-SML controller can ensures robustness of the controlled system even in presence of small resistance load.

4 Neural Control Synthesis

Grid -q- Current (A)

72

(a)

1

i

0.5

i

gq gq estimate

0

-0.5 0

10

20

30

40

50

60

70

80

90

Current -q- Weights

t [Sec] (b)

3 2

3,1

1

3,2

0 0

3

10

20

30

40

50

60

70

80

90

t [Sec]

DC-link Voltage (V)

Fig. 4.35 Grid −q− current identification and weights with N-SML

(a)

100

0 0

Udc Udc ref

10

20

30

40

50

60

70

80

90

Grid Power Factor

t [Sec] (b) 1

0.8 0

f f

10

20

30

40

50

60

70

80

g g ref

90

t [Sec] Fig. 4.36 DC voltage and grid power factor with N-SML

• Robustness to Grid Disturbances The objective to examine the capabilities of the proposed N-SML for ensuring robustness and stability of the controlled dynamics in presence of grid disturbances. The same fault grid conditions are selected as given Fig. 4.13. Figure 4.44 presents the dynamics of the DC voltage (a) and the grid power factor (b) as controlled by the N-SML. The obtained results illustrate the effectiveness of the proposed N-SML controller to ensure robustness and stability even in presence of grid disturbances, thanks to the neural identifier, which is used to obtain an adequate model of the

Grid -d- Current (A)

4.4 Simulation Results

73

(a)

3

i

2

gd

1 0 0

10

20

30

40

50

60

70

80

90

Grid -q- Current (A)

t [Sec]

(b)

0.5

i

gq

0

-0.5 0

10

20

30

40

50

60

70

80

90

t [Sec]

Active Power (W)

Fig. 4.37 Grid d − q currents with N-SML

(a)

150 100 50 0 0

Pg

10

20

30

40

50

60

70

80

90

Reactive Power (VAr)

t [Sec]

(b)

20 0

-20 0

Qg

10

20

30

40

50

60

70

80

90

t [Sec]

Fig. 4.38 Grid active and reactive power with N-SML

DC-link under different grid scenarios and helps the proposed controller to remove the disturbance influences on the controlled dynamics.

4 Neural Control Synthesis

Control -d- Signal (V)

74

(a)

200

u

gdc

100 0 0

10

20

30

40

50

60

70

80

90

Control -q- Signal (V)

t [Sec]

(b)

50

u

gqc

0

-50 0

10

20

30

40

50

60

70

80

90

t [Sec]

Fig. 4.39 Control d − q signals with N-SML

DC-link Voltage (V)

(a)

100

0 0

Udc Udc ref

10

20

30

40

50

60

70

80

90

Grid Power Factor

t [Sec]

(b) 1

0.8 0

f

g

fg ref

10

20

30

40

50

60

70

80

90

t [Sec] Fig. 4.40 Influence of r g changes on N-SML

4.4.2 RSC Controller The proposed controller (N-SML), the SM-L and the PI-FOC have been implemented using SimPower toolbox of Matlab. The objective is to evaluate the performance of the proposed controller and compare it with the SM-L and the PI-FOC controllers taking into account tracking test, robustness to parameter variations, sensitivity to speed changing, and the behavior of the controlled system in presence of grid distur-

DC-link Voltage (V)

4.4 Simulation Results

75

(a)

100

0 0

U

dc

Udc ref

10

20

30

40

50

60

70

80

90

Grid Power Factor

t [Sec] (b) 1 f f

0.8 0

10

20

30

40

50

60

70

g g ref

80

90

t [Sec]

DC-link Voltage (V)

Fig. 4.41 Influence of l g changes on N-SML

(a)

100

0 0

U

dc

Udc ref

10

20

30

40

50

60

70

80

90

Grid Power Factor

t [Sec] (b) 1

0.8 0

f

g

fg ref

10

20

30

40

50

60

70

80

90

t [Sec] Fig. 4.42 Influence of rl changes on N-SML: 10 M

bances. All controllers are applied to the DFIG prototype connected to a 220/60 Hz infinite bus through transmission lines. • Time-Varying Reference Tracking The purpose of this test is to compare trajectory tracking of the three control schemes above motioned to track a time-varying reference, while the DFIG parameters are maintained at their nominal values and the DFIG speed is fixed at its synchronous speed, which is equal to 189 rad/s. In addition, normal grid conditions are consid-

76

4 Neural Control Synthesis

DC-link Voltage (V)

(a)

100

Udc U

0 0

10

20

30

40

50

60

70

dc ref

80

90

Grid Power Factor

t [Sec]

(b) 1 fg fg ref

0.8 0

10

20

30

40

50

60

70

80

90

t [Sec]

DC-link Voltage (V)

Fig. 4.43 Influence of rl changes on N-SML: 10 

(a)

160 150 140

U

dc

Udc ref

Grid Power Factor

Remaining time

t = 5s to t = 7s

0.9

t = 15s to t = 17s

Phase-to-phase Fault

3-Phase-to-ground Fault

(b)

1.1 1

t = 10s to t = 12s

fg fg ref Balanced grid voltages

Phase-to-ground Fault

Fig. 4.44 Grid faults influences on N-SML

ered as presented in Fig. 4.1. Figure 4.45 presents the identification results for the DFIG rotor −q− currents (a) and the neural identifier weights (b). The DFIG rotor −d− identification and the neural identifier weights are displayed in Fig. 4.46a and b respectively. From the identification results, it clear to see that the proposed neural identifier provides a good identification for real variable state of the DFIG and also all the RHONN identifier weights are bounded. Figure 4.47 presents the DFIG d − q currents trajectories as controlled by the N-SML controller such that, the rotor −q− current is forced to track time-varying reference; its amplitude is changed from −1

Rotor -q- Current (A)

4.4 Simulation Results

(a)

0

irq

-0.5

irq estimate

-1

-1.5 0

Current -q- Weights

77

10

20

30

40

50

60

70

80

90

60

70

80

90

50

60

70

80

90

50

60

70

80

90

t [Sec]

(b)

1 2,1

0.5

2,2 2,3

0

2

0

10

20

30

40

50

t [Sec]

Grid -d- Current (A)

Fig. 4.45 DFIG rotor −q− currents identification for N-SML

(a)

1

0.5

i i

Grid -q- Current (A)

0 0

rd rd estimate

10

20

30

40

t [Sec]

(b)

3 2 1 0

-1 0

1,1 1,2 1,3 1

10

20

30

40

t [Sec]

Fig. 4.46 DFIG rotor −d− currents identification for N-SML

to −0.2 A, and the rotor −d− currents is fixed at constant value equal to 0.6 A. The desired trajectories of the rotor currents is defined from the stator active and reactive power required dynamics. Figure 4.48 the stator active power (a) and the stator reactive power generated by the DFIG as controlled by the N-SML (red), the SML (green) and the PI-FOC (blue) controllers. The zoom of the generated active and reactive power is presented in Fig. 4.49. The DFIG stator d − q currents as generated by the DFIG are illustrated in Fig. 4.50. The control signals u rq and u r d are plotted in

4 Neural Control Synthesis

Rotor -q- Current (A)

78

(a)

0

-1

i

-2 0

Rotor -d- Current (A)

i

10

20

30

40

50

60

70

rq rq ref

80

90

t [Sec]

(b)

1

0.5 0 0

i

rd

ird ref

10

20

30

40

50

60

70

80

90

t [Sec]

Fig. 4.47 DFIG rotor d − q currents with N-SML Table 4.4 MSE and STD values for reference tracking Case Tracking Active power with PI-FOC Reactive power with PI-FOC Active power with SM-L Reactive power with SM-L Active power with N-SML Reactive power with N-SML

MSE STD MSE STD MSE STD MSE STD MSE STD MSE STD

0.3732 0.6063 0.052 0.2263 0.5370 2.3245 1.3274 2.6883 1.055e − 4 0.01070 6.16e − 5 0.0086

Fig. 4.51a and b respectively. It possible to see that all controllers reach the tracking objective with a strong coupling for PI-FOC and SM-L controllers, and a sluggish response time is observed in the stator active power controlled by PI-FOC controller. Whereas, the proposed N-SML controller ensures a high decoupling between the control axes and the controlled dynamics achieves adequately the purpose of this test. From these results, we can conclude that the proposed controller (N-SML) has better tracking performance in presence of reference trajectory variations. Table 4.4 presents statical characteristics of MSE and STD for this test.

Active Power (W)

4.4 Simulation Results

79

(a)

200 100 0

P s with N-SML P s with SM-L P s with PI-FOC P s ref

0

10

20

30

40

50

60

70

80

90

60

70

80

90

Reactive Power (VAr)

t [Sec]

(b)

30 20 10 0 0

Q s with N-SML Q s with SM-L Q s with PI-FOC Q sref

10

20

30

40

50

t [Sec]

Active Power (W)

Fig. 4.48 PI-FOC, SM-L, and N-SML controllers trajectory tracking

(a)

100 P s with N-SML

50

P s with SM-L P s with PI-FOC

0 39.8

39.85

39.9

39.95

40

40.05

40.1

40.15

40.2

40.05

40.1

40.15

40.2

Reactive Power (VAr)

t [Sec]

(b)

30 20 10

Q s with N-SML Q s with SM-L Q s with PI-FOC Q sref

0 39.8

39.85

39.9

39.95

40

Fig. 4.49 N-SML controllers trajectory tracking zoom

• Robustness to Parameter Variations The purpose of this test is to evaluate the performances of the N-SML, the SM-L, and the PI-FOC controllers in presence of the DFIG parameters variations. To do that, the mechanical speed is fixed at its synchronous speed (189 rad/s), and normal grid conditions are considered. Figure 4.52 illustrates the stator active (a) and reactive (b) power as controlled by N-SML (red), the SM-L (green), the PI-FOC controller, and their references (black) when the stator resistance rs is increased by 200% of its nominal value. The zoom of these results is illustrated in Fig. 4.53. Figure 4.54a–

4 Neural Control Synthesis

Stator -q- Current (A)

80

(a)

2

isq

1 0

Stator -d- Current (A)

-1 0

10

20

30

40

50

60

70

80

90

t [Sec]

(b)

1

isd

0

-1 0

10

20

30

40

50

60

70

80

90

t [Sec]

Control -q- Signal (V)

Fig. 4.50 The DFIG stator d − q currents with N-SML

0 u

-50 0

Control -q- Signal (V)

(a)

50

10

20

30

40

50

60

70

80

rq

90

t [Sec] (b) 20 0 urd

-20 0

10

20

30

40

50

60

70

80

90

t [Sec]

Fig. 4.51 The DFIG control signals for N-SML

b presents the stator active and reactive power, respectively, controlled by the NSML (red), the SM-L (green) and the PI-FOC (blue) when the stator inductance ls is augmented by 100% of its nominal value. The zoom of the stator inductance variations id plotted in Fig. 4.55. Figure 4.56 displays the behavior of the stator active and reactive power as controlled by the N-SML (red), the SM-L (green) and the PIFOC (blue) when the rotor inductance lr is changed by 100% of its nominal value. Figure 4.57 shows the zoom of the obtained results for rotor inductance lr variations

Active Power (W)

4.4 Simulation Results

81

(a)

200 100 0

P s with N-SML P s with SM-L P s with PI-FOC P sref

0

10

20

30

40

50

60

70

80

90

60

70

80

90

Reactive Power (VAr)

t [Sec] (b)

30 20 10 0 0

Q s with N-SML Q s with SM-L Q s with PI-FOC Q sref

10

20

30

40

50

t [Sec]

Active Power (W)

Fig. 4.52 Influence of rs change on N-SML controller

(a)

100 50

P s with N-SML P s with SM-L P s with PI-FOC P sref

0 39.8

39.85

39.9

39.95

40

40.05

40.1

40.15

40.2

40.05

40.1

40.15

40.2

Reactive Power (VAr)

t [Sec] (b)

30 20 10

Q s with N-SML Q s with SM-L Q s with PI-FOC Q sref

0 39.8

39.85

39.9

39.95

40

t [Sec]

Fig. 4.53 rs change influence zoom

influences. The behavior of the stator active and reactive power when the mutual inductances lm is decreased by 90% of its nominal value is presented in Fig. 4.58, and their zoom is exposed in Fig. 4.59. From this test, it is very clear to see that parameter variations have a significant impact on the stator powers reference tracking when the SM-L and the PI-FOC controllers are used. Whereas, those effects are eliminated by the N-SML controller, which means that the proposed controller is robust in presence of parameter variations. In addition, decoupling is ensured by the N-SML controller,

4 Neural Control Synthesis

Active Power (W)

82

(a)

200 100 0

P s with N-SML P s with SM-L P s with PI-FOC P s ref

0

10

20

30

40

50

60

70

80

90

60

70

80

90

Reactive Power (VAr)

t [Sec]

(b)

30 20 10 0 0

Q s with N-SML Q s with SM-L Q s with PI Q sref

10

20

30

40

50

t [Sec]

Active Power (W)

Fig. 4.54 Influence of ls change on N-SML controller

(a)

100 50

P s with N-SML P s with SM-L P s with PI-FOC P s ref

0 39.8

39.85

39.9

39.95

40

40.05

40.1

40.15

40.2

40.05

40.1

40.15

40.2

Reactive Power (VAr)

t [Sec]

(b)

30 20 10

Q s with N-SML Q s with SM-L Q s with PI Q sref

0 39.8

39.85

39.9

39.95

40

t [Sec]

Fig. 4.55 ls change influence zoom

while a strong coupling appears for the other controllers. Table 4.5 presents statistical characteristics of MSE and Standard STD for this tracking test. • Sensitivity to Wind Speed Variations The objective of this test is to analyze the impact of WT speed changing on the stator reactive power. For that, we force the controllers to track constant stator powers references, while the DFIG parameters are maintained at their nominal values, and normal grid conditions are considered. the mechanical speed is changed at instant

Active Power (W)

4.4 Simulation Results

83

(a)

200 100 0

P s with N-SML P s with SM-L P s with PI-FOC P s ref

0

10

20

30

40

50

60

70

80

90

60

70

80

90

Reactive Power (VAr)

t [Sec] (b)

30 20 10 0 0

Q s with N-SML Q s with SM-L Q s with PI Q s ref

10

20

30

40

50

t [Sec]

Active Power (W)

Fig. 4.56 Influence of lr change on N-SML controller

(a)

100 50

P s with N-SML P s with SM-L P s with PI-FOC P s ref

0 39.8

39.85

39.9

39.95

40

40.05

40.1

40.15

40.2

40.05

40.1

40.15

40.2

Reactive Power (VAr)

t [Sec] (b)

30 20 10

Q s with N-SML Q s with SM-L Q s with PI Q s ref

0 39.8

39.85

39.9

39.95

40

t [Sec]

Fig. 4.57 lr change influence zoom

9.6 s from sub-synchronous to super-synchronous regime (170 rad/s → 200 rad/s), and reversing it at instant 10 s. Figure 4.60 gives the dynamics the stator reactive power as controlled by the N-SML (a), the SM-L (b), and the PI-FOC (c) controllers. From the obtained results, we can conclude that the speed change from nominal to sub-synchronous or super-synchronous speeds have a clear effect on the stator reactive power controlled by the SM-L and the PI-FOC controllers, whereas this impact is largely reduced by the N-SML controller. Table 4.6 presents the MSE and the STD for this test.

84

4 Neural Control Synthesis

Table 4.5 MSE and STD values for parameter variations Case Tracking – Rs Ls Active power with PI-FOC Reactive power with PI-FOC Active power with SM-L Reactive power with SM-L Active power with N-SML Reactive power with N-SML

Lr

Lm

MSE

0.3387

0.1159

1.5371

0.2637

STD MSE

0.5784 0.0417

0.3406 0.1378

1.2394 0.5927

0.5142 0.2684

STD MSE

0.205 5.4426

0.0552 6.7268

0.7691 5.1849

0.5161 6.04

STD MSE

5.5544 1.0162

6.0436 0.1544

3.1276 0.1506

2.4395 0.1052

STD MSE

4.4694 1.026e − 4

1.3537 1.84e − 5

0.3600 1.62e − 5

0.9837 1.57e − 4

STD MSE

0.0232 4.66e − 4

0.0097 6.52e − 6

0.0057 5.25e − 6

0.0247 1.12e − 5

STD

0.0235

0.0104

0.0142

0.0138

Table 4.6 MSE and STD values for speed variations Case Speed variation Active power with PI-FOC Reactive power with PI-FOC Active power with SM-L Reactive power with SM-L Active power with N-SML Reactive power with N-SML

MSE STD MSE STD MSE STD MSE STD MSE STD MSE STD

0.4077 0.6400 0.4147 0.6367 5.1528 6.0615 0.1662 1.3185 2.32e − 4 0.0304 1.12e − 4 0.0173

Active Power (W)

4.4 Simulation Results

85

(a)

200 100 0

P s with N-SML P s with SM-L P s with PI-FOC P s ref

0

10

20

30

40

50

60

70

80

90

60

70

80

90

Reactive Power (VAr)

t [Sec] (b)

30 20 10 0 0

Q s with N-SML Q s with SM-L Q s with PI-FOC Q s ref

10

20

30

40

50

t [Sec]

Reactive Power (VAr) Active Power (W)

Fig. 4.58 Influence of lm change on N-SML controller

(a)

100 50

P s with N-SML P s with SM-L P s with PI-FOC P s ref

0 39.8

39.85

39.9

39.95

30 20 10

40

40.05

40.1

40.15

40.2

40.05

40.1

40.15

40.2

t [Sec] (b) Q s with N-SML Q s with SM-L Q s with PI-FOC Q s ref

0 39.8

39.85

39.9

39.95

40

t [Sec]

Fig. 4.59 lm change influence zoom

• Robustness to Grid Disturbances The objective of this test is to examine the performance of the proposed controller (N-SML) under fault grid conditions and to validate its capabilities for improving the LVRT capacity of the DFIG. To achieve this objective, the same fault grid conditions are selected as Fig. 4.13. Figure 4.61 presents the dynamics of stator active (a) and reactive (b) powers generated by the DFIG under selected fault grid conditions for the N-SML. Figure 4.62a and b displays the DFIG stator active and reactive power dynamics as controlled by SM-L controller under fault grid conditions. The behavior

86

4 Neural Control Synthesis

(a)

30

Reactive Power [VAr]

20

Qs with N-SML

10

Qsref

0 9

9.2

9.4

9.6

9.8

10

(b)

25

10.2

10.4

10.6

10.8

11

Q with SM-L

20

s

Qsref

15 9

9.2

9.4

9.6

9.8

10

(c)

25

10.2

10.4

10.6

10.8

11

Qs with PI-FOC

20

Q

15 9

9.2

9.4

9.6

9.8

10

10.2

10.4

10.6

sref

10.8

11

Fig. 4.60 Speed variation effects on N-SML

(a)

Active Power [W]

120

110

Reactive Power [VAr]

100

Ps with N-SML Psref Remaining time

t = 5s to t = 7s

t = 10s to t = 12s

t = 15s to t = 17s

Phase-to-phase Fault

3-Phase-to-ground Fault

(b)

8 7 6

Qs withN-SML Qsref Balanced grid voltages

Phase-to-ground Fault

Fig. 4.61 Stator powers with N-SML under fault conditions

of the DFIG stator active and reactive power as controlled by the PI-FOC as controller in presence of grid disturbances is displayed Fig. 4.63. From the obtained results, we can observe that the proposed controller shows a high performance under grid disturbances, which confirm the purpose of the proposed controller for improving the LVRT capacity and ensuring stability of the DFIG under different grid scenarios.

4.5 Neural Inverse Optimal Control

87

Fig. 4.62 Grid fault influences on SM-L

Fig. 4.63 Stator powers with PI-FOC under fault conditions

4.5 Neural Inverse Optimal Control The proposed DFIG control scheme includes two parts: (a) The GSC controller, which allows to regulate the DC-link voltage and the grid power factor. (b) The RSC controller, which permits to control independently the DFIG stator active and reactive powers.

88

4 Neural Control Synthesis

4.5.1 GSC Controller Its objective is to keep the capacitor voltage Udc,k at desired values Udcr e f,k , and to maintain the grid power factor f g at the unit value independently. In order to simplify the DC-link voltage control, the Voltage Oriented Control (VOC) approach is applied [42]. The DC-link NN model (4.14)–(4.16) can be rewritten as χ2,k+1 = fˆ2 (x2,k ) + Bˆ2 u 2 (x2,k ) yˆ2,k = hˆ 2 (x2,k )

(4.58) (4.59)

where χ2,k = [Uˆ dc,k , iˆgd,k , iˆgq,k ]T ∈ n is the DC voltage and the grid d − q currents T  estimated state vector, x2,k = Udc,k , i gd,k , i gq,k ∈ n is the DC-voltage and the T  grid currents state vector, u 2,k = u gcd,k , u gcq,k ∈ m is the control input, yˆ2,k ∈    p is the output vector to be controlled, Bˆ 2 = diag 2,2 , 2,3 is the control matrix, and the vector fˆ2 (•), and hˆ2 (•) are defined as follows fˆ21 (x2,k ) = w11 S(Udc,k ) + w12 S(Udc,k )S(i gq,k ) + 13 i gd,k

w21 S(i gd,k ) + w22 S(i gq,k ) + w23 S(Udc,k ) fˆ22 (x2,k ) = w31 S(i gq,k ) + w32 S(i gd,k ) For control design, first the proposed controller is carried out for the DC voltage trajectory tracking. In fˆ21 (x2,k ), it is clear to observe that the DC voltage is directly controlled by the grid −d− current. For that, a cascade controller is proposed and its output defines the desired value of the grid −d− current. The DC voltage tracking error at k + 1 is obtained as e21,k+1 = χ21,k+1 − x21,r e f,k+1 = fˆ21 (x21,k ) + Bˆ 21 i dg,k − x21,r e f,k+1

(4.60)

By following the same steps as in Sect. 2.3, then, the control law of the DC voltage is calculated as follows i dgr e f,k = −

  −1 T 1 R21 + Q 21,k Bˆ 21 P21 fˆ21 (x2,k ) − x21,r e f,k+1 2

(4.61)

T where Q 21,k = 21 Bˆ 21 P21 Bˆ 21 , R21 , and P21 are positive constants, Bˆ 21 = 21 , and x21,r e f,k = Udcr e f,k is the DC voltage desired value. The grid −q− current reference is calculated using the required grid power factor value as follows [42]

 i gqr e f,k = −i gd,k

1 − f gr2 e f f gr e f

(4.62)

4.5 Neural Inverse Optimal Control

89

Let define χ22,k = [iˆgd,k , iˆgq,k ]T , the tracking error at k + 1 is obtained as e22,k+1 = χ22,k+1 − x22,r e f,k+1 = fˆ22 (x2,k ) + Bˆ 22 u 2 (x2,k ) − x22,r e f,k+1

(4.63)

and the N-IOC for the grid currents is determined as follows u 2,k = −

  −1 1 T R22 + Q 22,k Bˆ22 P22 fˆ22 (x2,k ) − x22,r e f,k+1 2

(4.64)

T where Q 22,k = 21 Bˆ 22 P2 Bˆ 22 is a symmetric and positive definite matrix and P22 ∈ 2x2 2x1 and R22 ∈  are positive definite matrices.

