Power Electronic Converters:PWM Strategies and Current Control Techniques 9781848211957

A voltage converter changes the voltage of an electrical power source and is usually combined with other components to c

525 128 19MB

English Pages 554 Year 2011

Report DMCA / Copyright

DOWNLOAD FILE

Polecaj historie

Power Electronic Converters:PWM Strategies and Current Control Techniques
 9781848211957

Citation preview

Power Electronic Converters

Power Electronic Converters PWM Strategies and Current Control Techniques

Edited by Eric Monmasson

First published 2011 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Adapted and updated from Commande rapprochée de convertisseur statique 1&2 published 2009 in France by Hermes Science/Lavoisier © LAVOISIER 2009 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

www.iste.co.uk

www.wiley.com

© ISTE Ltd 2011 The rights of Eric Monmasson to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. ____________________________________________________________________________________ Library of Congress Cataloging-in-Publication Data Commande rapprochee de convertisseur statique. English Power electronic converters : PWM strategies and current control techniques / edited by Eric Monmasson. p. cm. Includes bibliographical references and index. ISBN 978-1-84821-195-7 1. Electric current converters. 2. Electric motors--Electronic control. I. Monmasson, Eric. II. Title. TK7872.C8C66 2011 621.3815'322--dc22 2010051719 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-195-7 Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne.

Table of Contents

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

Chapter 1. Carrier-Based Pulse Width Modulation for Two-level Three-phase Voltage Inverters . . . . . . . . . . . . . . . . . . . . . . . . . Francis LABRIQUE and Jean-Paul LOUIS

1

1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Reference voltages varef, vbref, vcref . . . . . . . . . . . . . . . . . . 1.3. Reference voltages Paref, Pbref, Pcref . . . . . . . . . . . . . . . . . 1.4. Link between the quantities va, vb, vc and Pa, Pb, Pc . . . . . . 1.5. Generation of PWM signals . . . . . . . . . . . . . . . . . . . . . 1.5.1. Reverse sawtooth wave. . . . . . . . . . . . . . . . . . . . . . 1.5.2. Conventional sawtooth carrier . . . . . . . . . . . . . . . . . 1.5.3. Triangular carrier . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.4. Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6. Determination of the reference waves varef k, vbref k, and vcref k from the reference waves varef k, vbref k, vcref k . . . . . . . . . . . . . . 1.6.1. “Sine” modulation . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2. “Centered” modulation . . . . . . . . . . . . . . . . . . . . . . 1.6.3. “Sub-optimal” modulation. . . . . . . . . . . . . . . . . . . . 1.6.4. “Flat top” and “flat bottom” modulation. . . . . . . . . . . 1.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

1 4 10 12 13 13 17 20 23

. . . . . . .

. . . . . . .

24 25 27 29 30 32 33

vi

Power Electronic Converters

Chapter 2. Space Vector Modulation Strategies . . . . . . . . . . . . . Nicolas PATIN and Vincent LANFRANCHI 2.1. Inverters and space vector PWM . . . . . . . . . . . . . 2.1.1. Problem description . . . . . . . . . . . . . . . . . . . 2.1.2. Inverter model . . . . . . . . . . . . . . . . . . . . . . 2.1.3. Space vector modulation . . . . . . . . . . . . . . . . 2.2. Geometric approach to the problem . . . . . . . . . . . 2.2.1. Degrees of freedom . . . . . . . . . . . . . . . . . . . 2.2.2. Extension to the full domain . . . . . . . . . . . . . 2.2.3. Space vector modulation . . . . . . . . . . . . . . . . 2.2.4. PWM spectrum. . . . . . . . . . . . . . . . . . . . . . 2.3. Space vector PWM and implementation . . . . . . . . 2.3.1. Implementation hardware and general structure . 2.3.2. Determination of working sector. . . . . . . . . . . 2.3.3. Some variants of space vector PWM . . . . . . . . 2.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . .

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

35 35 36 40 48 48 50 55 56 58 58 62 63 68 69

Chapter 3. Overmodulation of Three-phase Voltage Inverters . . . Nicolas PATIN and Eric MONMASSON

71

3.1. Background. . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Comparison of modulation strategies . . . . . . . . . 3.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. “Full-wave” modulation . . . . . . . . . . . . . . . 3.2.3. Performance of standard modulation strategies 3.3. Saturation of modulators . . . . . . . . . . . . . . . . . 3.4. Improved overmodulation . . . . . . . . . . . . . . . . 3.5. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . . . . .

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

. . . . .

. . . . . . . .

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

93

. . . . .

. . . . . . . .

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

Chapter 4. Computed and Optimized Pulse Width Modulation Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vincent LANFRANCHI, Nicolas PATIN and Daniel DEPERNET . . . . .

. . . . . . . .

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

71 72 72 73 74 78 81 91

. . . . .

. . . . . . . .

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

. . . . . . . .

4.1. Introduction to programmed PWM. . . . . 4.2. Range of valid frequencies for PWM . . . 4.3. Programmed harmonic elimination PWM 4.4. Optimized PWM . . . . . . . . . . . . . . . . 4.4.1. Introduction . . . . . . . . . . . . . . . . .

. . . . . . . .

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

35

. . . . .

. . . . . . . .

. . . . .

. . . . .

93 95 97 100 100

Table of Contents

4.4.2. Minimization criteria . . . . . . . . . . . . . . 4.4.3. Applying optimization results . . . . . . . . 4.4.4. Principles of real-time generation . . . . . . 4.5. Calculated multilevel PWM . . . . . . . . . . . . 4.5.1. Introduction . . . . . . . . . . . . . . . . . . . . 4.5.2. Calculated three-level PWM . . . . . . . . . 4.5.3. Calculated PWM with independent levels. 4.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . 4.7. Bibliography . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

100 103 107 108 108 108 113 114 115

Chapter 5. Delta-Sigma Modulation . . . . . . . . . . . . . . . . . . . . . Jean-Paul VILAIN and Christophe LESBROUSSART

119

5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Principle of single-phase Delta-Sigma modulation. . . 5.2.1. Open-loop or closed-loop operation. . . . . . . . . . 5.2.2. Frequency characteristics . . . . . . . . . . . . . . . . 5.2.3. Influence of reference amplitude on the spectrum. 5.2.4. Influence of command signal frequency on spectral content . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5. Absence of short pulses . . . . . . . . . . . . . . . . . 5.2.6. Decisional element . . . . . . . . . . . . . . . . . . . . 5.2.7. Asynchronous and synchronous DSM . . . . . . . . 5.3. Three-phase case: vector DSM . . . . . . . . . . . . . . . 5.3.1. Criteria for selecting the new vector . . . . . . . . . 5.3.2. Three level three-phase inverter . . . . . . . . . . . . 5.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

vii

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

119 120 122 122 124

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

125 126 126 126 128 130 137 138 139

Chapter 6. Stochastic Modulation Strategies . . . . . . . . . . . . . . . Vincent LANFRANCHI and Nicolas PATIN

141

6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Spread-spectrum techniques and their applications 6.3. Description of stochastic modulation techniques . . 6.3.1. Deterministic basis of PWM . . . . . . . . . . . . 6.3.2. Variable-frequency stochastic PWM . . . . . . . 6.3.3. Random pulse position PWM. . . . . . . . . . . . 6.3.4. Stochastic PWM in three-phase inverters . . . . 6.3.5. General remarks . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

141 142 144 144 145 146 146 147

viii

Power Electronic Converters

6.4. Spectral analysis of stochastic modulation . . 6.4.1. Effects on voltage spectra . . . . . . . . . . 6.4.2. Impact on load current spectra . . . . . . . 6.4.3. Impact on DC bus current . . . . . . . . . . 6.4.4. Impact on machine noise and vibrations . 6.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . 6.6. Bibliography . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

147 147 149 150 151 155 156

Chapter 7. Electromagnetic Compatibility of Variable Speed Drives: Impact of PWM Control Strategies . . . . . . . . . . . . . . . . Bertrand REVOL

159

7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Objectives of an EMC study . . . . . . . . . . . . . . . . . . . . . . 7.3. EMC mechanisms in static converters. . . . . . . . . . . . . . . . 7.3.1. General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2. EMC standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3. Standardized measurement and simulation . . . . . . . . . . 7.4. Time-domain simulation . . . . . . . . . . . . . . . . . . . . . . . . 7.5. Frequency-domain modeling: a tool for the engineer . . . . . . 7.5.1. Objectives of modeling. . . . . . . . . . . . . . . . . . . . . . . 7.5.2. Modeling of disturbance sources . . . . . . . . . . . . . . . . 7.5.3. Frequency domain representation of the inverter . . . . . . 7.6. PWM control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1. Carrier-based PWM . . . . . . . . . . . . . . . . . . . . . . . . . 7.7. Comparison of sources for different carrier-based PWM strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.1. Sinusoidal intersective PWM . . . . . . . . . . . . . . . . . . . 7.7.2. Harmonic injection control . . . . . . . . . . . . . . . . . . . . 7.7.3. Limiting commutation rates: DeadBanded PWM control . 7.8. Space vector PWM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9. Structure for minimizing the common mode voltage . . . . . . 7.10. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

159 161 162 162 164 165 167 169 169 169 176 178 179

. . . . . . . .

190 190 191 192 193 199 200 200

Chapter 8. Multiphase Voltage Source Inverters . . . . . . . . . . . . . Xavier KESTELYN and Eric SEMAIL

203

8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Vector modeling of voltage source inverters. . . . . . . . . . . . .

203 204

Table of Contents

8.2.1. n-leg structure: terminology, notation, and examples. 8.2.2. Mean value control: PWM . . . . . . . . . . . . . . . . . 8.3. Inverter as seen by the multiphase load . . . . . . . . . . . . 8.3.1. Load topology and associated degrees of freedom . . 8.3.2. Worked example: three-phase case . . . . . . . . . . . . 8.3.3. Worked example: five-phase load . . . . . . . . . . . . . 8.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

204 209 221 223 227 232 237 238

Chapter 9. PWM Strategies for Multilevel Converters. . . . . . . . . Thierry MEYNARD and Guillaume GATEAU

243

9.1. Introduction to multilevel and interleaved converters . . . 9.2. Modulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1. Recap: two-level modulators . . . . . . . . . . . . . . . . 9.2.2. Multilevel modulators . . . . . . . . . . . . . . . . . . . . 9.3. Examples of control signal generators for various multilevel structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1. “3-point” inverters (Neutral Point Clamped Inverter) 9.3.2. Flying capacitor inverters . . . . . . . . . . . . . . . . . . 9.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

ix

. . . .

. . . .

. . . .

. . . .

243 252 252 255

. . . . .

. . . . .

. . . . .

. . . . .

274 274 275 280 283

Chapter 10. PI Current Control of a Synchronous Motor. . . . . . . Mohamed Wissem NAOUAR, Eric MONMASSON, Ilhem SLAMA-BELKHODJA and Ahmad Ammar NAASSANI

287

10.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2. Model of a synchronous motor . . . . . . . . . . . . . . . . . . . 10.2.1. Model of a synchronous motor in a fixed coordinate system based on the stator . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2. Model of a synchronous motor in a common coordinate system (d, q) aligned with the rotor winding axis of the motor. . 10.2.3. Expression for electromagnetic torque . . . . . . . . . . . . 10.3. Typical power delivery system for a synchronous motor . . . 10.4. PI current control of a synchronous motor in the fixed three-phase coordinate system of the stator . . . . . . . . . . . . . . . 10.4.1. Tuning of PI controllers in a fixed three-phase coordinate system aligned with the stator . . . . . . . . . . . . . . . 10.4.2. PI control structure in a fixed three-phase coordinate system aligned with the stator. . . . . . . . . . . . . . . . . . . . . . .

. .

287 288

.

288

. . .

294 299 300

.

303

.

306

.

309

x

Power Electronic Converters

10.5. PI current control for a synchronous motor in a rotating coordinate system (d, q) . . . . . . . . . . . . . . 10.5.1. Tuning of PI controllers in the (d, q) frame . . . . 10.5.2. PI control structure in the (d, q) reference frame. 10.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

Chapter 11. Predictive Current Control for a Synchronous Motor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mohamed Wissem NAOUAR, Eric MONMASSON, Ilhem SLAMA-BELKHODJA and Ahmad Ammar NAASSANI 11.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2. Minimum-switching-frequency predictive control strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3. Limited-switching-frequency predictive control strategies . 11.4. Limited-switching-frequency predictive current control strategies for a synchronous motor . . . . . . . . . . . . . . . . . . . . 11.4.1. Predictive current control for a synchronous motor with variable, limited switching frequency . . . . . . . . . . . . . 11.4.2. Predictive current control with fixed switching frequency for a synchronous motor . . . . . . . . . . . . . . . . . . 11.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

319

..

319

.. ..

320 321

..

322

..

322

.. .. ..

329 333 334

Chapter 12. Sliding Mode Current Control for a Synchronous Motor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ahmad Ammar NAASSANI, Mohamed Wissem NAOUAR, Eric MONMASSON and Ilhem SLAMA-BELKHODJA 12.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2. Sliding mode current control for a DC motor . . . . . . . . 12.2.1. Direct sliding mode current control of a DC motor . . 12.2.2. Indirect sliding mode current control for a DC motor. 12.3. Sliding mode current control of a synchronous motor . . . 12.3.1. Direct sliding mode control of the stator current vector in an induction motor . . . . . . . . . . . . . . . . . . . . . 12.3.2. Indirect sliding mode control of the stator current vector in an induction motor . . . . . . . . . . . . . . . . . . . . . 12.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

311 311 314 316 317

. . . . .

335

. . . . .

335 336 339 342 350

...

353

... ... ...

363 369 370

Table of Contents

xi

Chapter 13. Hybrid Current Controller with Large Bandwidth and Fixed Switching Frequency . . . . . . . . . . . . . . . . . . . . . . . . 371 Serge PIERFEDERICI, Farid MEIBODY-TABAR and Jean-Philippe MARTIN 13.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2. Main types of discrete-output current regulators . . . . . . . . 13.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2. Hysteresis regulator . . . . . . . . . . . . . . . . . . . . . . . . 13.2.3. Fixed-frequency hysteresis regulator . . . . . . . . . . . . . 13.2.4. Turn-on triggered current regulator . . . . . . . . . . . . . . 13.2.5. Turn-off triggered controller . . . . . . . . . . . . . . . . . . 13.2.6. Turn-on or turn-off triggered regulator . . . . . . . . . . . . 13.2.7. Principles of a hybrid modulated hysteresis regulator . . 13.3. Tools for limit cycle analysis. . . . . . . . . . . . . . . . . . . . . 13.3.1. Introduction to dynamic systems; concept of bifurcation 13.3.2. Concept of bifurcation of a dynamic system . . . . . . . . 13.3.3. Poincaré cross-section and bifurcation diagram . . . . . . 13.3.4. Application to electrical engineering . . . . . . . . . . . . . 13.3.5. Analysis of limit cycles in nonlinear current regulators . 13.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

371 374 374 374 375 378 383 385 388 392 392 395 396 397 401 414 414

Chapter 14. Current Control Using Self-oscillating Current Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jean-Claude LE CLAIRE

417

14.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2. Operating principle of the self-oscillating current controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1. Dual-purpose local loop . . . . . . . . . . . . . . . . . . 14.2.2. Local control loop for switching frequency control . 14.2.3. Local low-frequency current control loop . . . . . . . 14.2.4. Stability of the modulator . . . . . . . . . . . . . . . . . 14.3. Improvements to the SOCC. . . . . . . . . . . . . . . . . . . 14.3.1. Reducing the static error . . . . . . . . . . . . . . . . . . 14.3.2. Controlling the switching frequency . . . . . . . . . . 14.3.3. Variants on the initial design . . . . . . . . . . . . . . . 14.4. Characteristics of the SOCC . . . . . . . . . . . . . . . . . . 14.4.1. Switching frequency. . . . . . . . . . . . . . . . . . . . . 14.4.2. Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . .

....

417

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

418 418 419 423 427 428 428 430 431 432 432 434

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

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

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

xii

Power Electronic Converters

14.4.3. Harmonic distortion . . . . . . . . . . . . 14.5. Extensions to the SOCC concept . . . . . . 14.5.1. Self-oscillating voltage control . . . . . 14.5.2. Three-phase SOCC . . . . . . . . . . . . 14.5.3. Three-phase SOVC . . . . . . . . . . . . 14.5.4. Emulation of high-power active loads 14.5.5. Analog-to-digital converter for the measurement circuit . . . . . . . . . . . . . . . . . 14.6. Conclusion . . . . . . . . . . . . . . . . . . . . 14.7. Bibliography . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

435 435 435 441 442 444

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

444 445 445

Chapter 15. Current and Voltage Control Strategies Using Resonant Correctors: Examples of Fixed-frequency Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joseph PIERQUIN, Arnaud DAVIGNY and Benoît ROBYNS

449

15.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2. Current control with resonant correctors . . . . . . . . . . . . . . 15.2.1. Control using Kessler’s symmetric optimum . . . . . . . . . 15.2.2. Application to power control: example of a wind turbine . 15.3. Voltage control strategy . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.2. Principle of power control . . . . . . . . . . . . . . . . . . . . . 15.3.3. Voltage control at the capacitor terminals . . . . . . . . . . . 15.3.4. Determination of reference voltages. . . . . . . . . . . . . . . 15.3.5. Power control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.6. Voltage control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.7. Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5. Appendix: transformer parameters . . . . . . . . . . . . . . . . . . 15.6. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

449 451 451 457 463 463 465 468 472 473 476 477 483 484 484

Chapter 16. Current Control Strategies for Multicell Converters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Guillaume GATEAU and Thierry MEYNARD

487

16.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2. Multilevel conversion topology . . . . . . . . . . . . . . . . . . 16.2.1. Main types of multilevel structure . . . . . . . . . . . . . . 16.2.2. Advantages and disadvantages of a multicell structure .

487 488 489 492

. . . .

. . . .

Table of Contents

16.2.3. Evolution of high-power multicell topologies: stacked multicell converters . . . . . . . . . . . . . . . . . . . . . . . 16.3. Modeling and analysis of degrees of freedom for control. . 16.3.1. Instantaneous modeling. . . . . . . . . . . . . . . . . . . . . 16.3.2. Mean value model . . . . . . . . . . . . . . . . . . . . . . . . 16.4. Analysis of degrees of freedom available to the control algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.1. Open-loop PWM modulation . . . . . . . . . . . . . . . . . 16.4.2. Degrees of freedom in the topology . . . . . . . . . . . . . 16.4.3. Objective of command rules. . . . . . . . . . . . . . . . . . 16.5. Classification of control strategies . . . . . . . . . . . . . . . . 16.6. Indirect control strategy for a single-phase leg . . . . . . . . 16.6.1. Principle of decoupled control . . . . . . . . . . . . . . . . 16.6.2. Linearization and non-interacting control . . . . . . . . . 16.6.3. Decoupling using exact input/output linearization. . . . 16.6.4. Control exploiting the phase shifts between the command signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.7. Direct control strategy for a single-phase leg . . . . . . . . . 16.7.1. Sliding mode control . . . . . . . . . . . . . . . . . . . . . . 16.7.2. Current mode control . . . . . . . . . . . . . . . . . . . . . . 16.8. Command strategy, three-phase approach . . . . . . . . . . . 16.8.1. Features of two-level inverters for three-phase systems 16.8.2. Features of a three-phase N-level system . . . . . . . . . 16.8.3. Analysis of degrees of freedom made available by the use of multilevel inverters . . . . . . . . . . . . . . . . . . . . . 16.8.4. Examples of use of the degrees of freedom made available by using multilevel inverters . . . . . . . . . . . . . . . . 16.9. Features of multicell converters: need for an observer . . . 16.10. Conclusions and outlook. . . . . . . . . . . . . . . . . . . . . . 16.11. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

. . . .

. . . .

494 495 495 497

. . . . . . . . .

. . . . . . . . .

497 497 498 499 500 501 501 502 506

. . . . . . .

. . . . . . .

509 513 513 517 521 521 522

..

526

. . . .

. . . .

527 530 531 533

List of Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

537

Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

541

Introduction

The present-day soul searching over modern society’s dependency on oil has resulted in considerable attention being given to alternative, renewable energy sources; this has emphasized how important electrical energy will be in the future, and its potential for reducing environmental impact. This is particularly true for electrical energy conversion, a field that has seen continual progress over the last 30 years. The primary reasons for this have been the development of power switches operating at increasingly higher speeds and ever-increasing power levels. At the same time digital systems, able to act as controllers for power electronics, have opened up new possibilities both in terms of ease of use and increased performance. Thus static converters and their accompanying controllers have become critical to modern power conversion devices. This fact has rapidly led controller designers to pay close attention to the electrical output from static converters (voltages and currents) since these have a direct effect on the quality of the higher-level control variables such as torque and velocity (in the case of an actuator) or active and reactive power flows (in the case of a generator connected to the grid). This observation has naturally led designers to organize their controllers in a hierarchical manner. The lowest level is known as the current and/or voltage controller. This inner-loop controller ensures that electrical quantities output from the converter are correctly regulated, thus ensuring a high quality of energy transfer between the source upstream of the converter and its downstream load. The next highest level accompanies the inner loop

xvi

Power Electronic Converters

controller. This second level is often referred to as “algorithmic control” because it is generally implemented in a microprocessor (DSP, RISC, etc.). This more sophisticated element of the control focuses on controlling the variables directly relevant to the final application (speed of a motor, etc.). The resultant servo loops are known as outer-loop controllers and their control quantities act as references to the inner-loop controllers. The current-voltage control of static converters is an important technical consideration since it is crucial to the correct functioning of the entire energy conversion system. The dynamic characteristics that are required span over a considerable range and the specifications are continually being tightened in response to technological advances. Thus, a great deal of research effort has been devoted to this topic, both in universities and in industry, and we will attempt to summarize that work in this book, discussing not only the reference algorithms but also the current trends in innovation within this field. In this book on current and voltage control of static converters, we will focus on the following two central themes: – Chapters 1 to 9 focus on pulse width modulation (PWM) techniques that enable a static converter to continuously generate variable output voltages in response to binary orders sent to the static converter, with both the output amplitude and frequency being controllable in terms of instantaneous mean values; – Chapters 10 to 16 focus on electric current control techniques. In order to better introduce PWM techniques, we should recall a few important characteristics of electronic power devices. These are highefficiency devices since they only operate in fully blocking states (zero current) or fully conducting states (zero voltage). The transition from a blocking state to a conducting state, and vice versa, occurs in response to a switching action. However, this switching method of energy conversion can only produce a limited number of different voltage levels, in other words, it is a quantized process. Consequently, in order to achieve acceptable accuracy for the

Introduction

xvii

amplitudes and frequencies of the resultant voltage waveforms we must modulate the duration of the voltage pulses applied to the gates of the power switches. The modulation will be more effective when its associated frequency is higher. However, this modulation frequency can only be increased up to a certain limit beyond which it leads to unacceptably high switching losses in the power switches. Another factor limiting increases in switching frequency is associated with an increase in conducted and radiated interference, which can cause damage to equipment near the static converter: this is a problem of electromagnetic compatibility. Thus, this inevitable compromise between an increase in the modulation frequency and the drawbacks associated with this increase has led researchers to develop a wide range of modulation techniques. The quality criteria commonly specified for a PWM system include minimization of the total harmonic distortion of the electric current, maximization of the linear range of the fundamental harmonic voltage, minimization of torque harmonics (in the case of motor control), reduction of losses within the static converter, and minimization of the common mode voltage that is produced. The aim of the first nine chapters of this book is to highlight the wide variety of PWM techniques that are available. Focus will be on the case of the voltage source inverter because of its importance in industrial applications. Chapters 1 and 2 can be treated as reference chapters. They discuss in depth the two main families of PWM strategies: carrier-based PWM strategies and space vector PWM strategies. In both these cases we will study a two-level voltage inverter intended to feed a three-phase inductive load such as an electric motor. We will emphasize the conceptual similarities between these two approaches despite their different implementations. The degree of freedom introduced by the addition of a zero-sequence component to the modulated voltage enables us to meet a range of challenges (e.g. maximization of the linear range and limitation of losses).

xviii

Power Electronic Converters

Chapter 3 considers overmodulation of three-phase voltage inverters, a very important mode of operation in case of variable speed drive applications. We will discuss modulation strategies for when the required voltage is close to or greater than the maximum possible value, with the main objective to maximize the total power while restricting the effects of low frequency harmonic components. Chapter 4 discusses high-power systems for which the modulation frequency is necessarily restricted. The idea here is to work with modulation frequencies that are synchronized with the fundamental harmonic, and to optimize the harmonic content of the modulated voltage waveforms by careful choice of the exact switching times. We will consider the case of a three-level inverter as well. We will also present an original configuration for a multi-level power supply using active filtering based on two, two-level inverters (one supplying the requisite power and the other operating as an active filter). This construction enables the harmonic content of the power supply to be optimized. Chapter 5 describes the Delta-Sigma modulation strategy. The main advantages of this modulation are its robustness, the possibility of reducing the ratio between the switching frequency and the modulation frequency, and the possibility of operating with either variable or fixed switching frequency. Chapter 6 considers stochastic modulation methods. The main advantage is that they can broaden the spectrum of the modulated signals, thus reducing electromagnetic and acoustic interference. The latter will be the subject of a detailed study at the end of the chapter. Chapter 7 continues from the previous chapter. It focuses on analyzing conducted electromagnetic interference produced by the modulated voltages output from a voltage source inverter driving an electric motor. Chapters 8 and 9 offer an introduction to the study of modulation strategies for energy conversion devices where the power delivery is distributed. There is a strong interest, presently, in designing energy conversion structures with multiple windings or multiple levels. These structures are inevitably more complex than a traditional three-phase motor driven by a two-level inverter. Indeed, such structures enable fault-tolerant architectures

Introduction

xix

to be developed, exploiting the redundancies present in such a system; they can also be used to distribute between multiple components the power that the device must deliver, thus reducing the stress on the power switches and increasing the lifetime of the equipment. Chapter 8 introduces an extension of the space vector PWM technique to multiphase systems through a formalism based on linear algebra. Chapter 9 provides a generic discussion of PWM applied to common multilevel converter topologies. In particular, we show how redundancies in the voltage levels can be exploited to optimize additional objectives such as voltage balancing across flying capacitors. Chapters 10 to 16 are devoted to current regulation techniques. It seems worthwhile recalling the main reasons that have led designers to integrate this type of regulation in a fairly systematic manner into the design of innerloop control systems for static converters. The main objectives are to ensure accurate control of the instantaneous current waveforms to protect the static converter from any potential current surges, to reject disturbances caused by the load, to be robust with regard to parametric variations and to nonlinearities within the converter, and also to offer excellent control dynamics. From a quantitative point of view these criteria result in a minimal static error, maximize the bandwidth, and offer an optimized modulation depth and a minimum level of distortion. These fundamental requirements for current regulation are often accompanied, where possible, with additional regulation requirements such as control over the switching frequency in the case of hysteresis-based control, balancing of intermediate voltages in the case of a multicell converter, etc. As we will see in this part of the book, current control structures are also tightly integrated with the applications with which they are associated. It is for this reason that we have included a range of studied examples although we make no claims of being exhaustive. These examples will of course include the combination of a voltage inverter and motor, a very popular case and one that has seen the greatest range of experimental work in terms of current control. Nevertheless, the current control techniques that are brought

xx

Power Electronic Converters

together in this volume are also relevant to other types of applications such as high performance power-in-the-loop emulation, the generation of electrical energy both for the electrical grid and for isolated networks, DC/DC power delivery, and high-power applications based on multilevel converters. In terms of their operating principles, current regulation methods for static converters can be divided into two main families: – direct control, also known as amplitude control, for which the outputs of the current regulators directly control the associated static converter. These control strategies are exclusively nonlinear. Their main advantage is that they ensure excellent system dynamics along with a high robustness in the face of parametric variations and model uncertainties. Their main drawbacks are variation in switching frequency and the emergence of limit cycles at steady state; – indirect control, also known as PWM control, for which the outputs of the current regulators act as inputs to a PWM modulator (Chapters 1 to 9). These control strategies may be either linear or nonlinear and offer the possibility of controlling the converter in a very accurate manner and at a fixed frequency, thus avoiding any risk of limit cycles appearing. However, the dynamics that can be achieved using such methods are generally not up to the standard of those obtained using direct command. Chapters 10 to 12 discuss a single application: current control for a synchronous motor supplied by a three-phase voltage inverter. For this application, control of the current is equivalent to control of the torque. Chapter 10 discusses indirect control using a PI controller in a rotating reference frame. The quantities being controlled are constant at steady state, which makes servo control of those quantities easier. This first control method is linear and will be treated as a reference method since it is so widely used in industry. Chapter 11 discusses direct and indirect predictive control, the principle of which involves calculating the most suitable voltage vector to apply in each sampling period. This control strategy is relatively demanding in terms of computation time but can be implemented in a highly parallel manner. It

Introduction

xxi

is therefore very well suited to implementation in an FPGA (Field Programmable Gate Array). Chapter 12 describes direct and indirect sliding mode control. The design principle behind these two sliding mode control methods is described in detail and their contrasting strengths in terms of dynamics and precision are clearly demonstrated. Chapter 13 discusses hysteresis-based control. The aim is to explain the fundamentals of this type of control using theoretical tools developed for the study of nonlinear systems. Without focusing on any specific application, this chapter offers a different perspective on the qualities of this direct control method focusing on the concept of modulated hysteresis control, which combines the excellent dynamic performance of hysteresis control with guaranteed fixed frequency operation. Chapter 14 discusses current and voltage control using a self-oscillating regulator, known as SOCC (self-oscillating current control) and SOVC (selfoscillating voltage control). This innovative direct control technique is protected by patents. It relies on self-oscillation within the control loop, a property that guarantees fixed frequency operation at the same time as promising excellent performance in terms of dynamics and robustness. Here, the application area is for power-in-the-loop emulation of electrical loads. Chapter 15 introduces the principle of resonant control at fixed frequency. This type of control makes it possible to introduce an infinite gain at a precisely known frequency, which results both in elimination of any tracking error and rejection of any disturbance at this specific frequency. This control strategy, which is a sensitive form of regulation, is very promising since it is ideally suited to distributed energy generation networks. Here the authors demonstrate the qualities of these resonant regulators through an example of control of a wind turbine able to operate both on a power grid and on an isolated network. This is an indirect, linear type of control. Finally, Chapter 16 presents a state of the art for current control of multicell converters. The number of degrees of freedom available in such

xxii

Power Electronic Converters

conversion structures is useful for studies into multi-objective current control (both tracking of commanded current references and balancing of internal voltages). The application area here is for high-power equipment. We also show how the principle control paradigms presented earlier can be adapted to the case of multilevel converters.

Eric MONMASSON February 2011

Chapter 1

Carrier-Based Pulse Width Modulation for Two-level Three-phase Voltage Inverters

1.1. Introduction Two-level three-phase voltage inverters are very widely used for feeding alternating current electrical machines serving as actuators with variable input voltages (controllable for amplitude and frequency). However, they are also increasingly being used as sinusoidal current absorption rectifiers. A chapter from an earlier book [LAB 04] has already introduced these topics from a modeling perspective. Figure 1.1 recalls the basic principles of a twolevel three-phase voltage inverter feeding a balanced three-phase load connected in a star configuration with isolated neutral; the diagram introduces the notations we will use; the input reference voltage is taken to be the mid-point between the direct current bus rails. We can present the problem of control via PWM in the following manner: – starting with the reference voltages va ref , vb ref , vc ref to be imposed on terminals of the different phases of load, the first step is to determine the voltages Pa , Pb , Pc produced by the legs of the inverter, suitable reference Chapter written by Francis LABRIQUE and Jean-Paul LOUIS.

2

Power Electronic Converters

Figure 1.1. Schematic diagram showing notations used

values Pa ref , Pb ref , Pc ref such that the actual output voltages Pa , Pb , Pc lead to the desired values of the voltages v a , vb , v c ; – the next step is to transform the reference signals Pa ref , Pb ref , Pc ref into binary (or PWM) signals x j ∈ [0,1], j ∈ [ a , b , c ] corresponding to switches Sj being closed (if xj = 1) or Sj* being closed (if xj = 0), and to the production of voltages Pj , j ∈ [a, b, c] taking the value +U/2 or −U/2 depending on whether xj = 1 or 0. Then, by dividing the time into intervals [tk −1 , tk ], k ∈ N during each interval the fraction of the interval for which Pj is +U/2 (and hence the fraction for which Pj is −U/2) is altered in such a way that over each interval the mean value < Pj > of Pj matches the value of Pj ref. In case of carrier-based modulation, which is the subject of this chapter, the transformation of the reference signals Pj ref into binary signals xj is achieved by comparing these signals to a carrier wave vp (triangular or sawtoothed) whose frequency determines the intervals over which we want to match Pj ref (Figure 1.2). We have xj equal to 1 and therefore:

Carrier-Based Pulse Width Modulation

Figure 1.2. Sawtooth carrier modulation

PJ = U/2 if:

Pjref > vp Alternatively, we have xj equal to 0 and therefore:

PJ = −U/2 if:

Pjref < vp

3

4

Power Electronic Converters

Carrier-based modulation also refers to any sort of modulation where the intervals where xj is equal to 1 and those where xj is equal to 0 are produced by a microcontroller or FPGA [MON 08] using a computation that emulates the intersection process between the reference and carrier as described earlier. We will show how the reference voltages va ref , vb ref , vc ref to be applied to the load can be represented, and then describe the conversion of these values to the reference voltages Pa ref , Pb ref , Pc ref for each leg, and finally show how these values are transformed into binary control signals (PWM signals) for the switches. We will see that this enables us to: – derive the various intersective modulation strategies described in the literature (sine-triangle modulation, sub-optimal modulation, centered modulation, and flat-top and flat-bottom modulation); – establish the similarities between certain types of intersective modulation and other modulation strategies such as space vector modulation. 1.2. Reference voltages varef, vbref, vcref Since we are assuming that the load fed by the inverter is a balanced three-phase load connected in a star configuration with isolated neutral, the constraint on the currents resulting from this connection pattern, which is ia + ib + ic = 0 , leads to an equivalent constraint on the voltages va + vb + vc = 0 . Since va ref , vb ref , vc ref are the desired values for the voltages

, it is convenient to impose the same constraint on these such that ; this implies that there are only two degrees of freedom that must be set to determine the necessary reference values. v a , vb , v c

v a ref + vb ref + vc ref = 0

Traditionally, when discussing intersective PWM [KAS 91, LAB 95, MOH 89, and SEG 04] it is assumed that the reference values have the form:  va ref   vb ref   vc ref 

Vref sin θ ref      =  Vref sin θ ref − 2π / 3       Vref sin θ ref − 4π / 3

( (

) )

  ,   

[1.1]

Carrier-Based Pulse Width Modulation

5

where Vref is the desired amplitude for the voltages and θ ref is an angular coordinate obtained by integrating the desired reference pulsation for the voltages: θref =

t

∫ 0 ωref

[1.2]

dt.

