Advances in Control Technologies for Brushless Doubly-fed Induction Generators [1st ed. 2022] 9811904235, 9789811904233

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Yi Liu Wei Xu

Advances in Control Technologies for Brushless Doubly-fed Induction Generators

Advances in Control Technologies for Brushless Doubly-fed Induction Generators

Yi Liu · Wei Xu

Advances in Control Technologies for Brushless Doubly-fed Induction Generators

Yi Liu State Key Laboratory of Advanced Electromagnetic Engineering and Technology School of Electrical and Electronic Engineering Huazhong University of Science and Technology Wuhan, Hubei, China

Wei Xu State Key Laboratory of Advanced Electromagnetic Engineering and Technology School of Electrical and Electronic Engineering Huazhong University of Science and Technology Wuhan, Hubei, China

ISBN 978-981-19-0423-3 ISBN 978-981-19-0424-0 (eBook) https://doi.org/10.1007/978-981-19-0424-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Foreword

This book involves with an emerging dual-electrical-port machine, the brushless doubly-fed induction generator (BDFIG). In comparison with its counterpart which is the doubly-fed induction generator (DFIG), the BDFIG has no brushes and slip rings attached to the rotor, so it is expected to have a longer lifespan, higher reliability and lower maintenance cost. Such machine has a great potential in a variety of industrial applications, e.g., wind power generation, ship shaft power generation, ship electric propulsion, hydroelectric generation, pumped-hydro energy storage and so on. The high-reliability and cost-effective electrical drive systems are the continuous pursuits of the industry community. Therefore, the research on the control technologies of this emerging BDFIG will become a popular direction in the near future. This book mainly presents advanced control methods and technologies of BDFIGs proposed by the authors for solving some typical problems in industrial applications with BDFIGs, including distorted voltage rejection, compensation control strategies under heavy load disturbance, predictive control and sensorless control. Especially, the book has done a very comprehensive and pioneering work in the distorted voltage suppression methods as well as sensorless control of BDFIGs, which make this book different from other books in the field. This book will be very helpful to researchers and engineers in industry who are interested in renewable power generation based on advanced BDFIGs as well as graduate students for their study. Finally, I would like to express my sincere congratulations to the two authors, Dr. Yi Liu and Prof. Wei Xu, for their high-level and diligent work. I believe that this book will be a milestone contribution to the BDFIG and related fields of applications. December 2021

Frede Blaabjerg Professor and IEEE Fellow Aalborg University Aalborg, Denmark

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Preface

After long-term theoretical research and practical industrial applications, the theory and technology of traditional electrical machines have become mature gradually. However, traditional electrical machines are difficult to meet the ever-increasing demand for high-performance industrial applications. Therefore, academia and industry are constantly conducting research and exploration of novel electrical machines. In this context, the brushless doubly-fed generator (BDFG) came into being, and it has become a research hotspot in the last decade. The BDFG is a kind of dual-electrical-port machine, which consists of two sets of stator windings with different pole pairs (i.e., power winding and control winding) and a specially designed rotor. Through the modulation of the rotor, the two rotating magnetic fields generated by the two stator windings can be indirectly coupled. This kind of machine can eliminate the inherent brushes and slip rings of the traditional doubly-fed induction generator (DFIG), with the wide speed range, long lifespan, high reliability and low maintenance cost. Therefore, it has broad application prospects in new energy fields such as wind power generation, ship shaft power generation, ship electric propulsion, hydroelectric generation, pumped-hydro energy storage and so on. The main objective of this book is to present advances in control technologies for efficient operation of brushless doubly-fed induction generator (BDFIG). For robust and low-cost operation of BDFIG, it is required to keep high-quality output voltage and eliminate the speed/position encoder under different loads and operation conditions. Some advanced control technologies, from the authors’ latest work on these topics, are presented to achieve this goal with simple and accurate texts, illustrations and tables. The qualified outcomes obtained from this book would assure the highperformance operation of BDFIG and also give the readers a straight insight toward challenges in this research area in the future. This book consists of eight chapters, which are mainly based on the authors’ academic and industrial projects. Each chapter can be briefly summarized as follows. Chapter 1 surveys the state of the art of control technologies for BDFG under different operation conditions and the progresses of sensorless control for BDFG. The classification and comparison are carried out to discover the advantages and vii

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disadvantages of these control technologies. Chapter 2 presents a detailed description of the presented power generation system based on the BDFIG. Chapter 3 presents the negative-sequence voltage compensator, low-order harmonic voltage compensator and dual-resonant controller (DRC) for minimizing the nonlinear and unbalanced impacts. In Chap. 4, two advanced control strategies for the standalone BDFIG under heavy load disturbance based on the single power converter and the dual power converters are developed, respectively. Chapter 5 handles the improvements of the robustness and dynamic response for the standalone BDFIG system by applying both the model predictive current control (MPCC) and the improved nonparametric predictive current control (NPCC) methods. Chapter 6 develops two rotor position observers based on the phase–axis relationship and the space-vector model of BDFIG, as well as one machine-parameter-free speed observer for BDFIG operating with unbalanced and nonlinear loads. Chapter 7 presents five model reference adaptive system (MRAS)-based sensorless control strategies based on the control winding power factor, the power winding flux in αβ and dq reference frames, and the control winding flux in αβ and dq reference frames, respectively. The book can be used as a reference for university researchers, scientists, professionals and graduate students who are interested in BDFIGs and wish to learn control technologies for BDFIGs, especially the unbalanced and low-order harmonic voltage rejection methods for standalone BDFIG systems, advanced control strategies for standalone BDFIGs with heavy load disturbance, predictive control, rotor position/speed observation, model reference adaptive system-based sensorless control, and the other technical challenges of BDFIG control and their application prospects. The research work introduced in this book was supported in part by the Excellent Youth Fund of Shandong Natural Science Foundation under Grant ZR2020YQ40, the National Natural Science Foundation of China under Grant 51877093 and 51707079, the National Key Research and Development Program of China under Grant 2018YFE0100200, the Key Technical Innovation Program of Hubei Province under Grant 2019AAA026, the Science, Technology and Innovation Commission of Shenzhen Municipality under Grant JCYJ20190809101205546 and the Fundamental Research Funds for the Central Universities of China under Grant 2021XXJS002. The authors would like to express their sincere thanks to Prof. Wu Ai, Prof. Bing Chen, Prof. Shenghua Huang, Prof. Jianguo Zhu, Prof. Frede Blaabjerg, Prof. Ion Boldea, Prof. Xuefan Wang, Prof. Teng Long, Dr. Fei Xiong, Dr. Xi Chen, Mr. Dixian Shu and Mr. Gang Zhi for their great support to the authors’ research work. And, the authors would like to thank Dr. Mohamed G. Hussien, Dr. Omer Mohammed Elbabo Mohammed, Dr. Alameen K. Ebraheem, Mr. Junjie Chen, Mr. Kailiang Yu, Mr. Jianping Gao, Mr. Yangsheng Zhang and Mr. Juncai Jiang who actively participated in the authors’ research projects and academic research. Also, the authors would like to acknowledge the contributions of Miss Yifan Lin and Mr. Maoxin Zhang on careful editing and proofreading on this book. Meanwhile, many thanks go to Dr. Yahong Chen, Mr. Jian Ge, Mr. Zhen Bao and Miss Yaping Zhang for their suggestions and comments on this work. Moreover, the authors would also like to thank their families who have given tremendous support all the time. Finally, the authors are extremely

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grateful to Springer and the editorial staff for the opportunity to publish this book and help in all possible manners. Wuhan, China

Yi Liu Wei Xu

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction to Brushless Doubly-Fed Generators . . . . . . . . . . . . . . . 1.2 Advances in Control Technologies for Grid-Connected Power Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Grid-Connected Power Generation Under Normal Grid . . . . 1.2.2 Grid-Connected Power Generation Under Faulty Grid . . . . . 1.3 Advances in Control Technologies for Standalone Power Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Standalone Power Generation with Normal Loads . . . . . . . . 1.3.2 Standalone Power Generation with Special Loads . . . . . . . . . 1.4 Advances in Control Technologies for Sensorless Control . . . . . . . . 1.4.1 Without Speed Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 With Speed Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Organization of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1

16 16 22 28 28 29 33 35

2 Mathematical Modelling, Operation Characteristics and Basic Control Method of BDFIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Dynamic Models of BDFIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 abc-Axis Model of BDFIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 dq0-Axis Dynamic Model of BDFIG . . . . . . . . . . . . . . . . . . . 2.3 Steady-state Models of BDFIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Conventional -type Steady-state Model . . . . . . . . . . . . . . . . 2.3.2 Steady-state Model with Simplified Inner Core . . . . . . . . . . . 2.3.3 Steady-state T-type Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Operation Characteristics of the BDFIG Drive System . . . . . . . . . . . 2.4.1 Construction of the BDFIG Drive System . . . . . . . . . . . . . . . 2.4.2 Operation Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Power Flow Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Basic Control Method for the Standalone BDFIG . . . . . . . . . . . . . . .

41 41 43 43 44 46 46 47 47 48 48 49 50 51

5 5 13

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2.5.1 CW Current Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Design of the DVC Control Strategy . . . . . . . . . . . . . . . . . . . . 2.5.3 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Unbalanced and Low-Order Harmonic Voltages Rejection for Standalone BDFIGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Analysis of the Sources of Unbalanced and Low-Order Harmonic Voltages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Analysis Under Unbalanced Load . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Analysis Under Three-Phase Nonlinear Load . . . . . . . . . . . . 3.2.3 Analysis Under Unbalanced Plus Nonlinear Load . . . . . . . . . 3.2.4 Analysis Under Single-Phase Nonlinear Load . . . . . . . . . . . . 3.3 Unbalanced Voltage Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Conventional Direct Voltage Control Scheme . . . . . . . . . . . . 3.3.2 Design of Negative-Sequence Voltage Compensator . . . . . . . 3.3.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Low-Order Harmonic Voltage Compensation . . . . . . . . . . . . . . . . . . . 3.4.1 Design of Low-Order Harmonic Voltage Compensator . . . . 3.4.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Dual-Resonant Controller for Compensating Unbalanced and Low-Order Harmonic Voltages . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Design of Dual-Resonant Controller . . . . . . . . . . . . . . . . . . . . 3.5.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Advanced Control Strategies for Standalone BDFIGs with Heavy Load Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Vector Control Strategy with Transient Current Compensation . . . . 4.2.1 Transient Feedforward Compensation of CW Current . . . . . 4.2.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Cooperative Compensation Strategy Based on Dual Power Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Compensation Strategy Based on MSC . . . . . . . . . . . . . . . . . . 4.3.2 Compensation Strategy Based on LSC . . . . . . . . . . . . . . . . . . 4.3.3 Cooperative Compensation Strategy Based on Dual Power Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 52 53 56 56 59 59 60 60 63 65 68 71 71 73 77 81 86 87 109 113 116 117 137 143 152 159 163 163 164 164 168 174 174 176 177 178

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4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 5 Predictive Control for Standalone BDFIGs . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Model Predictive Current Control (MPCC) for Standalone BDFIGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Design of MPCC for CW Current . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Nonparametric Predictive Current Control (NPCC) for Standalone BDFIGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 CW Current Prediction W ithout Machine Parameters . . . . . . 5.3.2 Implementation of NPCC-Based Control Scheme . . . . . . . . . 5.3.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Rotor Position and Speed Observers of BDFIGs . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Rotor Position Observer Based on the Phase-Axis Relationship of BDFIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Rotor Position Observer Based on the Space-Vector Model of BDFIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Improved Rotor Speed Observer for Standalone BDFIG with Unbalanced and Nonlinear Loads . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Scheme Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Parameters Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Model Reference Adaptive System Based Sensorless Control for BDFIGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 MRAS Sensorless Control Based on CW Power Factor . . . . . . . . . . 7.2.1 PW Field-Oriented Control for Sensorless Voltage Control of BDFIGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Design Procedure of the Proposed Control Method . . . . . . . . 7.2.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

203 203 204 205 210 210 212 218 223 229 237 239 241 241 243 244 246 249 249 252 255 259 271 271 278 280 287 287 289 289 291 291 292 294

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7.2.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 MRAS Sensorless Control Based on PW Flux . . . . . . . . . . . . . . . . . . 7.3.1 Control Scheme Based on αβ-Axis PW Flux . . . . . . . . . . . . . 7.3.2 Control Scheme Based on dq-Axis PW Flux . . . . . . . . . . . . . 7.3.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 MRAS Sensorless Control Based on CW Flux . . . . . . . . . . . . . . . . . . 7.4.1 Control Scheme Based on αβ-Axis CW Flux . . . . . . . . . . . . . 7.4.2 Control Scheme Based on Dq-Axis CW Flux . . . . . . . . . . . . . 7.4.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

299 303 303 307 310 321 337 337 341 345 359 369 372

8 Conclusions and Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 8.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

About the Authors

Dr. Yi Liu (M’14-SM’20) received his B.E. and M.E. degrees in Automation and Control Engineering from the Wuhan University of Science and Technology, Wuhan, China, in 2004 and 2007, respectively; and his Ph.D. degree in Mechatronic Engineering from the Huazhong University of Science and Technology, Wuhan, China, in 2016. From March 2016 to June 2016, he was a Senior R&D Engineer at the Fourth Academy of China Aerospace Science and Industry Group, Wuhan, China. From July 2016 to October 2019, he was a Postdoctoral Research Fellow at the State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, where he has been a Lecturer since January 2020. He is the Vice Chair for IEEE IES Wuhan Chapter, the Associate Editor for IEEE Transactions on Industry Applications. His current research interests include multi-port electrical machines and drive systems. He has received one IEEE Prize Paper Award in 2020. Till now, Dr. Liu has published more than 50 high-quality SCI-indexed international journal papers, held over 20 granted/pending invention patents. Dr. Liu has been invited to give tutorials on brushless DFIG for two conferences, i.e., 2019 IEEE International Conference on Electrical Machines and Systems (ICEMS2019), 23rd China Power Supply Society Conference (CPSSC2019). And, he has organized special sessions on multi-port electrical machines for several conferences, e.g., ICEMS2019, ECCE Asia2020, ICEM2020 and IEEE-PEMC2020. xv

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About the Authors

Prof. Wei Xu (M’09-SM’13) received the double B.E. and M.E. degrees from Tianjin University, Tianjin, China, in 2002 and 2005, and the Ph.D. from the Institute of Electrical Engineering, Chinese Academy of Sciences, in 2008, respectively, all in electrical engineering. His research topics mainly cover design and control of linear/rotary machines. From 2008 to 2012, he made Postdoctoral Fellow with University of Technology Sydney, Vice Chancellor Research Fellow with Royal Melbourne Institute of Technology, Japan Science Promotion Society Invitation Fellow with Meiji University, respectively. Since 2013, he has been full professor with State Key Laboratory of Advanced Electromagnetic Engineering in Huazhong University of Science and Technology, China. He has more than 120 papers accepted or published in IEEE Journals, two edited books published by Springer Press, one monograph published by China Machine Press and more than 170 Invention Patents granted or in pending, all in the related fields of electrical machines and drives. He is Fellow of the Institute of Engineering and Technology (IET). He is the General Chair for 2021 International Symposium on Linear Drives for Industry Applications (LDIA 2021) and 2023 IEEE International Conference on Predictive Control of Electrical Drives and Power Electronics (PRECEDE 2023) in Wuhan, China, respectively. He has served as Associate Editor for several leading IEEE Transactions Journals, such as IEEE Transactions on Industrial Electronics, IEEE Transactions on Vehicular Technology, IEEE Transactions on Energy Conversion and so on.

Abbreviations

BDFG BDFIG BDFRG CDFIG CW DESRIM DFIG DFR DPC DRC DSDBFIG DSF DSOGI DSVM DTC DVC FCS-MPC HDN ISC LPF LSC LVRT MIMO MPC MPCC MPFSO MPTC MRAS MSC MSOGI NPCC

Brushless doubly-fed generator Brushless doubly-fed induction generator Brushless doubly-fed reluctance generator Cascade doubly-fed induction generator Control winding Double-excited slip ring induction machine Doubly-fed induction generator Dual frequency resonance Direct power control Dual-resonant controller Dual-stator BDFIG Dual synchronous frame Dual second-order generalized integrators Discrete space-vector modulation Direct torque control Direct voltage control Finite control set model predictive control Harmonic decoupling network Indirect stator-quantities control Low-pass filter Line side converter Low-voltage ride-through Multiple-input multiple-output Model predictive control Model predictive current control Machine-parameter-free speed observer Model predictive torque control Model reference adaptive system Machine side converter Multiple second-order generalized integrators Nonparametric predictive current control xvii

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PCC PI PIR PLL PNSC PR PW QSG RPO RSO RW SDR SFRF SSF SSM SVM THD VC VSCF

Abbreviations

Point of common coupling Proportional–integral Proportional–integral-resonant Phase-locked loop Positive–negative signal calculator Proportional resonant Power winding Quadrature signal generator Rotor position observer Rotor speed observer Rotor winding Series dynamic resistor Slip frequency rotating frame Single synchronous frame Super-twisting sliding mode Space vector modulation Total harmonic distortion Vector control Variable speed constant frequency

Nomenclature

p1 , p2 ω1 , ω2 ωa , θa θ 1, θ 2 ωr , θ r θ0 ωN θ rv u, i, ψ u, i,ψ U, I P1 , P2 , Pout R1 , R2 , Rr L1 , L2 , Lr L 1r L 2r σ2 Ts D s

Pole pair numbers of PW and CW Angular frequencies of PW and CW Angular speed of the rotating dq frame Position of the d-axis of the rotating dq frame Positions of PW voltage and CW current vectors Rotor speed and rotor position Initial phase difference between the A-phase axis of PW and the A-phase axis of CW Natural synchronous rotor speed Virtual rotor position Voltage, current and flux scalars Voltage, current and flux phasors Amplitudes of voltage and current Active powers of PW and CW, and output active power of the BDFIG Resistances of PW, CW and RW Self-inductances of PW, CW and RW Coupling inductance between the PW and the rotor winding Coupling inductance between the CW and the rotor winding Leakage constant of the CW Sampling period Disturbance term Differential operator, i.e., d/dt Variation quantity

Superscripts * ˆ, ~ T

Reference value Estimated value Matrix transpose xix

xx

+, – 3, 5, 7

Nomenclature

Positive and negative reference frames 3rd, 5th and 7th harmonic reference frames

Subscripts 1, 2, r a, b, c α, β d, q f +, – 3, 5, 7

PW, CW and rotor Phases a, b and c of the three-phase voltage or current Stationary α- and β-axes Synchronous rotating d- and q-axes Filtered quantity Positive- and negative-sequence components 3rd, 5th and 7th harmonic components

List of Figures

Fig. 1.1 Fig. 1.2 Fig. 1.3 Fig. 1.4 Fig. 1.5

Fig. 1.6

Fig. 1.7 Fig. 1.8 Fig. 1.9 Fig. 1.10

Fig. 1.11 Fig. 1.12

Fig. 1.13

The topology of the BDFG-based ac power generation system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The BDFIG application in wind power generation [12] . . . . . . . . The BDFIG application in ship shaft power generation [10] . . . . The BDFIG application in hydropower generation [12] . . . . . . . . The SFRF-VC method: a Structure of SFRF, b Obtained PW and CW current vectors by the SFRF-VC method proposed in [39] and [40] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure of the proposed SSM-DPC strategy for BDFIG: a Schematic diagram of the super-twisting sliding mode control, b Overall DPC strategy with the super-twisting sliding mode control, where F and D are two matrices determined the time-derivative formula of sliding surface [42] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DPC diagram of the open-winding BDFIG [43] . . . . . . . . . . . . . . The block diagram of the duty ratio modulation DTC for BDFIG proposed in [51] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Block diagram of the indirect stator-quantities control system with the reactive power controller for BDFIG [54] . . . . . Summarized control structures presented in [55–57] for the BDFIG under unbalanced grid with PI/PIR controllers, respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Different kinds of LVRT methods for the grid-connected BDFIG systems [61–65] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The block diagram of the crowbarless LVRT control strategy based on flux linkage tracking for BDFIG under symmetrical voltage dips proposed in [66] . . . . . . . . . . . . . The stator-flux-oriented control and the direct voltage control based on CW current PI controller without decoupling [10, 67] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 3 4 5

9

10 11 12 13

14 15

16

17

xxi

xxii

Fig. 1.14

Fig. 1.15 Fig. 1.16

Fig. 1.17 Fig. 1.18

Fig. 1.19

Fig. 1.20 Fig. 1.21 Fig. 1.22

Fig. 1.23 Fig. 1.24 Fig. 1.25 Fig. 1.26 Fig. 2.1 Fig. 2.2 Fig. 2.3 Fig. 2.4

List of Figures

Simulated and experimental results of the CW current PI controller without decoupling: a At the sub-synchronous speed of 400 rpm, b At the super-synchronous speed of 600 rpm [10] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The CW current control loop with cross feedforward compensation proposed in [68] . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulated and experimental results of the CW current control loop with cross feedforward compensation: a At the sub-synchronous speed of 600 rpm, b At the super-synchronous speed of 1200 rpm [68] . . . . . . . . . . . . . . The CW current control loop with the decoupling network proposed in [70] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental results of the CW current control loop with the decoupling network, where DN and FF indicate decoupling network and feedforward, respectively: a At the sub-synchronous speed of 350 rpm, b At the super-synchronous speed of 650 rpm [70] . . . . . . . . . . . . . . . Different compensation methods based on MSC for standalone BDFIG under special loads: a Indirect calculation of CW voltage compensation components-Type I, b Indirect calculation of CW voltage compensation components-Type II, c Direct calculation of CW voltage compensation components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compensation method based on LSC for standalone BDFIG under special loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compensation method based on the collaborative control of MSC and LSC for standalone BDFIG under special loads . . . Experimental results of the compensation method based on the collaborative control of MSC and LSC with the weight factor reduction from 1 to 0.2: a Weight factor, b CW current, c PW line voltage, d Amplitude of CW current, e LSC current and f Electromagnetic torque [79] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The actual and reference PW voltage vectors in the rotating dq reference frame [80] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The structure of the linearized control loop for tracking the PW voltage phase proposed in [80] . . . . . . . . . . . . . . . . . . . . . Flowchart of the direct calculation approach for the BDFIG rotor position as developed in [84] and [85] . . . . . . . . . . . . . . . . . Structure of the proposed CW-P/Q/X MRAS speed observers for standalone BDFIGs [98–100] . . . . . . . . . . . . . . . . . Typical configuration of the BDFIG system . . . . . . . . . . . . . . . . . The steady-state -type model of BDFIG . . . . . . . . . . . . . . . . . . Equivalent circuit of the simplified inner core model [29] . . . . . . Equivalent circuit of the steady-state T-type model [31] . . . . . . .

18 19

20 20

21

24 25 26

27 28 29 30 32 42 46 46 47

List of Figures

Fig. 2.5 Fig. 2.6

Fig. 2.7 Fig. 2.8 Fig. 2.9

Fig. 2.10

Fig. 2.11

Fig. 2.12

Fig. 3.1 Fig. 3.2 Fig. 3.3 Fig. 3.4 Fig. 3.5 Fig. 3.6 Fig. 3.7 Fig. 3.8 Fig. 3.9

Fig. 3.10

Fig. 3.11

Fig. 3.12

System structure of the standalone BDFIG [39] . . . . . . . . . . . . . . The power flow of the standalone BDFIG system, a under sub-synchronous speed, b under super-synchronous speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control loop for CW current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conceptual diagram of the DVC strategy for the standalone BDFIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Performance test under the condition of the step load change from the full load to the no load with the rotor speed of 600 rpm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental test under the condition of the step load change from the no load to 42 kW with the rotor speed of 400 rpm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test analysis under the condition of speed variation from the super-synchronous to sub-synchronous speed with the load of 42 kW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental analysis under the condition of speed variation from the sub-synchronous to super-synchronous speed with the load of 42 kW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vector diagram describing the relationship among various reference frames under unbalanced loads . . . . . . . . . . . . . . . . . . . Vector diagram showing the relationship among different frames under nonlinear loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vector diagram showing the relationship among different frames under the unbalanced plus nonlinear load . . . . . . . . . . . . . Vector diagram showing the relationship among different frames under single-phase nonlinear loads . . . . . . . . . . . . . . . . . . Conventional DVC control scheme . . . . . . . . . . . . . . . . . . . . . . . . Block diagram of the negative-sequence voltage compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overall control scheme for unbalanced voltage compensation of PW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Block diagram of DSOGI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results of the PW voltage amplitude with the conventional method at 600 rpm: a balanced load, b unbalanced load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results of the PW voltage with the conventional control strategy at 600 rpm: a balanced load, b unbalanced load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results of the amplitude of the negative-sequence PW voltage with the conventional method at 600 rpm: a balanced load, b unbalanced load . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results of the CW current with the conventional strategy at 600 rpm: a balanced load, b unbalanced load . . . . . . .

xxiii

49

50 52 53

54

55

55

56 61 63 66 69 72 74 75 76

78

79

80 81

xxiv

Fig. 3.13

Fig. 3.14 Fig. 3.15

Fig. 3.16 Fig. 3.17

Fig. 3.18

Fig. 3.19

Fig. 3.20

Fig. 3.21

Fig. 3.22

Fig. 3.23

List of Figures

Simulation results of the PW voltage amplitude with the conventional method at 900 rpm: a balanced load, b unbalanced load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results of the PW voltage with the conventional strategy at 900 rpm: a balanced load, b unbalanced load . . . . . . . Simulation results of the amplitude of the negative-sequence PW voltage with the conventional method at 900 rpm, a balanced load, b unbalanced load . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results of the CW current with the conventional strategy at 900 rpm: a balanced load, b unbalanced load . . . . . . . Simulation results of the conventional strategy with balanced and unbalanced loads under the variable rotor speed: a amplitude of the PW voltage, b PW voltage, c CW current, d amplitude of the negative-sequence PW voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results at 600 rpm with the unbalanced load: a PW voltage, b CW current, c amplitude of the negative-sequence PW voltage, d PW current, e harmonic spectrum of the PW current (the subscripts 1 and 2 indicate the conventional and proposed strategies, respectively) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results at 900 rpm with the unbalanced load: a PW voltage, b CW current, c amplitude of the negative-sequence PW voltage, d PW current, e THD of the PW current (the subscripts 1 and 2 indicate the conventional and proposed strategies, respectively) . . . . . . . . Simulation results during the proposed strategy startup and under the speed variation with the unbalanced load: a PW voltage, b CW current, c amplitude of the negative-sequence PW voltage . . . . . . . . . . . . . . . . . . . . . . Experimental results at 600 rpm with the three-phase unbalanced load: a PW voltage, b CW current, c amplitude of the negative-sequence PW voltage, d PW current, e harmonic spectrum of the PW current (the subscripts 1 and 2 indicate the conventional and proposed strategies, respectively) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental results at 900 rpm with the three-phase unbalanced load: a PW voltage, b CW current, c amplitude of the negative-sequence PW voltage, d harmonic spectrum of the CW current, e PW current, f harmonic spectrum of the PW current (the subscripts 1 and 2 indicate the conventional and proposed strategies, respectively) . . . . . . . . Dynamic performance test with the three-phase unbalanced load: a PW voltage, b CW current, c amplitude of the negative-sequence PW voltage . . . . . . . . . . . . . . . . . . . . . .

82 83

84 85

86

88

91

94

95

99

103

List of Figures

Fig. 3.24

Fig. 3.25 Fig. 3.26 Fig. 3.27 Fig. 3.28 Fig. 3.29 Fig. 3.30

Fig. 3.31 Fig. 3.32 Fig. 3.33 Fig. 3.34

Fig. 3.35 Fig. 3.36

Fig. 3.37

Fig. 3.38

Experimental results with the single-phase load: a PW voltage, b PW current, c CW current, d amplitude of the negative-sequence PW voltage, e expanded view of (a), f expanded view of (a), g expanded view of (b), h expanded view of (c), i harmonic spectrum of the CW current in (h) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Block diagram of the low-order harmonic voltage compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Block diagram of MSOGI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overall control scheme for low-order harmonic voltage compensation of PW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results of the PW voltage with the conventional strategy at 650 rpm: a linear load, b nonlinear load . . . . . . . . . . . The harmonic spectrum of the PW voltage at 650 rpm in simulation: a linear load, b nonlinear load . . . . . . . . . . . . . . . . Simulation results of the amplitudes of the 5th and 7th harmonic components of the PW voltage with the conventional method at 650 rpm: a linear load, b nonlinear load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results of the CW current with the conventional method at 650 rpm: a linear load, b nonlinear load . . . . . . . . . . . Simulation results of the PW voltage with the conventional strategy at 850 rpm: a linear load, b nonlinear load . . . . . . . . . . . The harmonic spectrum of the PW voltage at 850 rpm in simulation: a linear load, b nonlinear load . . . . . . . . . . . . . . . . Simulation results of the amplitudes of the 5th and 7th harmonics of the PW voltage with conventional method at 850 rpm: a linear load, b nonlinear load . . . . . . . . . . . . . . . . . . Simulation results of the CW current with the conventional method at 850 rpm: a linear load, b nonlinear load . . . . . . . . . . . Simulation results at 650 rpm under the nonlinear load: a PW voltage, b CW current, c amplitudes of the 5th and 7th harmonics of the PW voltage, d harmonic spectrum of the PW voltage (the subscripts 1 and 2 indicate the conventional and proposed strategies, respectively) . . . . . . . . Simulation results at 850 rpm under the nonlinear load: a PW voltage, b CW current, c amplitudes of the 5th and 7th harmonics of the PW voltage, d harmonic spectrum of the PW voltage (the subscripts 1 and 2 indicate the conventional and proposed strategies, respectively) . . . . . . . . Simulation results under the speed change with the nonlinear load: a PW voltage, b CW current, c amplitudes of the 5th and 7th harmonics of the PW voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xxv

104 107 108 109 110 111

112 113 114 115

116 117

118

121

124

xxvi

Fig. 3.39

Fig. 3.40

Fig. 3.41

Fig. 3.42 Fig. 3.43 Fig. 3.44

Fig. 3.45

Fig. 3.46

Fig. 3.47

Fig. 3.48

Fig. 3.49

List of Figures

Experimental results at 650 rpm under the three-phase nonlinear load: a PW voltage, b PW current, c amplitudes of the 5th and 7th harmonics of the PW voltage, d harmonic spectrum of the PW voltage (the subscripts 1 and 2 indicate the conventional and proposed strategies, respectively) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamics performance test of the proposed strategy during startup: a PW voltage, b CW current, c amplitudes of the 5th and 7th harmonics of the PW voltage . . . . . . . . . . . . . . Experimental results under the speed change with the three-phase nonlinear load: a PW voltage, b CW current, c expanded view of the PW voltage at 850 rpm, d expanded view of the PW voltage at 650 rpm, e amplitudes of the 5th and 7th harmonics of the PW voltage . . . . . . . . . . . . . . Structure of the dual-resonant controller (DRC) . . . . . . . . . . . . . . Overall control scheme for compensating unbalanced and nonlinear loads based on DRC . . . . . . . . . . . . . . . . . . . . . . . . Simulation results at 675 rpm with the single-phase load: a PW voltage, b CW current, c amplitude of the negative-sequence PW voltage (the subscripts 1 and 2 indicate the conventional and proposed strategies, respectively) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results at 675 rpm with the nonlinear load: a PW voltage, b CW current, c amplitudes of the 5th and 7th harmonics of the PW voltage, d harmonic spectrum of the PW voltage (the subscripts 1 and 2 indicate the conventional and proposed strategies, respectively) . . . . . . . . Dynamic performance test with the combination of the unbalanced and nonlinear loads: a PW voltage, b CW current, c extended view of (a) between 0.2 and 0.4 s, d extended view of (a) between 0.5 and 0.7 s . . . . . . . . . . . . . . . . Simulation results with the speed change under the combination of the unbalanced and nonlinear loads: a PW voltage, b CW current, c amplitude of the negative-sequence PW voltage, d amplitudes of the 5th and 7th harmonics of the PW voltage . . . . . . . . . . . . . . Experimental results at 675 rpm with the unbalanced load: a PW voltage, b CW current, c amplitude of the negative-sequence PW voltage (the subscripts 1 and 2 indicate the conventional and proposed strategies, respectively) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic performance test at 675 rpm with the unbalanced load: a PW voltage, b CW current, c amplitude of the negative-sequence PW voltage . . . . . . . . . . . . . . . . . . . . . .

125

128

129 137 138

139

141

144

145

147

149

List of Figures

Fig. 3.50

Fig. 3.51

Fig. 3.52

Fig. 3.53

Fig. 4.1 Fig. 4.2 Fig. 4.3

Fig. 4.4

Fig. 4.5

Fig. 4.6

Fig. 4.7

Experimental results at 675 rpm under the three-phase nonlinear load: a PW voltage, b CW current, c amplitudes of the 5th and 7th harmonics of the PW voltage, d harmonic spectrum of the PW voltage (the subscripts 1 and 2 indicate the conventional and proposed strategies, respectively) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic performance test at 675 rpm under the three-phase nonlinear load: a PW voltage, b CW current, c amplitudes of the 5th and 7th harmonics of the PW voltage, d harmonic spectrum of the PW voltage . . . . . . . . . . . . . . . . . . . . . . Effectiveness test for the proposed strategy under the unbalanced plus nonlinear load: a PW voltage, b CW current, c amplitude of the negative-sequence PW voltage, d expanded view of (a) between 3.2 and 3.4 s, e expanded view of (a) between 6.1 and 6.3 s, f harmonic spectrum of the PW voltage depicted in (d), g harmonic spectrum of the PW voltage depicted in (e) . . . . . . . . . . . . . . . . . Effectiveness test for the proposed strategy under the single-phase nonlinear load: a PW voltage, b CW current, c amplitude of the negative-sequence PW voltage, d expanded view of (a) between 0 and 0.15 s, e expanded view of (a) between 0.32 and 0.47 s, f harmonic spectrum of the PW voltage depicted in (d), g harmonic spectrum of the PW voltage depicted in (e) . . . . . . . . . . . . . . . . . The CW current vector control loop . . . . . . . . . . . . . . . . . . . . . . . The block diagram of the proposed vector control strategy with CW transient current compensation . . . . . . . . . . . . . . . . . . . Experimental results for the 90-kVA standalone BDFIG feeding a 7.5-kW three-phase induction motor at 600 rpm without the CW transient current compensation: a PW voltage, b PW current, c CW current . . . . . . . . . . . . . . . . . . . . . . . Experimental results for the 90-kVA standalone BDFIG feeding a 7.5-kW three-phase induction motor at 600 rpm with the CW transient current compensation: a PW voltage, b PW current, c CW current . . . . . . . . . . . . . . . . . . . . . . . Experimental results for the 90-kVA standalone BDFIG feeding a 15-kW three-phase induction motor at 900 rpm without the CW transient current compensation: a PW voltage, b PW current, c CW current . . . . . . . . . . . . . . . . . . . . . . . Experimental results for the 90-kVA standalone BDFIG feeding a 15-kW three-phase induction motor at 900 rpm with the CW transient current compensation: a PW voltage, b PW current, c CW current . . . . . . . . . . . . . . . . . . . . . . . Compensation strategy based on MSC . . . . . . . . . . . . . . . . . . . . .

xxvii

150

153

154

157 165 168

170

171

172

173 174

xxviii

Fig. 4.8 Fig. 4.9 Fig. 4.10 Fig. 4.11

Fig. 4.12

Fig. 4.13

Fig. 4.14

Fig. 4.15

Fig. 4.16

Fig. 4.17

List of Figures

The positive directions of the three-phase currents of PW, load and LSC (i1abc , ilabc and isabc ) . . . . . . . . . . . . . . . . . . . . . . . . . Compensation strategy based on LSC . . . . . . . . . . . . . . . . . . . . . . Cooperative compensation strategy based on dual power converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental results of the control strategy without compensation at the sub-synchronous speed of 900 rpm: a PW voltage amplitude, b CW d-axis current, c CW q-axis current, d PW line voltage, e CW phase current, f Load phase current, g dc bus voltage . . . . . . . . . . . . . . . . . . . . . . . Experimental results of the control strategy based on the MSC compensation at the sub-synchronous speed of 900 rpm: a PW voltage amplitude, b CW d-axis current, c CW q-axis current, d PW line voltage, e CW phase current, f Load phase current, g dc bus voltage . . . . . . . . . . . . . . Experimental results of the control strategy based on the LSC compensation at the sub-synchronous speed of 900 rpm: a PW voltage amplitude, b CW d-axis current, c CW q-axis current, d PW line voltage, e CW phase current, f Load phase current, g dc bus voltage . . . . . . . . . . . . . . Experimental results of the control strategy based on the dual-converter cooperative compensation at the sub-synchronous speed of 900 rpm: a PW voltage amplitude, b CW d-axis current, c CW q-axis current, d PW line voltage, e CW phase current, f Load phase current, g dc bus voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental results of the control strategy without compensation at the super-synchronous speed of 1100 rpm: a PW voltage amplitude, b CW d-axis current, c CW q-axis current, d PW line voltage, e CW phase current, f Load phase current, g dc bus voltage . . . . . . . . . Experimental results of the control strategy based on the MSC compensation at the super-synchronous speed of 1100 rpm: a PW voltage amplitude, b CW d-axis current, c CW q-axis current, d PW line voltage, e CW phase current, f Load phase current, g dc bus voltage . . . . . . . . . Experimental results of the control strategy based on the LSC compensation at the super-synchronous speed of 1100 rpm: a PW voltage amplitude, b CW d-axis current, c CW q-axis current, d PW line voltage, e CW phase current, f Load phase current, g dc bus voltage . . . . . . . . .

176 177 178

180

182

184

187

190

192

195

List of Figures

Fig. 4.18

Fig. 5.1 Fig. 5.2 Fig. 5.3

Fig. 5.4

Fig. 5.5

Fig. 5.6

Fig. 5.7

Fig. 5.8

Fig. 5.9

Fig. 5.10

Experimental results of the control strategy based on the dual-converter cooperative compensation at the super-synchronous speed of 1100 rpm: a PW voltage amplitude, b CW d-axis current, c CW q-axis current, d PW line voltage, e CW phase current, f Load phase current, g dc bus voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The flow chart of the MPCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The overall control scheme of the standalone BDFIG . . . . . . . . . Simulation results at 900 rpm with the load added at 1.5 s: a dq-axis CW currents with the MPCC, b dq-axis CW currents with the PI controller, c Amplitude of the PW phase voltage (the upper figure is under the PI control and the lower one is under the MPCC), d Frequency of the PW voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results under the speed change from 600 to 900 rpm with the constant load: a dq-axis CW currents with the MPCC, b dq-axis CW currents with the PI controller, c Amplitude of the PW phase voltage (the upper figure is under the PI control and the lower one is under the MPCC), d Frequency of the PW voltage, e Three-phase CW current (the upper figure is under the PI controller and the lower one is under the MPCC) . . . . . . . . . . . . . The value of each term in (5.18) with various operating states: a Value of A2 i 2d (k − 2), b Value of A1 i 1d (k − 2), c Value of u 2d (k − 2), d Detailed view of u 2d (k − 2) between 2.104 and 2.106 s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The NPCC strategy for the CW current control: a Whole control system, b Main procedures of the NPCC implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The minimum value of u 2d (k − 2) at different rotor speeds: a CW frequency 30 Hz at 1200 rpm, b CW frequency 10 Hz at 600 or 900 rpm, c CW frequency 1 Hz at 735 or 765 rpm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results under the variable load at the rotor speed of 600 rpm: a PW voltage, b CW current, c Reference and actual values of dq-axis CW currents . . . . . . . . . . . . . . . . . . . Simulation results under the variable load at the rotor speed of 900 rpm: a PW voltage, b CW current, c Reference and actual values of dq-axis CW currents . . . . . . . . . . . . . . . . . . . Simulation results under the constant load and variable rotor speed: a PW voltage, b CW current, c Reference and actual values of dq-axis CW currents . . . . . . . . . . . . . . . . . . .

xxix

198 209 209

211

213

216

219

222

224

225

226

xxx

Fig. 5.11

Fig. 5.12

Fig. 5.13

Fig. 5.14

Fig. 5.15

Fig. 5.16

Fig. 5.17

Fig. 5.18

Fig. 5.19

Fig. 5.20

Fig. 6.1 Fig. 6.2

List of Figures

Simulation results under the CW selfand mutual-inductances change from 100 to 80% at the rotor speed of 600 rpm: a Three-phase CW current with MPCC, b Reference and actual values of dq-axis CW currents with MPCC, c Three-phase CW current with NPCC, d Reference and actual values of dq-axis CW currents with NPCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results under PW self- and mutual-inductances change from 100 to 80% at the rotor speed of 600 rpm: a Three-phase CW current with MPCC, b Reference and actual values of dq-axis CW currents with MPCC, c Three-phase CW current with NPCC, d Reference and actual values of dq-axis CW currents with NPCC . . . . . . . . . Experimental results under the variable load at the rotor speed of 600 rpm: a PW voltage, b CW current, c Reference and feedback values of the d-axis CW current, d PW current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental results under the constant load and variable rotor speed: a PW voltage, b CW current, c Reference and feedback values of the d-axis CW current . . . . . . . . . . . . . . . Experimental results under the CW resistance change with MPCC: a Reference and feedback values of the d-axis CW current, b CW current, c PW voltage . . . . . . . . . . . . . . . . . . . Experimental results under the CW resistance change with NPCC: a Reference and feedback values of the d-axis CW current, b CW current, c PW voltage . . . . . . . . . . . . . . . . . . . Experimental results under the CW self-inductance change with MPCC: a Reference and feedback values of the d-axis CW current, b CW current, c PW voltage . . . . . . . . . . . . . . . . . . . Experimental results under the CW self-inductance change wiht NPCC: a Reference and feedback values of the d-axis CW current, b CW current, c PW voltage . . . . . . . . . . . . . . . . . . . Experimental results under the generator overload with MPCC: a Reference and feedback values of the d-axis CW current, b CW current, c PW current, d PW voltage . . . . . . . Experimental results under the generator overload with NPCC: a Reference and feedback values of the d-axis CW current, b CW current, c PW current, d PW voltage . . . . . . . Phase-axis relationship of the BDFIG for the sensorless control based on RPO_1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Main block diagram of the sensorless control method based on the RPO_1 for the standalone BDFIG . . . . . . . . . . . . . .

227

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232

233

234

235

236

238

239 243 244

List of Figures

Fig. 6.3

Fig. 6.4 Fig. 6.5 Fig. 6.6 Fig. 6.7

Fig. 6.8

Fig. 6.9

Fig. 6.10

Fig. 6.11

Fig. 6.12

Simulation results of the sensorless control of the standalone BDFIG based on the RPO_1 under speed variation (started with 900 rpm and then decelerated to 600 rpm) followed by the load change condition (started with 11.6 kW and then reduced to 9.7 kW): a Estimated and actual rotor positions, b three-phase PW voltages, c PW voltage amplitude and frequency, d CW dq-axis currents, e three-phase PW and CW currents . . . . . . . . . . . . . . . . Phase-axis relationship of the BDFIG for the sensorless control based on RPO_2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Main block diagram of the sensorless control method based on the RPO_2 for the standalone BDFIG . . . . . . . . . . . . . . Flowchart of the RPO_2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relationship of the phase-axes for the BDFIG with the fictitious frame and the consideration of the estimated-position error . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results of the sensorless control of the standalone BDFIG based on the RPO_2: a Estimated and actual rotor positions, b PW voltage amplitude and frequency, c CW dq-axis currents, d three-phase PW and CW currents . . . . . . . . . Performance test of the observer under 130% change in the machine inductances (L 1 ,L 1r ,L r ): a Estimated and actual rotor positions, b PW voltage amplitude and frequency, c CW dq-axis currents, d three-phase PW and CW currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Performance test of the proposed observer under 130% change in PW resistance: a Estimated and actual rotor positions, b PW voltage amplitude and frequency, c CW dq-axis currents, d three-phase PW and CW currents . . . . . . . . . Experimental results of the RPO_2 under the start-up operation: a Estimated and actual rotor positions at start up, b estimated and actual rotor positions at steady state, c rotor position error, d CW q-axis current, e PW phase voltage, f extended view of e, g PW phase current, h extended view of g, i CW phase current, j extended view of i at 600 rpm . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental results of the RPO_2 under the load variation (50% reduction): a Estimated and actual rotor positions under load change, b PW phase voltage, c PW phase current, d CW phase current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xxxi

247 250 250 253

253

256

259

261

263

266

xxxii

Fig. 6.13

Fig. 6.14

Fig. 6.15 Fig. 6.16 Fig. 6.17

Fig. 6.18

Fig. 6.19

Fig. 6.20

Fig. 6.21

List of Figures

Experimental results of the RPO_2 under the speed change (600–700 rpm): a Estimated and actual rotor positions (at 600 rpm), b estimated and actual rotor positions (at 700 rpm), c rotor position error, d CW q-axis current, e PW phase voltage, f extended view of e, g PW phase current, h extended view of g, i CW phase current, j extended view of i at 700 rpm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental results of the RPO_2 under the BDFIG parameter change (130% uncertainty): a Estimated and actual rotor positions, b PW phase voltage, c PW phase current, d CW phase current . . . . . . . . . . . . . . . . . . . . . . . . Structure of the basic RSO [12] . . . . . . . . . . . . . . . . . . . . . . . . . . . The whole control structure of the improved RSO, where PSC denotes the positive-sequence calculator . . . . . . . . . . The characteristics with magnitude–frequency of the presented LPF where the frequency is represented by a normalized scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental results of basic and improved RSOs the under the unbalanced resistive load (25, 100, and 100

in phases a, b, and c): a PW voltage (1 p.u. = 500 V) and CW current (1 p.u. = 50 A), b expanded view of a, c rotor speed estimated by the basic RSO, d speed error using the basic RSO, e rotor speed estimated by the improved RSO, f speed error using the improved RSO . . . . . . . . . . . . . . . . Experimental results of basic and improved RSOs under the nonlinear load (a diode-rectifier with a 25

resistor at the dc side): a PW voltage (1 p.u. = 500 V) and CW current (1 p.u. = 50 A), b expanded view of a, c rotor speed estimated by the basic RSO, d speed error using the basic RSO, e rotor speed estimated by the improved RSO, f speed error using the improved RSO . . . . . . . . . . . . . . . . Experimental results of basic and improved RSOs under both the unbalanced and nonlinear loads: a PW voltage (1 p.u. = 500 V) and CW current (1 p.u. = 50 A), b expanded view of a, c rotor speed estimated by the basic RSO, d speed error using the basic RSO, e rotor speed estimated by the improved RSO, f speed error using the improved RSO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The expanded view and the harmonic spectrum of the estimated rotor speed under the unbalanced load: a Expanded view of the rotor speed observed by basic RSO, b harmonic spectrum of the rotor speed observed by basic RSO, c expanded view of the rotor speed observed by improved RSO, d harmonic spectrum of the rotor speed observed by improved RSO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

267

270 272 277

280

282

283

284

285

List of Figures

Fig. 6.22

Fig. 6.23

Fig. 7.1 Fig. 7.2 Fig. 7.3 Fig. 7.4 Fig. 7.5

Fig. 7.6

Fig. 7.7

Fig. 7.8

The expanded view and the harmonic spectrum of the estimated rotor speed under the nonlinear load: a Expanded view of the rotor speed observed by basic RSO, b harmonic spectrum of the rotor speed observed by basic RSO, c expanded view of the rotor speed observed by improved RSO, d harmonic spectrum of the rotor speed observed by improved RSO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The expanded view and the harmonic spectrum of the estimated rotor speed under both unbalanced and nonlinear loads: a Expanded view of the rotor speed observed by basic RSO, b harmonic spectrum of the rotor speed observed by basic RSO, c expanded view of the rotor speed observed by improved RSO, d harmonic spectrum of the rotor speed observed by improved RSO . . . . . . . . . . . . . . . Basic MRAS observer structure . . . . . . . . . . . . . . . . . . . . . . . . . . . Relationship of the BDFIG phase-axis for the sensorless control strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Block diagram of the proposed sensorless control strategy for the adopted BDFIG system . . . . . . . . . . . . . . . . . . . . . . . . . . . . Main block diagram of the adopted CW power factor MRAS observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Performance test of the proposed sensorless control strategy based on CW power factor MRAS observer: a Actual and estimated rotor positions. b PW three-phase voltage and PW frequency. c PW dq-axis flux. d Three-phase PW and CW currents . . . . . . . . . . . . . . . . . . . . . . . . . The proposed sensorless system based on the CW power factor MRAS observer under load change: a Actual and estimated rotor positions. b PW three-phase voltage and PW frequency. c PW dq-axis flux. d Three-phase PW and CW currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The proposed sensorless system based on the CW power factor MRAS observer under the increased CW resistance (1.3 R2 ): a Actual and estimated rotor positions. b PW three-phase voltage and PW frequency. c PW dq-axis flux. d Three-phase PW and CW currents . . . . . . . . . . . . . . . . . . . . . . . Experimental results of the CW power factor-based rotor position observer under the start-up operation and speed variation: a Actual and estimated rotor positions at 600 rpm. b Actual and estimated rotor positions at 700 rpm. c PW phase voltage. d CW phase current at 600 rpm. e CW phase current at 700 rpm. f PW phase current . . . . . . . . . . . . . . .

xxxiii

286

287 290 291 292 293

295

296

298

300

xxxiv

Fig. 7.9

Fig. 7.10

Fig. 7.11 Fig. 7.12

Fig. 7.13 Fig. 7.14

Fig. 7.15

Fig. 7.16

Fig. 7.17

Fig. 7.18

List of Figures

Experimental results of the CW power factor-based rotor position observer under the load change condition: a Actual and estimated rotor positions. b PW phase voltage. c PW phase current. d CW phase current . . . . . . . . . . . . . . . . . . . Experimental results of the CW power factor-based rotor position observer under the case of BDFIG parameter change (130% uncertainty): a Actual and estimated rotor positions. b PW phase voltage. c PW phase current. d CW phase current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure of the αβ-axis PW flux MRAS observer . . . . . . . . . . . . Structure of the proposed sensorless control method based on the αβ-axis PW flux MRAS observer for the standalone BDFIG system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure of the dq-axis PW flux MRAS observer . . . . . . . . . . . . Structure of the proposed sensorless control method based dq-axis PW flux MRAS observer for the standalone BDFIG system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results under the speed ramp change and the constant load with the αβ-axis PW flux based MRAS sensorless control: a Actual and estimated speeds. b PW three-phase voltages. c PW dq-axis voltage. d PW three-phase currents. e CW three-phase currents. f PW αβ-axis flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results under the load variation and the constant speed with the αβ-axis PW flux based MRAS sensorless control: a Actual and estimated speeds. b PW three-phase voltages. c Detailed PW three-phase voltages between 0.9 and 1.1 s. d PW dq-axis voltage. e PW three-phase currents. f CW three-phase currents. g PW αβ-axis flux . . . . . . . Simulation results under 130% variation in the PW resistance with the αβ-axis PW flux based MRAS sensorless control: a Actual and estimated rotor speeds. b PW three-phase voltages. c Detailed view of PW three-phase voltages. d dq-axis PW voltage. e PW three-phase currents. f CW three-phase currents. g αβ-axis PW flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results under 150% variation in the whole inductance with the αβ-axis PW flux based MRAS sensorless control: a Actual and estimated rotor speeds. b PW three-phase voltages. c Detailed view of PW three-phase voltages. d dq-axis PW voltage. e PW three-phase currents. f CW three-phase currents. g αβ-axis PW flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

302

304 305

306 308

308

311

314

317

319

List of Figures

Fig. 7.19

Fig. 7.20

Fig. 7.21

Fig. 7.22

Fig. 7.23

Fig. 7.24

Fig. 7.25

Simulation results under the speed ramp change and the constant load with the dq-axis PW flux based MRAS sensorless control: a Actual and estimated speeds. b PW three-phase voltages. c Detailed PW three-phase voltages. d CW three-phase currents. e PW three-phase currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results under the load variation and the constant speed with the dq-axis PW flux based MRAS sensorless control: a Actual and estimated speeds. b PW three-phase voltages. c Detailed PW three-phase voltages between 0.9 and 1.1 s. d PW three-phase currents. e Detailed PW three-phase currents between 0.9 and 1.1 s. f CW three-phase currents. g dq-axis PW flux . . . . . . . . . . . . . . . . . . . . Simulation results under 130% variation in the PW resistance with the dq-axis PW flux based MRAS sensorless control: a Actual and estimated speeds. b PW three-phase voltages. c Detailed PW three-phase voltages between 0.9 and 1.1 s. d PW three-phase currents. e Detailed PW three-phase currents between 0.9 and 1.1 s. f CW three-phase currents. g dq-axis PW flux . . . . . . . . . . . . . . . . Simulation results under 150% variation in the whole inductance with the dq-axis PW flux based MRAS sensorless control: a Actual and estimated speeds. b PW three-phase voltages. c Detailed PW three-phase voltages between 0.9 and 1.1 s. d PW three-phase currents. e Detailed PW three-phase currents between 0.9 and 1.1 s. f CW three-phase currents. g dq-axis PW flux . . . . . . . . . . . . . . . . Experimental results under variable speed (from 700 to 600 rpm) with the dq-axis PW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b Rotor speed error. c PW phase voltage. d Overall CW phase current. e Detailed CW phase current between 2 and 3 s. f Detailed CW phase current between 43 and 44 s . . . . . Experimental results under variable load at the rotor speed of 600 rpm under the dq-axis PW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b Rotor speed error. c Overall PW phase voltage. d Detailed PW phase voltage between 3.55 and 3.68 s. e Overall CW phase current. f Detailed CW phase current between 0 and 1.8 s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental results under variable speed and variable load with 1.5 L variation under the dq-axis PW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b Rotor speed error. c PW phase voltage. d CW phase current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xxxv

320

322

325

328

330

331

333

xxxvi

Fig. 7.26

Fig. 7.27

Fig. 7.28 Fig. 7.29

Fig. 7.30 Fig. 7.31

Fig. 7.32

Fig. 7.33

Fig. 7.34

List of Figures

Experimental results under PW resistance variation (1.3 R1) and the speed of 600 rpm with 1.5 L variation under the dq-axis PW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b Rotor speed error. c Overall PW phase voltage. d Detailed PW phase voltage between 3.2 and 3.35 s. e Overall CW phase current. f Detailed CW phase voltage between 6 and 8 s . . . . . . . Experimental results under the speed change from 700 to 600 rpm and load change from 50 to 25

under the dq-axis PW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b Rotor speed error. c PW phase voltage. d Overall CW phase current. e Detailed CW phase current between 1 and 2 s. f Detailed CW phase current between 49 and 50 s . . . . . . . . . . . . . Structure of the αβ-axis CW flux MRAS observer . . . . . . . . . . . . Structure of the proposed sensorless control method based on the αβ-axis CW flux MRAS observer for the standalone BDFIG system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure of the dq-axis CW flux based MRAS observer . . . . . . . Structure of the proposed sensorless control method based on the dq-axis CW flux MRAS observer for the standalone BDFIG system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results under the speed ramp change and the constant load with the αβ-axis CW flux based MRAS sensorless control strategy: a Actual and estimated speeds. b PW three-phase voltage. c dq-axis PW voltage. d CW three-phase current. e PW three-phase current. f αβ-axis CW flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results under the load variation and the constant speed with the αβ-axis CW flux based MRAS sensorless control strategy: a Actual and estimated speeds. b PW three-phase voltage. c Detailed PW three-phase voltage between 0.9 and 1.1 s. d dq-axis PW voltage. e CW three-phase current. f PW three-phase current. g αβ-axis CW flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results under 130% variation in the PW resistance with the αβ-axis CW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b PW three-phase voltage. c Detailed view of PW three-phase voltage. d dq-axis PW voltage. e CW three-phase current. f PW three-phase current. g αβ-axis CW flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

335

336 337

338 342

342

346

348

350

List of Figures

Fig. 7.35

Fig. 7.36

Fig. 7.37

Fig. 7.38

Fig. 7.39

Fig. 7.40

Simulation results under 150% variation in the whole inductance with the αβ-axis CW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b PW three-phase voltage. c Detailed view of PW three-phase voltage. d dq-axis PW voltage. e CW three-phase current. f PW three-phase current. g αβ-axis CW flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results under the speed ramp change and the constant load with the dq-axis CW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b PW three-phase voltage. c Detailed PW three-phase voltage between 0.5 and 0.56 s. d Detailed PW three-phase voltage between 2 and 2.06 s. e CW three-phase current. f dq-axis CW flux . . . . . . . . . . . . . . . . . . . . . Simulation results under load change at the constant rotor speed with the dq-axis CW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b PW three-phase voltage. c Detail view of PW three-phase voltage. d CW three-phase current. e Detail view of CW three-phase current. f PW active power. g dq-axis CW flux . . . . Simulation results under 130% variation in the PW resistance with the dq-axis CW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b PW three-phase voltage. c detailed view of PW three-phase voltage. d PW three-phase current. e Detailed view of PW three-phase current. f PW active power. g dq-axis CW flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results under 150% variation in the whole inductance with the dq-axis CW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b PW three-phase voltage. c Detailed view of PW three-phase current. d CW three-phase current. e Detailed view of CW three-phase current. f PW active power. g dq-axis CW flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental results under the variable speed from 700 to 600 rpm with the dq-axis CW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b Speed percentage error. c PW phase voltage. d Overall CW phase current. e Detailed CW phase current between 1 and 2 s. f Detailed CW phase current between 16 and 17 s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xxxvii

352

354

356

358

360

362

xxxviii

Fig. 7.41

Fig. 7.42

Fig. 7.43

Fig. 7.44

Fig. 7.45

Fig. A.1 Fig. A.2

Fig. A.3 Fig. A.4 Fig. A.5 Fig. A.6 Fig. A.7

List of Figures

Experimental results under variable load at the rotor speed of 600 rpm with the dq-axis CW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b Speed percentage error. c Overall PW phase voltage. d Detailed PW phase voltage between 4 and 4.45 s. e Overall CW phase current. f Detailed PW phase current between 3.5 and 5 s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental results under variable speed, load and inductance with the dq-axis CW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b Speed percentage error. c PW phase voltage. d CW phase current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental results under PW resistance variation at the speed of 700 rpm with the dq-axis CW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b Speed percentage error. c Overall PW phase voltage. d Detailed PW phase voltage between 3.8 and 4.2 s. e Overall PW phase current. f Detailed PW phase current between 1.5 and 2 s . . . . . . . . . . . . . . . . . . . . . . . . . Experimental results under inductance variation at the speed of 700 rpm with the dq-axis CW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b Speed percentage error. c Overall PW phase voltage. d Detailed PW phase voltage between 2.8 and 3.5 s. e Overall PW phase current. f Detailed PW phase current between 2.9 and 3.5 s . . . . . . . . . . . . . . . . . . . . . . . . Experimental results under the changed speed, constant load and varied inductance with the dq-axis CW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b Speed percentage error. c PW phase voltage. d Overall CW phase current. e Detailed CW phase current between 0 and 5 s. f Detailed CW phase current between 45 and 50 s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure of the used BDFIG experimental platforms . . . . . . . . . Photograph of the 60-kVA BDFIG experimental platform (used as the ship shaft power generation system in a container vessel) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Photograph of the first 30-kVA BDFIG experimental platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Photograph of the second 30-kVA BDFIG experimental platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Photograph of the 90-kVA BDFIG experimental platform . . . . . . Photograph of the 3-kVA BDFIG experimental platform . . . . . . . Photograph of the 5-kVA BDFIG experimental platform . . . . . . .

364

366

367

369

371 380

381 382 382 383 384 385

List of Tables

Table 1.1 Table 1.2 Table 1.3 Table 1.4 Table 1.5 Table 1.6 Table 3.1 Table 3.2 Table 3.3 Table 3.4 Table 3.5 Table 4.1 Table 5.1 Table 6.1 Table A.1 Table A.2 Table A.3 Table A.4 Table A.5

Comparison of strengths and weaknesses between DFIG and BDFG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classification of SSF-VC methods according to utilized synchronous frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classification of SSF-VC methods based on the stator synchronous frame according to applicable machine types . . . . Classification of SSF-VC methods according to frame orientation styles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison among compensation methods based on MSC, LSC, and dual power converters . . . . . . . . . . . . . . . . . . Comparison of different sensorless control methods . . . . . . . . . . Frequencies in the rotor and stator of BDFIG with unbalanced loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequencies in the rotor and stator of BDFIG with nonlinear loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequencies in the rotor and stator of BDFIG with the unbalanced plus nonlinear load . . . . . . . . . . . . . . . . . . . Frequencies in the rotor and stator of BDFIG with single-phase nonlinear loads . . . . . . . . . . . . . . . . . . . . . . . . . The proportional and integral gains tuned by the Ziegler-Nichols strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of the amplitude drop and settling time of the PW voltage with different compensation strategies . . . . . Values of uα and uβ at various IGBT switch states of MSC . . . . Control parameters and settling time for the improved RSO . . . Main parameters of the 60-kVA BDFIG . . . . . . . . . . . . . . . . . . . Main parameters of the 30-kVA BDFIG . . . . . . . . . . . . . . . . . . . Main parameters of the 90-kVA BDFIG . . . . . . . . . . . . . . . . . . . Main parameters of the 3-kVA BDFIG . . . . . . . . . . . . . . . . . . . . Main parameters of the 5-kVA BDFIG . . . . . . . . . . . . . . . . . . . .

2 6 6 8 27 33 61 63 66 69 76 200 221 281 380 381 383 384 385

xxxix

Chapter 1

Introduction

Abstract The brushless doubly-fed generator (BDFG) is a kind of dual-electricalport field-modulated electrical machine containing two sets of stator windings with different pole pairs. The stator power winding undertakes most of the power inflow and outflow, and the stator control winding connected to the power converter can be utilized to regulate the machine operation status. Consequently, the rotor no longer needs to be supplied by the power converter, so that the brushes and slip rings can be removed. In comparison with its counterpart doubly-fed induction generator (DFIG), the BDFG is with longer lifespan, higher reliability and lower maintenance cost. The BDFG can be used for grid-connected power generation as well as standalone power generation. This chapter surveys recent advances of control technologies for BDFGs under different operation conditions, e.g., grid-connected power generation with normal and faulty grids and standalone power generation with normal and special loads. Besides, the progresses of sensorless control technologies for BDFG-based power generation systems are also discussed. The classification and comparison are carried out to discover the similarities and differences between theses control technologies. Keywords Brushless doubly-fed generator (BDFG) · Grid-connected power generation · Standalone power generation · Sensorless control

1.1 Introduction to Brushless Doubly-Fed Generators The high reliability structure gives the superiority to the brushless doubly-fed generator (BDFG) in the adjustable drive systems. The fixed stator frame of the BDFG contains two windings separated and isolated with different number of poles [1]. One of the two windings is specified as the main winding of the machine, which is denoted as the power winding (PW). The other winding is for control purpose and known as the control winding (CW). The two stator windings are with different pair poles to prevent the direct coupling between them. The role of the rotor is to couple the two fixed windings through indirect magnetic coupling. Unlike the conventional doubly-fed induction generator (DFIG), the rotor no longer needs to be supplied by

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Y. Liu and W. Xu, Advances in Control Technologies for Brushless Doubly-fed Induction Generators, https://doi.org/10.1007/978-981-19-0424-0_1

1

2 Table 1.1 Comparison of strengths and weaknesses between DFIG and BDFG

1 Introduction Metrics of interest

DFIG

BDFG

Size and mass

Small (+)

Large (−)

Efficiency

High (+)

Low (−)

Torque density

High (+)

Low (−)

Mechanical reliability

Low (−)

High (+)

Maintenance cost

High (−)

Low (+)

Manufacture cost

Low (+)

High (−)

LVRT capability

Low (−)

High (+)

the power converter, so that the brushes and slip rings associated with the rotor can be removed [2]. A comparison between the DFIG and BDFG is depicted in Table 1.1 to state the typical strengths and weaknesses of the two kinds of generators. The symbol “ + ” indicates the strength and the symbol “-” stands for the weakness. From Table 1.1, the BDFG is worse than the DFIG in terms of size and mass, efficiency, torque density and manufacture cost. Fortunately, the BDFG’s weaknesses can be compensated by its higher mechanical reliability, lower maintenance cost, and stronger low-voltage ride-through (LVRT) capability, due to the absence of brushes or slip-rings and the improved performance under faulty grids. These strengths of the BDFG are very important for practical industrial applications. The current BDFGs can be divided into two main categories, i.e., the single-stator BDFGs and the dual-stator BDFG. The single-stator BDFGs mainly include the brushless doubly-fed induction generator (BDFIG) with nested-loop rotor [3, 4], the brushless doubly-fed reluctance generator (BDFRG) [5, 6], the BDFIG with wound rotor [7], and the BDFG with hybrid rotor [8]. The dual-stator BDFG mainly refers to the dual-stator BDFIG (DSBDFIG) proposed in [9]. The topology of the BDFG-based power generation system can be seen in Fig. 1.1.The employed power converter is composed of two back-to-back voltage source converters with a common dc bus, i.e., the machine side converter (MSC) and Fig. 1.1 The topology of the BDFG-based ac power generation system

BDFG PW

CW Prime mover

Rotor p2 pole pairs Stator p1 pole pairs

MSC

dc bus

LSC Filter

Power converter

ac load or ac grid

1.1 Introduction to Brushless Doubly-Fed Generators

3 Wind turbine

20 kW BDFIG

Fig. 1.2 The BDFIG application in wind power generation [12]

the line side converter (LSC). This topology can be used for both grid-connected and standalone power generation. The difference between the grid-connected and standalone power generation systems is that the point of common coupling (PCC) of the grid-connected system is connected to grid, while the PCC of the standalone system is connected to loads. The grid-connected BDFG-based power generation system can be applied to variable speed constant frequency (VSCF) wind power and hydropower generation. The aim of this system is to regulate the active and reactive power of PW, which can be achieved by MSC. The LSC is employed to maintain the dc bus voltage constant regardless of power flow direction through MSC. Also, the LSC can be utilized to assist the fault ride-through under faulty grids. The application scenarios of the standalone power generation system include wind power and hydro power in remote locations, portable variable speed power generation, ship shaft power generation, and so on. Unlike the grid-connected BDFG system, the control objective of the standalone BDFG is to make the amplitude and frequency of PW voltage maintain constant under variable rotor speeds and loads. In the standalone system, the CW should be controlled by the MSC to build up a constant PW voltage to support the loads [10]. The main role of LSC is to keep the dc bus voltage stable, which is similar to the grid-connected system. The LSC can also assist in stabilizing the voltage fluctuation at the point of common coupling by supplying or absorbing reactive current as discussed in [11]. The BDFG has been applied to wind power, ship shaft power and hydropower generation. According to the literature survey, in 2009, the BDFIG with nestedloop rotor was applied to wind power generation for the first time [12]. As shown

4

1 Introduction

in Fig. 1.2, the BDFIG is mechanically coupled to the wind turbine via a speedincreasing gearbox. The tested BDFIG is with the rated power of 20 kW, the PW pole pair of 2 and the CW pole pair of 4. With the proposed control algorithm in [12], the efficiency of the tested BDFIG can be up to 80%. The BDFIG can also be applied to the ship power system. In general, the power redundancies of commercial ships have a range of approximately 10%-15% from the main engine for shipping safety. During the cruise, the ship shaft generator can use this redundant power of the main engine for power generation, so that the efficiency of the main engine can be significantly improved and the use of diesel generator sets can be greatly reduced. As shown in Fig. 1.3, a typical ship shaft power generation system based on a 60-kW wound-rotor BDFIG has been tested in a 325 TEU container ship from the Changjiang National Shipping Group of China between 2011 and 2014, which can save 30% of the fuel cost of the ship at most [10]. A typical case of the BDFIG application in hydropower generation can be seen in [13]. As illustrated in Fig. 1.4, the wound-rotor BDFIG is mechanically coupled to a hydraulic turbine without gearbox. The applied BDFIG is with the rated power of 800 kW, and with the PW and CW rated voltages of 6.3 kV and 400 V, respectively.