4.5.2 RSC Controller Its objective is to track desired trajectories for the DFIG stator active (Ps ) and reactive (Q s ) power via the DFIG rotor currents ir d and irq . The desired rotor currents dynamics ir dr e f and irqr e f are defined form required stator active Psr e f and reactive Q sr e f power dynamics. In order to simplify the control algorithm synthesis, the FOC approach is applied [36, 64]. The proposed neural model for the DFIG is given as [42] χ1,k+1 = fˆ1 (x1,k ) + Bˆ1 u 1 (x1,k ) yˆ1,k = hˆ 1 (x1,k )

(4.65) (4.66)

T  where χ1,k ∈ n is the DFIG rotor currents estimated state vector, x1,k = ir d,k , irq,k T  ∈ n is the DFIG rotor currents state vector, u 1,k = u r d,k , u rq,k ∈ m is input the input, yˆ1,k ∈  p is the estimated output vector to be controlled, Bˆ1 = diag   1,1 , 1,2 is control matrix composed of fixed weights, and the vector fˆ1 (•), and hˆ 1 (•) are smooth and bounded vector field defined as fˆ11 (x1,k ) = w11,k S(ir d,k ) + w12,k S(ir d,k )S(irq,k ) + w13,k S(irq,k ) fˆ12 (x1,k ) = w21,k S(irq,k ) + w22,k S(ir d,k )S(irq,k ) + w23,k S(ir d,k Then, the DFIG rotor currents tracking error is expressed as follows e1,k = χ1,k − x1,r e f,k

(4.67)

90

4 Neural Control Synthesis

 T where x1r e f,k = ir dr e f,k , irqr e f,k is the desired rotor currents trajectories. By evaluating (4.67) at k + 1 and using (4.65), we obtain e1,k+1 = χ1,k+1 − x1,r e f,k+1 = fˆ1 (x1,k ) + Bˆ 1 u 1 (x1,k ) − x1,r e f,k+1

(4.68)

Considering the same steps as in Sect. 2.3 applied to (4.68), the N-IOC for the DFIG rotor currents is calculated as follows u 1,k = −

 −1 T  1 R1 + Q 1,k Bˆ1 P1 fˆ1 (x1,k ) − x1,r e f,k+1 2

(4.69)

T where Q 1,k = 21 Bˆ1 P1 Bˆ1 is a positive definite matrix, P1 ∈ 2x2 , and R1 ∈ 2x1 are positive definite matrices.

4.6 Simulation Results

DC Voltage (V)

The N-IOC proposed controller as applied to the DFIG has been simulated using SimPower System tools of Matlab. The DFIG prototype is connected to an infinite bus through three-phase transmission lines. (a)

200 100 0 0

Udc Udc estimate

10

20

30

40

50

60

70

80

90

DC Voltage Weights

t [Sec] (b)

2

11

0

-2 0

12 1

10

20

30

40

50

t [Sec]

Fig. 4.64 DC voltage identification and weights with N-IOC

60

70

80

90

Grid -d- Current (A)

4.6 Simulation Results

91

(a)

3 2

i

1 0 0

gd

igd estimate

10

20

30

40

50

60

70

80

90

Current -d- Weights

t [Sec] (b)

3 2

21

1

22

0

23

-1 0

2

10

20

30

40

50

60

70

80

90

t [Sec]

Grid -q- Current (A)

Fig. 4.65 Grid −d− current identification and weights with N-IOC

(a)

2 0

-2 0

i

gq

igq estimate

10

20

30

40

50

60

70

80

90

Current -q- Weights

t [Sec] (b)

2

31

1 0 0

32 3

10

20

30

40

50

60

70

80

90

t [Sec]

Fig. 4.66 Grid −q− current identification and weights with N-IOC

4.6.1 GSC Controller • Time-Varying Reference Tracking The capabilities of the proposed controller to track a time-varying desired trajectories of the DC-link voltage and the grid power factor are examined. First, The DC-link parameters are maintained at their nominal values, the load resistance is considered constant, and normal grid conditions are selected, see Fig. 4.1. The identification

4 Neural Control Synthesis

DC-Link Voltage (V)

92

(a)

200 100 0 0

Udc Udc ref

10

20

30

40

50

60

70

80

90

Grid Power Factor

t [Sec] (b) 1

0.8 0

f

g

fg ref

10

20

30

40

50

60

70

80

90

t [Sec] Fig. 4.67 DC voltage and grid power factor with N-IOC

results by RHONN identifier for the DC-link are presented as follows: Fig. 4.64a illustrates the DC voltage real (blue) and estimated (red) dynamics; the corresponding neural identifier weights are presented in Fig. 4.64b. The grid d − q currents real, estimated dynamics, and their corresponding neural identifier weights are presented in Figs. 4.65 and 4.66 respectively. These result illustrate that the proposed neural identifier for the DC-link dynamics provides a good identification. In addition, the respective neural identifier weights are bounded. Figure 4.67 gives the DC-link voltage (a) and the grid power factor (b) when controlled by the N-IOC. From the obtained results, it is clear that the proposed control scheme (N-IOC) ensures the desired trajectories tracking, which are a time-varying desired trajectory for the DClink voltage (150 to 90 V) and the unit value for the grid power factor. The grid d − q currents and the gird active and reactive power dynamics flow between the grid and the RSC are illustrated in Figs. 4.68 and 4.69 respectively. Figure 4.70 presents the control u gdc and u gqc signals as applied to the GSC. The obtained results illustrate that the grid active power (Pg ) and the grid current on axis −d− are related to the DC voltage variations; while, the grid reactive power (Q g ) and the grid current on axis −q− are changed considering the grid power factor f g trajectory. In addition, the proposed controller forces the DC-link voltage to track a desired time-varying trajectory and keeps at unit value the grid power factor. Moreover, the controlled dynamics and the control signals u gcd and u gcq are bounded. It is possible to conclude that the proposed controller ensures adequate tracking performances for the DC-link dynamics and assures the decoupling between the control axes.

Grid -d- Current (A)

4.6 Simulation Results

93

(a)

3

igd

2 1 0 0

10

20

30

40

50

60

70

80

90

Grid -q- Current (A)

t [Sec] (b)

3

igq

2 1 0

-1 0

10

20

30

40

50

60

70

80

90

t [Sec]

Active power (W)

Fig. 4.68 Grid d − q currents with N-IOC

(a)

200

Pg

100 0 0

10

20

30

40

50

60

70

80

90

Reactive power(Var)

t [Sec] (b)

10

Q

g

0

-10 0

10

20

30

40

50

60

70

80

90

t [Sec]

Fig. 4.69 Grid active and reactive power with N-IOC

• Robustness to Parameter variations The objective of this experiment is to examine the proposed N-IOC performances in presence of the DC-link parameter variations and load resistance different values. The obtained results are presented as follows: Fig. 4.71 presents the DC-link voltage (a) and the grid power factor (b) dynamics controlled by the N-IOC when the grid resistance r g is increased by the 200% of its nominal value. Figure 4.72 displays the dynamics of the voltage at the output of the DC-link (a) and the grid power factor (b)

4 Neural Control Synthesis

Control -d- Signal (V)

94

(a) ugdc

100

50 0

10

20

30

40

50

60

70

80

90

Control -q- Signal (V)

t [Sec]

(b)

50

ugqc

0

-50 0

10

20

30

40

50

60

70

80

90

t [Sec]

DC-Link Voltage (V)

Fig. 4.70 Control d − q signals with N-IOC

(a)

200 100 0 0

Udc Udc ref

10

20

30

40

50

60

70

80

90

Grid Power Factor

t [Sec]

(b) 1

0.8 0

fg fg ref

10

20

30

40

50

60

70

80

90

t [Sec] Fig. 4.71 Influence of r g changes on N-IOC

controlled by the N-IOC when the grid inductance l g is augmented by the 100% of its nominal value. Figures 4.73 and 4.74 offers the dynamics of the DC voltage (a) and the grid power factor (b) controlled by the N-IOC when the load resistance rl is variated form 10 M to 100  respectively. From the obtained results, we observe that parameter variations and load resistance changes have no effects on the DClink dynamics controlled by N-IOC. In addition, the proposed controller ensures decoupling between the d − q axes.

DC-Link Voltage (V)

4.6 Simulation Results

95

(a)

200 100

U U

0 0

10

20

30

40

50

60

70

dc dc ref

80

90

Grid Power Factor

t [Sec]

(b) 1 f f

0.8 0

10

20

30

40

50

60

70

g g ref

80

90

t [Sec]

DC-Link Voltage (V)

Fig. 4.72 Influence of l g changes on N-IOC

(a)

200 100 0 0

U

dc

Udc ref

10

20

30

40

50

60

70

80

90

Grid Power Factor

t [Sec] (b) 1

0.8 0

f f

10

20

30

40

50

60

70

80

g g ref

90

t [Sec] Fig. 4.73 Influence of rl changes on N-IOC: 10 M

• Robustness to Grid Disturbances The objective to examine the capabilities of the proposed N-IOC for ensuring robustness and stability of the controlled dynamics in presence of grid disturbances. The same fault grid conditions are selected as given Fig. 4.13. Figure 4.75 presents the dynamics of the DC voltage (a) and the grid power factor (b) controlled by the N-IOC. The obtained results illustrate the effectiveness of the proposed controller, robustness and stability are assured even in presence of grid disturbances.

4 Neural Control Synthesis

DC-Link Voltage (V)

96

(a)

200 100

Udc U

0 0

10

20

30

40

50

60

70

dc ref

80

90

Grid Power Factor

t [Sec] (b) 1

0.8 0

fg f

10

20

30

40

50

60

70

80

g ref

90

t [Sec] Fig. 4.74 Influence of rl changes on N-SML: 100 

Fig. 4.75 Grid faults influences on N-IOC

4.6.2 RSC Controller The N-IOC proposed controller as applied to the DFIG has been simulated using SimPower System tools of Matlab. The DFIG prototype is connected to an infinite bus through three-phase transmission lines. In this section, a comparison for the DFIG stator active and reactive power as controlled by the proposed control scheme and the decoupled PI-FOC controller is done considering time-varying references tracking, robustness to the DFIG parameters variations, and LVRT capacity improvement.

Rotor -q- Current (A)

4.6 Simulation Results

97

(a)

0

-0.5 irq

-1

i

Current -q- Weights

-1.5 0

10

20

30

40

50

rq estimate

60

70

80

90

60

70

80

90

t [Sec] (b)

2 2,1

1 0 0

2,2 2,3 2

10

20

30

40

50

t [Sec] Fig. 4.76 DFIG rotor −q− currents identification for N-IOC

Note that the DC voltage is considered to be constant at 120 V for all simulations test. • Time-Varying Reference Tracking The purpose of this experiment is to examine the proposed controller capabilities to track time-varying required dynamics, while normal grid conditions are applied Fig. 4.1, the DFIG parameters are maintained at their nominal values, and the DFIG speed is kept at its synchronous speed (189 rad/s). Figure 4.76 presents the identification results for the DFIG rotor −q− current (a) and its corresponding neural identifier weights (b). The identification results for the DFIG rotor −d− current and its respective neural weights are displayed in Fig. 4.77. From the identification results, it is very clear to observe that the proposed RHONN identifiers for the DFIG rotor currents approximate adequately their dynamics. The DFIG rotor −d− identification and the neural identifier weights are displayed in Fig. 4.46a and b respectively. From the identification results, it clear to see that the proposed neural identifier provides a good identification for real variable state of the DFIG and also all the RHONN identifier weights are bounded. Figure 4.78 illustrates the DFIG d − q currents trajectories controlled by the NIOC. The rotor −q− current reference is chosen as a time-varying trajectory; its amplitude is variated from −1 to −0.2 A, and the rotor −d− currents reference is selected as constant value equal to 0.6 A. The desired trajectories of the rotor currents is defined from the stator powers desired trajectories.

4 Neural Control Synthesis

Grid -d- Current (A)

98

(a)

1

0.5

ird i

0 0

rd estimate

10

20

30

40

50

60

70

80

90

60

70

80

90

Grid -q- Current (A)

t [Sec] (b)

6 4 2 0 -2 0

1,1 1,2 1,3 1

10

20

30

40

50

t [Sec]

Rotor -q- current (A)

Fig. 4.77 DFIG rotor −d− currents identification for N-IOC

irq irq ref

0

-2 0

Rotor -d- current (A)

(a)

2

10

20

30

40

50

60

70

80

90

t [Sec] (b)

1

0.5 0 0

ird ird ref

10

20

30

40

50

60

70

80

90

t [Sec]

Fig. 4.78 DFIG rotor d − q currents with N-SML

Figure 4.79 presents the stator active power (a) and the stator reactive power controlled by the N-IOC. Figure 4.80 displays the zoom of the stator active and reactive power trajectory tracking response as controlled by PI-FOC (blue), N-IOC (green) controllers. The stator d − q currents are given in Fig. 4.81. The N-IOC signals u rq and u r d are displayed in Fig. 4.82a and b respectively. From these results, it is possible to see that all controllers successfully force the controlled dynamics to reach the desired trajectories with a sluggish response time for the

Active power (W)

4.6 Simulation Results

99

(a)

200 100

P P

0 0

10

20

30

40

50

60

70

s s ref

80

90

Reactive power (Var)

t [Sec] (b)

100 0

-100

Qs Q

0

10

20

30

40

50

60

70

80

s ref

90

t [Sec]

Fig. 4.79 N-IOC controllers trajectory tracking

Fig. 4.80 N-IOC trajectory tracking zoom

PI-FOC controller. In addition, we can observe that the proposed N-IOC ensures a perfect decoupling between the control axes; nevertheless a strong coupling is presented for the other one. As results of this test, we can conclude that the proposed control has better tracking performance in presence of trajectory variations with perfect decoupling between the control axes as compared with the PI-FOC controller. Table 4.7 presents statical characteristics of MSE and STD for this test.

4 Neural Control Synthesis

Stator -q- current (A)

100

0

-5 0

Stator -d- current (A)

(a)

5

isq

10

20

30

40

50

60

70

80

90

t [Sec] (b)

5 0

-5 0

isd

10

20

30

40

50

60

70

80

90

t [Sec]

Control -q- Signal (V)

Fig. 4.81 The DFIG stator d − q currents with N-IOC

(a)

50 0

u

-50 0

10

20

30

40

50

60

70

80

rq

90

Control -d- Signal (V)

t [Sec] (b)

20 0

-20 0

urd

10

20

30

40

50

t [Sec] Fig. 4.82 The DFIG control signals for N-SML

60

70

80

90

4.6 Simulation Results

101

Table 4.7 MSE and STD values for reference tracking Case Tracking Active power with PI-FOC Reactive power with PI-FOC Active power with N-IOC Reactive power with N-IOC

MSE STD MSE STD MSE STD MSE STD

0.3732 0.6063 0.052 0.2263 1.122e − 5 0.0030 8.72e − 4 0.0387

• Robustness to Parameter Variations To test the proposed control scheme robustness, the DFIG parameters are changed from their nominal values, while a normal grid conditions are considered and the mechanical speed is maintained at it synchronous value (189 rad/s). Figure 4.83 illustrates the stator active (a) and reactive (b) power controlled by N-IOC when the stator resistance rr is increased by 200% of its nominal value. Figure 4.85a–b presents the stator active and reactive power, respectively, controlled by the N-IOC when the stator inductance ls is augmented by 100% of its nominal value. Figure 4.87 displays the behavior of the stator active and reactive power controlled by the N-IOC when the rotor inductance lr is changed by 100% of its nominal value. The behavior of the stator active and reactive power when the mutual inductances lm is decreased by 90% of its nominal value is presented in Fig. 4.89. Zooms of the active and reactive power response for parameter variations are included as follows: Fig. 4.84 corresponds to the rotor resistance rr when its value is augmented by 200% of its nominal value. Figure 4.86 is related to the stator inductance ls when its value is increased by 100% of its nominal value. Figure 4.88 displays the effects of the rotor inductance lr when its value is enlarged by 100% of its nominal value, and finally Fig. 4.90 corresponds to the mutual inductance lm when its value is decreased by 90% of its nominal value. Note that, the obtained results for the N-IOC are presented in green color, for the PI-FOC in color blue, and the reference trajectory in red color. From the obtained results, it is clear to observe that parameter variations have a significant impact on the controlled dynamics using PI-FOC controller such that a strong coupling and a large response time are presented. However, the by using the proposed controller (N-IOC), an adequate performance are ensured. Table 4.8 presents statistical characteristics of MSE and Standard STD for this tracking test. • Sensitivity to Wind Speed Variations The objective of this test is to analyze the impact of WT speed changing on the stator reactive power. For that, the stator reactive power is forced to track constant reference, while the DFIG parameters are maintained at their nominal values, and normal grid

4 Neural Control Synthesis

Active power (W)

102

(a)

200 100

P P

0 0

10

20

30

40

50

60

70

s s ref

80

90

Reactive power (Var)

t [Sec] (b)

100 0

Q Q

-100

0

10

20

30

40

50

60

70

s s ref

80

90

t [Sec]

Active power (W)

Fig. 4.83 Influence of rr change on N-IOC

(a)

Reactive power (Var)

100

P with N-IOC s

50

Q with PI-FOC s

P

0 39.8

39.85

39.9

39.95

40

s ref

40.05

40.1

40.15

40.2

40.05

40.1

40.15

40.2

t [Sec] (b)

40 Q with N-IOC s

30

Q with PI-FOC s

Qs ref

20 39.8

39.85

39.9

39.95

40

t [Sec]

Fig. 4.84 rr changes influence zoom

conditions are considered. the mechanical speed is changed at instant 9.6 s from subsynchronous to super-synchronous regime (170 rad/s → 200 rad/s), and reversing it at instant 10 s. Figure 4.91 gives the dynamics the stator reactive power as controlled by the N-IOC (a), and the PI-FOC (b). From the obtained results, we can conclude that the speed change from nominal to sub-synchronous or super-synchronous impact is largely reduced by the N-IOC. Table 4.9 presents the MSE and the STD for this test.

Active power (W)

4.6 Simulation Results

103

(a)

200 100

Ps Ps ref

0 0

10

20

30

40

50

60

70

80

90

Reactive power (Var)

t [Sec] (b)

100 0

-100

Q

s

Qs ref

0

10

20

30

40

50

60

70

80

90

40.1

40.15

40.2

40.25

40.3

40.1

40.15

40.2

40.25

40.3

t [Sec]

Active Power [W]

Fig. 4.85 Influence of ls changes on N-IOC

Reactive Power [VAr]

100 50

(a) Ps with N-IOC Ps with PI-FOC Psref

0 39.8

39.85

39.9

39.95

40

40.05

t [Sec] 30 25 20 39.8

(b) Qs with N-IOC Q with PI-FOC s

Q

s ref

39.85

39.9

39.95

40

40.05

t [Sec]

Fig. 4.86 ls changes influence zoom

• Robustness to Grid Disturbances The purpose of this section is to illustrate the effectiveness of the proposed control scheme in presence of grid disturbances and compare it to the PI-FOC. To do that, the same fault grid conditions are selected as Fig. 4.13. Figures 4.92 and 4.93 offered the

4 Neural Control Synthesis

Active power (W)

104

(a)

200 100

P P

0 0

10

20

30

40

50

60

70

s s ref

80

90

Reactive power (Var)

t [Sec] (b)

100 0

Q Q

-100

0

10

20

30

40

50

80

s s ref

60

70

90

40.1

40.15

40.2

40.25

40.3

40.1

40.15

40.2

40.25

40.3

t [Sec]

Active Power [W]

Fig. 4.87 Influence of lr changes on N-IOC

Reactive Power [VAr]

100

(a) P with N-IOC s

50

Ps with PI-FOC P

s ref

0 39.8

39.85

39.9

39.95

40

40.05

t [Sec] 30

(b) Q with N-IOC s

25

Q with PI-FOC s

Q

20 39.8

s ref

39.85

39.9

39.95

40

40.05

t [Sec]

Fig. 4.88 lr changes influence zoom

DFIG powers dynamics in presence of grid disturbances controlled by the N-IOC and the PI-FOC respectively. These results confirm effectiveness of the proposed N-IOC for ensuring stability, improving LVRT capacity, and assuring continuous power generation under different grid scenarios.