We will introduce the rotation matrices P(θ) and the Clarke submatrix C32 [SEM 04]:  cosθ − sin θ  P(θ ) =    sin θ cos θ   1  C32 =  −1/ 2   −1/ 2

  + 3 / 2  − 3 / 2 0

Equation [1.1] can then be written as:  va ref   vb ref   vc ref 

  0   = Vref C32 P (θ ref )   = C32 P (θ ref   −1   

0 )   −Vref

  

[1.3]

or be represented by the diagram shown in Figure 1.3.

Figure 1.3. Generation of reference waves v a

ref

, vb

re f

, vc

re f

from the desired amplitude V ref and pulsation ω ref

r We can consider Vref and θref to represent a vector Vref rotating with speed ω ref and whose projection onto three axes mutually separated by

2π / 3 gives the reference voltages

va

ref

, v b re f , v c re f (Figure 1.4).

6

Power Electronic Converters

Vref

b Figure 1.4. “Vector” representation of the generation of reference waves v a vb

re f

, vc

ref

,

re f

In the steady state case Vref and ω ref have fixed values. In the transient case they may vary as a function of time. This classical approach takes the two degrees of freedom required to fix the reference voltages v a , vb , v c to be their amplitude Vref and their pulsation ω ref (equal to 2π times their reference frequency f ref ). There are however many applications where the load is active and includes sources of pulsation ω 0 . In this case, ω ref must be equal to ω 0 in the steady state case, and the difference between that value and ω 0 in the transient case [ ∆ω ref = (ω ref − ω 0 ) ] can be treated, after the integration relating θ ref to ω ref , as a phase shift ϕ ref added to an angle θ 0 equal to: θ0 =

t

∫ 0 ω0 dt

[1.4]

This is equivalent to taking (Figure 1.5a): θref =

t

t

∫ 0 ω0 dt + ∫ 0 (ωref − ω0 ) dt = θ0 + ϕref

[1.5]

Carrier-Based Pulse Width Modulation

7

It is then more helpful to consider Vref and ϕ ref as the control parameters (Figure 1.5b) since, in contrast to ∆ ω ref , ϕ ref is not required to be zero in the steady state regime but only to have a constant value.

a)

b)

Figure 1.5. Generation of reference waves v a ref , v b re f , v c re f when their steady state pulsation is determined by the load; a) expression [1.4]; b) expression [1.5]

On the basis of Figure 1.5b we can write:  va ref   vb ref   vc ref 

  0   = Vref C32 P(θ ref )   = Vref C32 P(θ 0 ) P (ϕref   −1  

0  ) ,  −1 

[1.6]

since a rotation by angle θ ref = θ 0 + ϕ ref is equivalent to a rotation by angle θ 0 followed by a rotation by angle ϕ ref .

8

Power Electronic Converters

Equation [1.6] can then be written as:  va ref   vb ref   vc ref 

   vd ref  +Vref sin ϕref   = C32 P(θ0 )   = C32 P (θ 0 )   −Vref cos ϕref   vq ref      

 ,  

[1.7]

so that finally:  va ref   vb ref   vc ref 

  Vref cos ϕ ref sin θ 0 + Vref sin ϕ ref cos θ 0      =  Vref cos ϕ ref sin(θ 0 − 2π / 3) + Vref sin ϕ ref cos(θ 0 − 2π / 3)       Vref cos ϕ ref sin(θ 0 − 4π / 3) + Vref sin ϕ ref cos(θ 0 − 4π / 3)    

Figure 1.6. Generation of reference waves va ref , vb ref , v c from their components dq

[1.8]

re f

Equation [1.7] can be observed to be equivalent to the diagram in Figure 1.6, where the reference quantities are taken to be: vd ref = Vref sin ϕref

and: v q ref = −V ref cos ϕ ref

instead of: V ref

Carrier-Based Pulse Width Modulation

9

and:

ϕref . since: t C32 C32 =

3 I, 2

where I is the 2 × 2 identity matrix, the voltages v d ref and v q ref can be determined from the desired voltages v a ref , v b re f , v c re f if we leftmultiply both sides of [1.7] by: 2 −1 t P (θ 0 )C 32 3

then:  vd ref   vq ref 

 va ref   2 −1 t  = P (θ 0 )C32  vb ref  3    vc ref 

  .   

[1.9]

Substituting v d ref = v ref sin ϕ ref and vq ref = − vref cos ϕ ref in [1.7] by their values as given by [1.9], we obtain the following equation:  va ref   vb ref   vc ref 

  va ref  2  t  = C32 C32  vb ref  3    vc ref  

  ,   

which shows that the matrix:  2 / 3 −1 / 3 −1 / 3  2   t C32 C32 =  −1 / 3 2 / 3 −1 / 3  3  −1 / 3 −1 / 3 2 / 3   

[1.10]

10

Power Electronic Converters

acts as an identity matrix on the vector va ref, vb ref, vc ref as long as the constraint v a ref + vb ref + vc ref = 0 is met (the homopolar component is zero). 1.3. Reference voltages Pa ref, Pb ref, Pc ref In contrast to the voltages va ref, vb ref, vc ref, the voltages Pa ref, Pb ref, Pc ref are not required to have a sum of zero. We therefore have three degrees of freedom in defining these voltages. If we introduce the homopolar component of Pa ref , Pb ref , Pc ref : P0 ref = ( Pa ref + Pb ref + Pc ref ) / 3

then the quantities: Pa ref − P0 ref = Pa − h ref Pb ref − P0 ref = Pb − h ref Pc ref − P0 ref = Pc − h ref

sum up to zero, just like the voltages va ref , vb ref , vc ref do. Making use of the definition of P0 r e f we can write:  Pa −h ref   Pb − h ref   Pc − h ref 

  2 / 3 −1/ 3 −1/ 3   Pa ref     =  −1/ 3 2 / 3 −1/ 3   Pb ref      −1/ 3 −1/ 3 2 / 3   Pc ref  

  Pa ref  2  t  = C32 C32  Pb ref  3    Pc ref  

     

[1.11]

t The matrix 2 / 3 C32C 32 , which acts as an identity matrix with respect to

three quantities whose sum is zero (Equation [1.10]), acts to eliminate the homopolar component when it is applied to three quantities that do not sum up to zero. Using a process similar to that used for the quantities va ref , vb ref , vc ref , we can represent the quantities Pa − h ref , Pb − h ref , Pc − h ref in terms of two reference quantities Pd ref , Pq re f (Figure 1.7):

Carrier-Based Pulse Width Modulation

 Pa − h ref   Pb − h ref   Pc − h ref 

   Pd ref  = C32 P (θ 0 )   Pq ref    

   

11

[1.12]

Pa − h ref Pb − h ref Pc − h ref

Figure 1.7. Generation of reference waves Pa − h ref , Pb − h ref , Pc − h ref from their dq components

By defining the matrix:

 1   C31 = 1 ,  1  

[1.13]

we can express the reference quantities Pa ref , Pb ref , Pc ref as a function of the reference quantities Pd ref , Pq ref , P0 ref (Figure 1.8):

Figure 1.8. Generation of reference waves Pa ref , Pb ref , Pc ref from their d-q-0 components

12

Power Electronic Converters

 Pa ref   Pb ref  Pc ref 

  Pd ref  C P ( θ ) =  32 0   Pq ref  

  + C31P0 ref 

[1.14]

1.4. Link between the quantities va , vb , vc and Pa , Pb , Pc Referring to the notations in Figure 1.1 we can write: Pa − Pb = v a − vb Pa − Pc = v a − vc

If

we

combine

these two equations we obtain , with the final equality reflecting the fact

2 Pa − Pb − Pc = 2 v a − vb − vc = 3v a

that va + vb + vc = 0 . Similarly, we can proceed to obtain v b and vc as a function of Pa , Pb , Pc , which leads to:  Pa   va   2 / 3 −1 / 3 −1 / 3   Pa        2 t   vb  =  −1 / 3 2 / 3 −1 / 3   Pb  = 3 C32 C32  Pb       P   vc   −1 / 3 −1 / 3 −2 / 3   Pc   c

[1.15]

Equation [1.15] states that the voltages v a , vb , vc correspond to the voltages Pa , Pb , Pc after their homopolar component P0 = ( Pa + Pb + Pc ) / 3 has been subtracted. We can therefore write: va = Pa − P0 vb = Pb − P0

[1.16]

vc = Pc − P0

Therefore, the difference between Pi and vi stems from the homopolar component: the Pi values include a homopolar component while the vi values do not.

Carrier-Based Pulse Width Modulation

13

1.5. Generation of PWM signals In order to determine the states x j , j ∈ [ a , b , c ] of the switches of each leg from the reference waves P j ref , we will consider sequentially: – the case where these waves are compared to a reverse sawtooth carrier; – the case where these waves are compared to a conventional sawtooth carrier; – the case where these waves are compared to a triangular carrier. We will assume that the carrier is normalized and varies between −1 and +1, and that the reference waves also vary in this manner since they are divided by U/2: [1.17]

P j ref , n = P j ref / (U / 2)

1.5.1. Reverse sawtooth wave Over each modulation period the carrier wave varies linearly from +1 to −1. It returns from −1 to +1 at the moment where one modulation period ends and the next begins. If Tp is the period of the carrier, over the (k+1) modulation period (from t k = kT p to tk +1 = ( k +1) Tp ) all the S ' j switches will

be closed at t k since at this moment the carrier takes the value of +1 and therefore has a value greater than the value of all the reference waves, implying that xa , xb , xc will be zero. Each leg then undergoes a transition from

S 'j

closed to S j closed at time

when the reference wave Pj ref , n intersects the carrier and takes a value greater than that of the carrier (Figure 1.9). t jk

The order in which the switches commutate depends on the order in which the carrier intersects the reference waves. There are six possible sequences: a, then b, then c; a, then c, then b; b, then c, then a; b, then a, then c; c, then a, then b; and c, then b, then a.

14

Power Electronic Converters

Pa ref ,n Pb ref ,n

Pa ref ,n

Figure 1.9. Modulation with a reverse sawtooth carrier

Carrier-Based Pulse Width Modulation

15

Each commutation causes one of the components of the vector to transition from 0 to 1; the vector starts with the value (0, 0, 0)

( x a , xb , x c )

at the start of the period, with Sa' , Sb' , Sc' all closed; it ends the period with the value (1, 1, 1). The voltage P j , j ∈ ( a, b, c), is −U/2 over the interval [t k , t jk ] , ' k = 1, 2,…., where x j is zero and where S j is closed. It is U/2 over the

interval [t jk , tk +1 ] where x j is 1 and where S j is closed. The voltages

vj ,

j ∈ (a, b, c) can be determined from the voltages P j through equation [1.15].

The time t jk , where following equation: Pj ref ,n (t jk ) = 1 − 2

P j ref , n

t jk − kT p Tp

intersects with the carrier is the root of the

;

The mean value < P j > k of t k +1 = t k + T p ) is therefore: < Pj > k =

j ∈ (a, b, c) Pj

over the modulation period

1 U  ( tk +1 − t jk ) − (t jk − t k )   Tp 2  1 U U 2 ( tk − t jk ) + T p  = Pj ref , n (t jk ) = Pj ref (t jk ) =  Tp 2 2

[1.18] (note:

[1.19]

(NOTE: Remember that P j ref , n is normalized using equation [1.17]). Equation [1.19] shows that the PWM process, over which P j takes the value −U/2 and then +U/2, results in the mean value of P j over the modulation period being equal to the value of P j re f at one particular point within an interval of this period: the point where the carrier intersects Pj ref , n . If the reference waves P j ref vary only slightly over a modulation period, the sequence of samples P j ref , n (t jk ) will provide a good representation of the reference waves. The same goes for the mean values < P j > k of the voltages P j .

16

Power Electronic Converters

Instead of the natural sampling of the P j waves that we have just described, an alternative synchronous sampling is possible, where the values of Pj ref , n for each period are based on their values P j ref , n k = P j ref , n ( kT p ) at the start of the period (Figure 1.10). We note that this synchronous sampling process for the reference waves will occur naturally if the modulation is performed numerically within a microprocessor using a calculation emulating the intersection process, or if the reference waves Pj ref , n are obtained by digital to analog conversion at the output of a computation unit whose sampling period is synchronized with the period of the carrier.

Figure 1.10. Modulation using a reverse sawtooth wave with synchronous sampling of the reference waves

For the rest of this section we will assume that the reference waves are sampled at the start of each modulation period, so that we have: <  < < 

Pa > k   U Pb > k  = 2 Pc > k 

 Pa ref , nk   Pb ref , nk   Pc ref , nk 

  Pa ref k    =  Pb ref k     Pc ref k  

   ; k = 1, 2,...   

[1.20]

Using equation [1.14] we can write:  Pa ref k     Pd ref k   + C31P0 ref k ,  Pb ref k  = C32 P(θ0 k )   Pq ref k       Pc ref k   

[1.21]

Carrier-Based Pulse Width Modulation

17

where P(θ0k ) , Pd ref k , Pq ref k are the values of their corresponding quantities at t k = kT p . Substituting [1.21] into [1.20] we obtain:  < Pa > k   Pd ref k    < Pb > k  = C32 P (θ 0 k )  P < P >   q ref k  c k

  + C31 P0 ref k  

[1.22]

Equation [1.15], which connects the voltages v a , vb , vc to the instantaneous values of the voltages Pa , Pb , Pc can also be applied to the mean values (over each modulation period) of these quantities. This gives us:  < Pa > k   < va > k     2 t   < vb > k  = 3 C32 C32  < Pb > k    < P >   < vc > k   c k

[1.23]

Substituting [1.22] into [1.23] we obtain:  < va   < vb < v  c

>k   Pd ref k   > k  = C32 P (θ 0 k )  ,  Pq ref k     >k 

[1.24]

given that: 3 2

t – C32 C32 = I , where I is (as mentioned earlier) the 2 × 2 identity

matrix: t t – C32C32 C31 = ( 0 0 0 ) .

1.5.2. Conventional sawtooth carrier Over each modulation period the carrier now varies linearly from −1 to +1 and returns from +1 to −1 at the boundary between one period and the next. Over the (k + 1)th modulation period from: tk = kTp

18

Power Electronic Converters

to:

t k +1 = ( k +1) Tp every switch S j will close at t k since at this moment the carrier takes the value −1 and therefore has a value smaller than that of each of the reference waves, implying that xa , xb and xc are equal to 1. Each leg then undergoes a transition from S j closed to S 'j closed at the time t jk when the reference wave

P j ref , n

intersects the carrier

vp

(Figure

1.11). Each transition causes one component of the vector ( x a , xb , xc ) to move from 1 to 0, starting from a value [1,1,1] at tk and finishing with a value [0,0,0] at the end of the period. The voltage

Pj

, j ∈ [a, b, c] is U/2 from tk to t jk over the interval where

x j is 1 and S j is closed. It is −U/2 from t jk to tk +1 over the interval where x j is zero and S 'j is closed. The voltages u j are linked to the voltages Pj

by equation [1.15]. The time t jk when

intersects the carrier is the solution of the

Pj

following equation: Pj ref n (t jk ) = −1 +

The mean value of < Pj >k = =

2(t jk − kT p )

Pj

(

Tp

;

j ∈ (a, b, c)

[1.25]

over the (k+1) th modulation period is therefore:

)

1 U t jk − tk − (tk +1 − t jk )   Tp 2 

(

)

1 U U −Tp + 2 t jk − tk  = Pj ref n (t jk ) = Pj ref (t jk )  2 Tp 2 

[1.26]

If we adopt a synchronous sampling scheme for the reference waves, we obtain:

Carrier-Based Pulse Width Modulation

Pa ref ,n Pb ref ,n Pc ref ,n

Figure 1.11. Modulation by a conventional sawtooth carrier

19

20

Power Electronic Converters

< Pj > k = Pj ref k = Pj ref (kT p ); k = 1, 2,...

[1.27]

When the reference waves are sampled at the beginning of the modulation period, equations [1.20] to [1.24] apply equally well to the case of a conventional sawtooth carrier as to the reverse sawtooth carrier. 1.5.3. Triangular carrier Modulation by a triangular carrier can be considered as equivalent to repeated modulation, first by a reverse sawtooth wave and then by a conventional sawtooth wave. The period of the carrier is twice the duration T p / 2 of each of the ramps (first decreasing and then increasing) that constitute the carrier (Figure 1.12). If the period starts with modulation by a decreasing ramp, at the start of the period all the switches S 'j are closed since at this point in time the carrier has a value greater than that of every reference wave. Each leg then undergoes a transition from S 'j closed to S j closed at the time when the corresponding reference wave crosses the carrier wave; by the end of the decreasing ramp all the switches S j are closed. During the increasing ramp each arm undergoes a transition from S j closed to S 'j closed, such that at the end of the period the situation is once again when all the S 'j switches are closed. There is no longer, as was the case with sawtooth waves, a moment where all the legs commutate simultaneously at the point between one modulation period and the next. If the reference waves are sampled, this may occur: – at the start of each modulation period, as was the case with sawtooth carriers (Figure 1.13);

Carrier-Based Pulse Width Modulation

Pa ref ,n Pb ref ,n Pc ref ,n

Figure 1.12. Modulation by a triangular carrier

21

22

Power Electronic Converters

Figure 1.13. Modulation by a triangular carrier with synchronous reference sampling at the start of each carrier period

– at the start of each sawtooth component of the carrier, which means that the waves P j match the mean values of their reference waves P j re f on the scale of every half-period of the modulation (Figure 1.14).

Figure 1.14. Modulation by a triangular carrier with synchronous reference sampling at the start of each half-period

Depending on the way the sampling is performed, we have either: – from t k = kT p to tk +1 = ( k +1) Tp ,

k = 1,2, …

< P j > k = P j ref k = P j ref [ kT p ]

– or from

t k = kT p / 2

to tk +1 = ( k +1) Tp / 2 ,

[1.28] k = 1,2, …

Carrier-Based Pulse Width Modulation < P j > k = P j ref

k

= P j ref [ kT p / 2]

23

[1.29]

Over each half-period of the carrier we have, as with sawtooth carriers, six possible switching sequences, depending on the values of Paref k , Pbref k , and Pc ref k over this half-period. For a half-period consisting of a decreasing ramp, the transitions from S 'j

closed to S j closed occur first on the leg whose reference voltage is

largest, and then on the leg with the intermediate reference voltage, and finally for the leg whose reference voltage has the smallest value. Thus, if Pa ref k > Pb ref k > Pc ref k the commutations will occur first on leg a, then on leg b, and finally on leg c and the vector [ xa , xb , xc ] representing the states ' of the switches on each leg ( S j closed for x j = 0; S j closed for x j = 1 )

moves from (0,0,0) to (1,0,0), then to (1,1,0), and finally to (1,1,1). A similar process can be used to determine the sequence of values of the vector ( xa , xb , xc ) and hence the states of the switches for each of the five other cases. For a half-period consisting of an upward ramp, the vector ( xa , xb , xc ) moves gradually from (1,1,1) to (0,0,0) with the transitions from S j closed '

to S j closed, acting first on the leg whose reference voltage is smallest, then on the one with the middle reference voltage, and finally on the leg with the largest reference voltage. It can be seen that the twelve switching sequences we have just defined are identical to those that are obtained using space vector modulation (Chapter 2 and reference [LAB 98]). Modulation by a triangular carrier has the property that it is indiscernible in terms of the switching sequences from space vector modulation. 1.5.4. Note A modulation based on a random carrier is sometimes used, selecting in a non-deterministic manner for each period, either a conventional or a reverse sawtooth.

24

Power Electronic Converters

1.6. Determination of the reference waves Pa ref k , Pbref k , and Pc ref k from the reference waves va ref k , vbref k , vc ref k As we saw in section 1.5, with PWM, P j will only match P j re f when averaged over a given period of modulation. The same clearly applies to the voltages v a , vb , vc with respect to the reference waves va ref , vb ref , vc ref . Here again, the problem is to determine over each modulation period the values of the waves Paref k , Pbref k , Pc ref k (or the Pd ref k , Pqref k , P0 ref k components of these waves) such that we obtain:  < va > k   va ref k     < vb > k  =  vb ref k  < v >    c k   vc ref k

     

[1.30]

Substituting [1.30] into [1.24] we obtain the equation that must be used to link the reference values for the voltages of each phase and the reference values for the dq components of the voltages in each leg:  va ref k   vb ref k   vc ref k 

   Pd ref k  = C32 P (θ 0 k )   Pq ref k    

   

[1.31]

If we multiply both sides of [1.31] by: 2 −1 t P (θ0 k )C32 3

we obtain:  va ref k     Pd ref k  2 −1 t   = P (θ 0 k )C32  vbref k  .  Pq ref k  3      vcref k   

[1.32]

Carrier-Based Pulse Width Modulation

25

We then substitute [1.32] into [1.21] to obtain:  Paref k   va ref k    2   t  Pbref k  = C32C32  vbref k  + C31 P0 ref k .   3    Pcref k   vcref k     

Since the matrix

2 t C32C32 is equivalent to an identity matrix for the 3

quantities varef k , vbref k , vcref reduced to:  Paref k   Pb ref k   Pc ref k 

[1.33]

  va ref k    =  vb ref k     vc ref k  

k

that sum up to zero, equation [1.33] can be

   + C31 P0 ref k   

[1.34]

Equation [1.34] shows that the reference values va ref , vb ref , vc ref fix the values of Paref k , Pbref k , Pc ref k except for their homopolar component P0 ref k , which is a remaining degree of freedom, which can be manipulated to optimize the modulation to match some desired quality criterion. This result is consistent with the statement given at the end of section 1.4. 1.6.1. “Sine” modulation “Sine” modulation is obtained if in equation [1.34] we take the homopolar component P0 r e f of the reference waves P j re f to have a value zero, which makes these waves equal to the reference waves v j ref :  Paref k   Pb ref k   Pc ref k 

  va ref k    =  vb ref k     vc ref k  

     

[1.35]

In the steady state case the P j re f waves then form a balanced three-phase system of sinusoidal voltages, just like the waves v j ref , and therefore the term “sine” modulation is given (Figure 1.15).

26

Power Electronic Converters

Figure 1.15. Sinusoidal modulation with triangular carrier

The amplitude of the P j re f waves, and hence the amplitude of the sinusoidal waves that can be produced in the steady state case at the threephase terminals of the load, cannot be greater than U/2 if we wish to avoid the emergence of an effect known as overmodulation1. Compared to full-wave control, which gives voltages at the three-phase 2U terminals of the load whose fundamental component has amplitude , π “sine” modulation incurs a reduction in amplitude of: U /2 =π /4 2U / π

[1.36]

or a reduction of 21%. This reduction in amplitude is known as “voltage drop due to pulse width modulation” [LAB 95].

1. This effect represents the absence of any intersection between the carrier and a reference wave for one or more modulation periods, because the value of a reference wave is greater than the maximum value of the carrier (or less than its minimum value). The voltage Pj is thus equal to +U/2 (or −U/2) over the entire interval. When overmodulation occurs, the equality k = Pjref k is not maintained. For a detailed analysis of overmodulation see Chapter 3.

Carrier-Based Pulse Width Modulation

27

1.6.2. “Centered” modulation “Centered” modulation is when a value for the homopolar component in equation [1.34] is taken to be equal to + minus half the sum of the largest and smallest of the reference waves va ref k , vb ref k , vc ref k . If max( v j ref k ) denotes the operation of selecting the largest of the reference waves va ref k , vb ref k , vc ref k and min( v j ref k ) the operation for selecting the smallest of the waves, we obtain:  Paref k   Pb ref k   Pc ref k 

  va ref k    =  vb ref k     vc ref k  

  1  − C31  max( P j ref k ) + min( P j ref k )   2  

[1.37]

It can be seen that the value of the homopolar component has the effect of causing the largest and smallest of the reference waves P j ref k to lie symmetrically on each side of the horizontal axis, and hence the term “centered” is given. In the case where the reference waves v j ref form a balanced three-phase system with sinusoidal values of amplitude V and pulsation ω: v a ref = V ref sin ω ref t vb ref = Vref sin(ω ref t − 2π / 3)

vcref = Vref sin(ωref t − 4π / 3),

equation [1.37] gives the following voltages P j re f (Figure 1.16): – between ω ref t = − π / 6 and ω ref t = π/6, the voltage vc ref is the most positive and vb ref is the most negative; we therefore have: 3 Paref = varef − 1/ 2(vbref + vcref ) = Vref sin ωref t 2

Pbref = vbref − 1/ 2(vbref + vcref ) = −

3 Vref cos ωref t 2

28

Power Electronic Converters

3 Vref cos ωref t 2

Pcref = vcref − 1/ 2(vbref + vcref ) = +

– between ωref t = π/6 and ωref t = π/2 , the voltage v a r e f

is the most

positive and vb ref is the most negative, such that: Paref = varef − 1/ 2(varef + vbref ) =

3 Vref cos(ωref t − π / 3) 2

Pbref = vbref − 1/ 2(varef + vbref ) = −

3 Vref cos(ωref t − π / 3) 2

Pcref = vcref − 1/ 2(varef + vbref ) = −

3 Vref sin(ωref t − π / 3) 2

and so on.

Figure 1.16. Centered modulation with a triangular carrier

The amplitude of the reference waves Paref , Pbref , Pcref is never greater than U/2 and there are no saturation effects as long as: 3 U Vref < 2 2

Carrier-Based Pulse Width Modulation

29

or when: Vref
1 MW) and moderate voltages (from around 1 to 10 kV) that could not be met with standard switching cell designs, led to a wide range of studies throughout the world.

PWM Strategies for Multilevel Converters

245

Many multilevel conversion structures have appeared and reappeared over the years. The NPC structure can be modified into an active NPC by introducing controllable power switches associated with the diodes [BRU 01, BRU 05a]. Another variant using flying capacitors for voltage balancing (Figure 9.3a) can be used as in [MEY 02a], has given rise to variants performing direct AC/AC conversion [MEY 02b], [TUR 02], can be adapted to operate in a soft-switching mode, can be stacked (Figure 9.3b) [DEL 01, DEL 03], and has characteristics that make it useful for fault tolerant applications [MEY 02c]. An ANPC can be transformed into a five-level device by adding a flying capacitor (Figure 9.3c) [BAR 03], [BAR 05] and still other combinations are possible to produce switching cells with improved performance [MEY 06]. It is however also possible to create multilevel waveforms by combining two isolated two-level converters (Figure 9.3d) or non-isolated converters (Figure 9.3e, f) [PEN 96]. Finally, it is possible to create devices involving combinations of more than one multilevel converter [ITU 08]. At the same time as this revolution was taking place in the field of highpower conversion, developments in microprocessors led to an increase in their power demands and a reduction in their working voltages. The huge increase in the current they draw, along with their unusual dynamic requirements, naturally led designers to suggest supplies based on a number of parallel conversion structures with interleaved control (Figure 9.4) [FOR 07a], [ZHA 08]. The existence of magnetic components within these interleaved structures also led to the development of isolated variants (Figure 9.5). Finally, taking advantage of the isolation of various electrical paths, four different variants can be imagined for each design involving series or parallel coupling of the primaries and secondaries, several different combinations of which have proved popular (Figure 9.6) [FOR 07b], [VIS 04]. Thus, while their aims and terminology were very different, the low voltage and medium voltage communities developed power conversion techniques, sometimes apparently in isolation from one another, which served to divide the power to be transferred among a number of “commutation cells” in the traditional sense of the term, to produce multilevel voltages or waveforms – the apparent frequency of which is higher than the effective operating frequency of the power switches.

246

Power Electronic Converters

a)

b)

c)

Figure 9.3. Series multilevel converters; a) multi-cell with flying capacitor; b) stacked multi-cell; c) Active five-level NPC; d) cascade inverter structure; e) three-level step-up chopper; f) three-level rectifier

PWM Strategies for Multilevel Converters

247

d)

e)

f)

Figure 9.3. (Continued)

248

Power Electronic Converters

a)

b)

Figure 9.4. Examples of parallel multilevel structures; a) inductances in a star configuration (coupled or uncoupled); b) with inter-cell transformers

PWM Strategies for Multilevel Converters

249

Figure 9.5. Examples of isolated parallel structures

The complexities of these structures are in many cases justified by the following desired performance requirements: – increased voltage or use of low voltage components, which are less expensive and faster; – reduction in filter volume;

250

Power Electronic Converters

Figure 9.6. Examples of hybrid series/parallel structures

– improved dynamics by increasing the apparent frequency, but primarily by reducing the energy stored in filters; – better distribution of heat sources by dividing up the work between more switches; – standardization of operating limits through the use of standardized modules; – increased reliability through division of work or redundancy. It should be noted that although multilevel converters can promise all these improvements, increased modularity, and above all reliability, is far from trivial to achieve in practice. The aim of this chapter is to demonstrate that the features common to all these structures (presence of several elementary switching cells where each switching operation leads to a variation in the instantaneous power transfer, either by varying the voltage applied to the current source or by varying the current flowing through the voltage source), can be used to develop general control principles that can be applied to any structure of this type.

PWM Strategies for Multilevel Converters

251

We will attempt to show that the control of these converters can be divided into two parts (Figure 9.7): – a completely generic modulator which makes suitable use of the different levels of energy transfer; it can be used to determine the sequence of levels that should be used for each phase in order to achieve any desired goal. This part of the control strategy is independent of the topology of the converter; it considers for example the need to make effective use of the three possible transfer levels in each leg of a three-phase inverter; this need is the same whether in a NPC converter, a multi-cell converter, or parallel inverters with or without coupled inductances. This aspect of the control strategy typically deals with control issues (response times, tracking of the reference, bandwidth, etc.) and spectral effects (harmonic distortion, EMC conformance, psopho-metric weighting, etc.). This is also the part where any redundancy within a multi-leg system is considered; – generators of control signals which are, in contrast, intimately tied in with the topology. These involve determining the commands to be sent to each switch in order to generate the required waveform, while fulfilling or optimizing constraints that are more directly connected to the topology (respecting the requirements of individual switching cells, the detailed relationship between switch state, and transfer levels, etc.), practical implementation of switching functions (minimization or suitable distribution of switching losses, distribution of energy fluxes between the various paths, minimum switching times, etc.), and management of stored energy (capacitative midpoint, flying capacitor, wiring inductance, etc.). Note that most multilevel topologies exhibit redundant states even on the level of a single phase (states that give the same level of energy transfer), which opens up the possibility of choosing between different strategies and optimizing some chosen criterion. As we saw in section 9.1, since all multilevel structures make use of additional reactive components, their corresponding state variables must be controlled, and it is generally these redundant states that enable this control to be performed either at the generator stage or the modulator stage. Without attempting an exhaustive description, we will first present the way in which the control of various types of multilevel structures can be approached, focusing on a single phase, by associating with each topology a

252

Power Electronic Converters

command generator that is designed to take into account the restrictions of that particular topology. Required multilevel signals

Phase reference voltages

V V V

+VDC/

M o d

G G Gé é enn né é

Modulator

Multilevel output voltages

Control signals

Control signal generator (1 per phase)

C Vdéc c C C V V Vdéc b VDC V V X X sw a N XN L VDC Multilevel converter (1 per phase)

Figure 9.7. Schematic showing control of a multilevel converter

We will then see how this control can be understood in the most generic manner possible. Nevertheless, the problem is rather different depending on the number of phases to consider and we will discuss the three most commonly encountered systems: power conversion systems with one, two, and three phases. 9.2. Modulators 9.2.1. Recap: two-level modulators A simple way of obtaining a digital (i.e. two-level) signal whose mean value over each switching period is proportional to a reference signal sampled at frequency f is to compare this reference signal with a triangular carrier wave of the same frequency f. In the absence of saturation, the switches will undergo exactly two transitions per carrier period and the switching frequency is therefore precisely controlled, which is important when evaluating inverter losses.

PWM Strategies for Multilevel Converters

253

Figure 9.8a. Two-level carrier-based modulation: the phase of the first harmonic of the switched voltage is constant over the entire modulation range

Another crucial characteristic of the resultant signal is the phase of the harmonics, and in particular, that of the component at the carrier frequency. As can be seen from Figure 9.8, which shows the signal produced by the carrier–reference comparison along with its component at frequency f, which here is determined by a simple second-order filter; this harmonic is modulated in amplitude but it has a fixed phase and it is in opposition with the carrier. S(t)

(Φ/2π).Τ

α.Τ

t T Figure 9.8b. Modulation harmonic

254

Power Electronic Converters

This can also be seen from the following expression, by observing that the nth harmonic of a square wave signal taking the values 0 and 1, of phase Φ and cyclic ratio α, is given by: Hn =

2 .sin(n.π .α ).e j.n.Φ n.π

The first voltage harmonic component, with a high amplitude and the lowest frequency, is the main cause of current variation in a two-level half bridge and we know that when we come to considering full bridge operation we will want to use carriers for each leg that are in phase with each other, in other words, a single carrier used for both. Modulation using these carriers will therefore have the effect of introducing harmonics on each leg that are in phase with one another. Harmonics which will therefore tend to disappear from the point of view of the load, which experiences the difference between these two voltages. In the case of a single-phase bridge, this compensation is perfect as long as the reference signals for each leg are symmetric (which is generally the case), as can be seen from the example in Figure 9.9.

