VSC-HVDC Power Transmission Control: A Case Study in Linear Controller Design for Nonlinear Plants Vom Promotionsausschuss der Technischen Universität Hamburg-Harburg zur Erlangung des akademischen Grades Doktor-Ingenieur (Dr.-Ing.) genehmigte Dissertation von Martyn Simon Durrant aus Cardiff, Wales 2007
1. Gutachter: Professor Dr. Herbert Werner, TUHH 2. Gutachter: Professor Dr.-Ing. habil. Edwin Kreuzer, TUHH 3. Gutachter: Dr. James Whidborne, Cranfield University, UK Tag der mündlichen Prüfung: 17.07.2007
Berichte aus der Steuerungs- und Regelungstechnik Martyn Durrant VSC-HVDC Power Transmission Control: A Case Study in Linear Controller Design for Uncertain Nonlinear Plants Shaker Verlag Aachen 2008
Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de. Zugl.: Hamburg-Harburg, Techn. Univ., Diss., 2007 Copyright Shaker Verlag 2008 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publishers. Printed in Germany. ISBN 978-3-8322-6933-3 ISSN 0945-1005 Shaker Verlag GmbH P.O. BOX 101818 D-52018 Aachen Phone: 0049/2407/9596-0 Telefax: 0049/2407/9596-9 Internet: www.shaker.de e-mail: info@shaker.de
Acknowledgements Many people encouraged and supported me in the writing of this thesis. Indeed, I would not even have begun a doctorate at all or moved to Hamburg without the persuasion and reassurances of my friends and family and my doctoral supervisor, Professor Dr. Herbert Werner. I first want to thank Professor Werner for supervising me during this thesis, and for his support, wisdom and stimulating discussions throughout my time at UMIST and TUHH. I am also grateful for the support and collaboration of Keith Abbott and others at Areva plc, particularly during the first year of this work while it was funded through an EPSRC Industrial CASE Studentship at UMIST. I would also like to thank Professor Werner, Professor Dr.-Ing. habil. Edwin Kreuzer and Dr. James Whidborne for taking the time to examine this thesis, and Professor Dr.-Ing. Wolfgang Meyer for chairing the doctoral committee. It was a great pleasure to work at the Institute of Control Systems at TUHH. I would like to thank everyone whose research projects and ideas contributed to this thesis, in particular my colleagues Dr.-Ing. des. Andreas Kwiatkowski, Dr.-Ing. Adel Farag and Dr. Saulat Chughtai, and students Jan Hendrik Wülbern, Birgit Menzel and Antje Paulmann. I also appreciate the time Andreas and Saulat took to make comments on how to improve this thesis. I would furthermore like to thank Andreas, Saulat and Dr.-Ing. Gerwald Lichtenberg for their friendship, support and enjoyable discussions; Birthe von Dewitz for her kindness and advice on living in Germany; Herwig Meyer, Klaus Baumgart and Uwe Jahns for their cordiality and technical support; and Sudchai Boonto for his help in solving problems encountered with the Latex text processor. Moving abroad is never easy, and I thank my friends and family in the UK and my friends in Germany for their encouragement and affection during my time here. I especially want to thank my wife Nela for the love and support she gave me during the writing of this thesis, and for her comments on how to improve it. Indeed I am immeasurably thankful for all that she has added to my life. Above all I am indebted to my parents for giving me support and encouragement throughout my life and providing me with the attributes necessary to complete this work. To thank them for their love and hard work, I dedicate this thesis to them. Martyn Durrant, Hamburg, January 2008
To my Parents
Abstract The problem of designing linear controllers for nonlinear plants is investigated in this thesis through a practical example from electrical transmission engineering, the control of a terminal of a VSC HVDC (voltage source converter high voltage direct current) transmission system. VSC HVDC is a technology for transmitting electrical energy between two AC systems in which high power transistors are switched at high frequency to control the voltagesandpowerflowsacrossthelink. An analytical nonlinear model of the plant is developed and validated against a rigorous time domain model which explicitly includes the plant s switching behaviour. From an assessment of the nonlinearity of this model it is concluded that linear controller design and analysis is a pragmatic approach to take for the plant. The nonlinear model is used to generate a linear model representing a nominal operating point of the plant, a discrete set of linear models representing the range of operation of the plant, and uncertain linear models. Issues relating to the selection and construction of each of these representations and the conservatism of the uncertain linear model are explored. A new result that makes explicit the connection between the degrees of freedom in an SVD based approach to uncertain model construction and the multipliers of robust controller synthesis is introduced. It is appropriate to pose the controller design objective for the VSC HVDC terminal as a simultaneous performance problem: that is, to find a controller that minimises worst case performance over a discrete set of operating points. Nominal, simultaneous performance, and robust controller synthesis procedures are applied to meet this objective and analysed. A new and efficient LMI based algorithm for simultaneous synthesis of low order controllers is introduced based on the ellipsoidal controller design framework. This approach is shown to provide similar or better controllers than a number of low order controller design approaches based on direct search, and which have better simultaneous performance and robustness than a nominal design. It is demonstrated that robust controller synthesis procedures can provide controllers with better performance than nominal designs; however the design process for robust synthesis is more difficult than for nominal or simultaneous synthesis. This is because the most appropriate uncertain linear model has to be selected and the conservatism inherent in robust controller synthesis has to be compensated for by, for example, using only a fraction of the associated uncertainty in the synthesis. The performance of the controllers designed is demonstrated on the rigorous model, and the performance and robustness characteristics are found to match those predicted by linear analysis. i