Ship engine

Power converter 64 kW BDFIG Fig. 1.3 The BDFIG application in ship shaft power generation [10]

1.1 Introduction to Brushless Doubly-Fed Generators

5

6.3 kV high-voltage grid

PW

High-voltage switchgear

BDFIG

Turbine

CW

MSC

LSC

Filter

Transfer

Filter

Fig. 1.4 The BDFIG application in hydropower generation [12]

Hence, the PW can be directly connected the 6.3 kV high-voltage grid, and the CW has to be indirectly linked the same grid through the low-voltage back-to-back power converters and the transfer. Consequently, the high-voltage BDFIG can be regulated by the low-voltage power converters. Lots of control technologies for BDFG-based power generation systems have been developed toward industry applications in wind power, ship shaft power, hydropower, and so on [14]. This chapter will introduce recent advances of these control technologies, and investigate the similarities and differences between them through the classification and comparison. Since the cascade DFIG has the same operation principle with the BDFIG, its control technologies will be also introduced.

1.2 Advances in Control Technologies for Grid-Connected Power Generation 1.2.1 Grid-Connected Power Generation Under Normal Grid BDFGs show commercial promise for grid-connected power generation systems, due to their lower operational costs and higher reliability as compared with DFIGs. Many control strategies have been developed for BDFGs over the years for gridconnected applications, among which the control strategies under normal grid can be summarized as follows.

6

1 Introduction

1.2.1.1

Vector Control

The most practical and preferred technique for BDFGs is the vector control (VC). The vector control methods of BDFG, in general, are derived from the dynamic mathematical models of BDFG based on different rotating reference frames. At present, the rotating reference frames used for BDFG mathematical modeling mainly includes: the single synchronous frame, the dual synchronous frames, and the slipfrequency rotating frame. Hence, the current vector control methods of BDFG can be divided into three categories, i.e., the single-synchronous-frame vector control (SSFVC), the dual-synchronous-frame vector control (DSF-VC), and the slip-frequencyrotating-frame vector control (SFRF-VC). A.

Single-synchronous-frame Vector Control (SSF-VC)

According to the utilized synchronous frames, the SSF-VC methods can be divided into two main types, i.e., the VC methods based on the rotor synchronous frame and the ones based on the stator synchronous frame, as shown in Table 1.2. The rotor-flux-orientation VC methods have been developed in [15–17]. Although the control method proposed in [17] is for cascaded DFIGs, it can be directly applied to BDFIG. A vector model under the rotor synchronous frame for the BDFIG with nested-loop rotor has been established in [18]. Afterwards, based on the vector model developed in [18], a generalized VC strategy without rotor flux orientation has been proposed in [19], which only realizes the speed control, but does not achieve the adjustment of the reactive power. And, a hysteresis controller has been employed to regulate the CW current, which makes the CW current contain significant harmonics. Based on the stator synchronous frame, quite a few SSF-VC methods have been proposed for different BDFGs as depicted in Table 1.3, which will be described in details in the following paragraphs. In [20], the BDFRG dynamic model under the single synchronous frame has been proposed, and a VC strategy based on the PW flux orientation has been developed, which realizes the decoupling control of torque and reactive power. In [21] and [22], the PW-flux-orientation and PW-voltage-orientation VC strategies for BDFRG Table 1.2 Classification of SSF-VC methods according to utilized synchronous frames

Table 1.3 Classification of SSF-VC methods based on the stator synchronous frame according to applicable machine types

Utilized synchronous frames

SSF-VC methods

Rotor synchronous frame

[15–19]

Stator synchronous frame

[20–33]

Applicable machine types

SSF-VC methods

BDFRG

[20–23]

Cascade DFIG

[24–28]

BDFIG with nested-loop rotor

[29–31]

BDFIG with wound rotor

[32, 33]

1.2 Advances in Control Technologies for Grid-Connected …

7

have been proposed and compared. The difference between these two strategies is that the PW-flux-orientation VC strategies require the PW resistance to estimate the flux linkage, while the PW-voltage-orientation VC strategies can be implemented independently of machine parameters. However, the resulting problem is that the PWvoltage-orientation strategies is slightly worse than the PW-flux-orientation strategies in the response to reactive power. As the power rating of the BDFRG increases, the PW resistance would decrease, so that the performance difference between these two control strategies would also be smaller. In [23], a PW-voltage-orientation gridsynchronization strategy has been proposed for BDFRG to ensure that the PW voltage has the same phase-sequence, frequency and amplitude as the grid voltage. And, in this paper, two PW-voltage-orientation VC strategies have also been developed to achieve two different targets, i.e., unity power-factor operation and maximum electromagnetic torque per converter current. A VC method based on the stator flux orientation of the power machine has been proposed for cascaded DFIGs in [24]. It can basically realize the decoupling control of active and reactive power, although there is some cross coupling between the d- and q-axis control loops. Subsequently, the same authors as in [24] proposed a VC method based on the orientation of the virtual combined flux of the power machine and control machine stators in [25]. However, due to the serious cross coupling, the control method developed in [25] is not suitable for the decoupling control of active and reactive power, but it is feasible for the speed control. Another stator-flux-orientation VC method for cascaded DFIGs has been proposed in [26] with a feedforward compensation for the decoupling of the stator current control loop of the control machine. The disadvantage of this method is that it requires the knowledge of the inductances of the power and the control machines, as well as the estimation of the stator and rotor fluxes of the power machine, which would degrade the system robustness under the mismatch of machine parameters. In [27] and [28], the other two stator-flux-orientation VC methods have also been investigated, which are with different feedforward compensation terms from the method in [26]. These feedforward compensation terms are also related to machine parameters, so that the system robustness against machine parameter variation would be deteriorated. Ref. [29] proposed a vector model based on the unified reference frame for the BDFIG with nested-loop rotor, in which both of the machine parameters and variables are referred to the PW side. Based on the model developed in [29], a VC strategy based on the PW flux orientation has been investigated in [30]. However, it additionally introduces a PW current control loop, which makes the control parameter tuning complicated. According to the vector model proposed in [28], the more concise PWflux-orientation VC strategy has been studied in [31], which removes the CW current loop and can directly control the reactive power and speed by regulating the d- and q-axis voltages of CW, respectively. The potential problem of this method is that the CW is with the risk of transient overcurrent when the speed and reactive power commands change rapidly due to the CW current being not controlled. For the BDFIG with wound rotor, the grid-flux-orientation VC strategy has been proposed to implement the decoupling control of active and reactive power with feedforward compensation [32]. Different from the grid voltage orientation, the grid

8

1 Introduction

Table 1.4 Classification of SSF-VC methods according to frame orientation styles

Frame orientation styles

SSF-VC methods

Rotor flux orientation

[15–17]

PW (Power machine stator winding) flux orientation

[20–22, 24, 26–31, 33]

Virtual combined flux orientation

[25]

PW voltage orientation

[21–23]

Grid flux orientation

[32]

Without orientation

[18, 19]

flux orientation is achieved by q-axis of the synchronous frame being aligned with the grid voltage vector. Besides, the PW-flux-orientation VC strategy for the woundrotor BDFIG has also been investigated in [33], which is without any feedforward compensation terms at all. The frame orientation style is a key element for VC methods. Based on the above analysis, the SSF-VC methods can be classified into six main categories according to frame orientation styles, as depicted in Table 1.4. From Table 1.4, it can be seen that most of the research works focus on the PW-flux-orientation VC methods for BDFGs and the VC methods based on stator winding flux orientation of the power machine for the cascade DFIG. The main reason is that the VC methods based on the flux orientation of PW or power machine stator winding can better achieve the decoupling of active power and reactive power. B.

Dual-synchronous-frame Vector Control (DSF-VC)

The BDFG can be regarded as the combination of PW and CW subsystems, and the BDFG dynamic model in the dual synchronous frames has been established in [34]. The dual synchronous frames contain the PW subsystem synchronous frame and the CW subsystem synchronous frame. A few DSF-VC methods have been proposed for BDFIG [35, 36] and BDFRG [37, 38]. In [35] and [36], the rotor flux orientation of BDFIG is carried out in both PW and CW subsystems, so that the input and output quantities of the two subsystems in the steady state are no longer time-variable but constant. Consequently, the CW flux can be controlled by the d-axis CW current, and the electromagnetic torque can be controlled by the q-axis CW current. However, in comparison with the SSFVC methods, the DSF-VC methods is more complicated in the estimation of the electromagnetic torque. The reason is that the d- and q-axis CW currents should be transformed to the PW subsystem according to the angle difference between the two synchronous frames when calculating the electromagnetic torque. The dynamic model of BDFRG is simpler than that of BDFIG, since there are no voltage and flux equations in the dynamic model of BDFRG. In [37], the dynamic model of BDFRG has been expressed in two separate reference frames, which makes the model of BDFRG very similar to that of the double-excited slip ring induction machine (DESRIM). Consequently, the existing control methods

1.2 Advances in Control Technologies for Grid-Connected …

9

Fig. 1.5 The SFRF-VC method: a Structure of SFRF, b Obtained PW and CW current vectors by the SFRF-VC method proposed in [39] and [40]

for DESRIM can be essentially applied to BDFRG, although the operating principles of the two machines are fundamentally different. In [38], based on the dualsynchronous-frame dynamic model, a multiple-input multiple-output (MIMO) robust controller has been proposed for the BDFRG-based wind energy conversion system. The MIMO controller can achieve two objectives at the same time, i.e., the wind energy conversion maximization and the machine copper losses minimization. C.

Slip-frequency-rotating-frame Vector Control (SFRF-VC)

In general, the PW is established on the stationary frame A1 B1 C1 . Through the magnetic coupling of the rotor, the CW current can induce an air gap magnetic field with the same frequency as the PW current. As a result, the angular frequency of the CW current vector relative to the frame A1 B1 C1 can be regarded as the PW current frequency ω1 . Since the actual frequency of the CW current is ω2 , a CW frame A2 B2 C2 rotating at the slip frequency (ω1 -ω2 ) can be constructed, in which the angular frequency of the CW current vector is still ω2 , as shown in Fig. 1.5a [39]. Based on the constructed frames A1 B1 C1 and A2 B2 C2 , the BDFRG mathematical model similar to DFIG has been established, and the corresponding control strategy has been proposed in [39] and [40]. The proposed control strategy employs only the CW current for magnetic excitation, while the PW current is used entirely to generate electromagnetic torque, as shown in Fig. 1.5b. Consequently, the greater dynamic electromagnetic torque can be provided, and the dynamic response of system can be significantly improved.

1.2.1.2 A.

Direct Control

Direct Power Control

The direct power control (DPC) can directly decouple and separately control the active and reactive power. The DPC has a simpler algorithm and less calculation and does not need to observe the flux amplitude, which can well solve the problem of

10

1 Introduction

the bad real-time of control system caused by the flux observer being sensitive to generator parameter variations. The DPC has been proposed for cascade DFIG in [41] as an alternative to the VC scheme. Such strategy has fast dynamic response, simple implementation and robustness. The DPC provides direct regulation of the machine power by selecting proper voltage vectors from the lookup-tables. However, the converter switching frequency varies with operating conditions, which results in large power ripple and current distortion. To improve such shortcomings, while keeping the advantages of DPC over the VC, a super-twisting sliding mode direct power control (SSM-DPC) strategy has been proposed in [42] for the BDFIGs, as given in Fig. 1.6. The SSM-DPC strategy can control active and reactive power directly without the need of phase locked loop. Moreover, its transient performance is similar to the conventional DPC and its steady-state performance is the same as the vector control. The proposed controller in

Fig. 1.6 Structure of the proposed SSM-DPC strategy for BDFIG: a Schematic diagram of the super-twisting sliding mode control, b Overall DPC strategy with the super-twisting sliding mode control, where F and D are two matrices determined the time-derivative formula of sliding surface [42]

1.2 Advances in Control Technologies for Grid-Connected …

11

Fig. 1.7 DPC diagram of the open-winding BDFIG [43]

[42] is robust to uncertainties toward parameter variations and achieves the constant converter switching frequency by using space vector modulation. In [43], the DPC based on the twelve sections has been adopted to implement the power tracking of the open-winding BDFIG system as illustrated in Fig. 1.7. In comparison with the typical BDFIG DPC system based on the six and twelve sections, the advantages of the proposed scheme lie in lower converter capacity and cost, simpler control structure, more flexible control mode, and better operation performance and fault-tolerant ability. B.

Direct Torque Control

Direct torque control (DTC) has features of simple structure, fast dynamicresponse torque, good robustness and low-reliability on machine parameters, which solve the problems about complicated structure, large amount of calculation and sensitive to parameter change of vector control [44–51]. For the traditional DTC, hysteresis comparator cannot distinguish the size of torque and flux linkage error. Controller applies voltage vector to the whole control cycle. In the cycle with smaller torque error, the voltage makes the torque to achieve a given value quickly in a short time, and unchanged switch state of inverter makes torque along the original direction, then huge torque ripple occurred. To reject the torque ripple in DTC, scholars from all over the world did a lot of research and put forward many improvement methods, such as using the voltage space vector modulation (SVM–DTC) [45], the discrete space voltage vector [46], fuzzy control [47, 48], predictive control [49], duty ratio modulation [50], combination of predictive control and duty ratio modulation [51]. The SVM–DTC algorithm in [45] can effectively reduce the torque ripple but requires a large amount of calculation and more parameters. The discrete space vector modulation (DSVM) in [46] has the advantage of good robustness in direct vector control, but the improvement of control accuracy is based on the premise of breakdown voltage vector which increased the complexity of the control system.

12

1 Introduction

Fig. 1.8 The block diagram of the duty ratio modulation DTC for BDFIG proposed in [51]

In [47], a fuzzy control algorithm with a good dynamic performance is presented, which can effectively reduce the torque ripple. But in the state machine, variable membership has uncertainty; if membership choice is not appropriate, the system performance will be deteriorated. In [48], the improved fuzzy DSVM–DTC control method is used to strengthen the control of torque and speed, but the control rules are very complex due to the fuzzy controller being with five inputs. The torque prediction is used in [49], which reduces torque ripple and improves the waveform of stator current, but needs a little huge computation. In [50], the DTC method based on duty ratio modulation is studied, in which the non-zero voltage vector only acts partial time in one control cycle and zero voltage vector acts on the rest time. In [51], a method of combining predictive control and duty ratio modulation has been proposed, as given in Fig. 1.8, to reduce the torque ripple. This control method can reduce the torque ripple and keep the advantages of fast torque response and simple control of the traditional DTC.

1.2.1.3

Indirect Control

The indirect stator-quantities control (ISC) is a new control strategy for BDFIG [52–54]. Compared to VC and DTC control schemes, there are some remarkable advantages to the ISC, such as not requiring rotating coordinate transformations and flux linkage orientation, less sensitivity to machine parameters, simpler system structure and the capability of current-limit. In [53], the control structure of the ISC strategy has been demonstrated by means of theoretical derivation as well as experimental results. The dynamic performance of the ISC controller has been tested over wide range of speeds from sub-synchronous to super-synchronous, and also under the conditions of sudden load change. The study in [54] has further developed the dynamic control of reactive power for the BDFIG with the ISC scheme shown in Fig. 1.9. Detailed theoretical analysis has

1.2 Advances in Control Technologies for Grid-Connected …

13

Fig. 1.9 Block diagram of the indirect stator-quantities control system with the reactive power controller for BDFIG [54]

been done to show the controller structure of the reactive power. As a result, both speed and the reactive power can be controlled simultaneously. These control features are achieved in a static reference frame, and rotating coordinate transformations and flux linkage orientation are unnecessary, so the ISC is simpler to implement in practice.

1.2.2 Grid-Connected Power Generation Under Faulty Grid 1.2.2.1

Under Unbalanced Grid

A few literatures have studied the operation of BDFIG under unbalanced grid conditions [41, 55–57]. When the grid is unbalanced, the PW current is unbalanced, torque and power will oscillate at twice the PW frequency, and hence the CW current is also distorted. In [41], the active and reactive power compensation terms are added to the reference values of active and reactive power, and then the DPC strategy is employed to control the power with eliminating the negative-sequence components of the power machine stator currents. Different from the work in [41] based on DPC, all the studies in [55–57] based on the VC method can be summarized in Fig. 1.10. The study in [55] has proposed the dual proportional-integral (PI) vector control, regulating the positive-sequence CW current controller for active and reactive power generation, and the negative-sequence CW current controller for three different control targets. The process of voltage and current decomposition not only made the model and algorithm complex, but also introduced delay by band-trap filters. The control scheme proposed in [56] is with the similar structure as that in [55], although it is based on the dual synchronous rotating frames. In [57], an improved VC has been proposed for BDFIG under unbalanced grid

14

1 Introduction Refs. [54] and [55] CW compensation current PI controller

Active and reactive power PI controllers

Basic CW current PI controller

CW compensation current reference calculation

Cdc + +

Space vector Pulses PWM generator

MSC

CW compensation current reference calculation

CW BDFIG

++

CW current PIR controller PW

Ref. [56]

Grid Side

Fig. 1.10 Summarized control structures presented in [55–57] for the BDFIG under unbalanced grid with PI/PIR controllers, respectively

conditions by introducing proportional-integral-resonant (PIR) controller in single synchronous reference frame. Compared to the existed solutions, the algorithm as proposed in [57] is easy to implement, fast in transient response performance, and also robust under parameter variations. Furthermore, no negative-sequence voltage and current decomposition is required in [57], which would simplify the algorithm and provide fast dynamic response. It can be noted that BDFIG torque ripple caused by the unbalanced grid has not attracted enough attention, as few papers have discussed this topic expect [57]. However, the torque ripple should be the most concerned issue in wind power system for its degradation in gear box and reduction on life span. Hence, more studies on this topic are greatly desirable in the future.

1.2.2.2

Low-Voltage Ride-Through

The low-voltage ride-through (LVRT) capability indicates that grid-connected wind turbines must remain connected and supply reactive current to the grid during grid voltage dips. The LVRT is the most important indicator in many national grid codes [58–65]. The LVRT methods can be divided into two categories, i.e., based on the hardware and software, as shown in Fig. 1.11. The hardware mainly includes the crowbar and series dynamic resistor (SDR). The use of a crowbar or SDR enables the BDFIG to ride through the most serious voltage dips. However, some unavoidable problems are caused by these hardware devices, such as increasing system cost and reactive power consumption [58–60]. The software-based methods, i.e. the crowbarless LVRT control methods, can be implemented in the rotating frame and the static frame. To date, the crowbarless LVRT

1.2 Advances in Control Technologies for Grid-Connected …

Crowbar

Control methods with rotating frame

Series dynamic resistor (SDR)

Control methods with static frame

LVRT methods based on hardware

LVRT methods based on software

15

Fig. 1.11 Different kinds of LVRT methods for the grid-connected BDFIG systems [61–65]

control schemes for BDFIGs developed in [61–65] are all essentially implemented in the rotating reference frame. The control strategies proposed in [61] and [62] can compensate for the couplings between the two axes of the CW current, with different compensation terms. The simulation results in [61] and [62] show that the BDFIG cannot ride through severe low-voltage dips with the proposed control strategies. In [63], a crowbarless control strategy has been proposed to enable the ride-through of the BDFIG under the 100% symmetrical grid voltage dip, and only the forward sequence has been controlled in the LVRT mode. However, under the 100% symmetrical grid voltage dip, there is still a zero-sequence component in the PW flux linkage. The ignorance of the zerosequence component may result in unsatisfactory LVRT behavior. In [64], the authors have presented a simple control strategy for riding through asymmetrical faults. The experimental results in [63] and [64] indicate that when the BDFIG rotor speed is 96% of the maximum rotor speed (100% is the most severe), the CW current peak is approximately 2.42 p.u. of the BDFIG CW current rating or 2 p.u. of the transient IGBT current rating. As can be seen from these experimental results, the LVRT performance got by the hardware-based methods in [63] and [64] is a bit worse than that obtained by the software-based method in [58]. In [65], in order to achieve LVRT under symmetrical grid voltage dips, the integral part of the CW current PI controller is removed and the proportional gain is optimized in the LVRT mode, so that the order of the control system can be reduced, and the d- and q-axis currents of CW and PW can quickly converge. Consequently, the overcurrent and oscillation of the CW and PW currents can be avoided during symmetrical grid voltage dips. In [66], a crowbarless LVRT control strategy has been proposed in the static reference frame, which is based on the flux linkage tracking under symmetrical voltage dips, as shown in Fig. 1.12. To both suppress the CW current and increase PW reactive current, a time-sharing control and CW current-first control solution have been proposed. In the most severe case, where the BDFIG rotor speed is 100% of the

16

1 Introduction

Fig. 1.12 The block diagram of the crowbarless LVRT control strategy based on flux linkage tracking for BDFIG under symmetrical voltage dips proposed in [66]

maximum rotor speed and under 100% symmetrical voltage dips, the experimental results have shown that the CW current peak could be limited to 2 p.u. of the CW current rating and the torque ripple be small during the fault.

1.3 Advances in Control Technologies for Standalone Power Generation The key control loop for the standalone ac BDFG power generation system is the CW current control loop. The currently proposed CW current controllers can be divided into three main categories, i.e., PI controllers without decoupling, PI controllers with decoupling, and predictive controllers.

1.3.1 Standalone Power Generation with Normal Loads 1.3.1.1

PI Controllers for CW Current Without Decoupling

The relationship between CW current and CW voltage can be expressed as [10] i 2d = (R2 + σ2 L 2 s)u 2d + (α1 i 2q + α2 i 1d + α3 i 1q )    Cross - coupling disturbance

(1.1)

1.3 Advances in Control Technologies for Standalone Power Generation

17

Fig. 1.13 The stator-flux-oriented control and the direct voltage control based on CW current PI controller without decoupling [10, 67]

i 2q = (R2 + σ2 L 2 s)u 2q + (α4 i 2d + α5 i 1d + α6 i 1q )   

(1.2)

Cross - coupling disturbance

where σ2 = 1 − L 22r /(L 2 L r ) is the leakage constant of the CW, the coefficients α 1 , α 2 , α 3 , α 4 , α 5 and α 6 are related to machine parameters, PW frequency and rotor speed. From and, ignoring the cross-coupling disturbance, the structure of the CW current controller can be designed as a proportional-integral (PI) control loop as shown in Fig. 1.13. Both of the stator-flux-oriented control proposed in [67] and the direct voltage control proposed in [10] are based on the inner PI control loop for CW current as shown in Fig. 1.13. The difference between the stator-flux-oriented control and the direct voltage control is the outer control loop. In the stator-flux-oriented control, the PW voltage is indirectly regulated by adjusting the equivalent PW magnetizing current in the outer control loop, and the PW flux orientation is achieved by regulating the q-axis CW current proportional to the q-axis PW current. Consequently, the feedback value of the equivalent PW magnetizing current and the reference value of the q-axis CW current depend on machine parameters, such as the mutual- and selfinductances, which would degrade the robustness of the control system. In order to address this issue, the direct voltage control is proposed, where the outer control loop is used to adjust the amplitude of PW voltage directly. And, the reference value of the q-axis CW current in the direct voltage control is set to zero, which means that the direct voltage control method is based on the CW current orientation. In [10], the CW current control loop shown in Fig. 1.13 has been tested on a 60-kVA BDFIG with the natural synchronous speed of 500 rpm. The test is carried

1 Introduction

i2d, i2q (A)

i2d, i2q (A)

18 35 30 25 20 15 10 5 0 -5 0

35 30 25 20 15 10 5 0 -5 0

i2d-exp i2d-sim i2d-ref i2q-exp i2q-sim i2q-ref 0.1

0.2 Time (s) (a)

0.3

0.4

i2d-exp i2d-sim i2d-ref i2q-exp i2q-sim i2q-ref 0.1

0.2 Time (s) (b)

0.3

0.4

Fig. 1.14 Simulated and experimental results of the CW current PI controller without decoupling: a At the sub-synchronous speed of 400 rpm, b At the super-synchronous speed of 600 rpm [10]

out at the sub- and super-synchronous speeds, respectively. The reference value of the d-axis CW current is around 15% of the rated current, and that of the q-axis CW current is zero. From the simulated and experimental results shown in Fig. 1.14, it can be seen that the settling time of CW current is 50 and 70 ms at the sub- and super-synchronous speeds, respectively, with the d-axis CW current overshoot of around 7.5%. However, it is difficult to obtain a fast response speed for CW current with small overshoot based on the control method shown in Fig. 1.13. The main reason for this problem is that the d- and q-axis control channels of the CW current loop is severely coupled as shown in and, which cannot be ignored for obtaining fast response.

1.3.1.2

PI Controllers for CW Current with Decoupling

In order to improve the dynamic performance of the CW current controller, the CW current control loop with cross feedforward compensation is proposed in [68], as shown in Fig. 1.15. Besides, the d- and q-axis PW currents are also the inputs of the feedforward compensation terms, which can get further benefits from the proposed feedforward compensation method, i.e., suppression over load disturbances, and so on.

1.3 Advances in Control Technologies for Standalone Power Generation

19

Fig. 1.15 The CW current control loop with cross feedforward compensation proposed in [68]

The CW current control loop with cross feedforward compensation proposed in [68] has been tested on a 30 kVA BDFIG under the sub-synchronous speed of 600 rpm and the super-synchronous speed of 1200 rpm, respectively. Similar to the test shown in Fig. 1.14, the reference value of the d-axis CW current is also around 15% of the rated current, and that of the q-axis CW current is zero. As can be seen from Fig. 1.16, with cross feedforward compensation, the settling time of CW current can be reduced to 26 and 38 ms under the similar overshoot of d-axis CW current (around 7.5%) at the sub- and super-synchronous speeds, respectively. Hence, the control method proposed in [68] can significantly enhance the dynamic performance of the CW current control loop in comparison with that proposed in [10]. Also, the similar feedforward compensation method has been proposed in [69]. Besides, in [70], a decoupling network is developed to eliminate the influence of the cross-coupling disturbance in the CW current control loop, as shown in Fig. 1.17. The models of the BDFIG and the load are integrated to a dual-input-dual-output system, in which the inputs are u2d and u2q , and the outputs are i2d and i2q . And then, by selecting appropriate transfer functions L 1 (s) and L 2 (s), the d- and q-axis control channels of the CW current loop can be completely decoupled. In general, L 1 (s) and L 2 (s) can be designed as the fifth-order transfer functions. This control

20

1 Introduction

Fig. 1.16 Simulated and experimental results of the CW current control loop with cross feedforward compensation: a At the sub-synchronous speed of 600 rpm, b At the super-synchronous speed of 1200 rpm [68]

Fig. 1.17 The CW current control loop with the decoupling network proposed in [70]

* i2d

i2d

+

-

++

PI

* u2d

L1(s)

BDFIG & Load

L2(s) * i2q

+i2q

PI

+ +

* u2q

Decoupling network

method has been verified on a 32 kW BDFIG with the natural synchronous speed of 500 rpm, in comparison with the feedforward compensation method proposed in [69] and the method without feedforward compensation. From some of typical experimental results depicted in Fig. 1.18, it can be seen that the CW current control loop with the decoupling network possesses the best dynamic performance among the three methods. However, the dependence of the decoupling network design on the load model may make this method difficult for industrial applications.

1.3 Advances in Control Technologies for Standalone Power Generation Fig. 1.18 Experimental results of the CW current control loop with the decoupling network, where DN and FF indicate decoupling network and feedforward, respectively: a At the sub-synchronous speed of 350 rpm, b At the super-synchronous speed of 650 rpm [70]

21

DN

i2d

FF

Without FF

i2q (a)

Without FF

i2d

FF DN

i2q (b) 1.3.1.3

Predictive Controllers for CW Current

In order to improve current tracking accuracy, the model predictive current control (MPCC) method has been developed for standalone BDFIG [71]. The predicted CW current expressions can be expressed as.   R2 Ts Ts Ts i 2d (k) U2d (k) − D2q i 2q (k + 1)+ 1 − σ2 σ2 σ2 ⎡ ⎤ 2 R L ω R2 + L 2 (ωr 1 −2rp21ωr ) Ts Ts r ⎣ i 2q (k + 1) = 1 − Ts ⎦i 2q (k)+ U2q (k) − D2q σ2 σ2 σ2

(1.3)

(1.4)

where σ 2 is the leakage constant of the CW, T s the sampling time, and D2d and D2q are the cross-coupling disturbance between d- and q-axis currents. Since the MPCC is designed to enhance the current tracking accuracy, the cost function of MPCC can be expressed as.

22

1 Introduction

g = [i 2dr e f − i 2d (k + 1)]2 + [i 2qr e f − i 2q (k + 1)]2

(1.5)

However, as can be seen from (1.3) and (1.4), the MPCC method heavily depends on machine parameters. When the BDFIG temperature rises or magnetic saturation occurs, the machine parameters will change, which would degrade the performance of the MPCC. In order to address this issue, the nonparametric predictive current control (NPCC) method for standalone BDFIG is proposed in [72]. With NPCC, the predicted currents at the (k + 2)th instant with delay compensation can be derived as i 2d (k + 2) = i 2d (k) + X 2d U2d (k) + X 2d U2d (k + 1) + 2[i 2d (k − 1) − X 2d U2d (k − 1)]

(1.6)

i 2q (k + 2) = i 2q (k) + X 2q U2q (k) + X 2q U2q (k + 1) + 2[i 2q (k − 1) − X 2q U2q (k − 1)]

(1.7)

where X 2d =

i 2q (k − 1) − i 2q (k − 2) i 2d (k − 1) − i 2d (k − 2) , X 2q = . U2d (k − 1) − U2d (k − 2) U2q (k − 1) − U2q (k − 2)

Actually, the coefficients X 2d and X 2q can approximately represent the parameter (T s /σ 2 ), which frequently appears in the current prediction expressions of MPCC. In other words, the NPCC can use the sampled BDFIG signals rather than machine parameters to predict the future current. Besides, the cost function of NPCC is the same as that of MPCC.

1.3.2 Standalone Power Generation with Special Loads Generally, standalone generation systems are susceptible to special loads, e.g., unbalanced loads and nonlinear loads. The unbalanced loads would generate negativesequence PW voltage, and the nonlinear loads can introduce low-order harmonic voltages into PW. These control strategies mentioned above do not take into account the stable operation under these special loads. To solve this problem, some compensation methods have been developed, which can be divided into two main categories, i.e., the compensation methods based on single power converter and those based on dual power converters.

1.3.2.1

Compensation Based on Single Power Converter

The voltage at the PCC of the standalone BDFIG system can be expressed as

1.3 Advances in Control Technologies for Standalone Power Generation

u PCC = E 1 − R1 i 1 − L 1 si 1 = E 1 − (R1 i 1 f + L 1 si 1 f ) − (R1 i 1h + L 1 si 1h )       Fundamental voltage drop

23

(1.8)

Harmonic voltage drop

where E 1 is the inductive electromotive force (EMF) of PW, i1f the PW fundamental current, and i1h the negative-sequence and harmonic currents. As can be seen from (1.8), the negative-sequence and hormonic components of the PW voltage can be eliminated by producing an additional compensation EMF E1h to compensate the harmonic voltage drop in the PW. The E1h can be obtained by MSC injecting compensation current to CW. Also, the negative-sequence and hormonic components of PW voltage can be suppressed by removing the negative-sequence and harmonic currents i1h, which can be achieved by LSC injecting compensation current to unbalanced and nonlinear loads. Hence, the compensation methods based on the single power converter can be classified as two types, i.e., the method based on MSC and the one based on LSC. According to the calculation mode of CW voltage compensation components, the compensation methods based on MSC can be divided into two main categories, as illustrated in Fig. 1.19. Figure 1.19a and b present the compensation method with indirect calculation of CW voltage compensation components, and Fig. 1.19c shows the one with direct calculation of CW voltage compensation components. In Fig. 1.19a, both the obtained CW current compensation components and the basic CW current reference are in the same rotating frame, so that the fundamental and compensation components of CW voltage can be calculated at the same time. The control schemes proposed in [73] and [74] can be classified into the control method shown in Fig. 1.19a. A negative-sequence voltage compensation scheme for unbalanced standalone BDFIGs has been proposed in [73], which is implemented in the MSC. The negative-sequence PW voltage is extracted by the dual second-order generalized integrators (DSOGI) and then regulated by the PI controller to obtain the compensation current of CW. Consequently, the reference value of the CW current in the positive-sequence frame is with both the dc and ac components, which causes that the PIR controllers must be utilized in the CW current control loop. Another control scheme based the similar control concept has also been developed in [74], which can handle both the unbalanced and nonlinear loads. The multiple second-order generalized integrators (MSOGI) is adopted to obtain the negative-sequence and 5th and 7th harmonic components of PW voltage, and the dual frequency resonance (DFR) controller has to be employed in the CW current control loop. The control structure shown in Fig. 1.19b is a little different from that shown in Fig. 1.19a. In Fig. 1.19b, the CW voltage fundamental component and the compensation components are obtained separately, and the latter ones have to be transformed to the fundamental frame. The negative-sequence voltage compensation scheme proposed in [75] belongs to the method depicted in Fig. 1.19b. In [75], the amplitudes of the positive-and negative-sequence PW voltages can be regulated separately with a robust rotor speed observer.

24

1 Introduction

PW fundamental voltage control Distortion components extraction of PW voltage

CW current feedback CW voltage Basic CW current Calculation of CW reference reference voltage fundamental ++ and compensation components Calculation of CW current compensation components (a)

PW fundamental voltage control

CW current feedback Basic CW current reference PI +

Distorted components extraction of PW voltage

Calculation of CW voltage compensation components

Calculation of CW current compensation components

CW voltage reference

++

CW voltage compensation

(b)

PW fundamental voltage control

CW current feedback Basic CW current reference PI +

Distorted PW voltage

Calculation of CW voltage compensation components (c)

++

CW voltage reference

CW voltage compensation

Fig. 1.19 Different compensation methods based on MSC for standalone BDFIG under special loads: a Indirect calculation of CW voltage compensation components-Type I, b Indirect calculation of CW voltage compensation components-Type II, c Direct calculation of CW voltage compensation components

The common feature of the compensation methods shown in Fig. 1.19a and b is that the CW current compensation components have to be obtained before calculating the CW voltage compensation components. Therefore, in essence, the two compensation methods are the same. However, the indirect obtaining of CW voltage compensation components results that too many PI or resonant controllers have to be employed, which causes that the control algorithm is too complicated for practical industry applications. In order to address this issue, the compensation method shown in Fig. 1.19c is developed, which employs only one compensator to calculate the CW voltage compensation components directly. Hence, it can be significantly simplified in comparison to the methods depicted in Fig. 1.19a and b. A typical case of the method shown in Fig. 1.19c has been introduced in [76], where an improved control strategy

1.3 Advances in Control Technologies for Standalone Power Generation

DC link voltage control Distortion components extraction of PW voltage

25

LSC current feedback LSC voltage Basic LSC Calculation of LSC reference current reference voltage fundamental ++ and compensation components Calculation of LSC current compensation components

Fig. 1.20 Compensation method based on LSC for standalone BDFIG under special loads

based on the dual-resonant controller (DRC) is proposed to minimize both the unbalance and nonlinear effects of PW voltage at the same time. The CW current controller is just a simple PI controller. The DRC composes of two parts: unbalance resonant controller with the resonant frequency of 2-time PW frequency and the harmonics resonant controller with the resonant frequency of 6-time PW frequency. The unbalance resonant controller is used to compensate the unbalanced and 3rd harmonic components in the PW voltage. And, the harmonics resonant controller is adopted to eliminate the 5th and 7th harmonics of PW voltage. It is not necessary to extract the negative-sequence and harmonic components of PW voltage. The compensation method based on LSC for standalone BDFIG under special loads can be briefly described as Fig. 1.20. It utilizes LSC to inject compensation current to unbalanced and nonlinear loads, so that the negative-sequence and hormonic components of the loads can be no longer supplied by PW. Reference [77] employs this method to successfully eliminate the harmonic voltages of PW caused by nonlinear loads. Generally, when using this kind of compensation method, the distorted components of PW voltage are extracted by the MSOGI.

1.3.2.2

Compensation Based on Dual Power Converters

Although these single power converter-based methods introduced in the last section can achieve voltage distortion compensation under special loads, they require a higher power converter capacity. However, in practical applications, the converter capacity is limited to some extent. In order to eliminate voltage distortion as much as possible under the limited converter capacity, the collaborative control of MSC and LSC has been developed to optimize the system performance, whose basic principle can be seen in Fig. 1.21. This method can dynamically distribute the distorted PW voltage to be compensated between MSC and LSC according to the limitations of the capacities, voltages, currents and IGBT junction temperatures of the two power converters. And then, the MSC and LSC still adopt the compensation methods shown in Figs. 1.19 and 1.20, respectively. In [78], considering the limitation of the rated voltage of the power converters, the MSC is responsible to eliminate the 7th harmonic components of PW voltage, and

26

1 Introduction Distorted voltage to be compensated by MSC Distorted PW voltage

Compensation method based on MSC

Distribution of distorted voltage to be compensated Distorted voltage to be compensated by LSC

Compensation method based on LSC

CW voltage reference

LSC voltage reference

Fig. 1.21 Compensation method based on the collaborative control of MSC and LSC for standalone BDFIG under special loads

the LSC is used to suppress the negative-sequence and 5th harmonic components of PW voltage, when the standalone BDFIG supplying unbalanced and nonlinear loads. In [79], a real-time and flexible assignment method of PW harmonic voltage elimination tasks between MSC and LSC has been developed. A weight factor is proposed to dynamically adjust the contribution of the two power converters on harmonics elimination, according to one or more constraint conditions, e.g., overheat, overcurrent, overvoltage, and so on. Hence, on the premise of ensuring safety, the capacity of each inverter is expected to be fully utilized to mitigate harmonics. If the weight factor is set to 1, all harmonics would be compensated by MSC. And, if the weigh factor is equal to 0, all harmonics would be compensated by LSC. Otherwise, both MSC and LSC would be utilized to suppress all the harmonics. The typical experimental results, carried out a 30 kVA BDFIG under nonlinear loads, can be seen in Fig. 1.22. The weight factor is reduced from 1 to 0.2 step by step to test the system characteristics. As illustrated in Fig. 1.22, with the decrease of the weight factor, both the harmonics and amplitude of the CW current are reduced, while those of the LSC current are with the opposite trend. In addition, the electromagnetic torque ripple goes down along with the reduction of the weight factor. In the future, it is necessary to develop optimization algorithms for automatically searching the optimal weight factor under different constraint conditions. Table 1.5 shows the attribute comparison of the different compensation methods based on single power converter (MSC or LSC) and dual power converters. From Table 1.5, it can be concluded that the compensation method based on dual power converters is one of the most promising methods for BDFG-based standalone ac power generation under unbalanced and nonlinear loads.

1.3 Advances in Control Technologies for Standalone Power Generation

27

Weight factor

CW current (A)

The amplitude limit of CW current

Time(s)

Time(s)

(a)

(b)

PW line voltage (A)

Amplitude of CW current (A)

The amplitude limit of CW current

Time(s)

Time(s)

(c)

(d)

LSC current (A)

Te (300Hz) = 23.07Nm Te (300Hz) = 2.83Nm

Time(s)

(e)

(f)

Fig. 1.22 Experimental results of the compensation method based on the collaborative control of MSC and LSC with the weight factor reduction from 1 to 0.2: a Weight factor, b CW current, c PW line voltage, d Amplitude of CW current, e LSC current and f Electromagnetic torque [79] Table 1.5 Comparison among compensation methods based on MSC, LSC, and dual power converters Attribute

MSC compensation

LSC compensation

Dual power converters compensation

Does it need extra load current sensors?

No

Yes

No

Can it eliminate the torque ripple?

No

Yes

Yes

Can it allocate of the harmonic mitigation task between the two power converters?

No

No

Yes

28

1 Introduction

1.4 Advances in Control Technologies for Sensorless Control 1.4.1 Without Speed Observer In [80], a sensorless direct voltage control (DVC) method without the speed observer has been developed for standalone BDFIGs, which utilizes the principle of the dqframe phase-locked loop (PLL) as proposed in [82]. It achieves the sensorless control function by adjusting the CW current frequency to track the PW voltage phase, so that it is not necessary to estimate the rotor speed or position. The principle of this method can be introduced as follows. In the control method proposed in [80], the d-axis of the rotating dq reference frame is aligned with the reference PW voltage vector, as shown in Fig. 1.23. Hence, the q-axis PW voltage can be expressed as u 1q = U1 sin(θ1 − θ1∗ )

(1.9)

where U 1 and U 1 * are the amplitudes of the actual and reference PW voltage vectors, θ 1 and θ 1 * are the phase angles of the actual and reference PW voltage vectors, respectively. Around the equilibrium point, Eq. (1.9) can be simplified as u 1q ≈ U1 (θ1 − θ1∗ )= −U1 θ1

(1.10)

where θ 1 is the difference between the reference phase angle and actual phase of the PW voltage. If the u1q is equal to zero, the actual and reference PW voltage vectors would be with the same phase angle. Besides, for the BDFIG, the CW current frequency can regulate the PW voltage frequency, thus adjusting the PW voltage phase. Hence, a PI controller can be employed to control u1q , and the output of the PI controller is the reference value of the CW current frequency. The linearized control loop for Fig. 1.23 The actual and reference PW voltage vectors in the rotating dq reference frame [80]

1.4 Advances in Control Technologies for Sensorless Control

29

Fig. 1.24 The structure of the linearized control loop for tracking the PW voltage phase proposed in [80]

regulating the PW voltage phase can be seen in Fig. 1.24, which is similar to the dq-frame PLL. The advantages of the sensorless DVC method presented in [80] are that the implementation process is simple and the machine parameters are not required. However, from the experimental results, it can be seen that the PW voltage phase is with steady-state error during the rapid change of the rotor speed. The sensorless DVC method has been successfully used in the sensorless phase control of PW voltage for standalone BDFIGs under unbalanced loads [81].

1.4.2 With Speed Observer The estimation approaches of the current BDFG speed observers mainly include the direct estimation approach and the closed-loop estimation approach, which will be introduced in details as follows.

1.4.2.1

Direct Calculation Approaches

The direct calculation approaches for BDFIG rotor position have been investigated in [83–85]. Generally, no PI controllers are required in these direct calculation approaches, and consequently a good starting performance can be obtained. Besides, there are no control parameters to be tuned so that the algorithm implementation is easy. However, the dependence on machine parameters will degrade the robustness of these approaches. In [83], the CW back-EMF is integrated to obtain the CW flux in αβ frame at first, and then the position angle of the CW flux can be obtained by

 θ2_ flux = tan−1 ψ2β /ψ2α

(1.11)

The PW flux in αβ frame can be also derived by the similar method. Afterwards, using a rotational transformation to the fixed rotor frame, the estimated rotor position can be obtained by

30

1 Introduction

θˆr =



∠ e−jθ2_ flux ψ 1αβ p1 + p2

(1.12)

Due to the usage of PW and CW flux, the accurate 3-phase voltages and currents of PW and CW need to be measured, and the PW and CW resistances should be known. Besides, the pure integrators are not directly suitable for experimental implementation of the PW and CW flux calculation, because of unavoidable dc-offsets in the measured voltages and currents. In [84] and [85], the proposed approach needs the knowledge of the PW flux, and the PW and CW currents. In order to eliminate the integration process for calculating the PW flux, the dq-axis PW flux is calculated at first and then transformed to the αβ-axis quantities. Finally, the rotor position can be obtained by

 

ψ − A p i 1α i 2β + ψ1β − A p i 1β i 1α ˆθr = tan−1 1α  

ψ1α − A p i 1α i 2α − ψ1β − A p i 1β i 2β

(1.13)

where A p = L 1 − L 21r /L r . It can be found that the calculation process shown in Fig. 1.25 is easy to implement in the digital signal processor. Furthermore, no integrators are required, which would significantly reduce the cumulative error of the calculated rotor position. Fig. 1.25 Flowchart of the direct calculation approach for the BDFIG rotor position as developed in [84] and [85]

Start

ReadIng u1abc , i1abc , i1abc , θ1 , ω1

Determine Ψ1dq

Determining Ψ1αβ using the dq/αβ transformation matrix based on Ψ1dq and θ1

Determining the estimated rotor position using (1.13)

End

1.4 Advances in Control Technologies for Sensorless Control

1.4.2.2

31

Closed-Loop Estimation Approach

The closed-loop estimation means that the estimated rotor speed/position needs to be fed back to the speed observer for improving the estimation accuracy. Generally, this kind of approach is with the higher estimation accuracy than the direct estimation approach. However, the starting performance of the closed-loop estimation approach is worse than that of the direct estimation approach. Since some control parameters have to be tuned, the algorithm implementation of the closed-loop estimation approach is more difficult than that of the direct estimation approach. A.

Speed Observers Dependent on Machine Parameters

In [86] and [87], a quasi-closed-loop estimation approach is proposed for the BDFRG, which is the combination of the direct estimation and the closed-loop estimation. At first, the raw value of the rotor position is derived by the direct estimation approach. And then, the raw rotor position is sent to a Luenberger type closed-loop PI observer for filtering out erroneous signals, which can obtain the accurate rotor speed. The sensorless control method with the principle of model reference adaptive system (MRAS) has attracted great attention from researchers for its high-performance operation. The MRAS speed observers have been applied to DFIGs [88–93] and also been extended to the applications of BDFGs with different control state variables [94-100]. According to the control state variables, these MRAS speed observers presented in [94–101] can be divided to three groups, i.e., the observers based on CW current, the ones based on CW flux, and the ones based on CW power. In [94] and [95], the MRAS speed observers have been developed with the CW current served as the control state variable. And, the stator flux is also required in these observers, so that the integration process has to be used. However, the integration process is susceptible to the dc-offsets in the sampled PW and CW back-EMFs, and would slow down the response speed at the system startup. In order to avoid the integration process for the flux estimation, the MRAS speed observers employing the dq-axis CW flux as the control state variable have been proposed for sensorless control of BDFRG and BDFIG in [96] and [97], respectively. Without the integration operation for the flux estimation, some other improved MRAS speed observers based on the CW active/reactive/fictitious power (CW-P/Q/X MRAS) as the control state variables, as shown in Fig. 1.26, have been investigated for standalone BDFIGs in [98–100]. The main innovation of these MRAS observers presented in [98–100] is to eliminate the integration process for the flux estimation and reduce the voltage sensors used to detect the CW dq-axis components. Moreover, the presented simulation and experimental results have ensured the functionality of the adopted MRAS method to effectively estimate the rotor position with a good tracking under various operating states of both speed and load changes. B.

Machine-parameter-free Speed Observers

All the closed-loop estimation approaches mentioned above are severely dependent on the machine parameters, which would degrade the system robustness against

32

1 Introduction

Fig. 1.26 Structure of the proposed CW-P/Q/X MRAS speed observers for standalone BDFIGs [98–100]

the machine parameter mismatch. In order to address this issue, some typical machine-parameter-free speed observers have been developed for standalone BDFIG [101–103]. It is well known that the mechanical rotor speed of BDFIG can be decided by. ( p1 + p2 )ωr = ω1 + ω2

(1.14)

Ignoring the initial value of the integral operation, the integration of (1.14) can be written as ( p1 + p2 )θr v = θ1 + θ2

(1.15)

where θ 1 and θ 2 are the phase angles of PW voltage and CW current, respectively, and θ rv can be regarded as the virtual rotor position, i.e., an intermediate variable for calculating the rotor speed. Based on the small signal analysis method, the difference between the actual and estimated virtual rotor positions, θ rv , can be expressed as ( p1 + p2 )θr v = ( p1 + p2 )(θr v − θˆr v ) = (θ1 + θ2 ) − ( p1 + p2 )θˆr v ≈ sin[(θ1 + θ2 ) − ( p1 + p2 )θˆr v ]

(1.16)

According to the principle of the αβ-frame PLL presented in [104], θ rv can converge to zero by using a PI controller, and the output of the PI controller would be the accurately estimated rotor speed. However, under unbalanced and nonlinear loads, the standalone BDFIG suffers from the distorted PW voltage and CW current, which results in the inaccurate rotor speed estimation. To solve this problem, the improved MPFSOs have been proposed in [101–103], with using appropriate filters to eliminate the distorted components in PW voltage and CW current generated by unbalanced and nonlinear loads.

1.4 Advances in Control Technologies for Sensorless Control

33

Table 1.6 Comparison of different sensorless control methods Attribute

Without speed observer

Direct speed/position estimation

Closed-loop speed/position estimation

Starting performance

Medium

High

Medium

Transient performance

Medium

Low

High

Estimation accuracy

Null

Medium

High

Robustness to machine parameters

Medium

Low

High

Ease of implementation

Medium

High

Low

In [101] and [103], the improved MPFSO utilizes the second-order generalized integrators (SOGIs) to filter the harmonics in PW voltage and CW current, and employs the positive-sequence calculator (PSC) to eliminate the negative-sequence components in PW voltage. Afterwards, the filtered PW voltage and CW current are sent to the basic MPFSO for deriving the accurate rotor speed. It can be noted that when the BDFIG operates at its natural synchronous speed, the CW frequency should be set to zero to keep constant PW frequency. Hence, the filtering performance of the low-pass filter (LPF) would be better than that of the SOGI for CW current around the natural synchronous speed. Consequently, the improved MPFSO presented in [103] employs LPFs to eliminate the impact of unbalanced and nonlinear loads on the CW current. In addition, the parameter tunning guideline for the improved MPFSO has been designed for various operation conditions. Under the most severe load condition (the unbalanced nonlinear load), the ripple in the rotor speed obtained by the improved MPFSO can be significantly reduced, with almost the same response speed as the basic MPFSO. The defect of the improved MPFSO is that the parameter tunning requires the knowledge of the speed range. Table 1.6 shows the attribute comparison of different sensorless control methods, in terms of starting performance, transient performance, estimation accuracy, robustness to machine parameters and ease of implementation. From this table, it can be concluded that the sensorless control methods based on closed-loop speed/position estimation have the best overall drive performance.

1.5 Organization of the Book This book consists of eight chapters, and the main content is based on the authors’ several research projects. Each chapter can be summarized as follows. Chapter 1 surveys the state-of-art of control technologies for BDFG under different operation conditions and the progresses of sensorless control for BDFG. The classification and comparison are carried out to discover the advantages and disadvantages

34

1 Introduction

of theses control technologies, and also reveals the the importance of the work in this book. Chapter 2 presents a detailed description of the power generation system based on the BDFIG at first. And then, the steady-state models and the power flow characteristics are fully analyzed. Furthermore, the basic control method for the standalone BDFIG has also been domenstrated with detailed operation characteristics analysis. Chapter 3 explains the adverse effects of unbalanced and nonlinear loads on the standalone BDFIG, and analyzes the dynamic behavior of the standalone BDFIG under the unbalanced load, three-phase nonlinear load, unbalanced plus nonlinear load, and single-phase nonlinear load, respectively. Afterwards, the unbalanced voltage compensator, low-order harmonic voltage compensator and dual-resonant controller are presented for minimizing the nonlinear and unbalanced impacts. Finally, the effectiveness of these presented control strategies is confirmed by simulation and experimental results. In Chap. 4, two advanced compensation control strategies for the standalone BDFIG under heavy load disturbance are developed based on the single power converter and on the dual power converters, respectively. The single power converterbased compensation control strategy utilizes the transient feedforward compensation of the CW current to supress the PW votltage drop. And, the dual power converterbased strategy adopts both the MSC and LSC to compensate for changes in the PW active and reactive currents, so that the redundant capacities of the two converters can be fully used to compensate for load disturbance. Chapter 5 handles the improvements of the robustness and dynamic response for the standalone BDFIG system by applying both MPCC and NPCC methods. Firstly, the MPCC is used to replace the traditional PI controller for the CW current regulation of the standalone BDFIG. However, the control behaviour of MPCC would be adversely affected by the machine parameter mismatch. And then, another new current control method based on NPCC is proposed to solve this problem, which is free of machine parameters. In Chap. 6, two rotor position observers with the direct estimation principle are developed, which are based on the phase-axis relationship and space-vector model of the BDFIG, respectively. However, both the two rotor position observers rely on the machine parameters. In order to enhance the robustness of the observer to the machine parameter mismatch, and then, an improved rotor speed observer with the closed-loop estimation principle is presented, which is based on PW voltage and CW current and free of electrical parameters of the BDFIG. Futhermore, the presented rotor speed observer can run well under unbalance and nonlinear loads. Chapter 7 presents three kinds of MRAS based sensorless control strategies for standalone BDFIGs. These control strategies are based on the CW power factor, PW flux and CW flux, respectively. Among the MRAS sensorless control strategies based on PW flux and CW flux, the ones based on the dq-axis flux have the better dynamic performance than those based on the αβ-axis flux, due to the absence of integrators in the former. Both simulation and experimental results verify the presented control strategies.

1.5 Organization of the Book

35

Chapter 8 concludes the whole book and proposes future research and development.

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42. Sadeghi R, Madani SM, Ataei M et al (2018) Super-twisting sliding mode direct power control of a brushless doubly fed induction generator. IEEE Trans Ind Electron 65(11):9147–9156 43. Jin S, Shi L, Zhu L et al (2016) Performance comparison of direct power control for brushless doubly-fed wind power generator with different control winding structure. In: 2016 IEEE transportation electrification conference and expo, Asia-Pacific (ITEC Asia-Pacific), Busan, pp 261–266 44. Brassfield WR, Spee R, Habetler TG (1996) Direct torque control for brushless doubly-fed machines. IEEE Trans Ind Appl 32(5):1098–1104 45. Ge Y, Wu N (2011) Direct torque control for brushless doubly-fed machine based on SVPWM. Small Spec Electr Mach 39(8):58–60 46. Casadei D, Serra G, Tani A (1998) Improvement of direct torque control performance by using a discrete SVM technique. In: 29th Annual IEEE power electronics specialists conference, Fukuoka, Japan, pp 997–1003 47. Fattahi SJ, Khayyat AA (2011) Direct torque control of brushless doubly-fed induction machines using fuzzy logic. In: 2011 IEEE ninth international conference on power electronics and drive systems, pp 619–624 48. Yao X, Shi L, Guo Q (2011) Brushless doubly-fed machine direct torque control based on fuzzy strategy. J Power Supply 11(6):54–58 49. Zhang A, Zhang Y (2007) Direct torque control for brushless doubly-fed machine based on torque predict control strategy. Electr Mach Control 11(4):326–330 50. Kang JK, Sul SK (1999) New direct torque control of induction motor for minimum torque ripple and constant switching frequency. IEEE Trans Ind Appl 35(5):1076–1082 51. Li B, Liu S, Long T et al (2017) Duty ratio modulation direct torque control of brushless doubly-fed machines. Automatika 58(4):479–486 52. Zhang A, Jia W, Zhou Z et al (2011) A study on indirect stator-quantities control strategy for brushless doubly-fed induction machine. In: Proceedings international conference electronics machine systems, pp 1–6 53. Zhang A, Wang X, Jia W et al (2014) Indirect stator-quantities control for the brushless doubly fed induction machine. IEEE Trans Power Electron 29(3):1392–1401 54. Zhao R, Zhang A, Ma Y et al (2015) The dynamic control of reactive power for the brushless doubly fed induction machine with indirect stator-quantities control scheme. IEEE Trans Power Electron 30(9):5046–5057 55. Shao S, Long T, Abdi E et al (2013) Dynamic control of the brushless doubly fed induction generator under unbalanced operation. IEEE Trans Ind Electron 60(6):2465–2476 56. Xu L, Cheng M, Wei X et al (2021) Dual synchronous rotating frame current control of brushless doubly fed induction generator under unbalanced network. IEEE Trans Power Electron 36(6):6712–6724 57. Chen J, Zhang W, Chen B et al (2016) Improved vector control of brushless doubly fed induction generator under unbalanced grid conditions for offshore wind power generation. IEEE Trans Energy Convers 31(1):293–302 58. Gholizadeh M, Tohidi S, Orae A et al (2018) Appropriate crowbar protection for improvement of brushless DFIG LVRT during asymmetrical voltage dips. International J Electr Power Energy Syst 95:1–10 59. Tohidi S, Oraee H, Zolghadri MR et al (2015) Influence of different series dynamic resistors on low-voltage ride-through of brushless doubly fed induction generator. Electric Power Compon Syst 43(8–10):995–1005 60. Tohidi S, Oraee H, Zolghadri M et al (2013) Analysis and enhancement of low-voltage ridethrough capability of brushless doubly fed induction generator. IEEE Trans Ind Electron 60(3):1146–1155 61. Tohidi S, Oraee H, Zolghadri MR et al (2016) A control scheme to enhance low voltage ride-through of brushless doubly-fed induction generators. Wind Energy 19(9):1699–1712 62. Huang J, Li S (2018) Analytical expression for LVRT of BDFIG with enhanced current control to CW and reactive power support from GSC. Int J Electr Power Energy Syst 98:243–255

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101. Liu Y, Xu W, Long T et al (2019) A new rotor speed observer for stand-alone brushless doubly-fed induction generators. In: IEEE energy conversion congress and exposition (ECCE), Cincinnati, OH, USA, pp 5086–5092 102. Liu Y, Xu W, Zhu J et al (2019) Sensorless control of standalone brushless doubly fed induction generator feeding unbalanced loads in a ship shaft power generation system. IEEE Trans Ind Electron 66(1):739–749 103. Liu Y, Xu W, Long T et al (2020) An improved rotor speed observer for standalone brushless doubly-fed induction generator under unbalanced and nonlinear loads. IEEE Trans Power Electron 35(1):775–788 104. Teodorescu R, Blaabjerg F (2004) Flexible control of small wind turbines with grid failure detection operating in stand-alone and grid-connected mode. IEEE Trans Power Electron 19(5):1323–1332

Chapter 2

Mathematical Modelling, Operation Characteristics and Basic Control Method of BDFIG

Abstract This chapter presents a detailed description of the presented power generation system based the brushless doubly-fed induction generator (BDFIG). The adopted system description and the main considerations of the BDFIG-construction are also illustrated. In addition, detailed abc-axis and dq0-axis dynamic models are also presented. This model can be used to investigate the dynamic behaviour of the BDFIG. Moreover, the detailed steady-state models and the power flow considerations are fully described. Furthermore, the configuration and operating characteristics of the BDFIG drive system are completely studied and analysed. Keywords Brushless doubly-fed induction generator (BDFIG) · Dynamic model · Steady-state model · Operating characteristics · Direct voltage control

2.1 Introduction For efficient reduction in the cost of electric power generating systems, the doubly-fed generator is considered with its associated fractional power-rating converter (approximately 30% of the machine ratings) [1–8]. However, the existence of brushes is the main demerit of this generator type due to the high maintenance cost. Over the years, different developments in the structure of the adopted doubly-fed machines have been addressed and investigated with different categories of construction configurations which can be classified into brush and brushless machines [8]. The reliable construction of the BDFIG promotes its notability for the industrial drive applications. The main structure of the BDFIG includes the two windings of the machine in the stator side for the doubly fed operation as shown in Fig. 2.1. One of the fixed two windings is used as the main side, as called the power winding (PW). Moreover, the other winding is used as a secondary side for the control purpose of the machine, as represented as the control winding (CW) [8–12]. Moreover, the role of the rotor side is to couple the two fixed windings through magnetic coupling. The structure of the rotor has been developed over the years, including the cage, reluctance, and the desired wound types [13–23]. This development in the rotor frame construction is to realize the recent challenges of the adopted BDFIG for a high-efficient power-generation systems. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Y. Liu and W. Xu, Advances in Control Technologies for Brushless Doubly-fed Induction Generators, https://doi.org/10.1007/978-981-19-0424-0_2

41

42

2 Mathematical Modelling, Operation Characteristics …

Fig. 2.1 Typical configuration of the BDFIG system

According to the analysis in [14], the BDFIG with the wound rotor has shown a great interest compared to the cage and reluctance types due to the great reduction in the harmonic effect of the rotor side. As a required necessity in the ship applications, it is to realize the loads demand of the electrical power generation for its various arrangements. Hence, the ship generator became a mandatory part in the generating system and this will in turn affect the overall efficiency from the view point of the whole fuel consumed. Therefore, the most recent trends in this research area are to optimize the whole consumed amount of fuel required for the ship operation. The commercial generator types used in the ship power generation systems are specified as the synchronous generator, the doubly-fed generator, and the BDFIG. For the generation systems based synchronous machines, the power rating of the required converter is the same as that of the generator, which would increase the overall cost [24, 25]. In order to reduce the overall system cost, the doubly-fed generator is developed with its merits of the fractional power-rating converters assisted in the system configuration [26–28]. However, the existence of the brushes in this type of shaft generators would increase the maintenance cost, and hence affects the overall efficiency of the ship’s energy system. Due to the effective merits of the adopted BDFIG with its reliable construction, this type of machine has assured the qualified behavior over the doubly-fed generators in the ship power generation systems [29–32]. Hence, the presented wound-rotor BDFIG would be more efficient and preferred in the ship applications [33–35].

2.2 Dynamic Models of BDFIG

43

2.2 Dynamic Models of BDFIG In the subsequent sections, a detailed derivation of the abc-axis and dq0-axis dynamic model of the BDFIG, including developed power and electromagnetic torque expressions, is presented.