Active power (W)

4.7 Conclusions

105

(a)

200 100

P

s

Ps ref

0 0

10

20

30

40

50

60

70

80

90

Reactive power (Var)

t [Sec] (b)

100 0

-100

Q

s

Qs ref

0

10

20

30

40

50

60

70

80

90

40.1

40.15

40.2

40.25

40.3

40.1

40.15

40.2

40.25

40.3

t [Sec]

DC-link Voltage (V)

Fig. 4.89 Influence of lm changes on N-IOC 100

(a) P with N-IOC s

50

Ps with PI-FOC P

sref

0 39.8

39.85

39.9

39.95

40

40.05

Grid Power Factor

t [Sec] (b)

30 20

Q with N-IOC

10

Qs with PI-FOC

s

Q

0 39.8

sref

39.85

39.9

39.95

40

40.05

t [Sec] Fig. 4.90 lm changes influence zoom

4.7 Conclusions In this chapter, a N-SMFOC, N-SML, and N-IOC schemes are proposed and tested via simulation for both DC-link and DFIG. The proposed controllers are used to force the DC voltage and the grid power factor to track their references controlled through the GSC; and to track the stator active and reactive power references through the control of the DFIG rotor currents using the RSC. The performances of these schemes are evaluated for time-varying tracking references, parameter variations and

106

4 Neural Control Synthesis

Table 4.8 MSE and STD values for parameter variations Case Tracking – Rr Ls Active power with PI-FOC Reactive power with PI-FOC Active power with N-IOC Reactive power with N-IOC

Lr

Lm

MSE

0.3387

0.1159

1.5371

0.2637

STD MSE

0.5784 0.0417

0.3406 0.1378

1.2394 0.5927

0.5142 0.2684

STD MSE

0.205 1.125e − 5

0.0552 1.2469e − 5

0.7691 1.213e − 5

0.5161 1.3412e − 5

STD MSE

0.0026 0.0098

0.0042 0.0033

0.0028 0.0020

0.0025 0.0082

STD

0.0376

0.0607

0.0597

0.0245

(a)

Reactive Power [VAr]

30 20

Qs with N-IOC Q

10 9

9.2

9.4

9.6

9.8

10

10.2

10.4

10.6

s ref

10.8

11

(b)

25

20 Q with PI-FOC s

15 9

Qsref

9.2

9.4

9.6

9.8

10

10.2

10.4

10.6

10.8

11

Fig. 4.91 Speed variation effects on N-SML

speed changing influence considering ideal grid conditions. In addition the proposed neural controllers performances for the DFIG are compared with conventional control algorithms as the PI-FOC. Moreover, all controllers are examined under different grid conditions in order to test the LVRT capacity of the DFIG. We can conclude that the proposed controllers based on their corresponding neural identifier presents excellent performances in presence of time-varying reference, parameters variations, and speed changing comparing with the other conventional

4.7 Conclusions

107

Table 4.9 MSE and STD values for speed variations Case Speed variation Active power with PI-FOC Reactive power with PI-FOC Active power with N-IOC Reactive power with N-IOC

MSE STD MSE STD MSE STD MSE STD

0.4077 0.6400 0.4147 0.6367 1.122e − 5 0.0035 0.0100 0.03605

Fig. 4.92 Stator powers with N-IOC under fault conditions

controllers. In addition, decoupling between control axes is ensured and response time is improved. Moreover,they enhance the LVRT capacity and ensure transit stability under different fault grid scenarios, which accomplish with modern grid requirements. The high performances of the proposed controllers are achieved thanks to the neural identifier, which permits to obtain an adequate models of the controlled systems in presence of parameter variation and/or grid disturbances, helping the controllers to reject their effects. For the proposed neural controllers comparison, all controllers almost have the same performances with a slight improvement in the MSE and the STD values for all experiments which have been with the N-SML and

4 Neural Control Synthesis

Active Power [W]

108

(a)

160 155 150

Ps with PI-FOC Psref

Reactive Power [Var]

t = 4s to t = 9s

t = 9s to t = 14s

(b)

-10

t = 14s to t = 20s

20 0

-10

Qs with PI-FOC Q

sref

Ground to Phase Fault

Phase to Phase Fault

Three-Phase to Ground Fault

Fig. 4.93 Stator powers with PI-FOC under fault conditions

the N-IOC controllers. Regarding these performance, the following controllers are selected for real-time validation in the next Chapter: SM-FOC, N-SMFOC, N-SML, and N-IOC.

Chapter 5

Experimental Results

This chapter discusses experimental implementations of the SM-FOC, N-SMFOC, and N-SML schemes for controlling the GSC and the RSC of 1/4 HP DFIG prototype. The proposed controller are tested to track desired trajectories of the DC voltage, the Grid power factor through the control of the GSC; and the stator active and reactive power via the DFIG rotor currents using the RSC. In addition, the proposed controllers are tested for variable wind speed profile to track a time-varying power reference and to extract the maximum power from the wind, under both balanced and unbalanced grid conditions.

5.1 Prototype Description The proposed control schemes performance are evaluated using the DFIG prototype of the Automatic Control Laboratory, Cinvestav, Guadalajara, Mexico. It comprises five major parts as displayed in Fig. 5.1: (a) a 1/4 HP three-phase DFIG, (b) to emulate a wind turbine a DC motor is used, (c) two Pulse Width-Modulation (PWM) converters for power stage, (d) a transmission line module to emulate the grid, and (e) to supervise and to generate the results a personal computer. Figure 5.2 displays the experimental control scheme. The control platform comprises: MatLab/Simulink R2008a the RT-LAB software, the dSPACE DS11041 Controller Board real-time interface including the dSPACE ControlDesk 3.3 software are installed on the personal computer. The DFIG series 8231-002 is provided by Lab-Volt; its rotor is linked

1 dSPACE 2 DFIG

DS1104. de 2011-2014, © Maplesoft, Inc. is a Lab-Volt commercial machine of 1/4 HP.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 E. N. Sánchez and L. Djilali, Neural Control of Renewable Electrical Power Systems, Studies in Systems, Decision and Control 278, https://doi.org/10.1007/978-3-030-47443-0_5

109

110

5 Experimental Results

Fig. 5.1 DFIG prototype

Fig. 5.2 DFIG prototype scheme

to the grid via a back-to-back converter and a step-up terminal transformer. In order to generate wind profiles, a DC motor model 8968-3, 3/4 HP of LabVolt3 is coupled with the DFIG rotor shaft which is used to emulate a small-scale wind turbine (three blade rotor, fixed pitch, 1.15 m diameter), for more details see [65]. To protect the prototype, electric protection devices are used to limit stator and rotor currents. To measure speed, voltages, and currents, sensors are adapted with the DS1104 controller Board. Both electronics converters are controlled using Space-Vector Pulse

3 DC

motor is a Baldor reliance industrial motor of 3/4 HP.

5.1 Prototype Description

111

Width Modulation (SV-PWM) pulses, which are generated on the DS1104 controller Board. The switching frequency is defined for the SVPWM as 4860 Hz for the GSC, and as 1620 Hz for the RSC.

5.2 Real-Time SM-FOC The purpose of this section is to evaluate in real-time the reference tracking for the DC voltage and the grid power factor controlled by SM-VOC through the control of the GSC and the stator active and reactive power as controlled by the SM-FOC via the control of the RSC, considering ideal and faulted grid conditions.

5.2.1 GSC Results

150 100

Grid power factor (fg)

DC Link Voltage [V]

The objective of this test is to force the DC voltage to track a constant desired trajectory and at the same time to kept the grid power factor at the unit value under normal grid conditions. Figure 5.3 presents the DC-link voltage Udc and the electric power factor f g at the step-up transformer controlled by the GSC; it is clear to see that the GSC controller maintains the DC-link voltage and the grid power factor at constant values. The grid active Pg and reactive powers Q g dynamics and the grid i gd , i gq currents are displayed in Figs. 5.4 and 5.5 respectively; we observe that the dynamics of the grid active power Pg and i gd current depend on the DC voltage; the

Udc

50 0

Udcref 0

10

20

30 t [Sec]

40

50

60

1.5 1 fg

0.5 0

fgref 0

10

20

30 t [Sec]

Fig. 5.3 DC-link voltage and grid power factor with SM-VOC

40

50

60

Reactive power [VAr]

Active power [W]

112

5 Experimental Results

100 50 0 0

Pg 10

20

30 t [Sec]

40

50

60 Qg

20 0 −20 0

10

20

30 t [Sec]

40

50

60

Grid q−current [A]

Grid d−current [A]

Fig. 5.4 Grid active and reactive power 2

igd

1.5

1 0

10

20

30 t [Sec]

40

50

0.2

60 igq

0

−0.2 0

10

20

30 t [Sec]

40

50

60

Fig. 5.5 Grid currents i gd and i gq

grid reactive power Q g and i gq current responses are related to the electric power factor at the step-up transformer. The control signals u gcd and u gcq of the DC-Link are displayed in Fig. 5.6.

Control ugcq (V)

Control ugcd (V)

5.2 Real-Time SM-FOC

113

65

u

gcd

60

55 0

10

20

30 t [Sec]

40

50

1

60 ugcq

0

−1 0

10

20

30 t [Sec]

40

50

60

Fig. 5.6 GSC control signals u gcd and u gcq

5.2.2 RSC Results The goal id to test the proposed control system capabilities to operate wind power generation; two cases are studied. For the first case, we apply a constant wind turbine speed profile, which is fixed at the DFIG synchronous speed (189 rad/s); the stator active power is forced to track a time-varying reference and the stator reactive power is kept at desired constant value. For the second case, the control scheme is modified in order to extract the maximum power from the wind speed using the Maximum Power Point Tracking (MPPT) algorithm. We note that the DC voltage at the output of the DC-link and the step-up transformer terminals electric power factor are maintained constants at their desired values by the GSC control scheme for all cases. • Constant Wind Turbine Speed Figure 5.7 presents a constant WT speed (189 rad/s), which is emulated by the DC motor. The tracking response for the rotor d − q currents are presented in Fig. 5.8; the rotor currents references are defined from specified stator active and reactive powers. Figures 5.9 and 5.10 display the stator active Ps and reactive Q s powers and the stator i sd , i sq currents, respectively generated from the DFIG. We observe that the dynamics of the active power Ps and the stator i sq currents depend on irq rotor current, and the dynamics of the reactive power Q s and the stator i sd currents depend on ir d rotor current. The RSC control dynamics u rq and u r d for this first case are presented in Fig. 5.11. It is possible to see that the control signals are bounded. In addition, the RSC controller ensures an adequate performances even in presence of reference variations.

114

5 Experimental Results 220

Rotor speed [rad/s]

200

180

160

140

ω

r

ω 120 0

rref

10

20

30

40

50 t [Sec]

60

70

80

90

100

Rotor d− current [A]

Rotor q− current [A]

Fig. 5.7 Constant wind turbine speed 2

irq real

DFIG start−up

irqref 0 Control start−up

−2 0

10

20

30 t [Sec]

40

50

60

2 1 0 ird real

−1 −2 0

irdref 10

20

30 t [Sec]

40

50

60

Fig. 5.8 Rotor d − q currents tracking with SM-FOC

• Maximum Power Extraction In this case, a time-varying wind turbine speed profile is used to generate the mechanical speed as presented in Fig. 5.12 which is emulated by the DC motor. The desired DFIG rotor speed profile is given by: ωr = 194 + sin(0.14t) + sin(0.27t + π/2) + sin(0.31t) + sin(1.3t − π/2) + sin(0.2t) + sin(0.7t) + sin(0.2t)

115

200

0 Ps real −200 0

Reactive power [VAr]

Active power [W]

5.2 Real-Time SM-FOC

Psref 10

20

30 t [Sec]

40

50

60

50 40 30 20

Q s real

10 0 0

Q sref 10

20

30 t [Sec]

40

50

60

Stator d current [A]

Stator q current [A]

Fig. 5.9 Stator active and reactive powers tracking with SM-FOC 2

isq

0

−2 0

10

20

30 t [Sec]

40

50

0.2

60 isd

0 −0.2 −0.4 0

10

20

30 t [Sec]

40

50

60

Fig. 5.10 Stator currents dynamics with SM-FOC

The proposed mechanical speed is changed randomly form sub-synchronous speed to super-synchronous speed (185 to 200 rad/s) in order to simulate a realistic wind turbine operation modes. The objective in this case is to examine the capabilities of the proposed control scheme to operate the wind turbine system to generate the maximum electrical power from the wind. To ensure a maximum power generation, the stator active power reference is calculated using the MPPT algorithm based on the following equation

Control urq (V)

116

5 Experimental Results 100

u rq

50 0 −50

Control urd (V)

0

10

20

30 t [Sec]

40

50

60

20

u rd

0

−20 0

10

20

30 t [Sec]

40

50

60

Fig. 5.11 RSC control signals u rq and u r d , first case 220

Rotor speed [rad/sec]

210 200 190 180 170 ωr

160 150

ωrref 0

10

20

30

40

50 t [Sec]

60

70

80

90

100

Fig. 5.12 Variable wind turbine speed

Pr e f = K opt 3mec (λ

,β

)

(5.1)

, and C pmax represents the maximum yield aerodywith K opt = pmax λ3opt opt ρ.π.r 2G 3 opt namics of the WT, βopt is the optimal angle blade, λopt is optimal relative speed. The proposed MPPT algorithm in this work is one of the most applied technique for WT generation because it is simple, fast and efficient as addressed in [57, 66]. The desired trajectories of the DFIG rotor d − q currents are defined from the desired stator active and reactive power such that the rotor −q− reference is calculated from C

5

Rotor d current [A]

Rotor q current [A]

5.2 Real-Time SM-FOC

117

1 0 −1 irq real

−2 −3 0

irqref 10

20

30 t [Sec]

40

60

50

1

0.5 ird real 0 0

irdref 10

20

30 t [Sec]

40

50

60

Active power [W]

Fig. 5.13 MPPT rotor current tracking with SM-FOC 200 100 Ps real 0

Reactive power [VAr]

0

Psref 10

20

30 t [Sec]

40

50

60

30 25 Qs real

20 15 0

Q sref 10

20

30 t [Sec]

40

50

60

Fig. 5.14 MPPT stator active powers with SM-FOC

the desired active power generated from the MPPT algorithm, while the rotor −d− reference is obtained from the stator reactive power reference value, which is maintained constant in order to ensure a desired stator power factor. Figure 5.13 presents tracking for the d − q rotor currents; The stator active and reactive powers are presented in Fig. 5.14. Figure 5.15 illustrates the stator d − q currents. The stator power factor and the electromagnetic torque are depicted in Fig. 5.16. It can be observed that the proposed control scheme has a good tracking performance which ensure the maximum power extraction and a desired power factor in the DFIG stator. In addition the variable speed and the stator active power changes have no significant effects on

5 Experimental Results

Stator d−current [A]

Stator q−current [A]

118 2

isq

1 0 −1 0

10

20

30 t [Sec]

40

50

60

0 isd −0.2 −0.4 0

10

20

30 t [Sec]

40

50

60

1.5

stator power factor

1 0.5 0 −0.5 0

Electomagnetic torque [N−m]

Stator power factor

Fig. 5.15 MPPT stator currents with SM-FOC

10

20

30 t [Sec]

40

50

3

60

Tem

2 1 0 0

10

20

30 t [Sec]

40

50

60

Fig. 5.16 MPPT stator power factor and electromagnetic torque with SM-FOC

the stator reactive power and the stator power factor, which means that the proposed control scheme ensures a perfect decoupling between the control d − q axes. Figure 5.17 illustrates the SM-FOC controller d − q signals as applied to the RSC.

5.2 Real-Time SM-FOC

119

50 0 −50 0

urq 10

20

30

40

50

10

20

30

40

50

60

50 0 −50 0

urd 60

Fig. 5.17 RSC control signals u rq and u r d , fault grid case

5.2.3 Robustness to Grid Disturbances The WT based on the DFIG is a very sensitive electric generator in presence of nonideal grid conditions, due to direct connection of its stator to the grid. For wind power generation system, voltage dip is not acceptable because it can produce considerable oscillations in the electromagnetic torque and the stator active and reactive power, which lead to disconnect the wind turbine. The objective of this experiment is to examine the capabilities of the proposed control scheme to allow the WT to stay connected under distorted and unbalanced stator voltage, to eliminate the oscillation in the stator active and reactive powers, and to maintain the DC voltage constant at the output of DC-link under fault grid conditions. In this experiment, the DFIG stator is connected to the infinite bus via a three-phase transmission lines module of Lab Volt (8329-00), which permits to adjust the grid impedance to four different values using a selector switch (0, 60, 120, 180 ), and a three-phase variac module is integrated between the infinite bus and the three-phase transmission line in order to produce drop voltage in the grid, Fig. 5.18. Figure 5.19 presents the three-phase stator currents and three-phase stator voltage as applied to the DFIG such as, a 15% two-phase drop voltage and harmonic perturbations occur from t = 15 s to t = 35 s; normal conditions are considered for all different time. The insertion of the three-phase transmission line module produces a symmetric threephase voltage drop and a harmonic perturbation in the grid voltages and currents. Figures 5.20 and 5.21 present the DC voltage at the output of DC-link and the grid power factor controlled by GSC, and the maximum stator active power and the stator reactive power controlled by RSC respectively. From the results, it is possible to observe that the proposed controller for both GSC and RSC ensures the desired values of the DC voltage and the stator active and reactive powers under normal and perturbed grid conditions. At t = 15 s, when the voltage dip appears, we observe

120

5 Experimental Results

stator Current [A]

Fig. 5.18 DFIG prototype for fault grid conditions 2

0

−2

15s−35s

stator Voltage [V]

0s−15 s

35s−60s

250

0

−250

Balanced voltages 0s−15 s

Unbalanced voltages 15s−35s

Balanced voltages 35s−60s

Fig. 5.19 Grid voltage under fault conditions

a small tracking error in the DC voltage, which causes a small tracking error in the stator active and reactive power trajectories. The controllers remove these errors before voltage dip vanishing, which mean that the proposed controllers are able to stabilize the GSC and the RSC under drop grid voltage. From this discussion, it is possible to see that the RSC and the GSC controllers ensure good performances even in presence of reference variations, it can ensure a maximum power extraction, and transit stability of the controlled dynamics in presence of fault grid conditions. In

Grid Power Factor DC−link Voltage [V]

5.2 Real-Time SM-FOC

121 udc

140

udcref 120 100 0

10

20

30 t [Sec]

40

50

60

1 0.8

fg

0.6 0

fgref 20

10

30

50

40

60

90

80

70

t [Sec]

Reactive power [VAr] Active power [W]

Fig. 5.20 DC voltage and grid power factor under fault grid conditions 200 100 Ps real 0 0

Psref 10

20

30 t [Sec]

40

60

50

30 25 20

Qs real Q sref

15 0

10

20

30 t [Sec]

40

50

60

Fig. 5.21 Active and reactive under fault grid conditions

addition, from the control signal figures, all the control signals generated from the different tests are less then grid based voltage (179.63) V, which ensure to avoid the PWM saturation.

5.3 Real-Time N-SMFOC The purpose of this section is to examine the N-SM-FOC controllers to track in realtime the reference tracking for the DC voltage and the grid power factor controlled by GSC and the stator active and reactive power through the control of the DFIG rotor currents controlled by RSC considering ideal and faulted grid conditions.

5 Experimental Results

(a) 180 160 140 120 100 80 0

DC Voltage Weights

DC−Link Voltage [V]

122

udc udc estimate 10

20

30

40

t [Sec]

50

60

70

80

90

(b) 1 ω1,1 ω1,2

0.5 0 0

ϖ1 10

20

30

40

t [Sec]

50

60

70

90

80

Fig. 5.22 DC voltage identification and weights

5.3.1 GSC and RSC Identification Results

Grid Current idg [A]

Real-time identification results include both GSC and RSC as follows: Fig. 5.22 illustrates the real and the estimate DC voltage state and the evolution of the neural weights according to (4.14). Figures 5.23 and 5.24 present grid d − q currents real and estimate states and the behavior of neural weights vectors considering (4.15)

(a) 3 2 1

idg

0 −1 0

idg estimate 10

20

30

40

t [Sec]

50

60

70

80

90

(b) idg weight

2 1

ω2,1

0

ω2,3

−1 0

ω2,2 ϖ2

10

20

30

50

40 t [Sec]

Fig. 5.23 Grid d current identification and weights

60

70

80

90

Grid Currents iqg [A]

5.3 Real-Time N-SMFOC

123

(a)

2 0 −2 0

iqg iqg estimate 10

20

30

40

t [Sec]

50

60

70

90

80

iqg weight

(b) 2

ω3,1 ω3,2

0 −2 0

ϖ3 10

20

30

40

t [Sec]

50

60

70

90

80

Stator ird Current [A]

Fig. 5.24 Grid q current identification and weights

(a)

1 0 −1 0

ird ird estimate 10

20

30

40

50

60

70

80

90

(b)

2 ird Weights

t [Sec]

ω

1,1

ω1,2

0 −2 0

ω1,3 ϖ

1

10

20

30

40

t [Sec]

50

60

70

80

90

Fig. 5.25 Rotor d current identification and weights

and (4.16) respectively. Figures 5.25 and 5.26 show the dynamics of the rotor d − q real and estimate state and the behavior of the neural weights vector according to the RHONN identifier equation cited in (4.33) and (4.34) respectively. From the identification results, it is very clear to see that the proposed RHONN identifier for both DC-link and DFIG achieves good identification for the real states; in addition all the neural weights are bounded.

5 Experimental Results

Stator irq Current [A]

124

(a)

2

irq i estimate rq

0 −2 0

10

20

30

40

t [Sec]

50

60

70

80

90

(b) irq Weights

4 ω

2,1

2

ω2,2 ω

0 −2 0

2,3

ϖ

2

10

20

30

40

t [Sec]

50

60

70

80

90

Fig. 5.26 Rotor q current identification and weights

5.3.2 GSC Controller Results

DC−Link Voltage [V]

In order to test the capabilities of the proposed controller to assure that the GSC tracks the desired reference and to ensure decoupling between the control axes, we force the DC voltage to track time-varying references and to keep the grid power factor at a nominal value. Figure 5.27 presents the response of the DC voltage dynamics

(a)

180 160 140

Udc

120 100 0

Udcref 10

20

30

40

t [Sec]

50

60

70

90

80

Grid Power Factor

(b) 1 0.9 0.8

fg

0.7 0

fgref 10

20

30

40

t [Sec]

50

60

Fig. 5.27 DC voltage and grid power factor tracking with N-SMVOC

70

80

90

5.3 Real-Time N-SMFOC

125

(a)

Grid Powers

200 100

Pg

0

Grid Currents

−100 0

Qg 10

20

30

40

t [Sec]

50

60

70

80

90

(b)

4 2

idg

0 −2 0

igq 10

20

30

40

t [Sec]

50

60

70

80

90

Fig. 5.28 Grid powers and currents with N-SMFOC

at the output of the DC-link (a) and the grid power factor at the step-up transformer (b) controlled by the GSC. Figure 5.28 displays the grid active and reactive powers behaviors as follows: (a) flow between the grid and the DFIG winding rotor terminal and (b) the grid d − q currents measurements. From these results, it is possible to see that the grid active power (Pg ) and the grid current on axis d are related to the DC voltage behavior; the grid reactive power (Q g ) and the grid current on axis q depend on the grid power factor f g dynamics. In addition, we can conclude that the GSC controller ensures good tracking performance for the DC voltage timevarying desired trajectory and for the grid power factor constant value. Moreover, the proposed controller assures a high decoupling between the control axes.