Figure 9.9. Compensation of the first family of switching harmonics in a single-phase inverter with two two-level inverter legs; above: switched voltage spectrum from one inverter leg; below: differential voltage spectrum as provided to the AC source

PWM Strategies for Multilevel Converters

255

Thus, once we transition from a half bridge to a correctly controlled full bridge, the dominant harmonic will become the 2f harmonic and the total weighted harmonic distortion, which is fairly representative of the variation in current, is reduced by roughly a factor of four: ∞

WTHD =



H



n  ∑  n.Fond 

2

2

In a similar way, in a three-phase inverter the use of the same carrier for all three phases ensures that the phases of the switching harmonics for all three output phases are the same, and so once again the voltage harmonics are reduced. Nevertheless, in the three-phase case the U and V phase references, for example, do not have equal amplitudes and opposite signs, as was the case in a single-phase bridge; as a result the amplitudes of the switching harmonics will be different and the compensation will only be partial. Certain strategies, the most developed of which is centered space vector modulation, enable a common-mode component to be inserted in order to maximize this compensation effect. 9.2.2. Multilevel modulators 9.2.2.1. Single-phase case In a multilevel conversion system, it is possible to achieve analogous compensation of switching harmonics within a single phase although there are certain differences: – The combination of switched signals takes the form of a sum rather than a difference, as was the case in a full bridge. This means that compensation between two harmonic components will now occur when their carriers are out of phase by 180° rather than 0°. – Most multilevel structures can be generalized to an arbitrary number of levels, and so there may be p switched signals to be added together. It is thus useful to vary the phase of the switched signals by 360° per p in order to

256

Power Electronic Converters

cancel out not only the harmonic components at f but also those at 2.f, …, (p-1). f since the nth order harmonic has the form: VSn =

p

2

∑ n.π .sin(n.π .α k ).e j.n.Φ

k

.

k =1

In order to achieve these out-of-phase switching signals, it is natural to use carriers that are out of phase by 2π/p and to add together the result of these comparisons (PS, or phase-shifted, modulation): p

Vdec _ total =

∑Vdec _ i k =1

The three-carrier example in Figure 9.10 shows that the use of a phase difference of 120° enables the first and second order harmonics to be almost completely eliminated, and more generally all families whose order is not a multiple of three.

Figure 9.10a. Four-level phase shifted modulator with three carriers out of phase by 120°; above: carriers and reference; below: switched voltage Vsw_total

PWM Strategies for Multilevel Converters

257

Figure 9.10b. Four-level phase shifted modulator with three carriers out of phase by 120°; above: voltage spectrum Vsw_1 ; below: voltage spectrum Vsw_total

9.2.2.2. Multilevel inverter with two-phase bridge The transposition of known results for two-level systems is trivial for the two-phase multilevel bridge; use of the same carriers for both phases ensures that harmonics are generated in each phase such that the first families cancel from the point of view of the differential voltage experienced by the load. This is shown in Figure 9.11, an example of a two-phase four-level system using three carriers out of phase by 120°. Within each phase, the 1st, 2nd, 4th, and 5th order families cancel, and the 3rd order family cancels when the voltage difference between two phases is considered. In the end, the only families present in the differential voltage are those families whose order is a multiple of six. 9.2.2.3. Multilevel inverter with three-phase bridge Since the transition from a single multilevel phase to two multilevel phases gives the same harmonic compensation as with two levels, we might expect a direct generalization to the case of a multilevel three-phase system. However, a slightly closer investigation of the harmonic behavior of the four-level waveform generated using phase-shifted carriers (Figures 9.12 and

258

Power Electronic Converters

Figure 9.11. Two-phase four-level inverter; above: voltage spectrum Vsw_1; center: voltage spectrum output from a single phase (four levels); below: voltage spectrum delivered to the AC source

9.13) reveals that when the reference transitions from a value greater than 1/3 to a value less than 1/3, the useful part of the carriers follows a triangular pattern that is shifted by half the period relative to what it previously was. This indicates that when this transition occurs, the phase of the switching harmonics changes by 180°. This rotation of the phase of the harmonics is not particularly important in the case of a half bridge and is effectively irrelevant in the case of a full bridge since the transitions will take place at the same time in both phases and so their effects will cancel out.

PWM Strategies for Multilevel Converters

259

Figure 9.12. Modulation with four-level phase-shifted carriers: the phase of the first switched voltage harmonic changes by 180° at certain points (reference = 1/3 and +1/3)

Conversely, in case of a three-phase inverter these transitions will occur at different moments in each of the three phases; the switching harmonics will sometimes be added and sometimes subtracted.

Figure 9.13. Modulation with four-level phase-shifted carriers: depending on the value of the reference relative to 1/3 and +1/3, the useful part of the carriers generates a signal at 3f with phase 0 or 180°

In order to stabilize the phase of the switching harmonics in multilevel converters, a strategy involving in-phase carriers was introduced and developed under the name of “phase disposition” (PD) [CAR 92]. In this

260

Power Electronic Converters

case the carriers are shifted by a DC component and their frequency is p times the frequency of the carriers used in the previous case in order to obtain the same number of switching operations when added up over all the comparison signals (Figure 9.14).

Figure 9.14. Four-level PD modulator: three carriers in phase

Using this approach with distributed phase-shifted carriers, it is also possible to reconstruct a modulator that is equivalent to the phase-shifted carrier modulator by reversing the phase of every other carrier (Figure 9.15). Such a modulator is referred to in the literature by the term “phase opposition disposition” (POD).

Figure 9.15. Modulator with vertically distributed carriers (above: PD; below: POD)

PWM Strategies for Multilevel Converters

261

Based on what we have seen so far, it is clear that the PD modulator is particularly promising for three-phase applications. The most spectacular and most directly observable effect is applied to the voltage between phases (Figure 9.16). In the case of phase-shifted carriers, areas are seen where the voltage is switched over several different levels within a single switching period, which is of course not optimal. In contrast, in the case where the carriers are in phase with each other, no more than two levels are ever used within a given switching period. This advantage results of course in an attenuation of the first family of voltage harmonics and a reduction in the amplitude of the variation in an RL load (Figure 9.17). The use of in-phase carriers still does not, however, result in optimal waveforms; it is simply a way of synchronizing the harmonics of the three phases in the way that occurs naturally in the two-level case with a threephase sinusoidal modulation and a single common carrier. In order to optimize the modulation, we still need to generalize the principle of centered space vector modulation. This can be achieved in a range of different ways. Without suggesting that it is the best method to use, we will demonstrate a construction based on carrier-based modulation with insertion of a suitable zero-sequence component, since this construction fits more naturally into the structure of this chapter. This zero-sequence component is generated in a process involving several stages [MCG 06], which we will describe here for the case of an inverter fed by a DC voltage VDC, intended to produce the voltages for phases Va, Vb, and Vc, which can in turn be calculated from the dq components. The first stage involves adding harmonics whose order is a multiple of three: Vk ' = Vk − [max(Va , Vb , Vc ) + min(Va , Vb , Vc )] / 2

for:

k = a, b, c.

262

Power Electronic Converters

Figure 9.16. Three-phase four-level modulation: carriers, reference, and voltage between phases; above: phase-shifted carriers (PS); below: vertically distributed carriers (PD)

PWM Strategies for Multilevel Converters

263

Figure 9.17a. Effects of modulation strategy on current fluctuations; above: phase-shifted carriers (PS) ; below: vertically distributed carriers (PD)

A modulo function is then used to ensure that the various vectors forming the sequence are centered over each switching period: Vk " = [Vk '+ ( N − 1)VDC / 2]mod(VDC ) .

These components are then combined as follows: Vref

_k

= Vk '+

VDC − [max(Va ", Vb ", Vc ") + min(Va ",Vb ", Vc ")] / 2 . 2

The effect of adding such a zero-sequence component can be seen in Figure 9.18. In the example shown there, the reference voltages Va, Vb, and Vc all intersect with the same carrier but this line of reasoning can be extended to the case where they intersect with different carriers, which are all in phase with one another.

264

Power Electronic Converters

Figure 9.17b. Influence of modulation strategy on current fluctuations. Vertically distributed carriers (PD)

Figure 9.18. Effect of adding a zero-sequence component

Between 0 and T where no zero-sequence component is present, the signals are all symmetric with respect to T/2 but all the signals will have a period of T and non-zero harmonics at the frequency 1/T.

PWM Strategies for Multilevel Converters

265

Conversely, from T to 2T, the insertion of a zero-sequence component enables the two vca pulses to be centered in the middle of the intervals [T/2 ; 3T/2] and [3T/2 ; 2T]. With this zero-sequence component, the period of the signal vca will be T/2 and the harmonic at frequency 1/T is zero. Attenuation of the harmonic of frequency 1/T can also be achieved for the voltages vab and vbc whose pulses are also close to the center of these same intervals, but in this case the attenuation is only partial. The improvements brought by using in phase carriers on the one hand, and adding an appropriate zero-sequence the other hand, are illustrated in Figure 9.19. As can be seen from this figure, the level utilization is improved and a significant reduction of the current ripple is obtained. Finally, we note that with vector modulation with in-phase carriers, all three levels of each phase are exploited in the case of low modulation depths, whereas there is an entire accessible region of operation, i.e. where only the low and medium levels are used. It turns out that if this situation of two levels per phase is imposed by adding a DC zero-sequence component to the references at low modulation depths, the distortion (WTHD) and the current fluctuations are slightly smaller. This can be generalized to an arbitrary number of levels [MCG 06]. The idea then is to use the minimum number of levels for each phase that is compatible with the modulation depth. With multilevel space vector modulation, as the modulation depth is increased a p-level converter will use: – 2, then 4, … and finally p levels if p is even; – or 1, then 3, … and finally p levels if p is odd. With injection of a suitable DC component, the p-level converter will use 1, then 2, then 3, etc., and finally p levels. Figure 9.20 gives an example showing the effects of inserting a DC zerosequence component into a five-level converter. It can be seen that the injection of the DC component occurs for the Va, Vb, and Vc signals, and that the voltage Vref_a shown here is significantly altered.

266

Power Electronic Converters

Figures 9.21a and 9.21b show the waveforms obtained for increasing modulation depths in case of converters with respectively 3 and 5 levels per phase.

Figure 9.19. Comparison of waveforms obtained using various modulators for a three-level converter (from 0 to 2.5 ms: POD-sine, from 2.5 to 5 ms: PD-sine, from 5 to 8 ms: PDCSV)

PWM Strategies for Multilevel Converters

267

Figure 9.20. Centered PD vector modulation for a five-level converter (modulation depth =0.6): from 0 to 2.5 ms, no DC component, from 2.5 to 5 ms, injection of zero-sequence DC component. From top to bottom: a) voltage Vref_a and carriers; b) decoupled voltage for phase a; c) voltage between phases ab; d) phase current

Table 9.1 shows the value of the DC component and the regions where this DC component must be injected. 9.2.2.4. Discontinuous strategies In the previous sections, the zero-sequence component was used to reduce the harmonic distortion of the inter-phase voltages without altering the carrier frequency.

268

Power Electronic Converters

Number of levels

Offset

Offset required for

2

0

never

3

1/2

0.00 ≤ M < 0.37

4

1/3

0.23 ≤ M < 0.60

5

1/4

0.00 ≤ M < 0.17 0.43 ≤ M < 0.73

6

1/5

0.15 ≤ M < 0.33 0.62 ≤ M < 0.83

7

1/6

0.00 ≤ M < 0.12 0.30 ≤ M < 0.53 0.68 ≤ M < 0.92

Table 9.1. Characteristics of the offset that must be injected to optimize the spectrum

It is also possible, as in the case of two-level converters, to use the zerosequence component not to reduce distortion but rather to produce saturation, or one of the three references without modifying their differential components. The result of this is to inhibit switching for the corresponding phase and thus reduce switching losses. Such modes of operation are not easy to compare in a generalized and fair manner with those that have been considered so far since they involve very different compromises between distortion and switching losses and a fair comparison requires switching losses to be modeled and the carrier frequency to be reconsidered. In a given application, or in the context of an explicit design specification, we can nevertheless compare the two approaches, for example by adapting the carrier frequencies to obtain equivalent losses and then comparing their respective distortions. Experience shows that in many cases discontinuous strategies have a great deal of potential.

PWM Strategies for Multilevel Converters

269

Figure 9.21a. Waveforms over the entire operating range (centered vector modulation, carriers in phase, DC injection). Three-level converter

270

Power Electronic Converters

Figure 9.21b. Waveforms over the entire operating range (centered vector modulation, carriers in phase, DC injection). Five-level converter

PWM Strategies for Multilevel Converters

271

Using the approach taken above, with a modulator and a generator of control signals, the various discontinuous two-level strategies set out in the literature, can be directly applied to multilevel converters. Figure 9.22 gives an example of a comparison between space vector control (9.22a) and discontinuous control (9.22b). Note in particular the phases during which switching is suspended and the resultant reduction in commutation losses. In this case the comparison is made with the assumption that losses are proportional to the current. The phases over which switching is suspended represent 33% of the total time, but since in this case that period coincides with the highest current values, the reduction in losses is much greater than 33%; in this example it might be possible to double the switching frequency and still obtain a level of losses comparable with space vector modulation. When the frequency of the carriers is doubled in order to return to a similar level of losses, the current spectrum is improved; the results for this are shown in Figure 9.23. This comparison is only given for illustrative purposes and clearly the results may vary depending on the operating point and on the model of how losses vary as a function of current. Conversely, it is also possible to use intermediate levels to temporarily suspend switching, which significantly increases the number of possibilities available [BRU 05b]. Nevertheless, it is the topology that determines whether it is convenient or awkward to exploit these intermediate levels over a sustained period of time (several carrier periods); in practice, most multilevel structures incorporate passive components whose state variables will be altered when these intermediate levels are used and the undesirable consequences of suspending switching for a given phase must be considered. In a structure with flying capacitors, the voltages across the flying capacitors may reach dangerously high values, while in interleaved structures it is the fluxes that will be affected and may reach saturation. Finally, in cascaded structures with isolated sources, it is the input currents that must be carefully controlled. These discontinuous control strategies can therefore only be considered if the specific details of the topology are taken into account; however, in certain cases they may prove advantageous.

272

Power Electronic Converters

Figure 9.22a. Waveforms over the full operational range. From top to bottom: current, phase voltage, inter-phase voltage, switching losses. Centered space vector modulation

PWM Strategies for Multilevel Converters

Figure 9.22b. Waveforms over the full operational range. From top to bottom: current, phase voltage, inter-phase voltage, switching losses. Discrete modulation (psi = 15°)

273

274

Power Electronic Converters

Figure 9.23. Phase current spectrum in a three-level inverter (Fsw= 20 kHz; E = 350 V, modulation depth 1.0, Rload =2.55 Ω, L = 80µH); a) centered vector modulation at 20 kHz; b) discrete modulation (psi = 15°) at 20 kHz; c) discrete modulation (psi = 15°) at 40 kHz

9.3. Examples of control signal generators for various multilevel structures 9.3.1. “3-point” inverters (Neutral Point Clamped Inverter) This four-switch circuit can act as the leg of an inverter capable of providing a voltage twice as high as a standard inverter leg using two switches with the same voltage characteristics. The voltages at the upper and lower switch terminals will never be greater than E/2, thanks to the diodes connected to the mid-point. This two-stage four-switch circuit can in theory take 24 different states. In practice, this power system must follow rules governing how the sources may be interconnected (no short-circuit of voltage sources, no open-circuit current sources). This significantly reduces

PWM Strategies for Multilevel Converters

275

the number of available possibilities. If we assume that the sign of the current is unknown, only three states satisfy all the constraints, and consequently it is only these three states that are usable (Figure 9.24). A1

A1

A1

A2

A2

A2

B2

B2

B2

B1

Vdec=0

B1

Vdec=v

B1

Vdec=v/2

Figure 9.24. The three possible states for a leg of an NPC inverter

This first structure is in fact a very special case since the number of states is equal to the number of levels. There is no redundancy and the control signal generator for this three-level leg is therefore simply a table mapping the required levels to commands: – high: 1100; – medium: 0110; – low: 0011. This absence of redundancy therefore results in a very simple control signal generator but it also leads to unavoidable restrictions. For example, when the modulation is in the “high” region (use of only the high and medium states), it is always the same two switches that commutate. Restrictions on the minimum conduction and isolation times for the switches have therefore a direct impact on the achievable switched waveforms and switching losses are not spread evenly between the four switches. 9.3.2. Flying capacitor inverters In this inverter arm (Figure 9.25), a flying capacitor is added to the four switches that are meant to have the bus voltage divided between them.

276

Power Electronic Converters

We can then identify two pairs of switches whose states must be opposite to one another in order to follow the rules governing how sources must be connected. The pair consisting of the central switches is a standard switching cell – the fact that the switches and current sources are connected to the same circuit node implies that these two switches cannot both be open at the same time and since the switches and the voltage source VC belong to the same mesh, the two switches must not conduct simultaneously. The pair formed by the upper and lower switches is also a switching cell in that the switches and the current source are connected via a cut set, which results in the same equations (a cut set is a curve encircling a short circuit, and this concept can be used to generalize the law applying to a circuit node: the sum of all currents across the cut set is zero), and where the switches belong to the same mesh as the voltage sources E and VC. We therefore have two switching cells which are respectively subject to constraints on (VC, I) and (E-VC, I). Thus, in this structure, if the voltage across the capacitor can be maintained close to half the bus voltage, the voltage will indeed be equally divided across the four switches in series.

Vdec

Figure 9.25. Inverter leg with flying capacitor

More generally, the control signals for A1 and A2 (denoted as SC1 and SC2) implicitly define the states of B1 and B2 ( B1 = A1 ; B2 = A2 ) and since the voltages across the terminals of the non-conducting switches are always V/2, we have at any given moment:

v B1 = SC1 ⋅

V 2

[9.1]

PWM Strategies for Multilevel Converters

Vdec=V

Vdec=0

Vdec=V/2

Vdec=V–V/2

277

Figure 9.26. Circuit diagram using complementary switches

V 2

[9.2]

vdec = v B1 + v B2

[9.3]

v

B2

= SC

2



⇒ vdec = ( SC1 + SC2 ) .

V 2

.

[9.4]

There are therefore four states for these two switching cells, corresponding to three switched voltage levels. There is a redundancy in the

278

Power Electronic Converters

medium level, which can be obtained for the states [SC1, SC2] = [0,1] and [SC1, SC2] = [1,0]. Furthermore, the current witnessed by the flying capacitor can be written as the difference of the currents crossing A1 and A2, and at all times it is therefore equal to the difference in the control signals multiplied by the current I: ⇒ iC = ( SC1 − SC 2 ) . I

[9.5]

This current is zero for the high and low levels and when the medium level is required, we can make use of the redundancy to select the sign of this current. Before even considering regulating this voltage, we must ensure that there are alternate charging and discharging phases for the capacitor. It may be tempting to achieve this alternation through shifted carriers and comparators that directly deliver the commands to the four switches (Figure 9.27).

Figure 9.27. Modulator with phase-shifted carriers

This extremely simple circuit is a very good solution in structures with one or even two phases and it is quite reasonable to use in practice since the self-balancing property of this circuit (Figure 9.27) eliminates any need for regulation of the voltage across the flying capacitor. Nevertheless, such a control strategy turns out to perform poorly for three-phase systems and the inability to identify a distinct modulator and control signal generator makes it impossible to appreciate certain subtleties

PWM Strategies for Multilevel Converters

279

in multilevel modulation. Because of this, we will instead use the representation in Figure 9.28, which is in this specific case entirely equivalent. It does however distinguish between the modulator and the control signal generator, which takes the form of a state machine.

Level Level

Figure 9.28. Control circuit with phase-shifted carriers consisting of a modulator and a control signal generator

This circuit can be used to drive a three-level inverter leg in an open loop configuration, relying on natural voltage balancing. The state machine can be extended to take account of additional transitions, which may enable the voltage across the flying capacitor to be managed (Figure 9.29).

Bal

11

11

2↓1 1↑2

10

2↓1 1↑2

Bal

01

1↓0 0↑1

00

Bal

1↓0 0↑1

00

Figure 9.29. Modified state machine enabling flying voltages to be correctly balanced

280

Power Electronic Converters

Thus, by adding the lower or upper Bal transition it is possible to transition between the two, 00 or 11 states. Introduction of this type of transition does not alter the actual state but does alter the next upcoming state and hence makes it possible to modify the future evolution of the voltage across the flying capacitor. These transitions have no cost in terms of additional switching, and their only purpose is to allow approximate rebalancing to take place. Activation of the central Bal transition does lead to an additional commutation but appropriate management of the moment at which it occurs does allow exact re-balancing to take place. None of these transitions alter the output voltage and therefore balancing takes place without affecting the output voltage and current spectra. Such a state machine may be combined with a generic optimized threephase modulator in order to obtain the waveforms shown in Figure 9.30b, for example. 9.4. Conclusion Multilevel converters were initially developed to access higher power by dividing the constraints between several semiconductors. They rapidly displayed promising potential in terms of their dynamic performance as well. Nevertheless, as more degrees of freedom are made available their complexity grows, which generally brings into play additional state variables that must be monitored or regulated (potential of the neutral point or voltage across flying capacitors, circulation current in inductive couplers, or magnetic flux in inter-cell transformers). These degrees of freedom are therefore rarely truly “free”, and an appropriate method is required to handle the control of these converters and for making best use of their exceptional capabilities. A two-part picture of the control has gradually taken shape in the relevant literature, consisting of a generic modulator and a command generator specific to each topology; which is probably one of the most important stages in the development of this method.

PWM Strategies for Multilevel Converters

281

Figure 9.30a. Three-phase three-level inverter with regulated floating capacitor. Control circuit

This chapter may seem incomplete in that it does not conclude with an “optimal” modulation strategy. If this is the case, it is because it would be unhelpful to highlight just one of these strategies based on waveform or even THD criteria only. For example, the choice between a continuous or discontinuous control strategy may involve considerations ranging from switching losses to the weight of input and output filters, as well as consideration of the common-mode voltage and dynamic performance.

282

Power Electronic Converters

Consequently, the choice of strategy must be made on a case-by-case basis after consideration of all the relevant factors; it is only then that one or the other strategy can be selected. This comment of course applies to all aspects of power electronics but it is probably true particularly in the case of multilevel converters, which are, and always will be, in competition with two-level converters, such that their use must always be justified in terms of a clear improvement to the system in question.

Figure 9.30b. Three-phase three-level inverter with regulated flying capacitor. Switched voltage waveforms for phase U, voltage between phases U and V, the three-phase currents, and finally the flying voltages

Finally we note that while recent years have seen spectacular breakthroughs in the field of multilevel converters, the future may see progress accelerate still further in that it is now an area of interest to the entire lowvoltage power electronics community, thanks to the challenges presented in feeding power to the microprocessors of the future.

PWM Strategies for Multilevel Converters

283

9.5. Bibliography [BAK 81] BAKER R.H., Bridge Converter Circuit, US patent, n° 4,270,163, 1981. [BAR 03] BARBOSA P., STEINKE J., STEIMER P., MEYSENC L., MEYNARD T., Converter Circuit for connecting a plurality of switching voltage levels, Patent WO 02005036719, 2005, PCT Patent 03405748.9, 2003. [BAR 05] BARBOSA P., STEIMER P., STEINKE J., WILKENKEMPER M., CELANOVIC N., “Active Neutral Point Clamped (ANPC) multilevel converter technology”, European Conference on Power Electronics and Applications, 2005. [BRU 01] BRUCKNER T., BERNET S., “Loss balancing in three-level voltage source inverters applying active NPC switches”, IEEE Power Electronics Specialists Conference, p. 1135–1140, Vancouver, Canada, 2001. [BRU 05a] BRUCKNER T., BERNET S., GULDNER H., “The active NPC converter and its Loss-Balancing Control”, IEEE Transactions on Industrial Electronics, vol. 52, n° 3, p. 858–868, 2005. [BRU 05b] BRUCKNER T., HOLMES D.G., “Optimal PWM for three-level inverters”, IEEE Transactions on Power Electronics, vol. 20, n° 1, p. 82–89, 2005. [CAR 92] CARRARA G., GARDELLA S., MARCHESONI M., SALUTARI R., SCIUTTO G., “A new PWM method: a theoretical analysis”, IEEE Transactions on Power Electronics, vol. 7, n° 3, p. 497–505, 1992. [DEL 01] DELMAS L., GATEAU G., MEYNARD T.A., FOCH H., “Stacked Multicell Converter (SMC): Properties and Design”, IEEE Power Electroics Specialists Conference, PESC, Vancouver, Canada, 2001. [DEL 03] DELMAS L., Convertisseurs multicellulaires superposés. Etude, commande et réalisation d’un prototype, PhD thesis, Institut national polytechnique de Toulouse, 2003. [FOC 06] FOCH H., METZ M., MEYNARD T., PIQUET H., RICHARDEAU F., “Des dipoles à la cellule de commutation”, Techniques de l’ingénieur, Rubrique “Electronique de Puissance”, vol. D3075, 2006. [FOR 07a] FOREST F., MEYNARD T., LABOURE E., COSTAN V., CUNIERE A., SARRAUTE E., “Optimization of the supply voltage system in interleaved converters using intercell transformers”, IEEE Transactions on Power Electronics, vol. 22, p. 934–942, 2007. [FOR 07b] FOREST F., MEYNARD T., LABOURE E., HUSELSTEIN J.J., “A multi-cell interleaved flyback using intercell transformers”, IEEE Transactions on Power Electronics, vol. 22, n° 5, p. 1662–1671, 2007. [ITU 08] ITURRIZ F., RICHARDEAU F., MEYNARD T., ELHALI E., Circuit et systèmes redresseurs de puissance, procédé associé, aéronef comprenant de tels circuit ou systèmes, French patent n° FR20080050622 (filed by Airbus-CNRS), 2008.

284

Power Electronic Converters

[LI 04] LI J., STRATAKOS A., SCHULTZ A., SULLIVAN C.R., “Using coupled Inductors to enhance transient performances of multi-phase buck converters”, IEEE Applied Power Electronics Conference and Exposition, vol. 2, p. 1289– 1293, 2004. [MCG 06] MC GRATH B., HOLMES G., MEYNARD T., “Reduced PWM harmonic distortion for multilevel inverters operating over a wide modulation range”, IEEE Transactions on Power Electronics, vol. 21, n° 4, p. 941–949, 2006. [MCG 07] MC GRATH B., HOLMES G., MEYNARD T., GATEAU G., “Optimal modulation of flying capacitor and stacked multicell converters using a state machine decoder”, IEEE Transactions on Power Electronics, vol. 22, n° 2, p. 508–516, 2007. [MEY 97] MEYNARD T., FADEL M., AOUDA N., “Modelling of multilevel converters”, IEEE Transactions on Industrial Electronics, vol. 44, n° 3, p. 356– 364, 1997. [MEY 02a] MEYNARD T., FOCH H., THOMAS P., COURAULT J., JAKOB R., NAHRSTAEDT M., “Multicell converters: basic concepts and industry applications”, IEEE Transactions on Industrial Electronics, vol. 49, n° 5, p. 955– 964, Special Issue on Multilevel Converters, 2002. [MEY 02b] MEYNARD T., FOCH H., FOREST F., TURPIN C., RICHARDEAU F., DELMAS L., GATEAU G., LEFEUVRE E., “Multicell converters: derived topologies”, IEEE Transactions on Industrial Electronics, vol. 49, n° 5, p. 978– 987, Special Issue on Multilevel Converters, 2002. [MEY 02c] MEYNARD T., TURPIN C., BAUDESSON P., RICHARDEAU F., FOREST F., “Fault management of multicell converters”, IEEE Transactions on Industrial Electronics, vol. 49, n° 5, p. 988–997, Special Issue on Multilevel Converters, 2002. [MEY 06] MEYNARD T., LIENHARDT A.M., GATEAU G., HAEDERLI C., BARBOSA P., “Flying capacitor multicell converters with reduced stored energy”, IEEE-ISIE 2006, Montréal, Canada, 2006. [NAB 81] NABAE A., TAKAHASHI I, AKAGI H., “A new neutral Point Clamped PWM inverter”, IEEE Transactions on Industry Applications, 1981. [PAR 97] PARK G., KIM S.I. “Modeling and analysis of multi-interphase transformers for connecting power converters in parallel”, in Park G., Seos Ik KIM, IEEE Power Electronics Specialists Conference, vol. 2, p. 1164–1170, 1997. [PEN 96] PENG F.Z., LAI J.S., MCKEEVER J.W., VANCOEVERING J., “A Multilevel Voltage Source inverter with separate DC sources for Static Var generation”, IEEE Transactions on Industry Applications, vol. 32, n° 5, p. 1130–1138, 1996.

PWM Strategies for Multilevel Converters

285

[QUI 07] QUINTERO J., BARRADO A., SANZ M., RAGA C., LAZARO A., “Bandwidth and dynamic response decoupling in a multi-phase VRM by applying linear-nonlinear control”, IEEE International Symposium on Industrial Electronics, p. 3373–3378, 2007. [TUR 02] TURPIN C., DEPREZ L., FOREST F., RICHARDEAU F., MEYNARD T., “A ZVS imbricated cells multilevel inverter with auxiliary resonant commutated pole”, IEEE Transactions on Power Electronics, vol. 17, n° 6, p. 874–882, 2002. [VIS 04] VISAIRO H., SANCHEZ A., RODRIGUEZ E., ARAU J., COBOS J.A., “MultiPhase VRM based on the symmetrical half-bridge converter”, IEEE Applied Power Electronics Conference and Exposition, vol. 2, p. 1275–1281, 2004. [ZHA 08] ZHANG X., LIU J., WONG P.L., CHEN J., WU H.P., AMOROSO L., LEE F.C., CHEN D.Y., “Investigation of Candidate VRM Topologies for future microprocessors”, IEEE Applied Power Electronics Conference and Exposition, vol. 1, p. 145–150, 1998.

Chapter 10

PI Current Control of a Synchronous Motor

10.1. Introduction This chapter discusses proportional integral (PI) current control of a synchronous motor. This machine forms a standard type of load: it forms a RLE (resistance, inductance, and EMF) type system, a type that is commonly encountered in electrical engineering. PI controllers are often used for current control of electrical machines since they make it possible to cancel the static error in the steady state regime and to influence the dynamics of current control. This chapter considers two methods of current control of a synchronous motor using PI controllers. The first method operates in a fixed three-phase coordinate system based around the stator. In this case, a PI controller ensures correct management of the amplitude and phase of the current in each stator phase of the synchronous motor. The second method works in a rotating reference frame (d, q), where the d axis is linked to the rotor winding axis of the synchronous motor. In this case the direct and transverse components of the stator current vector are constants in the steady state regime and the PI controller only imposes the amplitude of the stator currents, while their phase is imposed through coordinate transforms. In order to implement a PI control system we first require a model of the Chapter written by Mohamed Wissem NAOUAR, Eric MONMASSON, Ilhem SLAMA-BELKHODJA and Ahmad Ammar NAASSANI.

288

Power Electronic Converters

synchronous motor. We will initially develop a model of a synchronous motor in a fixed three-phase coordinate system and in a two-phase rotating coordinate system. We will then discuss the use of a PI controller to impose the required static and dynamic performance. 10.2. Model of a synchronous motor In order to control a synchronous motor we need to develop a mathematical model describing its response. With this in mind, the present section will develop suitable mathematical models that can be used in the design and implementation of current control systems for a synchronous motor. In our model of a synchronous motor we will make the following simplifying assumptions [LOU 04]: – the synchronous motor is coupled in a star topology with isolated neutral; – saturation effects are ignored, which implies that the individual and mutual inductances are independent of current; – the three phases are symmetric, in other words the resistance and inductance of each phase is the same; – the harmonics generated by the windings will be ignored since we will assume that the spatial distribution of the windings is perfectly sinusoidal; – our synchronous motor is assumed to be free from damping and to have a wound rotor. 10.2.1. Model of a synchronous motor in a fixed coordinate system based on the stator We will base our work on the schematic diagram of a synchronous motor shown in Figure 10.1. In the figure, the phase of the rotor is rotated from the first phase of the stator by a mechanical angle θm. This angle represents the angular position of the rotor relative to the stator. The stator is fixed, whereas the rotor turns with an angular mechanical velocity Ω given by the following equation: dθ m =Ω dt

[10.1]

PI Current Control of a Synchronous Motor

289

Sb isb



Vsb

r

Φsb

θm Φsa Φr Vr

Vsc isc

ir

isa

Sa

Vsa

Φsc

Sc

Figure 10.1. Schematic diagram of a synchronous motor with a wound rotor

The angular electrical position of the rotor along with its electrical rotation speed are given by the following equations: θ = pθ m

[10.2]

ω = pΩ .

[10.3]

Since the windings of the three stator phases are fixed, the rotation of the rotor winding (the inductor) exposes the stator windings to a varying field, and consequently induces an electromotive force (EMF) at the terminals of each stator phase. Thus, if a three-phase voltage [Vsa, Vsb, Vsc]t is applied to the stator terminals, the following voltage equations are obtained: Vsa (t ) = Rs i sa (t ) +

dΦ sa (t ) dt

[10.4]

Vsb (t ) = Rs i sb (t ) +

dΦ sb (t ) dt

[10.5]

Vsc (t ) = Rs i sc (t ) +

dΦ sc (t ) dt

[10.6]

290

Power Electronic Converters

The rotor voltage equation is written as follows: Vr (t ) = Rr ir (t ) +

dΦ r (t ) . dt

[10.7]

Since the axis of the rotor winding forms an electrical angle θ with the axis of the first stator phase, the stator fluxes Φsa, Φsb and Φsc for each stator phase of the synchronous motor can be written as: Own flux Flux produced by other phases 678 644474448 Φ sa = La isa (t ) + M ab isb (t ) + M ac isc (t ) Flux

+

produced

by rotor 64 4744 8 M sr cos(θ )ir

winding

Own flux Flux produced by other phases 678 644474448 Φ sb = Lb isb (t ) + M ba isa (t ) + M bc isc (t ) Flux

+

produced by rotor winding 644 47444 8 2π M sr cos(θ − )ir 3

Own flux Flux produced by other phases 678 644474448 Φ sc = Lc isc (t ) + M ca isa (t ) + M cb isb (t ) Flux

+

produced by rotor winding 644 47444 8 4π M sr cos(θ − )ir 3

[10.8]

[10.9]

[10.10]

The expression for the rotor flux Φr is given by the following equation: Own flux 678 Φ r (t ) = Lr ir (t ) Flux produced by stator winding 644444444444 47444444444444 8 2π 4π + M rs cos(θ )is1 (t ) + M rs cos(θ − )is 2 (t ) + M rs cos(θ − )is 3 (t ) , 3 3

[10.11]

where Li(i=a,b,c) is the self-inductance of the winding in the ith stator phase, Mij is the mutual inductance between the windings of stator phases i and j, Msr and Mrs are the maximum mutual inductances between the stator and rotor windings, and Lr is the self-inductance of the rotor winding.