2.2.1 abc-Axis Model of BDFIG The modelling of the BDFIG in the abc-axis frame can be described below. The voltages of the PW, CW and rotor can be expressed as [36] ⎫ d ψ 1abc ⎪ ⎪ ⎪ ⎪ dt ⎪ ⎬ d = diag[R2 ](i 2abc ) + ψ 2abc ⎪ dt ⎪ ⎪ ⎪ ⎪ d = diag[Rr ](i rabc ) + ψ rabc ⎭ dt

u1abc = diag[R1 ](i 1abc ) + u2abc urabc

(2.1)

where ⎡

R1 ⎣ diag[R1 ] = 0 0 ⎡ R2 diag[R2 ] = ⎣ 0 0 ⎡ Rr diag[Rr ] = ⎣ 0 0

⎤ 0 0 R1 0 ⎦ 0 R1 ⎤ 0 0 R2 0 ⎦ 0 R2 ⎤ 0 0 Rr 0 ⎦ 0 Rr

The flux of the stator and rotor windings can be written as ⎫ ψ 1abc = L abc1−abc1 i 1abc + L abc1−abcr i rabc ⎬ ψ 2abc = L abc2−abc2 i 2abc + L abc2−abcr i rabc ⎭ ψ rabc = L abcr −abcr i rabc + L abcr −abc1 i 1abc + L abcr −abc2 i 2abc

(2.2)

where u, i and ψ are the voltage, current and flux phasors, respectively; Labc1-abc1 , Labc2-abc2 and Labcr-abcr are the mutual inductance matrixes of PW, CW and RW, respectively; Labc1-abcr and Labcr-abc1 are the mutual inductance matrixes between

44

2 Mathematical Modelling, Operation Characteristics …

PW and RW; Labc2-abcr and Labcr-abc2 are the mutual inductance matrixes between CW and RW. All the mutual inductance matrixes can be expressed as [37] ⎡

L abc1−abc1

L abc2−abc2

L abcr −abcr

L σ 1 + L m1 = ⎣ −0.5L m1 −0.5L m1 ⎡ L σ 2 + L m2 = ⎣ −0.5L m2 −0.5L m2 ⎡ L σ r + L mr = ⎣ −0.5L mr −0.5L mr

⎤ −0.5L m1 −0.5L m1 L σ 1 + L m1 −0.5L m1 ⎦ −0.5L m1 L σ 1 + L m1 ⎤ −0.5L m2 −0.5L m2 L σ 2 + L m2 −0.5L m2 ⎦ −0.5L m2 L σ 2 + L m2

⎤ −0.5L mr −0.5L mr L σ r + L mr −0.5L mr ⎦ −0.5L mr L σ r + L mr

L abc1−abcr ⎡

⎤ cos( p1 θr − 4π/3) cos( p1 θr − 2π/3) cos( p1 θr ) = L pr ⎣ cos( p1 θr − 2π/3) cos( p1 θr ) cos( p1 θr − 4π/3) ⎦ cos( p1 θr ) cos( p1 θr − 4π/3) cos( p1 θr − 2π/3)

L abc2−abcr ⎡

⎤ cos( p2 θr + θ0 − 4π/3) cos( p2 θr + θ0 − 2π/3) cos( p2 θr + θ0 ) ⎣ = L cr cos( p2 θr + θ0 − 2π/3) cos( p2 θr + θ0 ) cos( p2 θr + θ0 − 4π/3) ⎦ cos( p2 θr + θ0 − 4π/3) cos( p2 θr + θ0 − 2π/3) cos( p2 θr + θ0 )

L abcr −abc1 = (L abc1−abcr )T L abcr −abc2 = (L abc2−abcr )T . In the above expressions, θ r indicates the rotor position, θ 0 is the initial phase difference between the A-phase axis of PW and the A-phase axis of CW, the superscript “T” represents the matrix transpose.

2.2.2 dq0-Axis Dynamic Model of BDFIG The dynamic model of the BDFRG in the abc-axis frame is of time-varying coefficients due to time variation of mutual inductances as given in (2.2). Once the associated variables of the three-phase model are transformed to a corresponding dq0-axis fictitious frame, the inductances with time variation can be eliminated. The transformation of variables from abc-axis to dq0-axis reference frame can be obtained as F dq0 = K ∗ F abc

(2.3)

2.2 Dynamic Models of BDFIG

45

where F denotes either voltage, current or flux phasor, K represents the abc-dq0 transformation matrix given by



⎞ 2π cos θ + cosθa cos θa − 2π a 3 3



2 ⎠ K = ⎝ −sinθa − sin θa − 2π − sin θa + 2π 3 3 3 1 1 1 ⎛

2

2

2

where θ a is the desired frame electrical angle. The character “0” refers to the zero-sequence component of the dq0-axis frame which can be omitted in some typical applications, e.g., the ship shaft power generation system. This is due to the three-phase three-wire load connection where there is no path for the zero-sequence currents. The complete modelling expressions of the presented BDFIG in the dq-axis frame [38] can be summarized as follows. The voltage equations of the BDFIG in the dq frame with the arbitrary angular speed ωa can be written as u 1d u 1q u 2d u 2q ur d u rq

= = = = = =

⎫ ⎪ R1 i 1d + sψ1d − ωa ψ1q ⎪ ⎪ ⎪ ⎪ R1 i 1q + sψ1q + ωa ψ1d ⎪ ⎪ ⎬ R2 i 2d + sψ2d − [ωa − ( p1 + p2 )ωr ]ψ2q R2 i 2q + sψ2q + [ωa − ( p1 + p2 )ωr ]ψ2d ⎪ ⎪ ⎪ ⎪ ⎪ Rr ir d + sψr d − (ωa − p1 ωr )ψrq ⎪ ⎪ ⎭ Rr irq + sψrq + (ωa − p1 ωr )ψr d

(2.4)

where s is the differential operator d/ dt. The dq-axis equations representing the flux-linkage components of the machine for all sides of PW, CW and rotor frame can be given as ψ1d ψ1q ψ2d ψ2q ψr d ψrq

= = = = = =

⎫ ⎪ L 1 i 1d + L 1r ir d ⎪ ⎪ ⎪ ⎪ L 1 i 1q + L 1r irq ⎪ ⎪ ⎬ L 2 i 2d + L 2r ir d ⎪ L 2 i 2q + L 2r irq ⎪ ⎪ ⎪ L r ir d + L 1r i 1d + L 2r i 2d ⎪ ⎪ ⎪ ⎭ L r irq + L 1r i 1q + L 2r i 2q

(2.5)

46

2 Mathematical Modelling, Operation Characteristics …

2.3 Steady-state Models of BDFIG 2.3.1 Conventional Π-type Steady-state Model In general, the steady-state operation of the BDFIG can be described with the -type model [31] shown in Fig. 2.2, in which all the parameters are referred to the PW with neglecting the iron losses. In Fig. 2.2, R1 , R2 and Rr are the PW, CW and rotor resistances, L m1 and L m2 the PW and CW magnetizing inductances, L σ 1 , L σ 2 and L σ r the PW, CW and rotor leakage inductances, U˙ 1 and U˙ 2 the PW and CW phase voltage vectors, and I˙1 , I˙2 and I˙r the PW, CW and rotor phase current vectors, respectively.

Fig. 2.2 The steady-state -type model of BDFIG

Fig. 2.3 Equivalent circuit of the simplified inner core model [29]

2.3 Steady-state Models of BDFIG

47

2.3.2 Steady-state Model with Simplified Inner Core In [29], McMahon et al. has presented a simplified inner core model for the steadystate operation of BDFIG, as illustrated in Fig. 2.3, with neglecting the inductances of magnetization and stator leakage, and also the resistances of all sides. It is obvious from the presented model in Fig. 2.3 that I˙1 = − I˙2 . However, this simplification is not suitable for the practical performance of the BDFIG. This is due to that, under the no load condition, the current I˙1 = 0 will result the CW current I˙2 = 0. In this condition, there is no exciting current for the CW side, and hence no voltage can be generated for the PW side. Therefore, this simplified inner core model cannot describe the steady-state operation of the BDFIG accurately.

2.3.3 Steady-state T-type Model According to the analysis given in [31], the equivalent circuit of the -type model, illustrated in Fig. 2.2, can be represented with its corresponding T-type model with the associated three impedances as shown in Fig. 2.4. Aided with the equivalent circuit in Fig. 2.4, considering the voltage and current of PW side as input and that of the CW side as output, the steady-state model of the BDFIG [31] can be expressed with a matrix form as 

U˙ 2 ss21 I˙2



 =

Z 2 +Z m Zm 1 Zm

 − Z1 + Z2 + m − Z 1Z+Z m

Z1 Z2 Zm

 

U˙ 1 I˙1



The expressions of the three impedances Z 1 , Z 2 , and Z m are given as

Fig. 2.4 Equivalent circuit of the steady-state T-type model [31]

(2.6)

48

2 Mathematical Modelling, Operation Characteristics …

⎧ Z 1 = jω1 (α1 + L σ 1 ) + R1 ⎪ ⎪ ⎨

s2 Z 2 = jω1 α2 + L σ 2 + R2 s1 ⎪ ⎪ ⎩ Z m = jω1 α3

(2.7)

where α1 =

L m1 L σ r L m1 + L σ r + L m2

α2 =

L m2 L σ r L m1 + L σ r + L m2

α3 =

L m1 L m2 L m1 + L σ r + L m2

It can be stated that the T-type model in Fig. 2.4 with its associated representation of the equivalent three impedances is simple for the description of the steady-state operation for the BDFIG, which would give an effective approach for analyzing the behavior of the presented power generation system.

2.4 Operation Characteristics of the BDFIG Drive System 2.4.1 Construction of the BDFIG Drive System In a BDFIG, the fixed two windings of the stator side are allocated in the same frame with different number of poles [9]. One of the two windings is considered as the primary side, and known as the power winding (PW) with p1 pair-pole. Moreover, the other winding is used as the secondary side and denoted as the control winding (CW) with p2 pair-pole. The number of poles of both the PW and CW must be different to avoid the direct transformer coupling between them. Furthermore, for avoiding the unbalanced magnetic pull on the rotor, a difference of more than one must be considered between the pole pairs of both windings [9]. The main conceptual diagram of the BDFIG is shown in Fig. 2.5 [39]. The main role of the PW is to directly feed the load demand. Furthermore, the control targets of the BDFIG can be realized with the aid of the CW through the bidirectional power converter between the side of CW and the load terminals. The bidirectional converter in Fig. 2.5 consists of two main parts, the machine-side converter (MSC) and the load-side converter (LSC) with a common dc-link between them. The essential role of the LSC is to regulate the voltage across the terminals of the dc-link [40].

2.4 Operation Characteristics of the BDFIG Drive System

49

Fig. 2.5 System structure of the standalone BDFIG [39]

For a high-efficient power-generation systems based on the BDFIG, the rotor structure and its reliability issues are the main challenges for the development of the rotor design including the cage, wound, and reluctance types [14]. With the study presented in [14], the type of wound rotor has assured its efficacy for the development of the promising BDFIG due to the attained high reduction for the effect of the harmonics in the rotor frame. The basic expression for the mechanical speed of the BDFIG can be described as [9] ωr =

ω1 + ω2 p1 + p2

(2.8)

where ω1 and ω2 denote the angular electrical frequency of the PW and CW sides, respectively.

2.4.2 Operation Modes The operating modes of the BDFIG can be categorized based on the excitation of CW side into three modes, namely synchronous, sub-synchronous, and super-synchronous modes. According to (2.8), the synchronous mode of the BDFIG occurs when the PW side is directly connected to the load-side terminals and the CW side is excited from a dc source. In other words, the synchronous speed of the generator, namely the natural speed ωN , is given by (ω1 /[ p1 + p2 ]) [41]. The common connection under the excitation from a dc source can be considered as two phases in parallel and one phase in series with them.

50

2 Mathematical Modelling, Operation Characteristics …

In addition, the sub-synchronous mode of the adopted generator (the low-speed operation with a speed below its natural value) can be obtained by feeding the CW terminals from the MSC, as shown in Fig. 2.5, with a negative command control frequency. On the other hand, the generator can operate in the super-synchronous mode (the high-speed operation with a speed above its natural value) when the control-winding terminals are fed from the MSC with a positive command control frequency.

2.4.3 Power Flow Characteristics By neglecting the copper losses, the relationship of the PW and CW active power, P1 and P2 , can be obtained by [36] ω1 P1 = P2 ω2

(2.9)

From (2.1), it can be noted that the CW frequency is a negative value under the sub-synchronous speed, and vice versa. Besides, according to the relation in (2.9), the power flow of the standalone BDFIG system can be described in Fig. 2.6. In other words, the active power of the PW is directly fed to the load. On the other side, the power direction of the CW side is related to the operation speed range, i.e. the sub-

Fig. 2.6 The power flow of the standalone BDFIG system, a under sub-synchronous speed, b under super-synchronous speed

2.4 Operation Characteristics of the BDFIG Drive System

51

and super-synchronous speeds, which determine whether the CW absorbs or extracts electrical power. For all the modes of operation, by ignoring the copper and iron consumptions, the mechanical loss, and the converter loss in the generator, the balance equation of active power for the BDFIG can be expressed as Pm = P1 + P2 = Pout

(2.10)

where Pm is the mechanical input power and denotes Pout the electrical output power. Furthermore, the mechanical input torque of the BDFIG can be given as Tm =

Pm ωr

(2.11)

2.5 Basic Control Method for the Standalone BDFIG 2.5.1 CW Current Control To improve the performance of operation of the control system, the CW current is controlled by directly orienting the d-axis component of the CW current, i2d * , to its total vector, I 2 * [42, 43]. Consequently, the other quantity of the CW current in its q-axis, i2q * , is adjusted to the consequent zero value, as shown in Fig. 2.7. The dq-axis current expressions of the CW side [43] can be given as 

i 2d = K d u 2d + Dd i 2q = K q u 2q + Dq

(2.12)

where Dd and Dq represent the cross-coupling between both sides of the PW and CW, as illustrated by [43] L 1r L 2r ρ L 1r L 2r [Rr ρ + L r ω1 (ω1 − p2 ωr )] i 1q i 1d + (R2 + σ2 L 2 ρ)L r (R2 + σ2 L 2 ρ)L r2 (ω1 − p2 ωr )

2 ω1 (ω1 − p2 ωr ) L r L 2 + L 22r L r − L 22r Rr ρ i 2q − (R2 + σ2 L 2 ρ)L r2 (ω1 − p2 ωr )

Dd =

L 1r L 2r [ω1 Rr − L r (ω1 − p2 ωr )ρ] ω1 L 1r L 2r i 1q − i 1d (R2 + σ2 L 2 ρ)L r2 (ω1 − p2 ωr ) (R2 + σ2 L 2 ρ)L r σ2 L 2 L r ω1 + i 2d (R2 + σ2 L 2 ρ)L 1

Dq = −

52

2 Mathematical Modelling, Operation Characteristics …

Fig. 2.7 Control loop for CW current

In addition, K d and K q are the direct relationship between the dq-axis quantities of voltage and current for the CW side, as given by 

K d = K q = 1/(R2 + σ2 L 2 ρ) σ2 = 1 − L 22r /(L 2 L r )

.

(2.13)

2.5.2 Design of the DVC Control Strategy With the rotor-position encoder, the complete analysis of the direct voltage control (DVC) strategy for the standalone BDFIG in the ship shaft power generation application has been investigated in [43] to attain the desired reference voltage profile of PW side as illustrated in Fig. 2.8. As seen from this picure, the voltage of PW side can be maintained fixed at its set value under the condition of load or speed changes by regulating the current of CW side, I 2 through adjusting the corresponding PI controller. Moreover, the LPF2 is required in the presented DVC system to efficiently realize the voltage control loopbased on the calculation process of the PW voltage amplitude. The main purpose of the LPF2 in the presented DVC scheme is to eliminate any noise in the actual voltage signal caused by the measuring sensors. Furthermore, the control target of PW frequency can be realized by adjusting the frequency of CW side with the aid of (2.8) along with the variations of the BDFIG rotor speed. It is obvious from Fig. 2.8 that the rotor speed, ωrm , is detected by differentiating the measured rotor-position signal, θ rm , which can be given from a mechanical encoder.

2.5 Basic Control Method for the Standalone BDFIG

53

Fig. 2.8 Conceptual diagram of the DVC strategy for the standalone BDFIG

Therefore, the low-pass filter (LPF1) is used to clear the speed signal from any noise resulted from the differentiation step. In the final stage of the control system, the resulted gating signals for the converter of CW side are realized depending on the adjusted values of the CW dq-axis currents, with the aid of the presented control loop of CW current shown in Fig. 2.7, to obtain the desired voltage and frequency of the PW side with the principle of DVC.

2.5.3 Performance Analysis A practical standalone variable speed constant frequency (VSCF) ship shaft power generation system, based on a 60-kVA BDFIG, has been built in a 325 TEU container vessel from the Changjiang National Shipping Group of China. The performance analysis is implemented on the 60-kVA BDFIG experimental platform, whose details can be seen in Section A.2, Appendix. The dynamic performance of the proposed generator system is verified by four different experiments, and the results are shown in Figs. 2.9, 2.10, 2.11 and 2.12 [43]. The first experimental case studies the performance of the BDFIG under a step load change from the full load to the no load condition with the same speed operation of 600 r/min as illustrated in Fig. 2.9. It can be dedicated that a rapid reduction

1000 500 0 -500 -1000 0

PW & CW currents (A)

Rms of output line voltage (V)

Fig. 2.9 Performance test under the condition of the step load change from the full load to the no load with the rotor speed of 600 rpm

2 Mathematical Modelling, Operation Characteristics … Output line voltage (V)

54

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1.0

0.8

1

600 500 400 300 200 0

200

PW current

100

CW current

0 -100 -200 0

0.2

0.4

Time (s)

0.6

occurs to the CW current from 79.2 to 28.3 A during the sudden change in the load. Furthermore, a sudden increase in the output voltage is detected during the transient period of load change with an acceptable percentage value of 8.8% of the rated voltage. In the second experimental case shown in Fig. 2.10, the dynamic behavior of the BDFIG is studied under a step load change from the no load condition to 42 kW with the same operating rotor-speed of 400 r/min. It is obvious that a reduction occurs in the output voltage to about 91% of the rated value within 300 ms and then the voltage recovers. Moreover, during this condition of the step load change, the current of CW side increases from 28.2 to 115 A as illustrated in Fig. 2.10. The third test condition represents the performance of the BDFIG under the case of speed variation with the same load of 42 kW as shown in Figs. 2.11 and 2.12, in which the output voltage profile is maintained constant at its desired reference level with an error of less than ±1% of the rated values. Furthermore, it can be dedicated from Figs. 2.11 and 2.12 that a change of the current sequence for the CW side occurs during the transition between the sub/super-synchronous speed modes considering the operation period of the natural speed.

500 0 -500

Rms of output line voltage (V)

200

Output line voltage (V)

600

PW & CW currents (A)

-1000 0

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1.0

0.6

0.8

1

500 400 300 200 0

100

PW current CW current

0 -100 -200 0

0.2

0.4

Time (s)

1000 500 0 -500 -1000 0

Rms of output line voltage (V)

Fig. 2.11 Test analysis under the condition of speed variation from the super-synchronous to sub-synchronous speed with the load of 42 kW

55

1000

0.5

1

1.5

2

0.5

1

1.5

2

1.5

2

500 450 400 350 300 0 200

CW current (A)

Fig. 2.10 Experimental test under the condition of the step load change from the no load to 42 kW with the rotor speed of 400 rpm

Output line voltage (V)

2.5 Basic Control Method for the Standalone BDFIG

Phase B

Phase A

100 0 -100 -200 0

0.5

1 Time (s)

1000 500 0 -500 -1000 0

Rms of output line voltage (V)

Fig. 2.12 Experimental analysis under the condition of speed variation from the sub-synchronous to super-synchronous speed with the load of 42 kW

2 Mathematical Modelling, Operation Characteristics … Output line voltage (V)

56

0.5

1

1.5

2

0.5

1

1.5

2

500 450 400 350 300 0

CW current (A)

200 Phase B

Phase A

100 0 -100 -200 0

0.5

1 Time (s)

1.5

2

2.6 Summary This chapter has presented a detailed description of the standalone wound-rotor BDFIG in ship shaft applications. The complete dynamic model of the BDFIG and the main construction considerations have been discussed in details. In addition, the dynamic and steady-state models of the BDFIG have been conducted and analyzed. Furthermore, the intended target of the voltage and frequency control for the presented standalone power generation system has been explained with the principle of the DVC strategy. Comprehensive simulation and experimental results have confirmed the operation characteristics of the BDFIG system with the DVC strategy under different speed and load conditions.

References 1. Ledesma P, Usaola J (2005) Doubly fed induction generator model for transient stability analysis. IEEE Trans Energ Convers 20(2):388–397 2. Lei Y, Mullane A, Lightbody G et al (2006) Modeling of the wind turbine with a doubly fed induction generator for grid integration studies. IEEE Trans Energ Convers 21(1):257–264 3. Liao Y, Ran L, Putrus GA et al (2003) Evaluation of the effects of rotor harmonics in a doubly-fed induction generator with harmonic induced speed ripple. IEEE Trans Energ Convers 18(4):508–515

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4. Hughes FM, Anaya-Lara O, Jenkins N et al (2006) A power system stabilizer for DFIG-based wind generation. IEEE Trans Power Syst 21(2):763–772 5. Hughes FM, Anaya-Lara O, Jenkins N et al (2005) Control of DFIG-based wind generation for power network support. IEEE Trans Power Syst 20(4):1958–1966 6. Xu L, Cartwright P (2006) Direct active and reactive power control of DFIG for wind energy generation. IEEE Trans Energ Convers 21(3):750–758 7. Anaya-Lara O, Hughes FM, Jenkins N et al (2006) Contribution of DFIG-based wind farms to power system short-term frequency regulation. IEE Proc Gene Trans Dist 153(2):164–170 8. Mousa MG (2016) Vector control of wind-driven brushless doubly-fed reluctance generators. Electrical Power and Machines Engineering Dept., Faculty of Engineering, Tanta Univ., Egypt 9. Liu Y, Xu W, Zhu J et al (2019) Sensorless control of standalone brushless doubly fed induction generator feeding unbalanced loads in a ship shaft power generation system. IEEE Trans Ind Electron 66(1):739–749 10. Mousa MG, Allam SM, Rashad EM (2015) Maximum power tracking of a grid-connected wind-driven brushless doubly-fed reluctance generator using scalar control. In: 8th IEEE GCC conference and exhibition (GCCCE), Muscat, Oman, pp 1–6 11. Mousa MG, Allam SM, Rashad EM (2017) A comparative study of vector-control strategies for maximum wind-power extraction of a grid-connected wind-driven brushless doubly-fed reluctance generator. Aus J Electr Electron Eng 14:1–11 12. Mousa MG, Allam SM, Rashad EM (2018) Sensored and sensorless scalar-control strategy of a wind-driven BDFRG for maximum wind-power extraction. J Control Deci 5:209–227 13. Zhang J, Wang X, Wu T et al (2008) The principle and harmonic analysis of a new BDFM with tooth harmonic wound rotor using as a generator. In: International conference on electrical machines and systems (ICEMS), pp 3622–3626 14. Xiong F, Wang X (2014) Design of a low-harmonic-content wound rotor for the brushless doubly fed generator. IEEE Trans Energ Convers 29(1):158–168 15. Ruviaro M, Rüncos F, Sadowski N et al (2012) Analysis and test results of a brushless doubly fed induction machine with rotary transformer. IEEE Trans Ind Electron 59(6):2670–2677 16. Oraee A, Abdi E, Abdi S et al (2015) Effect of rotor winding structure on the BDFM equivalent circuit parameters. IEEE Trans Energ Convers 30(4):1660–1669 17. McMahon R, Malliband P, Tatlow M et al (2015) Rotor parameter determination for the brushless doubly fed induction machines. IET Electr Power Appl 9(8):549–555 18. Abdi S, Abdi E, Oraee V et al (2014) Equivalent circuit parameters for large brushless doubly fed machines (BDFMs). IEEE Trans Energ Convers 29(3):706-715 19. Wang X, McMahon RA, Tavner PJ (2007) Design of the brushless doubly-fed (induction) machine. In: IEEE international electric machines and drives conference, pp 1508–1513 20. Abdi S, Abdi E, Oraee A et al (2015) Optimization of magnetic circuit for brushless doubly fed machines. IEEE Trans Energ Convers 30(4):1611–1620 21. Abdi S, Grace A, Abdi E et al (2017) A new optimized rotor design for brushless doubly fed machines. In: 20th international conference on electrical machines and systems (ICEMS), pp 1–6 22. Fengge Z, Guangjian P, Fengxiang W et al (2009) Study on design of brushless doubly-fed machine with a new type rotor. In: IEEE international conference on industrial technology, pp 1–3 23. Chen X, Wang X (2015) A novel wound rotor type for brushless doubly fed induction generator. J Electr Eng Tech 2(2):595–602 24. Li S, Yang XP, Liu LF et al (2012) Research on grid-connected operation of novel variable speed constant frequency (VSCF) shaft generator system on modern ship. In: Proceedings of 15th international conference on electrical machines and systems (ICEMS), pp 1–5 25. Gully BH, Webber ME, Seepersad CC et al (2011) Shaft motor-generator design assessment for increased operational efficiency in container ships. In: Proceedings of ASME 5th international conference on energy sustainability, pp 1813–1819 26. Xu XY, Ye YZ, Chen C et al (2009) On control of power supply process of marine shaft generator. In: IEEE international conference on system, man and cybernetics, pp 4776–4779

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27. Peng L, Li YD, Chai JY et al (2007) Vector control of a doubly fed induction generator for stand-alone ship shaft generator systems. In: International conference on electrical machines and systems (ICEMS), pp 1033–1036 28. Phan VT, Lee HH, Chun TW (2010) An improved control strategy using a PI-resonant controller for an unbalanced stand-alone doubly-fed induction generator. J Power Electr 10(2):194–202 29. McMahon RA, Roberts PC, Wang X et al (2006) Performance of BDFM as generator and motor. IEE Proc Electr Power Appl 153(2):289–299 30. Poza J, Oyarbide E, Roye D et al (2006) Unified reference frame dq model of the brushless doubly fed machine. IEE Proc Electr Power Appl 153(5):726–734 31. Liu Y, Xu W, Zhi G et al (2017) Performance analysis of a stand-alone brushless doubly-fed induction generator using a new T-type steady-state model. J Power Electron 17(4):1027–1036 32. Shao S, Abdi E, McMahon R (2012) Low-cost variable speed drive based on a brushless doublyfed motor and a fractional unidirectional converter. IEEE Trans Ind Electron 59(1):317–325 33. Chen X, Wang X (2014) A novel wound rotor BDFIG for ship shaft generator application— control and prototype tests. In: 17th international conference on electrical machines and systems (ICEMS), pp 1196–1200 34. Liu G, Wang X (2015) Design and performance analysis of a 700 kW wound-rotor BDFIG for ship shaft generator application. IEEJ Trans Electr Electron Eng 11:112–123 35. Chen X, Wang X, Xiong F (2013) Research on excitation control for stand-alone wound rotor brushless doubly-fed generator system. In: International conference on electrical machines and systems (ICEMS), pp 663–667 36. Roberts PC (2004) A study of brushless doubly-fed (induction) machines. Dissertation, Cambridge University 37. Xu W, Liu Y (2020) Advanced control technologies for brushless doubly-fed induction machine. In: China Machine Press, pp 5–9 38. Liu Y, Ai W, Chen B et al (2016) Control design and experimental verification of the brushless doubly-fed machine for stand-alone power generation applications. IET Electr Power Appl 10(1):25–35 39. Allam SM (2009) Analysis and control characteristics of a brushless doubly-fed induction machine. Electrical Power and Machines Engineering Dept, Faculty of Engineering, Tanta University, Egypt 40. Wang X, Lin H, Wang Z (2017) Transient control of reactive current for line-side converter of brushless doubly fed induction generator in stand-alone operation. IEEE Trans Power Electron 32(10):8193–8203 41. Xu W, Hussien MG, Liu Y et al (2020) Sensorless control of ship shaft stand-alone BDFIGs based on reactive-power MRAS observer. IEEE J Emerg Sel Topics Power Electron 42. Xu W, Hussien MG, Liu Y et al (2020) Sensorless voltage control schemes for Brushless Doubly-Fed Induction Generators in stand-alone and grid-connected applications. IEEE Trans Energ Convers 35(4) 43. Liu Y, Ai W, Chen B et al (2016) Control design of the brushless doubly-fed machine for stand-alone VSCF ship shaft generator systems. J Power Electron 16(1):259–267

Chapter 3

Unbalanced and Low-Order Harmonic Voltages Rejection for Standalone BDFIGs

Abstract The brushless double-fed induction generator (BDFIG) is anticipated to be one of the important generators especially in the standalone generation mode in the future, due to the distinctive design of BDFIG without the slip rings and brushes. The stability of the amplitude and frequency of the PW voltage is the main control target in the standalone generation mode. Moreover, the standalone mode is oversensitive under abnormal operation conditions, especially under nonlinear and unbalanced loads. The nonlinear and unbalanced loads can cause severe distortion and unbalance for the PW current and voltage. In the PW voltage, the negativesequence component represents the effect of unbalanced loads, and the 3rd, 5th and 7th harmonic components indicate the effect of nonlinear loads. This chapter presents the negative-sequence voltage compensator, low-order harmonic voltage compensator and dual-resonant controller (DRC) for minimizing the nonlinear and unbalanced impacts. These control strategies presented in this chapter have been fully verified on a 30-kVA BDFIG test platform. Keywords Standalone operation · Negative-sequence voltage · Low-order harmonic voltage · Unbalanced load · Nonlinear load

3.1 Introduction In the standalone mode, the BDFIG is oversensitive to abnormal operation conditions, particularly unbalanced and nonlinear loads. With unbalanced loads, the negativesequence component causes severe unbalanced impact on the PW current and voltage. With three-phase nonlinear loads, the harmonic components mainly contain the 5th and 7th harmonics, which causes significant nonlinear effect of the PW current and voltage. Moreover, single-phase nonlinear loads would lead to negative-sequence and 3rd harmonic components, which indicates unbalanced and nonlinear influences of the PW current and voltage. The research motivations in this chapter can be briefly clarified as follows: 1.

Recognizing the significant effects on the standalone BDFIG system under nonlinear and unbalanced loads.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Y. Liu and W. Xu, Advances in Control Technologies for Brushless Doubly-fed Induction Generators, https://doi.org/10.1007/978-981-19-0424-0_3

59

60

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

2.

Analyzing the mathematical model of the BDFIG under nonlinear and unbalanced loads for finding an appropriate methodology to design control strategies that compensate for these adverse effects. Designing an effective control strategy for the standalone BDFIG operating under unbalanced loads to reduce the unbalance effect of the PW voltage. Designing an effective control strategy for the standalone BDFIG working with nonlinear loads to decrease the nonlinear influence on the PW voltage. Investigating a comprehensive control strategy for the standalone BDFIG to minimize both the nonlinear and unbalanced impacts of the PW voltage.

3. 4. 5.

3.2 Analysis of the Sources of Unbalanced and Low-Order Harmonic Voltages With unbalanced loads, the PW voltage and current contains many negative-sequence components, and the electromagnetic torque, power and dc bus voltage are with significant ripples [1]. Under nonlinear loads, the odd harmonics appear in the whole system [2]. Thus, the BDFIG may suffer from high current, severe vibration, and overheating, which can reduce the efficiency and lifetime of the BDFIG [3–6]. Also, the lifespan of the dc bus capacitor in the power converter would be decreased, due to the fluctuating currents flowing through the dc-bus capacitor [7, 8]. Besides, either unbalanced or nonlinear loads may lead to deterioration of the electrical apparatuses normally linked to the system.

3.2.1 Analysis Under Unbalanced Load The mathematical model of BDFIG can be expressed as [9] u 1 = R1 i 1 + sψ1 + jω1 ψ1

(3.1)

u 2 = R2 i 2 + sψ2 − j[ω1 − ( p1 + p2 )ωr ]ψ2

(3.2)

u r = Rr ir + sψr + j (ω1 − p1 ωr )ψr

(3.3)

ψ1 = L 1 i 1 + L 1r ir

(3.4)

ψ2 = L 2 i 2 + L 2r ir

(3.5)

ψr = L r ir + L 1r i 1 + L 2r i 2

(3.6)

3.2 Analysis of the Sources of Unbalanced and Low-Order …

61

Table 3.1 Frequencies in the rotor and stator of BDFIG with unbalanced loads PW frequency

Rotor frequency

CW frequency

Positive component

ω1

ω1 − p1 ωr

ω2

Negative component

−ω1

−ω1 − p1 ωr

2ω1 + ω2

where u, i and ψ represent the voltage, current and flux vectors, respectively; R and L represent the resistance and self-inductance, respectively; L 1r and L 2r represent the mutual inductance between the PW and rotor winding and the mutual inductance between the CW and rotor winding, respectively; s indicates derivative term. According to (2.8), the frequencies of the positive and negative components of the rotor and CW can be derived from the positive- and negative-sequence PW frequencies, respectively, which are listed in Table 3.1. In order to derive the dynamic vector model of the BDFIG with unbalanced loads, the positive and negative reference frames have to be established. The vector diagram for the positive and negative reference frames has been shown in Fig. 3.1, which describes the relationship among the PW stationary frame α 1 β 1 , the fundamental positive frame dq+ rotating at the angular speed of ω1 , and the negative frame dq− rotating at the angular speed of −ω1 . From Fig. 3.1, it can be noted that the negative frame rotates at the angular speed of −2ω1 relative to the corresponding positive one. Consequently, under unbalanced loads, the voltage, current and flux of the BDFIG can be split into the positive and negative components as follows [10, 11]: F+ =

F++  postive component

+

F−+  negative component

=

F++  postive component

+ F−− e− j2ω1 t   

(3.7)

negative component

where F symbolizes the flux, current and voltage of the PW, CW and rotor; the subscripts “−” and “+” indicate the negative and positive components, respectively; Fig. 3.1 Vector diagram describing the relationship among various reference frames under unbalanced loads

62

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

the superscripts “−” and “+” represent the negative and positive reference frames, respectively. From (3.7), the flux, voltage and current vectors of PW in the positive reference frame can be calculated as ⎧ + + − − j2ω1 t ⎪ ⎨ u 1 = u 1+ + u 1− e + − − j2ω1 t (3.8) i 1+ = i 1+ + i 1− e ⎪ ⎩ + + − − j2ω1 t ψ1 = ψ1+ + ψ1− e By substituting (3.8) into (3.1) and (3.4), the negative and positive components of the PW voltage equations in the corresponding reference frames can be expressed as + + + u+ 1+ = R1 i 1+ + sψ1+ + jω1 ψ1+

(3.9)

− − − u− 1− = R1 i 1− + sψ1− − jω1 ψ1− .

(3.10)

Similarly, the negative and positive components of the PW flux equations can be presented by + + = L 1 i 1+ + L 1r ir++ ψ1+

(3.11)

− − ψ1− = L 1 i 1− + L 1r ir−− .

(3.12)

A similar procedure can also be used to derive the negative- and positive-sequence flux and voltage equations of the rotor and CW. Thus, the dynamic vector model of the BDFIG with unbalanced loads can be divided into two sets of equations as follows: ⎧ + + + + u 1+ = R1 i 1+ + sψ1+ + jω1 ψ1+ ⎪ ⎪ ⎪ + + ⎪ ⎪ ψ1+ = L 1 i 1+ + L 1r ir++ ⎪ ⎪ ⎪ ⎨ u + = R i + + sψ + + j[ω − ( p + p )ω ]ψ + 2 2+ 1 1 2 r 2+ 2+ 2+ + + + ⎪ ψ = L i + L i 2 2+ 2r r + ⎪ 2+ ⎪ ⎪ + + + + ⎪ ⎪ u = R i + sψ r r+ ⎪ r+ r + + j (ω1 − p1 ωr )ψr + ⎪ ⎩ + + + ψr + = L r ir++ + L 1r i 1+ + L 2r i 2+ ⎧ − − − − u 1− = R1 i 1− + sψ1− − jω1 ψ1− ⎪ ⎪ ⎪ ⎪ ψ− = L i− + L i− ⎪ 1 1− 1r r − ⎪ 1− ⎪ ⎪ ⎨ u − = R i − + sψ − + j[−ω − ( p + p )ω ]ψ − 2 2− 1 1 2 r 2− 2− 2− − − − ⎪ ψ = L i + L i 2 2r ⎪ 2− r− 2− ⎪ ⎪ ⎪ ⎪ u r−− = Rr ir−− + sψr−− + j (−ω1 − p1 ωr )ψr−− ⎪ ⎪ ⎩ − − − ψr − = L r ir−− + L 1r i 1− + L 2r i 2−

(3.13)

(3.14)

3.2 Analysis of the Sources of Unbalanced and Low-Order …

63

where (3.13) and (3.14) are the positive and negative component equations, respectively.

3.2.2 Analysis Under Three-Phase Nonlinear Load With three-phase nonlinear loads, the PW terminal voltage of the standalone BDFIG include severe harmonic components, which would affect the operation of the other normal loads connected to this system. Among these harmonic components, the 5th and 7th harmonic components are the most significant ones, which rotate at the angular speeds of −5ω1 and 7ω1 , respectively [12]. The frequencies of the 5th and 7th harmonic components of the rotor and CW can be derived from the corresponding PW frequencies, which are listed in Table 3.2. Similarly, the fundamental positive frame and the 5th and 7th harmonic frames are established to derive the dynamic vector model of the BDFIG with nonlinear loads. The vector diagram of the fundamental and harmonic frames has been shown in Fig. 3.2, which presents the relationship among the PW stationary frame α 1 β 1 , the fundamental positive frame dq+ rotating at the angular speed of ω1 , the 7th harmonic frame dq7 rotating at the angular speed of 7ω1 , and the 5th harmonic frame dq5 rotating at the angular speed of −5ω1 . Table 3.2 Frequencies in the rotor and stator of BDFIG with nonlinear loads PW frequency

Rotor frequency

CW frequency

Fundamental positive component

ω1

ω1 − p1 ωr

ω2

5th harmonic component

−5ω1

−5ω1 − p1 ωr

6ω1 + ω2

7th harmonic component

7ω1

7ω1 − p1 ωr

−6ω1 + ω2

Fig. 3.2 Vector diagram showing the relationship among different frames under nonlinear loads

64

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

From Fig. 3.2, it can be inferred that the 5th harmonic frame rotates at the angular speed of −6ω1 relative to the fundamental positive frame, and the 7th harmonic frame rotates at the angular speed of 6ω1 relative to the fundamental positive frame. Consequently, under three-phase nonlinear loads, the voltage, current and flux of the BDFIG can be split into the fundamental positive component and the 5th and 7th harmonic components as follows [13]: F+ = =

F++ 

F5+ 

+

fundamental postive component

5th harmonic component

F++

F55 e− j6ω1 t

+

 fundamental postive component





F7+ 

+



7th harmonic component

+

5th harmonic component

F 7 e j6ω1 t  7  

(3.15)

7th harmonic component

where F represents the flux, current and voltage of the PW, CW and rotor; the subscripts “+”, “5” and “7” indicate the fundamental positive-sequence component, the 5th harmonic component and the 7th harmonic component, respectively; the superscripts “+”, “5” and “7” represent the fundamental positive frame, the 5th harmonic frame and the 7th harmonic frame, respectively. From (3.15), the PW voltage, current and flux vectors in the fundamental positive frame can be illustrated as ⎧ + + 5 − j6ω1 t + u 71_7 e j6ω1 t ⎪ ⎨ u 1 = u 1+ + u 1_5 e + 5 7 . (3.16) i 1+ = i 1+ + i 1_5 e− j6ω1 t + i 1_7 e j6ω1 t ⎪ ⎩ + + 5 7 ψ1 = ψ1+ + ψ1_5 e− j6ω1 t + ψ1_7 e j6ω1 t By substituting (3.16) into (3.1) and (3.4), the fundamental positive and harmonic voltage equations of PW in the corresponding frames can be obtained by + + + u+ 1+ = R1 i 1+ + sψ1+ + jω1 ψ1+

(3.17)

5 5 5 u 51_5 = R1 i 1_5 + sψ1_5 + j (−5)ω1 ψ1_5

(3.18)

7 7 7 u 71_7 = R1 i 1_7 + sψ1_7 + j7ω1 ψ1_7 .

(3.19)

Similarly, the fundamental positive and harmonic flux equations of PW can be given by + + = L 1 i 1+ + L 1r ir++ ψ1+

(3.20)

5 5 ψ1_5 = L 1 i 1_5 + L 1r ir5_5

(3.21)

7 7 ψ1_7 = L 1 i 1_7 + L 1r ir7_7 .

(3.22)

3.2 Analysis of the Sources of Unbalanced and Low-Order …

65

The fundamental positive and harmonic voltage and flux equations for the rotor and CW can be calculated with the same way for the PW. Finally, the dynamic vector model of the BDFIG with nonlinear loads can be split into three sets of equations as follows: ⎧ + + + + u 1+ = R1 i 1+ + sψ1+ + jω1 ψ1+ ⎪ ⎪ ⎪ + + ⎪ ⎪ ψ1+ = L 1 i 1+ + L 1r ir++ ⎪ ⎪ ⎪ ⎨ u + = R i + + sψ + + j[ω − ( p + p )ω ]ψ + 2 2+ 1 1 2 r 2+ 2+ 2+ (3.23) + + + ⎪ ψ2+ = L 2 i 2+ + L 2r ir + ⎪ ⎪ ⎪ ⎪ ⎪ u r++ = Rr ir++ + sψr++ + j (ω1 − p1 ωr )ψr++ ⎪ ⎪ ⎩ + + + ψr + = L r ir++ + L 1r i 1+ + L 2r i 2+ ⎧ 5 5 5 ⎪ u 51_5 = R1 i 1_5 + sψ1_5 + j (−5)ω1 ψ1_5 ⎪ ⎪ ⎪ ⎪ 5 5 ⎪ ψ1_5 = L 1 i 1_5 + L 1r ir5_5 ⎪ ⎪ ⎪ ⎪ ⎨ u 5 = R2 i 5 + sψ 5 + j[−5ω1 − ( p1 + p2 )ωr ]ψ 5 2_5 2_5 2_5 2_5 (3.24) 5 5 5 ⎪ ⎪ ⎪ ψ2_5 = L 2 i 2_5 + L 2r ir _5 ⎪ ⎪ ⎪ ⎪ u r5_5 = Rr ir5_5 + sψr5_5 + j (−5ω1 − p1 ωr )ψr5_5 ⎪ ⎪ ⎪ ⎩ ψ5 = L i5 + L i5 + L i5 r _5

r r _5

1r 1_5

2r 2_5

⎧ 7 7 7 7 u 1_7 = R1 i 1_7 + sψ1_7 + j7ω1 ψ1_7 ⎪ ⎪ ⎪ ⎪ 7 7 ⎪ ψ1_7 = L 1 i 1_7 + L 1r ir7_7 ⎪ ⎪ ⎪ ⎪ ⎨ u 7 = R2 i 7 + sψ 7 + j[7ω1 − ( p1 + p2 )ωr ]ψ 7 2_7 2_7 2_7 2_7 7 7 7 ⎪ ψ = L i + L i ⎪ 2_7 2 2_7 2r r _7 ⎪ ⎪ ⎪ ⎪ u 7 = Rr i 7 + sψ 7 + j (7ω1 − p1 ωr )ψ 7 ⎪ r _7 r _7 r _7 r _7 ⎪ ⎪ ⎩ 7 7 7 7 ψr _7 = L r ir _7 + L 1r i 1_7 + L 2r i 2_7

(3.25)

where (3.23)–(3.25) represent the mathematical equations of the fundamental positive, 5th harmonic and 7th harmonic components, respectively.

3.2.3 Analysis Under Unbalanced Plus Nonlinear Load Under the nonlinear plus unbalanced load, both of unbalanced and nonlinear effects would appear in the BDFIG at the same time. Table 3.3 shows the frequencies in the stator and rotor of the BDFIG under the unbalanced plus nonlinear load. The vector diagram with four different reference frames shown in Fig. 3.3 presents the relationship among the PW stationary frame α 1 β 1 , the fundamental positive frame dq+ , the negative frame dq− , the 7th harmonic frame dq7 and the 5th harmonic frame dq5 .

66

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Table 3.3 Frequencies in the rotor and stator of BDFIG with the unbalanced plus nonlinear load PW frequency

Rotor frequency

CW frequency

ω1

ω1 − p1 ωr

ω2

Negative component

−ω1

−ω1 − p1 ωr

2ω1 + ω2

5th harmonic component

−5ω1

−5ω1 − p1 ωr

6ω1 + ω2

7th harmonic component

7ω1

7ω1 − p1 ωr

−6ω1 + ω2

Fundamental positive component

Fig. 3.3 Vector diagram showing the relationship among different frames under the unbalanced plus nonlinear load

Combining (3.7) and (3.15), the voltage, flux and current vectors of the BDFIG can be divided into the fundamental positive component, the negative component and the 5th and 7th harmonic components as follows: F+ =

F++ 

+

fundamental postive component

F5+

+

F7+  7th harmonic component

5th harmonic component

F++ 

=

negative component

+



+ F−− e− j2ω1 t   

fundamental postive component

F55 e− j6ω1 t

+

 5th



F−+ 



harmonic component

+

negative component

F 7 e j6ω1 t  7   7th

.

(3.26)

harmonic component

From (3.26), the voltage, flux and current vectors of PW in the fundamental positive frame can be illustrated as

3.2 Analysis of the Sources of Unbalanced and Low-Order …

⎧ + + − − j2ω1 t + u 51_5 e− j6ω1 t + u 71_7 e j6ω1 t ⎪ ⎨ u 1 = u 1+ + u 1− e + − − j2ω1 t 5 7 . i 1+ = i 1+ + i 1− e + i 1_5 e− j6ω1 t + i 1_7 e j6ω1 t ⎪ ⎩ + + − − j2ω1 t 5 − j6ω1 t 7 j6ω1 t ψ1 = ψ1+ + ψ1− e + ψ1_5 e + ψ1_7 e

67

(3.27)

By substituting (3.27) into (3.1) and (3.4), the PW positive, negative and harmonic voltage equations in the corresponding frames can be expressed as + + + u+ 1+ = R1 i 1+ + sψ1+ + jω1 ψ1+

(3.28)

− − − u− 1− = R1 i 1− + sψ1− − jω1 ψ1−

(3.29)

5 5 5 u 51_5 = R1 i 1_5 + sψ1_5 + j (−5)ω1 ψ1_5

(3.30)

7 7 7 u 71_7 = R1 i 1_7 + sψ1_7 + j7ω1 ψ1_7 .

(3.31)

And, the PW positive, negative and harmonic flux equations can be derived as + + = L 1 i 1+ + L 1r ir++ ψ1+

(3.32)

− − ψ1− = L 1 i 1− + L 1r ir−−

(3.33)

5 5 ψ1_5 = L 1 i 1_5 + L 1r ir5_5

(3.34)

7 7 ψ1_7 = L 1 i 1_7 + L 1r ir7_7 .

(3.35)

The similar derivation process can also be used to obtain the positive, negative and harmonic flux equations of the CW and rotor. Consequently, the dynamic vector model of the BDFIG with the unbalanced plus nonlinear load can be expressed by four sets of equations as follows: ⎧ + + + + u 1+ = R1 i 1+ + sψ1+ + jω1 ψ1+ ⎪ ⎪ ⎪ + ⎪ ⎪ ψ + = L 1 i 1+ + L 1r ir++ ⎪ ⎪ 1+ ⎪ ⎨ u + = R i + + sψ + + j[ω − ( p + p )ω ]ψ + 2 2+ 1 1 2 r 2+ 2+ 2+ + + + ⎪ ψ2+ = L 2 i 2+ + L 2r ir + ⎪ ⎪ ⎪ ⎪ ⎪ u r++ = Rr ir++ + sψr++ + j (ω1 − p1 ωr )ψr++ ⎪ ⎪ ⎩ + + + ψr + = L r ir++ + L 1r i 1+ + L 2r i 2+

(3.36)

68

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

⎧ − − − − u 1− = R1 i 1− + sψ1− − jω1 ψ1− ⎪ ⎪ ⎪ − − − ⎪ ⎪ ψ1− = L 1 i 1− + L 1r ir − ⎪ ⎪ ⎪ ⎨ u − = R i − + sψ − + j[−ω − ( p + p )ω ]ψ − 2 2− 1 1 2 r 2− 2− 2− − − − ⎪ ψ = L i + L i 2 2r ⎪ 2− r− 2− ⎪ ⎪ ⎪ ⎪ u r−− = Rr ir−− + sψr−− + j (−ω1 − p1 ωr )ψr−− ⎪ ⎪ ⎩ − − − ψr − = L r ir−− + L 1r i 1− + L 2r i 2− ⎧ 5 5 5 ⎪ u 51_5 = R1 i 1_5 + sψ1_5 + j (−5)ω1 ψ1_5 ⎪ ⎪ ⎪ ⎪ 5 5 ⎪ ψ1_5 = L 1 i 1_5 + L 1r ir5_5 ⎪ ⎪ ⎪ ⎪ ⎨ u 5 = R2 i 5 + sψ 5 + j[−5ω1 − ( p1 + p2 )ωr ]ψ 5 2_5

2_5

2_5

2_5

5 5 ⎪ ψ2_5 = L 2 i 2_5 + L 2r ir5_5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u r5_5 = Rr ir5_5 + sψr5_5 + j (−5ω1 − p1 ωr )ψr5_5 ⎪ ⎪ ⎪ ⎩ ψ5 = L i5 + L i5 + L i5 r r _5 1r 1_5 2r 2_5 r _5 ⎧ 7 7 7 7 u 1_7 = R1 i 1_7 + sψ1_7 + j7ω1 ψ1_7 ⎪ ⎪ ⎪ ⎪ 7 7 7 ⎪ ⎪ ⎪ ψ1_7 = L 1 i 1_7 + L 1r ir _7 ⎪ ⎪ ⎨ u 7 = R2 i 7 + sψ 7 + j[7ω1 − ( p1 + p2 )ωr ]ψ 7 2_7 2_7 2_7 2_7 7 7 7 ⎪ ψ2_7 = L 2 i 2_7 + L 2r ir _7 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u r7_7 = Rr ir7_7 + sψr7_7 + j (7ω1 − p1 ωr )ψr7_7 ⎪ ⎪ ⎩ 7 7 7 ψr _7 = L r ir7_7 + L 1r i 1_7 + L 2r i 2_7

(3.37)

(3.38)

(3.39)

where (3.36)–(3.39) are the mathematical equations of the fundamental positive, negative, 5th harmonic, and 7th harmonic components, respectively.

3.2.4 Analysis Under Single-Phase Nonlinear Load The single-phase nonlinear load under consideration is a two-pulse diode rectifier (i.e. full-wave center-tapped rectifier) [14]. According to [15], the characteristic harmonic currents are the negative-sequence (kn − 1) and positive-sequence (kn + 1) orders, where k is the pulse number of the rectifier and n an integer. When n = 1, it can get the fundamental component in negative sequence (2 * 1 − 1 = 1) and the 3rd harmonic in positive sequence (2 * 1 + 1 = 3), which are the most significant harmonics to be eliminated. Table 3.4 presents the frequencies in the rotor and stator of BDFIG with single-phase nonlinear loads. The fundamental positive frame, 3rd harmonic frame and negative frame are built to derive the dynamic vector model of the BDFIG with single-phase nonlinear loads. The vector diagram for fundamental and harmonics frames, as shown in Fig. 3.4,

3.2 Analysis of the Sources of Unbalanced and Low-Order …

69

Table 3.4 Frequencies in the rotor and stator of BDFIG with single-phase nonlinear loads PW frequency

Rotor frequency

CW frequency

ω1

ω1 − p1 ωr

ω2

Negative component

−ω1

−ω1 − p1 ωr

2ω1 + ω2

3rd harmonic component

3ω1

3ω1 − p1 ωr

−2ω1 + ω2

Fundamental positive component

Fig. 3.4 Vector diagram showing the relationship among different frames under single-phase nonlinear loads

reveals the relationship among the PW stationary frame α 1 β 1 , the fundamental positive frame dq+ rotating with at the angular speed of ω1 , the 3rd harmonic frame dq3 rotating at the angular speed of 3ω1 , and the negative frame dq− rotating at the angular speed of −ω1 . From Fig. 3.4, it can be concluded that the negative frame and 3rd harmonic frame rotate at the angular speeds of −2ω1 and 2ω1 relative to the fundamental positive frame, respectively. Consequently, under single-phase nonlinear loads, the voltage, current and flux of the BDFIG can be described by the fundamental positive component, the negative component and the 3rd harmonic component as follows: F + = F++ + F−+ + F3+ = F++ +

F − e− j2ω1 t +  −   Negative component

F 3 e j2ω1 t  3   3rd

.

(3.40)

harmonic component

From (3.40), the voltage, flux and current vectors of PW in the fundamental positive frame can be illustrated as ⎧ + + − − j2ω1 t + u 31_3 e j2ω1 t ⎪ ⎨ u 1 = u 1+ + u 1− e + − − j2ω1 t 3 . i 1+ = i 1+ + i 1− e + i 1_3 e j2ω1 t ⎪ ⎩ + + − − j2ω1 t 3 j2ω1 t ψ1 = ψ1+ + ψ1− e + ψ1_3 e

(3.41)

By substituting (3.41) into (3.1) and (3.4), the PW positive, negative, and 3rd harmonic voltage equations in the corresponding frames can be presented by

70

3 Unbalanced and Low-Order Harmonic Voltages Rejection … + + + u+ 1+ = R1 i 1+ + sψ1+ + jω1 ψ1+

(3.42)

− − − u− 1− = R1 i 1− + sψ1− − jω1 ψ1−

(3.43)

3 3 3 u 31_3 = R1 i 1_3 + sψ1_3 + j3ω1 ψ1_3 .

(3.44)

Similarly, the positive, negative and 3rd harmonic flux equations of PW can be given by + + = L 1 i 1+ + L 1r ir++ ψ1+

(3.45)

− − ψ1− = L 1 i 1− + L 1r ir−−

(3.46)

3 3 ψ1_3 = L 1 i 1_3 + L 1r ir3_3 .

(3.47)

The derivation method for (3.42)–(3.47) can also be used to derive the positive, negative and 3rd harmonic voltage and flux equations for the CW and rotor. As a result, the dynamic vector model of the BDFIG with single-phase nonlinear loads can be split into three sets of equations as follows: ⎧ + + + + u 1+ = R1 i 1+ + sψ1+ + jω1 ψ1+ ⎪ ⎪ ⎪ ⎪ ψ+ = L i+ + L i+ ⎪ 1 1+ 1r r + ⎪ 1+ ⎪ ⎪ + + + + ⎨ u 2+ = R2 i 2+ + sψ2+ + j[ω1 − ( p1 + p2 )ωr ]ψ2+ + + + ⎪ ⎪ ψ2+ = L 2 i 2+ + L 2r ir + ⎪ ⎪ ⎪ ⎪ u r++ = Rr ir++ + sψr++ + j (ω1 − p1 ωr )ψr++ ⎪ ⎪ ⎩ + + + ψr + = L r ir++ + L 1r i 1+ + L 2r i 2+ ⎧ − − − − u 1− = R1 i 1− + sψ1− − jω1 ψ1− ⎪ ⎪ ⎪ − − ⎪ ⎪ ψ = L 1 i 1− + L 1r ir−− ⎪ ⎪ 1− ⎪ ⎨ u − = R i − + sψ − + j[−ω − ( p + p )ω ]ψ − 2 2− 1 1 2 r 2− 2− 2− − − − ⎪ ψ2− = L 2 i 2− + L 2r ir − ⎪ ⎪ ⎪ ⎪ ⎪ u r−− = Rr ir−− + sψr−− + j (−ω1 − p1 ωr )ψr−− ⎪ ⎪ ⎩ − − − ψr − = L r ir−− + L 1r i 1− + L 2r i 2−

(3.48)

(3.49)

3.2 Analysis of the Sources of Unbalanced and Low-Order …

⎧ 3 3 3 3 u 1_3 = R1 i 1_3 + sψ1_3 + j3ω1 ψ1_3 ⎪ ⎪ ⎪ ⎪ 3 3 ⎪ ψ1_3 = L 1 i 1_3 + L 1r ir3_3 ⎪ ⎪ ⎪ ⎪ ⎨ u 3 = R2 i 3 + sψ 3 + j[3ω1 − ( p1 + p2 )ωr ]ψ 3 2_3 2_3 2_3 2_3 3 3 3 ⎪ ψ2_3 = L i + L i ⎪ 2 1r 2_3 r _3 ⎪ ⎪ ⎪ 3 3 3 3 ⎪ ⎪ u = R i + sψ r r _3 r _3 r _3 + j (3ω1 −1 ωr )ψr _3 ⎪ ⎪ ⎩ 3 3 3 ψr _3 = L r ir3_3 + L 1r i 1_3 + L 2r i 2_3

71

(3.50)

where (3.48)–(3.50) represent the mathematical equations of the positive, negative, and 3rd harmonic components, respectively.

3.3 Unbalanced Voltage Compensation When the BDFIG works with an unbalance load, the negative component causes a serious unbalanced impact on the current and voltage of PW. The research motivation of this section is to find an effective control method to decrease the unbalance impact on the PW voltage for the BDFIG under unbalanced loads [16]. The proposed strategy contains two parts. The first part is the conventional DVC method to regulate the amplitude and frequency of the PW voltage in positive sequence. The second part is the proposed negative-sequence compensator to decrease the unbalanced impact on the PW voltage via the MSC. The negativesequence component of the PW voltage is extracted by the dual second-order generalized integrator (DSOGI) filter. The PI controller is employed to calculate the CW current reference value that can compensate for the unbalanced PW voltage. The PIR controller is used to adjust the CW current in the fundamental positive frame.

3.3.1 Conventional Direct Voltage Control Scheme Because of its simplicity and robustness, the DVC is usually adopted in the standalone mode to keep the amplitude and frequency of the PW voltage stable under the variation of the rotor speed and load [17, 18]. The conventional DVC scheme is presented in Fig. 3.5 [19], in which the amplitude of the PW voltage is adjusted by the CW current reference value in d axis with setting the CW current reference value in q axis to zero. The actual PW voltage amplitude |U 1 | can be calculated as |U1 | =

u 21d + u 21q .

(3.51)

72

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.5 Conventional DVC control scheme

The CW frequency must be varied instantly to keep the frequency of the PW voltage equal to 50 Hz when the rotor speed is changed. The dc bus capacitor voltage is regulated via the LSC. The LSC has two functions: controlling the voltage of the dc bus capacitor and realizing bidirectional power flow for CW. The vector control based on the PW voltage orientation represents the common control strategy used in the LSC [20]. The reactive power of the LSC is adjusted via the q-axis current, while dc bus voltage is adjusted by the d-axis current. The basic control strategy is depicted in Fig. 3.5.

3.3 Unbalanced Voltage Compensation

73

3.3.2 Design of Negative-Sequence Voltage Compensator 3.3.2.1

Basic Theory

The proposed strategy is based on the linear relationship between the CW current and PW voltage in the negative sequence, which can be derived from the BDFIG model as follows. From (3.1)–(3.6), the relationship between the CW and PW currents in the positive sequence can be given by i 2 = α2 i 1 − α4 ψ1 − j (α1 i 1 − α3 ψ1 )/(ω1 − p1 ωr )

(3.52)

α1 = (Rr L 1 )/(L 1r L 2r )

(3.53)

α2 = (L 1 L r σ1 )/(L 1r L 2r )

(3.54)

α3 = Rr /(L 1r L 2r )

(3.55)

α4 = L r /(L 1r L 2r )

(3.56)

σ1 = 1 − L 21r /L r L 1 .

(3.57)

From (3.14), the relationship between the CW and PW currents in the negative sequence can be derived as − − − − − = α2 i 1− − α4 ψ1− + j (α1 i 1− − α3 ψ1− )/(ω1 + p1 ωr ) i 2−

(3.58)

Arranging (3.58), the negative-sequence PW current can be expressed by a function of the negative-sequence CW current as follows:  − − − = i 2− + α4 ψ1− +j i 1−

 − α3 ψ1− α1 / α2 + j . ω1 + p1 ωr ω1 + p1 ωr

(3.59)

From (3.14), the negative-sequence d- and q-axis PW voltage can be expressed as −dq

−dq

−dq

u 1− = R1 i 1− + s(L 1 i 1

−dq

−dq

+ L 1r ir−dq ) − jω1 (L 1 i 1− + L 1r ir − ).

(3.60)

− − where F dq represents (F d + jF q ), and the variable F indicate u − 1− , i 1− and i r . When the derivative terms are equal to zero in steady state, Eq. (3.60) can be simplified as −dq

−dq

−dq

−dq

u 1− = R1 i 1− − jω1 (L 1 i 1− + L 1r ir − ).

(3.61)

74

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Substituting (3.59) and (3.12) into (3.61), the negative-sequence PW voltage can be obtained by

−dq

−dq

−dq

α3 (L 1 i 1− +L 1r ir − ) −dq −dq −dq (R1 − jω1 L 1 ) i 2− + α4 (L 1 i 1− + L 1r ir − ) + j ω1 + p1 ωr

u 1− =

α

α2 + j ω + p1 ωr 1 1

 −dq

− jω1 L 1r ir − .

(3.62)

From (3.62), if the negative-sequence rotor current and PW current are considered as additional disturbances, the negative-sequence PW voltage can be controlled via adjusting the negative-sequence CW current according to the their linear relationship.

3.3.2.2

Controller Design and Parameters Tuning

Figure 3.6 presents the block diagram of the proposed negative-sequence voltage compensator, where the DSOGI filter is employed to extract the negative-sequence PW voltage [21], and the PI controller can produce the additional compensation current for CW to minimize the unbalanced influence of the PW voltage. The CW reference current in the negative frame can be calculated by  Ki  0 − u− 1d− s   Ki  − u 1q− − 0 = Kp + s

− i 2d− =

− i 2q−



Kp +

(3.63) (3.64)

where K i is the integral gain and K p the proportional gain. The obtained CW compensation current in the negative frame has to be transformed to the fundamental positive frame via employing twice PW phase angle 2θ 1 . Afterwards, the CW reference currents for controlling the PW positive and negative voltages can add up to obtain the total CW reference current. The total reference current of CW in the positive frame is given by

Fig. 3.6 Block diagram of the negative-sequence voltage compensator

3.3 Unbalanced Voltage Compensation

75

Fig. 3.7 Overall control scheme for unbalanced voltage compensation of PW +∗ +∗ +∗ i 2dq = i 2dq+ + i 2dq− =

+∗ i 2dq+    Postive-sequence component

+

−∗ i 2dq− e− j2ω1 t   

.

(3.65)

Negative-sequence component

The overall control scheme for unbalanced voltage compensation of PW is presented in Fig. 3.7. The CW current control loop in Fig. 3.7 utilizes the PIR controller, where the R controller regulates the CW current in the negative sequence to eliminate the negative-sequence PW voltage, and the PI controller addresses the CW current in the positive sequence to control the amplitude of the PW voltage. The PI parameters are tunned with the classic Ziegler-Nichols strategy [22]. This strategy begins via setting the integral gain (K i ) to be zero, then increases the proportional gain (K p ) until the system becomes unstable, and finally get the oscillation at the frequency of f 0 . The value of K p at the point of instability is called the K max . And then, the K p is reduced to a predefined value, and the K i is set as a function of f 0 . In general, the values of K p and K i can be tuned according to Table 3.5. Figure 3.8 depicts the block diagram of the DSOGI filter, which consists of two quadrature signal generators (QSGs) based on the SOGI and positive–negative signal

76

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Table 3.5 The proportional and integral gains tuned by the Ziegler-Nichols strategy

Parameter

Kp

Ki

Value

0.45K max

1.2f 0

Fig. 3.8 Block diagram of DSOGI

calculator (PNSC). From Fig. 3.8, the characteristic of the transfer functions of the SOGI can be calculated as follows: D(s) =

kω1 s u 1 (s) = 2 u 1 (s) s + kω1 s + ω12

(3.66)

Q(s) =

kω12 qu 1 (s) = 2 u 1 (s) s + kω1 s + ω12

(3.67)

where qu  and u 1 represent the filtered quadrature and direct values of u 1 , respectively. In addition, k indicates the damping factor. From (3.66) and (3.67), Q(s) is a low-pass filter, and D(s) is a band-pass one. The instantaneous symmetrical component (ISC) theory based on the αβ-axis frame represents the working principle of the DSOGI as illustrated as follows. The Clark transformation is employed to transform the PW voltage from the three-phase (abc) frame to the stationary (αβ) frame. And then, two SOGI-QSGs are employed to generate the filtered quadrature and direct versions of uα and uβ , i.e. u 1α , u 1β , qu 1α , and qu 1β , respectively. The PNSC block deals with these signals as

3.3 Unbalanced Voltage Compensation

77

inputs and obtains the negative- and positive-sequence αβ components based on the ISC theory. The output signals of the PNSC can be calculated as   u+ 1α+ = u 1α − qu 1β

(3.68)

  u+ 1β+ = qu 1α + u 1β

(3.69)

  u− 1α− = qu 1β + u 1α

(3.70)

  u− 1β− = u 1β − qu 1α

(3.71)

where q = e− j (π/2) represents a phase-shift operator in time-domain to acquire quadrature-phase waveform, superscript  is an indication to filtered signals. − The output signals u − 1α− and u 1β− are sent to the Park transformation block to − calculate the corresponding signals in the negative fame, i.e. u − 1d− and u 1q− . Finally, − both u − 1d− and u 1q− are input to the PI controllers to produce the CW current compensation commands. The disturbance rejection capability, the stability margin, and the transient response are the main factors that are considered into the design procedure of the DSOGI [23].

3.3.3 Simulation Results The simulation is carried out on a 30-kVA BDFIG, whose parameters can be seen in Sect. A.3, Appendix. The unbalanced load (12, 12 and 6  in three phases) is employed to realize the unbalanced impact.

3.3.3.1

Conventional Method

Three typical tests are employed to verify the performance of the conventional method with balanced and unbalanced loads. Firstly, the effectiveness of the conventional strategy is investigated at the sub-synchronous speed of 600 rpm. Figures 3.9 and 3.10 clarify the PW voltage amplitude and three-phase PW voltage with the balanced and unbalance loads, respectively. The unbalance effect of the PW voltage is completely apparent in Figs. 3.9b and 3.10b once the unbalanced load is added to the system. Figure 3.11 illustrates the amplitude of the negative-sequence PW voltage. With the balanced load, the amplitude of the negative-sequence PW voltage is around 1 V, as calculated by

78

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.9 Simulation results of the PW voltage amplitude with the conventional method at 600 rpm: a balanced load, b unbalanced load

 − − U  = (u )2 + (u − )2 . 1− 1q− 1d−

(3.72)

With the unbalanced load, the amplitude of the negative-sequence PW voltage is increased to 25 V. In Fig. 3.12, the distortion of the CW current is clear after adding the unbalanced load to the system. Secondly, the effectiveness of conventional strategy is investigated at the supersynchronous speed of 900 rpm. Figures 3.13 and 3.14 clarify the PW voltage amplitude and three-phase PW voltage, respectively. The influence of unbalanced of PW voltage is very clear in Figs. 3.13b and 3.14b after adding three-phase unbalanced load to the system. Figure 3.15 illustrates the amplitude of the negative-sequence PW voltage. Under the balanced load, the PW voltage amplitude in negative sequence is around 0.5 V. And, it reaches about 23 V under the unbalanced load. In Fig. 3.16, the distortion of the current of CW is clear when adding the unbalanced load to the system.