5.3.3 RSC Controller Results The objective of this test is to examine the abilities of the RSC controller; two cases are considered: time-varying power reference tracking and maximum power extraction tracking. Note that for both cases the DC voltage at the DC-link and the grid electric power factor at the step-up transformer are maintained constant at 120 V and at nominal value respectively. In addition, a time-varying wind speed profile is applied on the DFIG rotor shaft, which is emulated by the DC motor as shown in Fig. 5.12. The proposed mechanical speed is changed randomly form sub-synchronous speed to super-synchronous speed (185 to 200 rad/s) in order to emulate a realistic wind turbine behavior. • Time-varying power reference

Stator ird Current [A]

Stator irq Current [A]

126

5 Experimental Results

(a)

2

irq irqref

0 −2 0

10

20

30

40

t [Sec]

50

60

70

(b)

3

90

80

ird irdref

2 1 0 −1 0

10

20

30

40

t [Sec]

50

60

70

90

80

(a) 200 100

Ps

0 −100 0 Stator reactive power [Var]

Stator active power [W]

Fig. 5.29 Rotor d − q tracking with N-SMFOC

Psref 10

20

30

40

t [Sec]

50

60

70

90

80

(b) 25 20 15 0

Qs Qsref 10

20

30

50

40

60

70

80

90

t [Sec]

Fig. 5.30 Stator active and reactive powers with N-SMFOC

Figure 5.29 shows the rotor q currents tracking (a), calculated from a desired stator active power using (4.37), and the rotor d current (b), obtained from a desired stator reactive power using (4.38). Figures 5.30 and 5.31 display the stator active (Ps ) and reactive (Q s ) powers, and the stator (i sd ), (i sq ) currents respectively generated by the DFIG. It is possible to see that the stator active power (Ps ) and the stator (i sq ) current depend on the rotor (irq ) dynamics, and the stator reactive power (Q s ) and the stator (i sd ) current are related to the rotor (irq ) behaviors. As results of this case, it is easy

5.3 Real-Time N-SMFOC

127

Stator isq Current [A]

(a) 2

isq

1

0 0

10

20

30

50

40

60

70

80

90

t [Sec]

Stator isd Current [A]

(b) 2

isd

0

−2 0

10

20

30

40

50

60

70

80

90

t [Sec]

Fig. 5.31 Stator d − q currents with N-SMFOC

to conclude that the RSC controller attains an adequate tracking performance for time-varying reference of the stator active power and the constant desired value of the stator reactive powers, which maintain a constant stator power factor. Moreover, the decoupling between the d − q control axis is ensured. • Maximum power extraction The objective is to test the capability of the proposed controller to track the maximum power reference obtained with the MPPT algorithm to ensure a maximum power generation. So, the stator active power reference is calculated using the MPPT algorithm as in (5.1). Figure 5.32a, b presents the tracking response for the rotor d − q currents. The rotor q current reference is defined from the MPPT stator active power using (4.37), which ensures an optimal stator active power. In addition, to assure a constant stator power factor, the rotor d current reference is calculated from a constant stator reactive power. Figure 5.33 displays the stator active (Ps ) and reactive (Q s ) powers respectively generated from the DFIG. The stator electric power factor and the electromagnetic torque are presented in Fig. 5.34a, b respectively. We can observe that the RSC controller presents a good tracking performance for a maximum power reference extracted from the wind and for a constant stator reactive power, which improves quantity and quality of the power generations. In addition, the controller ensures a high decoupling between the control axes components. Moreover, the speed variation has no effect on the stator reactive power controlled by the N-SMFOC.

Rotor irq Current [A]

128

5 Experimental Results

(a)

1 0 -1

irq

-2

irqref

-3 0

10

20

30

40

50

60

70

80

90

Rotor ird Current [A]

t [Sec]

(b)

2 0

i rd irdref

-2 0

10

20

30

40

50

60

70

80

90

t [Sec]

Active power [W]

Fig. 5.32 MPPT rotor d − q currents with N-SMFOC

(a)

300 200 100

Ps Psref

0 -100

0

10

20

30

40

50

60

70

80

90

Reactive power [VAr]

t [Sec]

(b)

25 20 15 0

Qs Qsref

10

20

30

40

50 t [Sec]

60

70

80

90

Fig. 5.33 MPPT stator active and reactive powers with N-SMFOC

5.3.4 Robustness to Grid Disturbances The purpose of this experiment is to examine the capabilities of the proposed controller to eliminate the ripple, which appears in the stator active and reactive power and in the DC voltage under non-ideal grid conditions when the classical controllers are used, to ensure stability and continuity of WT power generation.

129

(a) 1 0.8 0.6 0

Stator power factor

10

20

30

40

50

60

70

80

60

70

80

90

t [Sec]

Electromagnetic Torque [N.m]

Stator power factor

5.3 Real-Time N-SMFOC

(b)

4 2 0 0

Tem

10

20

30

40

50

90

t [Sec]

Grid Current [A]

Fig. 5.34 MPPT stator power factor and electromagnetic torque with N-SMFOC 2

0

Grid Voltage [V]

-2 0-30 Sec

30-40 Sec

40-50 Sec

Real grid condition

Line transmission impedance 60 Ω

Line transmission impedance 120 Ω

60-70 Sec

200 0 -200 Line transmission impedance 60 Ω and unbalance voltage

Fig. 5.35 Grid currents and voltages under fault conditions

For that, the same electric scheme is used as in Fig. 5.18. Figure 5.35 presents the different scenarios of the three-phases currents and voltages as applied to the DFIG prototype such as: • At t = 30 s, a symmetric voltage dips and distortions are provoked by adjusting the selector switch of the transmission lines module form 0 to 60 . • At t = 40 s, additional symmetric voltage dips and distortions are created by adjusting the selector switch of the transmission lines module form 60 to 120 .

5 Experimental Results

(a)

160

udc udcref

140

120 0

Grid Power Factor

DC-Link Voltage [V]

130

10

20

30

40 50 t [Sec]

60

70

80

90

(b) 1 0.9

fg

0.8 0

f 10

20

30

40 50 t [Sec]

60

70

80

gref

90

Fig. 5.36 DC voltage and grid power factor under fault conditions

• At t = 50 s, a sudden changing in the transmission lines impedance is incepted by adjusting the selector switch of the three-phases transmission line module form 120 to 60  and reversing it. • From t = 60 s to 70 s, an asymmetric (two-phases) voltage dip of 15% is considered using the three-phase variac module. • Normal grid conditions are considered for the remaining time. Figures 5.36 and 5.37 present the DC voltage at the output of the DC-link and the grid power factor controlled by the GSC, and the maximum stator active power and the stator reactive power controlled by the RSC respectively. In order to compare real-time results for the proposed controller (N-SMFOC), the same test is done with the prototype and under the same condition for the SM-FOC controller. The obtained results for the stator active and reactive power controlled by the SM-FOC are presented in Fig. 5.38. From these results, it is possible to observe that the proposed controller for both the GSC and the RSC ensures tracking of the desired values for the DC voltage and for the stator active and reactive powers under normal and disturbance grid conditions. In addition, it removes oscillations in the stator active and reactive power appearing under fault grid condition. Whereas, the SM-FOC does not eliminate ripples produced by grid disturbances on the stator active and reactive power and on the stator currents when an addition symmetric voltage dip is provoked at the instant 40 s, which lead to instability. The DFIG stator currents for SM-FOC and N-SMFOC are presented in Fig. 5.39a, b respectively.

5.4 Real-Time N-SML Control

131

Active power [W]

(a) 200 100 0 0

Ps Psref

10

20

30

40

50

60

70

80

90

Reactive power [VAr]

t [Sec]

(b)

30

Qs

25

Qsref

20 15 0

10

20

30

40

50

60

70

80

90

t [Sec]

Stator active power [W]

Fig. 5.37 Stator powers with N-SMFOC under fault conditions

(a)

400

Ps Psref

200 0 −200 0

10

20

30

40

50

60

70

80

100

90

Stator reactive power [Var]

t [Sec]

(b)

100

Qs Qsref

50 0 −50 0

10

20

30

40

50

60

70

80

90

100

t [Sec]

Fig. 5.38 Stator powers with SM-FOC under fault conditions

5.4 Real-Time N-SML Control The objective of this section is to evaluate in real-time the N-SML controller under the same scenario as before.

132

5 Experimental Results

(a)

Stator current [A]

6

High stator current pick

4

The DFIG prototype is disconnected from the Grid

2 0 −2

(b) Stator current [A]

3 2 0 −2 −3 Nominal values of the grid impedances

Sudden variation in the grid impedances at t=45s

Nominal values of the grid impedances

Fig. 5.39 Stator currents with SM-FOC and N-SMFOC

5.4.1 Identification Figure 5.40 portrays the behavior of the neural identifier for the DC-link according to the neural model presented in (4.14)–(4.16). The identification results are included as follows: Fig. 5.40—(a) presents the DC-voltage estimation, the DC voltage value is 120 V; Fig. 5.40—(b) shows the grid −d− current estimation, its value is equals to 1.5 A; and Fig. 5.40—(c) displays the grid −q− current estimation, its value is null. Figure 5.41 presents the evolution of the weights vector for DC-link RHONN identifier; each vector is composed by adjustable weights and a fixed weight as follows: Fig. 5.41—(a) displays the behavior of W1 = [w11 w12 1 ]; Fig. 5.41— (b) presents the behavior of W2 = [w21 w22 w23 2 ]; Fig. 5.41—(c) appears the behavior of W3 = [w31 w32 3 ]. Figure 5.42 offers identification results for DFIG according to the neural identifier presented in (4.33)–(4.34). Figure 5.41—(a) displays the rotor −d− current estimation; and Fig. 5.41—(d) arises the rotor −q− current estimation. Figure 5.43 presents the evolution of weights, such as each vector is composed by three adjustable weights and one fixed one. Figure 5.43—(a) displays the behavior of W1 = [w11 w12 w13 1 ]; Fig. 5.43—(b) offers the behavior of W2 = [w21 w22 w23 2 ]. It is clear to see that the proposed neural model provides a good identification for variable state of both the DC-link and the DFIG and all the NN weights are bounded.

Dc−link Voltage [V]

5.4 Real-Time N-SML Control

133

(a) 100 Udc

50 0 0

Udc estimate

10

20

30

40

50

60

70

80

100

90

Grid current i gq

Grid current i gq

(b) 2 i gd

1 0 0

i gd estimate

10

20

30

40

50

60

70

80

100

(c)

2 0

i gq

−2 0

90

i gq estimate

10

20

30

40

50 t [Sec]

60

70

80

100

90

(a)

2

w11

1 0 0

w12 ϖ

1

10

20

30

40

Weight vector W

50

60

70

80

90

w21

2

w

22

w23

1 0 0

100

(b)

3

ϖ

4

10

20

30

40

50

60

70

80

90

100

(c)

3

Weight vector W2

Weight vector W1

Fig. 5.40 RHONN state identification of DC-link

5

w31

0 −5 0

w32 ϖ

3

10

20

30

40

50 t [Sec]

Fig. 5.41 Weights vector of DC-link

60

70

80

90

100

5 Experimental Results

Rotor -d- current [A]

Rotor -q- current [A]

134

(a)

2

irq irq estimate

0 -2 0

10

20

30

40

50 t [Sec]

60

70

80

90

100

(b)

1

ird ird estimate

0.5 0

10

20

30

40

50 t [Sec]

60

70

80

90

100

Weights vector W2

Weights vector W1

Fig. 5.42 RHONN state identification of DFIG rotor currents

(a)

2

w w

0 -2 0

w

12 13 1

10

20

30

40

50 t [Sec]

60

70

80

90

100

(b)

2

w11 w

0 -2 0

11

w

12 13 4

10

20

30

40

50 t [Sec]

60

70

80

90

100

Fig. 5.43 Weights vector of DFIG

5.4.2 GSC Controller Results Figure 5.44—(a) and (b) presents the DC-link voltage Udc and the electric power factor f g at the step-up transformer controlled by the GSC. It can be seen that the GSC controller reaches the control objective in which the DC-link voltage is forced to achieve its desired values which is 120 V and the grid power factor is maintained at the unit value. Figure 5.45a presents the grid active (Pg , blue, 100 W) and reactive powers (Q g , red, 0V ar ). The grid i gd , i gq currents dynamics are displayed in Fig. 5.45b

5.4 Real-Time N-SML Control

135

100 50 0 0

Udc Udcref 10

20

30

40

50 t [Sec]

60

70

80

90

100

(b)

1.5

Grid power factor

DC−link Voltage [V]

(a)

1 fg

0.5 0 0

fgref 10

20

30

40

50 t [Sec]

60

70

80

90

100

Grid g powers (Pg ,Q g )

Fig. 5.44 DC-link voltage and grid power factor with N-SML

(a)

150 100 50 0

Pg

−50

Qg

0

10

20

30

40

50

60

70

80

90

100

t [Sec] Grid currents (igd,igq )

(b) 2 1 igd

0 −1 0

igq 10

20

30

40

50

60

70

80

90

100

t [Sec]

Fig. 5.45 Grid powers and currents with N-SML

(blue, 1.5 A–red, 0 A) respectively. The obtained results illustrate that the grid active power Pg depends on the grid i gd current, which is related to the DC voltage; and the grid reactive power Q g depends on the grid i gq current, which is related to the electric power factor at the step-up transformer. Note that: (1) the DC voltage desired value is selected for providing enough active power to the RSC controller in order

Control signal u gcd [V]

136

5 Experimental Results

(a)

100

u gcd

0

−100 0

10

20

30

40

50

60

70

80

90

100

Control signal ugcq [V]

t [Sec]

(b)

20

ugcq

0

−20 0

10

20

30

40

50

60

70

80

90

100

t [Sec]

Fig. 5.46 Grid control signals with N-SML

to control the stator active and reactive power. (2) the unit value of the grid power factor helps to improve the quality of the supplied power for the RSC and it allows to maintain at a desired value the grid reactive power, which has a large effect on the security of the WT power systems because it affects the voltage throughout the system. Figure 5.46a, b presents the GSC control signals u gcd , u gcq respectively. It is possible to see that the control signals are bounded and non-saturated.

5.4.3 RSC Controller Results To test the proposed control system capabilities to obtain wind power generation, two cases are studied: For the first case, we apply a constant wind turbine speed profile, Fig. 5.8 with a time-varying reference stator active power and constant reference stator reactive power; the objective of this case is to examine the reference stator active and reactive power tracking. For the second case, a varying-time wind turbine speed profile, (Fig. 5.12), is applied in order to operate the wind system to extract the maximum power from the wind. We note that: (1) the DC motor is used to generate the DFIG rotor speed where the desired DFIG rotor speed profile is applied as a reference speed for the DC motor (0 to 2500r/min, [65]). (2) the DC voltage at the output of GSC and the step-up transformer terminals electric power factor are maintained constants, 120 V and at the unit value, by the GSC control scheme for both of these cases. • Constant Wind Turbine Speed

5.4 Real-Time N-SML Control

137

Rotor −q− current [A]

(a) 2

irq irqref

0

−2 0

10

20

30

40

50

60

70

80

90

100

t [Sec] Rotor −d− current [A]

(b) 0.8

i

rd

i

rdref

0.7 0.6 0.5 0

10

20

30

40

50

60

70

80

90

100

t [Sec]

Fig. 5.47 Rotor currents tracking with N-SML

The tracking response for the rotor d − q currents are presented in Fig. 5.47, the rotor currents references are defined from desired stator reactive and active powers. The rotor −q− current is forced to track a time-varying reference, its amplitude can vary from −1 to −0.2 A; while, the rotor −d− current is maintained at a desired value equals to 0.6 A. Figure 5.48 displays the trajectory of stator active Ps and reactive Q s powers obtained from the DFIG stator. The RSC controller fulfilled the control objectives, where the stator active power achieves the desired trajectory tracking, which is a time-varying signal changing from 40 to 150 W, and the stator reactive power is maintained constant at a desired value equal to 21V ar . Figure 5.49 offers the stator i sd , i sq currents, respectively generated by the DFIG. From these results, we can see that the dynamics of stator i sq currents and stator active power Ps depend on rotor irq current, and the dynamics of stator i sd currents and stator reactive power Q s depend on rotor ir d current. The RSC control dynamics u rq and u r d are presented in Fig. 5.50a, b. It is possible to see that the RSC control signals are bounded and non-saturated. In addition, the RSC controllers have a good performance even in presence of reference variations. • Maximum Power extraction The objective of this case is to test the capability of the proposed controller to track the maximum power reference obtained from the MPPT algorithm to ensure a maximum power generation. So, the stator active power reference is calculated using the MPPT algorithm as in (5.1). The tracking response for the rotor d − q currents are presented in Fig. 5.51, the rotor −q− current reference is defined from the active power reference generated by MPPT and the rotor −d− current reference is defined in

Stator active power [W]

138

5 Experimental Results

(a) 200

0 Ps

−200 0

Psref

10

20

30

40

50

60

70

80

90

100

Stator reactive power [Var]

t [Sec]

(b) Qs

25

Qsref

20 15 0

10

20

30

50

40

60

70

80

90

100

t [Sec]

stator −q− current [A]

Fig. 5.48 Stator active and reactive powers tracking with N-SML

(a)

4

isq

2 0 −2 0

10

20

30

40

50

60

70

80

90

100

Stator −d− current [A]

t [Sec]

(b)

0.2

isd

0 −0.2 −0.4 0

10

20

30

40

50

60

70

80

90

100

t [Sec]

Fig. 5.49 Stator currents dynamics with N-SML

order to obtain a constant stator reactive power. Figures 5.52 and 5.53 show the stator active Ps and reactive Q s powers and the stator i sd , i sq currents respectively. We can observe that the proposed control scheme has a good tracking of the maximum power reference extracted from the wind. In addition, the stator reactive power is maintained constant ensuring a good decoupling between the powers axes. Figure 5.54 shows the stator electric power factor and the electromagnetic torque. It is possible to see that

[V]

5.4 Real-Time N-SML Control

139

(a)

100

Control Signal u

rq,k

urq,k

50 0 −50 0

10

20

30

40

50

60

70

80

100

90

t [Sec]

Control Signal u

rd,k

[V]

(b) urd,k

40 20 0 −20 0

10

20

30

40

50

60

70

80

90

100

t [Sec]

Rotor −q− current [A]

Fig. 5.50 RSC control signals with N-SML, case 1

(a)

1

irq irqref

0 −1 −2 0

10

20

30

40

50

60

70

80

100

90

Rotor −d− current [A]

t [Sec]

(b) ird

1

irdref

0.5 0 0

10

20

30

40

50

60

70

80

90

100

t [Sec]

Fig. 5.51 Rotor currents tracking with N-SML

the variation in the rotor speed has no significant effects on the stator electric power factor. Figure 5.55 illustrates the dynamics of rotor voltages u rq and u r d , which are applied to the DFIG rotor in order to control stator active and reactive power in this case.

Stator active power [W]

140

5 Experimental Results

(a)

200

100 Ps

0 0

Psref

10

20

30

50

40

60

70

80

100

90

Stator reactive power [Var]

t [Sec]

(b)

26

Q

s

Q

sref

24 22 20 0

10

20

30

40

50

60

70

80

90

100

t [Sec]

Stator q−current [A]

Fig. 5.52 Stator active and reactive powers tracking with N-SML

(a)

2

isq

1 0 −1 0

10

20

30

40

50

60

70

80

90

100

Stator d−current [A]

t [Sec]

(b)

0

isd

−0.2 −0.4 0

10

20

30

50

40

t [Sec]

Fig. 5.53 Stator currents dynamics with N-SML

60

70

80

90

100

Stator power factor

5.4 Real-Time N-SML Control

141

(a)

1.5

stator power factor

1 0.5 0 −0.5 0

10

20

30

50

40

60

70

80

100

90

Electomagnetic torque [N−m]

t [Sec]

(b)

3

T

em

2 1 0 0

10

20

30

50

40

60

70

80

100

90

t [Sec]

Control Signal u

rq,k

[V]

Fig. 5.54 Stator power factor and electromagnetic torque with N-SML

(a)

60

u

rq,k

40 20 0 −20 0

10

20

30

50

40

60

70

80

100

90

Control Signal urd,k [V]

t [Sec]

(b)

50

u

rd,k

0 −50 0

10

20

30

40

50 t [Sec]

Fig. 5.55 RSC control signals for N-SML, case 2

60

70

80

90

100

Stator active power [W]

142

5 Experimental Results

(a)

200

100 Ps P

0 0

sref

10

20

30

40

50

60

70

80

90

100

Stator reactive power [Var]

t [Sec]

(b) Qs

40

Q

sref

20 0 0

10

20

30

40

50

60

70

80

90

100

t [Sec]

Fig. 5.56 Stator powers with N-SM-L under grid disturbances

5.4.4 Robustness to Grid Disturbances The objective of this experiment is to examine the capabilities of the proposed control scheme for ensuring the wind turbine system stability when the transmission lines reactance changes suddenly. The transient stability question is: will the controller force the system back into a steady-state condition after the transient period [67]? To reach this objective, the same electric scheme is used as in Fig. 5.18 and the same scenarios are applied as in Sect. 5.3.4. Figure 5.56 illustrates the results of stator active and reactive powers controlled by N-SM-L controller when we apply this fault (parameter variations) on the transmission lines. The same test is applied for SM-FOC controller and the real-time results are presented in Fig. 5.57. Figure 5.58 displays the stator currents for both controller SM-FOC (a) and N-SM-L (b). From these results, it is possible to observe that the proposed control algorithm stabilizes the disturbed system and it has good robustness in presence of transmission lines impedance variations. Whereas, The SM-FOC which is implemented on the same prototype shows that the transmission lines impedance variations produce a high pick stator currents which destabilize the system and turn the prototype off, (Fig. 5.58a).

Stator active power [W]

5.4 Real-Time N-SML Control

143

(a)

400

Ps P

sref

200 0 −200 0

10

20

30

40

50

60

70

80

100

90

Stator reactive power [Var]

t [Sec]

(b)

100

Qs Qsref

50 0 −50 0

10

20

30

40

50

60

70

80

90

t [Sec]

Stator current [A]

Stator current [A]

Fig. 5.57 Stator powers with SM-FOC under grid disturbances

(a)

6

High stator current pick

4

The DFIG prototype is disconnected from the Grid

2 0 −2

(b)

3 2 0 −2 −3 Nominal values of the grid impedances

Sudden variation in the grid impedances at t=45s

Fig. 5.58 DFIG stator currents with SM-FOC under grid disturbances

Nominal values of the grid impedances

100

144

5 Experimental Results

5.5 Real-Time N-IOC Scheme The purpose of this section is to validate experimentally the proposed N-IOC for reference tracking of the DC voltage and the grid power factor, through the control of the GSC, and the stator active and reactive power, via the control of the RSC, considering ideal and faulted grid conditions.