PI Current Control of a Synchronous Motor

291

The three-phase variables (voltage, current, and flux) can be represented r graphically using direction vectors. The relationship between the vector X for a stator variable and the corresponding variables for each phase (Xa, Xb and Xc) is given by the following equation: 2π 4π r j j X = K(X a + e 3 X b + e 3 X c )

[10.12]

where K is the coefficient of amplitude or power conservation. In the steady state, the variables Xa, Xb , and Xc form a balanced sinusoidal three-phase system: – if K = 2/3 then the transformation conserves amplitude. Thus, the r amplitude of a vector X is the same as that of the Xa, Xb, and Xc components in the steady state regime; – if K = 2 / 3 then the transformation conserves power. Rewriting equations [10.4], [10.5], and [10.6] in vector form, the expression for the stator voltage vector can be written as follows: r r r dΦ s Vs = Rs is + . dt

[10.13]

Now rewriting equations [10.8], [10.9], [10.10], and [10.11] in vector r form, the expressions for the stator flux vector Φ s and the rotor flux vector Φr are given by the following equations: r

r

Φ s = [ Ls ]is + [ M sr ]ir r

Φ r = [ M rs ]is + Lr ir .

[10.14] [10.15]

For salient pole synchronous motors, the expression for the stator inductance includes terms that are a function of position θ. According to

292

Power Electronic Converters

[LOU 04], the stator inductances of a salient pole synchronous motor can be written as follows: 2π 4π r j j X = K(X a + e 3 X b + e 3 X c )

 La (θ )

[ Ls ] =  M ba (θ )  M ca (θ )

 Ls 0 =  M s 0  M s 0

M s0 Ls 0 M s0

M ab (θ ) M ac (θ )  Lb (θ ) M bc (θ )  M cb (θ ) Lc (θ )  2π 2π  [10.16]  cos(2θ − ) cos(2θ + )  cos(2θ ) 3 3  M s0    2π 2π M s 0  + Lsv  cos(2θ − ) cos(2θ + ) cos(2θ )    3 3 Ls 0    2 2 π π  cos(2θ + ) cos(2θ ) cos(2θ − )  3 3 

The Ls0, Ms0, and Lsv terms are constants dependent on the machine design. The expressions for the elements of the mutual matrices [Msr] and [Mrs] connecting the stator and the rotor of a synchronous motor are given by the following equations:

[ M sr (θ )] = M sr

   cos(θ )     cos(θ − 2π )   3    4  cos(θ − π )  3  

[ M rs (θ )] = [ M sr (θ )]T .

[10.17]

[10.18]

It should be noted that for a non-salient pole synchronous motor the selfinductance and mutual inductances are independent of position θ and depend only on the construction parameters of the machine. In this case, for a constant gap width (the poles are flush with the mounting), the selfinductance and mutual inductances of the synchronous motor satisfy the following equations: La = Lb = Lc

[10.19]

M ab = M ba = M ac = M ca = M bc = M cb

[10.20]

PI Current Control of a Synchronous Motor

293

Equation [10.12] can be used to determine the real and imaginary components of Xα and Xβ, respectively from the direction vector r r X ( X = X α + jX β ) in the fixed two-phase coordinate system (α, β). This results in the following transformations: – in the case of amplitude conservation, the result is the Clarke transformation, given by the following equation: X a  Xα  X  [ ] = C X   b  β  X c 

1  1 − 2 2 where [C ] =  3 3 0  2

1  2   3 − 2  −

[10.21]

– in the case of power conservation, the resultant transformation is the Concordia transform, given by the following equation: X a  Xα   X  where [ ] = T X   b  β  X c 

1  1 − 2 2  3 3 0  2

[T ] =

1  2   3 − 2  −

[10.22]



isβ Φsβ



Vsβ

r θm Φsα ir

Φr Vr

isα



Vsα

Figure 10.2. Schematic diagram of a wound rotor synchronous motor in a two-phase coordinate system (α, β)

294

Power Electronic Converters

The Clarke or Concordia transform can be used to derive a two-phase basis system for the motor that is equivalent to the three-phase system shown in Figure 10.1. In Figure 10.2 the three-phase windings are replaced by equivalent right-angle windings for a two-phase system. 10.2.2. Model of a synchronous motor in a common coordinate system (d, q) aligned with the rotor winding axis of the motor A common coordinate system can be used to represent both the stator windings and the rotor winding of the synchronous motor in a common coordinate system (d, q). Such a coordinate system can be used to eliminate the variable coupling caused by variation in the angle θ that appears in the self-inductance and mutual inductances even in the case of a salient pole synchronous motor. The (d, q) coordinate system is the result of a rotation by an angle θdq of the two-phase system given by Figure 10.2. Figure 10.3 gives a schematic diagram of the common coordinate system (d, q), where the d axis is aligned with the rotor winding axis. Note that in this figure it is the electrical angles that are used and the angle θdq represents the electrical angle between the axis of the rotor winding and the Sα axis. In this case the angle θdq satisfies the following equation: θdq = θ

[10.23]

The complex space vectors representing a particular quantity expressed in r the common coordinate system (d, q) ( X dq ) and in the fixed reference frame r

associated with the stator ( X s ) are related in the following manner: r r − jθ X dq = X s e dq

[10.24]

PI Current Control of a Synchronous Motor

295

ωdq= ω d=r

isd

Φsd

q

isq

Vsq

Vsd

θdq= θ

Φsq ir



Φr Vr

Figure 10.3. Schematic diagram of a synchronous motor with a wound rotor in a rotating coordinate system (d, q)

Taking equation [10.24] and comparing its real and imaginary parts, we can determine the direct component Xd as well as the transverse component r Xq of the direction vector X dq using components Xα and Xβ of the space r

vector X s using the following equation: X d  Xα   X  = R(θ dq )   where X β   q

[

]

θ dq ) [R(θ dq )] = cos( sin(θ ) 

dq

− sin(θ dq ) , [10.25] cos(θ dq ) 

where [R(θdq)] is a rotation matrix. Based on equations [10.21], [10.22], and [10.25], we can determine the direct and transverse components of a space vector expressed in the common coordinate system (d, q) using equations [10.26] and [10.27]. These equations represent the Park transformation. The Park transformation given by equation [10.26] is an amplitudeconserving transformation, whereas the one given by equation [10.27] is a power conserving transformation.

296

Power Electronic Converters

Xa  Xd     X  =  R (θ dq )  [C ]  X b  q    X c 

[10.26]

Xa  Xd     X  =  R (θ dq )  [T ]  X b  .  q  X c 

[10.27]

As mentioned earlier, the synchronous motor model we are using in this section is based on a common coordinate system (d, q), where the direct axis d is parallel to the axis of the rotor winding (Figure 10.3). In this case, the rotor components (voltage, flux, and current) are equal to their direct components as stated in the following equation:  Vr = Vrd   ir = ird Φ = Φ rd  r

[10.28]

Equation [10.7] therefore becomes: Vrd = Rr ird +

dΦ rd . dt

[10.29]

The electrical angular rotation velocity ωdq of the common coordinate system (d, q) is given by the following equation: dθ dq dt

= ωdq .

[10.30]

Making use of equations [10.13] and [10.24], the voltage equation becomes: r r d r jθ jθ jθ Vsdq e dq = Rs isdq e dq + (Φ sdq e dq ) dt r . r d (θ dq ) jθ dq r r jθ dq jθ dq jθ dq d (Φ sdq ) Vsdq e = Rs isdq e +e +j (e )Φ sdq dt dt

[10.31]

PI Current Control of a Synchronous Motor

Canceling terms in e

jθdq

297

, this result can be simplified to:

r r dΦ sdq r r + jωdqΦ sdq . Vsdq = Rs isdq + dt

[10.32]

r

In equation [10.32], the Rs isdq term represents the voltage drop due to Joule losses. The

r dΦ sdq dt

term represents the electromotive transformation

force obtained from Lenz’s Law. The

r jωdqΦ sdq

term represents the

electromotive rotation force obtained from Laplace’s Law. By equating the real and imaginary parts of equation [10.32], the direct and transverse components of the voltage vector can be expressed in the rotating coordinate system (d, q) as follows: Vsd = Rs isd +

Vsq = Rs isq +

dΦ sd − ωdqΦ sq dt dΦ sq dt

+ ωdqΦ sd .

[10.33]

[10.34]

In terms of the stator and rotor flux equations given by equations [10.14] and [10.15], their direct and transverse components can be determined in the common coordinate system (d, q) by applying one of the Park transformations given by equations [10.26] and [10.27]. Application of the Park transformation eliminates the variable coupling due to variation in the angle θ that appears in equation [10.16]. The expression for the d and q components of the stator flux vector, expressed in the common coordinate system (d, q), is then equal to: Φ sd = Lsd isd + KΦ M sr ird

[10.35]

Φ sq = Lsq isq .

[10.36]

298

Power Electronic Converters

The constant KΦ is equal to 1 if the variables in the (d, q) reference frame are obtained using an amplitude-conserving transformation and equal to 1.5 if the variables in the (d, q) reference frame are obtained using a power conserving transformation. The direct and transverse inductances Lsd and Lsq are constant and depend on the Ls0, Ms0, and Lsv parameters. This dependence is as follows: 3   Lsd = ( Ls 0 − M s 0 ) + 2 Lsv  L = (L − M ) − 3 L s0 s0 sv  sq 2

[10.37]

The expression for the d and q components of the rotor vector flux, expressed in the common coordinate system (d, q), is equal to: Φ rd = Lrd ird + KΦ M sr isd

[10.38]

Φ rq = 0 .

[10.39]

To summarize, Figure 10.4 shows the Park model for a salient pole synchronous motor with no damping, in the common coordinate system (d, q) aligned with the rotor, with the d axis aligned with the axis of the rotor winding of the synchronous motor. q isq (Rs,Lsq)

Vsq

Msr ωdq (Rr,Lrd)

(Rs,Lsd) d ird

Vrd

isd Vsd

Figure 10.4. Park model for a synchronous motor

PI Current Control of a Synchronous Motor

299

10.2.3. Expression for electromagnetic torque The apparent instantaneous power S(t) of the synchronous motor can be written as follows: r r* S (t ) = K s (Vsdq isdq ).

[10.40]

The “*” represents the complex conjugate of a quantity: r

r

– Ks = 1 if the Vsdq and isdq vectors are obtained using an amplitudeconserving transformation; r

r

– Ks = 3/2 if the Vsdq and isdq vectors are obtained using a power conserving transformation. For the remainder of this section we will use the first of these values. The instantaneous active stator power of the synchronous motor can be written as follows: r r* Ps (t ) = Re[ S (t )] = Re[Vsdq isdq ].

[10.41]

r

If we replace the vector Vsdq in this equation with its expression as given by equation [10.32] and multiply by dt, we obtain an expression for the elementary stator energy: r r r r r r* dws = Ps (t )dt =  Rs isdq .isdq* + Re[dΦ sdq isdq* ] + ωdq Im[isdqΦ sdq ] dt .  

[10.42]

Equation [10.42] shows that the electrical energy provided to the stator in a time interval dt can be divided into three terms: – the first term represents energy losses due to Joule heating; – the second term represents variation in the magnetic energy of the stator; – the third term represents the energy dwsr transferred from the stator to the rotor across the air gap by means of the rotating field.

300

Power Electronic Converters

Thus we have: r r* r r* dwsr = ωdq Im[isdqΦ sdq ]dt = ω Im[isdqΦ sdq ]dt .

[10.43]

For a synchronous motor, the energy dwsr crossing the air gap is equal to the mechanical energy dwmec . The mechanical power is thus given by the following equation: Pmec =

r r* dwmec dwsr  = Ω Cem = ω Cem = = ω Im isdqΦ sdq   dt dt p

[10.44]

The expression for the electromagnetic torque Cem is therefore given by the following equation: r r* . Cem = p Im isdqΦ sdq  

[10.45]

It should be noted that in the case of amplitude conservation (Ks=3/2) and following a similar line of reasoning, the equation for the torque becomes: Cem =

r r* 3  p Im  isdqΦ sdq   2

[10.46]

Expanding equation [10.45], the expression for the electromagnetic torque of a synchronous motor is equal to: Cem = p (Φ sd isq − Φ sq isd ) = p(( Lsd − Lsq )isd isq + M sr isq ird ) .

[10.47]

For amplitude conserving transformations, this equation becomes: Cem =

3 3 p (Φ sd isq − Φ sq isd ) = p (( Lsd − Lsq )isd isq + M sr isq ird ) . 2 2

[10.48]

10.3. Typical power delivery system for a synchronous motor AC three-phase motors including synchronous motors are generally intended for variable speed operation. The incorporation of variable speed electrical power trains requires the use of power converters able to generate

PI Current Control of a Synchronous Motor

301

variable voltages and frequencies, such as three-phase voltage inverters. Such a structure is presented in Figure10.5 and can be used to produce a three-phase voltage system with adjustable frequency and amplitude from a DC voltage source. Synchronous motor Rectifier

Voltage source inverter

Filter

Van

E/2 C 3-phase supply

a

o

b

c

n

E/2 C

Vcn

Vbn Sa

Sb

Sc

Figure 10.5. Typical structure of a voltage inverter supplying a synchronous motor

Figure 10.6 shows a schematic diagram of voltage control of one leg of a three-phase voltage inverter using PWM based on a carrier wave. A PWM modulation strategy does not attempt to impose the phase voltages Vin(i=1,2,3) to the machine’s terminals at every time instant but aims to impose their mean values when averaged over the PWM switching period. Carrier

Reference

Vio*[k]

E/2 o

i

E/2

R -E/2

Vno* Vsi*=Vin*

++

Vsi

Vio* +

+E/2

Si

-

E/2

L Si

e

ton/2

TMLI toff

ton/2

-E/2

1 0 +E/2

n

Vio(t)-E/2

(k+1)TMLI

kTMLI

-E/2

a)

b)

Figure 10.6. a) Circuit diagram showing PWM of one voltage source inverter leg; b) operating principles of PWM control

302

Power Electronic Converters

Over the course of the kth switching period, the mean values of the simple voltages Vsi(k)(i=1,2,3) can be expressed as a function of the duty cycles ai(k)(i=1,2,3) as follows: Vsa (k ) Van (k )  2 − 1 − 1 a a (k ) V (k ) = V (k )  = E − 1 2 − 1  a (k )   sb   bn  3   b  Vsc (k )  Vcn (k )  − 1 − 1 2   ac (k ) 

[10.49]

The aim of PWM is to determine these duty cycles ai(k) such that: Vin (k ) = Vin * (k ) (i = a, b, c)

[10.50]

where Vin*(k)(i=a,b,c) are the simple reference voltages equal to the reference voltages Vsi*(k)(i=a,b,c). In order to determine suitable duty cycles such that equation [10.50] is satisfied, we consider the voltages Vio(k)(i=a,b,c), which can be written as follows: Vio (k ) =

E (2ai (k ) − 1) (i = a, b, c) 2

[10.51]

By inverting equation [10.42] and writing the reference voltages Vio*(k)(i=1,2,3) as a function of the simple reference voltages Vin*(k)(i=1,2,3), the duty cycles ai(k)(i=1,2,3) can be determined as follows: ai (k ) =

1 1 (Vin* (k ) + Vno* (k )) + (i = 1, 2,3) , 2 E

[10.52]

where Vno*(k) is the mean value of the zero-sequence component for the kth switching period. Thus, in order to determine the duty cycles that will result in the voltages Vin*(k)(i=1,2,3) being applied to the load terminals while also meeting the constraints on the duty cycles (which must lie between 0 and 1), we must

PI Current Control of a Synchronous Motor

303

determine the component Vno*(k) that must be added. A number of solutions are possible, including: – PWM without injection of zero-sequence signal (Vno*(k) = 0). An initial solution involves selecting a zero-sequence signal Vno*(k) of zero. This results in the reference voltages Vio*(k)(i=a,b,c) being equal to the voltages Vin*(k)(i=a,b,c). The maximum fundamental amplitude attainable by the simple voltages in this case is equal to only E/2 [LOU 97, MON 97]; – PWM with injection of zero-sequence signal (Vno*(k) ≠ 0). By adding a particular zero-sequence signal [LOU 97, MON 97], it is possible to achieve a fundamental amplitude equal to E/√3 for the simple voltages, which represents a gain of 15% compared with PWM without injection of a zero sequence component. This matches the performance of space vector PWM (Chapter 2). 10.4. PI current control of a synchronous motor in the fixed three-phase coordinate system of the stator This section discusses PI current control of a synchronous motor in a fixed three-phase coordinate system associated with the stator. We will assume that the synchronous motor has non-salient poles so that the selfinductances and mutual inductances are independent of position θ. The case of a synchronous motor with salient poles is considered in the next section (section 10.5), where a common coordinate system (d, q) will be used so that the inductances are independent of position θ. Since the neutral point of the synchronous motor is not connected, the sum of the three stator currents is zero: i sa (t ) + i sb (t ) + i sc (t ) = 0

[10.53]

We can use equations [10.8], [10.9], [10.10], [10.19], [10.20], and [10.53] to show that the stator fluxes for a synchronous motor with non-salient poles can be written as follows: Φ sa = ( La − M ab )i sa (t ) + M sr cos(θ )i r

[10.54]

304

Power Electronic Converters

Φ sb = ( La − M ab )i sb (t ) + M sr cos(θ −

2π )ir 3

[10.55]

Φ sc = ( La − M ab )i sc (t ) + M sr cos(θ −

4π )i r 3

[10.56]

Taking Ls=(L1–M12) to be a cyclic inductance and using equations [10.4], [10.5], and [10.6], the voltages applied to the stator windings of a synchronous motor with non-salient poles can be written as: esa (t )

di (t ) 6447448 Vsa (t ) = Rs i sa (t ) + Ls sa − M sr ω sin(θ )i r dt sb ( t ) 6444e7 4448 di sb (t ) 2π − M sr ω sin(θ − )ir Vsb (t ) = R s i sb (t ) + Ls dt 3

sc ( t ) 6444e7 4448 di sc (t ) 4π − M sr ω sin(θ − )i r V sc (t ) = R s i sc (t ) + Ls dt 3

[10.57]

[10.58]

[10.59]

In equations [10.57], [10.58], and [10.59], the first terms [Rsisi(t)](i=1,2,3) represent the voltage drops caused by the Joule effect, and the second terms:   di (t )     Ls si  ,  dt  (i =1,2,3)  

represent the force transformation electromotive force resulting from Lenz’s Law and the third terms represent the rotational electromotive force resulting from Laplace’s Law. Figure 10.7 shows the equivalent electrical circuit for a phase i (i=a,b,c) of a synchronous motor with non-salient poles, based on equations [10.57], [10.58], and [10.59]. It should be noted that just as in the case of a DC motor, the equivalent circuit consists of an input voltage Vsi, a resistance Rs, an inductance Ls, and an induced voltage esi representing the EMF.

PI Current Control of a Synchronous Motor Rs

305

Ls

isi(t)

esi(t)

Vsi(t)

Figure 10.7. Equivalent circuit for the ith phase of a synchronous motor with non-salient poles

The instantaneous power Pem(t) can be written as follows: Pem (t ) = e sa (t )i sa (t ) + e sb (t )i sb (t ) + e sc (t )i sc (t )

[10.60]

In addition, the instantaneous electromagnetic torque is related to the instantaneous power by the following equation: Pem (t ) = Cem (t )Ω (t ) .

[10.61]

We can then use equations [10.57], [10.58], [10.59], [10.60], and [10.61] to determine an expression for the electromagnetic torque: C em =

Pem Ω

π

π

π 4π 2π = pM sr ir (i sa cos(θ + ) + i sb cos(θ + − ) + i sc cos(θ + − )) 2 2 3 2 3

[10.62]

Based on these equations, it can be seen that a constant electromagnetic torque can be obtained without any oscillations by injecting currents of the following form: π  i sa = I s cos(θ + 2 − ψ )  π 2π  −ψ ) i sb = I s cos(θ + − 2 3  π 4π  i sc = I s cos(θ + 2 − 3 − ψ ) 

[10.63]

In this case the expression for the torque becomes: Cem =

3 pM sr ir I s cos(ψ ) 2

[10.64]

306

Power Electronic Converters

Note that the angle ψ represents the phase shift between the EMF esi(t) and the current isi(t) of a given phase i. Cem 1.5pMsrirIs

Is

ψ

0

-1.5pMsrirIs -π

-π/2

0

π/2

π

Figure 10.8. Variation of electromagnetic torque as a function of phase shift between the EMF and the current of a given phase

Figure 10.8 shows the variation of the electromagnetic torque as a function of the phase shift ψ. It can be seen that for a given current amplitude Is, the electromagnetic torque is a maximum when the EMF and the current are in phase or are 180° out of phase. When operating with a positive rotation velocity (positive electromagnetic torque) the torque is a maximum when the EMF and the current are in phase (ψ = 0). When operating with a negative rotation velocity (negative electromagnetic torque) the torque is a maximum when the EMF and the current are in phase opposition (ψ = π). 10.4.1. Tuning of PI controllers in a fixed three-phase coordinate system aligned with the stator If we apply the Laplace transform to equations [10.57], [10.58], and [10.59], equations: i sa =

1 (Vsa − e sa ) Rs + Ls s

[10.65]

PI Current Control of a Synchronous Motor

307

i sb =

1 (Vsb − e sb ) Rs + Ls s

[10.66]

i sc =

1 (Vsc − e sc ) . Rs + Ls s

[10.67]

These three equations can be used to establish the block diagram shown in Figure 10.9, which represents the electrical model for a given phase (i = a,b,c) of a synchronous motor with non-salient poles. esi Vsi

+-

1 R s + Ls s

isi

Figure 10.9. Equivalent electrical circuit for phase i (i=a,b,c) of a synchronous motor with non-salient poles

The currents isi(i=1,2,3) are controlled by the PI regulators described by equations [10.68], [10.69], and [10.70]: PIis1 ( s ) = Kpis1 +

Kiis1 s

[10.68]

PIis 2 ( s ) = Kpis 2 +

Kiis 2 s

[10.69]

PIis3 ( s ) = Kpis 3 +

Kiis 3 s

[10.70]

Figure 10.10 shows how the PI control loop controls the currents isi(i=a,b,c). In this figure the voltages VsiL*(i=a,b,c) correspond to the reference voltages produced by the PI controllers for the three stator phases. As mentioned in section 10.3, PWM control of the voltage inverter enables a mean value for the voltages Vsi(i=a,b,c) to be applied that matches the reference voltages Vsi*(i=a,b,c). In what follows, we will compensate for the inducted EMF terms esi(i=a,b,c), assume that the period of the carrier is very small compared with the electrical time constants of the synchronous motor,

308

Power Electronic Converters

and take the mean values of the voltages VsiL(i=a,b,c) to be equal to the reference voltages VsiL*(i=a,b,c). The transfer function H given by Figure 10.9 is then reduced to the identity operator. esi isi*

+

VsiL* +

Kpisi(s+Kiisi/Kpisi) s

-

+

esi Vsi

*

PWM V si + + Inverter

PIisi

VsiL

1 Ls(s+Rs/Ls)

isi

H

Control

Electrical model of phase i

Figure 10.10. Control loop for one current phase isi(i=a,b,c) of a synchronous motor

Figure 10.11 gives a simplified form for the control loops for the currents isi(i=1,2,3). isi*

+

-

Kpisi(s+Kiisi/Kpisi) VsiL s

1 Ls (s+Rs/Ls)

isi

PIisi

Figure 10.11. Simplified control loop for the currents isi(i=a,b,c)

It can be seen from Figure 10.11 that the transfer function for the control loop acting on isi(i=a,b,c) is: 1 Kpisi ( s + Kiisi / Kpisi ) s Ls ( s + Rs / Ls ) = * isi ( s ) 1 + 1 Kpisi ( s + Kiisi / Kpisi ) s Ls ( s + Rs / Ls ) isi ( s )

[10.71]

In order to determine the parameters for PI controllers, the method of dominant pole compensation is used to determine the gains of the PI controllers. This method involves setting the ratio Kiisi/Kpisi equal to Rs/Ls. In this case, the transfer function for the control loops acting on the currents isi(i=a,b,c) becomes:

PI Current Control of a Synchronous Motor

isi ( s ) isi* ( s )

=

1 1 , = Ls Tisi s + 1 s +1 Kpisi

309

[10.72]

where Tisi is the closed loop time constant that is to be imposed. The transfer functions for the control loops acting on the currents isi(i=a,b,c) are therefore first-order transfer functions whose time constant Tisi is given by the following expression: Tisi =

Ls . Kpisi

[10.73]

Equation [10.73] shows that in order to improve the current control dynamics the gain Kpisi must be increased. The choice of the closed loop time constant then enables the parameters for the PI controllers to be determined. It follows that: Kpisi =

Ls R R and Kiisi = Kpisi s = s Tisi Ls Tisi

[10.74]

This proposed control strategy has the advantage of being simple. Nevertheless, if there is any delay introduced by the computation time of the microcontroller, current filtering, or the PWM, and this is not small relative to the closed-loop time constant, then these delays must be taken into account in the design of the corrector in order to avoid any risk of instability. 10.4.2. PI control structure in a fixed three-phase coordinate system aligned with the stator The three reference current command values are calculated from equation [10.63], which in turn makes use of the measured position θ, the amplitude Is of the current, and the intended phase shift ψ that is to be applied between the EMF and the current within each phase. Referring to Figure 10.10, the current control loop operates by comparing the reference currents isi*(t)(i=1,2,3) with the measured currents isi(t)(i=1,2,3) and compensating for any resultant current error using PI controllers. These then enable suitable reference voltages Vsi*(t)(i=1,2,3) to be determined. Finally, these references are applied to the terminals of the synchronous motor using a PWM inverter.

310

Power Electronic Converters

Figure 10.12 shows the resultant control structure in the fixed three-phase coordinate system aligned with the stator. Figure 10.13 shows the evolution of the currents isd and isq, along with the stator currents isa, isb, and isc in response to the application of a step change to the reference current isq*.

Figure 10.12. PI current control in a synchronous motor using a fixed three-phase coordinate system aligned with the stator

Isn 20ms

isq

isd 0

Isn

isb

0

isc -Isn

isa

Figure 10.13. Response to a step change in the reference current isq*

PI Current Control of a Synchronous Motor

311

10.5. PI current control for a synchronous motor in a rotating coordinate system (d, q) 10.5.1. Tuning of PI controllers in the (d, q) frame This section discusses PI current control for a synchronous motor in the (d, q) reference frame, where the d axis is aligned with the rotor winding axis of the synchronous motor. The control strategy discussed here is valid for a synchronous motor with either salient or non-salient poles since the inductances of a synchronous motor modeled in the (d, q) plane are independent of the position θ regardless of the variant of synchronous motor considered. If we assume that the excitation current ird is constant (and hence has a derivative of zero) and apply the Laplace transform to equations [10.33] and [10.34], the currents isd and isq can be expressed as follows: e

sd 6474 8 1 (Vsd +ωdqΦ sq ) isd = Rs + Lsd s

[10.75]

esq

isq

6474 8 1 (Vsq −ωdqΦ sd ) , = Rs + Lsq s

[10.76]

where the esd and esq terms represent the terms of the induced EMF along the d and q axes. Equations [10.75] and [10.76] can be used to deduce the block diagrams shown in Figures 10.14a and 10.14b which show the electrical model of a synchronous motor along the d and q axes, respectively. -ωdqΦsq Vsd

+-

ωdqΦsd 1 R s + L sd s

a)

isd

Vsq

+-

1 R s + L sq s

b)

Figure 10.14. Electrical model of a synchronous motor: a) along the d axis; b) along the q axis

isq

312

Power Electronic Converters

Control of the isd and isq components is managed by PI controllers whose operation is described by the following two equations: K PIid = K pid + iid s

PIiq = K piq +

Kiiq s

[10.77]

[10.78]

.

esd isd*

isq*

-

+

Kpid(s+Kiid/Kpid) VsdL*++ s (PIid)

+

-

esd Vsa

Vsd*

*

Vsa

dq

VsdL

1 Lsd(s+Rs/Lsd)

isd

VsqL

1 Lsq(s+Rs/Lsq)

isq

dq Vsb*

* * Kpiq(s+Kiiq/Kpiq) VsqL Vsq* abc Vsc + s + (PIiq) esq θdq

Control

Vsd+

-

PWM Vsb + Inverter

Vsc abc Vsq +

θdq

-

esq

Hdq

Electrical model of the synchronous motor

Figure 10.15. Control loops for the currents isd and isq

Figure 10.15 shows a schematic diagram representing the PI current control loops acting on the currents isd and isq. In this figure the voltages VsdL* and VsqL* are the reference voltages generated by the PI controllers for the d and q axes. Just as in the case of PI current control in a fixed threephase coordinate system aligned with the stator, we can compensate for the induced electromotive force terms and assume that the PWM period is small compared to the electrical time constants of the synchronous motor. If we do this, the mean values of the voltages VsdL and VsqL are equal to the reference voltages VsdL* and VsqL*. The transfer function Hdq representing Figure 10.15 then becomes the identity operation. Having made this observation, Figure 10.16 shows a simplified form for the current control loops acting on the currents isd and isq.

PI Current Control of a Synchronous Motor

isd*

+

-

1 Kpid(s+Kiid/Kpid) VsdL s Lsd(s+Rs/Lsd) (PIid)

313

isd

a) isq*

+

-

1 Kpiq(s+Kiiq/Kpiq) VsqL s Lsq(s+Rs/Lsq) (PIiq)

isq

b)

Figure 10.16. Simplified current control loops: a) isd; b) isq

The transfer functions for the current control loops acting on the currents isd and isq, as shown in Figure 10.16a, b are given by the following equations: 1 K pid ( s + Kiid / K pid ) isd ( s ) s Lsd ( s + Rs / Lsd ) = * 1 K pid ( s + Kiid / K pid ) isd ( s ) 1+ s Lsd ( s + Rs / Lsd )

[10.79]

1 K piq ( s + Kiiq / K piq ) isq ( s ) s Lsq ( s + Rs / Lsq ) = . * 1 K piq ( s + K iiq / K piq ) isq ( s ) 1+ s Lsq ( s + Rs / Lsq )

[10.80]

In order to establish the parameters for the PI controllers, we will use the dominant pole compensation method to determine the gains of the PI controllers. This method involves setting the ratio Kiid/Kpid equal to Rs/Lsd and the ratio Kiiq/Kpiq equal to Rs/Lsq. In this case, the transfer functions for the current control loops acting on the currents isd and isq become: isd ( s ) *

isd ( s )

=

1 1 = Lsd Tisd s + 1 s +1 K pid

[10.81]

314

Power Electronic Converters

isq ( s )

1

=

isq* ( s )

Lsq K piq

= s +1

1 , Tisq s + 1

[10.82]

where Tisd and Tisq are the time constants that are to be applied to the currents isd and isq. The transfer functions for the current control loops acting on the currents isd and isq are thus first-order transfer functions whose respective time constants Tisd and Tisq are given by the following equations: Tisd =

Tisq =

Lsd K pid

Lsq K piq

[10.83]

[10.84]

.

These two equations show that in order to improve the dynamics of the current control for the currents isd and isq, the gains Kpid and Kpiq must be increased while taking care to avoid saturation effects. Having selected the closed-loop time constants, the parameters for the PI controllers acting along the d and q axes can then be determined. They are as follows: K pid =

K piq =

Lsd R R and Kiid = K pid s = s Tisd Lsd Tisd Lsq Tisq

and Kiiq = K piq

Rs R = s . Lsq Tisq

[10.85]

[10.86]

10.5.2. PI control structure in the (d, q) reference frame Figure 10.15 can be used to determine the PI control structure in the (d, q) reference frame. This control structure is shown in Figure 10.17.

PI Current Control of a Synchronous Motor

315

Figure 10.17. PI current control of a synchronous motor in the (d, q) plane

In the structure shown in Figure 10.17, the measured stator currents isa, isb, and isc are modified by the direct Park transform (abc-to-dq) in order to obtain the isd and isq components of the stator current vector in the coordinate system rotating at the synchronous velocity. Each component of the vector stator current is then regulated by a PI controller. The PI controllers then generate the reference voltages VsdL* and VsqL*, to which the compensation terms VsdNL* and VsqNL* are added. The resultant voltages Vsd* and Vsq* are then transformed. This inverse Park transform (dq-to-abc) is used to generate the three-phase reference voltages Vsa*, Vsb* and Vsc*. These reference voltages are then applied (as mean values) to the terminals of the stator phases of the synchronous motor by means of the voltage controller via the PWM block. Figure 10.18 shows the responses of the stator currents isa, isb, and isc along with the isd and isq components of the vector stator current in response to step changes in the references isd* and isq*. These responses show that the evolution of the currents isd and isq in the transient regime correspond to the response of a first-order system whose dynamics are determined by the parameters of the PI controllers.