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Contents Notation, Symbols and Abbreviations vii 1 Introduction 1 1.1 The VSC HVDC Plant.............................. 1 1.2 Representations of the VSC HVDC Terminal for Control Analysis and Synthesis 2 1.3 LTI Control System Analysis and Synthesis using LTI and LFT Uncertain Models....................................... 4 1.4 LTI Controller Synthesis and Analysis for Uncertain Linear and Nonlinear Plants..................... 6 1.5 Controller Synthesis for Simultaneous Performance............................ 7 1.6 The Central Issues Addressed by this Thesis.................. 8 1.7 Thesis Summary................................. 8 1.8 Contributions of this Thesis........................... 10 2 Robustness, Performance, and Robust Performance Analysis 11 2.1 Introduction.................................... 11 2.2 Sources of Plant Uncertainty........................... 12 2.3 Representation of Plant Uncertainty by LFT Structures..................................... 12 2.4 Closed Loop Sensitivity Functions........................ 13 2.5 System Analysis using LMIs........................... 14 2.6 Robustness Analysis of LFT Uncertain Systems using the H Norm................................ 16 2.7 Representation of Unknown and Uncertain Dynamics by an Uncertain Plant Model..................... 17 2.8 Generalised Sensitivity.............................. 18 2.9 Measuring the Distance between Plant Models................. 20 2.10 Multipliers or Scaling Matrices in Robustness Analysis...................................... 21 iii
2.11 Generalised Multipliers.............................. 22 2.12 Analysis Results with Reduced Conservatism.................. 24 2.13 Robust Performance Analysis.......................... 24 2.14 Frequency Domain Robustness Analysis with Structured Uncertainty.............................. 27 2.15 Conclusions.................................... 29 3 The Representation of Plant Parametric Uncertainty 30 3.1 Introduction.................................... 30 3.2 Representation of Model Uncertainty through an LFT Uncertain Model... 31 3.3 Symbolic Generation of LFT Uncertain Models with Diagonal Uncertainty. 33 3.4 Generation of Full Block LFT Uncertain Models................ 37 3.5 The Relationship between the Generalised Plant and Generalised Multipliers 42 3.6 Approximate Representations of Uncertain Linear Models and Nonlinear Models as Diagonal LFT Uncertain Models................... 45 3.7 Construction of quasi-lpv Plant Models.................... 46 3.8 Conclusions.................................... 47 4 LTI Controller Design 48 4.1 Introduction.................................... 48 4.2 Nominal H Synthesis.............................. 49 4.3 Nominal LQG and H 2 Synthesis......................... 53 4.4 Robust Controller Synthesis........................... 55 4.5 Low Order Controller Synthesis......................... 58 4.6 Simultaneous Performance Synthesis...................... 59 4.7 The Ellipsoidal Set Framework for Controller Synthesis...................................... 60 4.8 Direct Search Approaches to Controller Synthesis............... 65 4.9 Literature Survey of Control Design using Genetic Algorithms........ 67 4.10 Controller Parameterisation for Direct Search................. 68 4.11 Design of Objective Functions for Direct Search................ 73 4.12 Conclusions.................................... 74 5 Modeling, Validation and Control Objective Specification for the VSC HVDC Terminal 76 5.1 Introduction.................................... 76 5.2 The Role of HVDC in Power Systems...................... 77 5.3 Published Models of VSC HVDC........................ 78 5.4 Development of an Improved Analytical Model................. 80 iv
5.5 Validation of the Weak System Model...................... 85 5.6 Control Objectives................................ 85 5.7 Gridding, Linearisation and Scaling of the Weak System Model........ 88 5.8 Open loop Assessment using the Grid of Models................ 88 5.9 Nonlinearity of the Weak System Model.................... 95 5.10 Construction of LFT Uncertain Models for the VSC HVDC Terminal.... 96 5.11 Representation of the VSC HVDC Terminal Model as Quasi-LPV Models........................... 98 5.12 Conclusions.................................... 101 6 Controller Design and Analysis for the VSC HVDC Terminal 103 6.1 Introduction.................................... 103 6.2 Literature Review of Control Design for VSC HVDC.................................... 105 6.3 The Design and Analysis Setup......................... 106 6.4 Decoupled PI Controller Design......................... 111 6.5 Mixed Sensitivity Design............................. 112 6.6 LSDP Design................................... 124 6.7 LQG and H 2 Design............................... 141 6.8 Robustness Analysis using LFT Uncertain Models and Robustness to Increased AC System Impedance..................................... 149 6.9 Controller Performance on the PSCAD Model................. 152 6.10 Conclusions and Comparison of Controllers................... 162 7 Conclusions and Future Directions 164 Appendices: A LMI Synthesis through Transformation of the Analysis Problem 169 A.1 Nominal H Synthesis.............................. 169 A.2 Robust H Synthesis............................... 171 A.3 Scaling of the Uncertainty............................ 171 B Genetic Algorithms 173 B.1 Setup of the Genetic Algorithm......................... 174 C VSC HVDC: Background and Modeling Details 176 C.1 Introduction.................................... 176 C.2 Representation of AC System Dynamics.................... 176 v
C.3 The Voltage Source Converter Circuit...................... 177 C.4 Assumptions and Limitations of the Strong System Model.......... 178 C.5 Control Objectives and Structures for VSC HVDC Control.......... 178 C.6 PSCAD Model Details.............................. 179 C.7 Frequency Response Details........................... 181 D Details of Nonlinearity Measures θ V and θ P 186 The Bibliography 188 Publications by the Author 198 Curriculum Vitae 200 vi