5.5.1 Normal Grid Conditions

Grid Currents [A]

As a first experiment, normal grid conditions are considered. The objective is to test the proposed control scheme effectiveness to track the desired trajectories of the DC voltage and the grid power factor controlled by the GSC and the stator powers controlled by the RSC. Figure 5.59 illustrates three-phase currents (a) and voltages (b) as measured at the transmission line module. The phase-to-phase current and voltage amplitude are 1.5 A and 220 V respectively. As it can be observed from the current and voltage dynamics, there are no faults applied to the DFIG prototype and voltages and currents are balanced. Figure 5.60 presents the dynamics of the DC voltage (a) and the grid power factor (b) controlled by the GSC. The grid active and reactive power flow between the grid and the DFIG rotor are presented in Fig. 5.61. From the obtained results, it is very clear to observe that the proposed GSC controller successfully reaches the control objectives in which the DC voltage is maintained constant at 120 V and the grid power factor is kept at the unit value independently. In order to examine the proposed control scheme capabilities to operate WT system in presence of wind speed variations, a time-varying mechanical speed is

2

(a)

0

-2

t [s]

Grid Voltages [V]

(b) 200 0 -200 t [s]

Fig. 5.59 Grid currents and voltages: normal grid conditions

DC Voltage [V]

5.5 Real-Time N-IOC Scheme

145

(a)

125 120

Vdc

115 110 0

Vdcref

20

40

60 t [s]

80

100

120

Power Factor

(b) 1 fg

0.98 0

fgref

20

40

60 t [s]

80

100

120

(a) 100

Reactive Power [Var]

Active Power [W]

Fig. 5.60 DC voltage and grid power factor: normal grid conditions

50 0 0

Pg

20

40

60 t [s]

80

100

120

(b)

10

Qg

0 -10 0

20

40

60 t [s]

80

100

120

Fig. 5.61 Grid powers: normal grid conditions

applied to the DFIG rotor, which is obtained from the DC motor. It’s value is changed randomly from sub-synchronous speed to super-synchronous one (185 → 200 rad/s) as presented in Fig. 5.62. To achieve this objective, two cases are considered for the RSC controller: (a) Firstly, we force the controlled dynamics to track time-varying reference for the stator active power and to keep at a constant value the stator reactive power. (b) Secondly, the stator active power desired trajectory is defined from the MPPT algorithm in order to ensure that WT system is optimally operated and also the stator reactive power is maintained at constant value. Figure 5.63 illustrates the obtained results for the DFIG rotor −q− current (a) and the DFIG rotor −d− current (b) dynamics controlled by the RSC proposed controller. As a first case, the

146

5 Experimental Results 200

Rotor Shaft Speed [rad/s]

190 180 170 160 150

ωr

140 0

ωrref

20

40

60 t [s]

80

100

120

Fig. 5.62 DFIG rotor shaft speed

(a)

irq [A]

0 -0.5

irq irq ref

-1 0

20

40

60 t [s]

80

100

120

(b)

1 ird [A]

i rd i rdref

0.5 0

20

40

60 t [s]

80

100

120

Fig. 5.63 DFIG rotor currents: normal grid conditions

rotor −q− current is forced to track a time varying reference defined form the stator active power required trajectory using (4.37), its amplitude changes −1 → −0.2 A. The rotor −d− current is fixed at a constant value (0.6) A calculated from the reactive power reference using (4.38). Figure 5.64 displays the stator active and reactive power trajectory tracking responses. From these results, it is possible to see that the proposed controller successfully forces the controlled dynamics to reach their desired trajectories with perfect decoupling on the control axes. In addition, speed changing has no effects on the generated stator active and reactive power. As a second case, the rotor −q− current (a) is calculated from the MPPT stator active power (5.1) while, The rotor −d− current (b) is maintained at a constant

Active Power [W]

5.5 Real-Time N-IOC Scheme

147

(a)

150

Ps

100

Psref

50 0 0

20

40

60

80

100

120

Reactive Power [Var]

t [s]

(b) 20 0

Qs

-20

Q

0

20

40

60

80

sref

100

120

t [s]

Fig. 5.64 DFIG stator powers: normal grid conditions

value (0.6) A. Figure 5.65 presents the DFIG rotor −q− current (a) and the DFIG rotor −d− current (b) dynamics. Figure 5.66 displays the tracking response for the stator active (a) and reactive (b) power, respectively, generated from the DFIG. From the obtained results, it is very clear to observe that the proposed GSC controller successfully reaches the control objectives in which the DC voltage is maintained constant at 120 V and the grid power factor is kept at the unit value independently. In addition, the obtained results for the RSC illustrate that the proposed controller

(a) irq [A]

-0.5 irq

-1 0

irqref

20

40

60 t [s]

80

100

120

(b)

1

ird [A]

i

rd

irdref

0.5 0

20

40

60 t [s]

Fig. 5.65 DFIG rotor currents: MPPT, normal grid conditions

80

100

120

5 Experimental Results

(a) Ps Psref

100

Reactive Power [Var]

Active Power [W]

148

50 0

20

40

60 t [s]

80

100

120

(b) 20 0

Qs

-20 0

Qsref

20

40

60 t [s]

80

100

120

Fig. 5.66 DFIG stator powers: MPPT, normal grid conditions

ensures adequate tracking performances for the stator active reference (for both cases) and keeps at a constant value the stator reactive power even in presence of time-varying wind speed.

5.5.2 Abnormal Grid Conditions The objective of this experiments is to validate the LVRT capacity of the DFIG controlled by the proposed control scheme. To do so, three fault grid types are considered: single-phase-to-ground, two-phase-to-ground, and three-phase-to-ground. Note that each type of grid disturbances is applied between 40 to 60 s, and for the remaining time normal grid conditions are considered. Single-Phase-to-Ground Fault Figure 5.67 presents the three-phase currents (a) and voltages (b) when a singlephase-to-ground fault is applied; a voltage dip of 55% from nominal value of 220 V appears on the faulted phase. The DC-link voltage and the grid power factor are presented in Fig. 5.68. Figure 5.69 displays the DFIG stator active and reactive. It is possible to see that the proposed controller for the GSC and the RSC tracks correctly the desired dynamics of the DC voltage, the grid power factor, and the stator active and reactive powers even in presence of single-phase-to-ground fault. Two-Phase-to-Ground Fault Figure 5.70 illustrates the three phase currents (a) and voltages (b) when two-phaseto-ground fault is incepted. Two phases voltage dip of 35% is created causing increas-

Grid Currents [A]

5.5 Real-Time N-IOC Scheme

149

(a)

2

0

-2

t =0 s to t=60 s

t =40 s to t=60 s

t =60 s to t=120 s

Grid Voltages [V]

(b) 200 0 -200 Balanced grid conditions

Phase-to-ground fault

Balanced grid conditions

DC Voltage [V]

Fig. 5.67 Grid currents and voltages: single-phase-to-ground fault

(a)

125 120

Vdc

115 110 0

Vdcref

20

40

60 t [s]

80

100

120

Power Factor

(b) 1 fg

0.98 0

fgref

20

40

60 t [s]

80

100

120

Fig. 5.68 DC voltage and grid power factor: single-to-ground fault

ing on the amplitude of the other two-phases. The DC-link voltage and the grid power factor are presented in Fig. 5.71a, b respectively. Figure 5.72 illustrates trajectory tracking results for the stator active and reactive power. From these results, it is clear that the proposed controller presents adequate performances for tracking the desired trajectories of the DC voltage, the grid power factor, and the generated stator powers even in presence of two-phase-to ground fault. However, a small tracking error is observed during this fault. However, the proposed controller illustrates its capability to remove this error after the fault vanishing and to ensure stability.

5 Experimental Results

(a) Ps Psref

100 50 0

Reactive Power [Var]

Active Power [W]

150

20

40

60 t [s]

80

100

120

(b) 20 0

Q

-20

Q

0

20

40

60 t [s]

80

100

s sref

120

Grid Currents [A]

Fig. 5.69 DFIG stator powers: MPPT, single-phase-to-ground fault

(a)

2

0

-2

t =0 s to t=60 s

t =40 s to t=60 s

t =60 s to t=120 s

Grid Voltages [V]

(b) 200 0 -200 Balanced grid conditions

Phase-to-ground fault

Balanced grid conditions

Fig. 5.70 Grid currents and voltages: two-phase-to-ground fault

Three-Phase-to-Ground Fault Figure 5.73 displays the dynamics of the three-phase currents (a) and voltages (b) when three-phase-to-ground fault grid is created. Symmetric voltage dip of 45% from nominal value of 220 V is inserted. The DC voltage, the grid power factor, and the stator active and reactive power behaviors are presented in Figs. 5.74 and 5.75 respectively. From the obtained results, the proposed controller for the GSC and the RSC successfully tracks the controlled dynamics desired trajectories. However, a slight tracking error is observed in the DC voltage dynamics at the same instant of the fault grid apparition at 40 s as illustrated in Fig. 5.74a. This error is removed

5.5 Real-Time N-IOC Scheme

151

(a)

DC Voltage [V]

130 120 110 100 0

20

40

60 t [s]

80

100

120

Power Factor

(b) 1 fg

0.98 0

fgref

20

40

60 t [s]

80

100

120

Reactive Power [Var]

Active Power [W]

Fig. 5.71 DC voltage and grid power factor: two-phase-to-ground fault

(a) Ps Psref

100 50 0

20

40

60 t [s]

80

100

120

(b) 20 0

Qs Qsref

-20 0

20

40

60 t [s]

80

100

120

Fig. 5.72 DFIG stator powers: MPPT, two-phase-to-ground fault

when the voltage system gets back to its nominal conditions. In addition, small amplitude ripples are observed in the stator active and reactive power at the same instant (t = 40) s. The proposed RSC controller rejects successfully those ripples at the next instants during the fault. From the experiment results, we conclude that the proposed controller achieves the control objectives and ensures stability of the controlled system under normal and

5 Experimental Results

Grid Currents [A]

152

(a)

2

0

-2

t =0 s to t=60 s

t =40 s to t=60 s

t =60 s to t=120 s

Grid Voltages [V]

(b) 200 0 -200 Balanced grid conditions

Three-phase to ground fault Balanced grid conditions

DC Voltage [V]

Fig. 5.73 Grid currents and voltages: three-phase to ground fault

(a)

125 120 115 110 0

Vdc Vdcref

20

40

60 t [s]

80

100

120

Power Factor

(b) 1 fg

0.98 0

fgref

20

40

60 t [s]

80

100

120

Fig. 5.74 DC voltage and grid power factor: three-phase to ground fault

abnormal grid conditions. In addition, the LVRT capacity of the DFIG is enhanced. Moreover, the proposed controller can reject grid disturbances and ensures permanent connection of the WT system to the power system, which fulfills the modern grid codes and ameliorates quantity of the generated power.

Reactive Power [Var]

Active Power [W]

5.6 Conclusions

153

(a) Ps

Ripples

Psref

100 50 0

20

40

60 t [s]

80

100

120

(b) 20 0

0

Qs

Ripples

-20 20

Qsref

40

60 t [s]

80

100

120

Fig. 5.75 DFIG stator powers: MPPT, three-phase to ground fault

5.6 Conclusions In this chapter, real-time experiments for SM-FOC, N-SMFOC and N-SML control schemes are implemented for a DFIG based Wind Turbine. The proposed controllers are used to control the DC voltage at the output of the DC-link, to maintain the electric power factor at nominal values, using the GSC; and to force the stator active and reactive powers generated from the DFIG to track desired values through rotor currents control using the RSC. For identification, an RHONN identifier trained on-line with an EKF based algorithm is used for both the DC-link and the DFIG. Real-time identification results show a good approximation for the real state vector. Base on such identification, the proposed controllers are designed, which help to reject the unknown dynamic and grid disturbances. The proposed control schemes are tested in real-time for tracking time-varying power references and for extracting the maximum power from the wind under ideal and no-ideal grid conditions. Real-time results illustrate that the proposed controllers based on neural identification assure better performances for both GSC and RSC even in presence of references variations, and wind turbine speed changes. In addition, it illustrates an adequate performance under fault grid conditions. Moreover, decoupling, stability, and convergence are achieved. However, the conventional controllers (SM-FOC) presents an acceptable performances under ideal grid conditions, but it cannot ensure stability in presence of grid disturbances. Finally, the proposed neural controllers present a simple control algorithm, which can be used in industrial wind turbine applications. Moreover, it can operate WTs to extract the maximum power from the wind under different fault scenarios, which ensure a continuous energy production and improve quality and quantity of power generation.

Chapter 6

Microgrid Control

The microgrid system presents an efficient solution for renewable energy resource exploration. To increase the penetration of the Distributed Energy Resource (DER) into the distribution system, renewable energy sources (RES) local controllers should provide Low Voltage Ride-Through (LVRT) capability to sustain power systems operations and ensure their stability in presence of grid disturbances. In this paper, Neural Sliding Mode Linearization (NSML) controller is proposed to control the generated active and reactive power of each DER. The developed controller is based on RHONN, online trained with an EKF algorithm. Based on neural identification an adequate model of the microgrid generation units is obtained in presence of grid disturbances, which helps the proposed controller to reject perturbations, to ensure stability, and to operate the RES under different grid scenarios. The proposed microgrid is composed of Wind Power System (WPS), Solar Power System (SPS), Battery Bank (BB) and programmable Load Demand (LD). In addition, it is coupled to IEEE 9-bus system. The whole system is real-time simulated using Opal-RT (OP5600).

6.1 Opal-RT Lab RT-LAB is OPAL-RT’s real-time simulation software combining performance and enhanced user experience. Fully integrated with MATLAB/Simulink, RT-LAB offers the most complex model-based design for interaction with real-world environments. It provides the flexibility and scalability to achieve the most complex real-time simulation applications in the automotive, aerospace, power electronics, and power systems industries. It supplies three types of functionalities which are: Real-time simulation, Hardware-In-Loop (HIL), and Rapid Control Prototyping (RCP), for different © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 E. N. Sánchez and L. Djilali, Neural Control of Renewable Electrical Power Systems, Studies in Systems, Decision and Control 278, https://doi.org/10.1007/978-3-030-47443-0_6

155

156

6 Microgrid Control

Applications Rapide Control Prototyping

Opal-RT Lab

Hardwardr-in-Loop Real-time Simulation

Fig. 6.1 Opal RT-Lab

applications, Fig. 6.1. The Opal-RT (OP5600) Simulator1 is used to simulate the proposed microgrid and the IEEE 9-bus system. The OP5600 has two primary sections: an upper section containing analog and digital I/O signal modules, and a bottom section containing the multi-core processor computer and FPGA capable of running the entire OPAL-RT suite of real-time simulation software. The OP5600 can be configured with up to 32 Intel Xeon E5 processing cores, and comes with a custom-designed Linux operating system, providing the best realtime performance on the market. The OP5600 also provides the option of userprogrammable I/O management, handled by a fast Xilinx Spartan-3 FPGA.

6.2 Considered Microgrid The selected microgrid under study is presented in Fig. 6.2. It is composed of: (1) Wind turbine is based on the DFIG where its rotor is linked to the grid through backto-back converter, while its stator is connected directly to the grid. (2) Solar panel is linked to the grid through two converters; the first one is a DC/DC boost converter which helps to maintain constant the DC voltage at the output of the DC link and a DC/AC converters which is used to control the active and reactive power injected 1 Opal-RT

(OP5600) Simulator ©Opal-RT Technologies, Inc.

6.2 Considered Microgrid

Battery

157

Buck-Boost Converter

25kV-575V

Wind Turbine

Back-to-back Converter

=

= = =

575V-25KV

=

Inverter 575V-25kV

= =

=

To IEEE 9-Bus

Inverter Boost Converter Solar Panels

PCC

AC Load

Fig. 6.2 Microgrid electric structure

into the grid. (3) Battery Bank is also coupled to the grid via a DC/DC buck-boost converter and DC/AC converter. The mode boost ensures the transfer of the current to the batteries bank in order to charge it, and the mode buck grantees the discharge of the batteries bank by transferring the current to the electric grid. (4) An AC electric programmable LD. In addition, This microgrid is linked to an IEEE 9-bus system [68, 69], which is consist of nine buses with four electric generators type synchronous, three AC electric load, transmission lines, and electric transformers. The structure of the IEEE 9-bus system is presented in Fig. 6.3.

Bus 7 230kV

Bus 2 A 20kV

Bus 8 230kV

Line 7-8

Line 8-9

Bus 3 20kV

Bus 9 230kV

C M2 900 MVA A

20kV-230kV

Line 5-7

100MW 35MVR

B C M4 900 MVA

A

B

B Line 6-9 AC Load

Bus 5 230kV

20kV-230kV

Bus 6 230kV

230kV-25kV

125MW Line 50MVR 4-5 AC Load Bus 4 230kV

Line 4-6

90MW 30MVR AC Load

From Microgrid B

C

A

230kV-20kV

M3 900 MVA

Fig. 6.3 IEEE 9-bus electric structure

C M3 900 MVA

230kV-20kV

Bus 1 20kV

158

6 Microgrid Control

6.3 Microgrid Modeling 6.3.1 Wind Power System The wind generation system in this study is based on the DFIG. As known, the wind system is composed of mechanical and electrical parts. The mechanical part is the aero-generator and it is used to extract the power form wind, and the electrical one, which is composed of DC link and DFIG is used to converter the generated mechanical power to the electrical energy. The electric configuration for WPS based DFIG is illustrated in Fig. 6.4. The WPS mathematical model is presented in Chap. 3.

6.3.2 Solar Power System The solar panel system is consist of photo-voltaic arrays based on solar cell, DC/DC boost converter, and DC/AC converter. Figure 6.5 presents the electric its configuration under study Photo-Voltaic Array Model The Photo-voltaic arrays is based on the solar cell, which is a p − n semiconductor junction, the following equation presents the V-I characteristic of a solar array [70]

Mechanical Power Conversion Aero-generator

Gearbox

Electrical Power Conversion Electrical Generator Power Converte

Transformer

Grid

DFIG Power Conversion Fig. 6.4 Wind power system configuration

Solar Power System

Boost Converter

Fig. 6.5 Solar power system configuration

Ppv , Qpv

PCC

575 V 60 Hz

=

= =

P dcpv

AC Converter

Grid

6.3 Microgrid Modeling

159

    q(V + Rs I ) V + Rs I −1 − I = Isc − I0 ex p nkTk Rsh

(6.1)

where V and I present the output voltage (V ) and current (A) of the PV array, respectively, Rs and Rsh are the series and shunt resistance () of the cell, q is the electric charge, Isc is the light-generated current, I0 is the reverse saturation current, n is a dimensionless factor, k is the Boltzman constant, Tk is the temperature in (0 K ). DC-DC Boost Converter Model The DC-DC boost converter has the characteristic that the output voltage is greater than the input voltage [71]. The discrete-time model of the boost converter using Euler discretization method is given as [71] x3,k+1 = f 3 (x3,k , k) + B3 u 3 (x3,k , k) y3,k = h 3 (x3,k )

(6.2) (6.3)

T  where x3,k = Udcpv,k I pv,k are the boost converter state vector, u 3 (x1,k , k) =  T u pvbo,k is the boost converter input, and y3,k = h 31 (x3,k ) h 32 (x3,k ) =  T  Vdcpv,k ; I pv,k is the variable to be controlled, and f 3 (x3,k , k) = f 31 (x3,k ) T f 32 (x3,k ) such as ts ts Udcpv,k − I pv ) rlpv c pv c pv 1 1 + ts ( U pv,k − Udcpv ) l pv l pv

f 31 (x3,k ) = Udcpv,k − f 32 (x3,k ) = I pv,k



and B3 =

ts I c pv pv

0

0 ts U l pv dcpv



with l pv and c pv are the inductance (H ) and the capacitor (F) of the DC/DC boost converter, rlpv is the resistance load (). where Udcpv,k is the boost output voltage (V ), I pv is the boost output current (A), U pv,k is the output solar panel array voltage (V ), and ts is the sample time. DC/AC Solar Inverter Model The Three-phase solar power inverters are usually used to convert the DC power generated from the solar panel system to the AC power and also to control the injected active and reactive power to the grid by controlling the current flows between the grid and the inverter. The discrete-time DC/AC converter in d − q reference frame is given by the following equations

160

6 Microgrid Control

x4,k+1 = f 4 (x4,k , k) + B4 u 4 (x4,k , k)

(6.4)

y4,k = h 4 (x4,k )

(6.5)

T T   where x4,k = i d,k i q,k are the inverter bus currents, u 4 (x4,k , k) = u cd,k u cq,k  T  T is the inverter input, and y4,k = h 41 (x4,k ) h 42 (x4,k ) = i d,k i q,k is the variable  T to be controlled, and f 4 (x4,k , k) = f 41 (x4,k ) f 42 (x4,k ) such as r 1 f 41 (x4,k ) = i d,k + ts (− Id,k + ωi q,k + u d,k ) l L r 1 f 42 (x4,k ) = i q,k + ts (−ωi d,k − i q,k + u q,k ) l l and B4 =

 ts l

0

0



ts l

where w is the synchronous frequency (rad/sec), i d,k and i q,k are the grid d − q currents, u cd,k , u cq,k , u d,k and u q,k , are d − q voltages at the converter and at the grid respectively, r is the grid line resistances (), l is the grid line inductance of (H ), ts is the sample time.

6.3.3 Batteries Bank Storage System Modeling The Battery bank system is a very important component in microgrid operation. It can decrease the operational cost of the system and improve the efficiency, performance and reliability of the system by matching the power demand with the generation sources [34]. The selected configuration of the battery Bank system is composed of a battery Bank, DC/DC buck-boost converter, and DC/AC converter, Fig. 6.6.