316

Power Electronic Converters Isn isd

isq

100ms

0

-Isn Isn

Isn isa isc

100m 0

10ms

0

isb -Isn

-Isn

Figure 10.18. Response step changes in the reference currents isd* and isq*

10.6. Conclusion This chapter discussed PI current control for synchronous motors. Moving beyond the case of a synchronous motor, such control can be applied in a similar manner to any RLE-type three-phase load. The first control method discussed in this chapter was for a synchronous motor with non-salient poles in a fixed three-phase coordinate system aligned with the stator. In this case the phase as well as the amplitude of the stator current vector are controlled by the PI controllers. The second control method discussed was that of a synchronous motor with salient poles in a rotating coordinate system (d, q). In this case, the direct and transverse components of the stator current vector are constant quantities in the steady state regime. The modulus of the stator current vector can therefore be controlled with a high precision since the integral component of the PI controller cancels out any error in the steady state regime. The phase of the stator current vector is correctly imposed via coordinate transforms such as the Park and inverse Park transforms. Thus, the second of these control methods is the one most widely used [BUL 77, BUL 86], particularly in applications where high static and dynamic performances are required.

PI Current Control of a Synchronous Motor

317

10.7. Bibliography [BÜH 86] BÜHLER H., Réglage par mode de glissement, Presses Polytechniques Romandes, Lausanne, 1986. [BÜH 77] BÜHLER H., “Einführung in die Theorie geregelter Drehstromantriebe”, Bd.1: Grundlagen, Bd.2: Anwendugen, Birkhäuser Verlag, Basel-Stuttgard, 1977. [LOU 97] LOUIS J.P., BERGMANN C., Commande numérique, régimes intermédiaires et transitoires, Techniques de l’ingénieur, traité Génie Electrique, D3 643, 1997. [LOU 04] LOUIS J.P., Modélisation des machines électriques en vue de leur commande, Hermes, Paris, 2004. [MON 97] MONMASSON E., FAUCHER J., “Projet pédagogique autour de la MLI vectorielle destinée au pilotage d’un onduleur triphasé”, Revue 3EI, n°8, p. 23– 36, 1997.

Chapter 11

Predictive Current Control for a Synchronous Motor

11.1. Introduction Predictive control is a very broad concept covering a range of different control strategies. A discussion on these different classes of control strategies can be found in [KEN 00]. Although the algorithmic content of predictive control is more complex than that of other control methods, it nevertheless represents a highly effective technique that can be used to satisfy a range of constraints associated with current control. The typical structure of a predictive current control strategy for a threephase electrical motor, as described in [KAZ 02], is shown in Figure 11.1. This strategy involves solving the mathematical equations describing the behavior of a motor in response to the voltage vector that is applied to it. This mathematical model of the machine can be used to estimate the actual state of the machine at a given moment. A prediction and optimization model can then be used to predict the evolution of the stator current vector and to choose, by way of some optimization procedure, the most appropriate voltage vector to apply.

Chapter written by Mohamed Wissem NAOUAR, Eric MONMASSON, Ilhem SLAMA-BELKHODJA and Ahmad Ammar NAASSANI.

320

Power Electronic Converters E Reference current vector

Prediction and optimization

3-phase inverter

Switching states

Current motor state

Motor model

Electrical motor

Figure 11.1. Typical structure for a predictive current control strategy

Predictive control strategies can be divided into two categories: – minimum-switching-frequency predictive control strategies; – limited-switching-frequency predictive control strategies. We will begin this chapter by discussing these two strategies one by one. We will then present two examples of predictive current control for a synchronous motor. 11.2. Minimum-switching-frequency predictive control strategies These strategies are based on the use of hysteresis correctors. These correctors impose a boundary limitation on the current vector error. For example, Figure 11.2 shows a current vector error limit in the form of a r circle centered on the tip of the reference current vector is* . r

When the actual current vector is crosses the limiting boundary imposed by the hysteresis correctors, there are seven possible new trajectories, each of which corresponds to one of the seven voltage vectors that the two-level voltage inverter can generate (six active vectors and one null vector). The seven possible current trajectories are predicted and an optimization procedure is used to select the voltage vector that minimizes the switching frequency. The chosen voltage vector is the one which will result in the longest time interval before the error on the current vector will again cross

Predictive Current Control for a Synchronous Motor

321

the boundary imposed by the hysteresis correctors. Note that in this case the form of the current vector error is independent of the chosen coordinate system. It is thus possible to apply this predictive control technique within either a stationary or a rotating coordinate system. Imaginary Boundary

dis dt is* is Real

Figure 11.2. Predictive control with minimum switching frequency (limiting boundary: circle)

11.3. Limited-switching-frequency predictive control strategies In the case of limited-switching-frequency predictive control strategies, the switching states are determined at each sampling period. As mentioned at the beginning of this chapter, a predictive control strategy is based on mathematical equations describing the motor. These equations are used r along with the assumption that the stator current vector is and the induced r electromotive force vector es for the three-phase AC motor are sampled with a sampling period Te. With these assumptions, the stator current vector corresponding to the (k+1)th sampling period can be predicted as a function of the current, voltage, and EMF vectors induced in the kth sampling period, as shown in the following equation: r r r r is [k + 1] = f (is [k ], Vs [k ], es [k ]) .

[11.1] r

r

In equation [11.1], the vectors Vs [k ] and es [k ] are assumed to be constant over a given sampling period Te. This assumption requires that the sampling period should be much smaller than the electrical time constants of the system.

322

Power Electronic Converters

More generally, equation [11.1] becomes even more exact as the sampling period is made shorter. There are seven possible different trajectories that can be applied using equation [11.1] depending on the voltage vector applied over the kth sampling period. Then, an optimization procedure is used to select a voltage vector from among the various vectors that the voltage inverter can produce. This vector is then applied for the kth sampling period. Note that in this case the switching frequency is variable but limited to half the sampling frequency of the control algorithm. Furthermore, it is possible to develop a predictive control strategy using a fixed switching frequency by determining an optimal voltage vector that tends to cause the instantaneous value of the current vector to approach the reference current vector. r

This is achieved by replacingr the current vector is [k + 1] in equation [11.1] with the reference current is*[k ] for the kth sampling period, enabling the optimal voltage vector to be determined that will correct the error on the current vector, as shown in equation [11.2]: r r r r Vsopt [k ] = f (is [k ], is* [k ], es [k ]) .

[11.2]

The voltage vector calculated from equation [11.2] is then imposed as the mean value for the PWM strategy. This then imposes a fixed switching frequency for the power switches in the voltage source inverter. 11.4. Limited-switching-frequency predictive current control strategies for a synchronous motor 11.4.1. Predictive current control for a synchronous motor with variable, limited switching frequency If we take equations [10.33] to [10.36] from Chapter 10 and assume that the current ird is constant, then the expressions for the time derivative of the currents isd and isq are given by the following matrix equation:

Predictive Current Control for a Synchronous Motor

 disd   − 1  dt   Tsd  =  disq   Lsd  dt   − L ωdq (t )    sq

Lsd   1 ωdq (t )   Lsq  isd  +  Lsd   isq   1 −    0 Tsq  

0 1 Lsq

323

  Vsd   V  ,   sq  ωdq (t )   i   rd  

0 −

M sr Lsq

[11.3] where Tsd=Lsd/Rs and Tsq=Lsq/Rs are the electrical time constants for the d and q axes. In order to develop our predictive control algorithm, we will simplify this matrix equation by assuming that the sampling period Te that we will use is much smaller than the electrical time constants Tsd and Tsq. As a result, over the kth sampling period Te, the rotational velocity and the angular position of the rotor of the synchronous motor can be assumed to be constant. Using these assumptions we can perform a first-order discretization of matrix equation [11.3] and define the following predictive equations: Te Te  isd [k + 1] = L (Vsd [k ] − esd [k ]) + (1 − T )isd [k ] sd sd  ,   i [k + 1] = Te (V [k ] − e [k ]) + (1 − Te )i [k ] sq sq sq  sq Lsq Tsq 

[11.4]

where: esd [k ] = − Lsqωdq [k ]isq [k ]  ,  = + e [ k ] L ω [ k ] i [ k ] M ω [ k ] i [ k ] sd dq sd sr dq rd  sq

where isd[k+1] and isq[k+1] are the direct and transverse components of the stator current vector predicted for the (k+1)th sampling period. The esd and esq terms are the induced EMF along the d and q axes. In order to predict the evolution of the isd and isq components over the (k+1)th sampling period, we must determine the stator voltage vector in the common coordinate system (d, q) for each switching state of the command signals for the voltage inverter. We can do this by determining the various voltage vectors r (V j = ([Vsαj Vsjβ ]t )( j =0..7) (for a given DC bus voltage E) written in the fixed coordinate system (α, β) for each of the possible combinations of command

324

Power Electronic Converters

signals, as given in Table 11.1. Then, the rotation matrix given by equation [11.5] can be substituted to determine the various voltage vectors: rj (Vsdq = ([Vsdj

Vsqj ]t )( j =0..7)

written in the common coordinate system (d, q). Of the eight possible switching states for the command signals, six produce a non-zero voltage r0 r7 rj vector (Vsdq )( j =1..6) , while two produce the zero voltage vector (Vsdq ,Vsdq ) . Sa

Sb

Sc

Vsαj

Vsβj

0

0

0

0

0

1

0

0

2E/3

0

1

1

0

E/3

E/√3

0

1

0

-E/3

E/√3

0

1

1

-2E/3

0

0

0

1

-E/3

-E/√3

1

0

1

E/3

-E/√3

1

1

1

0

0

r Vs r V0 r V1 r V2 r V3 r V4 r V5 r V6 r V7

rj Vsdq Vsdq0 Vsdq1 Vsdq2 Vsdq3 Vsdq4 Vsdq5 Vsdq6 Vsdq7

Table 11.1. Voltage vectors as a function of the switching states

V j   cos(θ dq ) sin(θ dq )  Vsαj    sd  =   Vsqj   − sin(θ dq ) cos(θ dq )  Vsβj     

[11.5]

Given these eight possible combinations for the switching states Sa, Sb, and Sc, equation [11.4] can be written for these various possible voltage r

j )( j =0..7) , as shown below: vectors (Vsdq

Te Te  j j (isd [k + 1] = L (Vsd [k ] − esd [k ]) + (1 − T )isd [k ])( j =0..7) sd sd  ,  T T  (i j [k + 1] = e (V j [k ] − e [k ]) + (1 − e )i [k ]) ( j = 0..7) sq sq sq  sq Lsq Tsq 

[11.6]

Predictive Current Control for a Synchronous Motor

325

where: j (isd [k + 1])( j =0..7)

and: j (isq [k + 1])( j =0..7)

are the direct and transverse components of the stator current vector predicted for the start of the (k+1)th sampling period when the voltage vector rj (Vsdq )( j =0..7) is applied for the kth sampling period. Equation [11.6] states that the direct and transverse components of the stator current vector at the start of the next sampling period can be predicted based on the voltage vector applied at the start of the current sampling period. The trajectories for r the currents (t j )( j =0..7) , corresponding to the application of the vectors rj (Vsdq )( j =0..7) , are defined as follows:

rj r r (t j [k ] = isdq [k + 1] − isdq [k ])( j =0..7) ,

[11.7]

where: rj (isdq [k + 1])( j =0..7)

is the stator current vector predicted for the start of the (k+1)th sampling rj period when the voltage vector (Vsdq )( j =0..7) is applied for the kth sampling period. In addition, it is also possible to predict the current error vector: r j (∆isdq [k + 1])( j =0..7)

expressed in the common coordinate system (d, q). Since this vector is equal to the difference between the reference stator current vector for the kth sampling period: r* * * isdq [k ] = [isd [k ] isq [k ]]t

326

Power Electronic Converters

and the stator current vector predicted for the start of the (k+1)th sampling period: rj (isdq [k + 1] = [isdj [k + 1] isqj [k + 1]]t )( j =0..7) r

j resulting from use of the stator voltage vector (Vsdq )( j =0..7) , we have:

r j r* rj (∆isdq [k + 1] = isdq [k ] − isdq [k + 1])( j =0..7) .

[11.8]

The components (∆isdj [k + 1])( j =0..7) and (∆isqj [k + 1])( j =0..7) of each current r

j [k + 1])( j =0..7) can be written as: error vector (∆isdq * (∆isdj [k + 1] = isd [k ] − isdj [k + 1])( j =0..7) j * (∆isq [k + 1] = isq [k ] − isqj [k + 1])( j =0..7) .

[11.9] [11.10]

Equations [11.6], [11.9], and [11.10] can be used to determine the modulus of the current error vector using the following equation: 2 r j ( ∆isdq [k + 1] = (∆isdj [k + 1])2 + (∆isqj [k + 1])2 )( j =0..7)

[11.11]

Figure 11.4 shows the principle of our predictive control strategy with a variable and limited switching frequency. The angular electrical rotation velocity ωdq for the synchronous motor can be determined by differentiating the angular position θdq. The currents isd and isq along with the induced EMF terms esd and esq can be calculated by using the measured value of the stator currents isa and isb, the measured position θdq, and the estimated velocity ωdq. A prediction module can use equations [11.6], [11.9], and [11.10] to predict r j the components of the various current error vectors (∆isdq [k + 1])( j =0..7) . An optimization procedure is then used to select the voltage vector that should be applied.

Predictive Current Control for a Synchronous Motor q j

∆isdq [k+1]

∆isdq1[k+1] isdq1[k+1]

tj[k]

t1

isdq*[k]

t6

t2 t0,7

isdq*[k]

t3

isdqj[k+1]

327

isdq[k]

t5 t4

isdq[k]

a)

b)

r

d

j

Figure 11.3. a) Prediction of current error vector ( ∆isdq [ k + 1])( j = 0..7) ; b) examples of various prediction options

This procedure can be used to choose the combination of switching states that results in the minimum value of the squared magnitude of the current error vector. r j 2 Min ( ∆isdq ).

( j = 0..7)

For example, in Figure 11.3b the switching states (Sa Sb Sc) are chosen to 1 be equal to (100) since the vector Vsdq leads to the smallest squared modulus

of the current error vector. It should be noted that if the minimum error results from the application of a zero voltage vector, the choice of the combination of switching states is made based on commutation state of the switching states in the previous sampling period in order to reduce the switching frequency. In this case, if the commutation state for the previous sampling period belongs to the set {(000), (001), (010), (100)}, then the chosen zero voltage vector is the one produced by the switching state (000). Otherwise, if the commutation state for the previous sampling period belongs to the set {(011), (101), (110), (111)}, then the chosen zero voltage vector is the one produced by the switching state (111).

328

Power Electronic Converters E isd isq

Sa

*

(∆isdqj)(j=0..7)

Prediction

*

Optimization

Three phase inverter

Sb Sc

esd

isd

esq

isq isa

isd EMF Estimation

isq ωdq

isb

abc-to-dq

d/dt

isc

θdq p

Synchronous motor

θm Vrd

Figure 11.4. Principle of predictive control with variable, limited switching frequency

Isn isq

isq

2ms

0

0 20ms

isd

-Isn

isd

0

0

isa 0

0

isa 2ms

100ms

isb 0

0

isb

Figure 11.5. Impulse response for the reference isq* (from +Isn to −Isn and from −Isn to +Isn) for predictive control with variable, limited switching frequency

Figure 11.5 shows the evolution in stator phase currents along with the direct and transverse components of the stator current vector after the application of a step change to the reference isq*. These are shown for predictive control with variable, limited switching frequency. The steps applied to isq* reveal transient values for the transition between –Isn and +Isn,

Predictive Current Control for a Synchronous Motor

329

where Isn is the nominal value of the stator current in one phase of the synchronous motor. These results show that predictive control with variable, limited switching frequency exhibits extremely good dynamics similar to an ON/OFF control. It should also be noted that the transient regime for the stator currents is1, is2, and is3 is smooth and does not include any current spikes. 11.4.2. Predictive current control with fixed switching frequency for a synchronous motor Predictive control with variable, limited switching frequency as described in the previous section cannot give an exact reproduction of the reference current. Given that the chosen voltage vector is applied over the entire sampling period, the method ensures that the measured current vector gives the best approximation to the reference value but it can never precisely reproduce that reference. However, by applying two adjacent voltage vectors along with the zero voltage vectors over carefully determined time ratios, it is possible to exactly reproduce the reference current vector [LIN 07]. This effect can be achieved with predictive control at a fixed commutation frequency based on the use of a PWM technique. If we assume that the sampling period Te is equal to the PWM time division period TPWM, we can rewrite the prediction equation [11.4] to obtain: TMLI TMLI  isd [k + 1] = L (Vsd [k ] − esd [k ]) + (1 − T )isd [k ] sd sd  ,  T T  isq [k + 1] = MLI (Vsq [k ] − esq [k ]) + (1 − MLI )isq [k ]  Lsq Tsq 

[11.12]

where: esd [k ] = − Lsq ωdq [k ]isq [k ]  .  esq [k ] = Lsd ωdq [k ]isd [k ] + M sr ωdq [k ]ird [k ]

Note that when PWM is used, it is possible to choose any sampling period that is a multiple of TPWM/2 [NAO 07].

330

Power Electronic Converters

Sa(t) Sb(t) Sc(t)

t0,7 t1 t6 t0,7 t0,7 t6 t1 t0,7 4 2 2 4 4 2 2 4 (k+1)TMLI kTMLI a)

r r r* isdq [kTMLI + t 0,7 + t1 + t 6 ] = i sdq [(k + 1)TMLI ] = isdq [k ]

r 3 isdq [kTMLI + t 0,7 + t1 + t 6 ] 4

r t 3 isdq [kTMLI + t 0,7 + 1 + t 6 ] 4 2

r t 0, 7 t 1 t 6 + + ] isdq [kTMLI + 4 2 2 r t 0,7 t1 isdq [kTMLI + + ] 4 2

r t t 3 isdq [kTMLI + t 0,7 + 1 + 6 ] 4 2 2

r r isdq [k ] = isdq [kTMLI ]

r t 0,7 isdq [kTMLI + ] 4

b)

Figure 11.6. a) Example of a sequence of commutation states for PWM control; b) corresponding evolution of the stator current vector in the plane (d, q)

The aim of predictive control at fixed switching frequency is for the predicted direct and transverse components isd[k+1] and isq[k+1] to be equal to the reference currents isd*[k] and isq*[k], as expressed by the following equations: * isd [k + 1] = isd [k ]

[11.13]

* isq [k + 1] = isq [k ]

[11.14]

By inverting equation [11.12] so that equations [11.13] and [11.14] are satisfied, we obtain the following equation:

Predictive Current Control for a Synchronous Motor

Lsd * T  opt (isd [k ] − (1 − MLI )isd [k ]) + esd [k ] Vsd [k ] = T Tsd MLI  ,  L  V opt [k ] = sq (i* [k ] − (1 − TMLI )i [k ]) + e [k ] sq sq sq  sq TMLI Tsq 

331

[11.15]

where Vsdopt and Vsqopt are the direct and transverse components of the optimal voltage vector to be applied as the mean value output by the voltage inverter for the stator current vector to match its reference value. Thus, for the kth switching period, a voltage vector whose direct and transverse components are given by equation [11.15] is applied during one PWM period TMLI. The stator current vector for the synchronous motor will then match its reference value at the end of the kth switching period. However, the voltage inverter is a discrete component, and the reference voltage will only be applied in mean value terms. Figure 11.6a shows an example of adjacent and null voltage vectors being applied over one PWM period. Figure 11.6b then shows the evolution of the stator current vector of the synchronous motor as it reaches its reference value.

Figure 11.7. Principle of predictive control at fixed switching frequency

Figure 11.7 shows the principle of a predictive control strategy at a fixed switching frequency. The electrical angular rotation speed ωdq of the synchronous motor is determined by differentiating the angular position θdq. The currents isd and isq, along with the induced EMF terms esd and esq are

332

Power Electronic Converters

computed by using the measured values of the stator currents isa and isb, the measured position θdq, and the estimated speed ωdq. The direct and transverse components of the optimal voltage vector are constructed using the induced EMF terms esd and esq, the currents isd and isq, and the reference currents isd* and isq*. An inverse Park transform (dq-to-abc) enables the three-phase references Vsaopt, Vsbopt, and Vscopt to be generated from the optimal reference voltage vector. PWM can then be used to apply these voltages, in mean value terms, to the terminals of the stator phases of the synchronous motor. Figure 11.8 shows the evolution of the stator phase currents, along with the direct and transverse components of the stator current vector after the application of a step change to the reference isq* in the case of predictive control with a fixed commutation frequency. The step change applied to isq* results in a transition from +Isn to −Isn. These results show that predictive control at a fixed switching frequency has good dynamic characteristics, with the transition from +Isn to –Isn requiring a time of the order of 2 ms. It should also be noted that in the steady state regime, predictive control at a fixed switching frequency gives very high quality current control. Isn

Isn isq

isq

50ms

isd

isd

0

0

-Isn

-Isn

Isn

2ms

50ms

isa

isb

Isn isb isa

0

0

-Isn

-Isn

2ms

Figure 11.8. Impulse response for the reference current isq* (from +Isn to −Isn and from −Isn to +Isn) for predictive control at a fixed switching frequency

Predictive Current Control for a Synchronous Motor

333

11.5. Conclusion This chapter has presented the theoretical underpinnings of predictive control of the isd and isq components of the stator current vector of a synchronous motor. For each predictive control strategy, a prediction procedure enables the most suitable voltage vector to be selected for application to the synchronous motor. In the case of predictive control at a variable, limited switching frequency, the voltage vector to be applied over each sampling period is determined using a prediction and optimization procedure. The prediction procedure first enables the various current trajectories to be predicted depending on the voltage vector applied by the voltage inverter. This prediction procedure is based on a mathematical model of a synchronous motor in the common coordinate system (d, q), where the d axis is parallel to the rotor winding axis of the field system of the synchronous motor. Next, the optimization procedure takes place, selecting the voltage vector that leads to the minimum error between the reference stator current vector and the voltage at the stator phase terminals of the synchronous motor. In this case, the switching states are unchanged over each sampling period. As a result, the switching frequency is variable but limited to half of the sampling frequency of the control algorithm. Predictive control at fixed switching frequency also relies on a prediction procedure in the common coordinate system (d, q). This prediction procedure enables an optimal voltage vector to be determined, one that ensures that the current vector at the stator phase terminals of the synchronous motor will exactly match its reference value. The optimum voltage vector is then applied, in a mean value sense, to the terminals of the stator phases of the synchronous motor through the intermediary of a PWM process. In this case the switching frequency is constant and is equal to the PWM frequency. Compared to predictive control at a variable switching frequency, predictive control at a fixed switching frequency ensures higher quality current regulation in the steady state regime, with a well-defined harmonic content at high frequencies.

334

Power Electronic Converters

The concepts of predictive control that we have discussed in this chapter in the context of synchronous motors can easily be extended to any type of three-phase RLE load. 11.6. Bibliography [KEN 00] KENNEL R., LINDER A., “Predictive control of inverter supplied electrical drives” Proc. IEEE PESC, vol. 2, p. 761–766, 2000. [KAZ 02] KAZMIERKOWSKI M.P., KRISHNAN R., BLAABJERG F., Control in Power Electronics: Selected Problems, Academic Press, New York, 2002. [LIN 07] LIN-SHI X., MOREL F., LLOR A.M., ALLARD B., RÉTIF J.M. “Implementation of hybrid control for motor drives”, IEEE Trans. Ind. Electron., vol. 54, n°4, p. 1946–1952, 2007. [NAO 07] NAOUAR M.W., MONMASSON E., NAASSANI A.A., SLAMA-BELKHODJA I., PATIN N., “FPGA-based current controllers for AC machine drives - a review”, IEEE Trans. Ind. Electron., vol. 54, n°4, p. 1907–1925, 2007.

Chapter 12

Sliding Mode Current Control for a Synchronous Motor

12.1. Introduction This chapter considers current control for a DC motor and a synchronous motor using sliding mode control theory. This theory is generally associated with variable structure systems. The principles of such systems were first studied in the Soviet Union [FIL 60], [UTK 77], and [UTK 78]. Subsequent research by a wide range of groups led to further theoretical development and studies into several possible applications, particularly in the field of electrical system control [BUH 86 and BUH 97]. Sliding mode control is a specific operational mode for variable structure systems. The use of this type of control was for a long time limited by oscillations caused by sliding effects and limitations on the commutation frequency of power switches. However, following advances in these technologies, particularly in terms of improved performance of electronic power components, this type of control has become more and more effective for electrical machine control applications.

Chapter written by Ahmad Ammar NAASSANI, Mohamed Wissem NAOUAR, Eric MONMASSON and Ilhem SLAMA-BELKHODJA.

336

Power Electronic Converters

This chapter will begin by introducing the theoretical basis of sliding mode control, through the example of sliding mode current control for a DC motor. Expanding upon this application, we will then consider sliding mode control of the d and q components of the stator current vector of a synchronous motor. For this type of current control, we will analyze two inverter control strategies: direct control using a suitable switching table and; indirect control that makes use of a PWM process. 12.2. Sliding mode current control for a DC motor A DC motor consists of two parts: a field system whose purpose is to produce a magnetic flux Φf surrounding the armature; and the armature itself, whose supply voltage can be controlled using a power converter. The underlying mathematical equation describing the relationship between the voltage Va at the armature terminals of the DC machine and the current ia flowing through it, is given by the following equation: Va = Ra ia + La

dia + kΦ f Ω , dt

[12.1]

where Ra and La are the resistance and inductance of the armature. The term kΦf Ω is the induced counter-EMF, which is proportional to the rotation speed Ω and the flux Φf, where k is a constant that depends on the construction of the machine. The underlying principle of a variable structure system controlled via sliding mode control is shown in Figure 12.1. This shows current control in the armature of a DC motor driven by a four-quadrant chopper, using sliding mode control. There are various structures that can be used for sliding mode current control of a DC motor.

Sliding Mode Current Control for a Synchronous Motor

337

Va E

ia*

Sliding mode control

ia

Ra

La

kΦfΩ

CH ia

Figure 12.1. Four-quadrant chopper driving the armature of a DC motor, with sliding mode control

In variable structure systems under sliding mode control, three main basic control structures can be used for synthesizing the different commands. Consider the following system to be controlled: dx = f ( x) + B( x)u , dt

[12.2]

where u is the m-dimensional input vector, x is the n-dimensional state vector of the system, f is the function describing the time evolution of the system, and B is an n × m matrix. To construct a sliding mode control structure, we must first define an mdimensional function S(x) known as the switching function: S ( x) = [ S1 ( x)...Sm ( x)]t ,

[12.3]

where Si(x) is the ith switching function of S(x). There are several different ways of defining the switching function S(x). The set of points where this function has a null value, in other words where all the switching functions Si(x)(i=1..m) are zero, is known as the switching surface or sliding surface. A sliding mode command can then be obtained using the switching function S(x) that has been defined. Various examples throughout this chapter will show how these switching functions can be defined and how sliding mode control commands can be generated.

338

Power Electronic Converters

Figure 12.2 introduces the first control structure. Depending on the sign of S(x), the control quantity u is given by u=−K1(x) if S(x)>0, and u=−K2(x) if S(x) 0

if

S ( x) < 0

[12.4]

Disturbances u

+

u

u

Controlled system

Output variable

-

x

State vector

Switching function S(x)

Figure 12.3. Switching-based control structure

Sliding Mode Current Control for a Synchronous Motor

339

The third control structure is shown in Figure 12.4. In this structure, an equivalent control vector ueq is introduced. This vector is the input vector u to be applied in the steady state. The equivalent control vector is accompanied by the attractive control vector uatt whose role is to control the system in the transient regime such that the quantities under control will tend toward their reference values. This type of control is characterized by indirect control of the converter. In this case, the total control vector u is equal to the sum of ueq and uatt. It is applied to the system through an intermediate PWM stage.

Calculation of uatt

uatt

uéq + u +

Disturbances Controlled system x

Output variable

State vector

Switching function S(x)

Figure 12.4. Control structure using equivalent control vector

Next, we will describe command synthesis for sliding mode control of the system shown in Figure 12.1. For this we will only consider the control structures shown in Figures 12.3 and 12.4, since these are the most commonly used structures. 12.2.1. Direct sliding mode current control of a DC motor The first sliding mode control method we will consider for the system shown in Figure 12.1 is based on direct control of the converter. This corresponds to the switch-based control structure shown in Figure 12.3. In this case, the voltage Va applied to the terminals of the armature of the DC motor is equal to −E or +E depending on the sign of the error between the reference current ia* and the measured current ia. Such a control structure can be described by the following equations: dia R 1 1 1 (− Ra ia + Va − kΦ f Ω ) = − a ia − (kΦ f Ω ) + Va , = dt La La La La

[12.5]

340

Power Electronic Converters

where: Va = + E if

S (i a ) > 0

Va = − E if

S (i a ) < 0

[12.6]

and: S (ia ) = ia* − ia .

[12.7]

In Figure 12.5, the sgn symbol represents the “sign” function. ia

kΦfΩ +E

Va +-

+

1/La

-

-E

Ra *

ia

1 s

ia

S(ia)0 t

sgn(S(ia)) a)

b)

Figure 12.5. a) Block diagram for direct sliding mode current control of the armature of a DC motor; b) regions defined by the switching function S(ia)

Figure 12.5a shows a block diagram of direct sliding mode current control of the armature of a DC motor. This representation is based on equations [12.5], [12.6], and [12.7]. In this case, the current ia represents the state vector and the function S(ia) given by equation [12.7] represents the chosen switching function. In this example, the switching function S(ia) is equal to the difference between the reference current ia* and the measured current ia. It represents a straight line (ia = ia*) that divides the plane (ia, t) into regions, where S(ia) takes different signs as shown in Figure 12.5b. This straight line represents a set of points known as the sliding surface, although the straight line (ia = ia*) does not represent a surface in the strict sense of the term. The voltage Va represents the input vector for the system. As shown in Figure 12.5a, the input vector Va is switched between −E and +E according

Sliding Mode Current Control for a Synchronous Motor

341

to the sign of the switching function S(ia). Thus, the system shown in Figure 12.5a is analytically described by the following equations: – for S(ia) > 0: dia R 1 1 E; (kΦ f Ω ) + = − a ia − dt La La La

[12.8]

– for S(ia) < 0: dia R 1 1 E. = − a ia − (kΦ f Ω ) − dt La La La ia

[12.9] ia

ia*

ia* Sliding mode

Sliding mode Attractive mode

Attractive mode t0

a)

t

t0

b)

t

Figure 12.6. Characteristic trajectory for direct sliding mode control: a) analog operation; b) discrete operation

For a given value of the reference current ia*, the current trajectory ia described by the system in Figure 12.5a is shown in Figure 12.6a. This trajectory, which starts from an initial value of zero, represents a special trajectory unique to the sliding mode. This motion generally consists of two stages: – the first stage represents the attractive mode, also known as the nonsliding mode. During this stage, the trajectory starts from an arbitrary initial point (which in this example is a value of zero) and travels toward the sliding surface, reaching this surface in a finite time (in our example, this time is equal to t0); – at this point the system enters the second mode - the sliding mode. In this stage the current trajectory remains on the sliding surface.

342

Power Electronic Converters

However, it should be noted that analog operation of this control structure causes an infinite switching frequency for the power switches, which is not a practical possibility. Because of this, a perfect reproduction of the trajectory shown in Figure 12.6a is in practice impossible due to the discrete operating mode of the chopper. In this case, the function S(ia) does not become zero when the sliding surface is reached (in the steady state regime) and the current ia oscillates around its reference value, as shown in Figure 12.6b. We can use this example to highlight a number of properties of variable structure systems controlled by direct sliding mode control. These are as follows: – the control system based on direct sliding mode control depends on the sign of the switching function S(ia); – direct sliding mode control is characterized by two different modes: - attractive mode; - and sliding mode; – direct sliding mode control is well suited to systems that must be controlled discontinuously, which is the case for static converters. 12.2.2. Indirect sliding mode current control for a DC motor 12.2.2.1. Equivalent control method A range of approaches have been used to describe the sliding mode as the trajectory of the system reaches the sliding surface (S(x)=0) [UTK 78]. In this section, we will follow the method devised by Utkin, known as equivalent control. This method describes the dynamics of the system along the sliding surface. It ensures that the variable under control remains on this surface by imposing the required steady state regime value for the input quantity. For a given system under control, the first stage in constructing a control structure based on equivalent control is to find an equivalent input vector ueq such that the trajectory of the state of the system under control remains on the pre-defined sliding surface. Once the equivalent control vector has been determined, the dynamics of the system can be described by substituting ueq into the equation of state [12.2]. According to Utkin’s method, this

Sliding Mode Current Control for a Synchronous Motor

343

equivalent vector can be determined by considering the following invariance conditions:  S ( x) = 0 &  S ( x) = 0

[12.10]

The equivalent control (or voltage) variable is then determined using the invariance conditions given by equation [12.10]. This makes it possible to ensure that the trajectory of the controlled variables remains confined to the sliding surface. However, this voltage does not ensure correct control outside the sliding surface. For this reason, we must introduce an additional condition in order to ensure that each controlled variable approaches and reaches its corresponding sliding surface. This new condition is the attractivity condition. This will be discussed in the next section. 12.2.2.2. Attractive control One of the most commonly used attractivity conditions is the one given in [UTK 78]. A given switching function S(x) must satisfy the following equation: S&i ( x) < 0 if & S i ( x) > 0 if

S i ( x) > 0 S i ( x) < 0

(i = 1..m)

[12.11]

The attractivity condition can be illustrated by applying it for current control of the armature of a DC motor, applying the attractivity condition [12.11] to the switching function S(ia). As shown in Figure 12.7, for a given reference value ia*, the current ia is only attracted to the sliding surface S(ia) = 0, when S& (ia ) S (ia ) < 0 . Case n°2: S(ia)0

S(ia)0

ia

ia*

S(ia)>0

S(ia) 0 if

Si ( x) ≠ 0 (i=1…m).