3.3 Unbalanced Voltage Compensation

79

Fig. 3.10 Simulation results of the PW voltage with the conventional control strategy at 600 rpm: a balanced load, b unbalanced load

The third simulation is implemented to verify dynamic performance of the conventional strategy with the speed variation from the sub- to super-synchronous speed. As shown in Fig. 3.17, the BDFIG begins operation under the sub-synchronous speed of 600 rpm with the balanced load, and the PW voltage is balance and the negativesequence PW amplitude is about 1 V. At 0.5 s, the unbalanced load is connected, resulting in a significant increase of the negative-sequence PW voltage amplitude from 1 to 25 V. And then, at 0.8 s, the BDFIG accelerates from 600 to 900 rpm. As can be seen in Fig. 3.17, the conventional strategy preserves the unbalanced PW voltage along with the changed speed.

3.3.3.2

Proposed Method

Three typical simulation cases are employed to verify the proposed strategy under the unbalanced load. Firstly, the effectiveness of the proposed strategy is verified at the sub-synchronous speed of 600 rpm in comparison with the conventional strategy. From Fig. 3.18, the PW voltage is unbalanced with the conventional strategy, and the amplitude of the negative-sequence PW voltage is about 25 V. Under the proposed strategy, the

80

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.11 Simulation results of the amplitude of the negative-sequence PW voltage with the conventional method at 600 rpm: a balanced load, b unbalanced load

amplitude of the negative-sequence PW voltage is significantly reduced to around 5 V. Thus, the PW voltage becomes almost balanced. The distortion of the CW current is significantly increased under the proposed strategy, due to the presence of a negative-sequence PW voltage compensation component. Figure 3.18d1 , e1 , d2 , e2 present the PW currents and their THDs under the conventional and proposed strategies. From Fig. 3.18e1 and e2 , the THD of the PW current is 2.83% before compensation and 2.01% after compensation. Hence, the proposed strategy can not influence the current of PW. Secondly, Fig. 3.19 illustrates the simulation results at 900 rpm. The PW voltage, CW current, amplitude of the negative-sequence PW voltage, PW current, and THD of the PW current are nearly identical to those at 600 rpm, which confirms the success of the proposed strategy at the super- and sub-synchronous speeds. The third simulation is performed to verify the dynamic performance of the proposed strategy during the startup process and speed variation. The result waveforms are presented in Fig. 3.20. The BDFIG begins with the conventional strategy at 900 rpm under the unbalanced load. The amplitude of the negative-sequence PW voltage is about 23 V, which means the large unbalance impact on the PW voltage. At 0.4 s, the proposed strategy is activated, resulting in a rapid decrease in the amplitude of the negative-sequence PW voltage from 23 to 3 V within 0.04 s. And then, at 0.6 s,

3.3 Unbalanced Voltage Compensation

81

Fig. 3.12 Simulation results of the CW current with the conventional strategy at 600 rpm: a balanced load, b unbalanced load

the BDFIG starts to decelerate from 900 to 680 rpm. The proposed strategy keeps the PW voltage balanced under the speed change. This test confirms the stability of the proposed strategy through the short transient state at the instant of the proposed method being activated, and also through the constant PW voltage under the changed speed.

3.3.4 Experimental Results 3.3.4.1

Experimental Setup

All the experiments are carried out on a 30-kVA BDFIG experimental platform, and the detailed parameters and photograph of the experimental platform can be seen in Sect. A.3, Appendix. Two kinds of loads are used in the experiments: the unbalanced load with the resistances of 12, 12 and 6  in the three-phase, and the single-phase load with the resistance of 12  connected between phase a and phase b of PW. A

82

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.13 Simulation results of the PW voltage amplitude with the conventional method at 900 rpm: a balanced load, b unbalanced load

single-phase load can produce a greater unbalanced influence than the three-phase unbalanced load.

3.3.4.2

Results with Three-Phase Unbalanced Load

Three typical experiments are applied to check the performance of the proposed strategy with a three-phase unbalanced load. Firstly, the effectiveness of the proposed strategy is checked at the subsynchronous speed of 600 rpm, as shown in Fig. 3.21. With the conventional strategy, the amplitude of the negative-sequence PW voltage reaches 20 V. Fortunately, with the proposed strategy, the amplitude of the negative-sequence PW voltage can be greatly reduced to around 5 V. However, the CW current is significantly distorted under the proposed strategy, which is caused by the compensation component for the negative-sequence PW voltage. From Fig. 3.21b1 , e1 , b2 and e2 , it can be noted that the PW currents before and after compensation are almost the same with the similar THD (5.76% before compensation and 7.64% after compensation). Hence, the PW current cannot be affected by the proposed strategy.

3.3 Unbalanced Voltage Compensation

83

Fig. 3.14 Simulation results of the PW voltage with the conventional strategy at 900 rpm: a balanced load, b unbalanced load

Secondly, Fig. 3.22 shows the experimental results at the super-synchronous speed of 900 rpm. With the proposed strategy, the amplitude of the negative-sequence PW voltage can be reduced to the similar value with that at the sub-synchronous speed of 600 rpm, which means that the proposed strategy is with good performance at both sub- and super-synchronous speeds. From Fig. 3.22b1 , d1 , b2 and d2 , the CW current is significantly distorted with the THD being increased from 18.51 to 73.60% under the proposed strategy, due to the injection of the compensated current for the negative-sequence of PW voltage. Figure 3.22e1 , f1 , e2 , f2 shows the PW currents and their harmonic spectrums under the conventional and the proposed strategies. The PW currents are with the similar THD values, 5.59 and 6.54%, under the conventional and proposed strategies, which means that the proposed strategy does not significantly increase the PW current distortion. The third experiment is carried out to check the dynamic performance of the proposed strategy. At first, the dynamic performance is tested during the startup process. At 3.15 s, the control strategy is switched from the conventional one to the proposed one. The amplitude of the negative-sequence PW voltage significantly decreases from 20 to 5 V within 0.25 s, as shown in Fig. 3.23c. Since the DSOGI in the negative-sequence voltage compensator serves as a filter, the CW compensation current can be smoothly increased and consequently the inrush current in CW current

84

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.15 Simulation results of the amplitude of the negative-sequence PW voltage with the conventional method at 900 rpm, a balanced load, b unbalanced load

would not be produced, as shown in Fig. 3.23b. And then, the dynamic performance is checked under the speed variation from the super-synchronous speed 900 rpm to the sub-synchronous speed 680 rpm within 5.6 s. From Fig. 3.23a, the proposed strategy can keep the balance of the PW voltage during the speed variation.

3.3.4.3

Results with Single-Phase Load

The single-phase load is employed to obtain a bigger unbalanced influence on the PW voltage than the three-phase unbalanced load. As can be seen from Fig. 3.24, the BDFIG starts up with the single-phase load at 900 rpm under the conventional strategy, and the PW voltage becomes unbalanced and the amplitude of the negativesequence PW voltage reaches around 80 V, as shown in Fig. 3.24a, d. At 2.0 s, the proposed strategy is inserted into the control system, and the amplitude of the negative-sequence PW voltage dramatically decreases from 80 to 20 V within 0.9 s, as shown in Fig. 3.24d. To test the dynamic performance during the speed variation, the BDFIG decelerates from 900 to 630 rpm between 4.7 and 6.0 s, and then accelerates

3.3 Unbalanced Voltage Compensation

85

Fig. 3.16 Simulation results of the CW current with the conventional strategy at 900 rpm: a balanced load, b unbalanced load

from 630 to 900 rpm between 7.9 and 9.1 s. From Fig. 3.24a and d, the proposed strategy keeps the PW voltage balanced under the changed speed. Figure 3.24e, f show the expanded views of the PW voltage under the conventional and proposed strategies, which clarifies the success of the proposed strategy over the conventional strategy to decrease the huge unbalanced impact on the PW voltage. From Fig. 3.24b and g, it can be noted that the PW current mainly contains two phases, and the third phase operates as a direct current with a small value, because of the influence of the single-phase load. Figure 3.24h, i show the expanded view and the harmonic spectrum of the CW current, respectively. As a result of the unbalanced PW voltage compensation, the CW current is with many harmonics. The fundamental frequency of the CW current is 10 Hz, with the main the harmonic frequency of 110 Hz, and the THD of the CW current is up to 112.87%.

86

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

3.4 Low-Order Harmonic Voltage Compensation With three-phase nonlinear loads, the harmonic components of the PW voltage and current mainly contain the 5th and 7th harmonics, which causes serious nonlinear influences on the BDFIG. The research motivation of this section is to investigate an

Fig. 3.17 Simulation results of the conventional strategy with balanced and unbalanced loads under the variable rotor speed: a amplitude of the PW voltage, b PW voltage, c CW current, d amplitude of the negative-sequence PW voltage

3.4 Low-Order Harmonic Voltage Compensation

87

Fig. 3.17 (continued)

effective control strategy for reducing the harmonic components of the PW voltage [24]. The proposed strategy contains two parts. The first part is the conventional DVC method to regulate the amplitude and frequency of the fundamental PW voltage. The second part is the proposed low-order harmonic voltage compensator to decrease the nonlinear impact on the PW voltage via MSC. The 5th and 7th harmonic components in the PW voltage is extracted via employing the multiple second-order generalized integrator (MSOGI) filter [25]. The PI controller is used to get the additional CW current reference value for eliminating the harmonic components of the PW voltage. The PIR controller is used to adjust the total CW current in the fundamental positive frame.

3.4.1 Design of Low-Order Harmonic Voltage Compensator 3.4.1.1

Basic Theory

The proposed strategy is derived from the linear relationship between the CW current and PW voltage in the 5th and 7th harmonics components, which can be obtained from the BDFIG model as follows. From (3.24), the relationship between the CW current and PW voltage in the 5th harmonic component can be expressed as 5 5 5 5 5 = α2 i 1_5 − α4 ψ1_5 + j (α1 i 1_5 − α3 ψ1_5 )/(5ω1 + p1 ωr ) i 2_5

(3.73)

From (3.73), the 5th harmonic PW current can be expressed by a function of the 5th harmonic CW current as follows:

88

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.18 Simulation results at 600 rpm with the unbalanced load: a PW voltage, b CW current, c amplitude of the negative-sequence PW voltage, d PW current, e harmonic spectrum of the PW current (the subscripts 1 and 2 indicate the conventional and proposed strategies, respectively)

3.4 Low-Order Harmonic Voltage Compensation

Fig. 3.18 (continued)

89

90

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.18 (continued)

 5 i 1_5

=

5 i 2_5

+

5 α4 ψ1_5

+j

5 α3 ψ1_5

5ω1 + p1 ωr

  / α2 + j

α1 . 5ω1 + p1 ωr

(3.74)

From (3.24), the d- and q-axis components of the 5th harmonic PW voltage can be depicted by 5dq

5dq

5dq

5dq

5dq

5dq

u 1_5 = R1 i 1_5 + s(L 1 i 1_5 + L 1r ir _5 ) − j5ω1 (L 1 i 1_5 + L 1r ir _5 ).

(3.75)

3.4 Low-Order Harmonic Voltage Compensation

91

Fig. 3.19 Simulation results at 900 rpm with the unbalanced load: a PW voltage, b CW current, c amplitude of the negative-sequence PW voltage, d PW current, e THD of the PW current (the subscripts 1 and 2 indicate the conventional and proposed strategies, respectively)

92

Fig. 3.19 (continued)

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

3.4 Low-Order Harmonic Voltage Compensation

93

Fig. 3.19 (continued) 5 where F dq indicates (F d + jF q ), and the variable F stands for u 51_5 , i 1_5 and ir5_5 . In steady state, the derivative terms in (3.75) can be ignored, and then (3.75) can be simplified as 5dq

5dq

5dq

5dq

u 1_5 = R1 i 1_5 − j5ω1 (L 1 i 1_5 + L 1r ir _5 ).

(3.76)

Substituting (3.74) and (3.21) into (3.76), the linear relationship between the CW current and PW voltage in the 5th harmonics component can be established by

94

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.20 Simulation results during the proposed strategy startup and under the speed variation with the unbalanced load: a PW voltage, b CW current, c amplitude of the negative-sequence PW voltage

5dq

u 1_5 =

  5dq 5dq α (L 1 i 1_5 +L 1r ir _5 ) 5dq 5dq 5dq (R1 − j5ω1 L 1 ) i 2_5 + α4 (L 1 i 1_5 + L 1r ir _5 ) + j 3 5ω 1 + p1 ωr α2 + j 5ω1 +α1p1 ωr 5dq

− j5ω1 L 1r ir _5 .

(3.77)

3.4 Low-Order Harmonic Voltage Compensation

95

Fig. 3.21 Experimental results at 600 rpm with the three-phase unbalanced load: a PW voltage, b CW current, c amplitude of the negative-sequence PW voltage, d PW current, e harmonic spectrum of the PW current (the subscripts 1 and 2 indicate the conventional and proposed strategies, respectively)

96

Fig. 3.21 (continued)

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

3.4 Low-Order Harmonic Voltage Compensation

Fig. 3.21 (continued)

97

98

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.21 (continued)

From (3.77), if the 5th harmonic rotor current and PW current are regarded as additional disturbances, the 5th harmonic PW voltage can be regulated by the 5th harmonic CW current. Afterwards, by using the similar derivation steps presented in (3.73)–(3.77), the equation for the linear relationship between the CW current and PW voltage in the 7th harmonic component can be obtained by

7dq

u 1_7 =

  7dq 7dq α (L 1 i 1_7 +L 1r ir _7 ) 7dq 7dq 7dq (R1 + j7ω1 L 1 ) i 2_7 + α4 (L 1 i 1_7 + L 1r ir _7 ) − j 3 7ω 1 + p1 ωr α2 − j 7ω1 +α1p1 ωr 7dq

+ j7ω1 L 1r ir _7

3.4.1.2

(3.78)

Controller Design and Parameters Tuning

Figure 3.25 presents the block diagram of the proposed low-order harmonic voltage compensator, in which the MSOGI filter is adopted to extract the 5th and 7th harmonic components of the PW voltage, and the PI controller is employed to generate the CW compensation current to minimize the harmonic components in the voltage of PW. The CW reference currents in the 5th and 7th harmonic frames can be obtained by

3.4 Low-Order Harmonic Voltage Compensation

99

Fig. 3.22 Experimental results at 900 rpm with the three-phase unbalanced load: a PW voltage, b CW current, c amplitude of the negative-sequence PW voltage, d harmonic spectrum of the CW current, e PW current, f harmonic spectrum of the PW current (the subscripts 1 and 2 indicate the conventional and proposed strategies, respectively)

100

Fig. 3.22 (continued)

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

3.4 Low-Order Harmonic Voltage Compensation

Fig. 3.22 (continued)

101

102

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.22 (continued)

 Ki  0 − u 51d_5 s   Ki  0 − u 51q_5 = Kp + s   Ki  0 − u 71d_7 = Kp + s 

5∗ i 2d_5 =

5∗ i 2q_5

7∗ i 2d_7

Kp +

(3.79) (3.80) (3.81)

3.4 Low-Order Harmonic Voltage Compensation

103

Fig. 3.23 Dynamic performance test with the three-phase unbalanced load: a PW voltage, b CW current, c amplitude of the negative-sequence PW voltage

 7∗ i 2q_7

=

 Ki  0 − u 71q_7 Kp + s

(3.82)

where K p and K i are the proportional and integral gains, respectively. Similar to the negative-sequence voltage compensator depicted in Fig. 3.6, all the calculated CW compensation current in the 5th and 7th harmonic frames has to be transformed to the fundamental positive frame with the aid of six times the PW angle

104

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.24 Experimental results with the single-phase load: a PW voltage, b PW current, c CW current, d amplitude of the negative-sequence PW voltage, e expanded view of (a), f expanded view of (a), g expanded view of (b), h expanded view of (c), i harmonic spectrum of the CW current in (h)

3.4 Low-Order Harmonic Voltage Compensation

Fig. 3.24 (continued)

105

106

Fig. 3.24 (continued)

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

3.4 Low-Order Harmonic Voltage Compensation

107

Fig. 3.25 Block diagram of the low-order harmonic voltage compensator

(6θ 1 ). Consequently, the CW reference currents for regulating the PW fundamental and harmonic voltages can be added together to obtain the total CW reference current in the positive frame, as given by +∗ +∗ +∗ +∗ = i 2dq+ + i 2dq_5 + i 2dq_7 i 2dq +∗ 5∗ 7∗ = i 2dq+ + i 2dq_5 e− j6ω1 t + i 2dq_7 e j6ω1 t .   

(3.83)

Harmonics components

Figure 3.26 illustrates the block diagram of MSOGI, which consists of the harmonic decoupling network (HDN), three DSOGIs and PNSC. The HDN is used to eliminate the mutual influence of different harmonics. In the PNSC, the fundamental, negative-sequence 5th and positive-sequence 7th harmonic components of the PW voltage in the αβ frame can be calculated by   u+ 1α+ = u 1α − qu 1β

(3.84)

  u+ 1β+ = qu 1α + u 1β

(3.85)

108

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.26 Block diagram of MSOGI

u 51α_5− = qu 1β_5 + u 1α_5

(3.86)

u 51β_5− = u 1β_5 − qu 1α_5

(3.87)

u 71α_7+ = u 1α_7 − qu 1β_7

(3.88)

u 71β_7+ = qu 1α_7 + u 1β_7 .

(3.89)

The methodology, used in tuning and designing the proposed negative-sequence voltage compensator in Sect. 3.3.2, can be also applied to the proposed low-order harmonic voltage compensator. Combining Figs. 3.5, 3.25 and 3.26, the overall control scheme can be illustrated in Fig. 3.27. The current loop in BDFIG control system contains (PI + R) controllers, where the R controller can regulate the CW compensation current for eliminating the 5th and 7th harmonic components of the PW voltage, and the PI controller can deal with the fundamental component of the CW current to adjust the amplitude of the PW voltage.

3.4 Low-Order Harmonic Voltage Compensation

109

Fig. 3.27 Overall control scheme for low-order harmonic voltage compensation of PW

3.4.2 Simulation Results The simulation is also carried out on a 30-kVA BDFIG, with the main parameters listed in Sect. A.3, Appendix. The nonlinear load used in the simulation is a threephase diode rectifier with a resistor of 25  in the dc side.

3.4.2.1

Conventional Method

Two typical simulation cases are implemented to verify the performance of the conventional method with linear and nonlinear loads. Firstly, the effectiveness of the conventional strategy is investigated at the sub-synchronous speed of 650 rpm. Figure 3.28a, b present the PW line voltages with linear and nonlinear loads, respectively. The nonlinear effect of the PW voltage is very clear in Fig. 3.28b. Figure 3.29a, b show the harmonic spectrum of the PW voltage. After adding the nonlinear load, the THD of the PW voltage is greatly increased from 0.76 to 15.25%. Figure 3.30 illustrates the PW voltage amplitudes in 5th and 7th harmonics. With the linear load, the amplitudes of the 5th and 7th harmonic components of the PW voltage are around 2 V, which is calculated by

110

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.28 Simulation results of the PW voltage with the conventional strategy at 650 rpm: a linear load, b nonlinear load

 5  5 5 2 2 U  = (u 1_5 1d_5 ) + (u 1q_5 )

(3.90)

 7  7 7 2 2 U  = (u 1_7 1d_7 ) + (u 1q_7 ) .

(3.91)

With the nonlinear load, the amplitudes of the 5th and 7th harmonic components are significantly increased to 60 and 33 V, respectively. Besides, from Fig. 3.31, the distortion of the CW current is very clear after adding the nonlinear load to the system. Secondly, the performance of the conventional strategy is investigated at the supersynchronous speed of 850 rpm. Similar to the simulation results obtained at the subsynchronous speed of 650 rpm, the harmonic in the PW voltage is very apparent after the nonlinear load is connected to the system, as shown in Fig. 3.32b. And, with the nonlinear load, the THD of the PW voltage is significantly increased from 0.29 to 15.39%, as presented in Fig. 3.33a, b. Figure 3.34 illustrates the amplitudes of the 5th and 7th harmonic components under different loads. The amplitudes of both 5th and 7th harmonics are 0.5 V under the linear load. However, they rise to 60 V and 33 V, respectively, under the nonlinear

3.4 Low-Order Harmonic Voltage Compensation

111

Fig. 3.29 The harmonic spectrum of the PW voltage at 650 rpm in simulation: a linear load, b nonlinear load

load. From Fig. 3.35, the distortion of the CW current is obvious with the nonlinear load.

3.4.2.2

Proposed Method

Three typical tests are employed to verify the proposed strategy under the nonlinear load.

112

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.30 Simulation results of the amplitudes of the 5th and 7th harmonic components of the PW voltage with the conventional method at 650 rpm: a linear load, b nonlinear load

Firstly, the effectiveness of the proposed strategy is verified at the sub-synchronous speed of 650 rpm. From Fig. 3.36, with the conventional strategy, the amplitudes of the 5th and 7th harmonics of the PW voltage reach 60 V and 33 V, respectively. With the proposed strategy, amplitudes of the 5th and 7th harmonics in the PW voltage are significantly reduced to about 10 V and 7 V, respectively. Thus, the PW voltage becomes almost linear. However, the distortion of the CW phase current is aggravated by the proposed strategy, because of the compensation component in the CW current for suppresing the 5th and 7th harmonics of the PW voltage. The harmonic spectrum and THD of the PW voltage are also presented in Fig. 3.36. The THD of the PW voltage can be drastically reduced from 15.25 to 5.47% with the proposed strategy. Secondly, Fig. 3.37 illustrates the experimental results at the super-synchronous speed of 850 rpm. The waveforms of PW voltage, CW current, amplitudes of the 5th and 7th harmonic PW voltage and THD of the PW voltage are nearly identical to those at 650 rpm, which confirms the effectiveness of the proposed strategy at the super-synchronous speed. Finally, the third simulation is performed to verify the dynamic performance of the proposed strategy under the speed variation. The results are presented in Fig. 3.38. The generator starts up with the proposed strategy at 850 rpm under the nonlinear

3.4 Low-Order Harmonic Voltage Compensation

113

Fig. 3.31 Simulation results of the CW current with the conventional method at 650 rpm: a linear load, b nonlinear load

load. At 1.05 s, the generator starts to decelerate from the super-synchronous speed 850 rpm to the sub-synchronous speed 650 rpm. From Fig. 3.38, the proposed strategy keeps the low 5th and 7th harmonic in the PW voltage under the changed speed.

3.4.3 Experimental Results Three typical tests, on a 30-kVA BDFIG experimental platform, are employed to verify the proposed control strategy. The main details of the experimental platform are introduced in Sect. A.3, Appendix. A three-phase diode rectifier connected with a resistor of 25  in the dc side serves as the three-phase nonlinear load. Firstly, the effectiveness of the proposed strategy is verified at the sub-synchronous speed of 650 rpm in comparison with the conventional strategy. From Fig. 3.39, the PW voltage is nonlinear with a conventional strategy, and the amplitudes of the 5th and 7th harmonic components of the PW voltage are about 60 V and 25 V, respectively. Fortunately, under the proposed strategy, the amplitudes of both the 5th and 7th harmonics in the PW voltage are greatly reduced to 15 V, so that the PW

114

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.32 Simulation results of the PW voltage with the conventional strategy at 850 rpm: a linear load, b nonlinear load

voltage becomes almost linear. Figure 3.39d1 , d2 present the harmonic spectrums of the PW voltage under the conventional and proposed strategies, with the THDs of 17.82% before compensation and 5.03% after compensation. From Fig. 3.39b1 and b2 , the proposed strategy does not affect the PW current. The second experiment is performed to verify the dynamic performance of the proposed strategy during startup. The experimental results are presented in Fig. 3.40. Before 0.49 s, the BDFIG operates with the conventional strategy under the nonlinear load at 650 rpm. The amplitudes of the 5th and 7th harmonics of the PW voltage are about 60 V and 25 V, respectively, which means the large nonlinear impact on the PW voltage. At 0.49 s, the proposed strategy is inserted, resulting in a significant decrease in the amplitudes of the PW 5th and 7th harmonic voltages to 15 V within 0.23 s. Hence, the good dynamic performance of the proposed strategy can be confirmed through the short settling time in the PW voltage and low surge current in the CW. Finally, the third experiment is performed to verify the dynamic performance via changing the speed from the super- to sub-synchronous speed. The experimental results are illustrated in Fig. 3.41. Before 1.1 s, with the nonlinear load, the BDFIG runs at the synchronous speed 850 rpm under the proposed strategy. At 1.1 s, the

3.4 Low-Order Harmonic Voltage Compensation

115

Fig. 3.33 The harmonic spectrum of the PW voltage at 850 rpm in simulation: a linear load, b nonlinear load

BDFIG starts to decelerate from 850 to 650 rpm. From Fig. 3.41, the proposed strategy keeps the sinusoidal PW voltage under the speed variation.

116

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

3.5 Dual-Resonant Controller for Compensating Unbalanced and Low-Order Harmonic Voltages The BDFIG in the standalone mode is sensitive to unusual working situations, particularly unbalanced and nonlinear loads. The unbalanced and nonlinear loads can cause significant unbalance and distortion for PW voltage and current. The negativesequence component of the PW voltage represents the unbalance influence, and the 5th and 7th harmonic components indicate the nonlinear influence. The research motivation of this section is to find an effective control strategy for the BDFIG working under the unbalanced load, three-phase nonlinear load and single-phase nonlinear load to reduce the unbalanced and nonlinear effects of the PW voltage. This section proposes the dual-resonant controller (DRC) for MSC to achieve the CW compensation voltage when the system is with nonlinear and unbalanced loads [26]. The proposed DRC composes of two resonant controllers, i.e., the unbalance resonant controller and the harmonics resonant controller. The advantages of applying DRC in the standalone BDFIG can be briefly summarized as follows:

Fig. 3.34 Simulation results of the amplitudes of the 5th and 7th harmonics of the PW voltage with conventional method at 850 rpm: a linear load, b nonlinear load

3.5 Dual-Resonant Controller for Compensating Unbalanced and …

117

Fig. 3.35 Simulation results of the CW current with the conventional method at 850 rpm: a linear load, b nonlinear load

1.

2. 3.

The filters to extract the negative-sequence and harmonic components can be removed, and the numbers of adopted PIR and PI controllers can be significantly reduced, which can improve the system response speed. The design methodology of the DRC is easier and the computational burden can be decreased. The ability of working under the unbalanced plus nonlinear load can be achieved.

3.5.1 Design of Dual-Resonant Controller 3.5.1.1

Basic Theory

The concept of the proposed methodology depends on the relation between the voltages of PW and CW derived from the BDFIG model as follows. In steady state, the differential terms are equal to zero, and the positive component equations of the PW and CW voltage in (3.13) can be simplified as + + u+ 1+ = R1 i 1+ + jω1 ψ1+

(3.92)

118

3 Unbalanced and Low-Order Harmonic Voltages Rejection … + + u+ 2+ = R2 i 2+ + j[ω1 − ( p1 + p2 )ωr ]ψ2+ .

(3.93)

Then, rearranging (3.92) and the PW flux equation in (3.13), the new expressions of the PW flux and rotor current can be derived as + + = (u + ψ1+ 1+ − R1 i 1+ )/jω1

(3.94)

Fig. 3.36 Simulation results at 650 rpm under the nonlinear load: a PW voltage, b CW current, c amplitudes of the 5th and 7th harmonics of the PW voltage, d harmonic spectrum of the PW voltage (the subscripts 1 and 2 indicate the conventional and proposed strategies, respectively)

3.5 Dual-Resonant Controller for Compensating Unbalanced and …

Fig. 3.36 (continued)

119

120

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.36 (continued)

+ + ir++ = (ψ1+ − L 1 i 1+ )/L 1r .

(3.95)

Substituting (3.94) into (3.95), the rotor current can be given by ir++ =

+ + (u + 1+ − R1 i 1+ )/( jω1 ) − L 1 i 1+ . L 1r

(3.96)

3.5 Dual-Resonant Controller for Compensating Unbalanced and …

121

Substituting (3.96) into the CW flux equation in (3.13), the CW flux can be obtained as + + = L 2 i 2+ + L 2r ψ2+

+ + (u + 1+ − R1 i 1+ )/( jω1 ) − L 1 i 1+ . L 1r

(3.97)

Fig. 3.37 Simulation results at 850 rpm under the nonlinear load: a PW voltage, b CW current, c amplitudes of the 5th and 7th harmonics of the PW voltage, d harmonic spectrum of the PW voltage (the subscripts 1 and 2 indicate the conventional and proposed strategies, respectively)

122

Fig. 3.37 (continued)

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

3.5 Dual-Resonant Controller for Compensating Unbalanced and …

123

Fig. 3.37 (continued)

Substituting (3.97) into (3.93), the relationship between the positive-sequence CW and PW voltages can be expressed as  + +  (u + 1+ − R1 i 1+ )/( jω1 ) − L 1 i 1+ + + + u 2+ = R2 i 2+ − j[ω1 − ( p1 + p2 )ωr ] L 2 i 2+ + L 2r . L 1r

(3.98) Hence, by neglecting the voltages drop of PW and CW, Eq. (3.98) becomes

124

3 Unbalanced and Low-Order Harmonic Voltages Rejection … + u+ 2+ = j[ω1 − ( p1 + p2 )ωr ][L 2r u 1+ /( jω1 L 1r )]

= u+ 1+ [ω1 − ( p1 + p2 )ωr ]L 2r /(ω1 L 1r ).

(3.99)

Rearranging (3.99), the positive-sequence PW voltage can be expressed as + u+ 1+ = u 2+ ω1 L 1r /[(ω1 +( p1 + p2 )ωr )L 2r ].

(3.100)

Fig. 3.38 Simulation results under the speed change with the nonlinear load: a PW voltage, b CW current, c amplitudes of the 5th and 7th harmonics of the PW voltage

3.5 Dual-Resonant Controller for Compensating Unbalanced and …

125

Since the change of the rotor speed is relatively slow in comparison with the change of the PW and CW voltages, the rotor speed in (3.100) can be considered as a constant. Hence, based on (3.100), it is seen that the relationship between the positive-sequence PW and CW voltages is linear. By adopting the similar mathematical derivation introduced in (3.92)–(3.99), the comprehensive equations for the

Fig. 3.39 Experimental results at 650 rpm under the three-phase nonlinear load: a PW voltage, b PW current, c amplitudes of the 5th and 7th harmonics of the PW voltage, d harmonic spectrum of the PW voltage (the subscripts 1 and 2 indicate the conventional and proposed strategies, respectively)

126

Fig. 3.39 (continued)

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

3.5 Dual-Resonant Controller for Compensating Unbalanced and …

127

Fig. 3.39 (continued)

linear relation between the PW and CW voltages under the unbalanced load, threephase nonlinear load and single-phase nonlinear load can be separately obtained as follows. • Under unbalanced load The comprehensive relation between the PW voltage and CW voltage under the unbalanced load consists of two parts, i.e. the positive-sequence part and the negativesequence part.

128

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.40 Dynamics performance test of the proposed strategy during startup: a PW voltage, b CW current, c amplitudes of the 5th and 7th harmonics of the PW voltage

Positive-sequence component part This part has been introduced previously in (3.92)–(3.100). Negative-sequence component part In steady state, the differential terms can be approximated to zero, so that the PW and CW voltage equations in (3.14) can be simplified as

3.5 Dual-Resonant Controller for Compensating Unbalanced and …

129

− − u− 1− = R1 i 1− − jω1 ψ1−

(3.101)

− − u− 2− = R2 i 2− + j[−ω1 − ( p1 + p2 )ωr ]ψ2− .

(3.102)

Fig. 3.41 Experimental results under the speed change with the three-phase nonlinear load: a PW voltage, b CW current, c expanded view of the PW voltage at 850 rpm, d expanded view of the PW voltage at 650 rpm, e amplitudes of the 5th and 7th harmonics of the PW voltage

130

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.41 (continued)

Then, rearranging (3.101) and the PW flux equation in (3.14), the negativesequence components of the PW flux and rotor current can be derived as − − = (u − ψ1− 1− − R1 i 1− )/(− jω1 )

(3.103)

− − ir−− = (ψ1− − L 1 i 1− )/L 1r .

(3.104)

Substituting (3.103) into (3.104), the negative-sequence rotor current can be given by ir−− =

− − [(u − 1− − R1 i 1− )/(− jω1 )] − L 1 i 1− . L 1r

(3.105)

Substituting (3.105) into the CW flux equation in (3.14), the negative-sequence CW flux can be clarified as − − ψ2− = L 2 i 2− + L 2r

− − [(u − 1− − R1 i 1− )/(− jω1 )] − L 1 i 1− . L 1r

(3.106)

3.5 Dual-Resonant Controller for Compensating Unbalanced and …

131

Substituting (3.106) into (3.102), the rapport between the negative-sequence CW and PW voltages can be obtained by − u− 2− = R2 i 2− + j[−ω1 − ( p1 + p2 )ωr ]  − [(u − 1− − R1 i 1 )/(− jω1 )] − L 1 i 1− − L 2 i 2− + L 2r . L 1r

(3.107)

With neglecting the internal voltage drops of PW and CW, Eq. (3.107) can be simplified as − u− 2− = j[−ω1 − ( p1 + p2 )ωr ][L 2r u 1− /(− jω1 L 1r )]

= u− 1− [ω1 +( p1 + p2 )ωr ]L 2r /(ω1 L 1r ).

(3.108)

Rearranging (3.108), the negative-sequence PW voltage can be expressed as − u− 1− = u 2− ω1 L 1r /[(ω1 +( p1 + p2 )ωr )L 2r ]

(3.109)

From (3.109), it can be noted that the relationship between the negative-sequence PW voltage and CW voltage is linear when the rotor speed is regarded as a constant. Hence, the negative-sequence PW voltage can be regulated by the negative-sequence CW voltage. Comprehensive relation Combining (3.7), (3.100) and (3.109), the comprehensive relation between the PW and CW voltages under the unbalanced load can be given by + − − j2ω1 t = u+ ω1 L 1r /[(ω1 +( p1 + p2 )ωr )L 2r ] u+ 1 = u 1+ + u 1− e    2+ Positive-sequence part

+

− j2ω1 t u− ω1 L 1r /[(ω1 +( p1 2− e





+ p2 )ωr )L 2r ] . 

(3.110)

Negative-sequence part

• Under three-phase nonlinear load The comprehensive relation between the PW voltage and CW voltage under the nonlinear load consists of three parts, i.e. the fundamental part, the 5th harmonic part, and the 7th harmonic part. Fundamental part This part has been introduced previously in (3.92)–(3.100). 5th harmonic part When the differential terms are ignored in steady state, the PW and CW voltage equations in (3.24) can be simplified as

132

3 Unbalanced and Low-Order Harmonic Voltages Rejection … 5 5 u 51_5 = R1 i 1_5 + j (−5)ω1 ψ1_5

(3.111)

5 5 u 52_5 = R2 i 2_5 + j[−5ω1 − ( p1 + p2 )ωr ]ψ2_5 .

(3.112)

Then, rearranging (3.111) and the PW flux equation in (3.24), the 5th harmonic components of the PW flux and rotor current can be expressed as 5 5 = (u 51_5 − R1 i 1_5 )/(− j5ω1 ) ψ1_5

(3.113)

5 5 ir5_5 = (ψ1_5 − L 1 i 1_5 )/L 1r .

(3.114)

Substituting (3.113) into (3.114), the 5th harmonic component of the rotor current can be given by ir5_5 =

5 5 [(u 51_5 − R1 i 1_5 )/(− j5ω1 )] − L 1 i 1_5

L 1r

.

(3.115)

Substituting (3.115) into the CW flux equation in (3.24), the 5th harmonic component of the CW flux can be clarified as 5 5 ψ2_5 = L 2 i 2_5 + L 2r

5 5 [(u 51_5 − R1 i 1_5 )/(− j5ω1 )] − L 1 i 1_5

L 1r

.

(3.116)

Substituting (3.116) into (3.112), the 5th harmonic component of the CW voltage can be rewritten as 5 + j[−5ω1 − ( p1 + p2 )ωr ] u 52_5 = R2 i 2_5   5 5 (u 51_5 − R1 i 1_5 )/(− j5ω1 ) − L 1 i 1_5 5 L 2 i 2_5 + L 2r . L 1r

(3.117)

With neglecting the internal voltage drops of PW and CW, Eq. (3.117) becomes u 52_5 = j[−5ω1 − ( p1 + p2 )ωr ][L 2r u 51_5 /(− j5ω1 L 1r )] = u 51_5 [5ω1 +( p1 + p2 )ωr ]L 2r /(5ω1 L 1r ).

(3.118)

Rearranging (3.118), the 5th harmonic component of the PW voltage can be given by u 51_5 = u 52_5 (5ω1 L 1r )/[(5ω1 +( p1 + p2 )ωr )L 2r ].

(3.119)

The linear relationship between the 5th harmonic component of the PW voltage and that of the CW voltage can be seen from (3.119), with the rotor speed being

3.5 Dual-Resonant Controller for Compensating Unbalanced and …

133

regarded as a constant. Hence, the 5th harmonic component of the PW voltage can be controlled by that of the CW voltage. 7th harmonic part In steady state, with ignoring the differential terms, the PW and CW voltage equations in (3.25) can be rewritten as 7 7 u 71_7 = R1 i 1_7 + j7ω1 ψ1_7

(3.120)

7 7 u 72_7 = R2 i 2_7 + j[7ω1 − ( p1 + p2 )ωr ]ψ2_7 .

(3.121)

Rearranging (3.120), the 7th harmonic component of the PW flux can be derived as 7 7 = (u 71_7 − R1 i 1_7 )/( j7ω1 ). ψ1_7

(3.122)

From the PW flux equation in (3.25), the 7th harmonic component of the rotor current can be expressed as 7 7 − L 1 i 1_7 )/L 1r . ir7_7 = (ψ1_7

(3.123)

Substituting (3.122) into (3.123), the 7th harmonic component of the rotor current can be rewritten as ir7_7 =

7 7 )/( j7ω1 ) − L 1 i 1_7 (u 71_7 − R1 i 1_7 . L 1r

(3.124)

Substituting (3.124) into the CW flux equation in (3.25), the 7th harmonic component of the CW flux can be given by 7 7 ψ2_7 = L 2 i 2_7 + L 2r

7 7 (u 71_7 − R1 i 1_7 )/( j7ω1 ) − L 1 i 1_7 . L 1r

(3.125)

Substituting (3.125) into (3.121), the relation between the 7th harmonic components of the CW voltage and PW voltage can be expressed as  7 + j[7ω − ( p + p )ω ] L i 7 + L u 72_7 = R2 i 2_7 1 1 2 r 2 2_7 2r

 7 )/( j7ω ) − L i 7 (u 71_7 − R1 i 1_7 1 1 1_7 . L 1r

(3.126) Without considering the internal voltage drops of PW and CW, Eq. (3.126) can be simplified as u 72_7 = j[7ω1 − ( p1 + p2 )ωr ][L 2r u 71_7 /( j7ω1 L 1r )]

134

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

= u 71_7 [7ω1 − ( p1 + p2 )ωr ]L 2r /(7ω1 L 1r ).

(3.127)

From (3.127), the 5th harmonic component of the PW voltage can be derived as u 71_7 = u 72_7 (7ω1 L 1r )/[(7ω1 − ( p1 + p2 )ωr )L 2r ].

(3.128)

The linear relationship between the 7th harmonic component of the PW voltage and that of the CW voltage can be seen from (3.128), with the rotor speed being considered as a constant. Hence, the 7th harmonic component of the PW voltage can be controlled by the corresponding CW voltage. Comprehensive relation Combining (3.15), (3.100), (3.119) and (3.128), the comprehensive relation between the PW and CW voltages under the three-phase nonlinear load can be described as + 5 − j6ω1 t + u 71_7 e j6ω1 t u+ 1 = u 1+ + u 1_5 e

= u+ ω1 L 1r /[(ω1 +( p1 + p2 )ωr )L 2r ]    2+ Fundamental part

+

u 52_5 e− j6ω1 t (5ω1 L 1r )/[(5ω1 +( p1

+

u 72_7 e j6ω1 t (7ω1 L 1r )/[(7ω1





+ p2 )ωr )L 2r ] 

5th harmonic part





− ( p1 + p2 )ωr )L 2r ] . 

(3.129)

7th harmonic part

• Under single-phase nonlinear load The comprehensive relation between the PW voltage and CW voltage under the single-phase nonlinear load consists of three parts, i.e. the fundamental part, the negative-sequence part, and the 3rd harmonic part. Positive sequence component part This part has been introduced previously in (3.92)–(3.100). Negative sequence component part This part has been introduced previously in (3.101)–(3.110). Third harmonic part In steady state, when the differential terms are equal to zero, the PW and CW voltage equations in (3.50) can be simplified as 3 3 u 31_3 = R1 i 1_3 + j3ω1 ψ1_3

(3.130)

3 3 u 32_3 = R2 i 2_3 + j[3ω1 − ( p1 + p2 )ωr ]ψ2_3 .

(3.131)

3.5 Dual-Resonant Controller for Compensating Unbalanced and …

135

And then, from (3.130) and (3.50), the 3rd harmonic components of the PW flux and rotor current can be rewritten as 3 3 = (u 31_3 − R1 i 1_3 )/( j3ω1 ) ψ1_3

(3.132)

3 3 ir3_3 = (ψ1_3 − L 1 i 1_3 )/L 1r .

(3.133)

Substituting (3.132) into (3.133), the rotor current can be given by ir3_3 =

3 3 )/( j3ω1 ) − L 1 i 1_3 (u 31_3 − R1 i 1_3 . L 1r

(3.134)

Substituting (3.134) into the CW flux equation in (3.50), the 3rd harmonic component of the CW flux can be derived as 3 3 ψ2_3 = L 2 i 2_3 + L 2r

3 3 (u 31_3 − R1 i 1_3 )/( j3ω1 ) − L 1 i 1_3 . L 1r

(3.135)

Substituting (3.135) into (3.131), the 3rd harmonic component of the CW voltage can be expressed as  3 + j[3ω − ( p + p )ω ] L i 3 + L u 31_3 = R2 i 2_3 1 1 2 r 2 2_3 2r

 3 )/( j3ω ) − L i 3 (u 31_3 − R1 i 1_3 1 1 1_3 . L 1r

(3.136) With neglecting the internal voltage drops of PW and CW, Eq. (3.136) can be simplified as u 32_3 = j[3ω1 − ( p1 + p2 )ωr ][L 2r u 31_3 /( j3ω1 L 1r )] = u 31_3 [3ω1 − ( p1 + p2 )ωr ]L 2r /(3ω1 L 1r ).

(3.137)

From (3.137), the 3rd harmonic component of the PW voltage can be obtained by u 31_3 = u 32_3 (3ω1 L 1r )/[(3ω1 − ( p1 + p2 )ωr )L 2r ].

(3.138)

Considering the rotor speed as a constant, the linear relationship between the 3rd harmonic component of the PW voltage and that of the CW voltage can be depicted by (3.138). Hence, the 3rd harmonic component of the PW voltage can be controlled by that of the CW voltage. Comprehensive relation Combining (3.40), (3.100), (3.109) and (3.138), the comprehensive relation between the PW and CW voltages under the single-phase nonlinear load can be described as

136

3 Unbalanced and Low-Order Harmonic Voltages Rejection … + − − j2ω1 t u+ + u 31_3 e j2ω1 t 1 = u 1+ + u 1− e

= u+ ω1 L 1r /[(ω1 +( p1 + p2 )ωr )L 2r ]  2+   Fundamental part

+

− j2ω1 t u− ω1 L 1r /[(ω1 +( p1 2− e

+

u 32_3 e j2ω1 t (3ω1 L 1r )/[(3ω1





+ p2 )ωr )L 2r ] 

Negative-sequence part





− ( p1 + p2 )ωr )L 2r ] . 

(3.139)

3rd harmonic part

3.5.1.2

Controller Design

According to the analysis in Sect. 3.5.1.1, the negative-sequence PW voltage u − 1− can be regulated by the negative-sequence CW voltage u − 2− . As can be seen from (3.110), under unbalanced loads, the negative-sequence PW voltage becomes the − j2ω1 t in the fundamental positive frame dq+ . Hence, 2nd harmonic component u − 1− e a resonant controller with the resonant frequency of 2ω1 can be adopted to regulate the negative-sequence PW voltage. The output signal of the resonant controller is the compensation reference command of the CW voltage, which is also with the frequency of 2ω1 . Similarly, from (3.129), it can be seen that under three-phase nonlinear loads the 5th and 7th harmonic components of the PW voltage become the 6th harmonic − j6ω1 t j6ω1 t and u − in the fundamental positive frame dq+ . Thus, components u − 1− e 1− e a resonant controller with the resonant frequency of 6ω1 can be used to control the 5th and 7th harmonic components of the PW voltage. From (3.139), it can be realized that under single-phase nonlinear loads the negative-sequence and 3rd harmonic components of the PW voltage become the 2nd − j2ω1 t j2ω1 t and u − in the fundamental positive frame harmonic components u − 1− e 1− e + dq . So, a resonant controller with the resonant frequency of 2ω1 can be adopted to adjust the negative-sequence and 3rd harmonic components of the PW voltage. Based on the above analysis, the DRC shown in Fig. 3.42 is proposed to reduce the nonlinear and unbalance influences of the PW voltage by generating the compensation reference command of the CW voltage, which can compensate the negativesequence component, 3rd, 5th, and 7th harmonics through the MSC. Compared with the unbalanced voltage and low-order harmonic voltage compensation strategies proposed in Sects. 3.3 and 3.4, the advantages of the DRC lie in that it can remove the SOGIs and MSOGIs for extracting the negative-sequence and harmonic components and reduce the numbers of the used PIR and PI controllers, which simplifies the control algorithm and improves the dynamic performance. The DRC composes of two parts, i.e. unbalance resonant controller and the harmonic resonant controller. The unbalance resonant controller is responsible for compensating the unbalanced effect and 3rd harmonic effect in the PW voltage, and

3.5 Dual-Resonant Controller for Compensating Unbalanced and …

137

Fig. 3.42 Structure of the dual-resonant controller (DRC)

the harmonic resonant controller is required for minimizing the nonlinear influence in the PW voltage. The transfer function of the unbalance resonant controller can be expressed as G Ru =

K r u ωc s s 2 + 2ωc s + (2ω1 )2

(3.140)

where K ru is the unbalance resonant gain, and ωc the cut-off frequency. The transfer function of the harmonics resonant controller can be clarified as G Rh =

s2

K r h ωc s + 2ωc s + (6ω1 )2

(3.141)

where K rh is the harmonics resonant gain. Combining (3.140) and (3.141), the transfer function of the DRC can be given by G D RC =

K r u ωc s K r h ωc s + 2 . s 2 + 2ωc s + (2ω1 )2 s + 2ωc s + (6ω1 )2

(3.142)

The range of ωc is generally from 5 to 15 rad/s for both preferable stability and rapid response [27]. Figure 3.43 depicts the overall control scheme with the DRC for compensating unbalanced and low-order harmonic voltages.

3.5.2 Simulation Results The adopted BDFIG in simulation is the 30-kVA BDFIG described in Sect. A.3, Appendix. A resistor of 12 , linked across the phases b and c of PW, is adopted to

138

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.43 Overall control scheme for compensating unbalanced and nonlinear loads based on DRC

simulate the single-phase load. This type of load can provide a significantly unbalanced influence in the system. The nonlinear load is a three-phase diode rectifier with a resistor of 25  at the dc side. Firstly, the performance of the proposed strategy is compared with that of the conventional strategy under the single-phase load at the constant speed of 675 rpm. From Fig. 3.44, the PW voltage is unbalanced with the conventional strategy, and the amplitude of the negative-sequence PW voltage is about 60 V. With the proposed strategy, the amplitude of the negative-sequence PW voltage is greatly reduced to around 10 V. Thus, the PW voltage becomes almost balanced. However, the distortion of the CW current is significantly increased with the proposed strategy, due to the presence of compensation component for the negative-sequence PW voltage. Secondly, the performance of the proposed strategy is compared with that of the conventional strategy under the nonlinear load at the constant speed of 675 rpm. From Fig. 3.45, the PW voltage is with significant harmonics under the conventional strategy. The amplitudes of the 5th and 7th harmonic components of the PW voltage is about 29 and 17 V, and can be greatly reduced to around 3 and 2 V under the

3.5 Dual-Resonant Controller for Compensating Unbalanced and …

139

proposed strategy, where the THD of the PW voltage can be sharply reduced from 17.15 to 6.06%, as illustrated in Fig. 3.45d1 , d2 . In order to compensate the harmonics in the PW voltage, some harmonic currents have to be injected to the CW, as shown in Fig. 3.45b1 . The third simulation is carried out to check the dynamic performance of the proposed strategy under the combination of the unbalanced and nonlinear loads, as

Fig. 3.44 Simulation results at 675 rpm with the single-phase load: a PW voltage, b CW current, c amplitude of the negative-sequence PW voltage (the subscripts 1 and 2 indicate the conventional and proposed strategies, respectively)

140

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.44 (continued)

illustrated in Fig. 3.46. Before 0.4 s, the BDFIG works with the conventional strategy. Both of the unbalanced and nonlinear effects exist in the PW voltage, which is very clearly depicted in Fig. 3.46c. At 0.4 s, the proposed method is activated. As a result, both of the two undesirable effects of the PW voltage can be significantly reduced as shown in Fig. 3.46d. This test confirms the good dynamic performance of the proposed strategy during the startup process through the short transient process of the PW voltage and CW current. The fourth simulation is carried out to check the performance of the proposed strategy under the variable speed. The corresponding simulation results are shown

3.5 Dual-Resonant Controller for Compensating Unbalanced and …

141

in Fig. 3.47. Before 0.2 s, the system runs under the combination of the unbalanced and nonlinear loads at the sub-synchronous speed of 675 rpm with the proposed method. Between 0.2 and 0.7 s, the BDFIG accelerates from the sub-synchronous speed 675 rpm to the super-synchronous speed 875 rpms. From Fig. 3.47, it can be seen that the proposed control strategy maintains the balanced and linear PW voltage during the speed variation.

Fig. 3.45 Simulation results at 675 rpm with the nonlinear load: a PW voltage, b CW current, c amplitudes of the 5th and 7th harmonics of the PW voltage, d harmonic spectrum of the PW voltage (the subscripts 1 and 2 indicate the conventional and proposed strategies, respectively)

142

Fig. 3.45 (continued)

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

3.5 Dual-Resonant Controller for Compensating Unbalanced and …

143

Fig. 3.45 (continued)

3.5.3 Experimental Results Four experiments under four types of loads are implemented on a 30-kVA BDFIG experimental platform, and the details of the experimental platform can be seen in Sect. A.3, Appendix. The adopted four types of loads are the unbalanced load, the three-phase nonlinear load, the unbalanced plus nonlinear load, and the single-phase nonlinear load. For the unbalanced situation, the resistor of 12  is linked across phases b and c of PW to achieve a single-phase load. This type of load provides a great unbalanced

144

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.46 Dynamic performance test with the combination of the unbalanced and nonlinear loads: a PW voltage, b CW current, c extended view of (a) between 0.2 and 0.4 s, d extended view of (a) between 0.5 and 0.7 s

3.5 Dual-Resonant Controller for Compensating Unbalanced and …

145

Fig. 3.47 Simulation results with the speed change under the combination of the unbalanced and nonlinear loads: a PW voltage, b CW current, c amplitude of the negative-sequence PW voltage, d amplitudes of the 5th and 7th harmonics of the PW voltage

146

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

influence in the system. For the nonlinear case, a three-phase diode rectifier with a resistor of 25  at the dc side is adopted to achieve the three-phase nonlinear load. For the unbalanced plus nonlinear load, the unbalanced load is a three-phase unbalanced load (25, 50, and 50  in the three phases), and the nonlinear load is simulated by the three-phase diode rectifier with a resistor of 25  at the dc side. For the single-phase nonlinear load, this situation is realized via adopting a single-phase diode rectifier with a resistor of 25  at the dc side. The constant speed of 675 rpm is adopted for all tests.

3.5.3.1

Under Unbalanced Load

Under the unbalanced case, Fig. 3.48 presents the experimental results of the proposed strategy, which are similar to the simulation results presented in Fig. 3.44. This validates the effectiveness of the proposed strategy. The second experiment is performed to verify the dynamic performance of the proposed strategy during the startup process. The results are presented in Fig. 3.49. Before 1.16 s, the system operates under the conventional method with unbalanced load, and the amplitude of the negative-sequence PW voltage is about 60 V, which means the large unbalanced impact of the PW voltage. At 1.16 s, the proposed strategy is inserted, resulting in a fast decrease of the amplitude of the negativesequence PW voltage from 60 to 22 V within 0.05 s. This test confirms the good dynamic performance of the proposed strategy during the startup process through the fast reduction of the negative-sequence PW voltage and the smooth transient process of the CW current.

3.5.3.2

Under Three-Phase Nonlinear Load

The experimental results of the proposed strategy under the three-phase nonlinear load are demonstrated in Fig. 3.50, which are similar to the simulation results shown in Fig. 3.45 and can confirm the effectiveness of the proposed strategy. Figure 3.51 illustrates another experiment to verify the dynamic performance of the proposed strategy during the startup process. Before 0.7 s, the BDFIG works under the conventional method with the nonlinear load. The amplitudes of the 5th and 7th harmonics of the PW voltage are around 32 and 17 V, which confirms the significant nonlinear impact of the PW voltage. At 0.7 s, the control strategy is switched to the proposed one, which leads the amplitudes of the 5th and 7th harmonics of the PW voltage rapidly decreasing to 7 V at the same time. This experiment validates the good dynamic performance of the proposed strategy during the startup process through the fast and smooth transient process of the PW voltage and CW current.

3.5 Dual-Resonant Controller for Compensating Unbalanced and …

3.5.3.3

147

Under Unbalanced Plus Nonlinear Load

In this experiment, both the unbalanced and nonlinear loads are applied to the system at the same time. The unbalanced load is the three-phase unbalanced load (25, 50 and 50  in the three phases), and the nonlinear load is the three-phase diode rectifier with a resistor of 25  at the dc side. Before 3.7 s, the BDFIG works with the conventional strategy. As a result, both the unbalanced and nonlinear effects exist in the PW voltage, as shown in Fig. 3.52d.

Fig. 3.48 Experimental results at 675 rpm with the unbalanced load: a PW voltage, b CW current, c amplitude of the negative-sequence PW voltage (the subscripts 1 and 2 indicate the conventional and proposed strategies, respectively)

148

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.48 (continued)

In addition, the average amplitude of the negative-sequence component of the PW voltage is up to 17 V as shown in Fig. 3.52c, and the THD of the PW voltage reaches 12.12% as shown in Fig. 3.52f. At 3.7 s, the proposed strategy is activated. Consequently, the THD of the PW voltage is significantly reduced to 4.7% as shown in Fig. 3.52g, and the amplitude of the negative-sequence component is reduced to 5 V as shown in Fig. 3.52c. Hence, both the unbalanced and nonlinear effects of the PW voltage are reduced, which makes the PW voltage almost balanced and sinusoidal, as shown in Fig. 3.52e. This test validates the effectiveness of the proposed strategy through the fast and stable suppression of the negative-sequence and harmonic components of the PW voltage.

3.5 Dual-Resonant Controller for Compensating Unbalanced and …

149

Fig. 3.49 Dynamic performance test at 675 rpm with the unbalanced load: a PW voltage, b CW current, c amplitude of the negative-sequence PW voltage

3.5.3.4

Under Single-Phase Nonlinear Load

In this experiment, the BDFIG works under the single-phase nonlinear load (i.e., the unbalanced nonlinear load). This situation is realized by adopting a single-phase diode rectifier with a resistor of 25  at the dc side. Before 0.3 s, the BDFIG operates under the proposed strategy. The amplitude of the negative-sequence component is around 35 V, and the THD of the PW voltage can be kept at a small value of 2.11%, as shown in Fig. 3.53. However, after the control strategy is switched to the conventional

150

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

one at 0.3 s, the amplitude of the negative-sequence component of the PW voltage immediately increases to 80 V and the THD rises to 6.43%. Hence, in comparison with the conventional strategy, the proposed strategy is with the good ability to reject the negative-sequence and harmonic components of the PW voltage caused by the single-phase nonlinear load.

Fig. 3.50 Experimental results at 675 rpm under the three-phase nonlinear load: a PW voltage, b CW current, c amplitudes of the 5th and 7th harmonics of the PW voltage, d harmonic spectrum of the PW voltage (the subscripts 1 and 2 indicate the conventional and proposed strategies, respectively)

3.5 Dual-Resonant Controller for Compensating Unbalanced and …

Fig. 3.50 (continued)

151

152

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.50 (continued)

3.6 Summary Firstly, this chapter explains the adverse effects of unbalanced and nonlinear loads on the standalone BDFIG, and analyzes the dynamic behavior of the standalone BDFIG under the unbalanced load, three-phase nonlinear load, unbalanced plus nonlinear load, and single-phase nonlinear load, respectively. Secondly, a negative-sequence voltage compensator in MSC is proposed to eliminate the unbalance influence of the PW voltage. Two kinds of loads (single-phase load and three-phase load) are adopted to carry out experiments under various work conditions. The performance of the proposed strategy has been compared with that of the conventional strategy. The simulation and experimental results confirm the effectiveness of the proposed strategy. And then, a low-order harmonic voltage compensator is proposed to enhance the quality of the PW voltage under three-phase nonlinear loads. Many harmonics with the frequencies of odd multiples of PW frequency are generated when the

3.6 Summary

153

Fig. 3.51 Dynamic performance test at 675 rpm under the three-phase nonlinear load: a PW voltage, b CW current, c amplitudes of the 5th and 7th harmonics of the PW voltage, d harmonic spectrum of the PW voltage

154

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.52 Effectiveness test for the proposed strategy under the unbalanced plus nonlinear load: a PW voltage, b CW current, c amplitude of the negative-sequence PW voltage, d expanded view of (a) between 3.2 and 3.4 s, e expanded view of (a) between 6.1 and 6.3 s, f harmonic spectrum of the PW voltage depicted in (d), g harmonic spectrum of the PW voltage depicted in (e)

3.6 Summary

Fig. 3.52 (continued)

155

156

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

Fig. 3.52 (continued)

BDFIG works with nonlinear loads. The 7th and 5th harmonic components are the most severe ones and have to be removed. The failure of the conventional strategy to address nonlinear loads has been proved. With the proposed strategy, the 5th and 7th harmonic components of the PW voltage can be reduced significantly. The experimental and simulation results demonstrate the effectiveness of the proposed method. Finally, a DRC control strategy in MSC is proposed to eliminate the distorted and unbalanced PW voltage simultaneously. In comparison with the former strategies, the DRC control strategy can work without any filter to acquire the harmonic and negative sequence components, and can decrease the numbers of PI, PIR, and PI controllers to boost the response speed of the system. Also, the design methodology of the DRC is simpler and the computational burden can be minimized. Comprehensive simulation and experiments are accomplished when the BDFIG operates under four types of loads (i.e., the unbalanced load, three-phase nonlinear load, unbalanced plus nonlinear load, and single-phase nonlinear load), which validates the strong drive performance of the proposed strategy.

3.6 Summary

157

Fig. 3.53 Effectiveness test for the proposed strategy under the single-phase nonlinear load: a PW voltage, b CW current, c amplitude of the negative-sequence PW voltage, d expanded view of (a) between 0 and 0.15 s, e expanded view of (a) between 0.32 and 0.47 s, f harmonic spectrum of the PW voltage depicted in (d), g harmonic spectrum of the PW voltage depicted in (e)

158

Fig. 3.53 (continued)

3 Unbalanced and Low-Order Harmonic Voltages Rejection …

3.6 Summary

159

Fig. 3.53 (continued)

References 1. Carrasco G, Silva CA, Peña R et al (2015) Control of a four-leg converter for the operation of a DFIG feeding stand-alone unbalanced loads. IEEE Trans Ind Electron 62(7):630–4640 2. Phan V, Nguyen D, Trinh Q et al (2016) Harmonics rejection in stand-alone doubly-fed induction generators with nonlinear loads. IEEE Trans Energy Convers 31(2):815–817 3. Pena R, Cardenas R, Escobar E et al (2007) Control system for unbalanced operation of stand-alone doubly fed induction generators. IEEE Trans Energy Convers 22(2):544–545 4. Wei F, Zhang X, Vilathgamuwa DM et al (2013) Mitigation of distorted and unbalanced stator voltage of stand-alone doubly fed induction generators using repetitive control technique. IET Electr Power Appl 7(8):654–663 5. Ataji AB, Miura Y, Ise T et al (2016) Direct voltage control with slip angle estimation to extend the range of supported asymmetric loads for stand-alone DFIG. IEEE Trans Power Electron 31(2):1015–1025 6. Muljadi E, Yildirim D, Batan T et al (1999) Understanding the unbalanced-voltage problem in wind turbine generation. In: Conference record of the 1999 IEEE industry applications conference. Thirty-forth IAS annual meeting, pp 1359–1365 7. Dehong X, Frede B, Wenjie C et al (2018) DFIG under unbalanced grid voltage. In: Advanced control of doubly fed induction generator for wind power systems, pp 237–258 8. Dehong X, Frede B, Wenjie C et al (2018) Analysis of DFIG under distorted grid voltage. In: Advanced control of doubly fed induction generator for wind power systems, pp139–165 9. McMahon RA, Roberts PC, Wang X et al (2006) Performance of BDFM as generator and motor. IEE Proc Electr Power Appl 153(2):289–299

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10. Xu H, Hu J, He Y (2012) Integrated modeling and enhanced control of DFIG under unbalanced and distorted grid voltage conditions. IEEE Trans Energy Convers 27(3):725–736 11. Cheng M, Jiang Y, Han P et al (2018) Unbalanced and low-order harmonic voltage mitigation of stand-alone dual-stator brushless doubly fed induction wind generator. IEEE Trans Ind Electron 65(11):9135–9146 12. Phan V, Lee H (2011) Control strategy for harmonic elimination in stand-alone DFIG applications with nonlinear loads. IEEE Trans Power Electron 26(9):2662–2675 13. Xu H, Hu J, He Y (2012) Operation of wind-turbine-driven DFIG systems under distorted grid voltage conditions: analysis and experimental validations. IEEE Trans Power Electron 27(5):2354–2366 14. Lu Z, Yang H, Zhang AJ (2014) New magnetic integration of full-wave rectifier with centertapped transformer. In: 2014 International power electronics and application conference and exposition, pp 609–613 15. Eggleston FJ (1985) Harmonic modelling of transmission systems containing synchronous machines and static convertors 16. Xu W, Mohammed OME, Liu Y et al (2020) Negative sequence voltage compensating for unbalanced standalone brushless doubly-fed induction generator. IEEE Tran Power Electron 35(1):667–680 17. Gonzalo A, Jesús L, Miguel R et al (2011) Stand-alone DFIM based generation systems. In: Doubly fed induction machine: modeling and control for wind energy generation applications, pp 537–578 18. Iwanski G, Koczara W (2007) Sensorless direct voltage control of the stand-alone slip-ring induction generator. IEEE Trans Ind Electron 54(2):1237–1239 19. Wei X, Cheng M, Wang W et al (2016) Direct voltage control of dual-stator brushless doubly fed induction generator for stand-alone wind energy conversion systems. IEEE Trans Magn 52(7), Article 8203804 20. Hu J, He Y (2009) Reinforced control and operation of DFIG-based wind-power-generation system under unbalanced grid voltage conditions. IEEE Trans Energy Convers 24(4):905–915 21. Rodriguez P, Luna A, Ciobotaru M et al (2006) Advanced grid synchronization system for power converters under unbalanced and distorted operating conditions. In: IECON 2006—32nd Annual conference on IEEE industrial electronics, pp 5173–5178 22. Ellis G (2004) Tuning a control system. In: Control system design guide, 3rd edn, Chap 3, pp 31–55 23. Golestan S, Monfared M, Freijedo FD (2013) Design-oriented study of advanced synchronous reference frame phase-locked loops. IEEE Trans Power Electron 28(2):765–778 24. Xu W, Mohammed OME, Liu Y et al (2018) Control design of stand-alone brushless doublyfed induction generator for supplying nonlinear loads. In: 21st International conference on electrical machines and systems (ICEMS), pp 1279–1284 25. Rodríguez P, Luna A, Candela I et al (2011) Multiresonant frequency-locked loop for grid synchronization of power converters under distorted grid conditions. IEEE Trans Ind Electron 58(1):127–138

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26. Xu W, Mohammed OME, Liu Y et al (2021) An improved control method for standalone brushless doubly-fed induction generator under unbalanced and nonlinear loads using dualresonant controller. IEEE Trans Ind Electron 68(7):5594–5605 27. Lascu C, Asiminoaei L, Boldea I et al (2009) Frequency response analysis of current controllers for selective harmonic compensation in active power filters. IEEE Trans Ind Electron 56(2):337–347

Chapter 4

Advanced Control Strategies for Standalone BDFIGs with Heavy Load Disturbance

Abstract The traditional control strategies for BDFIG operating in standalone mode are without satisfactory dynamic performance under heavy load disturbance. In this chapter, two different control strategies are proposed to address this issue. The first one, carried out in the MSC, tilizes the transient feedforward compensation of the CW current to supress the PW votltage drop. The second control strategy, i.e. the dualconverter cooperative compensation strategy, adopts the MSC and LSC to compensate for changes in the PW active and reactive currents, respectively. As a result, the dual-converter cooperative compensation strategy can make full use of the redundant capacity of MSC and LSC to compensate for the load disturbance. The proposed two control strategies are verified on the 90 and 3 kVA BDFIG experimental platforms, respectively. Keywords Vector control · Heavy load disturbance · Transient current compensation · Cooperative compensation

4.1 Introduction The BDFIG is anticipated to be one of the most important generators especially in the standalone mode in the next years, due to the distinctive structure of the BDFIG without slip rings and brushes enhancing the durability and reliability. The configuration of the BDFIG has been presented in Chap. 2. The BDFIG can produce electricity with the stable frequency under variable rotor speeds, which makes it operate in both standalone and grid-connected generation modes. The control strategies of BDFIG employed in wind power generation under grid-connected mode have been developed in [1–3]. In general, the wind generator in grid-connected mode needs to control reactive and active power, while in standalone mode it requires the stability of the amplitude and frequency of PW voltage when the load or rotor speed changes. Thus, the control strategies in grid-connected mode cannot be directly used for the standalone mode.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Y. Liu and W. Xu, Advances in Control Technologies for Brushless Doubly-fed Induction Generators, https://doi.org/10.1007/978-981-19-0424-0_4

163

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4 Advanced Control Strategies for Standalone …

The CW current orientation control of the standalone BDFIG has been presentedand analyzed [4, 5]. The DVC method of the standalone BDFIG for wind power systems has been investigated in [6]. The sensorless DVC method of the standalone BDFIG has been introduced in [7]. All the control methods introduced in [4, 6] and [7] are applied in MSC, with low dynamic performance particularly under heavy load disturbance. The reactive current transient control for LSC in BDFIG drive systems has been presented to enhance PW voltage with significant inductance-resistance load fluctuation via absorbing or supplying reactive current of loads [8]. Nevertheless, the current control methods for MSC are without high dynamic performance under large load fluctuation. In this chapter, two different control strategies are proposed to improve the dynamic performance of the standalone BDFIG under heavy load disturbance. The first control strategy is achieved by the MSC, which employes the transient feedforward compensation of the CW current to enhance the dynamic response of the PW voltage regulation. The second control strategy is realized by both the MSC and LSC, so that it can make full use of the redundant capacity of the two power converters to compensate for the load disturbance. The proposed two control strategies are verified on the 90 and 3 kVA BDFIG experimental platforms, respectively, at the super- and sub-synchronous speeds for feeding three-phase induction motors. The experimental tests prove good dynamic performance of the two control strategies.