Solar Power System

Buck-Boost Converter

Fig. 6.6 Storage system configuration

Pbtt , Qbtt

PCC

575 V 60 Hz

=

= =

P dcbtt

AC Converter

Grid

6.3 Microgrid Modeling

161

DC-DC Buck-Boost Converter Model The DC-DC converter for the battery bank includes boost converter, which is used under charge conditions and buck converter is used under discharge condition. The switching between both converters is performed by the Insulated-Gate Bipolar Transistor (IGBT) [72]. To obtain the mathematical model of the buck converter (discharging mode), we consider that the I G BTboost equal to zero and we apply the commutation law to I G BTbuck . Using the Euler discretization method, the DC-DC buck converter model is obtained as follows x5,k+1 = f 5 (x5,k , k) + B5 u 5 (x5,k , k) y5,k = h 5 (x5,k )

(6.6) (6.7)

T  where x5,k = Udcbtt,k Ibtt,k are the buck converter vector state, u 5 (x5,k , k) =  T  T [0 u bttbu ]T is the input vector, and y5,k = h 51 (x5,k ) h 52 (x5,k ) = Udcbtt,k Ibtt,k  T is the variable to be controlled, and f 5 (x5,k , k) = f 51 (x5,k ) f 52 (x5,k ) such as ts ts Udcbtt,k + Ibtt rlbtt cbtt cbtt 1 + ts Ubtt,k lbtt

f 51 (x5,k ) = Udcbtt,k − f 52 (x5,k ) = Ibtt,k and



0 0 B5 = ts Udcbtt 0 − lbtt



with lbtt and cbtt are the inductance (H ) and the capacitor (F) of the DC/DC buckboost converter, rlbtt is the resistance load (). where Udcbtt,k is the buck-boost output voltage (V ), Ibtt is the buck-boost output current (A), Ubtt,k is the battery bank voltage (V ), and ts is the sample time. To analyze the charge condition, the mathematical model of the boost converter can be obtained by considering that the I G BTbuck equal to zero and applying the commutation law to I G BTboost . Using the Euler discretization method, the DC-DC boost converter is obtained as follows x6,k+1 = f 6 (x6,k , k) + B6 u 6 (x6,k , k) y6,k = h 6 (x6,k )

(6.8) (6.9)

 T where x6,k = Udcbtt,k Ibtt,k are the boost converter vector state, u 6 (x6,k , k) = T  T  u bttbo is the input vector, and y6,k = h 61 (x6,k ) h 62 (x6,k ) = Udcbtt,k Ibtt,k is  T the variable to be controlled, and f 6 (x6,k , k) = f 61 (x6,k ) f 62 (x6,k ) such as

162

6 Microgrid Control

ts ts Udcbtt,k − Ibtt rlbtt cbtt cbtt 1 ts + ts Ubtt,k + Udcbtt lbtt lbtt

f 61 (x6,k ) = Udcbtt,k − f 52 (x5,k ) = Ibtt,k and B6 =

 −ts

I cbtt btt 0

0 ts U lbtt dcbtt



DC-AC Battery Inverter Model The Three-phase storage system inverter is usually installed to control the injected active and reactive power to the grid by controlling the current flows between the grid and the inverter. The discrete-time model of the battery inverter is considered as the same as the solar inverter in (6.4) and (6.5).

6.4 Microgrid Neural Controllers The RHONN identifier is proposed for microgrid subsystems to improve robustness of the proposed control algorithm in presence of parameter variations and grid disturbances; it is assumed that all the state variables are measured.

6.4.1 Wind Power System Controller The proposed RHONN identifiers and controllers for the DC-link and the DFIG are presented in Sect. 4.3. Remark The desired values of the DFIG rotor currents, in this chapter, are obtained by using the following expressions instead of (4.37)and (4.38) ir dr e f,k = kdq eq,k + kiq

ts eq,k z−1

(6.10)

with eq,k = Q sr e f,k − Q s,k the error of the desired and real dynamics of the stator reactive power, kdq and kiq are the proportional and integral controller gains respectively. ts e p,k (6.11) irqr e f,k = kdp e p,k + ki p z−1 with e p,k = Psr e f,k − Ps,k the error of the desired and real trajectory of the stator active power.

6.4 Microgrid Neural Controllers

163

6.4.2 Solar Power System Controller Boost Converter The objective is to force the DC voltage at the output of the boost converter to keep its desired constant value. In order to do so, the control scheme is composed of two controllers in cascade. The first one is used to generate the reference of the boost converter output current and the second one is utilized to track the obtained current reference. The proposed RHONN identifier for this converter is selected as follows χ3,k+1 = fˆ3 (x3,k ) + Bˆ 3 u 3 (x3,k ) yˆ3,k = = hˆ 3 (x3,k )

(6.12) (6.13)

T  where x3,k = Udcpv,k I pv,k is the solar power boost converter state vector, χ3,k = T

Uˆ dcpv,k Iˆpv,k is the vector estimated by the neural identifier, u 3 = u pvbo,k is the T

T

control vector, yˆ3 = hˆ 31 (x3,k ) hˆ 32 (x3,k ) = Uˆ dcpv,k Iˆpv,k is the output vector

T to be controlled and fˆ3 (x3,k ) = fˆ31 (x3,k ) fˆ32 (x3,k ) is as fˆ31 (x3,k ) = w11,k S(Udcpv,k ) + w12,k S(I pv,k ) + w13,k S(Udcpv,k )S(I pv,k ) + 3,1 Iˆpv,k fˆ32 (x3,k ) = w21,k S(I pv,k ) + w22,k S(Udcpv,k ) + w23,k S(U pv,k ) + w24,k S(I pv,k )S(Udcpv,k ) with Udcpv,k the boost output voltage (V ), I pv the boost output current (A), U pv,k the output solar panel array voltage (V ). Applying Theorem 4.2, the neural linearization control of the DC voltage at the boost converter output is calculated as follows I pvr e f,k =

−1 ˆ f 31 (x3,k ) + v3,1 3,1

(6.14)

The neural linearization control law for the boost converter output current is obtained as −1 ˆ (6.15) u pvbo = f 32 (x3,k ) + v3,2 3,2 The decoupled control part for the DC voltage v3,1 and current v3,2 boost converter are define as in (4.9), where the sliding surface of the boost converter DC voltage is defined as s3n,1,k = Udcpvr e f,k − Uˆ dcpv,k , and the sliding surface for the boost converter current is selected as s3n,2,k = I pvr e f,k − Iˆpv,k .

164

6 Microgrid Control

DC/AC Inverter The objective of this controller is to regulate the injected active and reactive power into the microgrid via control of the currents flow between the grid and the inverter. The proposed RHONN identifier for the DC/AC converter is selected as follows χ4,k+1 = fˆ4 (x4,k ) + Bˆ 4 u 4 (x4,k ) yˆ4,k = = hˆ 4 (x4,k )

(6.16) (6.17)

T

T  where x4,k = i d,k i q,k is the solar panel inverter state vector, χ4 = iˆd,k iˆq,k is T  the vector estimated by the neural identifier, u 4,k = u d,k u q,k is the control vecT

T

tor, yˆ4 = hˆ 41 (x4,k ) hˆ 42 (x4,k ) = iˆd,k iˆq,k is the output vector to be controlled,

T   Bˆ 4 = diag 4,1 , 4,2 , and fˆ4 (x4,k ) = fˆ41 (x4,k ) fˆ42 (x4,k ) is as fˆ41 (x4,k ) = w11,k S(i d,k ) + w12,k S(i q,k ) + w13,k S(i d,k )S(i d,k ) fˆ42 (x4,k ) = w21,k S(i q,k ) + w22,k S(i d,k ) + w23,k S(i q,k )S(i d,k ) with i d,k and i q,k are the grid d − q currents, u d,k and u q,k are the converter side d − q voltages. Applying Theorem 4.2, the neural linearization control law of the grid currents is obtained as 

ud uq



    ˆ v4,1 −1 − f 41 (x 4,k ) ˆ = B4 + v4,2 − fˆ42 (x4,k )

(6.18)

with v4,1 , v4,2 the grid currents decoupled controllers. The sliding surface for the grid d − q currents s4n,1,k and s4n,2,k are defined as s4n,1,k = i dr e f,k − iˆd,k ands4n,2,k = i qr e f,k − iˆq,k , respectively. The desired values of the grid currents is obtained by using a discrete-time PI controller as in (6.10), (6.11).

6.4.3 BB Neural Controller DC-DC Buck-Boost The control objective is to allow for charging and discharging operation modes (Buck or boost mode). For the currents flow direction between the converter and the battery, the positive sign means that the BB is discharging, while the negative sign means that it is charging. Taking into account the adaptive nature of RHONN identifier, and the similitude between the buck and boost converter models, a single RHONN identifier is proposed for both cases as

6.4 Microgrid Neural Controllers

165

χ5,k+1 = fˆ5 (x5,k ) + Bˆ 5 u 5 (x5,k , k) yˆ5,k = = hˆ 5 (x5,k )

(6.19) (6.20)

T  where x5,k = Udcbtt,k Ibtt,k is the buck-boost power converter state vector, χ5,k =

T Uˆ dcbtt,k Iˆbtt,k is the vector estimated by the neural identifier, u 5 = u c,k is the input   signal, yˆ5 = hˆ 52 (x5,k ) = Iˆbtt,k is the output to be controlled, Bˆ 5 = diag 0, 5,2 ,

T and fˆ5 (x5,k ) = fˆ51 (x5,k ) fˆ52 (x5,k ) is as fˆ51 (x5,k ) = w11,k S(Udcbtt,k ) + w12,k S(Ibtt,k ) + w13,k S(Udcbtt,k )S(Ibtt,k ) + 5,1 Ibtt,k ˆ f 52 (x5,k ) = w21,k S(Ibtt,k ) + w22,k S(Udcbtt,k ) + w23,k S(Ibtt,k )S(Udcbtt,k ) with Udcbtt,k the buck-boost output voltage (V ), Ibtt the buck-boost output current (A), Ubtt,k the battery bank voltage (V ). The BB current reference is calculated using the following relation Pbttr e f,k (6.21) Ibttr e f,k = Udcbtt,k where Pbttr e f is the BB power reference. The neural linearization control part of the current flow through the buck-boost converter inductor is obtained as follows u c,k =

1 ˆ − f 52 (x5,k ) + v5 5,2

(6.22)

with 5,2 = 0 is a fixed control weight and v5 is the buck-boost converter output current decoupled control, its sliding surface is defined as s5n,k = Ibttr e f,k − Iˆbtt,k . DC-AC Inverter The discrete-time RHONN identifier of the storage system inverter is considered as the same as the solar panel [(6.16) and (6.17)]. Considering charge and discharge operating modes, the control objective of the DC/AC converter is divided as follows: (a) for charging operation mode, the inverter controller objective is to maintain constant the DC-bus voltage, so the controller is the same as the one applied to the GSC. (b) for discharging operation mode, the purpose is to control the active and reactive power injected into the grid, so the same controller as the one used to the solar panel inverter.

166

6 Microgrid Control

6.5 Real-Time Simulation Results The proposed local controller, the AC microgrid, and the selected IEEE 9-bus network are simulated using SimPower System toolbox of Matlab. In addition, real-time simulations using an Opal-RT (O P 5600) Simulator2 have been done for the whole system. To determine the size of the WPS, the SPS, the BB and the power converters, different factors are considered as addressed in [73, 74]. Considering that this present paper is focused on resilience enhancing of grid-connected microgrid, only the variability of the renewable energy resources (wind speed and solar irradiation for the considered site), and the load demand are considered. The expected maximum power generated by the microgrid is 2.86%(10.5 MW) of the system total power. The maximum generated power by the WPS and SPS is 85% (9 MW) and 15% (1.5 MW) of the microgrid power respectively. The BB is used to maximize the usage of the renewable energy and to minimize the power injection from the power system by storing energy whenever the supply from the WPS and SPS exceeds the load demand. The maximum power that can be injected to the microgird is 500 KW. In addition, a resistive load with value of 10 MW is installed in order to maintain a balanced power flow. Moreover, the microgrid design allows to supply the power system by generated power from the DERs. To evaluate the capabilities of the proposed DER local controller in presence of grid disturbances, three different fault grid conditions: one-phase-to-ground, twophase-to-ground, and three-phase-to-ground are applied at two different locations. The first location is selected between bus 6 and bus 9 (near to bus 6, Fig. 6.7), whereas the second one is placed between the Point of Common Coupling (PCC) bus and the microgrid. Table 6.1 presents the parameters of the microgrid components (WPS, SPS, and the BB system). The electric power system under study is simulated under ideal grid conditions as a first experiment in order to test the performance of the proposed microgrid local controllers to track their set-points. In the second experiment, three different fault grid conditions: Phase-to-ground, two-phase-to-ground, and three-phase-to-ground are applied in two different locations in the electrical network. The first location is selected between bus six and bus nine (near to bus six), while the second one is chosen between the Point of Common Coupling (PCC) bus and microgrid. Table 6.1 presents the parameters of the microgrid components (wind power system, solar panel system, and the battery bank system). Table 6.2 illustrates the parameters of power generations in each bus including the microgrid. For more details about the IEEE 9-bus parameters see [68, 69, 75].

2 Opal-RT

(OP5600) Simulator ©Opal-RT Technologies, Inc.

6.5 Real-Time Simulation Results

167

Matlab/simulink (Opal-RT preccesor)

Matlab/simulink (computer preccesor)

AC microgrid Connected to IEEE-9-bus

Visualization scopes V, I, P, Q, and others

Three-phase fault block

Three-phase Fault selector

Local Controller

Controller parameters References system

Master subsystem

Console subsytem

Fig. 6.7 The master and console subsystem in Opal-RT Table 6.1 The parameters of the Microgrid subsystems Meanings Units UGS Transformer 1 Transformer 2 Transmission line Wind system (Pn ), (Vn ), (Fn ) DFIG stator (Rs ), (L s ), (L m ) DFIG rotor (Rr ), (L r ) Back-to-Back (Pn ), (Vn ), (Fn ) DC-link (Rg ), (L g ), (C) Solar system (Pn ), (Vn ), (Fn ) PV (irr, V dc, Pdc) Solar inverter (R), (L) Solar boost (L pv ), (C pv ) Inverter (P, V, F) Boost (P, V, I) Storage system (Pn ), (Vn ), (Fn ) DC/DC Buck-boost (L btt ), (Cbtt ) Buck boost (Pn ),(Vn ),(In )

120e3V , 60 Hz 120e3/25e3V R = 0.08/30, L = 0.08 pu 25e3/575V R = 0.025/30, L = 0.025 pu 30 km 9 MW, 575 V, 60 Hz 0.012, 0.0137H ,0.013H 0.021, 0.0136H 1.5 MW, 750V AC, 60 Hz 0.3666, 0.0031H , 0.0022F 1.5 MW, 575 V, 60 Hz 103 W/m2 , 481 V, 235 kW 0.3666,0.76e − 3H 0.76e − 3H , 470e − 6F 500 KW, 750V AC, 60 Hz 300K W p , 103 V DC, 600 A 500 kW, 575 V, 60 Hz 0.76e − 3H ,470e − 6F 600K W p , 103 V DC, 1145A

168

6 Microgrid Control

Table 6.2 The IEEE-9-bus power generation Bus P (MW) Q (Mvar) 1 2 3 4 5 6 7 8 9

72.19 163 85 0 0 0 47 0 10

26.8 6.69 −10.78 0 0 0 0 0 0

Load (MW)

Load (Mvar)

0 0 0 0 125 90 0 100 0

0 0 0 0 50 30 0 35 0

6.5.1 Normal Grid Conditions The objective of this experiment is to test the proposed local controller performances for each DER under ideal grid conditions. These microgrid operation conditions are considered: • The generated active from each DER is forced to track a time-varying trajectory in order to test the proposed control scheme in presence of reference changes. • In real applications, these reference changes cause that WPS and SPS power generation fluctuation due to fast wind speed and cloud transients. • The Maximum Power Point Tracking (MPPT) is not considered for any the WPS or the SPS. • The generated reactive power from each DER is kept at zero in order to ensure power factor equals to 1. • The load demand is fixed at 10 MW during the whole simulation lapse. • From t = 0 s to t = 3 s and t = 6 s to t = 10 s, the generated power from the microgrid satisfies the load demand and the exceeds power is injected in the power system. • From t = 3 s to t = 6 s, the generated power from the microgrid is not enough to satisfy the load demand and the power is injected from the power system. Figure 6.8 illustrates the three-phase voltages (a) and the three-phase currents (b) as measured at PCC bus where the microgrid is connected to the IEEE 9-bus system. The voltages and currents amplitude at PCC are given in per unit (pu) where the nominal real power value which is expected to generate by the microgrid is 10 MW under nominal voltage of 575 V. As can be observed from the voltages and currents dynamics are balanced. The control tracking results of each DER system which are composed the microgrid are presented in the following figures. Figure 6.9 displays the DC voltage (a) generated at the output of the DC link controlled by the GSC and the stator active and reactive powers (b) and (c) obtained from the DFIG stator, which are controlled by the RSC. The DC voltage is forced to track a constant value of 1.5 kV,

6.5 Real-Time Simulation Results

169

PCC Voltages (pu)

(a) 1 0 −1 5.1

5.08

5.06

5.04

5.02

5

5.12

5.14

5.16

5.18

5.2

5.12

5.14

5.16

5.18

5.2

t [Sec] PCC Currents (pu)

(b) 2

0

−2

5

5.02

5.04

5.06

5.08

5.1

t [Sec]

(a) 2000 1500 1000

Udc Udcref

0

Q

DFIG

(Var)

PDFIG(W)

DC link Voltage (V)

Fig. 6.8 Three-phase voltages and current: Ideal grid conditions

1 10

2

2

3

5

6

7

8

9

6

0.003 s 5.998

0

1

2

3

6

4

Ps

6.002 6.004

5

Psref

6

7

8

9

Qs

0 -1

10

(c)

106

1

10

(b)

1 0

4

Q

0

1

2

3

4

5

6

7

8

9

sref

10

t [Sec]

Fig. 6.9 WPS Controlled dynamics under ideal grid conditions

while the stator active power is forced to track a time-varying reference and the stator reactive power is fixed at zero in order to obtain an unit stator power factor. Note that the results in Fig. 6.9 consider the control of one wind turbine, whereas 10 wind turbines are installed with total generated power of 9 MW; all of them present the same behavior and are controlled by means of the same control scheme presented in Sect. 6.4.1. Figure 6.10 displays the tracking response of the DFIG d − q rotor currents. The rotor currents are obtained from the PI controller (outer loop), which

170

6 Microgrid Control

Rotor−q− Current (A)

(a) 0

i

rq

−1000

irqref

−2000 −3000 −4000 −5000

0

1

2

3

4

5

6

7

8

10

9

Rotor −d− Current (A)

t [Sec] (b) 1000 0

i

rd

i −1000

rdref

0

1

2

3

4

5 t (Sec]

6

7

8

9

10

Fig. 6.10 DFIG d − q rotor currents under ideal grid conditions

is used to control the stator active and reactive power, (see Sect. 6.4.1). The rotor −q− current tracks a time-varying reference obtained from the outer control loop of stator active power (a) and the rotor −d− current is forced to be at a constant value, which is defined from the reactive power outer control loop. The NSML controller is used in the inner control loop to force the rotor currents to track their desired references. In order to highlight the benefits of the proposed scheme for DERs dynamics control, a comparison with the PI controller, which is the most used one, is done. As an example, it is used to control the WPS active and reactive power. Figure 6.11 presents the obtained WPS dynamics when controlled by the PI scheme. Figure 6.12 illustrates the single-side amplitude analyses and the Total Harmonic Distortion (THD) of the voltages and currents, respectively, as measured at the PCC for the NSML (Fig. 6.12a T H D = 0.8362%, Fig. 6.12b T H D = 3.1880%) and for the PI (Fig. 6.12c, T H D = 2.8111% Fig. 6.12d T H D = 5.6481%) controllers. It is clear to see that both controllers achieve good WPS dynamics trajectories tracking with notable improvement in THD voltages and currents values at PCC when the proposed scheme is utilized. In addition, the response time of the proposed controller is 0.003 s (Fig. 6.9b), which ensures the WPS trajectories tracking even in presence of power generation fluctuations caused by fast wind changes (≥0.003 s). Figure 6.13 portrays the DC voltage (a) at the output of the solar panel boost converter and the solar panel inverter active and reactive power, (b) and (c) respectively. The NSML controller is used to force the output of the DC-DC boost converter to track a constant value of 1 kV, whereas the active power at the solar panel inverter is forced to track a time-varying reference, where its amplitude can vary from 125 to 250 kW. The solar panel inverter reactive power is maintained constant in order to improve the

171

(a)

2000 1500 1000

QDFIG(Var)

PDFIG (W)

DC Voltage(V)

6.5 Real-Time Simulation Results

0

1 10

2

2

3

4

5

6

7

8

9

10

6

7

8

9

10

6

7

8

9

10

(b)

6

0 -2

0

1

2

3

4

(c)

106

1

5

0 -1

0

1

2

3

4

5

t [Sec] Fig. 6.11 WPS controlled dynamics: PI, normal grid conditions

quality of the generated power and to keep the inverter power factor at the unit value. The nominal value of the generated power from the solar panels is 1.5 MW. The tracking response of the solar panel inverter d − q currents are presented in Fig. 6.14 where the output of the PI controller, which is installed to control the solar panel inverter active and reactive powers and is used to generate the inverter currents desired dynamics, (see Sect. 6.4.2). The solar panel inverter −q− current is forced to track a time-varying trajectory (a), while the solar panel inverter −d− current tracks a constant value. The proposed controller (NSML) is used to track the desired solar panel inverter d − q currents references. From the obtained results, it is possible to conclude that the proposed controller for SPS ensures good trajectory tracking even in presence of reference changes. In addition, the response time of the proposed controller is 0.0025 s (Fig. 6.13b), which ensures SPS stability even in presence of power generation fluctuations caused by fast cloud transients (≥0.0025 s). Figure 6.15 illustrates the direct power (a) and the direct current (b) flow between the BB and the buck-boost converter for charging and discharging modes. The BB state of charge, which is obtained from the Matlab/simulink BB block, is presented in Fig. 6.15c. In charging operating mode, t = 0 to t = 5 s, the boost converter mode is selected. For this mode, a negative direct power desired values is applied and the Eq. (6.21) is used to calculate the desired trajectory of the direct current. In addition, the DC voltage at the output of the BB inverter is controlled by means of the same controller as applied to the GSC. In discharging operating mode, t = 5 s to t = 10 s, the buck converter mode is chosen by switching the IGBTs to obtain a positive direct power (charging mode), and the same Eq. (6.21) is used to determine the reference of the direct current. As can be seen, the proposed controller ensures charging and discharging operating modes of the BB. In order to examine the battery bank inverter

172

6 Microgrid Control Single-Sided Amplitude Spectrum (NSML) (b)

(a)

PCC Currents

PCC voltages

1 THD=0.8362 %

1 THD=3.1880%

0.5

0.5 0 0

100

200

300

400

500

0 0

100

PCC Currents

PCC voltages

Single-Sided Amplitude Spectrum (PI)

(c)

1

THD=2.8111 %

0.5 0 0

100

200

200

300

400

500

f (Hz)

f (Hz)

300

400

1

THD=5.6481 %

0.5

500

0 0

(d)

100

f (Hz)

200

300

400

500

f (Hz)

(a) 1500

U

dcpv

1000 500

Q

SPS

(Var)

PSPS (W)

DC Voltage (V)

Fig. 6.12 THD for the NSML and the PI controllers

Udcpvref

0

1

2

3

4

6

7

8

9

0.0025s 5.998

0

0

1 10

5

2

3

6

4

Ppv

6.002 6.004

5

Ppvref

6

7

8

9

10

(c)

4

Q

0 -5

10

(b)

105

5

5

Q

0

1

2

3

4

5

6

7

8

9

pv pvref

10

t [Sec] Fig. 6.13 SPS controlled dynamics under ideal grid conditions

active and reactive power control that are injected to the grid, the discharging mode is selected. Figure 6.16 gives the DC voltage at the output of the DC/DC buck-boost converter in discharging operating mode (a) which is forced by the buck converter controller to track a desired value equal to 1000 V. The battery bank inverter active and reactive power which are injected to the grid are presented in Fig. 6.16b and c, respectively such as the same control strategy as applied for the solar panel inverter

6.5 Real-Time Simulation Results

173

Solar Inverter I (A)

(a) 0

i

qref

−500

−1000

0

1

2

3

4

5 t [Sec]

6

7

8

9

10

(b) 200

i

d

d

Solar Inverter I (A)

q

iq

i

100

dref

0 −100

1

0

2

3

4

6

5 t [Sec]

7

8

10

9

Fig. 6.14 Solar inverter d − q currents: Ideal grid conditions (a)

6

x 10

Charging Operation Mode

0

Discharging Operation Mode Pbtt

P

btt

(W)

1

−1

P

bttref

0

1

2

3

4

5

6

7

8

9

10

1000 I

btt

0

Ibttref

I

btt

(A)

(b)

Charge State

−1000

0

1

2

3

4

5

6

7

8

9

10

(c) 100

State

50 0

0

1

2

3

4

5 t [Sec]

6

7

8

9

10

Fig. 6.15 BB charging and discharging operating modes

is used to control the battery bank inverter. The maximum power that can be injected to the grid form the battery bank is 500 kW. From these results, it is very clear to conclude that the proposed local controller for each microgrid subsystem allows to obtain tracking of the desired trajectories for the active and reactive powers, which are generated by the WPS, SPS, and BB respectively. In addition, the proposed controller assures charging and discharging

174

6 Microgrid Control (a) Vbtt

1000

Vbttref

V

btt

(V)

1500

Q

btt

Inverter (Var)

Pbtt Inverter (W)

500

0

1

2

3

4

6

7

8

x 10

btt inv

P

btt inv ref

1

0

2

3

4

5

6

7

8

x 10

10

Q

btt inv

0 −1

9

(c)

5

1

10

P

5 0

9

(b)

5

10

5

Q

btt inv ref

0

1

2

3

4

5

6

7

8

9

10

t [Sec]

Fig. 6.16 BB controlled dynamics: Ideal grid conditions

operation modes of the BB. Moreover, it presents an adequate performances to track time-varying active and reactive power.