[12.15]

We can integrate the expression for any given function g(S) in order to determine the trajectory of the switching function S(x). The function g can then be selected in such a way so as to specify the dynamics of the attractive mode and ensure that it moves from any given initial point toward the sliding surface. The choice of the Q and K coefficients then determines the various speeds for the switching function S(x). A number of different structures for the g(S) function can be found in [GAO 93]. Of these structures, the following are worthy of mention here: – constant speed attraction (g (S)=0): S& ( x) = −Q sgn( S ( x)) .

[12.16]

Sliding Mode Current Control for a Synchronous Motor

345

This law forces the state variable trajectory to approach the sliding surface at a constant speed that depends on the value of Q, where Q must be selected in such a way so as to avoid too slow an attraction time (the terms in the Q matrix are too small) and large oscillations of the controlled variables (the terms in the Q matrix are too large); – constant speed attraction with proportional action (g (S)=S): S& ( x) = −Q sgn( S ( x)) − KS ( x)

[12.17]

The introduction of the –KS(x) term means that the state trajectory is forced to approach the sliding surface at a greater speed when S is large. The larger K is, the faster is the attraction time, while a small value for Q reduces the risk of oscillations; – attraction with zero Q coefficient and ( g ( S ) = S α sgn( S ); 0 < α < 1 ): α S& ( x) = − K S ( x) sgn( S )

[12.18]

With this structure, the attraction speed is higher when the state trajectory is far from the sliding surface. Conversely, close to the sliding surface this speed is considerably reduced. Furthermore, the absence of the Q sgn (S(x)) term means that oscillations are almost completely eliminated once the sliding surface is reached. The chosen attraction scheme can be expressed in terms of an attractive tension uatt. This tension will be particularly active during the transient regime and will determine the dynamics of the system outside the sliding surface. 12.2.2.3. Indirect sliding mode control The sliding mode control law must simultaneously satisfy the twin conditions of invariance and attractivity. To do this, the switching function S(x) must satisfy the conditions given by equation [12.19], which is a combination of conditions [12.10] and [12.11]:

346

Power Electronic Converters

 S& ( x) = 0 if &  S i ( x) < 0 if S& ( x) > 0 if  i

S ( x) = 0 S i ( x) > 0 (i = 1..m)

[12.19]

S i ( x) < 0

Thus, the conditions given by equation [12.19] lead us to determine a new control law that takes account of these two conditions of invariance and attractivity. The expression for this control law is given by equation [12.20]: u* = ueq + uatt .

[12.20]

This control law consists of two terms: – the first is the equivalent control specifying the control required for the system to remain on the sliding surface; – the second is the attractive control that handles the control of the system outside the sliding surface. It also determines the dynamics of the system from its initial point until the sliding surface is reached. Returning to our example of current control in the armature of a DC motor, for a constant reference current ia* the time derivative of the switching function S(ia) can be written as follows: dS (ia ) di 1 = − a = − (Va − Ra ia − kΦ f Ω ) . dt dt La

[12.21]

This equation shows that the evolution of the armature current ia depends on: – the value of the current ia; – the applied voltage vector; – the counter-EMF term kΦfΩ; – the armature parameters Ra and La. As mentioned earlier, in order for the trajectory of the current ia to remain on its sliding surface S(ia) = 0, an equivalent voltage Vaeq must be applied. This can be calculated by considering the following invariance conditions: di S (ia ) = (ia* − ia ) = 0 and S& (ia ) = − a = 0 dt

[12.22]

Sliding Mode Current Control for a Synchronous Motor

347

The equivalent voltage Vaeq that the four-quadrant chopper must apply in this case can be determined in the following manner: ia* = ia  S (ia ) = (ia* − ia ) = 0  ⇒  dS (ia )  1 & = − (Vaeq − Ra ia − kΦ f Ω ) = 0   S (ia ) = 0 La  dt [12.23] Vaeq = Ra ia* + kΦ f Ω ⇒ Vaeq = Ra ia + kΦ f Ω

Equation [12.23] defines the analog equivalent control that ensures that the current ia will remain on the sliding surface. However, the four-quadrant chopper is a discrete device. As a result, the derivative of the switching function S& (ia ) never reaches zero. In addition, the equivalent voltage given by equation [12.23] does not enable ia to be controlled outside the sliding surface. For these reasons, we must also consider the time derivative S& (ia ) of the switching function in our control formula. Given that the use of PWM enables us to impose a mean value for the voltage Va equal to the reference voltage Va* over each switching period, we can determine this reference voltage from equation [12.21] as follows: Va* = Ra ia + kΦ f Ω − La

dS (ia ) dS (ia ) = Vaeq − La = Vaeq + Vaatt dt dt

[12.24]

The reference Va* consists of two terms: The first is the equivalent reference Vaeq, while the second term (Vaatt = −La(dS(ia)/dt)) involves the time derivative S& (ia ) of the switching function, whose purpose is to steer the trajectory of the controlled variable toward the sliding surface. The derivative of the switching function is chosen in such a way that the attraction conditions are satisfied. By choosing a structure with constant speed attraction and proportional action, similar to that given by equation [12.17], the attractive voltage Vaatt can then be written as follows: Vaatt = − La (− q sgn( S (ia )) − kS (ia ))

where q and k are real, positive numbers.

[12.25]

348

Power Electronic Converters

We can use equations [12.23], [12.24], and [12.25] to determine the expression for the reference Va* to be applied. This is given by the following equation: Va* = Vaeq + Vaatt = R a i a* + kΦ f Ω + L a (q sgn( S (i a )) + kS (i a ))

[12.26]

To ensure that the current ia approaches its sliding surface S(ia) = 0, the attraction condition S (ia ) S& (ia ) < 0 must be satisfied. By substituting the expression for the reference voltage given by equation [12.26], we can write the expression for the product of the switching function S(ia) and its time derivative as follows: S (ia ) S& (ia ) = − S (ia )

dia 1 S (ia )(Va − kΦ f Ω − Ra ia ) =− dt La

=−

1 S (ia )(Va* − kΦ f Ω − Ra ia ) La

=−

Ra ( S (ia ))2 − qS (ia ) sgn( S (ia )) − k ( S (ia ))2 La

[12.27]

.

This expression for the product S (ia ) S& (ia ) consists of three negative terms. This product itself is therefore also negative. The attraction condition given by equation [12.11] is therefore satisfied regardless of the sign of the switching function S(ia). Figure 12.8 shows a block diagram of indirect sliding mode current control in the armature of a DC motor.

ia*

kΦfΩ ++

Ra +

S(ia)

Vaeq + Va*

k

-

+ Sgn q

+

+

La

Vaatt

Indirect sliding mode control

PWM + Chopper

Va + -

+ -

1/La

1 s

kΦfΩ Ra

DC motor armature

Figure 12.8. Block diagram for indirect sliding mode current control in the armature of a DC motor

ia

Sliding Mode Current Control for a Synchronous Motor

349

For a given value of the reference current ia*, the trajectory of the current ia produced by the system in Figure 12.8 is shown in Figure 12.9a. ia

ia

ia*

ia* Sliding mode

Sliding mode

Attractive mode t0

Attractive mode t

a)

t0

t b)

Figure 12.9. Characteristic trajectory for indirect sliding mode control: a) analog operation; b) discrete operation

In a way similar to direct sliding mode control, and starting from a given initial point (in this case, the starting point is equal to zero), the resultant trajectory consists of two stages. The first stage corresponds to the attractive mode while the second stage corresponds to the sliding mode. Figure 12.9a represents analog operation of the system shown in Figure 12.8. However, given the discrete nature of the power converter, and given that the PWM switching frequency is limited, it is impossible to obtain such operation in practice. It is for this reason that when the sliding mode is reached, the switching function S(ia) does not reach zero and the actual current oscillates around its reference value as shown in Figure 12.9b. The amplitude of the current oscillations in this case depend largely on the PWM switching frequency, although they also depend on the choice of k and q coefficients. Our consideration of this application also enables us to comment on a number of properties of variable structure systems controlled by indirect sliding mode control: – a control system based on indirect sliding mode control not only depends on the sign of the switching function S(ia) but also on its value;

350

Power Electronic Converters

– the dynamics of the system under indirect sliding mode control depend largely on the choice of coefficients for the K and Q matrices, as well as the structure of the function governing the attractive mode; – the use of indirect sliding mode control requires an intermediate PWM stage to be incorporated; – indirect sliding mode control is characterized by two modes: - the attractive mode, - and the sliding mode; – the theory of indirect sliding mode control can easily be adapted to systems requiring discontinuous control, such as static converters. 12.3. Sliding mode current control of a synchronous motor This section discusses sliding mode control of the stator current vector in a synchronous motor, working in the rotating synchronous reference frame (d, q). The underlying mathematical equations that describe the synchronous motor model in a (d, q) reference frame (where the d axis is aligned with the rotor winding axis of the synchronous motor) are given by equations [10.33] to [10.39] in Chapter 10. Figure 12.10 shows the reference stator current vector and its measured value in the (d, q) rotating reference frame. β

q

isdq* isq*

Sisq

isdq Sisd

isq

isd

d isd*

θdq α

Figure 12.10. Reference and measured stator current vectors

r

r

* vectors refer respectively to the In Figure 12.10, the isdq and isdq

measured and reference stator current vectors expressed in the rotating reference frame (d, q). The angle θdq represents the phase shift between the d and α axes.

Sliding Mode Current Control for a Synchronous Motor

351

Since the aim of sliding mode control is to control the d and q components of the stator current vector, two switching functions Sisd and Sisq are defined as follows: * Sisd = iisd − isd

[12.28]

* Sisq = iisq − isq ,

[12.29]

where isd* and isq* are the reference stator current vectors along the d and q axes, respectively. These two switching functions define two sliding surfaces (Sisd = 0) and (Sisq= 0). Equations [12.30] and [12.31] are the equations obtained when the trajectories of the currents isd and isq reach their sliding surfaces: * Sisd = 0 ⇒ isd = isd

[12.30]

Sisq = 0 ⇒ isq = isq* .

[12.31]

For constant reference currents isd* and isq*, the time derivatives of the switching functions Sisd and Sisq can be used to analyze the evolution of the isd and isq current components. The time derivative of the switching function Sisd is given by the following equation: dSisd di = − sd . dt dt

[12.32]

But, equation [10.35] states that the isd can be written as follows: isd =

Φ sd − M sr ird Lsd

.

[12.33]

If the reference current ird is kept constant, the time derivative of the isd component becomes: disd 1 dΦ sd = . dt Lsd dt

[12.34]

352

Power Electronic Converters

Equations [10.33], [12.32], and [12.34] can be used to determine the expression for the time derivative of the switching function Sisd: dSisd 1 dΦ sd 1 (Vsd − Rs isd + ωdqΦ sq ) . =− =− dt Lsd dt Lsd

[12.35]

Equation [12.35] describes the evolution of the d component of the stator current, which depends on: – the rotation speed of the synchronous motor; – the applied voltage vector; – the current isd and the flux Φsq; – the resistance and direct cyclic inductance of the synchronous motor. Moving on to the isq component, which we can obtain by differentiating the switching function Sisq with respect to time. Since isq* is assumed to be constant, the time derivative of Sisq is: dSisq dt

=−

disq dt

.

[12.36]

Equation [10.36] states that the isq component is: isq =

Φ sq Lsq

.

[12.37]

The time derivative of the isq component can therefore be written as: disq dt

=

1 dΦ sq . Lsq dt

[12.38]

Equations [10.34], [12.36], and [12.38] then give the following expression for the time derivative of the switching function Sisq: dSisq dt

=−

1 dΦ sq 1 =− (Vsq − Rs isq − ωdqΦ sd ) . Lsq dt Lsq

[12.39]

Sliding Mode Current Control for a Synchronous Motor

353

Equation [12.39] describes the evolution of the q component of the stator current vector, which depends on: – the rotation speed of the synchronous motor; – the applied voltage vector; – the current isq and the flux Φsd; – the resistance and transverse cyclic inductance of the synchronous motor. In the next sections we will discuss sliding mode control of the stator current vector in the (d, q) reference frame for a synchronous motor. We will use the control structures shown in Figures 12.3 and 12.4. The first type of sliding mode control involves direct control of the voltage inverter, while the second type involves indirect control of the voltage inverter. 12.3.1. Direct sliding mode control of the stator current vector in an induction motor The locus of the time derivatives of the switching functions Sisd and Sisq as described by equation [12.35] and [12.39] is shown in Figure 12.11. This locus is given for θdq = 0 and for both rotation directions (positive and negative) of the synchronous motor. We will use this locus to develop our description of sliding mode control of the stator current vector of a synchronous motor. Sisq

Sisq Sisd 0

V6

V5

Sisd 0

Sisd >0 Sisq >0

V6

V5

Sisd >0 Sisq >0 Sisd

V1

V4

V0,7

V1

V4 V0,7

Sisd Sisd 0 ), Figure 12.11a indicates that three different voltage vectors can be applied for the time derivative S&isd to be r

negative and the time derivative S&isq to be positive. These are the vectors V1 , r r V6 and V0,7 .

Sliding Mode Current Control for a Synchronous Motor

357

For a negative rotation speed ( ωdq < 0 ), Figure 12.11b indicates that just one voltage vector can be applied for the time derivatives S&isd to be negative r

and the time derivative S&isq to be positive. This is vector V6 . 12.3.1.3. Case where Cd = 0 and Cq=1 Equations [12.42] and [12.43] indicate that this logical state for the signals Cd and Cq requires that the switching function Sisd should be negative and the switching function Sisq should be positive. To steer the currents isd and isq toward their respective sliding surfaces Sisd = 0 and Sisq= 0, the time derivative S&isd must be positive and the time derivative S&isq must be negative. For a positive rotation speed ( ωdq > 0 ), Figure 12.11a indicates that only one voltage vector can be applied for the time derivative S&isd to be negative r

and the time derivative S&isq to be positive. This is vector V3 . For a negative rotation speed ( ωdq < 0 ), Figure 12.11b indicates that two different voltage vectors can be applied for the time derivatives S&isd to be r

negative and the time derivative S&isq to be positive. These are vectors V3 r

and V4 . 12.3.1.4. Case where Cd =1 and Cq=1 Equations [12.42] and [12.43] indicate that this logical state for the signals Cd and Cq requires that both switching functions Sisd and Sisq should be positive. To steer the currents isd and isq toward their respective sliding surfaces Sisd = 0 and Sisq= 0, the time derivatives S&isd and S&isq must both be negative. For a positive rotation speed ( ωdq > 0 ), Figure 12.11a indicates that just one voltage vector can be applied for the time derivatives S&isd and S&isq to be r

positive. This is the vector V2 .

358

Power Electronic Converters

For a negative rotation speed ( ωdq < 0 ), Figure 12.11b indicates that three different voltage vectors can be applied for the time derivatives S&isd and r r r S&isq to be positive. These are the vectors V1 , V2 , and V0,7 . i = 2 & ωdq > 0

i = 2 & ωdq < 0

i = 1 & ωdq > 0

Sisq

Sisq

Sisq

V2 V3

V1

V2 V6

V0,7 V4

Sisd

V1

V0,7

V3

V5

V1

Sisd

V2

V6

V4

Sisq V6

V1 V5

V0,7 V3

V5

i = 1 & ωdq < 0

Sisd

i = 0 & ωdq > 0

Sisq

V3

V2 V1

V0,7 V5

V6

V4

V0,7

i=0 V0,7

V1

d θdq

V5

V5

V3 Sisq

V6

V5

V0,7

V1

V6

V6

V3

b) i = 4 & ωdq < 0

i = 5 & ωdq > 0

Sisq

Sisq

Sisq

V6

Sisd V4

V2

i = 4 & ωdq > 0 V3

V4 V2

V0,7

Sisd

i = 0 & ωdq < 0

e)

V4

V4

V2

V1

i=5

a)

V5

V0,7

Sisd V1

V5

V6

i=1

i=4

V2

Sisq

V2

i=3

Sisq

V4

q

i=2

Sisd

i = 3 & ωdq < 0 V3

V4

c)

i = 3 & ωdq > 0

V4

Sisd V5

V3

d)

V3

V0,7

V2

V4

V6

Sisd

V1

V5

V0,7 V6

f)

V3

Sisd V2

V1

V5 V6

Sisq V4

V5 V3

V0,7 V1

i = 5 & ωdq < 0

Sisd

V6

V2

V4

V0,7 V1

Sisd V3

V2

g)

Figure 12.13. a) Decomposition into six sectors (i = 0..5) of the locus of the evolution of the d axis. Locus of the time derivatives of the switching functions when the d axis lies in the middle of: b) sector i = 0 (θdq = 0); c) sector i = 1 (θdq = π/3); d) sector i = 2 (θdq = 2π/3); e) sector i = 3 (θdq = π); f) sector i = 4 (θdq = 4π/3); g) sector i = 5 (θdq = 5π/3)

For a fixed rotation speed, as the angle θdq varies the resultant locus of the derivatives of the switching functions Sisd and Sisq takes the form of an ellipse, centered around the point obtained by applying a voltage vector of

Sliding Mode Current Control for a Synchronous Motor

359

zero. In addition, in response to a variation ∆θdq in the angle θdq, the ellipse is rotated by an angle ∆θdq. This rotation is centered on that same point, obtained by applying a voltage vector of zero, as shown in Figure 12.14. This figure shows the case where the angle θdq lies inside sector 0. Sisq

V6

V6

Sisq

V6

V5

V6

V5 V5

V5

Sisd

V1

V1

V1

∆θdq V0,7

V1

V4 V4

∆θdq V0,7

Sisd

V4 V4

V2

V2 V2

V3 a)

V3

V2

V3

V3

b)

Figure 12.14. Evolution of the locus of the time derivatives of the switching functions in response to a variation in the angle θdq: (a) ωdq = 0.5ωn; (b) ωdq = −0.5ωn

Thus, a natural periodicity appears as the indexing order of the active voltage vectors is changed. As a result, it is sufficient to consider a 60° sector for θdq and then to rotationally transform the resultant solution obtained for this sector according to the evolution of θdq. It is for this reason that the (α, β) reference plane is divided into six sectors i (I = 0.5), as shown in Figure 12.13a. These sectors are distributed such that each active voltage vector lies in the middle of its own sector. These sectors determine which region the angle θdq falls within. Thus, it is possible to build up a table containing the voltage vectors to be applied for each rotation direction. The switching table (Table 12.1), was built up by analyzing Figure 12.13. This table ensures correct control of the d and q components of the stator current vector and is valid for both directions of rotation. It also gives a general idea of the voltage vectors to be applied at each point in time in order that the d and q components of the stator current vector should be steered toward their reference values. However, in its present form this table

360

Power Electronic Converters

cannot be directly used in a control algorithm. In order to do this, it is crucial to refine the switching table to take into account the criteria imposed by the specification of the application in question. Cd Cq i=0 i=1 i=2 i=3 i=4 i=5

ωdq > 0 ωdq < 0 ωdq > 0 ωdq < 0 ωdq > 0 ωdq < 0 ωdq > 0 ωdq < 0 ωdq > 0 ωdq < 0 ωdq > 0 ωdq < 0

0 0

r r V4 , V5 r V5 r r V5 , V6 r V6 r r V1 , V6 r V1 r r V1 , V2 r V2 r r V2 , V3 r V3 r r V3 , V4 r V4

1

0

0

1

r r r V1 , V6 , V0,7 r V6 r r r V1 , V2 , V0,7 r V1 r r r V2 , V3 , V0,7 r V2 r r r V3 , V4 , V0,7 r V3 r r r V4 , V5 , V0,7 r V4 r r r V5 , V6 , V0,7 r V5

r V3

r r V3 , V4 r V4 r r V4 , V5 r V5 r r V5 , V6 r V6 r r V1 , V6 r V1 r r V1 , V2 r V2 r r V2 , V3

1 1

r V2 r r r V1 , V2 , V0,7 r V3 r r r V2 , V3 , V0,7 r V4 r r r V3 , V4 , V0,7 r V5 r r r V4 , V5 , V0,7 r V6 r r r V5 , V6 , V0,7 r V1 r r r V1 , V6 , V0,7

Table 12.1. Switching table for each rotation direction, for control of the d and q components of the stator current vector

We will propose a simple strategy that favors simultaneous control of the d and q components of the stator current vector, which enables the switching table to be simplified. This strategy can be illustrated by considering the case where the d axis lies in sector (i = 0) and when the logical variables Cd and r Cq take their high values. For positive rotation, the vector V2 is an r

r

r

appropriate choice. Conversely, the vectors V1 , V2 , and V0,7 are all r

appropriate choices for a negative rotation. In this case, the vector V2 is chosen for our table since it is valid for both rotation directions. This same reasoning can be applied to all the sectors, resulting in the simplified switching table (Table 12.2). This table is valid for both directions of rotation.

Sliding Mode Current Control for a Synchronous Motor Cd

0

1

0

Cq

0

0

1

r V5 r V6 r V1 r V2 r V3 r V4

i=0 i=1 i=2 i=3 i=4 i=5

r V6 r V1 r V2 r V3 r V4 r V5

361

1 1

r V3 r V4 r V5 r V6 r V1 r V2

r V2 r V3 r V4 r V5 r V6 r V1

Table 12.2. Simplified switching table valid for both rotation directions, for control of the d and q components of the stator current vector E isd

*

isq*

+ + -

Sisd

Cd

Table

Cq

Sisq

i

Sa Three-phase inverter

Sb Sc

Sector 3

21 45

0

isa

isd isq

abc-to-dq

isb isc

θdq p

synchronous motor

θm Vrd

Figure 12.15. Block diagram of direct sliding mode control of the stator current vector of an induction motor

The general form for direct sliding mode control of the stator current vector in a synchronous motor in the (d, q) reference frame is shown in Figure 12.15. Two sign comparators are used to determine the logical state of the variables Cd and Cq. The input to the sign comparators are the switching functions Sisd and Sisq. These functions are determined by calculating the difference between the reference and measured components of the stator current vector, expressed in the (d, q) reference frame. The switching table for the control structure is addressed using the logical

362

Power Electronic Converters

variables Cd and Cq, along with the number of the sector in which the d axis lies. The number of this sector is determined by measuring the angle θdq. Then, the switching table generates the switching states Sa, Sb, and Sc to be applied to the three-phase inverter. Figure 12.16 shows how the stator currents isa, isb, and isc vary, along with the direct and transverse components of the transverse stator current vector in response to a step change in the reference isq* in the case of direct sliding mode control. The step changes applied to isq*, reveal the transient behavior between –Isn and +Isn, where Isn is the nominal value of the stator phase current of the synchronous motor. These results show that direct sliding mode control offers extremely good dynamic performance for the isq component (of the order of 2 ms) for both rotation directions. We also note that the transients for the stator currents isa, isb, and isc do not involve any interference or current spikes. Sliding mode

0

isq

Attractive mode

isq

0

2ms

2ms

Sliding mode

isd 0

Isn

Isn isq

50ms

50ms 0

0

-Isn

isd

0

isd

isq isd

-Isn 0

0

isa 0

0 100ms 0

5ms

is1

isb 0

is2

Figure 12.16. Impulse response for the reference current isq* (from +Isn to −Isn and from −Isn to +Isn) for direct sliding mode control of the stator current vector of a synchronous motor

Sliding Mode Current Control for a Synchronous Motor

363

12.3.2. Indirect sliding mode control of the stator current vector in an induction motor Indirect sliding mode control of the stator current vector of a synchronous motor can be used to determine the direct and transverse components of the reference voltage vector expressed in the (d, q) plane. These components are then applied to the stator phase terminals of the synchronous motor through an intermediate PWM stage. As mentioned in section 12.2.2.3, such a control law must be formulated so as to satisfy the twin conditions of invariance and attraction as given by equation [12.19]. Furthermore, this control law can be developed by drawing on the illustrative example of sliding mode current control of a DC motor. By considering the –ωdqΦsq and ωdqΦsd terms as induced counter-EMF terms along the d and q axes, the expressions for the time derivatives S&isd and S&isq of the switching functions, given by equations [12.35] and [12.39], are analogous to those for the time derivative of the switching function S& (ia ) given by equation [12.21]. Thus, indirect sliding mode control of the direct and transverse components of the stator current vector can be achieved by applying the same approach to the d and q axes as was used in our formulation of indirect sliding mode current control in a direct current motor. As a result, the two components Vsd* and Vsq* of the reference voltage r

vector Vsdq* (expressed in the (d, q) reference frame) each consist of two terms, as stated in equations [12.44] and [12.45]. The first term corresponds to the equivalent voltage vector (Vsdeq for the Vsd* component and Vsqeq for the Vsq* component); this term is active in the steady state regime. The second term corresponds to the attractive voltage vector (Vsdatt for the Vsd* component and Vsqatt for the Vsq* component); active in the transient regime: Vsd * = Vsdeq + Vsdatt

[12.44]

Vsq * = Vsqeq + V sqatt .

[12.45]

364

Power Electronic Converters

For the trajectories of the currents isd and isq to remain on their respective sliding surfaces (Sisd = 0) and (Sisq = 0), the Vsdeq and Vsqeq components of the equivalent voltage vector must be applied to the d and q axes, respectively. These components can be deduced by observing the following invariance conditions: S isd = 0 and

di S&isd = − sd = 0 dt

S isq = 0 and

S& isq = −

di sq dt

[12.46] [12.47]

=0

The invariance conditions given by these two equations can be used to determine the Vsdeq and Vsqeq components as follows: i sd * = i sd S isd = (i sd * − i sd ) = 0  ⇒  dS isd  1 & (Vsdeq − Rs i sd + ω dq Φ sq ) = 0 [12.48] =− S isd = 0  dt L sd  ⇒ V sdeq = Rs i sd − ω dq Φ sq = Rs i sd * − ω dq Φ sq i sq * = i sq S isq = (i sq * − i sq ) = 0  ⇒  dS isq  1 & (Vsqeq − Rs i sq − ω dq Φ sd ) = 0 =− S isq = 0  Lsq  dt

[12.49]

⇒ Vsqeq = Rs i sq + ω dq Φ sd = Rs i sq * + ω dq Φ sd

By considering the time derivatives S&isd and S&isq of the switching functions in the control formula, the new components Vsd* and Vsq* along the d and q axes can be determined from equations [12.35] and [12.39] using the following calculations: Vsd * = Rs i sd * − ω dq Φ sq − Lsd V sq * = R s i sq * + ω dq Φ sd − Lsq

dS isd = Vsdeq + V sdatt dt dS isq dt

= V sqeq + V sqatt .

[12.50] [12.51]

Sliding Mode Current Control for a Synchronous Motor

365

Equation [12.50] (and [12.51]) shows that the attractive voltage Vsdatt (and Vsqatt) for the direct component Vsd* (and transverse component Vsq*) involves the time derivative of the switching function S&isd (and S&isq ). If we select a constant-velocity, proportional action attraction structure such as that given in equation [12.17], the expressions Vsdatt and Vsqatt are given by the following equations: Vsdatt = − Lsd (−qd sgn( Sisd ) − kd Sisd )

[12.52]

Vsqatt = − Lsq (−qq sgn( Sisq ) − kq Sisq ) ,

[12.53]

where qd, kd, qq , and kq are real, positive numbers. Equations [12.48] to [12.53] can be used to determine the expression for the d and q components of the reference voltage vector expressed in the (d, q) reference frame: Vsd *  V sdeq  Vsdatt  i *  − ω dq Φ sq  + = Rs  sd *  +   * =     Vsq  V sqeq  Vsqatt  i sq   ω dq Φ sd  0  q d 0  sgn( S isd ) k d 0   S isd   L sd −  (−   −  )  0 Lsq   0 q q  sgn( S isq )   0 k q   S isq 

[12.54]

By substituting in this expression for the reference voltage vector, we can obtain an expression for the product of each of these switching functions Sisd and Sisq with their respective derivatives: R Sisd S&isd = − s Sisd 2 − qd Sisd sgn( Sisd ) − kd Sisd 2 Lsd

[12.55]

Rs Sisq 2 − qq Sisq sgn( Sisq ) − kq Sisq 2 . Lsq

[12.56]

Sisq S&isq = −

Each of these products Sisd S&isd and Sisq S&isq consist of the sum of three negative terms. The products Sisd S&isd and Sisq S&isq themselves are therefore also negative, and the attraction condition given by equation [12.11] is satisfied regardless of the sign of the switching functions Sisd and Sisq. In this

366

Power Electronic Converters

case, the currents isd and isq will therefore be steered toward their respective sliding surfaces (Sisd= 0) and (Sisq= 0). Figure 12.17 shows a block diagram illustrating how the Vsd* and Vsq* components of the reference voltage vector can be determined using equation [12.54]. -ωdqΦsq *

isd

+ isd -

++

Rs Sisd

ωdqΦsd Vsdeq +

kd +

Sgn

+ qd a)

+

Lsd

Vsdatt

isq

*

Vsd* isq -+

++

Rs Sisq

Vsqeq +

kq +

Sgn qq

+

+

Lsq

Vsq*

Vsqatt

b)

Figure 12.17. Construction of the Vsd* and Vsq* components of the reference voltage vector

We should also note that the coefficients qd, kd, qq, and kq can be chosen on the basis of a range of criteria: – the modulus of the reference voltage vector must not exceed the maximum amplitude that the PWM-controlled voltage inverter can output. For example, the maximum amplitude is equal to E/2 in the case of sinetriangle PWM and E/√3 in the case of space vector PWM, where E denotes the voltage level at the input of the three-phase inverter; – the chosen coefficients must offer a suitable robustness with regard to parametric variations. The larger the values of the coefficients that are chosen, the more robust the system will be in the face of parametric variations; – the chosen coefficients must not lead to strong current oscillations. These current oscillations will be larger if the coefficients themselves are larger. This is due to the fact that higher values of these coefficients lead to an increase in the amplitude of the reference voltage vector. This increase results in more sensitive current dynamics, which can lead to current oscillations.

Sliding Mode Current Control for a Synchronous Motor

367

The general structure of indirect sliding mode control of the stator current vector of a synchronous motor in the (d, q) reference frame is shown in Figure 12.18. isd isq

Indirect sliding mode control

*

-ωdqΦsq ωdqΦsd

isd

Vsq*

E

Vsa*

Vsd*

*

dq-to-abc

Vsb* Vsc*

Sa PWM

Sc

θdq

isq

isa

isd EMF estimation

isq ωdq

Three phase inverter

Sb

abc-to-dq

isb isc

θdq

d/dt p

Synchronous motor

θm Vrd

Figure 12.18. Indirect sliding mode control of the stator current vector of a synchronous motor Vsd*

Vsa*

0

0 5ms

20ms

Vsb*

Vsq* 0

0

a)

b)

0

Vsa*

0 20ms

10ms

Vsa

Vsb

0

0

Vsb

*

c)

Figure 12.19. Variation of reference voltages (isq*= Isn, isd*= 0) (kd = 300, qd = 300, qq= 300, qq = 300); a) variation of Vsd* Vsq*; b) variation of Vsa* and Vsb* ; c) variation of Vsa and Vsb

368

Power Electronic Converters

Figures 12.19a and 12.19b show the variation in the reference voltages Vsd* and Vsq* along with the single-phase reference voltages Vsa*, Vsb*, and Vsc*. Note that these voltages have an oscillating form. This is due to the change in sign of the switching function, which affects the values of the reference voltages (as stated by equation [12.54]. Figure 12.19c then shows the variation in the single-phase voltages supplied by the voltage inverter to the terminals of the synchronous motor . isa

isa

0

10ms 50ms

0

isb 0

a)

b)

isa

isa

0

10ms 50ms

0

isb 0

c)

d)

Figure 12.20. Variation of stator currents (isq*= Isn, isd*= 0) (kd = 300, qd = 300, kq = 300, qq= 300); a-b) PWM frequency = 1.5 kHz; c-d) PWM frequency=3 kHz

Figure 12.20 shows the variation in the stator currents isa and isb obtained for a reference current isq* equal to the nominal current value, with the reference current isd* equal to zero. These variations are obtained for two different frequencies of the carrier, 1.5 KHz and 3 KHz. Note that the quality of the current control achieved by indirect sliding mode control operating at a PWM frequency of a few kHz is better than that obtained using direct sliding mode control, in spite of the latter’s considerably higher sampling frequency. Figure 12.21 shows the response of the isd and isq components of the stator current vector along with that of the stator currents isa and isb in response to a step change in the reference isq*(from +Isn to −Isn and from −Isn to +Isn). These responses illustrate that the transient dynamic behavior

Sliding Mode Current Control for a Synchronous Motor

369

between −Isn and +Isn is not as good as that obtained with direct sliding mode control (Figure 12.16). Sliding mode isq 0

5ms

isd

isd 0

Isn isq

0

50ms 0

50ms

-Isn

5ms

0

Sliding mode

0

Isn

isq

Attractive mode

isq isd

0

-Isn

isd

0

Figure 12.21. Response to a step change in the reference current isq* (from +Isn to −Isn and from −Isn to +Isn) for indirect sliding mode control of the stator current vector of a synchronous motor

12.4. Conclusion This chapter has introduced the theoretical basis of direct and indirect sliding mode control of the isd and isq components of the stator current vector of a synchronous motor. In each of these two cases, the currents isd and isq are compared with their respective reference values isd* and isq* in order to obtain the switching functions Sisd and Sisq. In the case of direct sliding mode control, a switching table that uses the six active voltage vectors of the voltage inverter was obtained using the theory of sliding mode control. This table enables simultaneous control of the currents isd and isq, and it is valid for both directions of rotation. Direct sliding mode control involves a variable switching frequency and has extremely good dynamic performance during transient regimes. We remark that this type of control is very similar to direct torque control (DTC). In the case of direct sliding mode control, the switching table can be determined analytically whereas in the case of DTC the switching table is determined intuitively.