4.2 Vector Control Strategy with Transient Current Compensation 4.2.1 Transient Feedforward Compensation of CW Current By substituting (3.4) into (3.5), the flux linkage of CW ψ 2 can be rewritten as without the rotor current: ψ2 = −

L 1 L 2r L 2r i1 + L 2i2 + ψ1 . L 1r L 1r

(4.1)

And then, substituting (4.1) into (3.2) combining (2.8), the CW voltage in dq reference frame can be calculated as u 2d = (R2 + L 2 s)i 2d + D2d

(4.2)

u 2q = (R2 + L 2 s)i 2q + D2q

(4.3)

4.2 Vector Control Strategy with Transient Current Compensation

165

Fig. 4.1 The CW current vector control loop

where s is the differential operator d/ dt, D2d and D2q can be considered as lowfrequency fluctuation, which indicate the cross-coupling effect between CW and PW. The detailed expressions for D2d and D2q can be illustrated as  L 1 L 2r   L 2r  sψ1d +ω2 ψ1q − si 1d +ω2 i 1q L 1r L 1r

(4.4)

 L 1 L 2r   L 2r  sψ1q − ω2 ψ1d − si 1q − ω2 i 1d . L 1r L 1r

(4.5)

D2d = ω2 L 2 i 2q + D2q = −ω2 L 2 i 2d +

With neglecting D2d and D2q , the term (R2 + L 2 s) becomes the first-order transfer function from i2dq to u2dq . Thus, two PI controllers can be used to adjust d and q-axis CW currents, respectively. The CW current control loop is illustrated in Fig. 4.1. From (3.1), the d and q-axis PW voltage can be expressed as u 1d = R1 i 1d + sψ1d − ω1 ψ1q

(4.6)

u 1q = R1 i 1q + sψ1q + ω1 ψ1d

(4.7)

The PW flux orientatin is adopted in this control strategy, so that the d and q-axis PW fluxes can be given by ψ1d = |ψ1 |, ψ1q = 0.

(4.8)

In general, the flux of PW can be considered as constant during one sampling period. Thus, by employing (4.8) and ignoring R1 , (4.6) and (4.7) can be simplified as u 1d ≈0, u 1q ≈U1 ≈ω1 ψ1d .

(4.9)

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4 Advanced Control Strategies for Standalone …

From (3.4), the d and q-axis PW fluxes can be expressed as ψ1d = L 1 i 1d + L 1r ir d

(4.10)

ψ1q = L 1 i 1q + L 1r irq .

(4.11)

From (3.3) and (3.6), the d and q-axis rotor currents can be derived as (ω1 − p1 ωr )2 L r (L 2r i 2d + L 1r i 1d ) + Dr d (Rr + L r s)2 + (ω1 − p1 ωr )2 L r2

(4.12)

(ω1 − p1 ωr )2 L r (L 2r i 2q + L 1r i 1q ) + Drq . (Rr + L r s)2 + (ω1 − p1 ωr )2 L r2

(4.13)

ir d = − irq = −

The detailed expressions for Drd and Drq can be illustrated as (Rr + L r s)(L 2r si 2d + L 1r si 1d ) (Rr + L r s)2 + (ω1 − p1 ωr )2 L r2 (Rr + L r s)(ω1 − p1 ωr )(L 2r i 2q + L 1r i 1q ) + (Rr + L r s)2 + (ω1 − p1 ωr )2 L r2 (ω1 − p1 ωr )L r (L 2r si 2q + L 1r si 1q ) − (Rr + L r s)2 + (ω1 − p1 ωr )2 L r2

Dr d = −

(Rr + L r s)(L 2r si 2q + L 1r si 1q ) (Rr + L r s)2 + (ω1 − p1 ωr )2 L r2 (Rr + L r s)(ω1 − p1 ωr )(L 2r i 2d + L 1r i 1d ) − (Rr + L r s)2 + (ω1 − p1 ωr )2 L r2 (ω1 − p1 ωr )L r (L 2r si 2d + L 1r si 1d ) + (Rr + L r s)2 + (ω1 − p1 ωr )2 L r2

(4.14)

Drq = −

(4.15)

Substituting (4.12) into (4.10) and substituting (4.13) into (4.11), the flux of PW can be calculated as (ω1 − p1 ωr )2 [L r L 1r L 2r i 2d + (L r L 21r − L 1 L r2 )i 1d ] + D1d (Rr + L r s)2 + (ω1 − p1 ωr )2 L r2

(4.16)

(ω1 − p1 ωr )2 [L r L 1r L 2r i 2q + (L r L 21r − L 1 L r2 )i 1q ] + D1q . (Rr + L r s)2 + (ω1 − p1 ωr )2 L r2

(4.17)

ψ1d = − ψ1q = −

where D1d and D1q can be considered as low-frequency disturbance. The detailed expressions for D1d and D1q can be illustrated as

4.2 Vector Control Strategy with Transient Current Compensation

167

D1d = L 1

(Rr + L r s)2 i 1d − L 1r Dr d (Rr + L r s)2 + (ω1 − p1 ωr )2 L r2

(4.18)

D1q = L 1

(Rr + L r s)2 i 1q − L 1r Drq (Rr + L r s)2 + (ω1 − p1 ωr )2 L r2

(4.19)

The PW flux orientation is achieved via adjusting ψ 1q to zero. According to (4.17), there is a linear relationship between ψ 1q and i2q . Based on (4.9), PW voltage amplitude can be controlled by ψ 1d , which is in linear relationship with i2d according to (4.16). With ignoring low-frequency disturbance D1d and D1q , fully considering (4.16) and (4.17), the variation of the PW flux with the changed load can be illustrated as (ω1 − p1 ωr )2 [L r L 1r L 2r i 2d + (L r L 21r − L 1 L r2 )i 1d ] (Rr + L r s)2 + (ω1 − p1 ωr )2 L r2

(4.20)

(ω1 − p1 ωr )2 [L r L 1r L 2r i 2q + (L r L 21r − L 1 L r2 )i 1q ] . (Rr + L r s)2 + (ω1 − p1 ωr )2 L r2

(4.21)

ψ1d = − ψ1q = −

To obtain a constant voltage of PW, the variation of the PW flux must be reduced to zero. Thus, by tuning ψ 1d and ψ 1q to zero, the transient feedforward compensation quantities of the CW current can be calculated as ∗ = i 2d

L 1 L r − L 21r L 1 L r − L 21r ∗ i 1d , i 2q = i 1q L 1r L 2r L 1r L 2r

(4.22)

According to the above derivation, the proposed control strategy can be illustrated in Fig. 4.2 with four main parts, i.e., the PW voltage amplitude control, the PW flux orientation control, the CW current control and the CW transient current compensation. In Fig. 4.2, the PW flux is calculated by a voltage-model-based estimator. In order to eliminate the influence of the dc offset and the initial value of the integral, this estimator uses an improved integrator based on the DSOGI [9]. Besides, the flux, current and voltage of PW and CW in the stationary reference frame should be transformed to those in the unified rotating reference frame according to the following expressions [10]: ∗

x αβ1 = e jθ1 x dq ∗

x αβ2 = e j (θg −θ1 ) x dq θg = ( p1 + p2 )θr − p2 γ

(4.23)

where γ is the mechanical angle between the PW and CW, the subscripts αβ 1 and αβ 2 represent the stationary reference frames of PW and CW, respectively, the subscript

168

4 Advanced Control Strategies for Standalone …

Fig. 4.2 The block diagram of the proposed vector control strategy with CW transient current compensation

dq represents the unified rotating reference frame, and the variable x stands for the flux, current or voltage.

4.2.2 Experimental Results All the experiments are carried out on a 90-kVA BDFIG experimental plarform with the structure as shown in Sect. A.1, Appendix. The detailed photograph of the experimental plarform and the main parameters of the adopted BDFIG are given in Sect. A.4, Appendix. For the standalone power gnenration system, the line-start induction motor is a typical and considerable load disturbance. And, the dynamic performance of the BDFIG is important at both super- and sub-synchronous speeds. Hence, two typical

4.2 Vector Control Strategy with Transient Current Compensation

169

experiments are carried out with different line-start induction motors at the super- and sub-synchronous speeds, respectively. The first experiment is conducted with a 7.5kW line-start three-phase induction motor at the the sub-synchronous speed. Since the load-carrying capacity of the standalone BDFIG is stronger at higher speeds [5], the second experiment is performed with a 15-kW line-start three-phase induction motor at the the super-synchronous speed.

4.2.2.1

At Sub-synchronous Speed

This experiment is carried out at 600 rpm, and the load is a 7.5-kW line-start threephase induction motor. Figure 4.3 presents the experimental results without the CW transient current compensation. At 4.77 s, the induction motor is connected to the generator, and the CW current amplitude is raised from 100.9 to 186.3 A. Nevertheless, the PW voltage amplitude is reduced by 38% and recovered within 40 ms, which cannot meet the requirements of the practical industry applicaions. Figure 4.4 shows the practical tests with the CW transient current compensation under the same rotor speed and load. The induction motor is inserted into the system at 2.62 s. The CW current amplitude increases fast from 103.1 to 334.1 A, and the PW voltage amplitude is decreased by 12% and gets back within 20 ms. Through a comparison between the experimental results as shown in Figs. 4.4 and 4.5, the CW transient current compensation can be attained to greatly enhance the dynamic performance of the standalone BDFIG under severe load fluctuation at the subsynchronous speed.

4.2.2.2

At Super-synchronous Speed

Figure 4.5 illustrates the experimental results without the CW transient current compensation. At 0.51 s, the-15 kW three-phase induction motor is connected to the BDFIG, and the CW current amplitude is raised from 104.9 to 168.2 A. Nevertheless, the PW voltage amplitude drops by 65.7% and comes back within 30 ms, which presents the poor dynamic performance under heavy load disturbance. Figure 4.6 shows the practical tests with the CW transient current compensation under the same operation condition as Fig. 4.5. The induction motor is started at 0.42 s. The CW current amplitude increases fast from 106.9 to 404.4 A, and the PW voltage amplitude is reduced by 28.5% and recovered within 150 ms due to the big start-up current of the 15-kW three-phase induction motor. Comparing the experimental results as shown in Figs. 4.6 and 4.7, the CW transient current compensation can significantly improve the dynamic performance of the standalone BDFIG under heavy load disturbance at the super-synchronous speed.

170

4 Advanced Control Strategies for Standalone …

Fig. 4.3 Experimental results for the 90-kVA standalone BDFIG feeding a 7.5-kW three-phase induction motor at 600 rpm without the CW transient current compensation: a PW voltage, b PW current, c CW current

4.2 Vector Control Strategy with Transient Current Compensation

171

Fig. 4.4 Experimental results for the 90-kVA standalone BDFIG feeding a 7.5-kW three-phase induction motor at 600 rpm with the CW transient current compensation: a PW voltage, b PW current, c CW current

172

4 Advanced Control Strategies for Standalone …

Fig. 4.5 Experimental results for the 90-kVA standalone BDFIG feeding a 15-kW three-phase induction motor at 900 rpm without the CW transient current compensation: a PW voltage, b PW current, c CW current

4.2 Vector Control Strategy with Transient Current Compensation

173

Fig. 4.6 Experimental results for the 90-kVA standalone BDFIG feeding a 15-kW three-phase induction motor at 900 rpm with the CW transient current compensation: a PW voltage, b PW current, c CW current

174

4 Advanced Control Strategies for Standalone …

Fig. 4.7 Compensation strategy based on MSC

4.3 Cooperative Compensation Strategy Based on Dual Power Converters 4.3.1 Compensation Strategy Based on MSC According to (4.12) and (4.13), the expressions of the d- and q-axis CW currents can be rewritten as   (ω1 − p1 ωr ) L 1r i 1q + L 2r i 2q M (4.24) ir d = − (L 1r i 1d + L 2r i 2d ) + (1 − M) Lr Rr + L r s irq = −

 M (ω1 − p1 ωr )(L 1r i 1d + L 2r i 2d ) L 1r i 1q + L 2r i 2q − (1 − M) Lr Rr + L r s

(4.25)

L 2 s 2 +L R s+L 2 (ω − p ω )2

r r 1 1 r r r where M = (L 2 2 2 . r s+Rr ) +L r (ω1 − p1 ωr ) Since the zeros and poles of M are very close, they can cancel each other. Equations (4.24) and (4.25) can be simplified to

(L 1r i 1d + L 2r i 2d ) Lr   L 1r i 1q + L 2r i 2q =− . Lr

ir d = −

(4.26)

irq

(4.27)

4.3 Cooperative Compensation Strategy Based on Dual Power Converters

175

Substituting (4.10), (4.11), (4.26) and (4.27) into (4.6) and (4.7), and ignoring the differential terms, the d- and q-axis PW voltages can be derived as  ⎫ L L i +L i u 1d = R1 i 1d − ω1 L 1 i 1q − 1r ( 1r 1qL r 2r 2q ) ⎬   . u 1q = R1 i 1q + ω1 L 1 i 1d − L 1r (L 1r i1dL r+L 2r i2d ) ⎭

(4.28)

Rearranging (4.28), it can be obtained that

L 21r ω1 L 1r L 2r − L 1 i 1q + i 2q Lr Lr

L 21r ω1 L 1r L 2r i 1d − + ω1 L 1 − i 2d . Lr Lr

u 1d = R1 i 1d + ω1 u 1q = R1 i 1q

(4.29)

From (4.29), it can be noted that i2q can be used to control u1d , and i2d can be employed to regulate u1q . In order to simplify the control scheme, the PW voltage orientation is adopted, which means that the d axis component has to be aligned with the PW voltage vector. Thus, the PW voltage amplitude can be controlled by i2q , and the frame orientation can be achieved by regulating i2d . Based on the small signal analysis on (4.29), the disturbances of the d- and q-axis PW voltages can be expressed as u 1d = R1 i 1d +ω1



u 1q = R1 i 1q + ω1

⎫  − L 1 i 1q + ω1 L 1rLLr 2r i 2q ⎬  L2 L 1 − L1rr i 1d − ω1 L 1rLLr 2r i 2d ⎭

L 21r  Lr

(4.30)

where u1d , u1q , i1d , i1q , i2d and i2q indicate the changes of d- and q-axis PW voltages, PW currents and CW currents within two adjacent instants of time, respectively. In order to obtain a constant PW voltage, the disturbance of the PW voltage should be reduced to zero. Hence, setting u1d and u1q in (4.30) to zero, the instantaneous feedforward compensation for the CW current can be obtained by L 1 L r −L 21r i 1q L 1r L 2r L r R1 i 1q ω1 L 1r L 2r

∗ i 2q = − ω1LLr1rRL1 2r i 1d +

∗ i 2d =

L 1 L r −L 21r L 1r L 2r

i 1d +

 (4.31)

According to (4.29) and (4.31), the compensation strategy based on MSC can be obtained, as shown in Fig. 4.7. The PW voltage orientation is used in the PW voltage control loop, and the feedforward compensation quantities calculated by (4.31) is added to the output of the PW voltage control loop. The reference value of the PW voltage phase angle is obtained by the integration of the PW reference frequency.

176

4 Advanced Control Strategies for Standalone …

Fig. 4.8 The positive directions of the three-phase currents of PW, load and LSC (i1abc , ilabc and isabc )

MSC

BDFIG

CW

PW i1abc ilabc

isabc

Load LSC

4.3.2 Compensation Strategy Based on LSC From the structure of the standalone BDFIG power generation system, the relationship among the PW, load and LSC currents can be expressed as i 1abc = ilabc + i sabc

(4.32)

where i1abc , ilabc , and isabc indicate the three-phase currents of PW, load and LSC, respectively. The positive directions of i1abc , ilabc and isabc are illustrated in Fig. 4.8. The d-axis and q-axis component expressions of (4.32) can be written as i 1d = ild + i sd

(4.33)

i 1q = ilq + i sq .

(4.34)

In the traditional LSC control strategy, isd is employed to adjust the dc bus voltage, and the reference value of isq is set to zero. It can be seen that the reactive current of LSC has not been fully utilized. If isq = −ilq (i.e., the reactive current of the load is fully supplied by LSC), according to (4.34), i1q would be zero. From (4.30), the disturbances of the d- and q-axis PW voltages can be simplified as

u 1q

 u 1d = R1 i 1d +ω1 L 1rLLr 2r i 2q   . L2 = ω1 L 1 − L1rr i 1d − ω1 L 1rLLr 2r i 2d

(4.35)

Hence, the influence of the load on the PW voltage can be reduced with the reactive current of the load being fully offered by LSC. Based on the LSC control strategy

4.3 Cooperative Compensation Strategy Based on Dual Power Converters

177

Fig. 4.9 Compensation strategy based on LSC

proposed in [11], the compensation strategy based on LSC is illustrated in Fig. 4.9, where the subscripts l and s indicate the load and LSC.

4.3.3 Cooperative Compensation Strategy Based on Dual Power Converters A large load disturbance will cause a significant change in the PW current. From (4.30), it can be seen that when the CW current is not compensated in time, the significant change of the PW current would cause the PW voltage to fluctuate greatly. When the PW current changes drastically, the instantaneous feedforward compensation quantities obtained by (4.31) would be very large, which may eventually cause the reference value of the CW current to exceed the rated current of the CW or MSC. When an induction motor is directly started, its instantaneous current will reach 5–7 times the rated current, of which the reactive current accounts for a larger proportion. If this induction motor serves as the load of the standalone BDFIG system, the reference value of the CW current obtained by the proposed MSC compensation strategy would be significantly greater than the rated currents of the CW and MSC. In order to avoid the system damage, the reference value of the CW current have to be limited to below the CW and MSC rated currents, which would in turn degrade the performance of the proposed compensation strategy.

178

4 Advanced Control Strategies for Standalone …

Fig. 4.10 Cooperative compensation strategy based on dual power converters

To solve the problem mentioned above, the LSC can be employed to compensate the PW reactive current (i.e. the q-axis current of PW). Thus, the influence of the q-axis current of PW on the MSC control can be ignored, and only the influence of the d-axis current of PW needs to be considered. Consequently, Eq. (4.31) can be simplified as ∗ = − ω1LLr1rRL1 2r i 1d i 2q ∗ i 2d =

L 1 L r −L 21r L 1r L 2r

i 1d

 (4.36)

Combining Figs. 4.8 and 4.10, the dual-converter cooperative compensation strategy can be obtained, as shown in Fig. 4.10, where the CW current compensation quantities in the MSC is calculated by (4.36). The dual-converter cooperative compensation strategy can make full use of the redundant capacity of MSC and LSC for compensating the load disturbance.

4.3.4 Experimental Results All the experiments are carried out on a 3-kVA wound-rotor BDFIG with the initial three-phase resistive load of 200 in each phase. A 3-kW induction motor mechanically coupled to the BDFIG is used as the prime mover, and a 0.55-kW induction

4.3 Cooperative Compensation Strategy Based on Dual Power Converters

179

motor serves as the added load of the standalone BDFIG system. The detailed photograph of the experimental platform and the main parameters of the BDFIG can be seen in Sect. A.5, Appendix. The comprehensive experiments are completed at the sub- and super-synchronous speeds with four different control strategies, i.e., the control strategy without compensation, the one with the MSC compensation, the one with the LSC compensation and the one with the dual-converter cooperative compensation, so as to fully realize the characteristics of different control strategies.

4.3.4.1

At Sub-synchronous Speed

All the experiments are carried out with the load of 0.55-kW induction motor being added at the sub-synchronous speed of 900 rpm. At first, the control strategy without compensation is implemented. And then, the three different compensation strategies are adopted separately. In Figs. 4.11, 4.12, 4.13 and 4.14, the sub-figures (a)–(g) present the PW voltage amplitude, CW d-axis current, CW q-axis current, PW line voltage, CW phase current, load phase current and dc bus voltage, respectively. The experimental results of the control strategy without compensation are illustrated in Fig. 4.11. At 3.36 s, the induction motor is started. The CW d-axis current does not change too much, and the CW q-axis current transiently rises from 3 to 9 A due to the load disturbance. The PW voltage amplitude and the dc bus voltage drop by 20% and 12%, respectively, along with the induction motor being connected the BDFIG system. The settling time of the PW voltage reaches 150 ms (i.e. 7.5 PW voltage cycles), as well as that of the dc bus voltage. The experimental results of the control strategy based on the MSC compensation are presented in Fig. 4.12. When the induction motor is started at 3.65 s, the CW q-axis current transiently sharply rises from 3 to 12 A with the instantaneous feedforward compensation, and the drops of the PW voltage amplitude and the dc bus voltage can be reduced to 12% and 7%, respectively. Both the settling times of the PW voltage and dc bus voltage can be decreased to 107 ms (about five PW voltage cycles). The experimental results shown in Fig. 4.12 demonstrate that the MSC compensation can suppress the PW voltage fluctuation caused by the heavy load disturbance to a certain extent. Figure 4.13 depicts the experimental results of the control strategy based on the LSC compensation. The BDFIG operation condition in this experiment is the same as that in the first two experiments. At 3.27 s, the induction motor is connected to the standalone BDFIG system. The drop of the PW voltage amplitude with the LSC compensation is very similar to that without compensation. Fortunately, the settling time can be reduced to less than four PW voltage cycles (around 74 ms). The experimental results in Fig. 4.12 indicate that the main role of the LSC compensation is to shorten the period of the PW voltage fluctuation. Figure 4.14 presents the experimental results of the control strategy based on the dual-converter cooperative compensation under the same rotor speed and load condition as before. The induction motor is started at 3.29 s. With the compensation from both the MSC and LSC, the drops of the PW voltage amplitude and the dc bus

4 Advanced Control Strategies for Standalone …

PW voltage amplitude(V)

180

150ms

311V

250V

Time (s)

CW d-axis current (A)

(a)

Time (s)

CW q-axis current (A)

(b)

Time (s) (c)

Fig. 4.11 Experimental results of the control strategy without compensation at the sub-synchronous speed of 900 rpm: a PW voltage amplitude, b CW d-axis current, c CW q-axis current, d PW line voltage, e CW phase current, f Load phase current, g dc bus voltage

PW line voltage (V)

4.3 Cooperative Compensation Strategy Based on Dual Power Converters

Time (s)

CW current (A)

(d)

Load current (A)

Time (s) (e)

Time (s) (f)

Fig. 4.11 (continued)

181

4 Advanced Control Strategies for Standalone …

DC voltage (V)

182

603V

533V

Time (s) (g)

PW voltage amplitude(V)

Fig. 4.11 (continued)

107ms

311V

270V

Time (s)

CW d-axis current (A)

(a)

Time (s)

(b) Fig. 4.12 Experimental results of the control strategy based on the MSC compensation at the subsynchronous speed of 900 rpm: a PW voltage amplitude, b CW d-axis current, c CW q-axis current, d PW line voltage, e CW phase current, f Load phase current, g dc bus voltage

CW q-axis current (A)

4.3 Cooperative Compensation Strategy Based on Dual Power Converters

Time (s)

PW line voltage (V)

(c)

Time (s)

CW current (A)

(d)

Time (s)

(e) Fig. 4.12 (continued)

183

4 Advanced Control Strategies for Standalone …

Load current (A)

184

DC voltage (V)

Time (s) (f)

603V

563V

Time (s) (g)

PW voltage amplitude(V)

Fig. 4.12 (continued)

74ms

311V

251V

Time (s) (a)

Fig. 4.13 Experimental results of the control strategy based on the LSC compensation at the subsynchronous speed of 900 rpm: a PW voltage amplitude, b CW d-axis current, c CW q-axis current, d PW line voltage, e CW phase current, f Load phase current, g dc bus voltage

CW d-axis current (A)

4.3 Cooperative Compensation Strategy Based on Dual Power Converters

Time (s)

CW q-axis current (A)

(b)

Time (s)

PW line voltage (V)

(c)

Time (s) (d)

Fig. 4.13 (continued)

185

4 Advanced Control Strategies for Standalone …

CW current (A)

186

Load current (A)

Time (s) (e)

DC voltage (V)

Time (s) (f)

603V

550V

Time (s) (g)

Fig. 4.13 (continued)

4.3 Cooperative Compensation Strategy Based on Dual Power Converters

187

voltage can be reduced to 10% and 5%, respectively. The CW q-axis current rises rapidly from 3 to 12 A, which is similar to the change of the CW current under the MSC compensation. The settling time of the PW and dc bus voltages can be reduced to 48 ms (less than 2.5 PW voltage cycles). Comparing the experimental waveforms illustrated in Figs. 4.11, 4.12, 4.13 and 4.14, it can be inferred that the dual-converter cooperative compensation strategy is with the best performance against the heavy load disturbance at the sub-synchronous speed.

4.3.4.2

At Super-synchronous Speed

PW voltage amplitude(V)

In this section, all the experiments are implemented at the super-synchronous speed of 1100 rpm with the load of 0.55-kW induction motor being added. The experimental results for the four different control strategies, i.e. the control strategy without compensation, the one with the MSC compensation, the one with the LSC compensation and the one with the dual-converter cooperative compensation, are presented in Figs. 4.15, 4.16, 4.17 and 4.18, respectively. The sub-figures (a)–(g) in Figs. 4.15, 4.16, 4.17 and 4.18 depict the PW voltage amplitude, CW d-axis current, CW q-axis current, PW line voltage, CW phase current, load phase current and dc bus voltage, respectively. Figure 4.15 shows the experimental results of the control strategy without compensation. At 3.63 s, the BDFIG start to supply the induction motor, and the PW voltage amplitude and the dc bus voltage drop by 19% and 8%, respectively. The settling time for the PW voltage and dc bus voltage is 87 ms (around 4.5 PW voltage cycles).

48ms

311V 279V

Time (s) (a)

Fig. 4.14 Experimental results of the control strategy based on the dual-converter cooperative compensation at the sub-synchronous speed of 900 rpm: a PW voltage amplitude, b CW d-axis current, c CW q-axis current, d PW line voltage, e CW phase current, f Load phase current, g dc bus voltage

4 Advanced Control Strategies for Standalone …

CW d-axis current (A)

188

Time (s)

CW q-axis current (A)

(b)

PW line voltage (V)

Time (s) (c)

Time (s) (d)

Fig. 4.14 (continued)

CW current (A)

4.3 Cooperative Compensation Strategy Based on Dual Power Converters

Load current (A)

Time (s) (e)

DC voltage (V)

Time (s) (f)

603V

570V

Time (s) (g)

Fig. 4.14 (continued)

189

4 Advanced Control Strategies for Standalone …

PW voltage amplitude(V)

190

87ms

311V

252V

Time (s)

CW d-axis current (A)

(a)

Time (s)

CW q-axis current (A)

(b)

Time (s)

(c) Fig. 4.15 Experimental results of the control strategy without compensation at the supersynchronous speed of 1100 rpm: a PW voltage amplitude, b CW d-axis current, c CW q-axis current, d PW line voltage, e CW phase current, f Load phase current, g dc bus voltage

PW line voltage (V)

4.3 Cooperative Compensation Strategy Based on Dual Power Converters

Time (s)

CW current (A)

(d)

Time (s)

Load current (A)

(e)

Time (s)

(f)

Fig. 4.15 (continued)

191

4 Advanced Control Strategies for Standalone …

DC voltage (V)

192

610V

559V

Time (s)

(g)

PW voltage amplitude(V)

Fig. 4.15 (continued)

100ms

311V

272V

Time (s)

CW d-axis current (A)

(a)

Time (s) (b)

Fig. 4.16 Experimental results of the control strategy based on the MSC compensation at the super-synchronous speed of 1100 rpm: a PW voltage amplitude, b CW d-axis current, c CW q-axis current, d PW line voltage, e CW phase current, f Load phase current, g dc bus voltage

CW q-axis current (A)

4.3 Cooperative Compensation Strategy Based on Dual Power Converters

Time (s)

PW line voltage (V)

(c)

CW current (A)

Time (s) (d)

Time (s)

(e)

Fig. 4.16 (continued)

193

4 Advanced Control Strategies for Standalone …

Load current (A)

194

Time (s)

DC voltage (V)

(f)

605V

580V

Time (s)

(g)

Fig. 4.16 (continued)

The CW d-axis current is with a minor change from 0 to 2 A. The CW q-axis current rises quickly from 2.5 to 9.5 A to supress the PW voltage fluctuation. The experimental results of the control strategy based on the MSC compensation are presented in Fig. 4.16. Under the same rotor speed and load condition, the MSC compensation can reduce the drops of the PW voltage and dc bus voltage to 13% and 4%, respectively. However, the settling time of the two voltages is slightly increased by about half one PW voltage cycle. The experimental results of the control strategy based on the LSC compensation are illustrated in Fig. 4.17. This experiment is carried out under the same rotor speed and load condition as before. In comparison to the experimental results obtained by the control strategy without compensation, the PW voltage amplitude drop in this experiment is not reduced, and the settling time is significantly decreased to 59 ms (less than three PW voltage cycles). Figure 4.18 presents the experimental results of the control strategy based on the dual-converter cooperative compensation under the same rotor speed and load condition as before. Due to the comprehensive compensation of MSC and LSC, the

4.3 Cooperative Compensation Strategy Based on Dual Power Converters

195

PW voltage amplitude(V)

PW voltage drop and the settling time can be significantly reduced to 10% and 25 ms (around one PW voltage cycle). By comparing the experimental results illustrated in Figs. 4.15, 4.16, 4.17 and 4.18, it can be noted that the control strategy based on the dual-converter cooperative compensation possesses the best transient performance under the heavy load disturbance at the super-synchronous speed. From Table 4.1, it can be seen that the MSC compensation can significantly reduce the drop of the PW voltage amplitude at both the sub- and super-synchronous speeds. However, the settling time of the PW voltage may not be shortened, because the PW voltage is regulated by the CW current and the dynamic performance of the CW current control loop is not changed by the MSC compensation strategy. The settling time of the PW voltage can be effectively reduced by the LSC compensation strategy at both the sub- and super-synchronous speeds. However, the drop of the PW voltage amplitude with the LSC compensation is almost the same as that

59ms

311V

252V

Time (s)

CW d-axis current (A)

(a)

Time (s)

(b) Fig. 4.17 Experimental results of the control strategy based on the LSC compensation at the supersynchronous speed of 1100 rpm: a PW voltage amplitude, b CW d-axis current, c CW q-axis current, d PW line voltage, e CW phase current, f Load phase current, g dc bus voltage

4 Advanced Control Strategies for Standalone …

CW q-axis current (A)

196

PW line voltage (V)

Time (s) (c)

Time (s)

CW current (A)

(d)

Time (s)

(e) Fig. 4.17 (continued)

197

Load current (A)

4.3 Cooperative Compensation Strategy Based on Dual Power Converters

DC voltage (V)

Time (s) (f)

605V

567V

Time (s) (g)

Fig. 4.17 (continued)

without compensation. The reason is that the PW current can directly affect the PW voltage drop according to (4.30), and the PW current with the LSC compensation is very similar to that without compensation at the moment of adding loads. The reactive current injected into the load by the LSC just gradually rises, and the reactive current of the PW gradually decreases to zero. The dual-converter cooperative compensation strategy combines the advantages of the MSC and LSC compensation, so that the PW voltage drop can be considerably reduced and the settling time can be significantly shortened at the same time at both the sub- and super-synchronous speeds.

4.4 Summary This chapter presents two different control strategies to enhance the dynamic performance of the standalone BDFIG under heavy load disturbance. The first control strategy is based on the single power converter, which is achieved by the MSC with

4 Advanced Control Strategies for Standalone …

PW voltage amplitude(V)

198

25ms

311V

280V

Time (s)

CW d-axis current (A)

(a)

Time (s)

CW q-axis current (A)

(b)

Time (s)

(c) Fig. 4.18 Experimental results of the control strategy based on the dual-converter cooperative compensation at the super-synchronous speed of 1100 rpm: a PW voltage amplitude, b CW d-axis current, c CW q-axis current, d PW line voltage, e CW phase current, f Load phase current, g dc bus voltage

199

PW line voltage (V)

4.4 Summary

Time (s)

CW current (A)

(d)

Time (s)

Load current (A)

(e)

Time (s)

(f) Fig. 4.18 (continued)

4 Advanced Control Strategies for Standalone …

DC voltage (V)

200

605V

575V

Time (s)

(g) Fig. 4.18 (continued)

Table 4.1 Comparison of the amplitude drop and settling time of the PW voltage with different compensation strategies BDFIG speed

Compensation strategies

Amplitude drops of PW voltage (%)

Settling time of PW voltage (PW voltage cycles)

Sub-synchronous speed

Without compensation

20

7.5

MSC compensation

12

5

LSC compensation 20

4

Dual-converter cooperative compensation

10

2.5

Without compensation

19

4.5

MSC compensation

13

5

LSC compensation 19

3

Dual-converter cooperative compensation

1

Super-synchronous speed

10

the CW transient current feedforward compensation. The second control strategy is realized by the dual power converters and compensates for the variation of the PW active and reactive currents caused by the load change through the MSC and LSC, respectively, which can make good use of the redundant capacity of the two power converters. The presented two control strategies are verified on the 90- and 3-kVA

4.4 Summary

201

BDFIG experimental platforms, respectively, at both the super- and sub-synchronous speeds with satisfactory dynamic performance.

References 1. Shao S, Abdi E, Barati F et al (2009) Stator-flux-oriented vector control for brushless doubly-fed induction generator. IEEE Trans Ind Electron 56(10):4220–4228 2. Shao S, Long T, Abdi E et al (2013) Dynamic control of the brushless doubly fed induction generator under unbalanced operation. IEEE Trans Ind Electron 60(6):2465–2476 3. Chen J, Zhang W, Chen B et al (2016) Improved vector control of brushless doubly fed induction generator under unbalanced grid conditions for offshore wind power generation. IEEE Trans Energ Convers 31(1):293–302 4. Liu Y, Ai W, Chen B et al (2016) Control design of the brushless doubly-fed machine for stand-alone VSCF ship shaft generator systems. J Power Electron 16(1):259–267 5. Liu Y, Xu W, Zhi G et al (2017) Performance analysis of the stand-alone brushless doubly-fed induction generator by using a new T-type steady-state model. J Power Electron 17(4):1027– 1036 6. Wei X, Cheng M, Wang W et al (2016) Direct voltage control of dual-stator brushless doubly fed induction generator for stand-alone wind energy conversion systems. IEEE Trans Magn 52(7). Article 8203804 7. Liu Y, Xu W, Xiong F et al (2017) Sensorless direct voltage control of the stand-alone brushless doubly-fed generator. In: 2017 international conference on electrical machines and systems (ICEMS), Sydney, Australia, pp 1–6 8. Wang X, Lin H, Wang Z et al (2017) Transient control of reactive current for line-side converter of brushless doubly fed induction generator in stand-alone operation. IEEE Trans Power Electron 32(10):8193–8203 9. Xin Z, Zhao R, Blaabjerg F et al (2017) An improved flux observer for field-oriented control of induction motors based on dual second-order generalized integrator frequency-locked loop. IEEE J Emerg Sel Topics Power Electron 5(1):513–525 10. Poza J, Oyarbide E, Roye D et al (2006) Unified reference frame dq model of the brushless doubly fed machine. Proc Inst Elect Eng-Elect Power Appl 153(5):726–734 11. Xu W, Liu Y (2020) Advanced control technologies for brushless doubly-fed induction machine. China Machine Press, China, pp 48–53

Chapter 5

Predictive Control for Standalone BDFIGs

Abstract A great attention has been given to predictive control due to its flexible principle, fast response and high accuracy. To obtain a better behaviour during both the steady-state and dynamic operation, a model predictive current control (MPCC) is handled in this chapter to replace the traditional PI controller for the current regulation on the CW of the BDFIG in the standalone applications. Simulation results under three typical conditions are presented to confirm that the MPCC can get a better CW current response than the PI controller, which can produce better effects on the output voltage of the PW. Due to the adverse effect of parameter mismatch on the control behaviour of MPCC, another new concept of nonparametric predictive current control (NPCC) is proposed for the drive system without any need of the machine parameters. The NPCC can predict the change of the CW current using the measured machine information instead of machine parameters. Simulation and experimental results show that the NPCC not only keeps the fast dynamic response performance of MPCC, but also is not affected by the change of machine parameters, which has demonstrated much stronger robustness than that of MPCC. Keywords Standalone power generation system · Model predictive current control (MPCC) · Nonparametric predictive current control (NPCC) · Machine parameter mismatch

5.1 Introduction A few classical control methods have been applied to BDFIGs, e.g., scalar control [1], vector control [2], direct torque control [3], and direct voltage control [4]. Scholars all over the world have carried out research on some new control strategies for BDFIG to further improve its operation performance. Among these control methods, predictive control has attracted much attention, since it is with fast dynamic response and intuitive principle and easy to understand and implement digitally [5, 6]. Predictive control strategies mianly include model predictive control (MPC), generalized predictive control and deadbeat control. The MPC has been widely employed in the industrial field by using the system model to predict the future behavior of control variables. Aided with the pre-defined optimization criteria, the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Y. Liu and W. Xu, Advances in Control Technologies for Brushless Doubly-fed Induction Generators, https://doi.org/10.1007/978-981-19-0424-0_5

203

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5 Predictive Control for Standalone BDFIGs

optimal switching mode of IGBTs can be determined by the predictive controller [7]. The MPC has been applied in power converters and motor drives [8], and also has been widely used for DFIGs [9, 10]. For a standalone BDFIG, the model predictive current control (MPCC) method has been proposed in [11] based on [9] and [10]. In [11], the MPCC has shown a better current tracking performance than the PI controller for the CW current control. However, a large current ripple is produced due to using only one voltage vector in the control cycle. Hence, a finite control set model predictive control (FCS-MPC) strategy has been presented in [12] with modulation algorithm, which would minimize the cost function with the selection of two adjacent active vectors in each sampling period and a zero vector. In comparison with the traditional FCS-MPC, the small current ripple and high control accuracy are the main advantages of this control method. Moreover, in [13, 14], the model predictive power control (MPPC) and model predictive torque control (MPTC) have been presented for BDFIGs. However, the presented MPC methods in literatures are very sensitive to the accuracy of machine model, such as the change of machine parameters under operation and the complexity of BDFIG characteristics. Therefore, the predictive control strategy without any dependence on the parameters is more attractive for BDFIG systems. Some research has been conducted for MPCC to improve the the robustness and dynamic response of the current prediction, which is based on the machine state information rather than the model parameters to predict the current. This can be realized by the improved nonparametric predictive current control (NPCC), which has been investigated for other electrical machines [15, 16]. This chapter addresses the improvement of the robustness and dynamic response for the standalone BDFIG system by applying both the improved MPCC and NPCC methods.

5.2 Model Predictive Current Control (MPCC) for Standalone BDFIGs The main characteristic of MPC is the use of the system model for predicting the future behavior of the controlled variables. This information is used by the controller to obtain the optimal actuation, according to a predefined optimization criterion. One important advantage of MPC is that the concepts are very simple and intuitive and the parameter is easy to adjust. In addition, nonlinearities of the system can be included in the model, avoiding the need of linearizing the model for a given operating point. And, it is very easy to include constraint conditions for some variables when the controller is designed [11]. In this section, one model predictive current control (MPCC) method is investigated for CW current regulation of standalone BDFIG systems.

5.2 Model Predictive Current Control (MPCC) for Standalone BDFIGs

205

5.2.1 Design of MPCC for CW Current 5.2.1.1

MPCC Scheme

In the standalone power generation system, the generator is not connected to the grid, so that its flux and voltage are susceptible to the variation of the rotor speed and load. Hence, the traditional PW flux or voltage orientation control method is difficult to achieve satisfactory performance in the standalone BDFIG. The control method presented in this section is based on the CW current vector orientation. The phase angle of the CW current can be obtained by integrating the reference frequency of the CW current, and its accuracy is much higher than the estimated vector angles of the PW flux and voltage. As a result, for standalone BDFIGs, the robustness of the CW current orientation-based control method is stronger than that of the PW flux and voltage orientation-based control methods. According to the above analysis, in the mathematical derivation for MPCC in this section, the angular frequency ωa of the dq frame in the BDFIG mathematical model (2.4) should be set as the CW current frequency ω2 . Thus, the voltage and flux equations of the CW and rotor can be written as 

u 2d = R2 i 2d + sψ2d − [ω2 − ( p1 + p2 )ωr ]ψ2q u 2q = R2 i 2q + sψ2q + [ω2 − ( p1 + p2 )ωr ]ψ2d  u r d = Rr ir d + sψr d − (ω2 − p1 ωr )ψrq u rq = Rr irq + sψrq + (ω2 − p1 ωr )ψr d  ψ2d = L 2 i 2d + L 2r ir d ψ2q = L 2 i 2q + L 2r irq  ψr d = L r ir d + L 1r i 1d + L 2r i 2d ψrq = L r irq + L 1r i 1q + L 2r i 2q

(5.1)

(5.2)

(5.3)

(5.4)

Since the CW current is regulated by the CW voltage, it is necessary to derive the relationship between the CW voltage and the CW current. It can be seen from (5.1) that the CW voltage expression includes the CW flux, and from (5.3) it can be seen that the calculation of CW flux requires the rotor current. However, the rotor current of the BDFIG cannot be directly measured. Hence, the rotor current in (5.3) has to be replaced by other measurable signals. Since the rotor terminal voltage of the BDFIG is zero, the rotor voltage expression (5.2) can be rewritten as 

0 = Rr ir d + sψr d − (ω2 − p1 ωr )ψrq 0 = Rr irq + sψrq + (ω2 − p1 ωr )ψr d

(5.5)

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5 Predictive Control for Standalone BDFIGs

Substituting (5.4) into (5.5), the rotor current can be expressed in terms of PW and CW currents as follows: ⎧   L r s 2 + Rr s + L r (ω1 − p2 ωr )2 (L 1r i 1d + L 2r i 2d ) ⎪ ⎪ ⎪ i = − rd ⎪ ⎪ (Rr + L r s)2 + L r2 (ω1 − p2 ωr )2 ⎪ ⎪ ⎪ ⎪ ⎪ Rr (ω1 − p2 ωr )(L 1r i 1q + L 2r i 2q ) ⎪ ⎪ − ⎪ ⎨ (Rr + L r s)2 + L r2 (ω1 − p2 ωr )2

⎪ L r2 s 2 + L r Rr s + L r2 (ω1 − p2 ωr )2 ω1 − p2 ωr ⎪ ⎪ (L 1r i 1d + L 2r i 2d ) 1− irq = ⎪ ⎪ ⎪ Rr + L r s (Rr + L r s)2 + L r2 (ω1 − p2 ωr )2 ⎪ ⎪

⎪ ⎪ ⎪ (L 1r i 1q + L 2r i 2q ) Rr L r (ω1 − p2 ωr )2 ⎪ ⎪ − s + ⎩ (Rr + L r s) (Rr + L r s)2 + L r2 (ω1 − p2 ωr )2 (5.6) However, Eq. (5.6) is too complicated to derive the MPCC control method and needs to be simplified reasonably. Since the poles and zeros of some terms in (5.6) are very close, they can cancle out each other. And then, ignoring some terms with the relatively small values, Eq. (5.6) can be further simplified as [17]  ⎧ Rr L 1r i 1q + L 2r i 2q L 1r i 1d + L 2r i 2d ⎪ ⎪ − ⎨ ir d = − Lr L r2 (ω2 − p2 ωr ) ⎪ ⎪ ⎩ irq = − L 1r i 1q + L 2r i 2q Lr

(5.7)

Afterwards, we substituts (5.7) into (5.3) to get the CW flux expressed by the PW and CW currents. And then, the obtained CW flux is taken into (5.1) to derive the new expression of CW voltage as follows: ⎧ 



L 22r ⎪ ⎪ ⎪ u si = R i + L − 2 2d 2 2d + α1 + α2 ⎨ 2d Lr 



⎪ Rr L 22r (ω2 − ( p1 + p2 )ωr ) L 22r ⎪ ⎪ si 2q + β1 + β2 i 2q + L 2 − ⎩ u 2q = R2 − L r2 (ω1 − p2 ωr ) Lr (5.8) where   2 L α1 = (ω2 − ( p1 + p2 )ωr ) L2rr − L 2 i 2q − α2 = β1 =

Rr L 22r si , L r2 (ω1 − p2 ωr ) 2q L 1r L 2r L 1r L 2r Rr L 1r L 2r − L r si 1d + (ω2 − ( p1 + p2 )ωr ) L r i 1q − L 2 (ω1 − p2 ωr ) si 1q , r   L2 (ω2 − ( p1 + p2 )ωr ) L 2 − L2rr i 2d ,

(ω2 −( p1 + p2 )ωr ) i 1q . β2 = −(ω2 − ( p1 + p2 )ωr ) L 1rLLr 2r i 1d − L 1rLLr 2r si 1q − Rr L 1r LL2r2 (ω 1 − p2 ωr ) r In (5.8), the α 1 and β 1 represent the disturbances of q-axis CW current on d-axis CW voltage and d-axis CW current on q-axis CW voltage, respectively. The α 2 and

5.2 Model Predictive Current Control (MPCC) for Standalone BDFIGs

207

β 2 are the additional d- and q-axis components of the CW voltage induced by the PW current, respectively. In order to facilitate the implementation of MPCC, it is necessary to simplify α 1 , α 2 , β 1 and β 2 . In this chapter, the reference value of the q-axis CW current is set to zero with the adopted control method for the standalone BDFIG, so that the terms related to the q-axis current in the expressions of α 1 , α 2 , β 1 and β 2 can be omitted. In addition, when the system is in steady state, the current in the dq frame would be a constant value. Hence, all the current differential terms in the expressions of α 1 , α 2 , β 1 and β 2 can be neglected. Through the above measures, Eq. (5.8) can be reasonably simplified as ⎧ 



L 22r ⎪ ⎪ ⎪ ⎨ u 2d = R2 i 2d + L 2 − L si 2d + D2d r (5.9) 



2 ⎪ L (ω − ( p1 + p2 )ωr ) R L 22r r 2 ⎪ 2r ⎪ si 2q + D2q i 2q + L 2 − ⎩ u 2q = R2 − L r2 (ω1 − p2 ωr ) Lr where D2d = (ω2 − ( p1 + p2 )ωr ) L 1rLLr 2r i 1q ,       L2 D2q = (ω2 − ( p1 + p2 )ωr ) L 2 − L2rr i 2d − (ω2 − ( p1 + p2 )ωr ) L 1rLLr 2r i 1d . D2d and D2q can be regarded as the disturbance terms caused by the q- and d-axis currents, respectively. With the Euler forward difference method, the discrete form of (5.9) can be expressed as 



⎧ L 22r i 2d (k + 1) − i 2d (k) ⎪ ⎪ u + D2d (k) (k) = R i (k) + L − 2d 2 2d 2 ⎪ ⎪ Lr Ts ⎪ ⎪ ⎪ 

⎨ Rr L 22r (ω2 − ( p1 + p2 )ωr ) i 2q (k) u 2q (k) = R2 − ⎪ L r2 (ω1 − p2 ωr ) ⎪ ⎪ 

⎪ ⎪ ⎪ L 22r i 2q (k + 1) − i 2q (k) ⎪ ⎩ + D2q (k) + L2 − Lr Ts

(5.10)

From (5.10), the prediction equation of the CW current can be derived as

 R2 Ts Ts i 2d (k + 1) = 1 − Ts i 2d (k) + U2d (k) − D2d (k) σ σ σ ⎡ ⎤ Rr L 22r (ω2 −( p1 + p2 )ωr ) R2 − ⎪ Ts Ts L r2 (ω1 − p2 ωr ) ⎪ ⎣1 − ⎪ Ts ⎦i 2q (k) + U2q (k) − D2q (k) (k + 1) = i ⎪ 2q ⎩ σ σ σ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

(5.11)

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5 Predictive Control for Standalone BDFIGs L2

where σ = L 2 − L2rr . By substituting the eight voltage vectors generated by the MSC into (5.11), the predicted CW current at the next instant produced by each voltage vector can be obtained. The maim objective of the presented MPCC method is to effectively track the reference value of the CW current with a limitation of the CW current amplitude, which can be checked by the cost function as follows: g = [i 2dr e f − i 2d (k + 1)]2 + [i 2qr e f − i 2q (k + 1)]2 + f [i 2d (k + 1), i 2q (k + 1)] (5.12) where the first and second terms represent the minimization of the error of CW current, and the last term is a nonlinear function for limiting the amplitude of the CW current as follows: ⎧  2 2 ⎪ ⎨ ∞ i 2d (k + 1) + i 2q (k + 1) > I2 max f [i 2d (k + 1), i 2q (k + 1)] = (5.13)  ⎪ ⎩ 0 i 2 (k + 1) + i 2 (k + 1) ≤ I2 min 2q 2d The I 2max and I 2min in (5.13) are the maximum and minimum amplitudes of the CW current, respectively. The flow chart of the MPCC is illustrated in Fig. 5.1. Because the mathematical model is built in the dq frame, the eight voltage vectors must be transformed to the same dq frame. After the optimal voltage vector is obtained, it should be transformed to the static reference frame.

5.2.1.2

Overall Control Scheme

The overall control scheme with the MPCC for the standalone BDFIG is shown in Fig. 5.2. The control scheme is made up with four main parts, i.e., the PW voltage amplitude calculation, the CW current frequency calculation, the PW voltage amplitude control and the CW current control. The frequency of PW voltage is regulated by controlling the frequency of CW current based on (2.8). The PW voltage amplitude is regulated by the CW current amplitude, whose reference value is obtained by PW voltage amplitude control part, as shown in Fig. 5.2. For simplifying the control scheme, the q-axis CW current is set to zero, and the daxis CW current is set to the CW current amplitude. The CW frequency can be calculated according to (2.8).

5.2 Model Predictive Current Control (MPCC) for Standalone BDFIGs Fig. 5.1 The flow chart of the MPCC

209 Start

Voltage vector transformation from αβ to dq frame

Sampling i2d(k) and i2q(k) j=0 j=j+1 CW current prediction 5.11

Waiting for the next instant

Cost function calculation 5.12 Storing the optimal voltage vector j=8? Y

The optimal voltage vector transformation from dq to αβ frame

Applying the optimal voltage vector

Fig. 5.2 The overall control scheme of the standalone BDFIG

N

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5 Predictive Control for Standalone BDFIGs

5.2.2 Simulation Results The simulation based on a 30-kVA BDFIG is carried out to verify the presented MPCC method under three operation conditions. The parameters of the 30-kVA BDFIG are shown in Section A.3, Appendix.

5.2.2.1

Load Step Change at Constant Speed

Before 1.5 s, the BDFIG operates at the rotor speed of 900 rpm with the 25 (11.6 kW) resistive load. The 30 (9.7 kW) resistive load is connected to the system at 1.5 s. The simulation results are illustrated in Fig. 5.3. The reference value of the d-axis CW current is the output of the PI controller for the PW amplitude control, so that it is inevitabl with an overshoot. When the load is added, the d- and q-axis CW current deviates from their reference values under the PI controller. Fortunately, the CW current can track their reference value very well when applying the MPCC. Besides, from Fig. 5.3c, it can be found that the drop of the PW voltage amplitude decreases with the MPPC. In Fig. 5.3d, it is clear that the frequency drop of the PW voltage with the MPCC is smaller that that with the PI controller. Based on the above analysis, the dynamic performance of the dq-axis CW currents can be significantly improved by the MPCC.

5.2.2.2

Variable Rotor Speed Under Constant Load

In this simulation, the BDFIG with the constant resistive load of 25 (11.6 kW). At 1.5 s, it begins to accelerate from 600 to 900 rpm with the rate of 300 rpm/s. The simulation results are shown in Fig. 5.4. From Fig. 5.4a, b, it can be seen that the CW current ripple is larger at the sub-synchronous speed with the PI controller. Fortunately, with the MPCC, the dq-axis CW currents can track their reference values very well and are not affected by the speed. From Fig. 5.4c, the ripple of the PW voltage amplitude under the MPCC is smaller than that under the PI controller. And, the similar results for the PW voltage frequency can be seen from Fig. 5.4d. The three-phase CW current is presented in Fig. 5.4e, which reflects the current ripple under the MPCC is a little bigger than that under the PI controller.

5.3 Nonparametric Predictive Current Control (NPCC) for Standalone BDFIGs Aided with the analysis presented in [10], the application of the MPCC is limited by the parameter sensitivity of the BDFIG. To address this issue, the nonparametric predictive current control (NPCC) strategy is investigated, which would reduce the

5.3 Nonparametric Predictive Current Control (NPCC) for Standalone …

211

Fig. 5.3 Simulation results at 900 rpm with the load added at 1.5 s: a dq-axis CW currents with the MPCC, b dq-axis CW currents with the PI controller, c Amplitude of the PW phase voltage (the upper figure is under the PI control and the lower one is under the MPCC), d Frequency of the PW voltage

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5 Predictive Control for Standalone BDFIGs

Fig. 5.3 (continued)

effect of parameter mismatch. Based on the measured signals, the optimization of the voltage vectors can be attained by the NPCC for the desired current track performance without using the machine parameters.

5.3.1 CW Current Prediction Without Machine Parameters From (5.11), the variation of the CW current at the (k)th instant can be expressed as

5.3 Nonparametric Predictive Current Control (NPCC) for Standalone …

213

Fig. 5.4 Simulation results under the speed change from 600 to 900 rpm with the constant load: a dq-axis CW currents with the MPCC, b dq-axis CW currents with the PI controller, c Amplitude of the PW phase voltage (the upper figure is under the PI control and the lower one is under the MPCC), d Frequency of the PW voltage, e Three-phase CW current (the upper figure is under the PI controller and the lower one is under the MPCC)

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5 Predictive Control for Standalone BDFIGs

PW voltage amplitude applying PI controller

PW voltage amplitude applying MPCC

(c)

PW voltage frequency applying PI controller

(d) a phase

b phase c phase

a phase

b phase c phase

(e)

Fig. 5.4 (continued)

PW voltage frequency applying MPCC

5.3 Nonparametric Predictive Current Control (NPCC) for Standalone …

i 2d (k) = i 2d (k + 1) − i 2d (k) =

215

Ts R2 Ts u 2d (k) − Ts i 2d (k) − D2q (k) σ2 σ2 σ2

(5.14)

Coonsequently, the variation of the CW current at the (k−1)th and (k−2)th instants can be written as i 2d (k − 1) = i 2d (k) − i 2d (k − 1) = −

Ts R2 u 2d (k − 1) − Ts i 2d (k − 1) σ2 σ2

Ts D2q (k − 1) σ2

i 2d (k − 2) = i 2d (k − 1) − i 2d (k − 2) = −

(5.15) Ts R2 u 2d (k − 2) − Ts i 2d (k − 2) σ2 σ2

Ts D2q (k − 2) σ2

(5.16)

Combining (5.15) and (5.16), the difference between the CW current changes at the two adjacent instants can be derived as i 2d (k − 1) − i 2d (k − 2) R2 Ts Ts [i 2d (k − 1) − i 2d (k − 2)] = [u 2d (k − 1) − u 2d (k − 2)] − σ2 σ2 Ts − [D2q (k − 1) − D2q (k − 2)] σ2

(5.17)

Substituting the expressions of D2d and D2q into (5.17), Eq. (5.17) can be rewritten as Ts i 2d (k − 1) − i 2d (k − 2) = [u 2d (k − 1) − u 2d (k − 2)] σ2

Ts L 22r R2 +[ω2 − ( p1 + p2 )ωr ](L 2 − − ) [i 2d (k − 1) − i 2d (k − 2)] σ2 Lr L 1r L 2r Ts [i 1d (k − 1) − i 1d (k − 2)] − [ω2 − ( p1 + p2 )ωr ] σ2 Lr Ts = [u 2d (k − 2)−A2 i 2d (k − 2)−A1 i 1d (k − 2)] σ2

(5.18)

L2

where A1 = [ω2 − ( p1 + p2 )ωr ] L 1rLLr 2r , A2 = R2 +[ω2 − ( p1 + p2 )ωr ](L 2 − L2rr ). Aided with the machine parameters specified in Section A.6 of Appendix, the values of the terms U2d (k − 2), A2 i 2d (k − 2) and A1 i 1d (k − 2) are given in Fig. 5.5 with various operating states, including the starting process at 0.5 s for the line converter, the speed variation between 1.5 and 2.5 s from 600 to 900 rpm, and the load change at 2.5 s. As shown in Fig. 5.5, the values of the last two terms are less than 10% that of the first term. Hence, the last two terms of (5.18) can be ignored.

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5 Predictive Control for Standalone BDFIGs

Moreover, the last two terms in (5.18) are with the peak value approximated to zero. As a result, it can be realized that i 2d (k − 1) ≈ i 2d (k − 2) and i 1d (k − 1) ≈ i 1d (k − 2). Afterwards, the relation D2q (k − 1) ≈ D2q (k − 2) can be obtained. Consequently, a simplification for (5.17) can be attained as Ts [u 2d (k − 1) − u 2d (k − 2)] σ2

(5.19)

Value of A2 Δi2 d (k − 2)

i 2d (k − 1) − i 2d (k − 2) ≈

Time (s)

Value of A1Δi1d (k − 2)

(a)

Time (s)

Value of ΔU 2 d (k − 2)

(b)

Time (s)

(c) Fig. 5.5 The value of each term in (5.18) with various operating states: a Value of A2 i 2d (k − 2), b Value of A1 i 1d (k − 2), c Value of u 2d (k − 2), d Detailed view of u 2d (k − 2) between 2.104 and 2.106 s

217

Value of ΔU 2 d (k − 2)

5.3 Nonparametric Predictive Current Control (NPCC) for Standalone …

Time (s)

(d) Fig. 5.5 (continued)

In order to simplify the following derivation, it is defined that  X 2d = Ts σ2

(5.20)

From (5.19) and (5.20), X 2d can be given by X 2d =

i 2d (k − 1) − i 2d (k − 2) u 2d (k − 1) − u 2d (k − 2)

(5.21)

Substituting (5.20) into (5.15), it can be obtained that i 2d (k − 1) − X 2d u 2d (k − 1)=X 2d [−R2 i 2d (k − 1) − D2q (k − 1)]

(5.22)

Consequently, the variation of CW current at the (k)th and (k + 1)th instants can be written as i 2d (k) − X 2d u 2d (k)sw=i =X 2d [−R2 i 2d (k) − D2q (k)] i 2d (k + 1) − X 2d u 2d (k + 1)sw=i = X 2d [−R2 i 2d (k+1) − D2q (k+1)]

(5.23) (5.24)

where the subscript sw = i represents the ith switch state of the IGBTs of the MSC. And then, the predictive values of the CW current at the (k + 1)th and (k + 2)th instants considering the delay compensation can be derived as i 2dp (k + 1) = i 2d (k) + i 2d (k)sw=i = i 2d (k) + X 2d u 2d (k)sw=i + X 2d [−R2 i 2d (k) − D2q (k)] i 2dp (k + 2) = i 2dp (k + 1) + i 2d (k + 1)sw=i

(5.25)

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5 Predictive Control for Standalone BDFIGs

= i 2d (k) + i 2d (k)sw= j + i 2d (k + 1)sw=i = i 2d (k) + X 2d u 2d (k)sw= j + X 2d u 2d (k + 1)sw=i + X 2d [−R2 i 2d (k) − D2q (k)] + X 2d [−R2 i 2d (k + 1) − D2q (k + 1)] (5.26) where the subscript sw = j denotes the jth switch state of the IGBTs of the MSC. Aided with the approximation i 2d (k−1) ≈ i 2d (k−2) and D2q (k−1) ≈ D2q (k−2), considering (5.22), Eq. (5.26) can be rewritten as i 2dp (k + 2) ≈ i 2d (k) + X 2d u 2d (k)sw= j +X 2d u 2d (k + 1)sw=i + 2[i 2d (k − 1) − X 2d u 2d (k − 1)]

(5.27)

Similarly, the prediction value of the q-axis CW current can be expressed as i 2qp (k + 2) ≈ i 2q (k) + X 2q u 2q (k)sw= j +X 2q u 2q (k + 1)sw=i + 2[i 2q (k − 1) − X 2q u 2q (k − 1)]

(5.28)

5.3.2 Implementation of NPCC-Based Control Scheme This section presents the NPCC to replace the traditional PI controller for the CW current control in the standalone BDFIG system. The overall control structure is shown in Fig. 5.6a, and the structure of the improved NPCC is shown in Fig. 5.6b. Based on Fig. 5.6, the NPCC can be implemented according to the following steps. • Step 1—Calculating the reference values of the d- and q-axis CW currents: ∗ The reference value of the d-axis CW current i 2d is obtained by the PW voltage ∗ control loop, and the reference value of the q-axis CW current i 2q is set to zero.