6.5.2 Abnormal Grid Conditions, First Location In this section, two-phase-to-ground grid fault type is applied in bus 6 (Fig. 6.3) in order to examine the performances of the proposed controllers. For the case of onephase-to-ground and three-phase-to-ground, the proposed NSML controller ensures the LVRT capacity for the DERs; the corresponding results are not included in order to fulfill the paper length requirements. Each type of the grid disturbance is applied for 5 s, which allows enough time to examine the proposed controller LVRT capacity according to I E E E1159 − 1995. One-Phase-to-ground Fault Case Figure 6.17 presents the three-phase voltages (a) and the three-phase currents (b) as obtained at the PCC, where a voltage dip of 50% appears at the faulted phase and the other phases are kept at nominal value 575 V. Figure 6.19 displays the DC voltage (a) at the output of the DC-link, DFIG stator active power (b), and DFIG stator reactive power (c) as generated by the WPS. The stator active power reference is selected to be constant at 1.5 MW and the stator reactive power set-point is zero. Figure 6.18 presents the WPS dynamics when the PI is used. It is clear that single-phase-toground fault causes high amplitude ripples on the WPS dynamics when controlled by the PI scheme, and may lead to instability of isolated wind system and perhaps

PCC Voltages (pu)

6.5 Real-Time Simulation Results

175 (a)

1 0 −1 From 3 s to 8 s

Remaining Time PCC Currents (pu)

(b) 2

0

−2

Phase−to−ground fault

Normal Grid Condition

(a) 4000 2000 0

Udc Udcref

0

Q

DFIG

(Var)

P

DFIG

(W)

DC Volatge (V)

Fig. 6.17 Three-phase voltages and current: Phase-to-ground fault

1

2

3

4

1

5

6

7

8

9

P

0 -1

P 0

1

10

1

2

3

4

5

6

7

8

s sref

9

10

(c)

6

Q

0 -1

10

(b)

107

s

Qsref 0

1

2

3

4

5

6

7

8

9

10

t [Sec]

Fig. 6.18 WPS Controlled dynamics: PI, single-phase-to-ground fault

the microgrid. While, this fault has no significant impact on WPS dynamics when controlled by the proposed scheme. The solar panel DC voltage boost converter and the solar panel inverter active and reactive power responses are presented in Fig. 6.20a, b and c respectively. The injected active power generated by the solar panel is maintained at 25 kW, whereas the inverter reactive power is kept constant at zero. These results illustrate that the proposed controller achieves adequately the control objectives. Figure 6.21 displays the DC voltage at the output of the BB buckboost converter at the discharging operating mode (a) and the BB inverter active and

DC link Voltage (V)

176

6 Microgrid Control (a) 2000

U

dc

1500 1000

Udcref 0

1

2

3

4

x 10

DFIG

(W) P

6

7

8

9

10

(b)

6

2

P

1

P

s sref

1

0

QDFIG (Var)

5

2

3

4

6

7

8

x 10

Qs

0 −5

10

9

(c)

5

5

5

Q

sref

1

0

2

3

4

5

6

7

8

10

9

t [Sec]

DC link Voltage (V)

Fig. 6.19 WPS Controlled dynamics: Phase-to-ground fault (a) 2000 1500

U

1000

U

dc dcref

1

0

2

3

4

PDFIG (W)

6

7

8

10

9

(b)

6

QDFIG (Var)

5

x 10 2

Ps

1

P

sref

1

0

2

3

4

6

7

8

x 10

Qs

0 −5

10

9

(c)

5

5

5

Qsref 0

1

2

3

4

5 t [Sec]

6

7

8

9

10

Fig. 6.20 SPS Controlled dynamics: Phase-to-ground fault

reactive power. The same results are obtained as the ones for the solar panel inverter, where the active power is operated to track a constant reference equals to 500 kW, while the reactive power is fixed at zero. Two-Phase-to-ground Fault Figure 6.22 shows the three-phase voltages (a) and the three-phase currents (b) as plotted at the PCC when a two-phase-to-ground fault is inserted. A voltage dip

6.5 Real-Time Simulation Results

177 (a) U

dcbtt

1000

U

U

btt

(V)

1500

Qbtt Inverter (Var)

dcbttref

1

0

2

3

4

5

6

7

8

10

9

(b)

5

10

x 10

5

P

0

P

P

btt

Inverter (W)

500

btt inv btt invref

0

1

2

3

4

6

7

8

x 10

10

Q

btt inv

0 −5

9

(c)

5

5

5

Qbtt invref 0

1

2

3

4

5 t [Sec]

6

7

8

9

10

PCC Voltages (pu)

Fig. 6.21 BB Controlled dynamics: Phase-to-ground fault (a) 1 0 −1

From 3 s to 8 s

Remaining Time PCC Currents (pu)

(b) 2

0

−2

Normal Grid Condition

2−Phase−to−ground fault

Fig. 6.22 Three-phase voltages and current: Two-phase-to-ground fault

of 45% from nominal value of 575 V appears at the two faulted phases. The DC link voltage, which is controlled by the GSC and the DFIG stator active and reactive power controlled by the RSC are presented in Fig. 6.23. The DC voltage is maintained constant at its desired value of 1500 V, the stator active and reactive power are kept at their selected values. Figure 6.24 illustrates the WPS dynamics when the PI controller is used. From these results, it is clear that the proposed control scheme ensures

DC link Voltage (V)

178

6 Microgrid Control (a) 2000 U

dc

1500 1000

Udcref 1

0

2

3

4

(W) DFIG

P

x 10

QDFIG (Var)

7

8

s

P

sref

1

2

3

4

5

6

7

8

9

10

(c)

5

x 10

Qs

0 −5

10

9

P 0

5

6

(b)

6

2.5 2 1.5 1

5

Qsref 0

1

2

3

4

5 t [Sec]

6

7

8

9

10

(a)

3000

Udc

2000

Udcref

1000

QDFIG (Var)

PDFIG (W)

DC Voltage (V)

Fig. 6.23 WPS Controlled dynamics: Two-phase-to-ground fault

0

1

2

3

4

6

7

8

9

P

s

Psref

0

1 10

2

2

3

4

5

6

7

8

9

10

(c)

6

Q

0 -2

10

(b)

106

3 2 1 0 -1

5

s

Qsref

0

1

2

3

4

5

6

t [Sec] Fig. 6.24 WPS Controlled dynamics: PI, two-phase-to-ground fault

7

8

9

10

DC voltage (V)

6.5 Real-Time Simulation Results

179 (a)

1200 U

dcpv

1000 800

U

dcpvref

0

1

2

3

4

PSP (W)

x 10

(Var) SP

7

8

9

10

Ppv P

pvref

0

1

2

3

4

5

5

6

7

8

9

x 10

Qpv

0 −5

10

(c)

4

Q

6

(b)

5

4 3 2 1

5

Qpvref 0

1

2

3

4

5 t [Sec]

6

7

8

9

10

Fig. 6.25 SPS Controlled dynamics: Two-phase-to-ground fault

stability of the WPS dynamics even in presence of two-phase-to-ground fault, while considerable ripples are presented when the PI controller is utilized. Figure 6.25 displays the obtained result of the solar panel dynamics such that, the DC voltage at the output of the DC/DC boost converter is presented in Fig. 6.25a, the injected active and reactive power at the output of the solar panel inverter are given in Fig. 6.25b and c respectively. From these results, it very clearly to see that the solar panel proposed local controller maintains constant the DC voltage at its desired value with a small error during the fault. For the solar panel injected active and reactive power also a small oscillation appears. The DC voltage at the output of the BB buck-boost converter and the active and reactive power calculated at the output of the BB inverter are presented in Fig. 6.26a, b and c respectively. The same results are obtained as explained for the solar panel inverter. Three-Phase-to-ground Fault Figure 6.27 displays the dynamics of the three-phase voltage (a) and the three-phase current (b) at the PCC. A symmetric voltage dip of 50% from nominal value of 575 V appears. Figure 6.28a presents the DC voltage at the output of the DC link controlled by the GSC. It is clear to see that the proposed controller for the GSC keeps constant the DC voltage at its desired value. The stator active and reactive power, which are controlled by the RSC also track their references with small tracking errors as illustrate in Fig. 6.28b, c. The proposed controller reject successfully these errors at the next instants during the fault grid and ensures the transition stability.

180

6 Microgrid Control (a) Udcbtt

1000

U

U

btt

(V)

1200 dcbttref

0

1

2

3

4

5

6

7

8

x 10

btt inv

P

btt invref

0

1

2

3

4

5

6

7

8

x 10

10

Qbtt inv

0 −5

9

(c)

5

5

10

P

5 0

9

(b)

5

10

Qbtt invref 0

1

2

3

Q

btt

Inverter (Var)

Pbtt Inverter (W)

800

4

5 t [Sec]

6

7

8

9

10

PCC Voltages (pu)

Fig. 6.26 BB Controlled dynamics: Two-phase-to-ground fault (a) 1 0 −1 From 3 s to 8 s

Remaining Time PCC Currents (pu)

(b) 2

0

−2

Normal Grid Condition

3−Phase−to−ground fault

Fig. 6.27 Three-phase voltages and current: corresponding to Three-phase-to-ground fault

Figure 6.29 illustrates the WPS dynamics when the PI controller is utilized. It can be seen that three-phase-to-ground fault causes considerable amplitude ripples on the WPS when controlled by the PI scheme, which might lead to instability of the microgird. Figures 6.30 and 6.31 display the solar panel inverter and BB inverter controlled dynamics.

DC link Voltage (V)

6.5 Real-Time Simulation Results

181 (a)

2000 Udc

1500 1000

U

dcref

0

1

2

3

4

(W) DFIG

P

1

2

3

4

9

5

6

7

8

9

x 10

10

Q

s

0 −5

10

(c)

5

(Var)

8

Psref

5

DFIG

7

Ps 0

Q

6

(b)

6

x 10

2.5 2 1.5 1

5

Qsref 0

1

2

3

4

5

6

7

8

9

10

t [Sec]

(a)

8000 6000 4000 2000 0

U

DFIG

P (Var) DFIG

2

3

4

5

6

7

8

9

Ps P

0

1

2

3

4

5

6

7

8

sref

9

10

(c)

106

Qs

0 -1

10

(b)

0

1

Q

1 106

5 -5

dc

Udcref

0

(W)

DC Volatge (V)

Fig. 6.28 WPS Controlled dynamics: Three-phase-to-ground fault

Q

0

1

2

3

4

5

6

t [Sec] Fig. 6.29 WPS Controlled dynamics: PI, three-phase-to-ground fault

7

8

9

sref

10

182

6 Microgrid Control

DC voltage (V)

(a) 1500 U

dcpv

1000

U

dcpvref

1

0

2

3

4

PSP (W)

x 10

7

8

10

9

P

pv

Ppvref 0

1

2

3

4

5

6

7

8

9

10

(c)

4

x 10

5

QSP (Var)

6

(b)

5

4 3 2 1

5

Qpv

0 −5

Qpvref 0

1

2

3

4

5 t [Sec]

6

7

8

9

10

Fig. 6.30 SPS Controlled dynamics: Three-phase-to-ground fault (a) Udcbtt

1000

U

U

btt

(V)

1500

Qbtt Inverter (Var)

Pbtt Inverter (W)

500

dcbttref

1

0

2

3

4

6

7

8

x 10

P

btt inv

5 0

P

btt invref

1

0

2

3

4

5

6

7

8

x 10

Qbtt inv

0 −1

10

9

(c)

6

1

10

9

(b)

5

10

5

Q

btt invref

0

1

2

3

4

5 t [Sec]

6

Fig. 6.31 BB Controlled dynamics: Three-phase-to-ground fault

7

8

9

10

6.5 Real-Time Simulation Results

183

6.5.3 Non-ideal Grid Conditions, Second Location For this second location, single-phase-to-ground and two-phase-to-ground faults are incepted between the PCC and the microgrid. The same results as for the first location case are obtained. The respective results are not included in order to fulfill the paper length requirements. From the results, we conclude that the proposed controller for each microgrid generation unit achieves the control objectives and ensures stability of the controlled system under normal and abnormal grid conditions. In addition, the LVRT capacity of the DERs is improved by using the proposed scheme as compared with the PI controller, which is unable to ensure stability in presence of abnormal grid conditions. Moreover, the proposed controller can reject grid disturbances, which appear at different grid locations.

6.6 Conclusions This chapter presents real-time simulation of a NSML controller for a grid-connected microgrid. The proposed local controller for each subsystem is used to control the active and reactive power, which is injected to the grid. Each proposed local controller is based on an RHONN identifier. The neural identifiers approximate the respective nonlinear dynamics and allow the controller to reject disturbances caused by parameter variation and/or abnormal grid conditions. Real-time simulation results illustrate the effectiveness of the proposed scheme to achieve trajectory tracking of the DER power references even in presence of grid disturbances. In addition, due to the fact that the identifier and controller are separately designed for each component, new elements can be easily integrated for a large power system. Moreover, the proposed control strategy presents an excellent solution for grid-connected microgrid local controller LVRT, which satisfies the modern grid conditions by ensuring uninterrupted power generation, achieving more flexibility in power sharing, and improving quantity of generated power. All these results allow to establish that the proposed control schemes improve substantially the respective microgrid resilience. As a future work, the proposed control scheme can be extended to regulate the generated power of other distributed resources. In addition, it is advisable to include tests for more complex power system with different grid fault scenarios.

Chapter 7

Conclusions and Future Work

In this book, the control of grid-connected DFIG based Wind Turbine and gridconnected microgrid under normal and abnormal grid conditions is investigated. In order to reach the main purposes, three control schemes are developed for the Doubly Fed Induction Generator Rotor Side Converter and Grid Side Converter which are: Neural Sliding Mode Field Oriented Control, Neural Sliding Mode Linearization Control, and Neural Inverse Optimal Control. A neural identifiers system is used to approximate the controlled dynamics and based on such identifiers, the controllers are synthesized. Firstly, the proposed controllers are used to control the DC voltage at the output of the DC-link, to maintain the electric power factor at nominal values, using the Grid Side Converter; and to force the stator active and reactive powers generated from the Doubly Fed Induction Generator to track desired values through the rotor currents control using the Grid Side Converter. The performances of the proposed controllers are examined using SimPower ToolBox of Matlab regarding the tracking of time-varying desired trajectories; the DC-link and the Doubly Fed Induction Generator parameters variations, and wind speed changing influences, considering ideal grid conditions. In addition, the proposed control schemes are evaluated via simulation under three type of grid disturbances: single-phase-to-ground, two-phaseto-ground, and three-phase-to-ground. Moreover, a comparison with the decoupled Proportional-Integral controller for the rotor side converter is done. From the obtained simulation results, we can conclude that the proposed Sliding Mode Field Oriented Control scheme shows adequate performances in presence of time-varying reference, parameters variations, and speed changing comparing with the standard ProportionalIntegral and the Sliding Mode Linearization controllers. In addition, the Sliding Mode Field Oriented Control scheme ensures an adequate decoupling between the control axes, however, a sluggish response time still appears in the stator active and reactive powers in presence of parameter variations for all controllers. Moreover, all those © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 E. N. Sánchez and L. Djilali, Neural Control of Renewable Electrical Power Systems, Studies in Systems, Decision and Control 278, https://doi.org/10.1007/978-3-030-47443-0_7

185

186

7 Conclusions and Future Work

conventional controllers cannot support the grid disturbances where a high amplitude ripples are presented in the stator active and reactive power. Whereas, the proposed neural controllers assure excellent performances in presence of all above experiments in comparing with the other conventional controllers. In addition, decoupling between control axes is ensured and response time is improved. Moreover, the neural controllers improves the LVRT capacity of the DFIG and ensures transit stability under different fault grid scenarios, which accomplish with modern grid requirements. Those high performances of the proposed controllers are achieved thanks to the neural identifier, which permits to obtain adequate models in presence parameter variation and/or grid disturbances, which helps the controllers to reject their effects. Taking into account the above discussion, real-time validation for the Sliding Mode Field Oriented Control, Neural Sliding Mode Field Oriented Control, Neural Sliding Mode Linearization Control, and nNeural inverse Optimal Control are implemented using a Simulink/MatLab and a DS1104 data acquisition board. Real-time identification results show a good approximation for real state vector. Base on such identification, the proposed controllers are implemented, which help to reject the unknown dynamic and grid disturbances. They are tested for the tracking of timevarying power references and for extracting the maximum power from the wind under ideal and no-ideal grid conditions. Real-time results illustrate that the proposed controllers based on neural identification ensure better performances for both Rotor Side Converter and Grid Side Converter, even in presence of references variations, and wind turbine speed changes. In addition, they has adequate performances under fault grid conditions. Moreover, decoupling, stability, and convergence are achieved. However, the conventional controller Sliding Mode Field Oriented Control presents an acceptable performances under ideal grid conditions, but it cannot ensure stability of the controlled dynamics in presence of grid disturbances. Secondly, a selected scheme, which is the Neural Sliding Mode Linearization control, is extended to regulate the generated power from each DER installed into grid-connected microgrid. The proposed microgrid is composed of a wind power system, a solar power system, a battery bank, and a programmable load demand. In addition, the microgrid under study is interconnected to an IEEE 9-bus system to evaluate its connection performance and response under grid disturbances, which are: single-phase-to ground, two-phase-to-ground and three-phase-to-ground faults. The whole system is real-time simulated using an Opal-RT (OP5600) simulator. Real-time simulation results illustrate effectiveness of the proposed control scheme to achieve trajectory tracking for DER active and reactive powers. In addition, the LVRT capability of the proposed control strategy is verified in presence of grid disturbances. All these results allow to establish that the proposed control schemes improve substantially the respective microgrid resilience. Finally, the proposed neural controllers present a simple control algorithm, which can operate WTs to extract the maximum power from the wind under different fault scenarios, and to ensure transient stability of distributed energy resources installed into grid-connected microgrid, and enhance their LVRT capacities. As future work, the following topics will be treated

7 Conclusions and Future Work

187

• To extended the proposed control schemes for regulating the generated power of other distributed resources. • To link the designed microgrid to more complex power systems and study their behaviors with different grid fault scenarios. • To study the impact and utilization of the renewable energy sources grid integration • To modify the proposed control schemes considering the microgrid island operation mode. • To investigate secondary control layer based neural networks for single and multiple microgrids.

Appendix A

Neural Controller Stability Analysis

Now, the stability analysis is required for the closed loop system in (2.5) controlled by the discrete-time N-SM or N-SML controllers. In order to analyze the proposed control algorithms stability, the separation principle in discrete-time is used [54], which divides the analysis in two parts [36]. 1. Minimization of the identification error. It can be achieved by on-line identification using EKF based algorithm as proved in Theorem 4.1 in [36]. As results of this theorem, the proposed RHONN model in (2.31) used to identify the class of system under study given in (2.5) ensures that the identification errors are semi globally uniformly ultimately bounded; moreover the RHONN weights remained bounded [36]. 2. The tracking error convergence to a small bound region. It is achieved by the proposed neural controllers as will be demonstrated in the following analysis.