370

Power Electronic Converters

Indirect sliding mode control is also formulated using sliding mode theory. In this case, a reference voltage vector is applied to the synchronous motor. This voltage vector consists of an equivalent voltage vector, valid on the sliding surface, and an attractive voltage vector valid outside the sliding surface (during the transient regime). In order to apply the reference voltage vector to the synchronous motor, an intermediate PWM process is required. In this case, the switching frequency is constant and is equal to the PWM frequency. The dynamic performance of indirect sliding mode control is not as good as that of direct sliding mode control. Nevertheless, it offers an improved quality of current control in the steady state regime, with a considerable reduction in current oscillations. 12.5. Bibliography [BUH 86] BÜHLER H., Réglage par mode de glissement, Presses Polytechniques Romandes, Lausanne, Switzerland, 1986. [BUH 97] BÜHLER H., Réglage de systèmes d’électronique de puissance, Presses Polytechniques Romandes, Lausanne, Switzerland, 1997. [GAO 93] GAO W., HUNG J.C., “Variable structure control of nonlinear systems: a new approach”, IEEE Trans. Ind. Electron., vol. 40, n°1, p. 45–55, 1993. [UTK 77] UTKIN V.I., “Variable structure systems with sliding modes”, IEEE Trans. on AC, vol. 22, n°2, p. 212–222, 1977. [UTK 78] UTKIN V.I., Sliding Modes and Their Application in Variable Structure Systems, MIR Publishers, Moscow, 1978.

Chapter 13

Hybrid Current Controller with Large Bandwidth and Fixed Switching Frequency

13.1. Introduction Current control in industrial electrical devices is one of the key issues in power converter control, and more generally in energy management within a system. The properties of current sources depend crucially on the current control technique that they use [KAZ 98]. Various techniques are in present-day use for regulating the current to a given reference value. The PWM technique uses regulators whose structure may be linear (PID or RST controllers, etc.) or non-linear (sliding mode, passivity-based, state feedback, etc.). The output of these regulators is then compared to a carrier signal, which is used to determine the commutation angles of the power switches. The frequency of the carrier signal therefore determines the switching frequency, if it is assumed that the output value of the regulators is slowly changing over the period of the carrier signal. This easy-to-use PWM technique therefore ensures fixed frequency operation with a harmonic content that is precisely known but this comes at the expense of reduced dynamic properties.

Chapter written by Serge PIERFEDERICI, Farid MEIBODY-TABAR and Jean-Philippe MARTIN.

372

Power Electronic Converters

One way of improving the dynamic properties of current regulators is to use an alternative class of current regulators, whose output in this case is discrete, in order to directly control the power switches. The most widely used of these regulators is known as a “hysteresis” controller and it is both simple to use and robust. This controller ensures good current regulation without requiring a perfect knowledge of the model or its parameters. Its weakness lies in the fact that its switching frequency is variable, which results in the generation of a wide range of harmonics in the current waveform. Considerable research has been performed into current regulators with discrete outputs operating at fixed frequencies and with dynamic properties that are as close as possible to that of a hysteresis regulator. One such approach involves varying the width of the hysteresis band as a function of the current system state in order to retain a fixed switching frequency. This is the case in [BOD 01], where the authors proposed to modify the width of the hysteresis band by requiring the current error signal to be zero at the start and the middle of each switching period. Similarly in [PAN 02], the authors proposed to supplement a standard hysteresis structure with a third-variable band which can be used to ensure fixed frequency operation, bearing in mind that this technique can only be applied to conversion structures that can output more than two voltage levels at the terminals of the inductive element (as is the case with an H-bridge chopper). In [ROD 97], the authors proposed to use predictive control to regulate the current to its reference value. The control algorithm was based on the fact that for a given power delivery structure, there are only a finite number of possible states of the switches. It is therefore possible, if a model of the system is known, to predict the behavior of the state variables as a function of the states of the switches and then to minimize an optimization criterion acting on the current error. Again, this method requires considerable computational power and parametric monitoring in order to allow for uncertainties in the model.

Hybrid Current Controller

373

Alternative approaches (Chapter 14) exist for the design of fixed switching frequency current controllers. In [HAU 92], [PIE 03], and [LEC 99], the authors developed a feedback loop leading to self-sustained oscillations in the system at a given frequency. This frequency is determined by using the Describing Functions method, and in particular the first harmonic equilibrium equation, the principle of which is discussed in [KHA 96]. It can then be seen that the oscillation pulsation of the system has very little dependency on the converter parameters but depends almost entirely on the structure of the feedback loop. Another class of discrete-output current controller with a nonlinear structure can also be used to ensure fixed switching frequency oscillations. These regulators are known as peak current regulators and their advantage is that they naturally produce fixed frequency operation due to periodic trigger and/or inhibition signals. Two disadvantages limit their use in electrical engineering. The first is that the structure of these regulators leads to a static error. The second is that in a particular region of parameter space, the limit cycle formed by the state trajectory of the system can be multiperiodic or even chaotic. Many articles have considered the emergence of chaotic cycles in converters using this type of regulator when the system parameters are changed [CAF 05], [CHA 96], [CHE 00a], [CHE 03], [IU 03], and [TSE 95]. The use of a compensation ramp, as demonstrated by the work of [RAS 99], can increase the region of parameter space within which correct system behavior occurs. In this chapter, we will mostly consider current controllers whose structure uses thresholds beyond which the conduction state of the power switches is modified. We will begin by presenting the main current controller structures that use this type of strategy. In the second part we will discuss the various tools required to study the high-frequency behavior of these controllers, and in particular how the current oscillation varies in response to changes in the system parameters whether these are control or load parameters.

374

Power Electronic Converters

13.2. Main types of discrete-output current regulators 13.2.1. Introduction A large number of discrete-output current regulators have been described over the last 30 years. In this section we will present four types of regulator in order to give a fairly precise picture of the behavior of this class of current controller. 13.2.2. Hysteresis regulator This regulator is the most widely used type of regulator in applications where fixed switching frequency operation is not required. The principles of its operation are described in Figure 13.1, and the waveforms typically obtained from such a structure are shown in Figure 13.2. When the current error (iref–imes) exceeds the value +Bh, which corresponds to the first hysteresis threshold, the control quantity is set to one and maintains this state until same error signal falls below a second hysteresis threshold that is set at –Bh. S

Q

+Bh iref

imes

-Bh

R

Figure 13.1. Block diagram of a hysteresis regulator

Iref+Bh iref Iref-Bh

Figure 13.2. Typical current waveform obtained with a hysteresis regulator

Hybrid Current Controller

375

This type of regulator can be used to control the current in most types of static converters. It is certainly the one with the best dynamic behavior and the best parametric robustness properties (current controller independent of system parameters). It however suffers from a major disadvantage; that the inverter switching frequency is variable, and depends on the system parameters. This property not only makes it very difficult to filter the high frequency current variations caused by the action of the semiconductor switches, but can also in some applications lead to unacceptably high switching losses that may lead to undesirable heating of the power electronics and even to its destruction. 13.2.3. Fixed-frequency hysteresis regulator The high level of robustness of a hysteresis regulator and its good dynamic performance have led a number of researchers to investigate ways of modifying the internal structure of the hysteresis regulator to make it operate at a fixed switching frequency, while attempting to retain its desirable features (parametric robustness and dynamic performance). Based on the observation that the variable frequency of this regulator results from switching commands produced by the intersection of the error signal with the high and low thresholds of the hysteresis regulator, the intention was to modify these thresholds as a function of the system state. This led to the controller design shown in Figure 13.3. The bandwidth is calculated so as to ensure fixed switching frequency operation in the steady state. We will take the example of a step-down chopper operating in continuous conduction mode, fed by a DC voltage from the output of a three-phase diode rectifier. The configuration we will consider is shown in Figure 13.4. In continuous conduction mode, the current variation ∆IL in the inductive element can easily be expressed as a function of the input voltage Ve, the output voltage Vs, the switching frequency F, and the value of the inductance L. It can be shown that: ∆IL =

(Ve − Vs ) L⋅F

V  ⋅ s   Ve 

[13.1]

376

Power Electronic Converters Measurements , parameters

bandwidth calculation

-Bh

+Bh

+

R

iref

-

u

Q

S

iL

Figure 13.3. Principle of a variable bandwidth hysteresis regulator

u

Three phase supply Veff = 230 V

L

IL

Ve

Vs

Figure 13.4. Buck converter fed by a three-phase rectifier bridge

In order to ensure fixed frequency operation of a hysteresis controller, we require that the hysteresis bandwidth Bh satisfies: Bh =

∆IL 2

=

(Ve − Vs )

V  ⋅ s  . 2 ⋅ L ⋅ F  Ve 

[13.2]

Hybrid Current Controller

377

Figure 13.5. Waveforms obtained using a fixed bandwidth hysteresis regulator

The results shown in Figure 13.5 can be used to compare the waveforms obtained using fixed bandwidth and variable bandwidth regulators (Figure 13.6). The switching frequency is 10 kHz. The bandwidth of the fixed bandwidth hysteresis regulator has been selected according to the following equation: Bh =

(Ve0 − Vs )

V  ⋅ s  , 2 ⋅ L ⋅ F  Ve0 

where: Ve0 =

3 ⋅V ⋅ 6

π

.

[13.3]

It can be seen that when a fixed switching frequency is used, this leads to a modulation in the high frequency envelope resulting from the switching action.

378

Power Electronic Converters

Conversely, the use of a fixed bandwidth hysteresis regulator leads to a fixed-amplitude current variation, but one whose frequency is modulated around the central switching frequency.

Figure 13.6. Waveforms obtained using a variable bandwidth hysteresis controller

13.2.4. Turn-on triggered current regulator 13.2.4.1. Principles Used for many years in DC-DC conversion, a turn-on triggered current regulator can be used to impose a fixed switching frequency, while also providing excellent dynamic properties for the current loop. The method is a form of single-band hysteresis control and the behavior of the current loop is illustrated in Figure 13.7.

Hybrid Current Controller

379

D.T

Upper threshold

Turn-off command Reference current I ref Measured current Imes Turn-on trigger signal Turn-off trigger signal

Compensating ramp

-mc m1

-m 2

Turn-off trigger signal

R

Turn-on trigger signal

S

Q

Power switch control

Figure 13.7. Operating principles of a turn-on triggered current controller

The system is switched on at the start of each switching period, and then when the current reaches its upper threshold, a switching order results in the power switching being turned off until the next trigger command. This threshold is the reference current value, which may be modified by a ramp signal known as the “compensation ramp”. The main advantages of this type of regulator are its fixed-frequency operation, its ease of implementation, and the fact that its maximum current value is precisely controlled. On the contrary, in the average value sense this current controller generates a static current error since only the maximum value of the current is controlled. Another property of this type of regulator is that for duty cycle values greater than 0.5 it generates multiperiodic or even chaotic limit cycles. In order to ensure normal operation of the regulator (in other words to ensure the presence of single-period stable limit cycles), it is then necessary to use a compensation ramp whose slope is calculated as a function of the current system parameters so as to ensure correct operation of the regulator over a given range of variations in the duty cycle. 13.2.4.2. Calculation of the average current value in the steady state For this calculation, we will assume that during normal operation the converter has two different sequences of operation, one corresponding to the

380

Power Electronic Converters

power switch being turned on and the other corresponding to it being turned off. The dynamic resistance of the components, their voltage drop in the conducting state, and the heating losses in the inductance of the converter are all assumed to be zero. By observing the waveforms shown in Figure 13.7, it is possible to calculate the average value of the current in the inductance as a function of the slopes m1, m2, mc and the duty cycle D of the inverter. This leads to two expressions: T 1  I L = I ref − m c ⋅  D ⋅ T −  − m1 ⋅ D ⋅ T 2 2  T 1  I L = I ref − m c  D ⋅ T −  − m ⋅ (1 − D ) ⋅ T 2 2 2 

[13.4]

Whatever the nature of the converter, it is possible to define the rise and fall slopes of the current as follows: m1 =

D'V off L

D V off , L

m2 =

[13.5]

where:

Voff = αVi + β Vo

α , β ∈ {0,1}

D' = 1 − D

Vi and Vo are the input and output voltages of the converter. Converter

α

β

buck

1

0

boost

0

1

buck--boost

1

1

Table 13.1. Values of coefficients α and β for buck, boost, and buck–boost converters

Hybrid Current Controller

381

Equations [13.4] and [13.5] can then be used to obtain a single expression for the mean value of the inductive current in the steady state. This is: T  D '⋅ D ⋅ T ⋅ Voff  . I L = I ref − m c ⋅  D ⋅ T −  − 2 2⋅ L 

[13.6]

Consider now a small disturbance to the system from its operating point. We will write the quantity in the disturbed regime as Xp, and use xˆ to represent its variation from its steady state value X. By taking the difference between the calculated average current value in the normal and disturbed states and then using a first-order Taylor series, we can obtain a first-order model of this converter. We find:  ( D' − D ) ⋅ V off ˆi = iˆ −  m + c L ref  2⋅L 

  T ⋅ dˆ − D ⋅ D' ⋅ T vˆ . off  2⋅L 

[13.7]

We can then express the variations in the duty cycle as a function of variations in the current error and in the voltage Voff. It follows that: dˆ =

D ⋅ D' ⋅ T 2⋅L 1   -iˆL  − . ⋅ vˆ  ˆi  off  T ⋅ Voff  2 ⋅ L ⋅ m 2⋅L   ref   c + 2  D' − 1    D' ⋅ Voff  

[13.8]

Examination of this equation reveals that a necessary condition for stability is that the denominator of equation [13.8] should be positive, which leads to the following condition: D'min ≥

1  2 ⋅ L ⋅ mc  + 2  D' ⋅ V off  

=

0,5 . mc 1+ m1

[13.9]

This last expression helps us to understand the impact of the compensation ramp. When the duty cycle is greater than 0.5, the compensation ramp must be added to the current reference in order to ensure correct operation of the regulator. If this ramp were not present, the proposed model predicts unstable behavior. In fact, when condition [13.9] is not satisfied, irregular structures in the current waveform can be observed,

382

Power Electronic Converters

which correspond to the creation of multiperiodic or even chaotic limit cycles (section 13.3). The proposed model is a simple model that can be used to better understand the role of the compensation ramp in this type of regulator. It is however very approximate, and does not take into account some relevant parameters (series resistance, nonlinear effects, etc.) that have a direct influence on the nature of the current waveforms. Only the formalism introduced in section 13.3 enables a detailed analysis of the nature of the current waveform. 13.2.4.2.1. Variants on this design It is possible to compensate for the static error introduced by this regulator. Various possible methods exist. It is possible to calculate the value of the current error in the steady state generated by this control and to compensate for it by adding an offset to the reference current given by the following equation: T  D '⋅ D ⋅ T ⋅ Voff  . offset = m c ⋅  D ⋅ T −  + 2 2⋅ L 

[13.10]

This easy-to-use technique is however reasonably sensitive to modeling errors and parametric fluctuations, in particular for the inductance whose value may vary significantly as a function of the level of induction. Another technique that is relatively robust in the face of parametric variations, involves modifying the structure of the triggered current regulator to obtain the structure shown in Figure 13.8. The generation of commands for the power switch takes place in a similar way to that discussed earlier. The switch is turned on in response to a signal generated by a clock. The control orders are sent when the variable S reaches its upper threshold, which is defined by the following equation: s(t) =  I L ( t ) − Iref ( t )  + K i ⋅

t

∫0 [ IL (τ ) − Iref (τ )] ⋅ dτ .

[13.11]

As before, the smoothness of the current waveform for these two variants on the triggered current regulator depends on the duty cycle, the given value of the compensation ramp, and the integral gain. It can be studied through the analysis of limit cycles, as will be discussed in section 13.3.

Hybrid Current Controller

383

D.T

Upper threshold

Turn-off command -m c

0

Compensating ramp

S

m1

Current IL

-m2

Turn on trigger signal Turn off trigger signal

IL

Ki (IL -I ref)(t).dt

S

Iref Generation of S

Turn off trigger signal

R

Q

Turn on trigger signal

S

Q

Power switch control

Figure 13.8. Schematic diagram for the variant of the switch-on regulator with integrator

13.2.5. Turn-off triggered controller 13.2.5.1. Principle The structure of this controller is similar to that described in section 13.2.4. Its principle is shown in Figure 13.9. It differs from the previous regulator design in that the periodic turn-off orders are sent to the power switch, which then remains in its non-conducting state until the arrival of a turn-on command. This command is sent when the current reaches a lower threshold that is set by the reference current and a possible additional compensation signal of slope mc. As with the standard triggered controller, the main advantage of this controller type is that it ensures operation at a fixed switching frequency, is simple to implement, and the minimum value of the current is precisely controlled. On the contrary, in mean value terms this current controller also introduces a static current error since only the minimum current value is controlled. Another feature of this type of controller is that, for duty cycles

384

Power Electronic Converters

below 0.5 it can produce multiperiodic or even chaotic limit cycles. In order to ensure current operation of the controller, it is necessary to use a compensation ramp like in the previous case. Turn-off trigger signal Turn-on trigger signal

Turn-off trigger signal

D.T

Current IL

m1

Reference current Iref

mc

Turn-on trigger signal

-m2

R

Q

S

Q

Power switch control Detection of turn-on and turn-off signal

Figure 13.9. Operating principles of a turn-off triggered current controller

13.2.5.2. Calculation of average steady state current value The assumptions for this calculation are identical to those in section 13.2.4.2. As before, the expression for the average current in the inductance can be written in the form: I

L

=I

ref

T  D '⋅ D ⋅ T ⋅ Voff  . + m c ⋅  D' ⋅ T −  + 2 2⋅ L 

[13.12]

If we calculate the difference between the undisturbed and disturbed model, retaining only the first-order terms, we obtain the following equation:  ( D' − D ) ⋅ V off ˆi = iˆ +  -m + c L ref  2⋅L 

  T ⋅ dˆ + D ⋅ D' ⋅ T vˆ . off  2⋅L 

[13.13]

The variation in the duty cycle can then be expressed as a function of variations in the current error and variations in the voltage Voff. We find: dˆ =

D ⋅ D' ⋅ T 2⋅L 1  -iˆL  + ⋅ vˆ  ˆi  off T ⋅ Voff  2 ⋅ L ⋅ m 2⋅L   ref   c + 2  D − 1    D ⋅ Voff  

  . [13.14] 

Hybrid Current Controller

385

The stability of the regulator requires, as before, that the denominator of equation [13.14] should be positive, which leads to the following condition: D min ≥

1 0,5 . = mc  2 ⋅ L ⋅ mc  1 + + 2  m2  D ⋅ V off 

[13.15]

When the duty cycle is less than 0.5, it is necessary to add a compensation ramp to the current reference in order to ensure correct operation of the regulator. If this is not present, the proposed average model predicts unstable behavior, which again results in an irregular structure for the current waveform. As before, analysis of the high-frequency behavior of this regulator relies on analysis of limit cycles, the techniques for which are described in section 13.3. 13.2.5.2.1. Variants on this regulator design It is possible to compensate for the static error introduced by this regulator. Both the methods described earlier can again be used. In the first case, the value of the offset that must be added to the current reference is given by the following equation: T  D '⋅ D ⋅ T ⋅ Voff  . offset = −m c ⋅  D' ⋅ T −  − 2 2⋅ L 

[13.16]

In the case of the method using an integral component, the variable S is defined in exactly the same way as before (equation [13.11]). The typical waveforms obtained using this regulator are shown in Figure 13.10. 13.2.6. Turn-on or turn-off triggered regulator 13.2.6.1. Principle This regulator is a hybrid of the two structures described earlier–turn-on and turn-off triggered regulators. A schematic is shown in Figure 13.11 along with the typical waveforms produced by this regulator. Turn-on and turn-off triggers spaced half a period apart are regularly sent, forcing a change in the conduction state of the power switch.

386

Power Electronic Converters Turn-off trigger signal Turn-on trigger signal

D.T m1

Current I L

-m 2

S 0

mc

L

Detection of turn-on and turn-off signal

Figure 13.10. Schematic of a variant on the turn-off triggered regulator that incorporates an integrator Detection of switch-off order Va

V-a

T/2

Detection of switch-on order

Signal Q’(t) : turn-off order

Signal Q(t) : turn-on order

Q’(t) Turn-off order Q(t) Turn-on order

R

Q

S

Q

Power switch control

Figure 13.11. Schematic of a turn-on or turn-off triggered regulator

Hybrid Current Controller

387

For duty cycles less than 0.5, the turn-off order is sent to the semiconductor at the moment when the current IL reaches its upper threshold equal to a constant Va, which may be modified by a compensation ramp of slope –mc. The current then falls until the semiconductor receives the conduction order given by the external signal Q(t) whose frequency determines the switching frequency. The signal Q’(t), phase-shifted by

T 2

relative to Q(t) (where T is the switching period), does not do anything in the example shown in this figure since the converter is already in the discharge phase for the inductance. For duty cycles greater than 0.5, the turn-on order is sent to the semiconductor when the current IL reaches is lower threshold –Va, which may again be modified by a compensation signal of slope mc. The current increases until the point when the semiconductor receives the turn-off command sent by the external signal Q’(t). Again, the Q(t) signal does not have any effect on the system state since the converter is already in the charging phase for the inductance. COMMENTS.– When the operating conditions of the system change (change in input or output voltage level, load conditions, etc.) in a way that would cause the duty cycle to change from less than 0.5 to greater than 0.5, the system will naturally switch from the mode D < 0.5 (intersection with the upper threshold) to the mode D > 0.5 (intersection with the lower threshold), with the transition happening when the duty cycle is equal to 0.5 (Figure 13.11). In general the levels V+a and V−a are the sum of two terms; the first is the current reference Iref and the second is an offset signal V0 (Va = Iref + V0, V−a = Iref − V0) whose value is selected such that the current variation will always be less than 2.V0. If this is not the case, the regulator will behave like a hysteresis regulator. Furthermore, the two compensation ramps are used to increase the stability range of the system (particularly for duty cycles close π to 0.5) and have a mutual phase shift of . As with the two other types of 2

regulator, this regulator operates at a fixed switching frequency but in a mean value sense it introduces a static error. It does however have the

388

Power Electronic Converters

advantage that it does not require a compensation ramp except when the system is operating with duty cycles that are close to 0.5. 13.2.6.1.1. Variants on the turn on or turn off triggered regulator It is again possible to compensate for the static error introduced by this regulator. Both the methods described earlier can again be used here. For the first one, where an offset is added to the upper and lower thresholds, the levels V+a and V−a are written in the following form: T  D '⋅ D ⋅ T ⋅ Voff  . Va = I ref + V0 + m c ⋅  D ⋅ T −  + 2 2⋅ L 

[13.17]

T  D '⋅ D ⋅ T ⋅ Voff  V−a = I ref − V0 − m c ⋅  D' ⋅ T −  − 2 2⋅ L 

[13.18]

For the method using an integral component, the definition of the variable S given in equation [13.11] still applies [LAC 08]. The typical waveforms obtained for the modes D < 0.5 and D > 0.5 are identical to those shown in Figures 13.9 and 13.11, except that the upper and lower thresholds are taken as V+a = +V0 and V−a = –V0 , respectively. The transition from one mode to the other still takes place at the fixed duty cycle of 0.5. The value of V0 is calculated such that the variation in the variable S is always less than 2.V0. 13.2.7. Principles of a hybrid modulated hysteresis regulator 13.2.7.1. Principle Figure 13.12 shows the first design for a regulator as described in [ALI 07a]. The principle of this regulator is to generate a modulated current reference Iref,m, obtained by adding a carrier signal of period T and amplitude Atr to the reference signal. The period T is the desired switching frequency for the converter. The modulated signal Iref,m is then compared to the current measurement and the resultant error signal is then connected to the input of a hysteresis regulator with a bandwidth of 2.Bh, which determines the switching commands.

Hybrid Current Controller

Iref -Bh Bh Atr -Atr

389

Power switch control

IL T

Figure 13.12. Block diagram for the initial version of this regulator as described in [ALI 07a]

An analysis of the steady state waveforms (ignoring the series resistance of the inductive elements) shown in Figure 13.13 leads us to the following equations: I max = m ⋅ t1 + Bh − Atr + I ref

[13.19]

I min = m ⋅ t2 − Bh − Atr + I ref ,

[13.20]

where: m=

4 ⋅ Atr , t1 + t2 = D '⋅ T . T T DT

t2

t1

Iref+Atr +Bh m Iref

m1

Imax m2 IL

I min I ref-Atr-Bh Figure 13.13. Typical waveform obtained in the steady state using a modulated hysteresis regulator

390

Power Electronic Converters

Equations [13.19] and [13.20] then enable us to calculate the average value IL of the current in the inductive element. We find: I L = I ref − Atr +

m ⋅ D '⋅ T = I ref + Atr ⋅ ( D '− D ) . 2

[13.21]

The normalized static error ε ∞ = ( I ref − I L ) / I ref , output by this regulator as a function of its parameters is: ε∞ =

Atr ⋅ ( D '− D ) I ref

[13.22]

The static current error does not therefore depend on the chosen hysteresis bandwidth Bh. Furthermore, it is independent of the chosen converter structure. In order to fix the steady state switching frequency, there must only be two intersections between the measured current and the modulated reference current during each period; the first with the upper limit of the hysteresis controller and the second with its lower limit. For a given carrier frequency, the controller parameters for this regulator type that are used to fix the switching frequency of the power switches are the amplitude Atr of the triangular signal and the hysteresis bandwidth Bh. If these parameters are not selected correctly, the switching frequency will be greater than or less than the required frequency, which is the frequency of the carrier signal. Examples of a poor choice of control parameters (Atr and Bh) are shown in Figures 13.14a and 13.14b. Only a study of the high-frequency current behavior enables correct values for the parameters Atr and Bh to be chosen by confirming the correct fixed-frequency operation (section 13.3). 13.2.7.1.1. Variant on the modulated hysteresis regulator In order to remove the static error, it is possible to modify the control structure of the regulator described in [ALI 07a] by introducing an integral term to the current error [ALI 07b].

Hybrid Current Controller

391

Figure 13.14. Example of waveforms obtained with poor parameter choice: a) switching frequency higher than required; b) switching frequency lower than required

Considering again the variable S defined in [13.11]. The signal S(t) is added to the aforementioned triangular carrier. The resultant modulated signal is connected to the input of the hysteresis regulator described earlier. A full schematic diagram of this controller is shown in Figure 13.15. The hysteresis bandwidth can be chosen to be relatively broad, leading to a more reliable rejection of parametric variations.

Ki (Iref -IL )(t).dt

Iref

IL

S

St -Bh Bh

Power switch control

Atr -Atr T

Figure 13.15. Principle of a modulated hysteresis regulator with an integral term

392

Power Electronic Converters

13.3. Tools for limit cycle analysis 13.3.1. Introduction to dynamic systems; concept of bifurcation 13.3.1.1. Definitions In this section, we will introduce some mathematical concepts pertaining to dynamic systems that will be required in order to understand the rest of this discussion. A dynamic system can be described either in terms of a differential equation: dx ≡ x& = f ( x,t,ν ) , dt

x ∈ U ⊆ ℜn ,

ν ∈V ∈ ℜ p

[13.23]

or in terms of a recurrence: x k +1 = f ( x k ,ν ) ,

x k ∈ U ⊆ ℜn ,

ν ∈V ∈ ℜ p ,

k = 1, 2,... ,

[13.24]

where ℜn and ℜ p are the phase space and parameter space, which will be defined below. Equation [13.23] is not autonomous since it depends on the time t. It would be an autonomous equation if the function f did not depend explicitly on the time t. In equation [13.24] the variable t does not vary in a continuous manner but consists of a set of integers k (discretized time). We will assume that in the case of equation [13.23], the existence and uniqueness conditions for the solution are satisfied for an initial condition x ( t0 ) = x0 . We will write such a solution as x = x ( x0 , t ) . This varies continuously with time. In a coordinate space x1 , x2, ..., xn known as phase space, this equation represents a curve passing through the initial point M 0 (with coordinates x0 ) and is known as the phase trajectory or orbit. Equation [13.24] represents a set of points M n with coordinates xk , k = 0,1, 2,... in a phase space x1 , x2, ..., xn . This set is known as the

Hybrid Current Controller

393

discrete-phase trajectory, with conditions on the existence and uniqueness of the solution associated with an initial point x0 defined for n = 0 . We will consider the autonomous system represented by: dx = x& = f ( x ) , dt

x ∈ U ⊆ ℜn .

[13.25]

13.3.1.1.1. Definition 1 Let x ( x0 , t ) be a solution to [13.25] for an initial condition x ( 0 ) = x0 . We will define the flow of equation [13.25] as the mapping Φt defined by: Φ t ( x0 ) = x( x0 , t ) . Φ t ( x0 ) has the following properties: Φ 0 ( x0 ) = x0 ; Φ t + s ( x0 ) = Φ t (Φ s ( x0 )) .

A bounding point a, is the bounding point ω of a trajectory x = x ( x0 , t ) if there exists a sequence tn → +∞ such that lim Φ tn = a . Similarly, a n→+∞

bounding point b is the bounding point α of a trajectory x = x ( x0 , t ) if there exists a sequence tn → −∞ such that lim Φ tn = b . n→+∞

The set of bounding points α (or ω) is written as α(x) (or ω(x)). The set α ( x ) ∪ ω ( x ) is known as the limit set of x = x ( x0 , t ) . A limit cycle α (or limit cycle ω) is a closed orbit Γ such that Γ ⊂ α ( x ) (or Γ ⊂ ω ( x ) ). 13.3.1.1.2. Definition 2 The fixed point of equation [13.25] is the point x* in phase space that is obtained by setting the right-hand side of equation [13.25] to zero. It is stable if: ∀ε > 0, ∃δ > 0 such that x(0) − x* < δ ⇒ x(t ) − x* < ε .

[13.26]

394

Power Electronic Converters

If there also exists δ 0 with 0 < δ 0 < δ such that: x(0) − x* < δ 0 ⇒

* lim x(t ) = x t → +∞

[13.27]

then x* is asymptotically stable. The stability of the equilibrium point is generally determined by considering the eigenvalues of the Jacobian matrix of f evaluated at the equilibrium point. Then: – if all the eigenvalues of the Jacobian matrix Df(0) have a negative real part, then the fixed point is asymptotically stable; – if one or more eigenvalues is pure imaginary and the other eigenvalues have a negative real part, then the fixed point is a center or an elliptical point (stable, but not asymptotically so); – if one of the eigenvalues has a positive real part then the fixed point is unstable; – if Df(0) has no eigenvalue that is zero or pure imaginary, the fixed point is a hyperbolic point; if this is not the case then it is non-hyperbolic; – if there exist i and j such that ℜ ( λi ) < 0 and ℜ ( λ j ) > 0 , then the fixed point is a saddle point; – if all the eigenvalues are real and have the same sign, the fixed point is a node. A stable node is a sink and an unstable node is a source. Till now we have discussed properties involving continuous dynamic systems. Moving on to discrete systems, these are stable if the Jacobian matrix Df ( x* ) does not have eigenvalues with modulus greater than 1, which

makes x* a hyperbolic fixed point. If the moduli of the eigenvalues of Df ( x* ) are equal to 1, then x* is an elliptical fixed point. The eigenvalues of the Jacobian matrix are known as its Floquet multipliers.

Hybrid Current Controller

395

13.3.2. Concept of bifurcation of a dynamic system The term “bifurcation” is generally associated with the concept of topological changes in the trajectory of a dynamic system in response to variations in one or more parameters that it depends on. We will only consider what are known as local bifurcations, in other words ones pertaining to the behavior of the trajectory of a fixed point, a case in which Taylor series expansions can be used. Consider the following 2D dynamic system: x& = f ( x, y, v) y& = g ( x, y, v)

[13.28]

Let ( x* , y* ) = ( x* (t0 ), y* (t0 )) be the fixed point of the system for v = v0 . This satisfies the existence condition of the fixed point stated earlier: 0 = f ( x* , y* , v0 ) and 0 = g ( x* , y* , v0 ) . If the fixed point is stable (or unstable) for ν > ν 0 and unstable (or stable) for ν < ν 0 then ν 0 is the bifurcation value of the system. To not overcomplicate this discussion, we will not include the mathematical proofs that define bifurcation. Four types of bifurcation can be found in the literature [CHE 00b] for continuous systems: saddle node bifurcations, transcritical bifurcations, pitchfork bifurcations, and Hopf bifurcations. These four types of bifurcations represent different ways in which the equilibrium points can evolve as a function of the bifurcation parameter. These concepts introduced here for continuous dynamic systems can be transferred to discrete dynamic systems. These have three types of single parameter bifurcations: saddle-node bifurcations, period-doubling bifurcations, and Neimark-Sacker bifurcations. There are also certain concepts unique to discrete systems, such as the p-cycle. There are therefore three ways in which a fixed point x* in a discrete dynamic system can lose or gain its stability (Figure 13.16): – when a real eigenvalue of Df ( x* ) enters or leaves the unit circle at the value –1, a period-doubling (or flip) bifurcation occurs;

396

Power Electronic Converters

– when a real eigenvalue of Df ( x* ) enters or leaves the unit cycle at the value +1, a saddle-node bifurcation occurs; – when two complex conjugate eigenvalues of Df ( x* ) enter or leave the unit circle simultaneously at λ1,2 = e±i⋅θ , a Neimark-Sacker bifurcation occurs.

Figure 13.16. Various ways for the fixed point to leave the unit circle

13.3.3. Poincaré cross-section and bifurcation diagram Consider the autonomous system [13.23] and assume that there exists a solution x( x0 , t0 , t ) = φt ( x0 ) of period T, in other words that φt +T ( x0 ) = φt ( x0 ) .