• Step 2—Sampling and storing the CW current and CW voltage: The sampled three-phase CW current should be transformed to the d- and q-axis components. And then, all the d- and q-axis CW currents at the the kth, (k−1)th and (k−2)th instants should be stored in the controller. Besides, the CW voltage during the NPCC implementation are the stored reference values of the d- and q-axis CW voltages at the kth, (k−1)th and (k−2)th instants. • Step 3—Calculating X 2d and X 2q : Firstly, the variation of the dq-axis CW currents can be calculated as

5.3 Nonparametric Predictive Current Control (NPCC) for Standalone …

219

Fig. 5.6 The NPCC strategy for the CW current control: a Whole control system, b Main procedures of the NPCC implementation

⎧ i 2d (k − 2) = i 2d (k − 1) − i 2d (k − 2) ⎪ ⎪ ⎪ ⎨ i (k − 1) = i (k) − i (k − 1) 2d 2d 2d ⎪ i (k − 2) = i (k − 1) − i 2q (k − 2) 2q 2q ⎪ ⎪ ⎩ i 2q (k − 1) = i 2q (k) − i 2q (k − 1)

(5.29)

And then, according to (5.21), the coefficients X 2d and X 2q can be attained as

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5 Predictive Control for Standalone BDFIGs

⎧ i 2d (k − 1) − i 2d (k − 2) ⎪ ⎪ ⎨ X 2d = u (k − 1) − u (k − 2) 2d 2d (k − 1) − i i ⎪ 2q 2q (k − 2) ⎪ ⎩ X 2q = u 2q (k − 1) − u 2q (k − 2)

(5.30)

• Step 4—Calculating the predicted CW current with delay compensation according to (5.27) and (5.28). • Step 5—Selecting the optimal voltage vector to minimize the cost function (5.12). • Step 6—Sending the reference value of the CW voltage to MSC to generate the required CW current. It should be noted that the value of u 2d (k − 2) (i.e., u 2d (k − 1) − u 2d (k − 2)) is sometimes very small as shown in Fig. 5.5d, leading to the value of X 2d being very large, which would cause the adverse effect on the system robustness. In order to solve this problem, in the Step 3 above, it can use (5.30) to calculate X 2d if |U2d (k − 1) − U2d (k − 2)| > 0.5. And, the value of X 2d can be set to the same as the value at the previous instant if |U2d (k − 1) − U2d (k − 2)| ≤ 0.5. The X 2q can be handled by the similar way. The NPCC is a discrete control method, who selects the optimal vector from eight voltage vectors. Using the SVPWM algorithm, any one of the eight voltage vectors can be accurately obtained, so that the reference value of the CW voltage can be regarded as the same as the actual one. Thus, in (5.30), the CW voltages in the denominator are the stored reference values of the d- and q-axis CW voltages at the (k−1)th and (k−2)th instants. In addition, in Step 3, when the update threshold of u 2d (k − 2) and u 2q (k − 2) is set to 0.05, the update stagnation can be greatly reduced without affecting the realization of the control algorithm. It can take the d-axis CW voltage as an example for analysis. The d-axis CW voltage at the (k−1)th instant can be expressed as   u 2d (k − 1) = u α (k − 1) ∗ cos θ2 (k − 1) + u β (k − 1) ∗ sin θ2 (k − 1) ∗u dc (5.31) where uα and uβ are per-unit values of α- and β-axis CW voltage, respectively, udc is the dc bus voltage, θ 2 is the phase angle of the CW current serving as the angle of the frame transformation. Table 5.1 presents the values of uα and uβ at various IGBT switch states of MSC. From (5.31), the difference between the CW voltages at the (k−1)th and (k−2)th instants can be derived as u 2d (k − 1) − u 2d (k − 2)   = u α (k − 1) ∗ cos θ2 (k − 1) + u β (k − 1) ∗ sin θ2 (k − 1) *u dc   − u α (k − 2) ∗ cos θ2 (k − 2) + u β (k − 2) ∗ sin θ2 (k − 2) *u dc

(5.32)

5.3 Nonparametric Predictive Current Control (NPCC) for Standalone … Table 5.1 Values of uα and uβ at various IGBT switch states of MSC

221

IGBT switch states

uα

uβ

S1

0

0

S2

2/3

S3

1/3

S4

−1/3

0 √ 1 3 √ 3 1

S5

−2/3

0

S6

−1/3

S7

1/3

√ −1 3 √ −1 3

S8

0

0

Under the normal operation condition, the dc bus voltage is high enough. The value of (5.32) would much greater than the threshold of 0.05 when the IGBT switch states at the (k−1)th and (k−2)th instants are different. Using trigonometric operations, Eq. (5.32) can be rewritten as u 2d (k − 1) − u 2d (k − 2) ⎡ θ2 (k − 1) − θ2 (k − 2) ⎤ θ2 (k − 1) + θ2 (k − 2) sin −2 ∗ u α (k − 1) ∗ sin ⎥ ⎢ 2 2 =⎣ ⎦*u dc θ2 (k − 1) − θ2 (k − 2) θ2 (k − 1) + θ2 (k − 2) sin +u β (k − 1) ∗ 2 ∗ cos 2 2 ⎡ θ2 (k − 1) + θ2 (k − 2) ⎤ θ2 (k − 1) − θ2 (k − 2) ⎢ u β (k − 1) ∗ cos ⎥ 2 = 2 sin ⎣ ⎦*u dc θ2 (k − 1) + θ2 (k − 2) 2 −u α (k − 1) ∗ sin 2  ≈ θ2 u β (k − 1) ∗ cos θ2 (k − 1) − u α (k − 1) ∗ sin θ2 (k − 1) *u dc (5.33) In order to verify the rationality of the threshold selection, it is necessary to compare the minimum value of (5.33) at different rotor speeds with the selected threshold value of 0.05. The speed range of the adopted BDFIG is 600–1200 rpm, in which the range of the CW frequency is 0–30 Hz. Hence, the minimum value of (5.33) is calculated at three typical CW frequencies of 30 Hz, 10 Hz and 1 Hz. The specific calculation process is given out as follows: In each sampling period, the CW current phase angle θ 2 is increased by θ 2 , and θ 2 is defined by 2πf 2 T s . The f 2 and T s is the CW frequency and sampling period, respectively. And then, the minimum value of (5.33) can be obtained by substituting the six non-zero voltage vectors listed in Table 5.1, and the calculation result is shown in Fig. 5.7. It can be seen from Fig. 5.7 that only when θ 2 is near the integer multiple of π/3, the update stagnation of u 2d (k − 2) will occur. When the frequency of the

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5 Predictive Control for Standalone BDFIGs

Fig. 5.7 The minimum value of u 2d (k − 2) at different rotor speeds: a CW frequency 30 Hz at 1200 rpm, b CW frequency 10 Hz at 600 or 900 rpm, c CW frequency 1 Hz at 735 or 765 rpm

CW voltage is high, as shown in Fig. 5.7a, b , the angular range of θ 2 where the update stagnation occurs is very small. When the CW frequency is 1 Hz, as shown in Fig. 5.7c, the probability of the minimum value of u 2d (k − 2) being less than the threshold 0.05 is about 20%. It is worth noting that, under the six non-zero voltage vectors, the probability of u 2d (k−2) reaching its minimum value is 1/6 at the current IGBT switch state. As a result, under the CW frequency of 1 Hz, the probability of

5.3 Nonparametric Predictive Current Control (NPCC) for Standalone …

223

u 2d (k − 2) being less than the threshold 0.05 is about 3.33%, which make the probability of the update stagnation of u 2d (k − 2) is very small. In addition, when the NPCC algorithm outputs the zero-voltage vector, the values of X 2d and X 2q at the current instant are the same as those at the last instant. Fortunately, under the NPCC, the zero-voltage vector does not appear at consecutive instants. Hence, the adverse effect of the non-zero voltage vector on control performance is limited.

5.3.3 Simulation Results In this subsection, the simulation based on a 30-kVA BDFIG is implemented to investigate the dynamic behavior of the NPCC method under different conditions, as well as the robustness to machine parameter mismatch in comparison with the MPCC method. The detailed parameters of the 30-kVA BDFIG is mentioned in Section A.3, Appendix.

5.3.3.1

Variable Load at Constant Rotor Speed

The simulation under the variable load is carried out at both sub- and the supersynchronous speeds. The initial load is a three-phase resistive load with the resistance of 25 per phase. And, the same load is added to the system at 2.5 s. The results in Figs. 5.8a and 5.9a confirm the effectiveness of the NPCC method to track the desired PW voltage under load change. Moreover, from Figs. 5.8c and 5.9c, the d-axis CW current tracks its reference value very well with a fast response under the load variation, and the q-axis CW current is maintained constant at zero. This ensures the effectiveness of the NPCC method for the desired current control loop of the CW current under different load situations with various speed cases.

5.3.3.2

Variable Rotor Speed Under Constant Load

The second simulation handles the implementation of the NPCC under speed change condition. As shown in Fig. 5.10, the BDFIG is started at the rotor speed of 600 rpm with a three-phase resistive load with the resistance of 25 per phase. And then, at 2 s, the rotor speed is accelerated with a rate of 300 rpm/s to reach 900 rpm at 3 s. It is can be seen from Fig. 5.10 that the frequency and amplitude of CW current is changed with the speed variation to maintain the frequency and amplitude of PW voltage constant. Moreover, the simulation result illustrates the good tracking of the dq-axis CW currents. This verifies that the NPCC method is effective for the standalone BDFIG system under the speed variation.

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Fig. 5.8 Simulation results under the variable load at the rotor speed of 600 rpm: a PW voltage, b CW current, c Reference and actual values of dq-axis CW currents

5.3.3.3

Robustness Comparison Under Machine Parameter Mismatch

In practical applications, if the temperature of the BDFIG rises, the winding resistance would increase. And, if the system is overloaded, the magnetic saturation may happen, which would decrease the self- and mutual-inductances of the windings. The MPCC method depends on the machine parameters, which will affect the

5.3 Nonparametric Predictive Current Control (NPCC) for Standalone …

225

Fig. 5.9 Simulation results under the variable load at the rotor speed of 900 rpm: a PW voltage, b CW current, c Reference and actual values of dq-axis CW currents

system robustness and even cause the BDFIG out of control. According to (5.11), the inductances of CW and PW has great influence on the calculation of the predictive current. Therefore, in order to highlight the good parameter robustness of the NPCC method, the simulation comparison between MPCC and NPCC is carried out under the following inductance parameters.

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5 Predictive Control for Standalone BDFIGs

Fig. 5.10 Simulation results under the constant load and variable rotor speed: a PW voltage, b CW current, c Reference and actual values of dq-axis CW currents

The simulation is carried out at the sub-synchronous speed of 600 rpm under a three-phase resistive loads with the resistance of 25 in each phase. The self- and mutual-inductances of CW and PW are reduced by 20%, respectively, at 0.4 s. The simulation results are shown in Figs. 5.11 and 5.12. It can be seen from Fig. 5.11a, b that when CW self- and mutual-inductances decrease, the d-axis CW current with

5.3 Nonparametric Predictive Current Control (NPCC) for Standalone … Fig. 5.11 Simulation results under the CW self- and mutual-inductances change from 100 to 80% at the rotor speed of 600 rpm: a Three-phase CW current with MPCC, b Reference and actual values of dq-axis CW currents with MPCC, c Three-phase CW current with NPCC, d Reference and actual values of dq-axis CW currents with NPCC

227

228

5 Predictive Control for Standalone BDFIGs

Time (s)

dq components of CW current (A)

(a)

Time (s)

(b)

CW current (A)

Fig. 5.12 Simulation results under PW self- and mutual-inductances change from 100 to 80% at the rotor speed of 600 rpm: a Three-phase CW current with MPCC, b Reference and actual values of dq-axis CW currents with MPCC, c Three-phase CW current with NPCC, d Reference and actual values of dq-axis CW currents with NPCC

CW current (A)

the MPCC method cannot track the current command, and finally a steady-state error is generated. Fortunately, in Fig. 5.11c, d, the NPCC method still keeps the good current tracking performance when the machine parameters are significantly changed, which is almost not affected by the change of the CW inductance. It can be seen from Fig. 5.12a, b that when the PW self- and mutual-inductances decrease, the d-axis CW current with the MPCC method increases greatly, and finally

Time (s)

(c)

Fig. 5.12 (continued)

229

dq components of CW current (A)

5.3 Nonparametric Predictive Current Control (NPCC) for Standalone …

Time (s)

(d)

a large steady-state error occurs. Meanwhile, the q-axis CW current recovers to its reference value slowly after a fluctuation. As shown in Fig. 5.12c, d, when the PW inductance is changed, the dq-axis CW currents with the NPCC method can quickly recover after a very small fluctuation. The simulation results domenstrate that the standalone BDFIG system with MPCC is greatly affected by machine parameter mismatch. When the machine parameters are changed, the CW current will not follow its reference value, which would lead to the PW voltage error for a long time. Fortunately, in the same case, the NPCC can prevent this problem. The dynamic response speed and current tracking accuracy of the NPCC are much better than that of the MPCC under the machine parameter mismatch. Thus, it can be concluded that the NPCC for standalone BDFIG does not rely too much on the machine parameters, and can significantly improve the system robustness against the machine parameter change.

5.3.4 Experimental Results In this subsection, three types of experiments are designed to verify the dynamic response and robustness of the NPCC method under different operation conditions. The experiments are carried out on a 5-kVA BDFIG experimental platform, whose main parameters and photograph can be seen in Section A.6 of Appendix.

5.3.4.1

Variable Load at Constant Rotor Speed

This experiment is carried out with a 1-kW three-phase resistive load with the resistance of 160  per phase at the sub-synchronous speed of 600 rpm. The experimental results are shown in Fig. 5.13. At 3.7 s, a 2-kW three-phase resistive load with the resistance of 80  per phase is added to the system, and it is removed at 4.8 s.

Time (s)

CW current (A)

(a)

Time (s)

Reference and feedback values of d-axis CW current (A)

(b)

Time (s)

(c)

PW current (A)

Fig. 5.13 Experimental results under the variable load at the rotor speed of 600 rpm: a PW voltage, b CW current, c Reference and feedback values of the d-axis CW current, d PW current

5 Predictive Control for Standalone BDFIGs

PW voltage (V)

230

Time (s)

(d)

5.3 Nonparametric Predictive Current Control (NPCC) for Standalone …

231

Apparently, it can be seen from Fig. 5.13 that under the constant speed and variable load condition, the standalone BDFIG with the NPCC method can not only maintain the stable frequency and amplitude of the PW voltage, but also can quickly track the CW current reference value, indicating that the NPCC is with good steady-state performance and dynamic response speed.

5.3.4.2

Variable Rotor Speed Under Constant Load

As shown in Fig. 5.14, the second experiment is implemented at 600 rpm with a three-phase resistive load of 160  per phase. At 14 s, the prime mover starts to accelerate with a ratio of 30 rpm/s, and then reaches the natural synchronous speed (750 rpm) at 19 s and the super-synchronous speed (900 rpm) at 24 s. It can be seen from Fig. 5.14 that under the constant load and variable speed condition, the standalone BDFIG can keep PW voltage constant frequency and amplitude, as well as the good CW current tracking performance. When the natural synchronous speed is reached, due to the change of the energy flow direction of the MSC, the CW current achieves smooth phase sequence switching, as shown near 18.5 s in Fig. 5.14.

5.3.4.3

Robustness Comparison Under Machine Parameter Mismatch

This subsection conducts comparative experiments of MPCC and NPCC under the changed resistance and inductance to verify the parameter robustness of the two methods. The following three experiments are all carried out at 600 rpm under a three-phase resistive load with the resistance of 25  per phase. • Under the variation of winding resistance When the MPCC method is adopted, a three-phase resistor with the resistance of 0.2  per phase is connected between the CW and MSC at 6.5 s, and it is short-circuited at 8.5 s. The experimental results are shown in Fig. 5.15. With the NPCC method, the same resistor is connected between the CW and MSC at 10 s, and it is short-circuited at 11.5 s. The experimental results are presented in Fig. 5.16. From Figs. 5.15 and 5.16, it can be noted that the d-axis CW current under both MPCC and NPCC can track its reference value without steady-state errors. • Under the variation of winding self-inductance In this experiment, an inductance is inserted between the CW and MSC to simulate the variation of the CW self-inductance. Under the MPCC method, the inductance of 20 mH is connected between the CW and MSC at 4.8 s and cut off at 8.1 s, with the experimental results illustrated in Fig. 5.17. When the NPCC method is employed, the same inductance is connected between the CW and MSC at 3.8 s and removed at 6.5 s, as shown in Fig. 5.18. From the experimental results presented in Figs. 5.17 and 5.18, the variation of the CW self-inductance, the d-axis CW current generated by the MPCC has a steady-state error, which leads

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5 Predictive Control for Standalone BDFIGs

Fig. 5.14 Experimental results under the constant load and variable rotor speed: a PW voltage, b CW current, c Reference and feedback values of the d-axis CW current

to a steady-state error in the PW voltage. Fortunately, under the same condition, the NPCC method can effectively avoid this problem. • Under the variation of winding magnetizing inductance

5.3 Nonparametric Predictive Current Control (NPCC) for Standalone …

233

Fig. 5.15 Experimental results under the CW resistance change with MPCC: a Reference and feedback values of the d-axis CW current, b CW current, c PW voltage

234

5 Predictive Control for Standalone BDFIGs

Fig. 5.16 Experimental results under the CW resistance change with NPCC: a Reference and feedback values of the d-axis CW current, b CW current, c PW voltage

5.3 Nonparametric Predictive Current Control (NPCC) for Standalone …

235

Fig. 5.17 Experimental results under the CW self-inductance change with MPCC: a Reference and feedback values of the d-axis CW current, b CW current, c PW voltage

236

5 Predictive Control for Standalone BDFIGs

Fig. 5.18 Experimental results under the CW self-inductance change wiht NPCC: a Reference and feedback values of the d-axis CW current, b CW current, c PW voltage

5.3 Nonparametric Predictive Current Control (NPCC) for Standalone …

237

In the last experiment, the change of the inductance is achieved by connecting an additional inductor in series outside the BDFIG, which in essence can only simulate the change of the leakage inductance of windings. In practical applications, the change of the BDFIG inductance is mainly the reduction of the magnetizing inductance caused by the electromagnetic saturation. In this experiment, the PW voltage closed-loop control is adopted to maintain the rms value of the PW line voltage at 70 V. The initial load is a three-phase resistive load with the resistance of 25  per phase, and the PW current is about 1.6 A. Later, an additional three-phase resistive load with the resistance of 4  per phase is incorporated. At this time, the PW current reaches 11.6 A (around 1.5 times the rated value of the PW current), which makes the BDFIG saturated and the magnetizing inductance reduced. When the MPCC method is used, the additional three-phase resistive load is connected to the BDFIG at 6.5 s and removed at 9 s. When the NPCC method is performed, the additional three-phase resistive load is added to the system at 4.6 s and cut off at 6.8 s. The specific results are shown in Figs. 5.19 and 5.20. In order to maintain a constant PW voltage, the reference value of the d-axis CW current will increase significantly as the load increases. From Fig. 5.19a, the MPCC cannot regulate the d-axis CW current to its reference value, resulting in a steady-state error. Fortunately, the current tracking performance of the NPCC is not affected by the variation of the magnetizing inductance, as shown in Fig. 5.20a. Based on the above experimental results, the NPCC method not only has good dynamic performance under various working conditions, but also can significantly improve the system robustness to machine parameter mismatch.

5.4 Summary In this chapter, two kinds of predictive current control methods are presented for standalone BDFIGs. Firstly, the MPCC method is suggested instead of the PI controller to improve the tracking performance of the CW current, thus enhancing the performance of the standalone BDFIGs. However, the performance of the MPCC is sensitive to the change of BDFIG parameters. Hence, for the better performance, the NPCC is investigated for an effective CW current control without machine parameters. All theoretical analysis and investigation process of the NPCC method is presented in details. The obtained simulation and experimental results have ensured the effectiveness of the NPCC for a good current tracking under different operation conditions of speed changes and load variations. Moreover, the simulation and experimental results have verified the robustness of the NPCC method to machine parameter mismatch through a comparative study with the MPCC method.

Time (s)

CW current (A)

(a)

Time (s)

PW current (A)

(b)

Time (s)

(c)

PW voltage (V)

Fig. 5.19 Experimental results under the generator overload with MPCC: a Reference and feedback values of the d-axis CW current, b CW current, c PW current, d PW voltage

5 Predictive Control for Standalone BDFIGs Reference and feedback values of d-axis CW current (A)

238

Time (s)

(d)

Fig. 5.20 Experimental results under the generator overload with NPCC: a Reference and feedback values of the d-axis CW current, b CW current, c PW current, d PW voltage

239

Reference and feedback values of d-axis CW current (A)

5.4 Summary

Time (s)

CW current (A)

(a)

Time (s)

PW current (A)

(b)

Time (s)

PW voltage (V)

(c)

Time (s)

(d)

References 1. Sarasola I, Poza J, Oyarbide E et al (2006) Stability analysis of a brushless doubly-fed machine under closed loop scalar current control. In: Proceedings of industrial electronics conference. Paris, France, pp 1527–1532

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2. Shao S, Abdi E, Barati F et al (2009) Stator-flux-oriented vector control for brushless doubly-fed induction generator. IEEE Trans Ind Electron 56(10):4220–4228 3. Jovanovi´c MG, Yu J, Levi E et al (2006) Encoderless direct torque controller for limited speed range applications of brushless doubly fed reluctance motors. IEEE Trans Ind Appl 42(3):712–722 4. Wei X, Cheng M, Wang W et al (2016) Direct voltage control of dual-stator brushless doubly fed induction generator for stand-alone wind energy conversion systems. IEEE Trans Magn 52(7), Article 8203804 5. Rivera M, Wheeler P, Olloqui A et al (2016) Predictive control in matrix converters—Part I: Principles, topologies and applications. In: Proceeding IEEE international conference industrial technology. Taipei, Taiwan, pp 1091–1097 6. Rivera M, Wheeler P, Olloqui A et al (2016) Predictive control in matrix converters-part II: control strategies, weaknesses and trends. In: Proceeding of IEEE international conference industrial technology, Taipei, Taiwan, pp 1098–1104 7. Rivera M (2015) A modulated model predictive control scheme for a two-level voltage source inverter. In: Proceedings IEEE International Conference Industrial Technology. Seville, Spain, pp 2224–2229 8. Rodriguez J, Kazmierkowski MP, Espinoza JR et al (2013) State of the art of finite control set model predictive control in power electronics. IEEE Trans Ind Electron 9(2):1003–1016 9. Wang X, Sun D et al (2017) Three-vector-based low-complexity model predictive direct power control strategy for doubly fed induction generators. IEEE Trans Power Electron 32(1):773–782 10. Rivera M, Elizondo J, Macias M et al (2010) Model predictive control of a doubly fed induction generator with an indirect matrix converter. In: IECON 2010–36th annual conference on IEEE industrial electronics society, pp 2959–2965 11. Xu W, Gao J, Liu Y et al (2017) Model predictive current control of brushless doubly-fed machine for stand-alone power generation system. In: IECON 2017–43rd annual conference on IEEE industrial electronics society, pp 322–327 12. Li X, Peng T, Dan H et al (2018) A modulated model predictive control scheme for the brushless doubly fed induction machine. IEEE J Emerg Sel Topics Power Electron 6(4):1338–1342 13. Wei X, Cheng M, Zhu J et al (2017) Model predictive power control of a brushless doubly fed twin stator induction generator. In: Proceeding IEEE energy conversion congress and exposition, pp 5080–5085 14. Bayhan S, Kakosimos P, Rivera M et al (2018) Predictive torque control of brushless doubly fed induction generator fed by a matrix converter. In: Proceeding 2018 IEEE 12th international conference on compatibility, power electronics and power engineering. Doha, pp 1–6 15. Fliess M, Join C (2013) Model-free control. Int J Ctrl 86(12):2228–2252 16. Lin CK, Liu TH, Yu JT et al (2014) Model-free predictive current control for interior permanentmagnet synchronous motor drives based on current difference detection technique. IEEE Trans Ind Electron 61(2):667–681 17. Liu Y, Ai W, Chen B et al (2016) Control design and experimental verification of the brushless doubly-fed machine for stand-alone power generation applications. IET Electr Power Appl 10(1):25–35

Chapter 6

Rotor Position and Speed Observers of BDFIGs

Abstract In this chapter, two rotor position observers and one rotor speed observer for the standalone BDFIG is investigated. The first rotor position observer (RPO_1) is based on the phase-axis relationship of the BDFIG to effectively estimate the rotor position and to improve the system performance, where a Luenberger-type PI observer is used to obtain the rotor speed. Furthermore, to assure the simplicity, the second rotor position observer (RPO_2) is based on the space-vector model of the BDFIG without any additional observers for obtaining the generator speed. Besides, an improved rotor speed observer (RSO) is presented, who is based on the PW voltage and CW current and free of machine parameters except the pole pairs. To eliminate the adverse impact of unbalanced and nonlinear loads on the RSO, second-order generalized integrators (SOGIs) and low-pass filters (LPFs) are introduced to prefilter the PW voltage and CW current, respectively. Comprehensive simulation and experimental results are obtained with a prototype 30-kVA BDFIG to demonstrate the validity of these presented rotor position and speed observers under different operation conditions. Keywords Rotor position observer · Rotor speed observer · Phase-axis relationship · Space-vector model · Unbalanced load · Nonlinear load

6.1 Introduction Most of the control strategies for BDFIGs require the information of the rotor speed or position, which is usually obtained by an encoder [1–6]. In recent years, the existence of the encoder in the control system has gradually become undesirable due to its associated problems such as the need for mechanical arrangements, the additional cost for the maintenance issues, and the reduction of the system reliability. Therefore, the sensorless control with the rotor position or speed observer is very essential for high-performance operation of BDFIG systems to improve the control reliability and reduce the overall cost.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Y. Liu and W. Xu, Advances in Control Technologies for Brushless Doubly-fed Induction Generators, https://doi.org/10.1007/978-981-19-0424-0_6

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6 Rotor Position and Speed Observers of BDFIGs

The rotor position and speed estimation approaches for the BDFG mainly include the direct estimation approach and the closed-loop estimation approach. In this chapter, two direct estimation approaches for the rotor position observer (RPO) of BDFIGs are presented, as well as one closed-loop estimation approach for rotor speed observer (RSO). The first rotor position observer (RPO_1) is based on the phase-axis relationship of the BDFIG to effectively estimate the rotor position, and a Luenberger-type PI observer is used to obtain the rotor speed. However, this observer (RPO_1) suffers from some disadvantages such as the bad performance and the slow convergence during the start-up. In addition, during the speed change, the larger amplitude error and frequency fluctuation in the PW voltage are produced, so that it takes more time to reach steady state and a high transient current can be detected in both the PW and CW currents. In order to address this issue, the second rotor position observer (RPO_2) is investigated with a simplified direct estimation procedure for the rotor position [7, 8]. Compored to the closed-loop estimation approach [9–11], the direct estimation principle is with the following merits: • The calculation complexity of the direct estimation method is low for detecting the rotor position signal. Hence, the fast implementation in the DSP chip can be achieved. • No PI controllers are required for the direct rotor position estimation, so that a better starting performance can be obtained. • No integrators are required, which would significantly reduce the cumulative error of the estimated rotor position. However, it is obvious that the above two rotor position observers with the direct estimation principle rely on the machine parameters. In order to enhance the robustness of the observer to the machine parameter mismatch, it is necessary to investigate the rotor position or speed observer free of machine parameters. In Liu et al. [12], a basic RSO based on PW voltage and CW current (the necessary sampled signals for the standalone BDFIG) is developed. In Dong et al. [13], the basic RSO has been successfully applied to the sensorless control system for the standalone BDFIG supplying balanced and linear loads. However, unbalanced and nonlinear loads can produce unbalanced and distorted PW terminal voltages, respectively, which further generate distorted CW current through the indirect coupling of the rotor. Using the basic RSO, the unbalanced/distorted PW voltage and distorted CW current can cause inaccurate rotor speed observation when the standalone BDFIG supplies unbalanced and nonlinear loads. Hence, it is necessary to improve the basic RSO to accurately observe the rotor speed for the standalone BDFIG feeding unbalanced and nonlinear loads. One new RSO has been proposed in [14] to enhance the observation accuracy of the rotor speed under unbalanced loads. However, the sensorless operation of the standalone BDFIG under nonlinear loads is not considered in [14].

6.1 Introduction

243

In Liu et al. [15], the effect of unbalanced and nonlinear loads on the basic RSO is analyzed in detail, and then the improved RSO is developed for the accurate rotor speed observation under both unbalanced and nonlinear loads. Afterwards, a comprehensive guideline for parameter tuning of the improved RSO is developed to ensure a similar dynamic performance with the basic RSO under different load conditions.

6.2 Rotor Position Observer Based on the Phase-Axis Relationship of BDFIG Figure 6.1 illustrates the relationship of the BDFIG phase-axis. It is obvious from Fig. 6.1 that the alignment of the q axis of the reference frame with the total PW voltage vector is attained to obtain the intended orientation of PW voltage. This can be realized by regulating the q-axis PW voltage to the total PW voltage (u1q = U 1 ) and the d-axis PW voltage to zero (u1d = 0). The main structure of the presented sensorless control method based on the RPO_1 for the standalone BDFIG is illustrated Fig. 6.2. The q-axis PW voltage tracks the reference magnitude U 1 * , as shown in Fig. 6.2, which can be realized by adjusting the d-axis CW current and giving the corresponding q-axis CW current (i2q * = 0) based on the CW current control loop. Furthermore, the d-axis PW voltage is set to zero by regulating the intended frame frequency of the PW voltage, ω1 . And then, the corresponding angle, θ 1 , is obtained to realize the desired orientation target of the PW voltage through the aligning the q axis of the reference frame with the PW voltage vector. Fig. 6.1 Phase-axis relationship of the BDFIG for the sensorless control based on RPO_1

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6 Rotor Position and Speed Observers of BDFIGs

Fig. 6.2 Main block diagram of the sensorless control method based on the RPO_1 for the standalone BDFIG

With the estimated rotor position θ r_est based on the rotor position observer presented in Sect. 6.2.1, the corresponding angle θ 2 * can be realized directly using the angle θ 1 * calculated from the integration of the reference PW frequency, as shown in Fig. 6.2.

6.2.1 Theoretical Analysis Based on the rotor winding dq-axis voltage and flux equations presented in Chap. 2, in the PW synchronous reference frame, the dq-axis rotor currents can be derived as

ir d =

irq =

  − L r s 2 + Rr s + ω2f r L r (L 1r i 1d + L 2r i 2d ) ω2f r L r2 + (Rr + L r s)2    − L r s 2 + Rr s + ω2f r L r L 1r i 1q + L 2r i 2q ω2f r L r2 + (Rr + L r s)2

where ω f r = ω1 − p1 ωr _est .

+

−

  Rr ω f r L 1r i 1q + L 2r i 2q ω2f r L r2 + (Rr + L r s)2 (6.1) Rr ω f r (L 1r i 1d + L 2r i 2d ) ω2f r L r2 + (Rr + L r s)2 (6.2)

6.2 Rotor Position Observer Based on the Phase-Axis Relationship …

245

The first term in (6.1) can be rewritten as   − s 2 + (Rr /L r )s + ω2f r (L 1r i 1d + L 2r i 2d )   . L r s 2 + 2(Rr /L r )s + ω2f r It is obvious that the poles and zeros in this term are very close, so that they can eliminate each other. Consequently, a simplified expression of the first term in (6.1) can be attained as −(L 1r i 1d + L 2r i 2d )/L r . Moreover, the denominator (Rr + L r s)2 in the second term of (6.1) can be ignored due to its small proportion. Therefore, a simplified form of (6.1) can be given as ir d

  Rr L 1r i 1q + L 2r i 2q −(L 1r i 1d + L 2r i 2d ) = + Lr ω f r L r2

(6.3)

Similarly, Eq. (6.3) can be simplified as   irq = − L 1r i 1q + L 2r i 2q /L r

(6.4)

Aided with the dq-axis PW voltage equations in Chap. 2 and considering the PW synchronous reference frame, the following assumptions can be obtained. Taking the actual parameters into consideration, the voltage drop on the resistance R1 can be neglected, and in the steady state Ψ 1 is constant so that the differentiation of the PW flux is approximately equal to zero. Meanwhile, assuming that the q axis of the reference frame is aligned to the PW voltage vector (u1d = 0, u1q = U 1 * ), the dq-axis PW flux can be modified as 

ψ1q ≈ 0 1 ≈ ψ1d ≈ u 1q /ω1∗

(6.5)

Substituting (6.5) into (2.4), and aided with (6.3) and (6.4), the dq-axis CW currents can be expressed as ⎧     L L −L 2 Rr L 1 ⎨ i 2d = i 1q + 1L 1rr L 2r 1r i 1d − ω L L f r 1r 2r   2 ⎩ i 2q = L 1 L r −L 1r i 1q L 1r L 2r

L r 1 L 1r L 2r

(6.6)

The angle of the CW current space vector (with respect to the dq-axis reference frame), ε, as shown in Fig. 6.1, can be determined by ε = tan−1

i 2q i 2d

(6.7)

246

6 Rotor Position and Speed Observers of BDFIGs

Then, the CW space-vector current angle (according to the natural axes of the machine), θ I2 , as shown in Fig. 6.1, can be derived with the aid of the actual threephase CW currents as

i2 =



4π 2 2 j 2π e 3 23 e j 3 3 3



i 2a i 2b i 2c

θ I2 = angle(i 2 )

T .

(6.8)

Finally, the BDFIG rotor position can be estimated by 

θ2∗ = θ I2 − ε . θr _est = θ1 + θ2∗

(6.9)

After obtaining the estimated rotor position, θ r_est , the speed signal can be easily detected by differential operation (ωr _est = dθr _est /dt). However, a fully noisy rotor-speed signal would be resulted, which would significantly affect the dynamic behaviour of the presented RPO_1. To overcome this issue, a Luenberger-type PI observer can be employed [16].

6.2.2 Simulation Results In this section, the simulation is carried out to verify the proposed sensorless control strategy based on the RPO_1. The detailed parameters of the adopted BDFIG in this simulation can be seen in Section A.3, Appendix. In the presented simulation results, the reference amplitude and frequency of the PW voltage are set at 311 V and 50 Hz, respectively. The capability of the RPO_1 is verified in this subsection under the speed change while the load is kept constant. Then, the dynamic performance of the presented BDFIG system is analysed under load changes while the speed is kept constant. Figure 6.3a–e illustrate the actual and estimated rotor position, three-phase PW voltage, PW voltage amplitude and frequency, dq-axis CW current, and three-phase PW and CW currents, respectively. Firstly, the BDFIG is started at super-synchronous speed 900 rpm with 11.6 kW resistive load. Then, it begins to decelerate from 900 to 600 rpm at t = 2 s under the same load 11.6 kW. Finally, at t = 4 s, the load is changed from 11.6 to 9.7 kW while the speed is maintained at 600 rpm. It is observed from Fig. 6.3b, c that the PW voltage (in terms of magnitude and frequency) tracks successfully the desired values (U 1 * = 311 V and f 1 * = 50 Hz). In addition, the results verify that the estimated rotor position is in a good accordance with the actual value as shown in Fig. 6.3a. This ensures the effectiveness of the proposed rotor position observer for sensorless DVC of standalone BDFIG.

Rotor Position (degree)

Rotor Position (degree)

6.2 Rotor Position Observer Based on the Phase-Axis Relationship … Starting-Up Actual Position

400

400

200

247 Speed Change

Estimated Position

200

0

0

0.05

0.1

0.15

0

0.2

2

2.05

2.1

400

2.15

2.2

2.25

2.3

Time (s)

Time (s) Load Change

200 0 3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

4.2

4.3

Time (s) (a)

PW three-phase voltages (V)

450 300 150 0 -150 -300 -450

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

3.5

4

4.5

5

5.5

3.5

4

4.5

5

5.5

PW Voltage Amplitude (V)

Time (s) (b) 450 300 150 0

0.5

1

1.5

2

2.5

3

PW frequency (Hz)

Time (s) 100 75 50 25 0

0

0.5

1

1.5

2

2.5

3

Time (s) (c) Fig. 6.3 Simulation results of the sensorless control of the standalone BDFIG based on the RPO_1 under speed variation (started with 900 rpm and then decelerated to 600 rpm) followed by the load change condition (started with 11.6 kW and then reduced to 9.7 kW): a Estimated and actual rotor positions, b three-phase PW voltages, c PW voltage amplitude and frequency, d CW dq-axis currents, e three-phase PW and CW currents

248

6 Rotor Position and Speed Observers of BDFIGs

DQ-axis CW currents (A)

600 500 400

D-axis Current

300 200 100

Q-axis Current

0 -100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Time (s) PW three-phase Currents (A)

(d)

300 150 0 -150 -300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

3.5

4

4.5

5

5.5

CW three-phase Currents (A)

Time (s) 800 400 0 -400 -800

0.5

1

1.5

2

2.5

3

Time (s) (e)

Fig. 6.3 (continued)

Moreover, Fig. 6.3d verifies that the q-axis CW current t tracks its reference value at zero. All the results ensure the simplicity and capability of the proposed sensorless control system to track the desired values of PW voltage (magnitude and frequency) for the BDFIG in standalone applications. On the other hand, it can be seen from Fig. 6.2 that the RPO_1 is fully dependent on the Luenberger-type PI observer to obtain the rotor speed and then use it as feedback for the algorithm of calculating the corresponding position. This Luenberger-type PI observer as presented in [16] is with the dependence on the PI controller. However, the PI controller has some disadvantages, such as slow convergence during the start-up, sensitivity to controller gains and sluggish response to sudden disturbances. These issues caused by the PI controller in the rotor position detection can be confirmed through the results given in Fig. 6.3 especially during both the start-up and speed change conditions.

6.2 Rotor Position Observer Based on the Phase-Axis Relationship …

249

In other words, it can be observed that the presented position estimation method (RPO_1) suffers from the bad performance and the slower convergence during the start-up. Specifically, under the speed change condition, the presented observer produces larger amplitude error and frequency fluctuation in PW voltage and takes much time to reach steady state, as shown in Fig. 6.3b, c. Furthermore, the high transient currents appear during speed variation in both the PW and CW threephase currents and their associated dq-axis components as illustrated in Fig. 6.3d, e respectively. Hence, the shortcoming of the RPO_1 affects its performance during both the start-up and speed change conditions, which is very important for practical industrial applications. Therefore, an improved approach will be proposed in the next subsection to directly detect the rotor position signal of the BDFIG without any need for additional observation methods such as the Luenberger-type PI observer. This will in turn eliminate the use of PI controller in the procedure of the position observer and hence, which will improve the starting performance of the proposed observer and also enhance the behaviour under speed variation.

6.3 Rotor Position Observer Based on the Space-Vector Model of BDFIG In general, the vector control is applied in this chapter for its high behaviour in which the control variables are simply to be controlled by using the rotating frames ω1 and ω2 of the machine. The corresponding phase-axis relationship of the BDFIG in the case of the applied scheme of PW voltage orientation is illustrated in Fig. 6.4. In the presented orientation method, the d-axis PW voltage is totally directed to the total PW voltage vector (u1d = U 1 ). The main schematic diagram of the presented sensorless voltage-control system with the rotor position observer based on the space-vector model of BDFIG (RPO_2) is illustrated in Fig. 6.5. As given in Fig. 6.5, the d-axis PW voltage, u1d , tracks the total reference voltage vector U 1 * , which can be achieved by adjusting the d-axis CW current. Moreover, the regulation process of the q-axis PW voltage can be obtained through the intended frequency of the associated frame, ω1 , and its corresponding angle, θ 1 . After obtaining the angle θ 1 * by integrating the PW reference frequency, the angle of the CW side, θ 2 * , can be realized with the estimated rotor position, θ r_est obtained by the RPO_2, as illustrated on Fig. 6.5.

6.3.1 Design Procedure During a short transient time, the PW flux can be approximated to be a constant value and hence, the zero value of the derivation terms of the dq-axis PW flux is

250

6 Rotor Position and Speed Observers of BDFIGs

Fig. 6.4 Phase-axis relationship of the BDFIG for the sensorless control based on RPO_2

Fig. 6.5 Main block diagram of the sensorless control method based on the RPO_2 for the standalone BDFIG

considered. With the dq-axis rotor voltage equations presented in (2.4), at steady state, the space-vector rotor flux Ψ r can be expressed as r = −

Rr Ir j (ω1 − p1 ωr )

(6.10)

where I r denotes the space-vector rotor current and j represents the unit imaginary.

6.3 Rotor Position Observer Based on the Space-Vector Model of BDFIG

251

Generally, in the actual BDFIG applications, the limitation of the CW frequency ω2 is often bounded to 30% of the PW frequency. According to the speed formula in (2.8), the term (p1 + p2 )ωr would be ranged between 70 and 130% of ω1 . With the typical pair poles of the BDFIG (p1 = 1 and p2 = 3), the term (ω1 − p1 ωr ) would be ranged as (67.5%–82.5%)ω1 . In general, the PW frequency ω1 is either 100π or 120π rad/s. In addition, the rotor current at steady state, I r , is bounded by the rated value, and hence the resulted value of the term [Rr I r /j(ω1 − p1 ωr )] would be very small. Consequently, the rotor flux, Ψ r , is so small that it can be approximately equals to zero. Based on the dq-axis flux relations in (2.5), the flux space-vector of the PW and rotor winding, Ψ 1 and Ψ r , are given as 

1PW = L 1 I1PW + L 1r IrPW rPW = L r IrPW + L 1r I1PW + L 2r I2PW

(6.11)

where I 1 and I 2 represent the space-vector currents of PW and CW, respectively. Moreover, the superscript PW denotes the PW αβ-axis frame. Using (6.11) and assuming that Ψ r ≈ 0, the current of rotor winding can be expressed as IrPW = −

L 1r PW L 2r PW I − I . Lr 1 Lr 2

(6.12)

Substituting (6.12) into (6.11), the space-vector current of CW is derived as I2PW =

1PW − A p I1PW Am

(6.13)

where A p = L 1 − L 21r /L r and Am = −L 1r L 2r /L r . The PW quantities can be transformed from the PW αβ-axis frame to the CW αβ-axis frame according to the following expression:  conj I2PW = e jθr I2CW

(6.14)

where the vector’s conjugate is denoted as conj. Furthermore, the superscript CW refers to the CW αβ-axis frame. From (6.13) and (6.14), the rotor position can be estimated as e jθr _est =

1PW − A p I1PW CW I2 |I2 |2 Am

(6.15)

where |I 2 | is the amplitude of the CW current. With the measured voltage and current quantities of PW and using the abc/dq transformation matrix, the corresponding rotating components in the dq-axis frame can be realized based on the angle θ 1 obtained from the PW voltage orientation

252

6 Rotor Position and Speed Observers of BDFIGs

control scheme. Consequently, with the assumption of a constant PW flux during one sampling period, the resulted flux components of PW in the dq-axis frame can be calculated by 

  ψ1d = u 1q − R1 i 1q /ω1 . ψ1q = (−u 1d + R1 i 1d )/ω1

(6.16)

Then, using the angle θ 1 and with the dq/αβ transformation, the αβ-axis PW flux linkage relations, ψ 1αβ , can be easily obtained without any voltage-integration to obtain the flux linkage. On the other hand, the αβ-axis PW and CW currents are also evaluated using the abc/αβ transformation of the measured PW and CW threephase currents. The flowchart of the rotor position observer based on the space-vector model of BDFIG (RPO_2) is presented in Fig. 6.6. Finally, based on (6.15) and using the αβ-axis components, the predicted rotor position can be represented as e jθr _est =

1 (N + j M) |I2 |2 Am

(6.17)

From (6.17), the estimated rotor position can be obtained by θr _est = tan−1

M N

(6.18)

where 

    M =  ψ1α − A p i 1α i 2β +  ψ1β − A p i 1β i 2α N = ψ1α − A p i 1α i 2α − ψ1β − A p i 1β i 2β

(6.19)

6.3.2 Stability Analysis The confirmation of the stability for the proposed sensorless position observer is very important issue to assure the strong observation of the presented position detection methodology. Starting the study of stability with the consideration of an estimated error between the real position, θ r , and the detected signal, θ r_est . Hence, the real frame specified in dq-axis allocated on the αβ-axis of PW, for attaining I 2 PW in (6.14), could not be localized. Consequently, a fictitious frame in d q-axis is given as shown in Fig. 6.7.

6.3 Rotor Position Observer Based on the Space-Vector Model of BDFIG

253

Start

Reading u1abc, i1abc, i2abc, θ1, ω1

Determining Ψ1dq using (6.16)

Determining Ψ1αβ using the dq/αβ transformation matrix based on Ψ1dq and θ1

Determining M & N using (6.19)

Determining the estimated rotor position using (6.18)

End Fig. 6.6 Flowchart of the RPO_2

Fig. 6.7 Relationship of the phase-axes for the BDFIG with the fictitious frame and the consideration of the estimated-position error

The estimated-position error is considered as the error difference between both the real and fictitious frames, ϕ = (θ r_est − θ r ). In addition, a coupling current I 2 PW is to be resulted in the q-axis due to the estimation error of rotor position. Hence, the coupling current should be adjusted to reach zero for a good tracking of the estimated rotor position. Aided with the fictitious frame in d q-axis, given in Fig. 6.7, the coupling current I 2 PW is obtained as

254

6 Rotor Position and Speed Observers of BDFIGs

conj  I2PW = (e jθr _est − e jθr ) I2CW

(6.20)

 conj I2PW = (e jϕ − 1)e jθr I2CW .

(6.21)

Therefore, aided with (6.13) I2PW =

1PW − A p I1PW jϕ (e − 1). Am

(6.22)

The estimated-speed error can be considered as ωr =

K o ( I2PW

1 + Ki

I2PW dt).

(6.23)

In addition, the differentiation of the estimated-position error can be attained as d 1 ϕ = ( p1 + p2 )K o ( I2PW + dt Ki

I2PW dt).

(6.24)

where K o and K i are the control factors for the proposed sensorless method. Aided with (6.22), (6.23), and (6.24), the associated state-space model can be assumed as      d σ 0 B1 f (ϕ) σ = B2 (6.25) B B f (ϕ) ϕ dt ϕ 2 1 Ki  PW −A I PW

where B1 = ( 1 Am p 1 ), B2 = ( p1 + p2 )K o .  With σ = I2P W dt, the function f(ϕ) can be expressed as f (ϕ) =

e jϕ − 1 ϕ

(6.26)

The expected domain of stability of (6.26) can be considered symmetrical at ϕ = 0. This is returned to the even continuous function in (6.26) and f (0) = 1. The representation of (6.26) as a multi model in (6.27) is used to confirm that the signal of large system is stable. x˙ = [μ1 (ϕ) · D1 + μ2 (ϕ) · D2 ]x. 

   0 B1 0 B1 f (ϕmax ) where D1 = B2 . , D2 = B2 B2 . B1 B2 B1 f (ϕmax ) Ki Ki  t For 0 ≤ |ϕ| ≤ ϕmax with x = σ ϕ , it will get

(6.27)

6.3 Rotor Position Observer Based on the Space-Vector Model of BDFIG



μ1 (ϕ) = μ2 (ϕ) =

f (ϕ)− f (ϕmax ) 1− f (ϕmax ) 1− f (ϕ) 1− f (ϕmax )

.

255

(6.28)

It is obvious that both (6.26) and (6.27) are appeared to be equivalent without applying approximation or linearization or simplification. In addition, aided with (6.28), the following relations in (6.29) can be realized: ⎧ ⎨ μ1 (ϕ) + μ2 (ϕ) = 1 . μ (ϕ) ≥ 0 ⎩ 1 μ2 (ϕ) ≥ 0

(6.29)

For the stability confirmation of the presented observer, using (6.27), the function of quadratic Lyapunov is expressed as 

V (x) = x t · M · x . M = Mt > 0

(6.30)

For a symmetrical matrix, M, the obtained analysis and using (6.26) and (6.27) assure that the presented observer is stable with the following equations: ⎧ ⎨M >0 (D t · M + M · D1 ) > 0 . ⎩ 1t (D2 · M + M · D2 ) > 0

(6.31)

With the inequalities of the linear matrix in (6.31), the maximum error of the estimated position, ϕ max , can be obtained to realize the domain convergence. Hence, the estimated convergence domain is detected as 0 ≤|ϕ|≤ 90 which could be considered to be the largest possible. This would verify the stability confirmation of the proposed sensorless control methodology.

6.3.3 Simulation Results In order to verify the RPO_2 and its capability for sensorless control of the standalone BDFIG, the complete investigation supported with simulation analysis is carried out in this subsection. The given analysis is obtained with a 30-kVA prototype woundrotor BDFIG, whose parameters are specified in Section A.3, Appendix. Furthermore, the desired amplitude and frequency of the PW voltage is set at 311 V and 50 Hz, respectively.

256

6 Rotor Position and Speed Observers of BDFIGs

6.3.3.1

Dynamic Performance Test

Rotor Position (degree)

Rotor Position (degree)

In this section, the dynamic performance test of the RPO_2 is attained to validate its efficacy for sensorless control of the standalone BDFIG under load and speed variation. Firstly, the generator is started with a small load and the rotor speed is accelerated. Then, the operation is continued with the same speed and the load is varied. The obtained analysis is illustrated in Fig. 6.8a–d which denotes the response of rotor position with actual and estimated values including the dynamic and steady state behaviour under the periods of start-up, load change, and speed variations, dqaxis voltages and frequency of PW, dq-axis CW current, and three-phase PW and CW currents, respectively.

400

Starting-Up

400 200

200 0

400

Speed Change

Estimated Position Actual Position

0.025

0.05

Time (s) Load Change

0 0.1 2.45

0.075

2.55

2.6

Time (s)

2.65

2.7

2.75

Steady State

400 200

200 0 5.45

2.5

5.5

5.55

0 6.5

5.6

6.55

6.6

6.65

6.7

6.75

Time (s)

Time (s)

PW frequency (Hz)

DQ-axis PW Voltages (V)

(a)

400 300 200

D-axis Voltage Q-axis Voltage

100 0 -100 51 50.5 50 49.5 49

1

2

3

1

2

3

4

5

6

7

4

5

6

7

Time (s) (b)

Fig. 6.8 Simulation results of the sensorless control of the standalone BDFIG based on the RPO_2: a Estimated and actual rotor positions, b PW voltage amplitude and frequency, c CW dq-axis currents, d three-phase PW and CW currents

6.3 Rotor Position Observer Based on the Space-Vector Model of BDFIG

257

DQ-axis CW currents (A)

80 60 40

D-axis Current

20

Q-axis Current

0 -20

1

2

3

4

5

6

7

Time (s) (c)

PW three-phase Currents (A)

0 -25

CW three-phase Currents (A)

50 25

100 50 0 -50 -100

-50

1

2

3

1

2

3

4

5

6

7

4

5

6

7

Time (s) (d)

Fig. 6.8 (continued)

With the rotor speed of 600 rpm, the BDFIG is started its operation with a resistive load of 11.6 kW. After that, the rotor speed is increased at t = 2.5 s with the rate of 300 rpm per second until it reaches 900 rpm at t = 3.5 s, without any load variation. In the final stage at t = 5.5 s, the load is reduced to 9.7 kW with the same operation speed of 900 rpm. The voltage response in Fig. 6.8b assures the good tracking of the PW voltage to the adjusted set values (U 1 * = 311 V and f 1 * = 50 Hz). In addition, the dq-axis voltages in Fig. 6.8b are maintained fixed at its desired values [(u1d * = U 1 * ) and (u1q * = 0)] set for the intended orientation target of PW voltage. Moreover, the analysis assures that the estimated signal of rotor position is in good accordance with the actual value as shown in Fig. 6.8a which validates the observability of the proposed position observer for sensorless DVC system of BDFIG.

258

6 Rotor Position and Speed Observers of BDFIGs

Furthermore, the q-axis current component of CW side tracked well its desired reference quantity (i2q * = 0) as shown in Fig. 6.8c to successfully attain the orientation purpose for the presented current-control loop of CW side.

6.3.3.2

Robustness Against Parameter Uncertainty

It can be stated from (6.15) and (6.16), that the observability of the detailed procedure for the proposed position observer is dependent on the control parameter, Ap and in terms of the generator inductances and also the resistance of PW side which affect the estimation of the associated PW flux, Ψ 1 PW . Hence, the sensitivity of the generator parameters is very essential to confirm the strong robustness of the proposed rotorposition observer and its capability for sensorless DVC system of BDFIG via any uncertainty issues. As shown in Fig. 6.9, the variation effect of the control parameter, Ap is studied with 130% change in the associated generator inductances (L 1 , L 1r , L r ) according to the design procedure of the proposed observer described in the subsection 6.3.1. In practical operation, the PW resistance can be changed with the winding temperature variations [8]. Generally, the formula for the BDFIG PW resistance variations under different temperatures can be expressed as Rnew = Rold

(T + t2 ) (T + t1 )

(6.32)

where Rold and Rnew are the measured and conversion resistances, respectively. In addition, the symbol T denotes the resistance temperature constant in Kelvin (235 for copper wire and 225 for aluminium wire). Moreover, the variables t 1 and t 2 represent the winding temperature and the conversion temperature, respectively. (The given parameters of the machine are measured at t 1 = 20 °C). Based on the design scheme of the adopted BDFIG prototype and aided with (6.32), the maximum variation of the PW resistance can be assumed as 130%, with considering the maximum temperature of windings to be about 180 °C. The given results shown in Fig. 6.9 and Fig. 6.10a–d represent the response of rotor position with estimated and actual values including the dynamic and steady state behaviour under the load and speed variation, dq-axis voltages and frequency of PW side, dq-axis CW current, and three-phase PW and CW currents, respectively. It is dedicated from Figs. 6.8, 6.9 and 6.10 that the changes in the generator parameters do not produce any effect on the observability of the RPO_2. This assures the strong robustness of the proposed rotor position observer and its effectiveness for the voltage control purpose of the BDFIG in standalone operation.

6.3 Rotor Position Observer Based on the Space-Vector Model of BDFIG

259

6.3.4 Experimental Results In order to verify the RPO_2, the comprehensive experiments are carried out on a 30-kVA prototype wound-rotor BDFIG, whose parameters are specified in Section A.3, Appendix.

6.3.4.1

Dynamic Performance Test

Rotor Position (degree)

Rotor Position (degree)

This part aims at investigating the proposed observer for rotor-position detection of the standalone BDFIG with a complete analysis supported by the obtained experimental results during different periods including the start-up case, the load variation,

400

Starting-Up

Estimated Position Actual Position 400 200

200 0

400

0.02

0.04

0.06

Time (s) Load Change

0.08

0.1

0 5.45

0 2.45 2.5 2.55 2.6 2.65 2.7 2.75

Time (s) Steady State

400 200

200 5.5

5.55

0 6.5

5.6

DQ-axis PW Voltages (V) PW frequency (Hz)

Time (s) 400 300 200 100 0 -100 51 50.5 50 49.5 49

Speed Change

6.55

6.6

6.65

6.7

6.75

Time (s) (a)

D-axis Voltage

Q-axis Voltage

1

2

3

1

2

3

4

5

6

7

4

5

6

7

Time (s) (b)

Fig. 6.9 Performance test of the observer under 130% change in the machine inductances (L 1 , L 1r , L r ): a Estimated and actual rotor positions, b PW voltage amplitude and frequency, c CW dq-axis currents, d three-phase PW and CW currents

260

6 Rotor Position and Speed Observers of BDFIGs

DQ-axis CW currents (A)

120 100

D-axis Current

80 60 40 20

Q-axis Current

0 -20

1

2

3

4

5

6

7

Time (s)

PW three-phase Currents (A) CW three-phase Currents (A)

(c)

50 25 0 -25

100 50 0 -50 -100

-50

1

2

3

1

2

3

4

5

6

7

4

5

6

7

Time (s) (d)

Fig. 6.9 (continued)

and the speed change conditions. This confirms the observability of the proposed estimation procedure for the rotor-position signal and its capability for the purpose of sensorless voltage control of BDFIG in the ship power generation systems. Furthermore, the desired voltage profile of PW side is set at 150 V and 50 Hz for the experimental work. Starting with a rotor speed of 600 rpm, the BDFIG is operated to fed a three-phase resistive with the resistance of 25  for each phase. Then, the generator continued its operation with the same speed and a reduction in each phase of the load side to its half power at t = 9.16 s. The obtained analysis with experimental results illustrated in Figs. 6.11, 6.12 and 6.13 gives the complete investigation of the new observer with the proposed procedure for rotor-position estimation under the operating conditions of the start-up, the load change, and the speed variations, respectively. The test performance in Fig. 6.11 denotes the response of rotor position with actual and estimated values for various periods of dynamic and steady-state operation in (a)

6.3 Rotor Position Observer Based on the Space-Vector Model of BDFIG

261

Rotor Position (degree)

and (b) under the start-up condition, the response of estimated position error in (c), the CW q-axis current in (d), the phase voltage of PW side in (e) and its detailed view in (f), the phase current of PW in (g) and detailed view in (h), the phase current of CW in (i) and detailed view in (j). In addition, the analysis under load change condition is given in Fig. 6.13a–d which represent the response of rotor position with the actual and estimated values, phase voltage of PW, PW and CW phase current, respectively. Moreover, the effectiveness of the proposed observer is studied under the speed change operation from 600 to 700 rpm as illustrated in Fig. 6.13 which denotes the rotor position signal during different periods with estimated and actual values in (a) and (b), the response of estimated position error in (c), the CW q-axis current in (d), the phase voltage of PW in (e) and its detailed view in (f), the phase current of PW side in (g) and expanded view in (h), the phase current of CW in (i) and detailed view in (j).

400

Starting-Up

400

200

200

Rotor Position (degree)

0

400

Speed Change

Estimated Position Actual Position

0.025

0.05

Time (s) Load Change

0.075

0 2.45

0.1

2.55

2.6

Time (s)

2.65

2.7

2.75

Steady State

400 200

200 0 5.45

2.5

5.5

5.55

0 6.5

5.6

6.55

6.6

6.65

6.7

6.75

Time (s)

Time (s)

PW frequency (Hz)

DQ-axis PW Voltages (V)

(a)

400 300 200

D-axis Voltage Q-axis Voltage

100 0 -100 51 50.5 50 49.5 49

1

2

3

1

2

3

4

5

6

7

4

5

6

7

Time (s) (b)

Fig. 6.10 Performance test of the proposed observer under 130% change in PW resistance: a Estimated and actual rotor positions, b PW voltage amplitude and frequency, c CW dq-axis currents, d three-phase PW and CW currents

262

6 Rotor Position and Speed Observers of BDFIGs

DQ-axis CW currents (A)

80 60

D-axis Current

40 20

Q-axis Voltage

0 -20

1

2

3

4

5

6

7

Time (s)

CW three-phase Currents (A)

PW three-phase Currents (A)

(c)

50 25 0 -25 -50 100 50 0 -50 -100

1

2

3

1

2

3

4

5

6

7

4

5

6

7

Time (s) (d)

Fig. 6.10 (continued)

The experimental results presented in Figs. 6.11, 6.12 and 6.13 confirm the good tracking between the voltage of PW side and its desired set profile for DVC purpose under different operations. Furthermore, a close correlation is successfully attained between the predicted and measured rotor position as shown in Figs. 6.11a, b, 6.12a and 6.13a, b under various operating states with the starting period, the load variation, and the speed changes, respectively. This assures the functionality of the proposed observer with the given rotorposition estimation procedure for sensorless voltage control of BDFIG in the ship power generation systems. The effectiveness of the presented CW current-control loop is also confirmed as obtained in Figs. 6.12d and 6.14d with a good tracking of the q-axis CW current to its adjusted value (i2q * = 0) to realize the desired orientation technique for CW current control.

6.3 Rotor Position Observer Based on the Space-Vector Model of BDFIG

Rotor Position (Degree)

450

263

Estimated Position Actual Position

400 350 300 250 200 150 100 50 0

0

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2

0.225

Time (s) (a)

Rotor Position (Degree)

450

Estimated Position Actual Position

400 350 300 250 200 150 100 50 0 3.5

3.525

3.55

3.575

3.6

3.625

Time (s)

3.65

3.675

3.7

3.725

3.75

(b)

Position Error (Degree)

15 10

Max. Position Error = 1.34 degree

5 0 -5 -10 -15

0

1

2

3

4

5

Time (s) (c)

Fig. 6.11 Experimental results of the RPO_2 under the start-up operation: a Estimated and actual rotor positions at start up, b estimated and actual rotor positions at steady state, c rotor position error, d CW q-axis current, e PW phase voltage, f extended view of e, g PW phase current, h extended view of g, i CW phase current, j extended view of i at 600 rpm

264

6 Rotor Position and Speed Observers of BDFIGs

Q-axis CW Current (A)

0.6 0.4

Start-Up of the Proposed Observer

0.2 0 -0.2 -0.4 -0.6 0

1

2

3

4

5

Time (s) (d)

PW Phase Voltage (V)

400 200 0 -200 -400 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

3.62

3.64

3.66

3.68

3.7

Time (s) (e)

PW Phase Voltage (V)

400 200 0 -200 -400 3.5

3.52

3.54

3.56

3.58

3.6

Time (s) (f)

Fig. 6.11 (continued)

6.3 Rotor Position Observer Based on the Space-Vector Model of BDFIG

265

PW Phase Current (A)

20

10

0

-10

-20

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

3.62

3.64

3.66

3.68

3.7

3

3.5

4

4.5

5

3.8

3.85

3.9

3.95

4

Time (s) (g)

PW Phase Current (A)

20

10

0

-10

-20 3.5

3.52

3.54

3.56

3.58

3.6

Time (s) (h)

CW Phase Current (A)

40

20

0

-20

-40

0

0.5

1

1.5

2

2.5

Time (s) (i) CW Phase Current (A)

40

20

0

-20

-40 3.5

3.55

3.6

3.65

3.7

3.75

Time (s)

(j)

Fig. 6.11 (continued)

266

6 Rotor Position and Speed Observers of BDFIGs

Rotor Position (Degree)

450

Estimated Position Actual Position

Load Change Condition

400 350 300 250 200 150 100 50 0

9.1

9.15

9.2

9.25

9.3

9.35

9.4

Time (s)

PW Phase Voltage (V)

(a)

400 200 0 -200 -400 9

9.05

9.1

9.15

9.2

9.25

9.3

9.35

9.4

9.45

9.5

9.3

9.35

9.4

9.45

9.5

9.3

9.35

9.4

9.45

9.5

Time (s) (b)

PW Phase Current (A)

20

10

0

-10

-20

9

9.05

9.1

9.15

9.2

9.25

Time (s) (c)

CW Phase Current (A)

40

20

0

-20

-40

9

9.05

9.1

9.15

9.2

9.25

Time (s) (d)

Fig. 6.12 Experimental results of the RPO_2 under the load variation (50% reduction): a Estimated and actual rotor positions under load change, b PW phase voltage, c PW phase current, d CW phase current

6.3 Rotor Position Observer Based on the Space-Vector Model of BDFIG

Rotor Position (Degree)

450

267

Estimated Position Actual Position

400 350 300 250 200 150 100 50 0 12

12.2

12.4

12.6

12.8

13

13.2

13.4

13.6

13.8

12.2

Time (s) (a)

Rotor Position (Degree)

450

Estimated Position Actual Position

400 350 300 250 200 150 100 50 0 22.8

22.82

22.84

22.86

22.9

Time(s)

22.92

22.94

22.96

22.98

23

(b)

30

Position Error (Degree)

22.88

20

Max. Position Error = 1.67 degree 10 0 -10 -20 -30 12

14

16

18

20

22

24

Time (s) (c)

Fig. 6.13 Experimental results of the RPO_2 under the speed change (600–700 rpm): a Estimated and actual rotor positions (at 600 rpm), b estimated and actual rotor positions (at 700 rpm), c rotor position error, d CW q-axis current, e PW phase voltage, f extended view of e, g PW phase current, h extended view of g, i CW phase current, j extended view of i at 700 rpm

268

6 Rotor Position and Speed Observers of BDFIGs

Q-axis CW Current (A)

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 12

14

16

18

20

22

24

20

22

24

Time (s) (d)

PW Phase Voltage (V)

450 300 150 0 -150 -300 -450 12

14

16

18

Time (s) (e)

PW Phase Voltage (V)

450 300 150 0 -150 -300 -450 20

20.1

20.2

20.3

20.4

20.5

Time (s) (f)

Fig. 6.13 (continued)

20.6

20.7

20.8

20.9

21

6.3 Rotor Position Observer Based on the Space-Vector Model of BDFIG

269

PW Phase Current (A)

15 10 5 0 -5 -10 -15 12

14

16

18

20

22

24

Time (s) (g)

PW Phase Current (A)

15 10 5 0 -5 -10 -15 20

20.1

20.2

20.3

20.4

20.5

20.6

20.7

20.8

20.9

21

Time (s) (h)

CW Phase Current (A)

20 15 10 5 0 -5 -10 -15 -20 12

14

16

18

20

22

24

Time (s) (i)

CW Phase Current (A)

20 15 10 5 0 -5 -10 -15 -20 20

20.2

20.4

20.6

20.8

21

Time (s) (j)

Fig. 6.13 (continued)

21.2

21.4

21.6

21.8

22

270

6 Rotor Position and Speed Observers of BDFIGs 450

Estimated Position Actual Position

Rotor Position (Degree)

400 350 300 250 200 150 100 50 0 5.3

5.32

5.34

5.36

5.38

5.4

Time (s)

5.42

5.44

5.46

5.48

5.5

5.3

5.35

5.4

5.45

5.5

5.3

5.35

5.4

5.45

5.5

5.3

5.35

5.4

5.45

5.5

(a)

PW Phase Voltage (V)

400 200 0 -200 -400 5

5.05

5.1

5.15

5.2

5.25

Time (s) (b)

PW Phase Current (A)

20

10

0

-10

-20

5

5.05

5.1

5.15

5.2

5.25

Time (s) (c)

CW Phase Current (A)

40

20

0

-20

-40

5

5.05

5.1

5.15

5.2

5.25

Time (s) (d)

Fig. 6.14 Experimental results of the RPO_2 under the BDFIG parameter change (130% uncertainty): a Estimated and actual rotor positions, b PW phase voltage, c PW phase current, d CW phase current

6.3 Rotor Position Observer Based on the Space-Vector Model of BDFIG

6.3.4.2

271

Robustness Against Parameter Uncertainty

Some of the experimental results are introduced as illustrated in Fig. 6.14 to validate the robustness of the proposed observer against the parameters uncertainty issues of the BDFIG system including the inductances variation which affect the control value, Ap and also the changes of PW resistance. The presented experimental results in Fig. 6.14a–d denote the repose of rotor position including the estimated and actual values, the PW phase voltage, the PW and CW phase current, respectively. In order to assure the robustness of the proposed position observer, the performance of the presented sensorless voltage-control system is studied via 130% uncertainty in the whole generator parameters related to (6.15). This can be attained by changing the parameter values of BDFIG in the associated control system. The obtained results in Fig. 6.14 confirm the observability of the proposed position detection procedure. The results also assure the efficacy of the sensorless system for voltage control against the parameter changes of the presented generator including the whole inductances of the machine and also the resistance of PW side in a similar comparative manner to the realized behaviour with simulation results, as observed in Figs. 6.9 and 6.10, under the same situation of parameter uncertainties. The good tracking between the PW voltage response with its desired set profile and also the close correlation between the estimated rotor position with its real value (as shown in Figs. 6.9, 6.10, and 6.14 for simulation and experiment results, respectively) under the change of generator parameters has proven the strong robustness and functionality of the proposed rotor-position observer for voltage control of the standalone BDFIGs. All the given analysis supported with the simulation and experimental work has verified the efficacy of the proposed observer to effectively estimate the rotor position of the promising BDFIG for the target of sensorless voltage control in the ship power generation systems.