A.1 N-SM Controller Due to identification error boundedness, there exists a bounded vector valued function i,k , which is smaller; then [76] xk = χk + i,k

i = 1, . . . , r

(A.1)

  with i,k  ≤ γi , γi > 0 and r the number of the state. The sliding surface at k + 1 is defined using xk as follows sk+1 = xk+1 − xr e f,k+1

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 E. N. Sánchez and L. Djilali, Neural Control of Renewable Electrical Power Systems, Studies in Systems, Decision and Control 278, https://doi.org/10.1007/978-3-030-47443-0

(A.2)

189

190

Appendix A: Neural Controller Stability Analysis

Using (4.1), the sliding surface in (A.2) is calculated as s(xk , k + 1) = fˆ(xk , k) + B(xk )u(xk , k) + (xk , k) − xr e f,k+1

(A.3)

and the equivalent control is determined as   u eq (xk , k) = −B(xk )−1 fˆxk , k + (xk , k) − xr e f,k+1

(A.4)

The discrete-time sliding mode controller proposed in (2.14) includes two cases:   • When u eqn (xk , k) ≤ u 0 The applied control law in this case is u n (xk , k) = u eqn (x, k, k) + u sn (x, k, k) then, by substituting u n (x, k, k) in (A.3), the sliding surface is rewritten as s(xk , k + 1) = kc s(xk , k) + (xk , k)

(A.5)

Let define the Lyapunov function candidate V (xk , k) = s(xk , k)T s(xk , k), then V (xk , k) = s(xk , k + 1)T s(xk , k + 1) − s(xk , k)T s(xk , k) = (kc s(xk , k) + (xk , k))T (kc s(xk , k) + (xk , k)) − s(xk , k)T s(xk , k) ≤ (kc  s(xk , k) + (xk , k))2 − s(xk , k)2 ≤ (kc  s(xk , k) + γ )2 − s(xk , k)2 ≤ kc 2 s(xk , k)2 + 2 kc  s(xk , k) γ + γ 2 − s(xk , k)2 ≤ −(1 − kc 2 ) s(xk , k)2 + 2 kc  s(xk , k) γ + γ 2 ≤ −(1 − θ1 )η1 s(xk , k)2 − θ1 η1 s(xk , k)2 + 2 kc  s(xk , k) γ + γ 2 ≤ −(1 − θ1 )η1 s(xk , k)2 + γ 2 + (−θ1 η1 s(xk , k) + 2 kc  γ ) s(xk , k)

with θ1 < 1, η1 = (1 − kc 2 ), kc  < 1, and in the region s(xk , k) ≥ obtain

2γ kc  , θ1 η1

we

V (xk , k) ≤ −(1 − θ1 )η1 s(xk , k)2 + γ 2 V (xk , k) ≤ −(1 − θ2 )η2 s(xk , k)2 − θ2 η2 s(xk , k)2 + γ 2 V (xk , k) ≤ −(1 − θ2 )η2 s(xk , k)2 with θ2 < 1, η2 = (1 − θ1 )η1 . So, for the region s(xk , k) ≥ √θγ β , yielding V (xk , k) ≤ 0 and the solution 2 1 of system (A.5) is ultimately bounded.   • When u eqn (xk , k) > u 0 u

(x ,k)

The following control law is applied u n (xk , k) = u 0 ||u eqn (xk ,k)|| . By substituting this eqn k control law in (A.3), the sliding surface is rewritten as

Appendix A: Neural Controller Stability Analysis

  u eq (xk , k) = B(xk )−1 s(xk , k) − xr e f,k + xk + fˆ(xk , k)   = B(xk )−1 s(xk , k) + fˆs (xk , k)

191

(A.6)

with fˆs (xk , k) = −s(xk , k) + fˆ(xk , k), and the sliding surface is calculated as   s(xk , k + 1) = s(xk , k) − xr e f,k + xk + fˆ(xk , k) + B(xk )u c (xk , k) + (xk , k)   = s(xk , k) + fˆs (xk , k)B(xk )u c (xk , k) + (xk , k) (A.7) Let define the Lyapunov function candidate V (xk , k) = s(xk , k)T s(xk , k), then V (xk , k) = s(xk , k + 1)T s(xk , k + 1) − s(xk , k)T s(xk , k)   = s(xk , k) + fˆs (xk , k)B(xk )u c (xk , k) + (xk , k) T  s(xk , k) + fˆs (xk , k)B(xk )u c (xk , k) + (xk , k) − s(xk , k)T s(xk , k)

    u eq (xk , k) ˆ  + (xk , k) = s(xk , k) + f s (xk , k) 1 − u 0  u eq (xk , k)

T    u eq (xk , k)  + (xk , k) s(xk , k) + fˆs (xk , k) 1 − u 0  u eq (xk , k) − s(xk , k)T s(xk , k)

 2   u 0    ˆ  + (xk , k) − s(xk , k)2 ≤ s(xk , k) + f s (xk , k) 1 −  u eq (xk , k)  2   u 0     + (xk , k) − s(xk , k)2 ≤ s(xk , k) + fˆs (xk , k) −   B(xk )−1 

Suppose that the control within the domain [77],  law u n (xk ) ≤ u 0 may vary     ˆ   f s (xk , k) + (xk , k) ≤ u 0 , and σ < u 0 with σ = f n,k + (xk , k), then 2 u 0   − s(xk , k)2 V (xk , k) ≤ s(xk , k) + σ  −   B(xk )−1 



2 u 0  u 0   − σ+  V (xk , k) ≤ 2 s(xk , k) σ +   B(xk )−1   B(xk )−1  

192

Appendix A: Neural Controller Stability Analysis

  σ ≤ u if  B(xk )−1    0  ≤ 2 s(xk , k) + σ holds then V (xk , k) ≤ 0. Hence U eq , both decreases monotonically. Therefore there will be a time s(xk , k) and r,k   k1 such that u eqn (xk , k) ≤ u 0  for k ≥ k1 . At that time, the control law u n (xk , k) is applied, yielding that the solution of the system (A.5) is ultimately bounded. 

A.2 N-SML Controller The DTSMC vn (xk ) is selected as (4.39). The convergence analysis of the N-SML control scheme is divided on neural identifier convergence and N-SML controller convergence, taking into account the discrete-time separation principle [54]. For the first analysis, it is achieved by using an EKF based algorithm to train the RHONN identifier as demonstrated in Theorem 4.1 of [36]. The tracking error convergence to a small bound region. It is achieved by the proposed controller as will be demonstrated in the following analysis Due to identification error boundedness, there exists a bounded vector valued function i,k , which is smaller; then xk = χk − i,k

i = 1, . . . , r

(A.8)

  with i,k  ≤ γi , γi > 0 and r the number of the state. The sliding surface at k + 1 is defined using xk as follows sn,k+1 = xr e f,k+1 − xk+1

(A.9)

Using (A.8) in (A.9), we obtain sn,k+1 = xr e f,k+1 − χk+1 + i,k

(A.10)

with xr e f,k+1 the desired dynamics to be tracked. Using the neural linearization control part (4.39), the sliding surface is expressed as follows sn,k+1 = xr e f,k+1 − vn (xk ) + i,k

(A.11)

The DTSMC in (4.42) includes two cases • When ||vcn (xk )|| ≤ u 0n , the control signal vcn,k is applied. Using (4.43) and (4.45) in (A.10), the sliding surface at k + 1 is sn,k+1 = kn sn,k + i,k

(A.12)

T sn,k , then its difference is given Let define the Lyapunov function candidate Vk = sn,k as

Appendix A: Neural Controller Stability Analysis

193

T T Vk = sn,k+1 sn,k+1 − sn,k sn,k T = (kn sn,k + i,k )T (kn sn,k + i,k ) − sn,k sn,k  2  2    ≤ (kn  sn,k  + i,k ) − sn,k   2   ≤ (kn  sn,k  + γi )2 − sn,k   2  2   ≤ kn 2 sn,k  + 2 kn  sn,k  γi + γi2 − sn,k   2   ≤ −(1 − kn 2 ) sn,k  + 2 kn  sn,k  γi + γi2  2  2 ≤ −(1 − θ1 )η1 sn,k  − θ1 η1 sn,k    +2 kn  sn,k  γi + γi2  2 ≤ −(1 − θ1 )η1 sn,k  + γi2  

  + −θ1 η1 sn,k  + 2 kn  γi sn,k 

  with 0 < θ1 < 1, η1 = (1 − kn 2 ) and η1 > 0, and for the region sn,k  ≥ we obtain

2γi kn  , θ1 η1

 2 Vk ≤ −(1 − θ1 )η1 sn,k  + γi2  2  2 Vk ≤ −(1 − θ2 )β1 sn,k  − θ2 β sn,k  + 2  2 Vk ≤ −(1 − θ2 )β1 sn,k    with 0 < θ2 < 1, β1 = (1 − θ1 )η1 and β1 > 0. Therefore Vk ≤ 0, ∀ sn,k  ≥  2 γi , and the solution of system (A.10) is ultimately bounded. θ2 β1 v

(x )

• When ||vcn (xk )|| > v0n , the following control law u 0 veqn (xk ) is applied. eqn

k

Let us define the equivalent control by imposing sn,k − xr e f,k + χk = 0. The equivalent control is defined as veqn,k = sn,k − xr e f,k + χk + xr e f,k+1 = sn,k + f n,k

(A.13)

with f n,k = −xr e f,k + χk + xr e f,k+1 . Then, the expression of the sliding mode surface is sn,k+1 = sn,k − xr e f,k + xk + xr e f,k+1 − vn (xk ) + i,k

veqn,k  + i,k = sn,k + f n,k − u o  veqn,k 

1  + i,k = (sn,k + f n,k ) 1 − u 0  veqn,k 

194

Appendix A: Neural Controller Stability Analysis

T Using the Lyapunov function candidate Vk = sn,k sn,k , then T T sn,k+1 − sn,k sn,k Vk = sn,k+1



T   1  + i,k ≤ sn,k + f n,k  1 − u 0   veqn,k 



  sn,k + f n,k  1 − u 0   1  + i,k veqn,k   2 − sn,k     2  2  ≤ sn,k + f n,k  − u 0  + i,k  − sn,k 

Suppose that the control law u n (xk ) ≤ u 0n  may varywithin the domain [77],   f n,k + i,k  ≤ u 0n , and σ < u 0n with σ =  f n,k + i,k  then

2  2   Vk ≤ sn,k  + σ − u 0n  − sn,k      ≤ sn,k  + σ − u 0  + sn,k      sn,k  + σ − u 0  − sn,k   

≤ 2 sn,k  + σ − u 0  (σ − u 0 )  

≤ − 2 sn,k  + σ − u 0  (u 0  − σ )

(A.14)

          If  f n,k + i,k  ≤ u 0n   ≤ 2 sn,k + f n,k + i,k holds, then Vk ≤ 0 [36,     78]. Hence sn,k and veqn,k , both decreases monotonically.  Therefore there will be a time k1 such that veqn,k  ≤ u 0  for k ≥ k1 . At that time, the control law vcn is applied, yielding that the solution of the system (A.12) is ultimately bounded. 

Appendix B

The Wind Turbine Modeling

B.1 Aerodynamic Model Consider the horizontal axis wind system shown in Fig. B.1 on which v1 is upstream wind speed of the turbine and v2 is downstream wind speed. The kinetic energy of the mass air particle m that moves with a velocity is given as E=

1 2 mv 2

(B.1)

with E is the kinetic energy (J), m is the mass of the air particle (kg) and v is the speed (or the velocity) of the air particle (m/s), such that m = ρV

(B.2)

with ρ is the air density (kg/m3 ) and V is the the volume (m3 ). Taking into account the density of the air is constant, the relative change of the mass is calculated as follows m˙ = ρ S x˙ (B.3) Knowing that the volume is obtained by multiplying the surface S by the length x. Only one dimension is considered because the amount of air varies along a single axis. Hence m˙ = ρ Sv (B.4) By assuming that the wind speed is constant, the power of the air movement will be calculated as 1 (B.5) Pw = E˙ = ρ Sv 3 2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 E. N. Sánchez and L. Djilali, Neural Control of Renewable Electrical Power Systems, Studies in Systems, Decision and Control 278, https://doi.org/10.1007/978-3-030-47443-0

195

196

Appendix B: The Wind Turbine Modeling

Hence the absorbed power by the aerogenerator equals the power difference of downstream and upstream power Paer o = Pw1 − Pw2 =

1 (ρ1 S1 v13 − ρ2 S2 v23 ) 2

(B.6)

Taking ρ1 = ρ2 and m˙ 1 = m˙ 2 , we obtain S1 v1 = S2 v2

(B.7)

Substituting (B.7) in (B.6), the absorbed aerogenerator power is expressed as follow Paer o =

1 ρ1 S1 v1 (v12 − v22 ) 2

(B.8)

In other hand and according to Newton’s second law F=

∂m ∂v ∂ (mv) = v+m ∂t ∂t ∂t

(B.9)

Considering the wind speed is constant and the mass quantity is variable, the force relation is rewritten as F = mv ˙ (B.10) Substituting (B.4) in (B.10), the applied force F on the wind turbine blade is expressed as F = ρ Sv 2 (B.11) The force variation between the two sides of the aerogenerator multiplied by the wind speed result the expression of absorbed power which is expressed by the following relation (B.12) Paer o = Fv = ρ S1 v1 (v1 − v2 )v Note that the power relationship is defined by the wind speed before and after the turbine and another wind speed named v. To define the value of the last one, we have to compare the two relationships of the wind turbine absorbed power in (B.8) and (B.12), we obtain 1 (B.13) v = (v1 + v2 ) 2 For HAWT, the air flow at wind turbine is presented in Fig. 3.4, with v1 is the upstream wind speed, v2 is the downstream wind speed, v is the speed of the wind passing through the aerogenerator, S1 and S2 are the upstream and the downstream sections of the air tube, S is the surface swept by the blade. The absorbed power by the aerogenerator in case of HAWT is calculated as follow

Appendix B: The Wind Turbine Modeling

197

1 ρ Sv(v12 − v22 ) 2

(B.14)

1 ρ S(v1 + v2 )(v12 − v22 ) 4

(B.15)

Paer o = Using (B.13) in (B.14), we get Paer o = and the wind power is given by

Pw =

1 ρ Sv13 2

(B.16)

The ratio between the extracted power from the wind and the available total power available is   2   2   v2  v2   Paer o 1  (B.17) Cp = = 1 −  1 +  Pw 2 v1   v1  Let define the relative speed λ as λ=

v2

t r = v1 v

(B.18)

with ρ is the air density, v is the wind speed, t is the rotational speed of the turbine (low shaft), and r is the radius of a blade. So the expression of the extracted power for the wind is defined as a function of the power coefficient C p as follows Paer o =

1 C p ρ Sv 3 2

(B.19)

The power coefficient C p represents the yield aerodynamics of the WT, which depends on the turbine characteristics defined by the following empirical formula for variable speed wind turbines [57]     −c6 1 x C p (λ, β) = c1 c2 − c3 β − c4 − c5 exp   with c1 = 0.5; c2 = 116; c3 = 0.4; c4 = 0; c5 = 5; c6 = 21 and 0.035 where λ is the relative speed, β is the blade angle. 1+β 3

(B.20) 1 

=

1 λ+0.08β

−

B.2 DC-Link Model In order to determine the mathematical model of the GSC, balanced grid conditions are considered. The three-phase model of the GSC is written as

198

Appendix B: The Wind Turbine Modeling

Fig. B.1 HAWT simplified scheme

Upstream

Downstream

v1

v2

Wind

dφga,t dt dφgb,t = −r g i gb,t + dt dφgc,t = −r g i gc,t + dt

u gca,t − u ga,t = −r g i ga,t +

(B.21)

u gcb,t − u gb,t

(B.22)

u gcc,t − u gc,t

(B.23)

and φga,t = l g i ga,t

(B.24)

φgb,t = l g i gb,t φgc,t = l g i gc,t

(B.25) (B.26)

abc are the three-phase grid currents (A), u abc where i g,t g,t are the three phase grid voltages abc (V), u gc,t are the three phase GSC voltages (V), which are present the control input of the DC-link voltage, r g is the grid line resistance ( ) and l g is the grid inductance (H ). Substituting (B.21)–(B.23) in (B.24)–(B.26), the GSC model can be rewritten as

di ga,t dt di gb,t = r g i gb,t + dt di gc,t = r g i gc,t + dt

u gca,t − u ga,t = r g i ga,t + l g

(B.27)

u gcb,t − u gb,t

(B.28)

u gcc,t − u gc,t

lg

abc di g,t

dt

abc abc = −r g i ga,t − u abc g,t + u gc,t

(B.29)

(B.30)

Appendix B: The Wind Turbine Modeling

199

The GSC converter can be expressed in d − q by using the following transformation K g−1

⎡ ⎤ cos(θg ) − sin(θg ) 1 3⎣ ) − sin(θg − 2π ) 1⎦ cos(θg − 2π = 3 3 2 2π cos(θg + 3 ) sin(θg + 2π ) 1 3

(B.31)

with θg is the grid angular position. Now, Applying the Park transformation to (B.30), we get d abc abc −1 abc l g K g−1 i g,t = −r g K g−1 i ga,t − K g−1 u abc (B.32) g,t + K g u gc,t dt then, lg

  d dq0 d dq0 dq0 dq0 dq0 i g,t + l g i g,t (K g ) (K g−1 ) = −r g i ga,t − u g,t + u gc,t dt dt

(B.33)

⎡ ⎤ 0 −ωg s

such that (K g ) dtd (K g−1 ) = Wg = ⎣ 0 ωg 0 ⎦, Hence, the GSC in d − q refer0 0 0 ence frame is obtained as r g dq0 d dq0 dq0 dq0 dq0 i g,t = − i g,t − Wg i g,t − u g,t + u gc,t dt lg

(B.34)

where di gd,t , di gq,t are the d − q component of the grid current, u gd,t , u gq,t are the grid voltage d − q component, and u gdc,t − u gqc,t are the GSC voltage d − q component. Neglecting the harmonics distortion due by the electronic switches, the GSC and electrical transformer losses, and the power balance between the ac and dc sides of the GSC, the DC voltage at the output of the DC link can be determined as [12]

3 dUdc,t = i gd,t u gd,t + i gq,t u gq,t (B.35) dt 2cg Udc,t

B.3 DFIG Model The three phase stator and rotor mathematical model of the DFIG is described as • The DFIG Stator Equations

abc dφs,t abc = u abc s,t − r s i s,t dt abc abc abc φs,t = L ss i s,t + Msr ir,t

(B.36) (B.37)

200

Appendix B: The Wind Turbine Modeling

• The DFIG Rotor Equations

abc dφr,t abc abc − rr ir,t = u r,t dt abc abc abc φr,t = L rr ir,t + Mr s i s,t

(B.38) (B.39)

with ⎡

⎡ ⎤ ⎤ L ls + L ss L lr + L rr Ms Ms Mr Mr ⎦, L rr = ⎣ ⎦, Ms L ls + L ss Ms Mr L lr + L rr Mr L ss = ⎣ Ms Ms L ls + L ss Mr Mr L lr + L rr

⎤ cos(θ ) cos(θ + 2π ) cos(θ − 2π ) 3 3 Msr = MrTs = Msr ⎣ cos(θ − 2π ) cos(θ ) cos(θ + 2π ) ⎦, 3 3 2π 2π cos(θ + 3 ) cos(θ − 3 ) cos(θ ) abc with u abc are the three-phase stator voltages (V), i s,t are the three-phase stator curs,t abc abc are the three-phase rotor rents (A), φs,t are the three-phase stator flux (W b), u r,t abc abc are the three-phase voltages (V), ir,t are the three-phase rotor currents (A), φr,t rotor flux (W b), rs is the stator winding per phase leakage resistance ( ), rr is the rotor winding per phase leakage resistance ( ), L ls is the stator winding per phase leakage inductance, L ss is the stator self inductance, L lr is the rotor winding per phase leakage inductance, L rr is the rotor self inductance, Msr is the stator to rotor mutual inductance, Mrr is the rotor to stator mutual inductance, Ms is the stator mutual inductance, and Mr is the rotor mutual inductance. The DFIG stator variables can be expressed in d − q reference frame by using the following transformation ⎡

⎤ cos(θs ) − sin(θs ) 1 3 ) − sin(θs − 2π ) 1⎦ = ⎣ cos(θs − 2π 3 3 2 2π ) 1 cos(θs + 3 ) sin(θs + 2π 3 ⎡

K s−1

d −1 dq0 dq0 dq0 K φs,t = K s−1 u s,t − rs K s−1 i s,t dt s dq0 dq0 dq0 K s−1 φs,t = K s−1 L ss i s,t + K s−1 Msr ir,t

(B.40)

(B.41) (B.42)

then,   d −1 d dq0 dq0 dq0 dq0 φs,t φs,t = u s,t − rs i s,t − K s K s dt dt

(B.43)

φs,t = K s L ss i K s−1 i s,t + K s Msr K s−1 ir,t

(B.44)

dq0

dq0

dq0

Appendix B: The Wind Turbine Modeling

201

⎡ ⎤ 0 −ωs

such that (K s ) dtd (K s−1 ) = Ws = ⎣ 0 ωs 0 ⎦, 0 0 0 d dq0 dq0 dq0 dq0 φs,t = u s,t − rs i s,t − Ws iφs,t dt dq0 dq0 dq0 φs,t = ls i s,t + lm ir,t

(B.45) (B.46)

with K s L ss i K s−1 = ls and K s Msr K s−1 = lm . The DFIG rotor variables can be expressed in d − q reference frame by using the same transformation as in () by changing (θs − θr ) d −1 dq0 dq0 dq0 K φr,t = K r−1 u r,t − rr K r−1 ir,t dt r dq0 dq0 dq0 K r−1 φr,t = K r−1 L rr i s,t + K r−1 Mr s i s,t

(B.47) (B.48)

then,   d d dq0 dq0 dq0 dq0 φr,t = u r,t − rr ir,t − K r K r−1 φr,t dt dt

(B.49)

φr,t = K r L rr i K r−1 ir,t + K r Mr s K r−1 i s,t

(B.50)

dq0

dq0

dq0

⎡ ⎤ 0 0 ωr − ωs

⎦, 0 such that (K r ) dtd (K r−1 ) = Wr = ⎣ 0 ωs − ωr 0 0 0 d dq0 dq0 dq0 dq0 φr,t = u r,t − rr ir,t − Wr φr,t dt dq0 dq0 dq0 φr,t = lr ir,t + lm i s,t with K r L rr i K r−1 = lr and K r Mr s K r−1 = lm .

(B.51) (B.52)

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