6.4 Improved Rotor Speed Observer for Standalone BDFIG with Unbalanced and Nonlinear Loads 6.4.1 Scheme Design 6.4.1.1

Working Principle

Under the doubly-fed mode, the mechanical rotor speed of BDFIG can be decided by ( p1 + p2 )ωr = ω1 + ω2

(6.33)

272

6 Rotor Position and Speed Observers of BDFIGs

Fig. 6.15 Structure of the basic RSO [12]

where p is the number of pole pairs, ω the angular frequency, and the subscripts 1, 2, and r indicate the PW, CW, and rotor, respectively. When the CW frequency ω2 is set to zero, the natural synchronous speed of the BDFIG can be obtained, above which the rotor speed is called the super-synchronous speed, and below which that is the sub-synchronous speed. When the rotor speed varies, in order to keep the PW frequency ω1 constant, the value of ω2 should be changed by ω2 = −ω1 +ωr ( p1 + p2 ).

(6.34)

The basic RSO has been developed in [12], with the structure illustrated in Fig. 6.15, where u1α and u1β are the α- and β-components of the PW voltage, i2α and i2β are those of the CW current, θ 1 and θ 2 are the phase angles of the PW voltage and CW current, respectively, and θ rv is the virtual rotor position serving as an intermediate variable for deriving the rotor speed. As can be seen from Fig. 6.15, the basic RSO employs the PW voltage and CW current to estimate the rotor speed. The values of sin(θ 1 + θ 2 ) and cos(θ 1 + θ 2 ) are calculated by trigonometric operation and then input to the αβ-frame PLL. If integrating both sides of (6.33) and ignoring the integration constant, the following expression can be obtained: ( p1 + p2 )θr v = θ1 + θ2 .

(6.35)

When the αβ-frame PLL is working in the quasi-locked state, based on (6.35), the difference between the actual and estimated virtual rotor positions, θ rv , can be expressed as ( p1 + p2 ) θr v = ( p1 + p2 )(θr v − θˆr v ) = (θ1 + θ2 ) − ( p1 + p2 )θˆr v

6.4 Improved Rotor Speed Observer for Standalone BDFIG …

≈ sin[(θ1 + θ2 ) − ( p1 + p2 )θˆr v ].

273

(6.36)

The θ rv can be regulated to zero by using a PI controller, and consequently the rotor speed can be accurately estimated. The tuning formula for the PI controller parameters of the basic RSO has been derived in [12], which can be expressed as k p_B R S O = 8/ts_B R S O , ki_B R S O = 16/(ξ B R S O ts_B R S O )2

(6.37)

where k p_BRSO and k i_BRSO are the proportional gain and integral gain, t s_BRSO and ξ BRSO are the settling time and damping factor, respectively.

6.4.1.2

Performance Analysis Under Unbalanced and Nonlinear Loads

Unbalanced loads and nonlinear loads can generate unbalanced and distorted PW terminal voltages, respectively, which further produce distorted CW current through the indirect coupling of the rotor. With the basic RSO, the unbalanced/distorted PW voltage and distorted CW current can cause inaccurate rotor speed observation, the reason of which is fully analyzed as follows. 1.

Analysis under Unbalanced Loads

According to the instantaneous symmetrical component method, the unbalanced three-phase voltage can be decomposed to balanced positive-, negative- and zerosequence components. In a low voltage power generation system used in ships, the three-phase three-wire mode is typically adopted, in which the path for the zero-sequence current component is absent. Hence, no steady state zero-sequence component of the PW exists and only positive- and negative-sequence components are considered. From (6.34), the angular frequency of the CW harmonic current induced by the negative-sequence PW current can be expressed by ω2−1 = −ω1−1 + ( p1 + p2 )ωr =ω1 + ( p1 + p2 )ωr

(6.38)

where ω1−1 and ω2−1 stand for angular frequencies of the PW negative-sequence voltage and the corresponding CW harmonic current, respectively. Similarly, the angular frequency of the positive-sequence CW current (i.e. CW fundamental current) can be expressed as ω2+1 = −ω1+1 + ( p1 + p2 )ωr = − ω1 + ( p1 + p2 )ωr

(6.39)

where ω1+1 and ω2+1 are angular frequencies of PW positive-sequence voltage and CW fundamental current, respectively. Under the constant rotor speed and PW frequency, the angular positions of positive- and negative-sequence PW voltage vectors and those of fundamental and harmonic CW current vectors can be given by

274

6 Rotor Position and Speed Observers of BDFIGs

 θ1+1 =  ω1+1 dt = ω1 t+ϕ1+1 θ2+1 = ω2+1 dt = −ω1 t + ( p1 + p2 )ωr t + ϕ2+1  −1  −1 θ1 =  ω1 dt = −ω1 t + ϕ1−1 θ2−1 = ω2−1 dt = ω1 t + ( p1 + p2 )ωr t + ϕ2−1



(6.40)

(6.41)

where θ is the real angular position, and ϕ the initial angular position. It is assumed that the amplitudes of positive- and negative-sequence PW voltage vectors are U1+1 and U1−1 , and those of fundamental and harmonic CW current vectors are I2+1 and I2−1 , respectively. Applying trigonometric operations, the angular positions of the real PW voltage and CW current vectors can be derived as ⎧ sin θ1 = K 1 sin θ1+1 + K 2 sin θ1−1 ⎪ ⎪ ⎨ cos θ1 = K 1 cos θ1+1 + K 2 cos θ1−1 ⎪ sin θ2 = K 3 sin θ2+1 + K 4 sin θ2−1 ⎪ ⎩ cos θ2 = K 3 cos θ2+1 + K 4 cos θ2−1

(6.42)

where K 1 = U1+1 /U1 , K 2 = U1−1 /U1 , K 3 = I2+1 /I2 , K 4 = I2−1 /I2 , U1 = I2 =



(U1+1 )2 + (U1−1 )2 + 2U1+1 U1−1 cos(2ω1 t + ϕ1+1 − ϕ1−1 ),



(I2+1 )2 + (I2−1 )2 + 2I2+1 I2−1 cos(−2ω1 t + ϕ2+1 − ϕ2−1 ).

The difference between the actual and estimated virtual rotor positions, θ rv , can be expressed as ( p1 + p2 ) θr v ≈ sin[(θ1 + θ2 ) − ( p1 + p2 )θˆr v ]   = K 1 K 3 sin Ar + ϕ1+1 + ϕ2+1   + K 2 K 3 sin −2ω1 t + Ar + ϕ1−1 + ϕ2+1   + K 1 K 4 sin 2ω1 t + Ar + ϕ1+1 + ϕ2−1   + K 2 K 4 sin Ar + ϕ1−1 + ϕ2−1

(6.43)

where Ar = ( p1 + p2 )ωr t −( p1 + p2 )θˆr v , and all the coefficients K 1 K 3 , K 2 K 3 , K 1 K 4 , and K 2 K 4 contain ac components at the frequency of 2ω1 . The detailed derivation process for (6.43) can be found in Appendix. The first and last terms in the right side of (6.43) can be eliminated by adjusting θˆr v to make (Ar + ϕ1+1 + ϕ2+1 ) and (Ar + ϕ1−1 + ϕ2−1 ) to zero. However, the other two terms in the right side of (6.43) both have ac components with the frequency of 2ω1 , which cannot be removed, because 2ω1 t and −2ω1 t cannot be canceled by adjusting θˆr v . The input of the PI controller of the basic RSO contains ac components

6.4 Improved Rotor Speed Observer for Standalone BDFIG …

275

at the frequency of 2ω1 , which results in fluctuation of the estimated rotor speed at the frequency of 2ω1 . 2.

Analysis under Nonlinear Loads

Under nonlinear loads, the most significant harmonics of the PW voltage are the 5th and 7th harmonics with frequencies of −5ω1 and 7ω1 . The corresponding CW harmonic currents can be generated by the harmonic components of PW current via the indirect coupling of the rotor, whose angular frequencies can be obtained by ω2−5 = −ω1−5 + ( p1 + p2 )ωr =5ω1 + ( p1 + p2 )ωr

(6.44)

ω2+7 = −ω1+7 + ( p1 + p2 )ωr = − 7ω1 + ( p1 + p2 )ωr

(6.45)

where ω1−5 and ω1+7 are angular frequencies of the 5th and 7th PW harmonic voltages, and ω2−5 and ω2+7 those of the corresponding CW harmonic currents induced by the PW harmonic currents, respectively. Similarly, under the constant rotor speed and PW fundamental frequency, the angular positions of the 5th and 7th PW harmonic voltage vectors and those of the corresponding harmonic CW current vectors can be calculated by  θ1−5 =  ω1−5 dt = −5ω1 t + ϕ1−5 θ2−5 = ω2−5 dt = 5ω1 t + ( p1 + p2 )ωr t + ϕ2−5  +7  +7 θ1 =  ω1 dt = 7ω1 t + ϕ1+7 . θ2+7 = ω2+7 dt = −7ω1 t + ( p1 + p2 )ωr t + ϕ2+7 

(6.46)

(6.47)

The amplitudes of the 5th and 7th PW harmonic voltage vectors are assumed to be U1−5 and U1+7 , and those of the corresponding CW harmonic current vectors to be I2−5 and I2+7 , respectively. Combining (6.40), (6.46) and (6.47), the angular positions of the real PW voltage and CW current vectors can be obtained by ⎧ sin θ1 = K 1 sin θ1+1 + K 2 sin θ1−5 + K 3 sin θ1+7 ⎪ ⎪ ⎨ cos θ1 = K 1 cos θ1+1 + K 2 cos θ1−5 + K 3 cos θ1+7 ⎪ sin θ2 = K 4 sin θ2+1 + K 5 sin θ2−5 + K 6 sin θ2+7 ⎪ ⎩ cos θ2 = K 4 cos θ2+1 + K 5 cos θ2−5 + K 6 cos θ2+7

(6.48)

where K 1 = U1+1 /U1 , K 2 = U1−5 /U1 , K 3 = U1+7 /U1 , K 4 = I2+1 /I2 , K 5 = I2−5 /I2 , K 6 = I2+7 /I2 ,   +1 2  (U1 ) + (U1−5 )2 + (U1+7 )2 + 2U1+1 U1−5 cos(6ω1 t + ϕ1+1 − ϕ1−5 )  U1 =  +2U1−5 U1+7 cos(−12ω1 t + ϕ1−5 − ϕ1+7 ) +2U1+1 U1+7 cos(−6ω1 t + ϕ1+1 − ϕ1+7 )

276

6 Rotor Position and Speed Observers of BDFIGs

  +1 2  (I2 ) + (I2−5 )2 + (I2+7 )2 + 2I2+1 I2−5 cos(−6ω1 t + ϕ2+1 − ϕ2−5 )  . I2 =  +2I2−5 I2+7 cos(12ω1 t + ϕ2−5 − ϕ2+7 ) +2I2+1 I2+7 cos(6ω1 t + ϕ2+1 − ϕ2+7 ) Substituting (6.48) to (6.36), the difference between the actual and estimated virtual rotor positions, θ rv , can be expressed as ( p1 + p2 ) θr v ≈ sin[(θ1 + θ2 ) − ( p1 + p2 )θˆr v ]   = K 1 K 4 sin Ar + ϕ1+1 + ϕ2+1   +K 1 K 5 sin 6ω1 t+Ar + ϕ1+1 + ϕ2−5   + K 1 K 6 sin −6ω1 t+Ar + ϕ1+1 + ϕ2+7   + K 2 K 4 sin −6ω1 t+Ar + ϕ1−5 + ϕ2+1   + K 2 K 5 sin Ar + ϕ1−5 + ϕ2−5   + K 2 K 6 sin −12ω1 t+Ar + ϕ1−5 + ϕ2+7   + K 3 K 4 sin 6ω1 t+Ar + ϕ1+7 + ϕ2+1   + K 3 K 5 sin 12ω1 t+Ar + ϕ1+7 + ϕ2−5   + K 3 K 6 sin Ar + ϕ1+7 + ϕ2+7

(6.49)

where Ar = ( p1 + p2 )ωr t − ( p1 + p2 )θˆr v , and all the coefficients K 1 K 4 , K 1 K 5 , K 1 K 6 , K 2 K 4 , K 2 K 5 , K 2 K 6 , K 3 K 4 , K 3 K 5 and K 3 K 6 contain ac components with frequencies of 6ω1 and 12ω1 . The first, fifth and ninth terms in the right side of (6.49) can be eliminated by regulating θˆr v to make (Ar + ϕ1+1 + ϕ2+1 ), (Ar + ϕ1−5 + ϕ2−5 ) and (Ar + ϕ1+7 + ϕ2+7 ) to zero. The other six terms in the right side of (6.49), containing ac components at the frequency of 6ω1 or 12ω1 , cannot be cleared, because ±6ω1 t and ±12ω1 t in these terms cannot be mitigated by adjusting θˆr v . As a result, ac components at frequencies of 6ω1 and 12ω1 are inevitable in the PI controller of the basic RSO, which results in fluctuation of the estimated rotor speed at the same frequencies. According to the analysis above, the basic RSO is unable to obtain the accurate rotor speed when the unbalanced or nonlinear load occurs.

6.4.1.3 1.

Design of Improved Rotor Speed Observer

Scheme Design In order to overcome the aforementioned problems, second-order generalized integrators (SOGIs) and low-pass filters (LPFs) are introduced to the RSO to pre-filter the PW voltage and CW current, respectively. The improved RSO is illustrated in Fig. 6.16.

6.4 Improved Rotor Speed Observer for Standalone BDFIG …

277

Fig. 6.16 The whole control structure of the improved RSO, where PSC denotes the positivesequence calculator

For a better observation of the estimated speed, the second-order generalized integrators (SOGIs) and low-pass filters (LPFs) are used for the improved RSO, as shown in Fig. 6.16, for prefiltering the PW voltage and CW current, respectively. It is obvious from Fig. 6.16 that the fundamental αβ-axis PW voltages (u1αf and u1β f ) and their corresponding quadratic components (qu1αf and qu1β f ) are realized by two SOGIs. Aided with the positive-sequence calculator (PSC) [17], the fundamental + positive-sequence αβ-axis PW voltages, u + 1α f and u 1β f , can be obtained as

  u+ 1α f = u 1α f − qu 1β f /2 u+ 1β f = qu 1α f + u 1β f /2

(6.50)

In order to make SOGIs frequency-adaptive, the PW frequency ωˆ 1 (represents the resonance frequency of two SOGIs) can be detected through the αβ-frame PLL + using the voltage components, u + 1α f and u 1β f given in (6.50). According to the speed formula in (2.8), during the operation of natural speed, the current frequency of CW side equals zero to attain the desired PW frequency and hence, the fundamental current of CW has a minimum frequency of zero in real applications. Therefore, the LPF is used to prefilter the harmonics in the αβ-axis CW + currents. Finally, the filtered components, u + 1α f , u 1β f , i 2α f and i 2β f , are used for the accurate detection of the estimated rotor speed based on the presented RSO. After the filtering process of the PW negative-sequence voltage and the resulted harmonic CW current, they can be fed to the structure of RSO and approximate the coefficients K 1 and K 3 in (6.42) as 1, K 2 and K 4 as 0. Hence, the error difference in the virtual rotor position, θ rv , can be simply given as

278

6 Rotor Position and Speed Observers of BDFIGs

  ( p1 + p2 ) θr v ≈ sin Ar + ϕ1+1 + ϕ2+1

(6.51)

6.4.2 Parameters Tuning For a high dynamic and steady-state behaviour of the proposed RSO, the optimal design of the considerable parameters for the PI-controller and the filters are very essential. The main target for attaining the optimal control parameters is to realize a speed response from the improved RSO to be not lower than that of the basic RSO with the effective verification of the filter’s role (i.e., SOGIs and LPFs), which can be investigated in the following three steps. • Step 1: Determination of SOGI Parameters The operation of the presented SOGIs in Fig. 6.16 can be represented with the following transfer functions as [17] u 1β f (s) u 1α f (s) 2ξ S OG I ωˆ 1 s = = 2 u 1α (s) u 1β (s) s + 2ξ S OG I ωˆ 1 s + ωˆ 12

(6.52)

qu 1β f (s) qu 1α f (s) 2ξ S OG I ωˆ 12 = = 2 u 1α (s) u 1β (s) s + 2ξ S OG I ωˆ 1 s + ωˆ 12

(6.53)

where ξ SOGI denotes the damping factor, and ωˆ 1 is the estimated PW frequency using the αβ-frame PLL (i.e., the resonance frequency). By considering the speed response and stability of the SOGI, the factor ξ SOGI is set to 0.707 [17]. Aided with (6.52) and (6.53), the settling time of SOGI is obtained as O1 ) ts_S OG I ≈ 4/(ξSOGI ω

(6.54)

The frequency ωˆ 1 can be considered as the nominal frequency of PW in the case that the speed response of the αβ-frame PLL is faster than that of the SOGI. Aided with (6.54), the time ts_S OG I is about 18 ms for the nominal frequency of PW to be equal to 50 Hz. • Step 2: Determination of PI Controller Parameters For the delay reduction of the αβ-frame PLL, its speed response should be faster than that of the SOGI. Moreover, the settling time of the SOGI and LPF should be less than that of the RSO to realize an improved RSO with a faster speed response. Consequently, the settling time of the main parts of the improved RSO should be bounded as follows: ts_P L L < ts_S OG I ≈ ts_L P F < ts_B R S O

(6.55)

6.4 Improved Rotor Speed Observer for Standalone BDFIG …

279

where t s_PLL , t s_SOGI , t s_LPF , and t s_BRSO represent the settling time of the αβ-frame PLL, SOGI, LPF, and basic RSO, respectively. Aided with (6.55) and considering that the time t s_SOGI is to be about 18 ms, hence the times t s_PLL and t s_BRSO can be approximated to be about 10 ms and 40 ms, respectively. The parameter tunning of the PI-controller can be achieved aided with (6.37). This is due to the similar control modelling of the αβ-frame PLL to that of the basic RSO. The damping factor in (6.37) can be adjusted to 0.707 by considering the tradeoff of the speed response and the stability. The gain parameters of the PI-controller in the basic RSO and the αβ-frame PLL can be obtained by 

k p_P L L = 800, ki_P L L = 80,000 k p_B R S O = 200, ki_B R S O = 5000

(6.56)

• Step 3: Determination of LPF Parameters For the presented LPF, it is needed to determine the cutoff frequency. The fundamental and harmonic current components of the CW side have a relationship for their angular frequencies, as illustrated by ⎧ −1 +1 ⎨ ω2 = ω2 + 2ω1 ω−5 = ω2+1 + 6ω1 ⎩ 2+7 ω2 = ω2+1 − 6ω1

(6.57)

According to the principles of the standalone BDFIG system, the power converter used for the control purpose has a frequency range of the fundamental CW current, ω2+1 , bounded to ±0.3ω1 (30% of the PW frequency). In order to employ a fractionally rated power converter to drive the standalone BDFIG system, the angular frequency of CW fundamental current, ω2+1 , is usually limited to 30% of the PW frequency (i.e., between −0.3ω1 and 0.3ω1 ). Hence, the maximum frequency of the fundamental CW current, |ω2f |max , is 0.3ω1 , and aided with (6.57), the minimum frequency of the harmonic CW current, |ω2h |min , is 1.7ω1 . The transfer function of the presented LPF is given as LPF(s) =

ωc s + ωc

(6.58)

where the cutoff frequency is represented by ωc . As shown in Fig. 6.17, the cutoff frequency should be greater than |ω2f |max and smaller than |ω2h |min to realize a weak decay of the fundamental component and a strong attenuation of the harmonic component. From what mentioned above, the optimal frequency of cutoff, ωc_opt , can be obtained as   ωc_opt |ω2f |max 1 |ω2h |min lg lg = + lg ωc_opt 2 ωc_opt ωc_opt

(6.59)

280

6 Rotor Position and Speed Observers of BDFIGs

Fig. 6.17 The characteristics with magnitude–frequency of the presented LPF where the frequency is represented by a normalized scale

Aided with (6.59), the frequency ωc_opt is given as ωc_opt =



|ω2f |max ∗ |ω2h |min ≈ 0.7ω1

(6.60)

Consequently, the real frequency of cutoff ωc should be adjusted to be about 2π × 35 rad/s according to a nominal frequency of PW side to be equal 50 Hz. Furthermore, considering an error at steady-state to be about 2%, the settling time of the presented LPF [16] is set as ts_L P F = 4/ωc ≈ 18 ms

(6.61)

Hence, it is obvious that the time t s_LPF can realize the boundary in (6.55).

6.4.3 Experimental Results 6.4.3.1

Dynamic Performance Test

In order to assure the observation of the improved RSO in comparison with its basic structure, comprehensive experimental results have been conducted on the 30-kVA BDFIG platform as illustrated in Section A.3, Appendix. The speed encoder is used to detect the real speed for only the comparison process with the estimated value. The main control parameters and settling time specified for the improved RSO can be given in Table 6.1, which are also used to carry out a comparative study with the basic RSO. • Experiment Under Unbalanced Load

6.4 Improved Rotor Speed Observer for Standalone BDFIG … Table 6.1 Control parameters and settling time for the improved RSO

281

Component

Control parameter

SOGI

ξ SOGI = 0.707

18

LPF

ωc = 2π35 rad/s

18

αβ-frame PLL k p_PLL = 800, k i_PLL = 80,000 Basic RSO

Settling time (ms)

10

k p_BRSO = 200, k i_BRSO = 40 5000

As illustrated in Fig. 6.18 and at t = 0.77 s, the presented system based the BDFIG is loaded with a three-phase unbalanced resistive load of 25, 100 and 100  in phases a, b, and c. Then, the speed operation is increased from 620 to 939 rpm within 4– 8.2 s, and then decreased from 939 to 606 rpm from 10 to 18.8 s. It is obvious from Fig. 6.19a, b that the CW current is distorted due to the generated unbalanced voltage of PW side with a resulted 14.1% unbalance factor (UF). In addition, the same speed response can be detected for both the improved RSO and the basic one is illustrated in Fig. 6.19c–f. Moreover, the speed error resulted from the basic RSO has an oscillation of about 12 rpm as given in Fig. 6.18d, which has dedicated that a great reduction with the speed error is attained using the improved RSO to be about 3 rpm as illustrated in Fig. 6.18f. Therefore, the observability of the improved RSO is significantly validated with a better dynamic behavior under the unbalanced load condition compared to the basic RSO. • Experiment Under Nonlinear Load In this second experimental case and at 2.43 s, the effectiveness of the improved RSO is verified under the nonlinear load condition (a diode-rectifier with a resistor of 25  at the dc side). As illustrated in Fig. 6.19, the rotor speed is increased from 597 to 928 rpm and then decreased to 694 rpm within the time period from 4.46 to 17.33 s. It is obvious from Fig. 6.20a, b that a distorted current of CW side is resulted due to the harmonics in the generated PW voltage with 9.1% total harmonic distortions (THD). Moreover, a speed error is appeared to be about 3.1% during the connection instant of the nonlinear load and then converges rapidly within 0.28 s to 0.5% as given in Fig. 6.20d, f. With a comparative study as illustrated in Fig. 6.20c–f, the observation of the estimated speed using the basic RSO suffers from an oscillation of about 20 rpm as obtained in Fig. 6.19d, which is reduced to about 3 rpm using the improved RSO as given in Fig. 6.19f. Consequently, the effectiveness of the improved RSO is confirmed with a better dynamic performance under the nonlinear load operation. • Experiment Under Combination Load In this situation, the BDFIG is loaded with both the unbalanced and nonlinear loads in which the unbalanced resistive load has a combination of 6, 12 and 12 

6 Rotor Position and Speed Observers of BDFIGs

PW voltage, CW current (p. u.)

PW voltage, CW current (p. u.)

282 2

P W v o lta ge C W c u rre nt

1 0 -1 -2

0

2

4

6

8

2

12

14

16

18

P W v o lta ge C W c u rre nt

1 0 -1 -2 0 .5

0 .6

0 .7

1000

Rotor speed (rpm)

10 T im e (s ) (a )

T im e (s ) (b )

0 .8

0 .9

1

900 800 700

O b s e rv e d b y b a s ic R S O M e a s u re d b y e nc o d e r

600

Rotor speed error (rpm)

0

2

4

6

8

50

Rotor speed (rpm)

12

14

16

18

R o to r s p e e d e rro r o f b a s ic R S O 1 2 rp m

0

-5 0

0

2

4

6

8

1000

10 T im e (s ) (d )

12

14

16

18

16

18

900 800 700

M e a s u re d b y e nc o d e r O b s e rv e d b y im p ro v e d R S O

600 0

Rotor speed error (rpm)

10 T im e (s ) (c )

2

4

6

8

50

10 T im e (s ) (e )

12

14

R o to r s p e e d e rro r o f im p ro v e d R S O 3 rp m

0

-5 0

0

2

4

6

8

10 T im e (s ) (f)

12

14

16

18

Fig. 6.18 Experimental results of basic and improved RSOs the under the unbalanced resistive load (25, 100, and 100  in phases a, b, and c): a PW voltage (1 p.u. = 500 V) and CW current (1 p.u. = 50 A), b expanded view of a, c rotor speed estimated by the basic RSO, d speed error using the basic RSO, e rotor speed estimated by the improved RSO, f speed error using the improved RSO

Rotor speed (rpm)

PW voltage, CW current (p. u.)

PW voltage, CW current (p. u.)

6.4 Improved Rotor Speed Observer for Standalone BDFIG … 2

P W v o lta ge

Rotor speed error (rpm) Rotor speed (rpm)

C W c u rre nt

1 0 -1 -2

0

2

4

6

8 10 T im e (s ) (a )

2

12

14

P W v o lta ge

1

16

18

C W c u rre nt

0 -1 -2 2 .2

2 .3

2 .4

T im e (s ) (b )

2 .5

2 .6

2 .7

900 800 700

O b s e rv e d b y b a s ic R S O M e a s u re d b y e nc o d e r

600 0

2

4

6

50

8 10 T im e (s ) (c )

12

14

16

18

R o to r s p e e d e rro r o f b a s ic R S O 2 0 rp m

0

-5 0

0

2

4

6

8 10 T im e (s ) (d )

12

14

16

18

16

18

900 800 700

M e a s u re d b y e nc o d e r O b s e rv e d b y im p ro v e d R S O

600 0

Rotor speed error (rpm)

283

2

4

6

50

8 10 T im e (s ) (e )

12

14

R o to r s p e e d e rro r o f im p ro v e d R S O 3 rp m

0

-5 0

0

2

4

6

8 10 T im e (s ) (f)

12

14

16

18

Fig. 6.19 Experimental results of basic and improved RSOs under the nonlinear load (a dioderectifier with a 25  resistor at the dc side): a PW voltage (1 p.u. = 500 V) and CW current (1 p.u. = 50 A), b expanded view of a, c rotor speed estimated by the basic RSO, d speed error using the basic RSO, e rotor speed estimated by the improved RSO, f speed error using the improved RSO

6 Rotor Position and Speed Observers of BDFIGs

PW voltage, CW current (p. u.)

PW voltage, CW current (p. u.)

284 2

CW current

PW voltage

1 0 -1 -2

0

2

1

2

3

5

6

CW current

7

8

PW voltage

1 0 -1 -2 3 .5

3 .5 2

3 .5 4

3 .5 6

3 .5 8

1000

Rotor speed (rpm)

4 T im e (s ) (a )

3 .6 3 .6 2 T im e (s ) (b )

3 .6 4

3 .6 6

3 .6 8

3 .7

900 800 700 600

Rotor speed error (rpm)

500

O b s e rv e d b y b a s ic R S O 0

1

2

3

50

Rotor speed (rpm)

M e a s u re d b y e nc o d e r 5

6

7

8

R o to r s p e e d e rro r o f b a s ic R S O

35 rpm

0

-5 0

0

1

0

1

2

3

1000

4 T im e (s ) (d )

5

6

7

8

900 800 700 600 500

Rotor speed error (rpm)

4 T im e (s ) (c )

M e a s u re d b y e nc o d e r 2

3

50

4 T im e (s ) (e )

O b s e rv e d b y im p ro v e d R S O 5

6

7

8

R o to r s p e e d e rro r o f im p ro v e d R S O

7 rpm

0

-5 0

0

1

2

3

4 T im e (s ) (f)

5

6

7

8

Fig. 6.20 Experimental results of basic and improved RSOs under both the unbalanced and nonlinear loads: a PW voltage (1 p.u. = 500 V) and CW current (1 p.u. = 50 A), b expanded view of a, c rotor speed estimated by the basic RSO, d speed error using the basic RSO, e rotor speed estimated by the improved RSO, f speed error using the improved RSO

6.4 Improved Rotor Speed Observer for Standalone BDFIG …

285

Fig. 6.21 The expanded view and the harmonic spectrum of the estimated rotor speed under the unbalanced load: a Expanded view of the rotor speed observed by basic RSO, b harmonic spectrum of the rotor speed observed by basic RSO, c expanded view of the rotor speed observed by improved RSO, d harmonic spectrum of the rotor speed observed by improved RSO

in three phases and the nonlinear load contains a diode-rectifier with a resistor of 50 . In addition, the speed is reduced from 894 to 630 rpm within the time period of 0.5–1.6 s and then again increased to 894 rpm between 5.5 and 6.97 s as illustrated in Fig. 6.20, with a high rate of change rather than the previous experimental cases in this subsection. It is obvious from Fig. 6.21a, b that a distortion current of the CW side is resulted due to the distorted PW generated voltage with 11.6% UF and 8.6% THD. Furthermore, it can be concluded from Fig. 6.21c, d that the estimated speed of the basic RSO suffers from an oscillation of about 35 rpm, which can be decreased to be about 7 rpm using the improved RSO. This would assure the capability of the improved RSO to effectively detect the rotor speed of the BDFIG under different loading conditions. All the obtained experimental results assure the observability of the improved RSO with a good dynamic performance, which would validate its effectiveness for the sensorless control system of BDFIG.

6.4.3.2

Steady-State Harmonic Analysis

The harmonic spectrum of the detected rotor speed and its expanded view, illustrated in Fig. 6.19 within 3–3.1 s, based on both the improved RSO and the basic one under the unbalanced load condition, are given in Fig. 6.21. It is clear that the oscillation issue appeared using the basic RSO is significantly eliminated using the improved RSO. In addition, according to the harmonic spectrum in Fig. 6.21, the rotor speed estimated by the basic RSO suffers from a twice-frequency (with respect to the frequency of PW voltage) harmonic component with approximately 3 rpm amplitude

286

6 Rotor Position and Speed Observers of BDFIGs

Fig. 6.22 The expanded view and the harmonic spectrum of the estimated rotor speed under the nonlinear load: a Expanded view of the rotor speed observed by basic RSO, b harmonic spectrum of the rotor speed observed by basic RSO, c expanded view of the rotor speed observed by improved RSO, d harmonic spectrum of the rotor speed observed by improved RSO

(0.47% of the fundamental component). The analysis confirms the effectiveness of the improved RSO with a significant reduction of the harmonics to be about 0.25 rpm (approximately 0.04%). Under the operation of the nonlinear load, the harmonic spectrum of the detected rotor speed and its expanded view, illustrated in Fig. 6.19 with the period from 3.7 to 3.8 s, based on both the two observers are given in Fig. 6.22. As clearly illustrated in Fig. 6.22, the harmonic component in the speed response is resulted with two frequencies 6ω1 and 12ω1 , using the basic RSO which have amplitudes of 0.7% and 0.25%, respectively. Furthermore, this harmonic content is significantly reduced to be about 0.07% and 0.02%, respectively, as given in Fig. 6.23c, d. With the third case under both loading of the unbalanced and nonlinear loads, the harmonic spectrum of the speed response and its expanded view, given in Fig. 6.20 with the period from 4.5 to 4.6 s, based on both the two observers are given in Fig. 6.23. As clearly illustrated in Fig. 6.23, the harmonic component in the speed response using the basic RSO is resulted with two frequencies 2ω1 generated by the unbalanced load and 6ω1 caused by the nonlinear load, which have amplitudes of 5.5% and 3.8%, respectively. It is obvious that under the nonlinear load in this operating case, the harmonic content with 12ω1 frequency is not appeared. This is due to the few harmonic voltage with 7th order of PW side generated in this experiment by the nonlinear load. Moreover, the harmonic contents are significantly reduced using the improved RSO to be about 0.28% and 0.19%, respectively.

6.5 Summary

287

Fig. 6.23 The expanded view and the harmonic spectrum of the estimated rotor speed under both unbalanced and nonlinear loads: a Expanded view of the rotor speed observed by basic RSO, b harmonic spectrum of the rotor speed observed by basic RSO, c expanded view of the rotor speed observed by improved RSO, d harmonic spectrum of the rotor speed observed by improved RSO

6.5 Summary This chapter investigates the rotor position and speed observers for sensorless control of standalone BDFIGs. The rotor position and speed estimation approaches for the BDFG mainly include the direct estimation approach and the closed-loop estimation approach. Firstly, two rotor position observers with the direct estimation principle are developed, which are based on the phase-axis relationship and space-vector model of the BDFIG, respectively. And then, the improved rotor speed observer with the closed-loop estimation principle is presented, which is based on PW voltage and CW current (the necessary sampled signals for the standalone BDFIG). To eliminate the adverse impact of unbalanced and nonlinear loads on the RSO, second-order generalized integrators (SOGIs) and low-pass filters (LPFs) are introduced to prefilt the PW voltage and CW current, respectively. Comprehensive simulation and experimental results are obtained with a prototype 30-kVA BDFIG to demonstrate the validity of these presented rotor position and speed observers under different operation conditions.

References 1. Barati F, McMahon R, Shao S et al (2013) Generalized vector control for brushless doubly fed machines with nested-loop rotor. IEEE Trans Ind Electron 60(6):2477–2485 2. Ademi S, Jovanovi´c MG (2015) Vector control methods for brushless doubly fed reluctance machines. IEEE Trans Ind Electron 62(1):96–104 3. Ademi S, Jovanovi´c MG, Hasan M (2015) Control of brushless doubly-fed reluctance generators for wind energy conversion systems. IEEE Trans Energy Convers 30(2):596–604

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4. Mousa MG, Allam SM, Rashad EM (2018) Maximum power extraction under different vectorcontrol schemes and grid-synchronization strategy of a wind-driven brushless doubly-fed reluctance generator. ISA Trans 72:287–297 5. Poza J, Oyarbide E, Sarasola I et al (2009) Vector control design and experimental evaluation for the brushless doubly fed machine. IET Elecrt Power Appl 3(4):247–256 6. Shao S, Abdi E, Barati F et al (2009) Stator-flux-oriented vector control for brushless doubly-fed induction generator. IEEE Trans Ind Electron 56(10):4220–4228 7. Xu W, Hussien MG, Liu Y et al (2020) Sensorless voltage control schemes for brushless doubly-fed induction generators in stand-alone and grid-connected applications. IEEE Trans Energy Convers 35(4):1781–1795 8. Hussien MG, Liu Y, Xu W (2019) Robust position observer for sensorless direct voltage control of stand-alone ship shaft brushless doubly-fed induction generators. CES Trans Electr Mach Syst 3(4):363–376 9. Yang J, Tang WY, Zhang GG et al (2019) Sensorless control of brushless doubly fed induction machine using a control winding current MRAS observer. IEEE Trans Ind Electron 66(1):728– 738 10. Jovanovic M (2009) Sensored and sensorless speed control methods for brushless doubly fed reluctance motor. IET Electric Power Appl 3(6):503–513 11. Ademi S, Jovanovi´c MG, Chaal H et al (2016) A new sensorless speed control scheme for doubly fed reluctance generators. IEEE Trans Energy Convers 31(3):993–1001 12. Liu Y, Xu W, Long T et al (2017) A new rotor speed observer for stand-alone brushless doubly-fed induction generators. In: IEEE energy conversion congress and exposition (ECCE), Cincinnati, OH, USA, pp 5086–5092 13. Dong D, Xu W, Liu Y et al (2017) Sensorless control of stand-alone BDFIG with compensation from rotor speed observer. In: 2nd international conference on electrical and electronic engineering (ICEEE), Rajshahi, Bangladesh, pp 1–4 14. Liu Y, Xu W, Zhu J et al (2019) Sensorless control of standalone brushless doubly fed induction generator feeding unbalanced loads in a ship shaft power generation system. IEEE Trans Ind Electron 66(1):739–749 15. Liu Y, Xu W, Long T et al (2020) An improved rotor speed observer for standalone brushless doubly-fed induction generator under unbalanced and nonlinear loads. IEEE Trans Power Electron 35(1):775–788 16. Ellis G (2004) Control system design guide, 3rd edn. Elsevier Academic Press, USA 17. Rodriguez P, Teodorescu R, Candela I et al (2016) New positive-sequence voltage detector for grid synchronization of power converters under faulty grid conditions. In: Proceedings of the 37th IEEE power electronics specialists conference, pp 1–7

Chapter 7

Model Reference Adaptive System Based Sensorless Control for BDFIGs

Abstract In this chapter, three kinds of model reference adaptive system (MRAS) based sensorless control strategies are presented for standalone BDFIGs. These control strategies are based on the CW power factor, PW flux and CW flux, respectively. Among the MRAS sensorless control strategies based on PW flux and CW flux, the ones based on the dq-axis flux have the better dynamic performance than those based on the αβ-axis flux, due to the absence of integrators in the former. In order to verify the effectiveness of the proposed control strategy, some simulation results are discussed in this chapter, which indicate that the estimated rotor speed can track the reference value at steady state with a good transient response, and the PW voltage amplitude and frequency are kept constant under the rotor speed change, load change, machine parameter change. Moreover, this chapter presents comprehensive experimental results to verify the proposed MRAS sensorless control strategies under the load change, speed change and machine parameter variation. Keywords Model reference adaptive system (MRAS) · Speed observer · Sensorless control · CW power factor · PW flux · CW flux

7.1 Introduction Because of the use of slip ring and brushes, the traditional doubly-fed induction generator (DFIG) suffers from various drawbacks, such as large size, low reliability, and high cost required for maintenance. On the other hand, the BDFIG has attracted a lot of attention due to its low maintenance cost and high reliability structure. The PW and CW with different numbers of pole-pairs to prevent the direct coupling between them are housed in the same stator of the BDFIG, and the rotor is specially designed to offer the cross-coupling between PW and CW [1–3]. The BDFIG can avoid all drawbacks of DFIG as it does not require any brushes and slip rings. The standalone BDFIG as a power generator has been widely used in variablespeed constant-frequency (VSCF) generation systems, e.g., ship shaft power generation applications [4]. The BDFIG as a standalone system should be controlled to keep the PW voltage constant to support the loads under variable rotor speeds [5]. In general, the rotor speed or position is detected by the encoder. However, the delicate © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Y. Liu and W. Xu, Advances in Control Technologies for Brushless Doubly-fed Induction Generators, https://doi.org/10.1007/978-981-19-0424-0_7

289

290

7 Model Reference Adaptive System …

Fig. 7.1 Basic MRAS observer structure

encoders are especially prone to damage due to the vibrations caused by the main energy source like a combustion engine or a wind turbine. Therefore, the sensorless control without using any physical speed and position sensors is with a very strong merit. In order to eliminate the use of the delicate speed or position sensor and to improve the reliability, a speed or position estimator can be used. The estimation approaches of the machine speed and position mainly include the direct estimation approach and the closed-loop estimation approach. The direct calculation approaches for BDFIG rotor position have been investigated in [6–8]. The closed-loop estimation means that the estimated rotor speed or position needs to be feed backed to the speed observer for improving the estimation accuracy. Generally, this kind of approach is with the higher estimation accuracy than the direct estimation approach. Due to its excellent performance, computational simplicity, and straightforward stability approach, the model reference adaptive system (MRAS) is a more and more popular strategy of sensorless control schemes [9, 10]. An MRAS observer is based on two models, i.e., a reference model and an adaptive model (see Fig. 7.1). The estimated speed is used to adjust the adaptive model, driving the error to zero, where the error is usually defined as the cross product between the reference and adaptive model. The error is used as the input of a PI controller, and the output as the estimation of the rotational speed is used to correct the adaptive model. The MRAS speed observers have been applied to DFIGs [11–16]. However, very few literatures involve with this issue for BDFIGs. In this chapter, three kinds of MRAS based sensorless control strategies are presented for standalone BDFIGs. Specifically, five MRAS speed observers are proposed, which are based on CW power factor, PW flux in both the stationary αβ frame and the rotary dq frame, CW flux in both the the stationary αβ frame and the rotary dq frame, respectively [17, 18].

7.2 MRAS Sensorless Control Based on CW Power Factor

291

7.2 MRAS Sensorless Control Based on CW Power Factor 7.2.1 PW Field-Oriented Control for Sensorless Voltage Control of BDFIGs From the view point of the vector control target, the controllable vector quantities appear as dc components based on the selection of ω1 and ω2 rotating frames. Figure 7.2 illustrates the relationship of the BDFIG phase-axis. It is obvious from Fig. 7.2 that the alignment of the reference frame d-axis with its total vector is employed to obtain the intended orientation of PW fluxe. Figure 7.3 shows the main block diagram of the proposed sensorless field-oriented vector control strategy for the standalone BDFIG. For getting better drive performance, the current-orientation method is applied for current control of the CW. To obtain this target, the reference q-axis CW current, i2q * , is adjusted to be equal zero. Hence, the set value of the d-axis CW current, i2d * , is directed to the desired current magnitude of CW, I 2 * . By measuring the PW three-phase voltages and currents, the dq-axis PW voltages and currents can be obtained using the abc/dq transformation with the intended frame angle, θ 1 , which will be obtained based on the presented PW field-orientation loop. Then, without any voltage-integration, the corresponding dq-axis PW flux can be easily calculated as    ψ1d = u 1q − R1 i 1q /ω1 ψ1q = (−u 1d + R1 i 1d )/ω1 Fig. 7.2 Relationship of the BDFIG phase-axis for the sensorless control strategy

(7.1)

292

7 Model Reference Adaptive System …

Fig. 7.3 Block diagram of the proposed sensorless control strategy for the adopted BDFIG system

The PW d-axis flux, ψ 1d , tracks the reference total PW flux, Ψ 1 * , which can be adjusted according to the desired PW voltage magnitude and frequency (Ψ 1 * = U 1 * /ω1 * ), as illustrated in Fig. 7.3. This can be attained through the adjustment of the reference CW current magnitude, i2d * , aided with the presented current-orientation of CW side by setting (i2q * = 0). Furthermore, the PW q-axis flux, ψ 1q , is set to zero by regulating the intended frame frequency of PW flux, ω1 . Moreover, the corresponding angle, θ 1 , is obtained to realize the desired orientation target of the PW flux through the alignment of the reference frame d-axis with the flux vector of the PW side. Aided with the estimated rotor position, θr _ est , which will be obtained in the next section based on the proposed new CW power factor MRAS observer, the corresponding angle θ 2 * can be realized directly using the angle θ 1 * calculated from the integration of the reference PW frequency, as shown in Fig. 7.3. Finally, the required pulses of the machine-side converter, shown in Fig. 7.3, can be attained based on the presented vector-control scheme of CW current to achieve the required flux orientation of PW side along with considering the intended set values of PW voltage.

7.2.2 Design Procedure of the Proposed Control Method The main design concept of the proposed rotor position observer based on CW power factor MRAS will be discussed in details as follows. The instantaneous active power of BDFIG CW-side is given by

7.2 MRAS Sensorless Control Based on CW Power Factor

P2 =

 3 u 2d i 2d + u 2q i 2q . 2

293

(7.2)

Based on the dq-axis model of BDFIG, the active power of BDFIG CW-side can be expressed as   3 L 2r L 1 2 + Pˆ2 = R2 i 2d (ω1 − ωr )i 1q i 2d . 2 L 1r

(7.3)

The instantaneous reactive power of BDFIG CW-side is expressed as Q2 =

 3 u 2q i 2d − u 2d i 2q . 2

(7.4)

Aided with the dynamic model of BDFIG, the expression of the BDFIG CW reactive-power can be rewritten as   3 L 2r L 1 L 2r 2 ˆ ψ1d i 2d − i 1d i 2d . Q 2 = (ω1 − ωr ) L 2 i 2d + 2 L 1r L 1r

(7.5)

The computation of the proposed new MRAS observer based on CW power factor as a functional quantity can be summarized as follows. The detailed structure of the proposed observer based on CW power factor MRAS is illustrated in Fig. 7.4. The power factor of BDFIG CW-side is obtained as (the reference model)

Fig. 7.4 Main block diagram of the adopted CW power factor MRAS observer

294

7 Model Reference Adaptive System …

p f2 =

P2 P22 + Q 22

.

(7.6)

Using the dynamic model of BDFIG, the expression of the BDFIG CW powerfactor can be expressed as (the adaptive model)

pf 2 =  1+

where 2 A1 = L 2 i 2d +

L 2r L 1r

ψ1d i 2d −



L 1 L 2r i i , L 1r 1d 2d

1 (ω1 −ωr )A1 A2 +(ω1 −ωr )A3

(7.7)

2

2 A2 = R2 i 2d , A3 =

L 2r L 1 i i . L 1r 1q 2d

7.2.3 Simulation Results In this subsection, the simulation based on a 30-kVA BDFIG is implemented to verify the CW power factor based MRAS control strategy. The detailed parameters of the 30-kVA BDFIG is mentioned in Section A.3, Appendix. Figure 7.5a–d denote the actual and estimated rotor positions, PW three-phase voltage and PW frequency, dq-axis PW flux, and PW and CW three-phase currents, respectively The operation of the adopted BDFIG is started under the synchronous speed with 750 rpm and then decreased to the low-speed condition (sub-synchronous mode) with 600 rpm at t = 1 s, while the applied resistive load is 11.6 kW. Finally, the generator speed accelerates to the synchronous mode at t = 2 s with 750 rpm and then increased to the high-speed condition (super-synchronous mode) with 900 rpm at t = 3 s, while the load is the same. On the other hand, Fig. 7.6 illustrates the dynamic performance of the proposed sensorless control system under load change condition to be decreased from 11.6 to 9.7 kW with the same speed of 600 rpm at t = 1.5 s. It is observed from Figs. 7.5b and 7.6b that the PW voltage (in terms of magnitude and frequency) successfully tracks the intended quantities for DVC purpose (U 1 * = 311 V and f 1 * = 50 Hz). In addition, it is obvious from Figs. 7.5c and 7.6c that the PW dq-axis flux linkage components are kept constant to attain the desired quantities [(ψ 1d * = Ψ 1 * = U 1 * /ω1 * ) and (ψ 1q * = 0)] adjusted for the intended orientation of the PW flux. Moreover, the estimated rotor position is verified as in a good accordance with the actual value, as shown in Figs. 7.5a and 7.6a. This ensures the capability of the proposed MRAS observer based on CW power factor to effectively estimate the rotor position of the adopted BDFIG. Meanwhle, the efficacy of the proposed new MRAS observer has been fully investigated for sensorless DVC of standalone BDFIG in ship shaft applications under different operating conditions of speed and load variations.

7.2 MRAS Sensorless Control Based on CW Power Factor

295

(a)

(b)

(c) Fig. 7.5 Performance test of the proposed sensorless control strategy based on CW power factor MRAS observer: a Actual and estimated rotor positions. b PW three-phase voltage and PW frequency. c PW dq-axis flux. d Three-phase PW and CW currents

296

7 Model Reference Adaptive System …

(d) Fig. 7.5 (continued)

(a)

(b) Fig. 7.6 The proposed sensorless system based on the CW power factor MRAS observer under load change: a Actual and estimated rotor positions. b PW three-phase voltage and PW frequency. c PW dq-axis flux. d Three-phase PW and CW currents

7.2 MRAS Sensorless Control Based on CW Power Factor

297

(c)

(d) Fig. 7.6 (continued)

The change effect of CW resistance on the whole calculations of the proposed observer is also studied under the case of 1.3 R2 , as shown in Fig. 7.7, where Fig. 7.7a–d denote the actual and estimated rotor position, PW three-phase voltages and PW frequency, dq-axis PW flux linkage components, and PW and CW three-phase currents, respectively. It can be concluded from Figs. 7.5, 7.6 and 7.7 that the effectiveness of the proposed rotor position observer and its capability for sensorless DVC strategy are not affected by any change in the CW resistance. This confirms and proves the functionality and robustness of the new rotor position MRAS observer based on CW power factor for sensorless DVC of the adopted ship shaft standalone BDFIG system. All the given results demonstrate the simplicity and capability of the proposed sensorless control system to track the desired values of the PW voltage for the promising BDFIG in standalone applications.

298

7 Model Reference Adaptive System …

(a)

(b)

(c) Fig. 7.7 The proposed sensorless system based on the CW power factor MRAS observer under the increased CW resistance (1.3 R2 ): a Actual and estimated rotor positions. b PW three-phase voltage and PW frequency. c PW dq-axis flux. d Three-phase PW and CW currents

7.2 MRAS Sensorless Control Based on CW Power Factor

299

(d) Fig. 7.7 (continued)

7.2.4 Experimental Results In this section, experimental results on a 30-kVA BDFIG platform are presented to validate the proposed rotor position observer under the start-up operation of the standalone BDFIG. In addition, the capability of the proposed observer is also confirmed under the whole operation period of the presented generating system through starting, speed variation, and load change conditions. The details of the adopted 30-kVA BDFIG experimental platform can be seen in Section A.3, Appendix. Firstly, the BDFIG is started at a mechanical rotor speed of 600 rpm and with a balanced three-phase resistive load of 25  per phase. Then, at 14 s, the generator speed is increased to 700 rpm until 28 s, at which the speed is again reduced to 600 rpm. On the other hand, in order to confirm the effect of load change, the generator is started at 600 rpm with the balanced three-phase resistive load and then the balanced load is suddenly increased to its double power in each phase at 4.95 s. Figures 7.8 and 7.9 show the experimental results with the proposed rotor position observer under the start-up operation, the speed variation and load change conditions, respectively. It is observed from Figs. 7.8 and 7.9 that the PW voltage tracks successfully the desired values for DVC under the whole operation period. Moreover, the results verify that the estimated rotor position is in a good accordance with the actual value as shown in Figs. 7.8 and 7.9 for both the starting condition speed variation state and load change condition, respectively. This ensures the effectiveness of the proposed new rotor position observer for sensorless DVC of the adopted standalone BDFIG system. To verify the robustness of the proposed rotor position observer under the case of parameter variation, the experiments are carried out with 130% uncertainties of the whole parameters that affect the calculations of the suggested algorithm. Figure 7.10a–d denote the actual and estimated rotor position, the PW phase voltage, the PW phase current, and the CW phase current, respectively. It is obvious from Fig. 7.10 that the validity of the proposed rotor position observer and its capability

300

7 Model Reference Adaptive System …

Fig. 7.8 Experimental results of the CW power factor-based rotor position observer under the startup operation and speed variation: a Actual and estimated rotor positions at 600 rpm. b Actual and estimated rotor positions at 700 rpm. c PW phase voltage. d CW phase current at 600 rpm. e CW phase current at 700 rpm. f PW phase current

7.2 MRAS Sensorless Control Based on CW Power Factor

301

(d)

(e)

(f)

Fig. 7.8 (continued)

for sensorless DVC are not affected by the uncertainty issue of the adopted BDFIG parameter. All presented experimental results ensure the effectiveness and capability of the proposed sensorless DVC strategy based a suggested new rotor position observer for the promising BDFIG in standalone applications.

302

7 Model Reference Adaptive System …

(a)

(b)

(c) Fig. 7.9 Experimental results of the CW power factor-based rotor position observer under the load change condition: a Actual and estimated rotor positions. b PW phase voltage. c PW phase current. d CW phase current

7.3 MRAS Sensorless Control Based on PW Flux

303

(d) Fig. 7.9 (continued)

7.3 MRAS Sensorless Control Based on PW Flux Referring to Fig. 7.11, it can be seen that two models, i.e. the reference and the adaptive models, are introduced to the proposed MRAS observer based on αβ–axis PW flux. Moreover, to acquire the error, the measured αβ-axis PW flux obtained from the αβ-axis currents and voltages of PW is compared to that obtained from the αβ-axis PW and CW currents. Then, the error is minimized utilizing a PI controller. Furthermore, the output signal of the controller is used to estimate the rotor speed. Then, this estimation of the rotational velocity is fed back to regulate the adaptive model. With reference to Fig. 7.12, it is possible to see the complete control structure of the proposed sensorless control strategy for the standalone BDFIG system based on αβ-axis PW flux MRAS observer.

7.3.1 Control Scheme Based on αβ-Axis PW Flux From the PW voltage Eq. (2.4) at ωa = 0, the reference model design can be obtained as  α  α ψ1 = u 1 − R1 i 1α  β (7.8) β β . ψ1 = u 1 − R1 i 1 Substituting the RW flux Eq. (2.5) to RW voltage Eq. (2.4) and setting ωa = 0, the adaptive model for the BDFIG can be got by αβ αβ αβ αβ αβ αβ αβ 0 = Rr ir + dtd (L r ir + L 2r i 2 + L 1r i 1 ) − ( j p1 ωr )(L r ir + L 2r i 2 + L 1r i 1 ). In order to simplify the derivation of rotor current, generally, s is employed to represent the differential operator d/dt. Hence, the rotor current can be obtained by

304

7 Model Reference Adaptive System …

irαβ =

αβ

αβ

−(s − j p1 ωr )(L 2r i 2 + L 1r i 1 ) . Rr + (s − j p1 ωr )L r

(7.9)

From the PW flux Eq. (2.5) at ωa = 0, it can further get αβ

i1 =

1 αβ L 1r αβ ψ − i . L1 1 L1 r

(7.10)

In order to employ a fractionally rated power converter to drive the standalone BDFIG system, the CW angular frequency ω2 is usually limited to 30% of the PW frequency. According to (2.8), the range of ωr would be between 70 and 130% of ω1 . In terms of the BDFIG with the typical pole pairs of 1 and 3 for PW and CW, the value of (ω1 − p1 ωr ) would be in the range of (67.5–82.5%)ω1 . In general, the value

(a)

(b) Fig. 7.10 Experimental results of the CW power factor-based rotor position observer under the case of BDFIG parameter change (130% uncertainty): a Actual and estimated rotor positions. b PW phase voltage. c PW phase current. d CW phase current

7.3 MRAS Sensorless Control Based on PW Flux

(c)

(d) Fig. 7.10 (continued)

Fig. 7.11 Structure of the αβ-axis PW flux MRAS observer

305

306

7 Model Reference Adaptive System …

Fig. 7.12 Structure of the proposed sensorless control method based on the αβ-axis PW flux MRAS observer for the standalone BDFIG system

of ω1 is 100π or 120π rad/s. Hence, the value of (ω1 − p1 ωr ) will be much greater than Rr , and then it can be concluded that the term Rr /[s + j (ω1 − p1 ωr )] can be approximately ignored at steady state. Similarly, Rr /(s − j p1 ωr ) can be ignored, Then, from (7.9), the rotor current can be expressed as irαβ =

αβ

αβ

−(L 2r i 2 + L 1r i 1 ) . Lr

(7.11)

Substituting (7.10) to (7.11), the rotor current can be rewritten as irαβ =

αβ

αβ

L 1 L 2r i 2 + L 1r ψ1 . (L 21r − L 1 L r )

(7.12)

Substituting (7.12) to PW flux Eq. (2.5) at ωa = 0, it can be derived that αβ

αβ

ψ1 = L 1 i 1 +

αβ

αβ

L 1 L 1r L 2r i 2 + L 21r ψ1 . (L 21r − L 1 L r ) αβ

(L 21r − L 1 L r ) − L 21r αβ L 1 i 1 (L 21r − L 1 L r ) + L 1 L 1r L 2r αβ ψ1 = i2 . 2 L 1r − L 1 L r L 21r − L 1 L r αβ

αβ

−L 1 L r L 1 (L 21r − L 1 L r )i 1 + (L 1 L 1r L 2r )i 2 αβ ψ1 = . 2 (L 1r − L 1 L r ) (L 21r − L 1 L r )

7.3 MRAS Sensorless Control Based on PW Flux αβ

ψ1 =

307

L 1 (L 21r − L 1 L r ) αβ (L 1 L 1r L 2r ) αβ i1 + i . −(L 1 L r ) −(L 1 L r ) 2

Finally, the PW flux can be expressed as αβ

ψ1 =

−L 21r + L 1 L r αβ L 1r L 2r αβ i1 − i . Lr Lr 2

(7.13)

It should be noted that the α- and β-axis CW currents in (7.13) are in the CW reference frame (αβ cw ), which have to be transformed to the quantites in the PW reference frame (αβ). The transformation from the CW reference frame (αβ cw ) to the PW reference frame (αβ) is given as [19] αβ

˜

˜

αβcw con j

X 2 = e j[ p1 (θr +δ1 )+ p2 (θr +δ2 )] (−X 2

)

.

(7.14)

Substituting (7.14) to (7.13), the estimation of the PW flux in the αβ reference frame can be expressed as αβ ψ˜ 1 =

−L 21r + L 1 L r αβ L 1r L 2r αβ i1 − i Lr Lr 2

(7.15)

Then, the rotor speed can be estimated by ω˜ r = (K p +

Ki β β )(ψ˜ 1α ψ1 − ψ1α ψ˜ 1 ). s

(7.16)

7.3.2 Control Scheme Based on dq-Axis PW Flux With regards to Fig. 7.13, the proposed MRAS observer, based on dq-axis PW flux, is formed of two models, the adaptive and the reference models. Furthermore, a comparison is made to find the error between the measured stator dq-axis PW flux taken from the voltages and currents of the PW to that of the dq-axis PW flux acquired from the PW and CW stator currents. Moreover, the error is nullified to zero through the use of a PI controller. Additionally, the output of the PI control scheme is adopted as the estimation of the rotational speed. Next, this estimate is fed back to update the adaptive model. Figure 7.14 presents the full control system of the proposed sensorless control method for the standalone BDFIG system utilizing the dq-axis PW flux MRAS observer. From the PW voltage Eq. (2.4) at ωa = ω1 and at steady state, a reference model design can be obtained as

308

7 Model Reference Adaptive System …

Fig. 7.13 Structure of the dq-axis PW flux MRAS observer

Fig. 7.14 Structure of the proposed sensorless control method based dq-axis PW flux MRAS observer for the standalone BDFIG system



u −R i

ψ1d = 1q ω1 1 1q . 1 i 1d ψ1q = − u 1d −R ω1

(7.17)

The adaptive model can be derived by the following procedure. Substituting the RW flux equation of (2.5) to RW voltage equation of (2.4) and setting ωa = ω1 , the rotor current can be derived by ir =

−[s + j (ω1 − p1 ωr )](L 2r i 2 + L 1r i 1 ) . Rr + [s + j (ω1 − p1 ωr )]L r

(7.18)

7.3 MRAS Sensorless Control Based on PW Flux

309

From the equation of the PW flux (2.5) at ωa = ω1 , it will get i1 =

1 L 1r ψ1 − ir . L1 L1

(7.19)

Substituting (7.19) to (7.18), and combining (7.11), the rotor current can be derived carefully as follows:

ir =

−[s + j (ω1 − p1 ωr )] L 2r i 2 +

L 1r L1

ψ1 −

L 21r i L1 r

Rr + [s + j (ω1 − p1 ωr )]L r

,

L2

ir −

ir =

ir =

[s + j (ω1 − p1 ωr )]( L1r1 )

Rr + [s + j (ω1 − p1 ωr )]L r −[s + j (ω1 − p1 ωr )](L 2r i 2 + LL1r1 ψ1 )

, R + [s + j (ω1 − p1 ωr )]L r  r  L2 ir [Rr + [s + j (ω1 − p1 ωr )]L r ] − [s + j (ω1 − p1 ωr )] L1r1

Rr + [s + j (ω1 − p1 ωr )]L r

−[s + j (ω1 − p1 ωr )] L 2r i 2 + LL1r1 ψ1 = , Rr + [s + j (ω1 − p1 ωr )]L r

−[s + j (ω1 − p1 ωr )] L 2r i 2 + LL1r1 ψ1

, L2 Rr + [s + j (ω1 − p1 ωr )]L r − [s + j (ω1 − p1 ωr )] L1r1

− L 2r i 2 + LL1r1 ψ1 L 1 L 2r i 2 + L 1r ψ1 ir = . 2 , ir = −L 1 Rr L Rr + (L 21r − L 1 L r ) + L r − L1r1 s+ j (ω1 − p1 ωr ) s+ j (ω1 − p1 ωr ) Finally, the RW current can be expressed as ir =

(L 1 L 2r i 2 + L 1r ψ1 ) . (L 21r − L 1 L r )

(7.20)

Substituting (7.20) to the equation of the PW flux (2.5) at ωa = ω1 ,it can be obtained that ψ1 = L 1 i 1 + L 1r

(L 1 L 2r i 2 + L 1r ψ1 ) . (L 21r − L 1 L r )

And then, the PW flux can be derived as ψ1 =

(L 1 L r − L 21r )i 1 − L 1r L 2r i 2 . Lr

(7.21)

310

7 Model Reference Adaptive System …

The adaptive model of the estimation on the PW flux in dq reference frame is dq ψ˜ 1 =

(L 1 L r − L 21r )i 1 − L 1r L 2r i 2 . Lr

(7.22)

The rotor speed can be observed by ω˜ r = (K p +

Ki )(ψ˜ 1d ψ1q − ψ1d ψ˜ 1q ). s

(7.23)

7.3.3 Simulation Results In order to confirm the capability of the proposed control strategies, some simulation results have been achieved on a 30-kVA prototype wound-rotor BDFIG, whose parameters are specified in Section A.3, Appendix.

7.3.3.1

Results with the αβ-Axis PW Flux Based MRAS Sensorless Control

Firstly, the αβ-axis PW flux based MRAS sensorless control strategy is verified under four different operation conditions, i.e., speed change, load change, PW resistance variation and inductance variation. Under Speed Change Figure 7.15 illustrates the response when the rotor speed decreases from 700 to 600 rpm between 1 and 2 s, and then is fixed at 600 rpm for the proposed αβ-axis PW flux MRAS observer under constant three-phase resistive load. Figure 7.16a–f illustrate the actual and estimated rotor speeds, the PW three-phase voltage, dq-axis PW voltage, PW three-phase currents, the CW three-phase currents, and the αβ-axis flux of the power winding, respectively. The control strategy can succeed in keeping the load voltage constant at the reference value as confirmed in Fig. 7.15b. The CW current under different speeds is shown in Fig. 7.15e. The frequency of CW current is changed with the variable speed, which is started with 3.33 Hz at the speed of 700 rpm and then changed to 10 Hz at the speed of 600 rpm, so that the PW frequency can be kept at its reference value 50 Hz as illustrated in Fig. 7.15e. This has fully indicated the effectiveness of the proposed sensorless control system under the speed variation. Under Load Change The generator is firstly operated at 600 rpm with the three-phase load of 25  per phase. Then, the terminal load is varied at 1 s from 25 to 50  per phase and the speed is kept constant at 600 rpm, as shown in Fig. 7.16. The actual and estimated

7.3 MRAS Sensorless Control Based on PW Flux

311

rotor speeds, three-phase PW voltages, detailed three-phase PW voltages, dq-axis PW voltages, three-phase PW currents, three-phase CW currents, and the αβ-axis PW flux are illustrated in Fig. 7.16a–g, respectively. The PW voltage is fixed to get the reference set value as shown in Fig. 7.16b. It can be illustrated from Fig. 7.16 that the fluctuation in the PW voltage amplitude is about 19 V. Also, the settling time is found as 0.03 s. This ensures the good transient performance of the control system under the load change condition. Meanwhile, it can be seen from Fig. 7.16c that the frequency of the PW voltage is maintained fixed at 50 Hz. Also, the frequency of CW current is not changed during the period of load change because the speed is kept constant so as to make the PW frequency approach to its reference value (50 Hz), as shown in Fig. 7.16f. Under PW Resistance Variation Figure 7.17 shows the response of BDFIG, under the same load change condition with constant speed 600 rpm, which starts with the three-phase load of 25  per phase and then varies at 1 s from 25 to 50  per phase. This is studied with 130% of PW resistance as for the PW resistance mismatch. The results in Fig. 7.17a–g illustrate the actual and estimated rotor speeds, three-phase PW voltages, detailed three-phase PW voltages between 0.9 and 1.1 s, dq-axis PW voltages, three-phase PW currents, three-phase CW currents, and the αβ-axis PW flux, respectively. The fluctuation in the PW voltage amplitude is around 21 V with the settling time of

(a)

(b) Fig. 7.15 Simulation results under the speed ramp change and the constant load with the αβ-axis PW flux based MRAS sensorless control: a Actual and estimated speeds. b PW three-phase voltages. c PW dq-axis voltage. d PW three-phase currents. e CW three-phase currents. f PW αβ-axis flux

312

7 Model Reference Adaptive System …

(c)

(d)

(e)

(f)

Fig. 7.15 (continued)

7.3 MRAS Sensorless Control Based on PW Flux

(a)

(b)

(c)

(d)

313

314

7 Model Reference Adaptive System …

Fig. 7.16 Simulation results under the load variation and the constant speed with the αβ-axis PW flux based MRAS sensorless control: a Actual and estimated speeds. b PW three-phase voltages. c Detailed PW three-phase voltages between 0.9 and 1.1 s. d PW dq-axis voltage. e PW three-phase currents. f CW three-phase currents. g PW αβ-axis flux

(e)

(f)

(g) Fig. 7.16 (continued)

7.3 MRAS Sensorless Control Based on PW Flux

315

0.032 s, as shown in Fig. 7.17c. In addition, the CW frequency is not changed under the constant speed operation to get a constant PW frequency. Under Inductance Variation Figure 7.18 shows the response of BDFIG, under the same load change condition, when the rotor speed is kept fixed at 600 rpm with 150% variation in the whole inductance values. The actual and estimated rotor speeds, the three-phase PW voltages, detailed PW three-phase voltages, dq-axis PW voltages, three-phase PW currents, three-phase CW currents, and αβ-axis PW fluxes are illustrated in Fig. 7.18a–g, respectively. The control method has the capability to maintain the PW voltage at its reference value with the fluctuation of 22 V and the settling time of 0.033 s. Also, according to the speed equation as shown in (2.13), to get the PW frequency fixed at its reference value, the CW frequency cannot be changed under the constant speed condition. It is seen from the aforementioned pictures that the proposed sensorless control method is not affected by any mismatch in the BDFIG parameters.

7.3.3.2

Results with the dq-Axis PW Flux Based MRAS Sensorless Control

And then, the dq-axis PW flux based MRAS sensorless control strategy is validated under the similar operation conditions with those in Sect. 7.3.3.1. Under Speed Change Figure 7.19 illustrates the response when the rotor speed decreases from 600 to 900 rpm between 1 and 2 s for the proposed dq PW-flux MRAS observer under constant three-phase resistive load. Figure 7.19a–e show the actual and estimated rotor speed, three-phase PW voltage, detailed three-phase PW voltage, three-phase CW currents, and three-phase PW currents, respectively. The reference voltage value is achieved in the PW side. Also, the CW frequency is changed with the variable speed, from 10 Hz at the sub-synchronous speed of 600 rpm to10 Hz at the super -synchronous speed of 900 rpm, in order to maintain the PW frequency fixed at its reference value (50 Hz). Under Load Change The generator is firstly operated at 600 rpm with the three-phase load of 50  per phase. Then, the terminal load is varied at 1 s from 50 to 25  per phase, where the speed is kept constant at 600 rpm, as shown in Fig. 7.20. The actual and estimated rotor speeds, the PW three-phase voltages, detailed PW three-phase voltages, PW three-phase currents, detailed PW three-phase currents, CW three-phase currents, and the dq-axis PW flux are illustrated in Fig. 7.20a–g, respectively. The PW voltage and its frequency are maintained fixed at the reference value under the case of load change, as shown in Fig. 7.20. The results show that the fluctuation in voltage is about 49 V. In addition, the settling time is found as 0.012 s, as illustrated in Fig. 7.20b.

316

7 Model Reference Adaptive System …

(a)

(b)

(c)

(d)

7.3 MRAS Sensorless Control Based on PW Flux

317

Fig. 7.17 Simulation results under 130% variation in the PW resistance with the αβ-axis PW flux based MRAS sensorless control: a Actual and estimated rotor speeds. b PW three-phase voltages. c Detailed view of PW three-phase voltages. d dq-axis PW voltage. e PW three-phase currents. f CW three-phase currents. g αβ-axis PW flux

(e)

(f)

(g) Fig. 7.17 (continued)

Under PW Resistance Variation Figure 7.21 shows the response of BDFIG, under the same load change condition as presented in Fig. 7.20 which starts with three-phase load of 50  per phase and then varied at 1 s from 50 to 25  per phase, where the speed is kept constant at 600 rpm with 130% mismatch in the PW resistance. The actual and estimated rotor speed, the three-phase PW voltages, the detailed three-phase PW voltages between 0.9 and

318

7 Model Reference Adaptive System …

(a)

(b)

(c)

(d)

7.3 MRAS Sensorless Control Based on PW Flux

319

Fig. 7.18 Simulation results under 150% variation in the whole inductance with the αβ-axis PW flux based MRAS sensorless control: a Actual and estimated rotor speeds. b PW three-phase voltages. c Detailed view of PW three-phase voltages. d dq-axis PW voltage. e PW three-phase currents. f CW three-phase currents. g αβ-axis PW flux

(e)

(f)

(g) Fig. 7.18 (continued)

320

7 Model Reference Adaptive System …

(a)

(b)

(c)

(d) Fig. 7.19 Simulation results under the speed ramp change and the constant load with the dq-axis PW flux based MRAS sensorless control: a Actual and estimated speeds. b PW three-phase voltages. c Detailed PW three-phase voltages. d CW three-phase currents. e PW three-phase currents

7.3 MRAS Sensorless Control Based on PW Flux

321

(e) Fig. 7.19 (continued)

1.1 s, the PW three-phase currents, the detailed PW three-phase currents between 0.9 and 1.1 s, the CW three-phase currents, and the dq-axis PW flux are shown in Fig. 7.21a–g, respectively. As illustrated in Fig. 7.21, the voltage of PW is kept fixed at its reference value under the load change condition with some fluctuation, and the settling time of 0.014 s. Under Inductance Variation Figure 7.22 shows the response of BDFIG, under the same load change condition when the rotor speed is kept constant at 600 rpm, as presented in Fig. 7.20, with 150% variation in the whole inductance values. The actual and estimated rotor speed, the three-phase PW voltages, the detailed three-phase PW voltages, the PW three-phase currents, the detailed PW three-phase currents, the CW three-phase currents, and the dq-axis PW flux are shown in Fig. 7.22a–g, respectively. It is illustrated from the results that the PW voltage is kept constant under the speed change condition with the settling time of 0.015 s and the amplitude fluctuation about 53 V. In addition, the CW frequency is not changed to keep the PW frequency at its reference value, as shown in Fig. 7.22. This ensures the effectiveness of the proposed control method. As shown by the presented results, the proposed sensorless control method is not affected by any mismatch in the BDFIG parameters.

7.3.4 Experimental Results The comprehensive experiments for the dq-axis PW flux based MRAS sensorless control strategy are carried out on a 30-kVA prototype wound-rotor BDFIG, whose parameters are specified in Section A.3, Appendix. Constant Load under Variable Speed Figure 7.23a shows the estimated speed by the proposed speed observer, which tracks the actual measured speed closely. The generator speed starts from 700 rpm under a three-phase load of 50  per phase, and then decreases at 5 s to 600 rpm, as shown in

322

7 Model Reference Adaptive System …

Fig. 7.23a. Finally, between 40 and 50 s, the speed is kept constant at 600 rpm with the same load condition. Furthermore, the error between the actual and estimated speeds is illustrated in Fig. 7.23b. The control strategy has successfully managed to regulate the load voltage at the reference RMS value of 150 V, as illustrated in Fig. 7.23c. The CW current response under various speeds is shown in Fig. 7.23d, and the estimated

(a)

(b)

(c) Fig. 7.20 Simulation results under the load variation and the constant speed with the dq-axis PW flux based MRAS sensorless control: a Actual and estimated speeds. b PW three-phase voltages. c Detailed PW three-phase voltages between 0.9 and 1.1 s. d PW three-phase currents. e Detailed PW three-phase currents between 0.9 and 1.1 s. f CW three-phase currents. g dq-axis PW flux

7.3 MRAS Sensorless Control Based on PW Flux

(d)

(e)

(f) Fig. 7.20 (continued)

323

324

7 Model Reference Adaptive System …

(g) Fig. 7.20 (continued)

speed can be confirmed by the frequency of CW current, as shown in Fig. 7.23e–f. The CW frequency and current are shown in Fig. 7.23d, and their extended views are shown in Fig. 7.23e–f. The proposed method can retain the PW frequency at its reference value (50 Hz) according to (2.13), as illustrated in Fig. 7.23c. The frequency of CW current is changed with the speed change, which is started with 3.33 Hz at the speed of 700 rpm and then changed to 10 Hz at the speed of 600 rpm. Moreover, it is evident that the speed error in Fig. 7.23b can be controlled to small value (less than 5 rpm) during the speed change. The effectiveness of the proposed sensorless control methods has been demonstrated by the results mentioned above. Constant Speed under Variable Load Initially, the generator speed is maintained at 600 rpm and the load resistance is set to 50  per phase. Afterward, it is reduced to 25  per phase at 0.8 s, then increased to 50  at 3.6 s. It is clear that both the actual and estimated speeds, as shown in Fig. 7.24a, are constant at 600 rpm. The error between estimated and measured speeds is shown in Fig. 7.24b. As illustrated in Fig. 7.24c, the output voltage of the PW is constant at 150 V RMS. Similarly, Fig. 7.24d represents the zoomed-in waveform of the PW voltage, in which the frequency is kept at 50 Hz. The CW current and the zoomed-in waveform are depicted in Fig. 7.24e and f, respectively. It can be shown from Fig. 7.24b that the fluctuation in the PW voltage amplitude is about 70 V. And, the settling time of the PW voltage is around 0.014 s. It can be observed from Fig. 7.24b that the speed error is still within a small range for the entire duration of the load change, which guarantees the effectiveness of the proposed sensorless approach in this work. Moreover, it can be concluded from Fig. 7.24f that the CW frequency is not affected during the load change, which

7.3 MRAS Sensorless Control Based on PW Flux

325

(a)

(b)

(c)

(d)

Fig. 7.21 Simulation results under 130% variation in the PW resistance with the dq-axis PW flux based MRAS sensorless control: a Actual and estimated speeds. b PW three-phase voltages. c Detailed PW three-phase voltages between 0.9 and 1.1 s. d PW three-phase currents. e Detailed PW three-phase currents between 0.9 and 1.1 s. f CW three-phase currents. g dq-axis PW flux

326

7 Model Reference Adaptive System …

(e)

(f)

(g) Fig. 7.21 (continued)

can keep the PW frequency constant at the reference value (50 Hz), as shown in Fig. 7.24d. Variable Load and Variable Speed under 1.5 L Change The third experiment test is carried out by changing the speed, changing the load resistance, and increasing all inductances. Figure 7.25a shows the actual and estimated rotor speed. Likewise, Fig. 7.25b illustrates the rotor speed error. Figure 7.25c provides the PW phase voltage. Similarly, Fig. 7.25d shows the CW phase current. The generator initially runs at 700 rpm with a three-phase load of 50  per phase. Then, the speed is decreased gradually to 600 rpm between 5 and 40 s. During the

7.3 MRAS Sensorless Control Based on PW Flux

(a)

(b)

(c)

(d)

327

328

7 Model Reference Adaptive System …

Fig. 7.22 Simulation results under 150% variation in the whole inductance with the dq-axis PW flux based MRAS sensorless control: a Actual and estimated speeds. b PW three-phase voltages. c Detailed PW three-phase voltages between 0.9 and 1.1 s. d PW three-phase currents. e Detailed PW three-phase currents between 0.9 and 1.1 s. f CW three-phase currents. g dq-axis PW flux

(e)

(f)

(g) Fig. 7.22 (continued)

7.3 MRAS Sensorless Control Based on PW Flux

329

(a)

(b)

(c)

(d)

330

7 Model Reference Adaptive System …

Fig. 7.23 Experimental results under variable speed (from 700 to 600 rpm) with the dq-axis PW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b Rotor speed error. c PW phase voltage. d Overall CW phase current. e Detailed CW phase current between 2 and 3 s. f Detailed CW phase current between 43 and 44 s

(e)

(f) Fig. 7.23 (continued)

reduction of the speed, the terminal load is changed from 50 to 25  per phase as demonstrated in Fig. 7.25a. Furthermore, the estimated speed tracks the actual speed closely during the load and speed change. The resulting error is approximately 0.7% of the reference speed, as shown in Fig. 7.25b. The control scheme has succeeded in keeping the load voltage fixed at the reference RMS value of 150 V, as depicted in Fig. 7.25c. The fluctuation in the PW voltage amplitude is around 20 V with the settling time of 0.01 s, as shown in Fig. 7.25c. Additionally, the frequency of CW current is changed with the variable speed to keep the PW frequency at its reference value, as illustrated in Fig. 7.25d. Load Change under PW Resistance Variation (1.3 R1 ) To show the effects of parameters changes, e.g. the PW resistance, the fourth experiment test is done at a fixed speed with the changed load. Firstly, the generator is operated at 600 rpm with a three-phase load of 25  per phase. At 3.2 s, the terminal load is changed from 25 to 50  per phase and the speed is kept constant

7.3 MRAS Sensorless Control Based on PW Flux

331

(a)

(b)

(c)

(d)

Fig. 7.24 Experimental results under variable load at the rotor speed of 600 rpm under the dq-axis PW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b Rotor speed error. c Overall PW phase voltage. d Detailed PW phase voltage between 3.55 and 3.68 s. e Overall CW phase current. f Detailed CW phase current between 0 and 1.8 s

332

7 Model Reference Adaptive System …

(e)

(f)

Fig. 7.24 (continued)

at 600 rpm, as shown in Fig. 7.26. Although the PW resistance varies, the proposed estimation strategy is capable of correctly estimating the speed, as illustrated in Fig. 7.26a. Furthermore, the error between the estimated and measured speed is plotted in Fig. 7.26b. The maximum value of the error is less than 0.3% of the reference speed. The overall and detailed PW voltages are illustrated in Fig. 7.26c and d, respectively. The control strategy has the capability to keep the PW voltage at its reference value with the fluctuation of 50 V and the settling time of 0.014 s. Overall, the impact of load change can be examined from the CW current, as shown in Fig. 7.26e and f (zoomed waveform). Since the speed is kept constant, the frequency of CW current is not changed under the load change. Speed Variation and Load Change The fifth experiment test is addressed with variable speed from 700 to 600 rpm and load variation from 50 to 25  per phase. All the experimental results are presented in Fig. 7.27a–f, clearly showing the actual and the estimated speeds, the error of speed tracking, the real-time and zoomed waveforms of the PW voltage and CW current, respectively. At first, the generator is run at 700 rpm with a three-phase load of 50  per phase. Next, the speed is decreased to 600 rpm between 5 and 40 s. Simultaneously, during the decline of the speed, the terminal load is also decreased from 50 to 25  per phase, as illustrated in Fig. 7.27a. Visibly, the estimated speed

7.3 MRAS Sensorless Control Based on PW Flux

333

(a)

(b)

(c)

(d) Fig. 7.25 Experimental results under variable speed and variable load with 1.5 L variation under the dq-axis PW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b Rotor speed error. c PW phase voltage. d CW phase current

334

7 Model Reference Adaptive System …

(a)

(b)

(c)

(d)

7.3 MRAS Sensorless Control Based on PW Flux

335

Fig. 7.26 Experimental results under PW resistance variation (1.3 R1) and the speed of 600 rpm with 1.5 L variation under the dq-axis PW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b Rotor speed error. c Overall PW phase voltage. d Detailed PW phase voltage between 3.2 and 3.35 s. e Overall CW phase current. f Detailed CW phase voltage between 6 and 8 s

(e)

(f) Fig. 7.26 (continued)

tracks the actual measured speed very well throughout the load and speed changes. The proposed control scheme has succeeded in keeping the load voltage fixed at its reference RMS value (150 V), as illustrated in Fig. 7.27c. Moreover, the frequency of CW current is adjusted with the speed change to keep the PW frequency at its reference value, as shown in Fig. 7.27d–f, respectively. The CW frequency is changed with the variable speed from 3.33 Hz at 700 rpm to 10 Hz at 600 rpm, in order to maintain the PW frequency fixed at its reference value (50 Hz). Also, the PW voltage and its frequency are maintained fixed at the reference value under the load change, as shown in Fig. 7.29. The results demonstrate that the fluctuation in PW voltage is about 53 V, and the settling time of PW voltage is around 0.01 s, as shown in Fig. 7.27c. As demonstrated by the above experimental results, the proposed sensorless control strategies cannot be affected by any mismatch in BDFIG parameters.

336

7 Model Reference Adaptive System …

(a)

(b)

(c)

(d)

Fig. 7.27 Experimental results under the speed change from 700 to 600 rpm and load change from 50 to 25  under the dq-axis PW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b Rotor speed error. c PW phase voltage. d Overall CW phase current. e Detailed CW phase current between 1 and 2 s. f Detailed CW phase current between 49 and 50 s

7.4 MRAS Sensorless Control Based on CW Flux

337

(e)

(f) Fig. 7.27 (continued)

7.4 MRAS Sensorless Control Based on CW Flux 7.4.1 Control Scheme Based on αβ-Axis CW Flux The proposed MRAS observer, based on αβ-axis CW flux as shown in Fig. 7.28, has two models, the adaptive model and the reference one. The αβ-axis CW flux obtained from the αβ-axis PW voltage and current is compared to that obtained from the αβ-axis PW and αβ-axis CW currents. The error is controlled to zero using a PI controller and the output of this controller is used as the estimation of the rotor speed.

Fig. 7.28 Structure of the αβ-axis CW flux MRAS observer

338

7 Model Reference Adaptive System …

Fig. 7.29 Structure of the proposed sensorless control method based on the αβ-axis CW flux MRAS observer for the standalone BDFIG system

Then, this estimated speed is fed back to adjust the adaptive model. Figure 7.29 shows the whole control structure of the proposed sensorless control method using αβ-axis CW-flux MRAS observer.

7.4.1.1

Adaptive Model Design

This section will give detailed introduction for designing the adaptive model. Substituting RW flux Eq. (2.5) at ωa = 0 to RW voltage Eq. (2.4) at ωa = 0, it can be obtained that αβ

αβ

0 =Rr irαβ + s(L r irαβ + L 2r i 2 + L 1r i 1 ) αβ

αβ

− j p1 ωr (L r irαβ + L 2r i 2 + L 1r i 1 ), αβ

αβ

0 =(Rr + s L r − j p1 ωr L r )irαβ + (s − j p1 ωr )(L 2r i 2 + L 1r i 1 ). The rotor current can be rewritten as irαβ =

αβ

αβ

−(s − j p1 ωr )(L 2r i 2 + L 1r i 1 ) . Rr + (s − j p1 ωr )L r

(7.24)

From the CW flux Eq. (2.5) at ωa = 0, the CW current can be rewritten as αβ

i2 =

1 αβ L 2r αβ ψ − i . L2 2 L2 r

(7.25)

7.4 MRAS Sensorless Control Based on CW Flux

339

Substituting (7.25) into (7.24), the RW current can be written as αβ

irαβ

=

−(s − j p1 ωr )( LL2r2 ψ2 −

L 22r αβ i L2 r

Rr + (s − j p1 ωr )L r irαβ

−

L 22r L2

(s − j p1 ωr )

Rr + (s − j p1 ωr )L r αβ

irαβ [Rr + (s − j p1 ωr )L r ] −

αβ

By ignoring

,

L 22r (s − j p1 ωr )irαβ L2

L 2r αβ αβ ψ + L 1r i 1 ), L2 2

αβ

( LL2r2 ψ2 + L 1r i 1 ) − s− jRpr1 ωr − L r +

Rr s− j p1 ωr

αβ

Rr + (s − j p1 ωr )L r

= −(s − j p1 ωr )(

,

irαβ

−(s − j p1 ωr )( LL2r2 ψ2 + L 1r i 1 )

=

irαβ =

αβ

+ L 1r i 1 )

L 22r L2

.

, the RW current can be simplified as irαβ =

αβ

αβ

L 2r ψ2 + L 2 L 1r i 1 . L 22r − L 2 L r

(7.26)

Substituting (7.26) into the CW flux Eq. (2.5) at ωa = 0, the CW flux can be obtained as L 22r L 2 L 1r L 2r αβ αβ αβ ψ2 =L 2 i 2 + 2 i , − L2 Lr L 2r − L 2 L r 1

αβ

ψ2 −

L 22r

αβ

αβ

αβ

αβ

αβ

αβ

αβ

αβ

αβ

(L 22r − L 2 L r )ψ2 − L 22r ψ2 =(L 22r − L 2 L r )L 2 i 2 + L 2 L 1r L 2r i 1 , −L 2 L r ψ2 =(L 22r − L 2 L r )L 2 i 2 + L 2 L 1r L 2r i 1 , αβ

ψ2 = αβ

ψ2 =

(L 22r − L 2 L r )L 2 i 2 + L 2 L 1r L 2r i 1 , −L 2 L r αβ

αβ

(−L 22r + L 2 L r )i 2 − L 1r L 2r i 1 . Lr

Hence, the CW flux can be expressed as αβCW

ψ2

=

L 2 L r − L 22r αβCW L 1r L 2r αβ i2 − i Lr Lr 1

(7.27)

It should be noted that the α- and β-axis CW currents in (7.27) are in the CW reference frame (αβ cw ). Substituting (7.14) to (7.27) to transform the CW current

340

7 Model Reference Adaptive System …

from the CW αβ-reference frame to the PW reference frame (αβ), the estimated CW flux can be expressed as αβ ψ˜ 2 =

7.4.1.2

L 2 L r − L 22r αβ L 1r L 2r αβ i2 − i Lr Lr 1

(7.28)

Reference Model Design

Moreover, the design of the reference model can be summarized as follows. From the RW flux Eq. (2.5) at ωa = 0, by ignoring the effect of rotor flux, as introduced in Sect. 7.3, the RW current can be obtained by irαβ = −

L 2r αβ L 1r αβ i2 − i . Lr Lr 1

(7.29)

From the PW flux Eq. (2.5) at ωa = 0, the RW current can be can be expressed as irαβ =

1 αβ L 1 αβ ψ1 − i . L 1r L 1r 1

(7.30)

From (7.29) and (7.30), the CW current can be written as 1 αβ L 1 αβ L 2r αβ L 1r αβ ψ − i =− i − i , L 1r 1 L 1r 1 Lr 2 Lr 1 1 αβ L 1 αβ L 1r αβ L 2r αβ ψ − i + i =− i , L 1r 1 L 1r 1 Lr 1 Lr 2 − − −

1 αβ L 1 αβ L 1r αβ L 2r αβ ψ + i − i = i , L 1r 1 L 1r 1 Lr 1 Lr 2

L r 1 αβ L 1 L r αβ L 1r L r αβ αβ ψ1 + i1 − i = i2 , L 2r L 1r L 1r L 2r L r L 2r 1

L r 1 αβ L r L 1 L r αβ L 1r L 1r L r αβ αβ ψ1 + i − i = i2 , L 2r L 1r L r L 1r L 2r 1 L 1r L r L 2r 1 −

Lr L r L 1 L r − L 1r L 1r L r αβ αβ αβ ψ1 + i1 = i2 . L 2r L 1r L r L 1r L 2r

Hence, the CW current can be obtained by −

Lr L r L 1 − L 21r αβ αβ αβ ψ1 + i1 = i2 . L 2r L 1r L 1r L 2r

(7.31)

7.4 MRAS Sensorless Control Based on CW Flux

341

Substituting (7.26) and (7.31) into the CW flux Eq. (2.5) at ωa = 0, the reference model of the CW flux can be given by αβ ψ2



Lr L r L 1 − L 21r αβ αβ =L 2 − ψ1 + i1 L 2r L 1r L 1r L 2r   αβ αβ L 2r ψ2 + L 2 L 1r i 1 + L 2r , L 22r − L 2 L r

αβ

ψ2 = −

L 2 L r αβ L 2 L r L 1 − L 2 L 21r αβ ψ + i1 L 2r L 1r 1 L 1r L 2r αβ

+ αβ



αβ

L 22r ψ2 + L 2 L 2r L 1r i 1 , L 22r − L 2 L r

L 22r − L 2 L r αβ ψ1 L 2r L 1r (L 2 − L 2 L r )(L r L 1 − L 21r ) + (L 1r L 2r )(L 2r L 1r ) αβ − 2r i1 , L r L 1r L 2r L 2 − L 2 L r αβ L 22r L r L 1 − L 2 L 1 L r2 + L 2 L r L 21r αβ = 2r ψ1 − i1 . L 2r L 1r L r L 1r L 2r

ψ2 =

αβ

ψ2

The CW flux can be derived as αβ

ψ2 =

L 22r − L 2 L r αβ L 2 L 1 L r − L 1 L 22r − L 2 L 21r αβ ψ1 + i1 . L 2r L 1r L 1r L 2r

From the PW voltage Eq. (2.4) at ωa = 0, the PW flux can be derived as 

 ψ1α = u α1 − R1 i 1α  β β β . ψ1 = u 1 − R1 i 1

(7.32)

The estimated speed can be defined as ω˜ r = (K p +

Ki β β )(ψ˜ 2α ψ2 − ψ2α ψ˜ 2 ). s

(7.33)

7.4.2 Control Scheme Based on Dq-Axis CW Flux As can be seen in Fig. 7.30, two models compose the proposed dq-axis CW flux based MRAS observer, the adaptive model and the reference model. Moreover, a comparison is made between the dq-axis CW flux obtained from the reference model and that from the adaptive model. Furthermore, the error is minimized to zero employing

342

7 Model Reference Adaptive System …

Fig. 7.30 Structure of the dq-axis CW flux based MRAS observer

Fig. 7.31 Structure of the proposed sensorless control method based on the dq-axis CW flux MRAS observer for the standalone BDFIG system

a PI controller, and the output of the controller is used as the estimation of the rotor speed. Afterwards, the estimated rotor speed is fed back to adjust and improve the adaptive model. Considering Fig. 7.31, it is possible to see the whole control structure of the proposed sensorless control system for the standalone BDFIG system using the dq-axis PW flux MRAS observer.

7.4.2.1

Adaptive Model Design

In order to design the adaptive model, the following procedure will be introduced in details. Substituting the RW flux equation of (2.5) to RW voltage equation of (2.4) and setting ωa = ω1 , we can get

7.4 MRAS Sensorless Control Based on CW Flux

343

0 =Rr ir + s(L r ir + L 2r i 2 + L 1r i 1 ) + j (ω1 − p1 ωr )(L r ir + L 2r i 2 + L 1r i 1 ). The RW current can be rewritten as ir =

−[s + j (ω1 − p1 ωr )](L 2r i 2 + L 1r i 1 ) . Rr + [s + j (ω1 − p1 ωr )]L r

(7.34)

From the CW flux Eq. (2.5) at ωa = ω1 , the CW current can be expressed as i2 =

1 L 2r ψ2 − ir . L2 L2

(7.35)

Substituting (7.35) into (7.34), the RW current can be written as ir =

−[s + j (ω1 − p1 ωr )]( LL2r2 ψ2 + L 1r i 1 ) Rr + [s + j (ω1 − p1 ωr )]L r

.

Divided by [s + j (ω1 − p1 ωr )] in the numerator and denominator, the RW current can be rewritten as ir =

( LL2r2 ψ2 + L 1r i 1 ) − s+ j (ωR1 −r p1 ωr ) − L r +

L 22r L2

.

(7.36)

In order to employ a fractionally rated power converter to drive the standalone BDFIG system, the CW angular frequency ω2 is usually limited to 30% of the PW frequency. According to (2.8), the range of ωr would vary between 70 and 130% of ω1 . In terms of the BDFIG with the typical pole pairs of 1 and 3 for both PW and CW, the value of (ω1 − p1 ωr ) would be in the range of (67.5–82.5%)ω1 . In general, the value of ω1 is 100π or 120π rad/s. Hence, the value of (ω1 − p1 ωr ) will become much greater than Rr , and then it can be concluded that the term Rr /[s + j (ω1 − p1 ωr )] can be approximately ignored at the steady state. Thereof, from (7.36), the RW current can be expressed as ir =

L 2r ψ2 + L 2 L 1r i 1 . L 22r − L 2 L r

(7.37)

Substituting (7.37) into the equation of CW flux (2.5) at ωa = ω1 , the CW flux can be obtained by  ψ2 = L 2 i 2 + L 2r ψ2 =

 L 2r ψ2 + L 2 L 1r i 1 , L 22r − L 2 L r

L 2 L r − L 22r L 1r L 2r i2 − i1 . Lr Lr

344

7 Model Reference Adaptive System …

The adaptive model of the estimation of CW flux in dq reference frame can be obtained by ψ˜ 2dq =

L 2 L r − L 22r L 1r L 2r i 2dq − i 1dq . Lr Lr

(7.38)

In the steady state,dψr /dt = 0. And then, from the RW voltage Eq. (2.4) at ωa = ω1 , the rotor flux at steady state can be derived as ψr = −

Rr ir . j (ω1 − p1 ωr )

(7.39)

Since the value of (ω1 − p1 ωr ) will become much greater than Rr in general, from (7.39), the rotor flux can be also ignored when ir is below or equivalent to the rated value.

7.4.2.2

Reference Model Design

The reference model can be designed according to the following process. From the equation of RW flux (2.5) at ωa = ω1 , by ignoring the effect of rotor flux, the RW current can be derived as ir = −

L 2r L 1r i2 − i1 . Lr Lr

(7.40)

From the equation of PW flux (2.5) at ωa = ω1 , it can be obtained that 1 L1 ψ1 − i 1 = ir . L 1r L 1r

(7.41)

Substituting (7.41) to (7.40), the CW current can be written as i2 = −

Lr L r L 1 − L 21r ψ1 + i1 . L 2r L 1r L 1r L 2r

(7.42)

Substituting (7.37) and (7.42) into the CW flux Eq. (2.5) at ωa = ω1 , the reference model of the CW flux can be given by ψ2 = − ψ2 =

L2 Lr L 2 L r L 1 − L 2 L 21r L 2 ψ2 + L 2 L 2r L 1r i 1 ψ1 + i 1 + 2r 2 , L 2r L 1r L 1r L 2r L 2r − L 2 L r

L 22r − L 2 L r ψ1 L 2r L 1r (L 2 − L 2 L r )[L r L 1 − L 21r ] + (L 1r L 2r )[L 2r L 1r ] i1 . − 2r (L r L 1r L 2r )

7.4 MRAS Sensorless Control Based on CW Flux

345

The CW flux can be obtained by L 22r − L 2 L r L2 L1 Lr ψ1d + L 2r L 1r L 22r − L 2 L r L2 L1 Lr = ψ1q + L 2r L 1r

ψ2d = ψ2q

− L 1 L 22r − L 2 L 21r i 1d L 1r L 2r − L 1 L 22r − L 2 L 21r i 1q . L 1r L 2r

(7.43)

The estimated rotor speed can be defined as ω˜ r = (K p +

Ki )(ψ˜ 2d ψ2q − ψ2d ψ˜ 2q ). s

(7.44)

7.4.3 Simulation Results All the simulation is carried out on a 30-kVA prototype wound-rotor BDFIG, whose parameters are specified in Section A.3, Appendix.

7.4.3.1

Results with the αβ-Axis CW Flux Based MRAS Sensorless Control

In order to confirm the capability of the MRAS sensorless control strategy based on the αβ-axis CW flux, some typical simulation results have been obtained as follows. Under Speed Change Figure 7.32 shows the response when the rotor speed reduces from 700 to 600 rpm between 1 and 2 s and then is fixed at 600 rpm for the suggested αβ-axis CW-flux MRAS observer under constant three-phase resistive load. The actual and estimated rotor speeds, the PW three-phase and dq-axis PW voltages, CW and PW three-phase currents, and the αβ-axis CW flux are illustrated in Fig. 7.32a−f, respectively. The load voltage is kept constant at its reference value with the proposed control method as shown in Fig. 7.32b. The frequency of CW is changed with the variable speed (to be 3.33 Hz at 700 rpm and then changed to 10 Hz at 600 rpm) to maintain the frequency of PW at its reference value (50 Hz), as shown in Fig. 7.32d. Under Load Change The generator is firstly operated at 600 rpm with the three-phase resistive load of 50  per phase. Then, the terminal load is varied at 1 s from 50 to 25  per phase and the speed is kept constant at 600 rpm as shown in Fig. 7.33. Figure 7.33 a-g show the actual and estimated rotor speeds, the PW three-phase voltage, detailed PW threephase voltage and dq-axis PW voltage, the CW and PW three-phase currents, and the αβ-axis CW flux, respectively. The amplitude of the PW voltage is fixed at its

346

7 Model Reference Adaptive System …

(a)

(b)

(c)

(d) Fig. 7.32 Simulation results under the speed ramp change and the constant load with the αβ-axis CW flux based MRAS sensorless control strategy: a Actual and estimated speeds. b PW three-phase voltage. c dq-axis PW voltage. d CW three-phase current. e PW three-phase current. f αβ-axis CW flux

7.4 MRAS Sensorless Control Based on CW Flux

347

(e)

(f) Fig. 7.32 (continued)

reference value with the fluctuation of about 45 V and the settling time of 0.02 s. The CW current frequency is not changed with the constant speed during the period of load change, which can keep the PW frequency at its reference value. Under PW Resistance Variation Figure 7.34 shows the response of BDFIG, under the same load change condition as presented in Fig. 7.33. The speed is kept constant at 600 rpm when the PW resistance varies. This is studied with 130% PW resistance for the PW resistance mismatch. The actual and estimated rotor’s angular speeds, the three PW phase voltages, detailed PW three-phase voltages between 0.9 and 1.1 s, dq-axis PW voltage, the CW threephase currents, the PW three-phase currents, and the αβ-axis flux of CW are given in Fig. 7.34a–g, respectively. The voltage of PW is kept around its reference setpoint value with the fluctuation of 41 V. Similarly, the settling time is found to be 0.02 s. During the period of load change, the CW frequency is not changed because of the constant speed according to (2.13), to keep the PW frequency at its reference value. Under Inductance Variation Figure 7.35 shows the response of BDFIG, under the same load change condition, as presented in Fig. 7.33, when the speed is kept constant at 600 rpm, with 150% variation of the whole inductance values. The results presented in Fig. 7.35a–g include the actual and estimated rotor speeds, the PW three-phase voltage, the detailed PW three-phase voltage, the dq-axis PW voltage, the CW three-phase current, the PW three-phase current, and the αβ-axis flux of the CW. Through the duration of the load variation, the CW frequency is not changed, due to the constant speed. The control

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7 Model Reference Adaptive System …

strategy can keep the load voltage at the reference value, as confirmed in Fig. 7.35b, with the fluctuation of about 48 V and the settling time of 0.02 s. As seen from the results mentioned above, the proposed sensorless control method cannot be affected by any mismatch in BDFIG parameters.

(a)

(b)

(c) Fig. 7.33 Simulation results under the load variation and the constant speed with the αβ-axis CW flux based MRAS sensorless control strategy: a Actual and estimated speeds. b PW three-phase voltage. c Detailed PW three-phase voltage between 0.9 and 1.1 s. d dq-axis PW voltage. e CW three-phase current. f PW three-phase current. g αβ-axis CW flux

7.4 MRAS Sensorless Control Based on CW Flux

(d)

(e)

(f)

(g)

Fig. 7.33 (continued)

349

350

7.4.3.2

7 Model Reference Adaptive System …

Results with the dq-Axis CW Flux Based MRAS Sensorless Control

Similarly, the simulation results on the MRAS sensorless control strategy based on the dq-axis CW flux also have been obtained. Under Speed and Load Change Figure 7.36 shows the response when the rotor speed reduces from 700 to 600 rpm between 1 and 2 s for the proposed dq-axis CW-flux MRAS observer under constant three-phase resistive load. In Fig. 7.36a–f, the measured and estimated rotor speeds, the PW three-phase voltage during different periods, the CW three-phase current,

(a)

(b)

(c) Fig. 7.34 Simulation results under 130% variation in the PW resistance with the αβ-axis CW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b PW three-phase voltage. c Detailed view of PW three-phase voltage. d dq-axis PW voltage. e CW three-phase current. f PW three-phase current. g αβ-axis CW flux

7.4 MRAS Sensorless Control Based on CW Flux

351

(d)

(e)

(f)

(g) Fig. 7.34 (continued)

and the dq-axis CW flux are depicted separately. The control strategy keeps the load voltage constant at the reference value as presented in Fig. 7.36b. According to (2.13), to keep the frequency of PW at its setpoint value (50 Hz), the CW frequency is adapted with the speed change (from 3.33 Hz at 700 rpm to 10 Hz at 600 rpm),

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7 Model Reference Adaptive System …

as illustrated in Fig. 7.36e. This ensures the effectiveness of the proposed sensorless control system under the speed fluctuation. Figure 7.37 shows the simulation results under load change at the constant rotor speed. The initial operation condition of the generator is under the rotational speed of 600 rpm with the load of 50  per phase. Subsequently, the terminal load resistance is varied at 1 s from 50 to 25  per phase while maintaining rotor speed at a constant speed of 600 rpm as shown in Fig. 7.37, which includes the actual and estimated rotor speeds, the PW and CW three-phase voltages during different periods, the PW active power, and the dq-axis CW flux, respectively. To reach the reference setpoint, the PW voltage is set to a fixed value. Additionally, it is noted that some fluctuation of roughly 56 V exists with a settling time around 0.01 s. The zoomed-in waveform

(a)

(b)

(c) Fig. 7.35 Simulation results under 150% variation in the whole inductance with the αβ-axis CW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b PW threephase voltage. c Detailed view of PW three-phase voltage. d dq-axis PW voltage. e CW three-phase current. f PW three-phase current. g αβ-axis CW flux

7.4 MRAS Sensorless Control Based on CW Flux

(d)

(e)

(f)

(g) Fig. 7.35 (continued)

353

354

7 Model Reference Adaptive System …

of the PW voltage is shown in Fig. 7.37c. It can be noted that the frequency of the PW voltage is fixed at 50 Hz. Additionally, the frequency of CW current remains unchanged throughout the period of load variation due to the constant speed, which keeps the PW frequency at its reference value. Under Machine Parameter Variation Figure 7.38 shows the response of BDFIG, under the same load change condition as presented in Fig. 7.37. This is studied with 130% PW resistance for the PW resistance mismatch. The real and estimated rotor speeds, the PW three-phase voltage and current during different periods, the PW active power, and the dq-axis CW flux are demonstrated in Fig. 7.38a–g, respectively. The output voltage of the PW is maintained at its setpoint. As shown in Fig. 7.38c, the zoomed waveform of the PW voltage indicates that the PW frequency can be maintained at 50 Hz. The fluctuation in

(a)

(b)

(c) Fig. 7.36 Simulation results under the speed ramp change and the constant load with the dq-axis CW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b PW three-phase voltage. c Detailed PW three-phase voltage between 0.5 and 0.56 s. d Detailed PW three-phase voltage between 2 and 2.06 s. e CW three-phase current. f dq-axis CW flux

7.4 MRAS Sensorless Control Based on CW Flux

355

(d)

(e)

(f) Fig. 7.36 (continued)

the PW voltage amplitude is around 56 V and the settling time is 0.01 s. Furthermore, the frequency of CW current is not changed during the period of load change, since the speed is kept constant, which allows the PW frequency to remain at its reference value. Figure 7.39 shows the response of BDFIG under the same load change condition as presented in Fig. 7.37, when the rotor speed is kept constant at 600 rpm with 150% variation of the whole inductance values. Figure 7.39a depicts the actual and estimated rotor speeds, while the PW three-phase voltage and CW three-phase current during different periods are shown in Fig. 7.40b–e. Figure 7.39f shows the PW active power and Fig. 7.39g shows the dq-axis CW flux. Evidently, the control strategy has succeeded in keeping the load voltage constant at the reference value as confirmed in Fig. 7.39b. Moreover, the CW current response under different speeds is shown in Fig. 7.39d. Similarly, the frequency of CW current is changed with the variable speed to keep the PW frequency at its reference value, as illustrated in Fig. 7.39d. The control method can keep the load voltage at its reference value, as approved in

356

7 Model Reference Adaptive System …

(a)

(b)

(c)

(d) Fig. 7.37 Simulation results under load change at the constant rotor speed with the dq-axis CW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b PW three-phase voltage. c Detail view of PW three-phase voltage. d CW three-phase current. e Detail view of CW three-phase current. f PW active power. g dq-axis CW flux

7.4 MRAS Sensorless Control Based on CW Flux

357

(e)

(f)

(g) Fig. 7.37 (continued)

Fig. 7.39b, with the fluctuation of around 61 V and the settling time of 0.01 s. This ensures the effectiveness of the proposed sensorless control system under the speed variation. From what mentioned above, it is known the proposed sensorless control method is not affected by any mismatch in BDFIG parameters.

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7 Model Reference Adaptive System …

(a)

(b)

(c)

(d)

Fig. 7.38 Simulation results under 130% variation in the PW resistance with the dq-axis CW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b PW three-phase voltage. c detailed view of PW three-phase voltage. d PW three-phase current. e Detailed view of PW three-phase current. f PW active power. g dq-axis CW flux

7.4 MRAS Sensorless Control Based on CW Flux

359

(e)

(f)

(g)

Fig. 7.38 (continued)

7.4.4 Experimental Results The comprehensive experiments for the dq-axis CW flux based MRAS sensorless control strategy are also carried out on a 30-kVA prototype wound-rotor BDFIG, whose parameters can be seen in Section A.3, Appendix. Variable Speed and Constant Load The generator speed is reduced from 700 to 600 rpm under the constant three-phase load of 50  per phase. Furthermore, Fig. 7.40a depicts the estimated and measured speeds. The output of the state observer tracks the actual speed well during the speed change. Figure 7.40b shows the error between the actual and the estimated speed. In general, the control strategy has succeeded in keeping the load voltage constant at its reference RMS value (150 V), as established in Fig. 7.40c. Additionally, the CW current response during the speed variation is evident in Fig. 7.40d, and their detailed illustrations are presented in Fig. 7.40e and f, respectively. It is noticed that the frequency of CW current is modified with the speed change, to keep the PW frequency at its reference value, as illustrated in Fig. 7.40d, e, and f, respectively. The

360

7 Model Reference Adaptive System …

(a)

(b)

(c)

(d) Fig. 7.39 Simulation results under 150% variation in the whole inductance with the dq-axis CW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b PW threephase voltage. c Detailed view of PW three-phase current. d CW three-phase current. e Detailed view of CW three-phase current. f PW active power. g dq-axis CW flux

7.4 MRAS Sensorless Control Based on CW Flux

361

(e)

(f)

(g)

Fig. 7.39 (continued)

CW frequency is varied with the speed variation (from 3.33 Hz at 700 rpm to 10 Hz at 600 rpm), as shown in Fig. 7.40d. All results mentined above have fully confirmed the effectiveness of the proposed sensorless control system under the variation of speed. Variable Load at Constant Speed The generator, in this scenario, is operated under a constant speed of 600 rpm with an initial three-phase load of 50  per phase. Afterwards, the terminal load resistance is varied from 50 to 25  per phase while maintaining the speed constant, as shown in Fig. 7.41. Both the real and estimated speeds are shown in Fig. 7.41a, and the difference between them is illustrated in Fig. 7.41b. As shown in Fig. 7.41c, the output voltage of PW is controlled at 150 V to obtain the reference value. Figure 7.41d shows the zoomed waveform of the PW voltage, which indicates that the frequency is held constant at 50 Hz. The results illustrate that the PW voltage is with the fluctuation of about 43 V and the settling time of 0.011 s, as illustrated in Fig. 7.41c. The CW current and the zoomed waveform are shown in Fig. 7.41e and f, respectively. The

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7 Model Reference Adaptive System …

frequency of CW current is not altered during the period of load change because the speed is kept constant, thus regulating the PW frequency at its reference value (50 Hz). Variable Speed and Variable Load under Inductance Variation This experiment test is implemented with variable speed, variable load, and 1.5 times increase in all inductances. Figure 7.42a shows the actual and estimated rotor speed, Fig. 7.42b the rotor speed error, Fig. 7.42c the PW phase voltage and Fig. 7.42d the CW phase current. The generator test begins with an initial speed of 700 rpm and a three-phase load of 50  per phase. Then, the speed is decreased to 600 rpm. While the speed is reduced, the terminal load is also changed from 50 to 25  per

(a)

(b)

(c) Fig. 7.40 Experimental results under the variable speed from 700 to 600 rpm with the dq-axis CW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b Speed percentage error. c PW phase voltage. d Overall CW phase current. e Detailed CW phase current between 1 and 2 s. f Detailed CW phase current between 16 and 17 s

7.4 MRAS Sensorless Control Based on CW Flux

363

(d)

(e)

(f) Fig. 7.40 (continued)

phase, as illustrated in Fig. 7.42a. The performance of estimated speed tracking to the actual speed is particularly well when changing both the load and speed. Moreover, the error due to the above variations is about 0.6% of the reference speed, as shown in Fig. 7.42b. The PW voltage and its frequency are maintained fixed at the reference value under the load change, as shown in Fig. 7.42. The results show that the fluctuation in PW voltage is about 32 V, and the settling time of PW voltage is 0.02 s, as shown in Fig. 7.42c. Also the CW frequency is changed with the variable speed from 3.33 Hz at the speed of 700 rpm to 10 Hz at the speed of 600 rpm, in order to keep the PW frequency fixed at its reference value (50 Hz). Noticeably, the control method has led to a successful load voltage control at the required reference RMS value of 150 V, as evidenced in Fig. 7.42c. Subsequently, the frequency of CW current is changed with the variable speed to keep the PW frequency at its reference value, as illustrated in Fig. 7.42d.

364

7 Model Reference Adaptive System …

Load Variation under PW Resistance Variation To demonstrate the consequences of parameter variations, e.g. the PW resistance, the following test is conducted at a constant speed with variable load. The generator initially runs at 700 rpm with a three-phase load resistance of 50  per phase. At 1.8 s, the terminal load resistance is decreased from 50 to 25  per phase as shown in Fig. 7.43. Although the PW resistance changes, the proposed estimation technique is capable to accurately estimate the actual speed as confirmed by Fig. 7.43a. The error between the estimated and measured speeds is plotted in Fig. 7.43b. As noticed, the maximum value of this error is less than 0.6% of the reference speed. The detailed PW voltage is illustrated in Fig. 7.43c and d. The effect of load change can be observed from the PW current response as demonstrated in Fig. 7.43e and f. Similarly, the frequency of CW current is not changed for the entire duration of load change due

(a)

(b)

(c) Fig. 7.41 Experimental results under variable load at the rotor speed of 600 rpm with the dq-axis CW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b Speed percentage error. c Overall PW phase voltage. d Detailed PW phase voltage between 4 and 4.45 s. e Overall CW phase current. f Detailed PW phase current between 3.5 and 5 s

7.4 MRAS Sensorless Control Based on CW Flux

365

(d)

(e)

(f) Fig. 7.41 (continued)

to the speed being kept constant. This enables controlling the PW frequency at its reference value (50 Hz). As shown in Fig. 7.43, the PW voltage is kept constant at its reference value under the load change with the settling time of 0.01 s and the fluctuation of around 61 V. Load Variation and Inductance Variation at Constant Speed In this section, the efficacy of the proposed MRAS observer is tested under the strong variation where all inductances are increased by 1.5 times of the rated values. These increments of inductances are set from the beginning of the process. Firstly, the experiment is studied at a constant speed of 700 rpm with a three-phase load of 50  per phase. Secondly, the load is changed from 50 to 25  per phase. Figure 7.44a illustrates the estimated and actual speeds through the load variation duration. Figure 7.44b plots the error of speed tracking. Figure 7.44c and d show the PW phase voltage, whose peak value is 212 V corresponding to 150 V RMS. The

366

7 Model Reference Adaptive System …

(a)

(b)

(c)

(d) Fig. 7.42 Experimental results under variable speed, load and inductance with the dq-axis CW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b Speed percentage error. c PW phase voltage. d CW phase current

7.4 MRAS Sensorless Control Based on CW Flux

367

output frequency can be seen at the detailed PW voltage graph, as shown in Fig. 7.44d. Moreover, the CW current and the zoomed plot are presented in Fig. 7.44e and f, respectively, to illustrate the effect of load change. However, the frequency of CW current remains fixed when the load changes, due to the speed being constant. It is known from the results that the PW voltage is kept fixed under the speed change with the settling time of 0.012 s and the amplitude fluctuation of about 64 V. Speed Change Under Inductance Variation To validate the proposed control strategy, the sixth experiment is presented in Fig. 7.45, with variable speed, fixed load, and 1.5 times increase in all inductances. The generator speed begins with 700 rpm under the three-phase load of 50  per phase. After that, it is reduced to 600 rpm while keeping the constant load resistance. All the experimental results are presented in Fig. 7.45a–f, illustrating the actual and

(a)

(b)

(c) Fig. 7.43 Experimental results under PW resistance variation at the speed of 700 rpm with the dq-axis CW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b Speed percentage error. c Overall PW phase voltage. d Detailed PW phase voltage between 3.8 and 4.2 s. e Overall PW phase current. f Detailed PW phase current between 1.5 and 2 s

368

7 Model Reference Adaptive System …

(d)

(e)

(f) Fig. 7.43 (continued)

the estimated speeds, the error of speed tracking, the real-time and zoomed waveforms of PW voltage and CW current, respectively. Ultimately, the estimated speed tracks the actual measured one closely during the load and speed change. Similarly, the frequency of CW current is varied with the speed variation (from 3.33 Hz at 700 rpm to 10 Hz at 600 rpm) to keep the PW frequency at its reference value, as illustrated in Fig. 7.45d–f, respectively. As determined from the experimental results, the proposed sensorless control technique remains good performance under the mismatch in BDFIG parameters.

7.5 Summary

369

7.5 Summary This chapter presents five MRAS speed observers based on CW power factor, PW flux in both the stationary αβ frame and the rotary dq frame, CW flux in both the the stationary αβ frame and the rotary dq frame for sensorless control of standalone BDFIGs. The presented control methods are fully investigated and verified by both simulation and experimental results, which have demonstrated excellent transient behavior under different rotor speeds, load variations and parameter changes by the proposed control methods based on the dq-axis CW power factor, αβ-axis PW flux, dq-axis PW flux, αβ-axis CW flux, dq-axis CW flux. Meanwhile, the proposed

(a)

(b)

(c) Fig. 7.44 Experimental results under inductance variation at the speed of 700 rpm with the dq-axis CW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b Speed percentage error. c Overall PW phase voltage. d Detailed PW phase voltage between 2.8 and 3.5 s. e Overall PW phase current. f Detailed PW phase current between 2.9 and 3.5 s

370

7 Model Reference Adaptive System …

(d)

(e)

(f) Fig. 7.44 (continued)

mehtods have benefited from the strong tracking ability between the estimated and actual speed response. Furthermore, it can be observed that the q-axis PW voltage tracks successfully the reference value set at zero, which has indicated the proposed vector control system can obtain the desired PW voltage orientation target. Based on both the αβ-axis PW flux and dq-axis PW flux MRAS observers, the proposed sensorless control strategy for the standalone BDFIG can enjoy very strong robustness capability.

7.5 Summary

371

(a)

(b)

(c)

(d) Fig. 7.45 Experimental results under the changed speed, constant load and varied inductance with the dq-axis CW flux based MRAS sensorless control strategy: a Actual and estimated rotor speeds. b Speed percentage error. c PW phase voltage. d Overall CW phase current. e Detailed CW phase current between 0 and 5 s. f Detailed CW phase current between 45 and 50 s

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7 Model Reference Adaptive System …

(e)

(f) Fig. 7.45 (continued)

References 1. McMahon RA, Roberts PC, Wang X et al (2006) Performance of BDFM as generator and motor. IEE Proc Electr Power Appl 153(2):289–299 2. Betz R, Jovanovic M (2002) Theoretical analysis of control properties for the brushless doubly fed reluctance machine. IEEE Trans Energy Convers 17(3):332–339 3. Xiong F, Wang X (2014) Design of a low-harmonic-content wound rotor for the brushless doubly fed generator. IEEE Trans Energy Convers 29(1):158–168 4. Liu Y, Ai W, Chen B et al (2016) Control design of the brushless doubly-fed machine for stand-alone VSCF ship shaft generator systems. J Power Electron 16(1):259–267 5. Liu Y, Ai W, Chen B et al (2016) Control design and experimental verification of the brushless doubly-fed machine for stand-alone power generation applications. IET Electr Power Appl 10(1):25–35 6. Shipurkar U, Strous TD, Polinder H et al (2017) Achieving sensorless control for the brushless doubly fed induction machine. IEEE Trans Energy Convers 32(4):1611–1619 7. Xu W, Hussien MG, Liu Y et al (2020) Sensorless voltage control schemes for brushless doubly-fed induction generators in stand-alone and grid-connected applications. IEEE Trans Energy Convers 35(4) 8. Hussien MG, Liu Y, Xu W et al (2019) Robust position observer for sensorless direct voltage control of stand-alone ship shaft brushless doubly-fed induction generators. CES Trans Electr Mach Syst 3(4):363–376 9. Tarchała G, Orłowska-Kowalska T (2018) Equivalent-signal-based sliding mode speed MRAStype estimator for induction motor drive stable in the regenerating mode. IEEE Trans Ind Electron 65(9):6936–6947 10. Pal A, Das S, Chattopadhyay AK (2017) An improved rotor flux space vector based MRAS for field-oriented control of induction motor drives. IEEE Trans Power Electron 33(6):5131–5141

References

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11. Iacchetti MF (2011) Adaptive tuning of the stator inductance in a rotor-current-based MRAS observer for sensorless doubly fed induction-machine drives. IEEE Trans Ind Electron 58(10):4683–4692 12. Cardenas R, Pena R, Proboste J et al (2004) Rotor current based MRAS observer for doubly-fed induction machines. Electron Lett 40(12):769–770 13. Pattnaik M, Kstha D (2012) Adaptive speed observer for a stand-alone doubly fed induction generator feeding nonlinear and unbalanced loads. IEEE Trans Energy Convers 27(4):1018– 1026 14. Cardenas R, Pena R, Asher G et al (2004) MRAS observer for doubly fed induction machines. IEEE Trans Energy Convers 19(2):467–468 15. Pena R, Clare J, Asher G et al (2008) MRAS observers for sensorless control of doubly-fed induction generators. IEEE Trans Power Electron 23(3):1075–1084 16. Cardenas R, Pena R, Proboste J et al (2005) MRAS observer for sensorless control of standalone doubly fed induction generators. IEEE Trans Energy Convers 20(4):710–718 17. Ebraheem AK, Xu W Liu Y (2018) Sensorless direct voltage control based on MRAS observer for the stand-alone brushless doubly-fed induction generator. In: 2018 21st international conference on electrical machines and systems (ICEMS), pp 1606–1611 18. Xu W, Ebraheem AK, Liu Y et al (2020) An MRAS speed observer based on control winding flux for sensorless control of standalone BDFIGs. IEEE Trans Power Electron 35(7):7271–7281 19. Poza J, Oyarbide E, Roye D et al (2006) Unified reference frame dq model of the brushless doubly fed machine. IEE Proc Electr Power Appl 153(5):726–734

Chapter 8

Conclusions and Future Works

Abstract This chapter concludes the book and gives some suggestions for future works. This book presents two types of control technologies for standalone BDFIGs. Firstly, the control technologies under special operation conditions are developed. The special operation conditions mianly include special loads (i.e., unbalanced and nonlinear loads) and heavy load disturbance. And then, some advanced control algorithms are proposed to enhance dynamic performance and robustness, including predictive control and sensorless control. Two typical current predictive control methods for control winding of BDFIG are presented, i.e., MPCC and NPCC methods. For the sensorless control, bothe the control methods dependent on and independent of machine parameters are developed. Finally, several research topics are recommended for future works, including identification of machine parameters, BDFIG-based dc power generation system, and applications to other promising industrial fields.

8.1 Conclusions This book presents two types of control technologies for standalone BDFIGs, i.e., control technologies under special operation conditions and some advanced control algorithms for enhancing dynamic performance and robustness. Each type of the control technologies mentioned above can be summarized as follows. (1)

Control technologies under special operation conditions: The special operation conditions mentioned in this book include special loads and heavy load disturbance.

• Control technologies under special loads In the standalone mode, the BDFIG is oversensitive to unbalanced and nonlinear loads. The unbalanced load and nonlinear load can cause unbalanced and harmonic PW voltage, respectively. In this book, the unbalanced voltage compensator and loworder harmonic voltage compensator are presented to minimize the unbalanced and harmonic impacts, respectively. And, the dual-resonant controller is developed to eliminate the influence of unbalanced and nonlinear loads at the same time. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Y. Liu and W. Xu, Advances in Control Technologies for Brushless Doubly-fed Induction Generators, https://doi.org/10.1007/978-981-19-0424-0_8

375

376

8 Conclusions and Future Works

• Control technologies under heavy load disturbance Two advanced compensation control strategies for the standalone BDFIG under heavy load disturbance are developed based on the single power converter and on the dual power converters, respectively. The single power converter-based compensation control strategy utilizes the transient feedforward compensation of the CW current to supress the PW votltage drop. And, the dual power converter-based strategy adopts both the MSC and LSC to compensate for changes in the PW active and reactive currents, so that the redundant capacities of the two converters can be fully used to compensate for load disturbance. (2)

Advanced control algorithms for enhancing dynamic performance and robustness: predictive control and sensorless control.

• Predictive control This book handles the improvements of the robustness and dynamic response for the standalone BDFIG system by applying both MPCC and NPCC methods. Firstly, the MPCC is used to replace the traditional PI controller for the CW current regulation of the standalone BDFIG. However, the control behaviour of MPCC would be adversely affected by the machine parameter mismatch. And then, another new current control method based on NPCC is proposed to solve this problem, which is free of machine parameters. • Sensorless control Two rotor position observers with the direct estimation principle are developed, which are based on the phase-axis relationship and space-vector model of the BDFIG, respectively. And, three kinds of MRAS based sensorless control strategiesare developed, which are based on the CW power factor, PW flux and CW flux, respectively. In order to enhance the robustness of the observer to the machine parameter mismatch, the improved rotor speed observer is also proposed, which is based on PW voltage and CW current and free of electrical parameters of the BDFIG.

8.2 Future Works With the ongoing development of BDFG control technologies during the past ten years and to the best knowledge of the authors, the following research topics should deserve more research efforts in the next step: (1)

(2)

Identification of machine parameters. The accurate machine parameters can significantly enhance the dynamic performance of the BDFG control system and improve the estimation accuracy and robustness of the rotor speed and position observers. BDFIG-based dc power generation. The BDFIG-based dc power generation system has some inherent advantages, e.g., relatively low power converter cost,

8.2 Future Works

(3)

377

simple control target, and so on. In this system, the frequency of PW voltage is a free variable and can be utilized to achieve the higher control objectives, which provides possibilities for speed range extension and efficiency optimization. Applications of BDFG to other promising industrial fields. Till now, BDFGs have been widely applied to grid-connected and standalone systems. The corresponding control technologies can be extended to some other promising industrial applications, such as the microgrid.

Appendix

A.1 Structure of BDFIG Experimental Platforms All the BDFIG experimental platforms used in this book are with the same structure as shown in Fig. A.1. The induction motor is used as the prime mover. The back-toback converter is connected between the PW and CW to drive the BDFIG. And, an initial charging circuit is installed to precharge the dc bus capacitors for the system startup. Since the dc bus is power-off before the start-up, the MSC cannot operate or provide CW with exciting current. Thus, a three-phase rectifier bridge is added into the system to precharge the dc bus. The current-limiting resistor ensures the the safety of charging process. When the dc bus voltage is stabilized, the Switch K2 is turned on to shorten the current-limiting resistor. Afterwards, the MSC starts to supply CW with exciting current so that the PW voltage rises to the set value. And then, the LSC begins to work. When the dc bus voltage goes up to the set value, the Switch K1 can be turned off and the entire startup process is complete.

A.2 60-kVA BDFIG Experimental Platform The 60-kVA BDFIG experimental platform is built in a 325 TEU container vessel from the Changjiang National Shipping Group of China, which has been successfully applied as the ship shaft power generation system. The BDFIG is coupled to the shaft of the vessel’s main engine without the gear box. The rated power and speed range of the vessel’s main engine are 518 kW and 375–700 rpm, respectively. The BDFIG is driven by a back-to-back power converter with the dc bus capacitors of 32 mF and the rated dc bus voltage of 800 V. The main parameters of the used 60-kVA BDFIG are presented in Table A.1, and the phtotgraph of this experimental platform is depicted in Fig. A.2. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Y. Liu and W. Xu, Advances in Control Technologies for Brushless Doubly-fed Induction Generators, https://doi.org/10.1007/978-981-19-0424-0

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380

Appendix

Initial charging circuit for dc bus capacitors Diode rectifier

K2 Current-limiting resistor

Three-phase K1 mains MSC

LSC

CW Induction motor

Filter

BDFIG

PW

Back-to-back converter

Load

Inverter

Three-phase mains

K3

Fig. A.1 Structure of the used BDFIG experimental platforms

Table A.1 Main parameters of the 60-kVA BDFIG Parameter

Value

Parameter

Value

PW pole pairs

4

R1

1.43 

CW pole pairs

2

R2

0.86 

PW rated PF

0.85 (lag)

Rr

0.1121 

Natural synchronous speed

500 rpm

L1

0.2238 H

Speed range

375–700 rpm

L2

0.2457 H

PW rated voltage

400 V

Lr

0.02162 H

PW rated current

92 A

L 1r

0.06371 H

CW voltage range

0–320 V

L 2r

0.1228 H

CW current range

0–150 A

Rotor type

Wound rotor

A.3 30-kVA BDFIG Experimental Platform The main parameters of the adopted 30-kVA BDFIG are listed in Table A. Based on the 30-kVA BDFIG, two experimental platforms are established, whose complete photographs are illustrated in Figs. A.3 and A.4, respectively. In each experimental platform, a 37-kW induction motor is used as the prime mover and controlled by a Siemens MM430 inverter, and the BDFIG is controlled by a back-to-back power converter with the dc bus capacitors of 13.62 mF.

Appendix

381

Back-to-back power converter

The ship’s main engine 60-kVA BDFIG

Fig. A.2 Photograph of the 60-kVA BDFIG experimental platform (used as the ship shaft power generation system in a container vessel)

Table A.2 Main parameters of the 30-kVA BDFIG Parameter

Value

Parameter

Value

PW pole pairs

1

R1

0.4034 

CW pole pairs

3

R2

0.2680 

PW rated PF

0.85 (lag)

Rr

0.3339 

Natural synchronous speed

750 rpm

L1

0.4749 H

Speed range

600–1200 rpm

L2

0.03216 H

PW rated voltage

380 V

Lr

0.2252 H

PW rated current

45 A

L 1r

0.3069 H

CW voltage range

0–350 V

L 2r

0.02584 H

CW current range

0–50 A

Rotor type

Wound rotor

A.4 90-kVA BDFIG Experimental Platform An experimental platform with a 90-kVA prototype BDFIG has been established as shown in Fig. A.5. A 120-kW three-phase induction motor serves as the prime mover. The BDFIG is controlled by a back-to-back power converter with the dc bus capacitors of 50 mF and the rated dc bus voltage of 800 V. The main parameters of the 90-kVA BDFIG can be seen in Table A.2 (Table A.3).

382

Appendix

Fig. A.3 Photograph of the first 30-kVA BDFIG experimental platform

MSC

Current sensors

LSC

Siemens MM430 DC bus capacitors Encoder

Voltage sensors

BDFIG

LCL filter

37-kW induction motor Oscilloscope

Fig. A.4 Photograph of the second 30-kVA BDFIG experimental platform

A.5 3-kVA BDFIG Experimental Platform The main parameters of the adopted 3-kVA BDFIG are given in Table A.4. A 3-kW three-phase induction motor, serving as the prime motor, is mechanically coupled to

Appendix

383

120-kW induction motor

MSC

90-kVA BDFIG LSC

Fig. A.5 Photograph of the 90-kVA BDFIG experimental platform

Table A.3 Main parameters of the 90-kVA BDFIG Parameter

Value

Parameter

Value

PW pole pairs

1

R1

0.0312 

CW pole pairs

3

R2

0.0412 

PW rated PF

0.85 (lag)

Rr

0.0373 

Natural synchronous speed

750 rpm

L1

0.0193 H

Speed range

600–1200 rpm

L2

0.0112 H

PW rated voltage

400 V

Lr

0.0312 H

PW rated current

150 A

L 1r

0.0186 H

CW voltage range

0–380 V

L 2r

0.0105 H

CW current range

0–220 A

Rotor type

Wound rotor

the BDFIG. The BDFIG is controlled by a back-to-back power converter with the dc bus capacitors of 1100 µF. The complete picture of the 3-kVA BDFIG experimental platform is illustrated in Fig. A.6.

A.6 5-kVA BDFIG Experimental Platform The main parameters of the used 5-kVA BDFIG are presented in Table A.5. An 11kW three-phase induction motor is connected to the BDFIG coaxially as the prime

384

Appendix

Table A.4 Main parameters of the 3-kVA BDFIG Parameter

Value

Parameter

Value

PW pole pairs

1

R1

4.21 

CW pole pairs

2

R2

3.10 

PW rated PF

0.85 (lag)

Rr

8.691 

Natural synchronous speed

1000 rpm

L1

1.510 H

Speed range

700–1200 rpm

L2

0.8928 H

PW rated voltage

380 V

Lr

2.314 H

PW rated current

4.5 A

L 1r

1.466 H

CW rated voltage

350 V

L 2r

0.5911 H

CW rated current

8A

Rotor type

Wound rotor

Back-to-back power converter

3-kVA BDFIG

3-kW induction motor

Fig. A.6 Photograph of the 3-kVA BDFIG experimental platform

motor. The BDFIG is controlled by a back-to-back power converter with the dc bus capacitors of 4000 µF. The complete picture of the 5-kVA BDFIG experimental platform can be seen in Fig. A.7.

Appendix

385

Table A.5 Main parameters of the 5-kVA BDFIG Parameter

Value

Parameter

Value

PW pole pairs

1

R1

2.43 

CW pole pairs

3

R2

0.571 

PW rated PF

0.8 (lag)

Rr

0.238 

Natural synchronous speed

750 rpm

L1

0.801 H

Speed range

600–1200 rpm

L2

0.0472 H

PW rated voltage

380 V

Lr

0.0769 H

PW rated current

8A

L 1r

0.273 H

CW rated voltage

120 V

L 2r

0.0281 H

CW rated current

24 A

Rotor type

Wound rotor

Fig. A.7 Photograph of the 5-kVA BDFIG experimental platform