Γ P(x) x x0 Σ

Figure 13.17. Poincaré cross-section

Hybrid Current Controller

397

The Poincaré cross-section is a hypersurface Σ of dimension n-1 that is transverse to the vector field f at x0 . Let x be a point in the neighborhood V ⊆ Σ of x0 . The Poincaré mapping P : V → Σ is then defined by: x1 = P ( x) = φτ ( x) ,

[13.29]

where τ = τ ( x ) is the time taken by the trajectory to reach this point starting from its initial point x on the surface (Figure 13.17). COMMENTS.– The time τ taken by the state trajectory to move from a point intersecting the Poincaré surface to the next point cannot be a constant. In many works on chaos in static converters, the surface used corresponds to a sampling of the value of the state vector with a period τ . The behavior of the system is then examined at this given time interval. The set of observed points then forms a subset of Rn of dimension n. This is therefore not in the strict sense a Poincaré surface, which is a surface of dimension n-1 and is time-independent. This surface, known as the time − τ surface or discrete time map [ALL 96], is often referred to as the Poincaré surface by certain authors [CAF 05], [CHE 00b], and [MAZ 03]. Once the Poincaré cross-section has been defined, the evolution of the nature of the cycle as a function of the bifurcation parameter can be visualized by plotting a bifurcation diagram. The abscissa of this diagram is the bifurcation parameter and the ordinate is a quantity representing the state of the system at the point where it intersects the Poincaré surface in the steady state regime (assuming this exists) for a given passage number (which is generally large). This may be a coordinate of the state vector, its norm, or any other property that depends on the state vector. 13.3.4. Application to electrical engineering 13.3.4.1. DC mode operation of current regulators We will mainly focus on the cycle induced by commutation of power switches in a static converter. We will assume that the switching order during one switching period is known. Let p be the number of operating

398

Power Electronic Converters

sequences and p-1 the number of conditions that can cause a transition from one sequence to another. Let T be the switching period of the power switches and αi .T the condition time for the ith sequence. Let us assume now that the state variables of the system satisfy: n ⋅ T ≤ t < n.T + α1 ⋅ T

 i −1   n ⋅T +  α j  ⋅T ≤ t < n ⋅T +      j =1  



x& = A1 ⋅ x + B1 (t ) i



j =1



∑α j  ⋅ T

[13.30]

x& = Ai ⋅ x + Bi (t ) for 2 ≤ i ≤ p, [13.31]

where:  p  S p =  α j  = 1 , x ∈ ℜm , Bi (t ) ∈ ℜm , Ai (t ) ∈ M m×m for 1 ≤ i ≤ p, t ≥ 0.    j =1 



If we assume that equations [13.30] and [13.31] are integrable over their respective intervals, it is possible to obtain an analytic expression linking the state vector at time n.T to that at time (n+1).T. We can define a mapping: f : ℜm × ℜ p × ℜ → ℜm × ℜ ,

such that:

[ x((n + 1) ⋅ T ), (n + 1) ⋅ T ]t

= f ( x (n ⋅ T ), α , nT ) ,

[13.32]

where: α = [α1,..,αp]t

if we know that the transition from the ith to the (i+1)th sequence obeys the following commutation law: Si ( x(n ⋅ T ), α1 ,.., αi , ⋅T ) = 0 for 1 ≤ i ≤ p.

[13.33]

Hybrid Current Controller

399

Equations [13.32] and [13.33] can be used to define a mapping P : ℜm × ℜ → ℜm × ℜ , such that:

[ x((n + 1) ⋅ T ), (n + 1) ⋅ T ]t

= P ( x (n ⋅ T ), nT ) ,

[13.34]

if we know that the terms in (αi)i=1 to p are defined by the p equations [13.33]. In most cases, the mapping P cannot be expressed explicitly. However, it is possible to find an explicit formulation for its Jacobian matrix, which can be used to study the stability of limit cycles formed by the state trajectory. This result can be deduced very easily from equations [13.32], [13.33], and [13.34]. Let (x, α, t) be a point in state space satisfying:

[ x, t + T ]t

= P ( x, t ) = f ( x, α , t ) ,

where n tends to infinity. A first-order Taylor expansion of the mappings f and P then gives us:  ∂f   ∂f   ∂f  ⋅ dx +   ⋅ dt +  ⋅ dα f ( x + dx, α + dα , t + dt ) ≈ f ( x, α , t ) +     ∂x  x,α ,t  ∂t  x,α ,t  ∂α  x,α ,t

[13.35]  ∂P   ∂P  P( x + dx, t + dt ) ≈ P ( x, t ) +   ⋅ dx +  ⋅ dt .  x ∂   x,α ,t  ∂t  x,α ,t

But, the control of the system ensures that: ∀i ∈ {1.. p} Si ( x + dx, α1 + dα1 ,.., α i + dα i , ⋅t + dt ) = 0  ∂S  ⇒0= i  ⋅ dx +  ∂x  x,α ,t

   ∂Si   ∂α j j =1   i



   ∂S  ⋅ dα j  +  i  ⋅ dt     ∂t  x,α ,t  x,α ,t 

[13.36]

400

Power Electronic Converters

If this equation is rewritten for every index I from 1 to p, we can obtain the following matrix equation: −1   ∂S    ∂S   ∂S  ⋅  ⋅ dx +   ⋅ dt  , dα = −      ∂α  x,α ,t   ∂x  x,α ,t  ∂t  x,α ,t 

[13.37]

where: S ( x, α , t ) =  S1 ( x, α , t ) .. S p ( x, α , t )   ∂S 

t

−1

 ∂S 

N.B.: The matrix   is non-singular since   is lower  ∂α  x,α ,t  ∂α  x,α ,t triangular with a non-zero determinant (the system is assumed to have p operating sequences). By using [13.35] and [13.37] and comparing them to the expression obtained in [13.36], it follows that the Jacobian for the mapping P is: −1  ∂f   ∂f   ∂S   ∂S  − ⋅ ⋅  J P x,t =      ∂x  x,α ,t  ∂α  x,α ,t  ∂α  x,α ,t  ∂x  x,α ,t −1

  ∂f   ∂f   ∂S   ∂S  − ⋅ ⋅        ∂t  x,α ,t  ∂α  x,α ,t  ∂α  x,α ,t  ∂t  x,α ,t 

.

[13.38]

It can be seen that although the mapping P does not have an implicit formulation, it is nevertheless possible to obtain an analytic expression for its Jacobian matrix. This can then be used to determine the value of the vectors x and α via Raphson–Newton type algorithms [TSE 95] or variablestep gradient algorithms when these eigenvalues are very close to zero. Once the point corresponding to the steady state situation has been identified, the eigenvalues of the Jacobian matrix can be calculated in order to investigate the stability of the limit cycle described by the trajectory. COMMENT ON AC MODE OPERATION.– In alternating mode, knowledge of the mapping P and its Jacobian matrix is not sufficient to study the stability

Hybrid Current Controller

401

of the cycle described by the state trajectory. Here, the mapping P depends explicitly on time. In order to analyze the stability of this cycle, it is then necessary to define a second-order Poincaré surface [MAZ 03] that will enable the cycle stability problem to be transformed into a fixed-point stability problem. For this, if we take ωr and ωs to be the pulsation of the reference signal and the switching respectively, we can identify three cases: – either the ratio ωs/ωr is an integer k, in which case the cycle’s stability can be determined by calculating the eigenvalues of the mappings Pok(x,t) with its Jacobian matrix being determined using equation [13.38]; – or ωs and ωr at least have a common divisor. In this case, the stability of the cycle can be determined by calculating the mapping Pok(x,t) as before along with its associated Jacobian matrix, where k = LCD(ωs, ωr); – if there is no common divisor, the stability is studied by interpolation [KAA 86]. 13.3.5. Analysis of limit cycles in nonlinear current regulators In order to study the high-frequency behavior of the current – and more generally of the state variables of the system – resulting from semiconductor switching, we will identify two different situations: DC-DC converters, where the current reference will assumed to be fixed, and AC-AC or DC-AC converters, where the reference is oscillating. 13.3.5.1. Application to DC mode In order to study the nature of cycles in the steady state regime, as described in sections 13.3.3 and 13.3.4, we will consider the behavior of the state vector in the steady state, with a sampling period equal to the switching period (discrete time map). We will assume that the mapping P and the control law (determining the order of the switching angles) are known (section 11.3). In DC mode, aside from the time-dependent component of the mapping P, the other components are not explicitly time-dependent. When we plot the eigenvalues (known as Floquet multipliers) of the mapping P, we will omit the unit eigenvalue due to its time-dependent component.

402

Power Electronic Converters

13.3.5.1.1. Example of limit cycle analysis Turn-on or turn-off triggered current regulators For hybrid turn-on or turn-off regulators with an integral component (Figures 13.9 and 13.11), the bifurcation parameters are the slope of the compensation signal and the integral gain Ki. Generally speaking, as we saw earlier, an increase in the slope of the compensation ramp can push back the appearance of multiperiodic or chaotic limit cycles. For an illustration, consider Figures 13.19 and 13.20, which show the cases of turn-on and turn-off triggered regulators whose respective cyclic ratios are less than or greater than 0.5. The converter in this case is a step-up converter (Figure 13.18). The input and output voltages are assumed to be ideal voltage sources. It can be seen that when the bifurcation parameter changes, the nature of the cycle changes from a T-periodic cycle to a 2T-periodic cycle. αT Switch-on order

-mc

S

n.T

αT

r L, L Ve

(n+1).T

IL

Switch-off order S

u

mc

VS n.T

(n+1).T

Figure 13.18. Step-up converter and typical waveform obtained for turn-on and turn-off current controllers (Vs>Ve)

For a closer analysis of the evolution of the cycle as a function of the compensation ramp slope bifurcation parameter, we can plot the bifurcation diagram for parameter values corresponding to the two examples earlier.

Hybrid Current Controller

403

To do this, we must know the expression for the Poincaré mapping P associated with these diagrams (section 12.4). 30.00

20.00

10.00

a)

0.00

0.00

0.10

0.20

0.30

0.00

0.10

0.20

0.30

25.00

20.00

15.00

10.00

b) 5.00

Figure 13.19. Experimental results (D = 0.4, Ki = 20,000, fd = 20 kHz, L = 0.1 mH, Vs=100 V, mc0 = Vs/L): (a) 2T-periodic cycle, mc /mc0 = 0.025 and; (b) single-period cycle mc /mc0 = 0.044 , current in A and time in ms

The system studied along with its controller is a second-order system 2 (m = 2). If we write: y (t ) =

t

∫ ( I L − Iref ) (τ ) dτ 0

404

Power Electronic Converters

then it satisfies the following differential equation over the course of each operating sequence: nT ≤ t ≤ n ⋅ T + α ⋅ T  r d  I L  − L = L dt  y    1

 V  0  I L   e   0  ⋅  + L ⋅+  ⇔ x& = A1 ⋅ x + B1   y     − I ref  0   0 

[13.39]

nT + α .T < t ≤ ( n + 1) ⋅ T  r d  I L  − L = L dt  y    1

 V − Vs   0  0  I L   e ⋅  + ⇔ x& = A2 ⋅ x + B2 [13.40] L ⋅+   − I ref   y  0   0 

These two equations can be integrated without difficulty and it is easy to obtain a formulation in which the state vector at time (n+1).T is expressed as a function of its value at time n.T. We then obtain: x((n + α ) ⋅ T ) = x(n ⋅ T ) ⋅ e A1⋅α ⋅T + A1−1 ⋅ (e A1 ⋅α ⋅T − I ) ⋅ B1 x((n + 1) ⋅ T ) = x((n + α ) ⋅ T ) ⋅ e A2 ⋅(1−α )⋅T + A2 −1 ⋅ (e A2 ⋅(1−α )⋅T − I ) ⋅ B2

The mapping f defined in [13.32] can be written for this mapping as: f ( x(n ⋅ T ), α , n ⋅ T ) =

(

)

 X (n ⋅ T ) ⋅ e A1 ⋅α ⋅T + A −1 ⋅ (e A1 ⋅α ⋅T − I ) ⋅ B ⋅ e A2 ⋅(1−α )⋅T + A −1 ⋅ (e A2 ⋅(1−α )⋅T − I ) ⋅ B  1 1 2 2    n ⋅T + T  

[13.41] The Poincaré mapping used to plot the bifurcation diagrams therefore has an identical form to that defined in [13.34]. The control law defined in [13.33] that gives the conduction duration α for the inverter can be written in the following form: – for the turn-on triggered controller: S ( x(n ⋅ T),α ,n ⋅ T ) = i ( t ) − i ref ( t ) 

+ K i ( y (nT +

nT +α T

∫nT

( i ( τ ) − i ref ( τ ) )  dτ − ca (t )

,

[13.42]

Hybrid Current Controller

405

30.00

25.00

20.00

15.00

10.00

a) 5.00

0.00

0.10

0.20

0.30

24.00

20.00

16.00

12.00

8.00

b) 4.00

0.00

0.10

0.20

0.30

Figure 13.20. Experimental results (D = 0.6, Ki = 20,000, fd = 20 kHz, L = 0.1 mH, Vs=100 V, mc0 = Vs/L): (a) 2T-period cycle, mc /mc0 = 0.025 and; (b) single-period cycle mc /mc0 = 0.044 , current in A and time in ms

where: ca ( t ) = Iref + mc T 2 − t ( mod ( T ) ) 

n ⋅ T ≤ t ≤ ( n + 1) ⋅ T

– for the turn-off triggered controller: S ( x(n ⋅ T),α ,n ⋅ T ) = i ( t ) − i ref ( t )  , ( n +1)⋅T −α ⋅T ( i ( τ ) − i ref ( τ ) )  dτ − ca (t ) + Ki ( y ( n ⋅ T ) + n⋅T



[13.43]

406

Power Electronic Converters

where: ca ( t ) = Iref + mc  -T 2 + t ( mod ( T ) ) 

n ⋅ T ≤ t ≤ ( n + 1) ⋅ T .

COMMENT.– The time origin for the turn-on triggered (or turn-off triggered) controller is the rising edge of a turn-on (or turn-off) trigger signal. As mentioned earlier, the Poincaré mapping P does not have an explicit representation. However, equations [13.38], [13.41], [13.42], and [13.43] can be used to obtain an explicit formulation for its Jacobian. It can be seen by considering the bifurcation diagrams shown in Figure 13.21 that with these chosen numerical values, the nature of the cycle can change completely as a function of the value of the bifurcation parameter. T-periodic cycles (operation at the desired switching frequency) and multiperiodic or even chaotic cycles (compensation slope value close to zero) can both be encountered. To study the stability of the cycles, it is possible to complement the bifurcation diagram plot by studying the behavior of the eigenvalues of the mapping P, which we will refer to as the Poincaré mapping even though this term is not strictly correct. Figure 13.22 shows the plot of the Floquet multipliers corresponding to the two cases considered earlier. Note that all the Floquet multipliers lie within the unit circle and that their curve approaches the point (−1,0) (flip bifurcation with period doubling) when the ratio mc / mc0 approaches 0.033 for a cyclic ratio of 0.4 and 0.028 for a cyclic ratio of 0.6. Modulated hysteresis current regulators This type of regulator has three control parameters: the value of the amplitude of the carrier signal Atr, the bandwidth Bh, and the integral gain. The bifurcation parameters to be considered will be the parameters Atr and Bh, and the gain Ki will be used to adjust the dynamic properties of the regulator.

12

14

11

13 amplitude (A)

amplitude (A)

Hybrid Current Controller

10 9 8

12 11 10 9

7 6

407

8 0

0.01

0.02 0.03 mc,a/mc0

0.04

0.05

0

0.01

0.02 0.03 mc/mc0

0.04

0.05

Figure 13.21. Bifurcation diagram (L = 0.1 mH, R = 0.0018 Ω, Ki 20,000, fd = 20 kHz); (a) D =0.4, Ve = 60 V, Vs = 100 V; (b) D = 0.6, Ve = 40 V, Vs = 100 V, mc0=Vs/L

In its normal operating mode, this regulator has three possible operating sequences. The time origin has been selected to coincide with a peak in the triangular signal (Figure 13.23). b)

. . . . . . . . . .

Imaginary part of poles

Imaginary part of poles

a)

.

.

. . Real part of poles

.

.

. . . . . . . . .

.

.

. . Real part of poles

.

Figure 13.22. Plot of Floquet multipliers for mc / mc 0 varying from 0.035 to 0.2 (L = 0.1 mH, R = 0.0018 Ω, Ki 20,000, fd = 20 kHz); (a) D =0.4, V1 = 60 V, V2 = 100 V, (b) D = 0.6, V1 = 40 V, V2 = 100 V

.

408

Power Electronic Converters

Figure 13.23. Four-quadrant chopper and typically observed waveform using a modulated hysteresis regulator

The differential equations satisfied by the state variables of the system (in other words the current in the inductance and the integrated variable) are listed here for:  −r t ∈ [ nT , nT + α1nT [ → x& =  L  1

 −ve  0 ⋅x+ L   0  − I ref

 −r & t ∈ [ nT + α1nT , nT + α1nT + DnT [ → x =  L  1

  = A ⋅x+B 1 1  

 ve  0 ⋅x+ L   0  − I ref

  = A ⋅x+B 2 2  

[13.44]

[13.45]

and:  −r t ∈ [ nT + α1nT + DnT , (n + 1)T [ → x& =  L  1

where: I  x =  L , y = y

t

∫ ( I L − Iref ) (τ ) dτ . 0

 −ve  0 ⋅x+ L   0  − I ref

  = A ⋅x+B , 3 3  

[13.46]

Hybrid Current Controller

409

As with the previous example, these equations can be integrated over each interval without any difficulty and it is easy to obtain a formulation in which the state vector at time (n+1)T can be expressed as a function of its state at time n.T. We find: x((n + α1n ) ⋅ T ) = x(n ⋅ T ) ⋅ e A1 ⋅α1n ⋅T + A1−1 ⋅ (e A1 ⋅α1n ⋅T − I ) ⋅ B1 x((n + α1n + Dn ) ⋅ T ) = x((n + α1n ) ⋅ T ) ⋅ e A2 ⋅Dn ⋅T + A2 −1 ⋅ (e A2 ⋅Dn ⋅T − I ) ⋅ B2 A ⋅ 1−α − D ⋅T A ⋅ 1−α − D ⋅T x((n + 1) ⋅ T ) = x((n + α1n + Dn ) ⋅ T ) ⋅ e 3 ( 1n n ) + A3−1 ⋅ (e 3 ( 1n n ) − I ) ⋅ B3

[13.47] The value of the state vector at time (n+1)T will therefore be a function of the value of the state vector at time nT and the durations α1nT and DnT. It can be written in a similar form to that given in [13.32] as: x((n + 1) ⋅ T ) = f ( x(n ⋅ T ), α1nT , DnT ) .

[13.48]

As in section 13.3.5, the values of α1n and Dn are solutions to two implicit equations S1 and S2 defined as follows: S1 ( x(nT ), α1n , nT ) = S (nT + α1nT ) − Sb (nT + α1nT ) = 0

[13.49]

S2 ( xn , α1n , Dn , nT ) = S (nT + α1nT + DnT ) − Sa (nT + α1nT + DnT ) = 0 ,

[13.50] where Sa(t) and Sb(t) correspond to the upper and lower hysteresis thresholds (Figure 13.23) with which the variable S is compared. They can be expressed in terms of Fourier series:   S a (t ) =    S (t ) =  b 

−8 Atr

π

2

−8 Atr

π2



1



1

n =1,3,5,... n n =1,3,5,... n

2

2

cos(nω t ) + Bh .

[13.51]

cos(nω t ) − Bh

The mapping P given in [13.34] can therefore be defined with the help of equations [13.48], [13.49], and [13.50].

410

Power Electronic Converters

Figure 13.24 shows experimental results for two types of system behavior. Figure 13.24a shows the behavior of the current for a T-periodic orbit (Atr=8, Bh=6, F=10 kHz), while Figure 13.24b shows the behavior of the current for multiperiodic orbits (Atr=5, Bh=6, F=10 kHz). The converter used is a H-bridge converter supplying a load R, L. To analyze these results, we can first plot a 3D bifurcation diagram (fixed value of integral gain, Ki=500) as shown in Figure 13.25. We can then deduce that for values of Atr between 0 and 10; the planar surface corresponds to the creation of a stable monoperiodic cycle. 12

Measured current (A)

12

10

10

Measured current (A )

8

8 Reference current (A )

6

4

2

2

-1

-0.5

a)

0 time (ms)

Reference current (A)

6

4

0

Ki=500 , Atr = 5, Bh = 6

14

ki=500 Atr=8 Bh=6

0.5

1

0

-5

b)

0

time (ms)

5

x 10-3

Figure 13.24. Experimental results for two choices of control parameter, where ve= 50 V, L=2.3 mH, R=0.01Ω: a) Atr=8, Bh=6, Ki=500, F=10 kHz; b) Atr=5, Bh=6, Ki=500, F=10 kHz

The bifurcation diagram (Figure 13.25) and the experimental results shown in Figure 13.24 can be compared in order to check the consistency of the results. To calculate the Floquet multipliers, only the first three harmonics of equation [13.51] have been considered in calculating the Jacobian. For this particular choice of parameters, the Floquet multipliers are all real. Figure 13.26 shows the evolution of the real part of the Floquet multipliers of the mapping P for various values of Atr and Bh=6. These all lie inside the unit circle as long as the value of the coefficient Atr is chosen to be greater than 5.4.

Hybrid Current Controller

411

Current (A) at (nT+t0)

ki=500 Ki=500Iref=10A , Iref=10A

20 10

0 10 8

0 6

Bh

Bh

2 4

4

6

2

8 0

Atr Atr

10

Figure 13.25. Three-dimensional bifurcation diagram obtained for F=10 kHz, ve=50 V, L=2.3 mH, r=0.01Ω

Figure 13.26. Evolution of the real part of the Floquet multipliers with F=10 kHz, ve=50 V, L=2.3 mH, r=0.01Ω, Ki=500, Bh=6 and six different values of Atr

The limit cycles generated by the state trajectory are therefore T-periodic stable cycles. 13.3.5.1.2. Analysis of parametric robustness Robustness is a concept that is always based on the same principle: insensitivity of the command strategy to external perturbations, parametric variations, and model uncertainties. Here, we will mainly focus on the

412

Power Electronic Converters

evolution of the nature of the cycle described by the state trajectory in response to changes in load parameters (value of inductance or resistance). The main tool that we will use to check the stability of DC-mode cycle will be calculation of the Floquet multipliers for a particular set of regulator and system parameters. Study of regulator robustness for turn-on or turn-off triggering The converter we will consider is a boost converter operating in continuous conduction mode as introduced in section 13.3.5.1.1 (Figure 13.18). Through an example, we will consider a variation in inductance (e.g. due to a short-circuit between coil turns). imaginary part of the poles

20

Average current value (A)

18 16 14 12 10 8 6 4 2

Real part of the poles

0

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Ratio L /L0

Figure 13.27. Evolution of (a) Floquet multipliers; and (b) mean current value as the inductance is varied from 50 % to 150 %, ( mc /mc0 = 0,1 , Ki = 20,000, L0 = 0.1 mH, r = 0.018 Ω, fs = 20 kHz, Iref = 15 A)

Figure 13.27 shows the evolution of the Floquet multipliers and the average value of the current in the steady state, as the value of the inductance varies from 50% to 150% of its nominal value. The ratio mc / mc0 is fixed at 0.1 and the current reference is fixed at 15 A. It can be seen that the Floquet multipliers remain within the unit circle regardless of the value taken by the inductance. The state trajectory therefore always takes the form of a Tperiodic cycle. The mean current is correctly held at 15 A. This all confirms the correct operation of the current regulator in spite of a considerable change to the inductance.

Hybrid Current Controller

413

Study of parametric robustness of a modulated hysteresis regulator Here, we will use the H-bridge converter shown in Figure 13.23, fed by a DC voltage source with a fixed voltage of 50 V. To investigate its robustness, we will take values for the pair (Atr, Bh) equal to (8, 6) and will fix the nominal load parameters at L0=2.3 mH and r0=0.01Ω. The resultant bifurcation diagram is shown in Figure 13.28. The resultant surface is a plane and shows that the cycles that will be obtained are stable T-periodic cycles. This result is confirmed by calculating the eigenvalues of the Poincaré map evaluated for all the values of the load parameters (Figure 13.29).

Current (A) at (nT+t0)

ki=500 f=0Hz Iref=10A Atr=8 Bh=6 K i=500 , iref=10A , Atr=8 , Bh=6

11 n oi 10.5 t a cr 10 uf i B 0.015

3.5 3 2.5

0.01

2

R(Ω) R(Ohm)

1.5 0.005

x 10

l(mH)

-3

L(H)

1

Figure 13.28. Bifurcation diagram obtained for F=10 kHz, Atr=8, Bh=6, v1=v2=50 V, e=0, l=2.3 mH, R=0.01Ω, Ki=500, iref =10 A

0.04

Imaginary Im

0.03 0.02 0.01 0 -0.01 -0.02 -0.03 -0.04 0.5

Real Real 0.6

0.7

0.8

0.9

1

Figure 13.29. Plot of Floquet multipliers for varying load, F=10 kHz,Atr=8, Bh=6, v1=v2=50 V, e=0, l=2.3 mH, R=0.01Ω, Ki=500, iref=10 A

414

Power Electronic Converters

13.4. Conclusion In this chapter we have discussed nonlinear current regulators operating using hysteresis at fixed or variable frequencies. These regulators can be used in most DC/DC, AC/DC, or DC/AC conversion structures. The use of current regulators operating at fixed-frequency output from a hysteresis type of control structure appears to be a useful compromise in terms of dynamics and parametric robustness. However, the high-frequency behavior of these regulators depends closely on the parameters of the regulator and the system. In order to study high-frequency current variations caused by semiconductor switching, it is necessary to make use of limit cycle analysis. 13.5. Bibliography [ALI 07a] ALI-SHAMSI NEJAD M., PIERFEDERICI S., MARTIN J.P., MEIBODY-TABAR F., “Modeling and design of non-linear hybrid current controller suitable for high dynamic current loops”, European Physical Journal Applied Physics, vol. 39, p. 51–65, 2007. [ALI 07b] ALI-SHAMSI NEJAD M., PIERFEDERICI S., MARTIN J.P., MEIBODY-TABAR F., “Study of an hybrid current controller suitable for DC/DC or DC/AC applications”, IEEE Transaction on Power Electronics, vol. 22, p. 2176–2186, 2007. [ALL 96] ALLIGOOD K.T., SAUER T.D., YORKE J.A., Chaos an Introduction to Dynamical Systems, Springer-Verlag, New York, 1996. [BOD 01] BODE G.H., HOLMES D.G., “Load independent hysteresis current control of a three level single phase inverter with constant switching frequency”, IEEE PESC 2001, 32nd Annual Power Electronics Specialists Conference, Vancouver, 17–21 2001. [CAF 05] CAFAGNA D., GRASSI G., “Experimental study of dynamic behaviors and routes to chaos in DC-DC boost converters”, Chaos Solitons and Fractals 2005, p. 499-507, Elsevier, Paris, 2005. [CHA 96] CHAN W.C.Y., TSE C.K., “Studies of routes to chaos for currentprogrammed DC/DC converters”, IEEE PESC 96, New York, USA, 1996. [CHE 00a] CHEN J.H., CHAU K.T., CHAN C.C., “Analysis of chaos in current-modecontrolled DC drive systems”, IEEE Transactions on Industrial Electronics, vol. 47, n° 1, 2000. [CHE 00b] CHEN G., MOIOLA J.L., WANG H.O., “Bifurcation control: theories, methods, and applications”, International Journal of Bifurcation and Chaos, vol. 10, n° 3, p. 511–548, 2000.

Hybrid Current Controller

415

[CHE 03] CHENG K.W.E., LIU M., WU J., “Chaos study and parameter-space analysis of the DC-DC buck-boost converter”, IEE Proc. Electr. Power Applications, vol. 150, n° 2, 2003. [HAU 92] HAUTIER J.P., “A pulse width modulation with synchronized selfoscillations of power electronic converter”, Report of the Academy of Sciences Paris, t. 314, series II, p. 1407–12, 1992. [IU 03] IU H.H.C., ROBERT B., “Control of chaos in a PWM current-mode H-bridge inverter using time-delayed feedback”, IEEE Trans. on Circuits and Systems, vol. 50, n° 8, 2003. [KAA 86] KAAS-PETERSEN C., “Computation of quasiperiodic solution of force dissipative systems”, J. Comput. Phys., vol. 58, p. 395–408, 1986. [KAZ 98] KAZMIERKOWSKI M.P., MALESANI L., “Current control techniques for three-phase voltage-source PWM converters: A Survey”, IEEE Transactions on Industrial Electronics, vol. 45, n° 5, p. 691–703, 1998. [KHA 96] KHALIL H., Nonlinear Systems, 2nd edition, Upper Saddle River, Prentice-Hall, Englewood Cliffs, NJ, 1996. [LAC 08] LACHICHI A., PIERFEDIRICI S., MARTIN J.P., DAVAT B., “Study of a hybrid fixed frequency current controller suitable for DC/DC applications”, IEEE Transactions on Power Electronics, vol. 23, n° 3, 2008. [LEC 99] LECAIRE J.C., SIALA S., SAILLARD J., LE DOEUF R., “A new pulse modulation for voltage supply inverter’s current control”, EPE99, Lausanne, Switzerland, 1999. [MAZ 03] MAZUMDER S.K., NAYFEH A.H., BOROYEVICH D., “An investigation into the fast- and slow-scale instabilities of a single phase bidirectional boost converter”, IEEE Tran. on Power Electronics, vol. 18, n° 4, p. 1063–1069, 2003. [PAN 02] PAN C.T., HUANG Y.S., JONG T.L., “A constant hysteresis band current controller with fixed switching frequency”, ISIE 2002, vol. 3, p. 1021–1024, 2002. [PIE 03] PIERQUIN J., ROBYNS B., HAUTIER J.P., “Self-oscillating current controllers: Principles and applications”, EPE 2003, Toulouse, France, 2003. [RAS 99] RAS A., GUINJOAN F., “Ramp-synchronized, sliding-mode hybrid control of buck converter”, EPE’99, Lausanne, Switzerland, 1999. [ROD 97] RODRIGUEZ J., PONTT J., SILVA C.A., CORREA P., LEZANA P., CORTÈS P., AMMANN U., “Predictive current control of a voltage source inverter”, IEEE Transaction on Industrial Electronics, vol. 54, n° 1, 2007. [TSE 95] TSE H.C., CHAN W.C.Y., “Instability and chaos in a current-mode controlled Cuk converter”, IEEE PESC 1995 26th Annual IEEE Power Electronics Specialists Conference, vol. 1, p. 608–613, Atlanta, USA, 1995.

Chapter 14

Current Control Using Self-oscillating Current Controllers

14.1. Introduction PWM techniques are used for current or voltage control at the output of power converters driving electrical loads [SEI 88]. These electrical loads may be DC motors, or single-phase or three-phase AC motors. In all these cases, the loads or filters at the output of the power converters act as inductive elements. Using PWM and nested control techniques, it is possible to control the output current from the converter that flows into these inductive elements. When high performances are required, local current control loops are best implemented in the analog domain in terms of their dynamics, robustness, etc., due to the rapid response time that it offers. Such performances can be obtained using hysteresis modulations [BOS 90], [ELS 94], and [MAL 97]. However, it is not easy to control the switching frequency of the power converter and the maximum frequency the converter can support must not be surpassed otherwise the power switches are liable to be destroyed. In order to restrict this frequency, various solutions have been devised. The hysteresis bandwidth can be adapted [BOS 90] and [MAL 97] over the Chapter written by Jean-Claude LE CLAIRE.

418

Power Electronic Converters

course of the current control. Nevertheless, control of the switching frequency still depends on the parameters of the system, which are sometimes not well known. In order to achieve high performance control and at the same time manage the maximum switching frequency of the power switches without detailed knowledge of the load parameters, a new current control technique and pulse generation method was studied and implemented [LEC 99a]. The resultant performance was remarkable: good dynamics, control over the maximum switching frequency, very weak influence of load parameters, very weak harmonic current distortion, and very good stability. The process we will describe in the next section has been patented [LEC 97], [LEC 02a], and [LEC 07a]. 14.2. Operating principle of the self-oscillating current controller 14.2.1. Dual-purpose local loop Figure 14.1 shows a minimal representation of the system [LEC 99a]. The converter delivers a signal u(t) to an inductive load. The transfer function of the load is denoted as F1(p). The current across it is measured using a current probe of transresistance RT. The feedback signal is fed through a second-order low-pass filter whose transfer function is denoted as F2(p). The output signal from this is compared to the reference in the error detection module. The sign of this error modifies the output state of the converter. If the error is positive, the converter outputs a steady positive voltage. If the error is negative, the converter outputs a steady negative voltage. Output current

Input current i ref (t)

RT

+

inverter -

u(t)

F1 (p)

i(t)

non-linear amplifier F2 (p)

Figure 14.1. Local current control loop

RT

Self-Oscillating Current Controllers

419

In the next section, we will show that such a device can be used to control both the maximum switching frequency of the power switches in the converter and also control the output current supplied to the inductive load. There are two modes operating simultaneously. The first operates at high frequencies and is associated with the feedback structure, which acts as an oscillator. The second mode acts at low frequencies and is also associated with the feedback structure, which involves a comparison with a reference value by means of a feedback system or a closed loop system. 14.2.2. Local control loop for switching frequency control Figure 14.2 is similar to the previous structure but the power converter is replaced by a linear amplifier [LEC 99a]. Input reference current iref (t)

RT

vref (t) + -

Linear amplifier gain A 0

u(t)

F (p) 1

Output current i(t)

current sensor RT F (p) 2

Figure 14.2. Local current control loop

The closed loop for the low-frequency signal applied at the input of the system also makes use of a “positive” feedback, which causes it to enter an oscillatory mode. The transfer functions are chosen deliberately to cause this mode to emerge spontaneously. For this to occur, we require that the transfer function F2(p) is a second-order low-pass filter since the transfer function F1(p) represents a first-order low-pass filter. If we assume that the amplifier gain is positive, other cascaded linear stages within the loop must result in a phase shift that equals zero. Since the error detector introduces a phase shift of 180º in the feedback signal, filters F1 and F2 must introduce a phase shift of −180º between them at the oscillation frequency. Since this frequency is fairly high compared to the characteristic frequency of filter F1, this filter produces a phase shift of close to −90º at the oscillation frequency. To achieve the desired oscillation, we

420

Power Electronic Converters

therefore require the filter F2 to introduce a phase shift of close to −90º. A second-order low-pass filter is ideal for introducing such a phase shift. Furthermore, with such a filter only one single oscillation frequency is possible. Under these conditions, the structure is simply that of a sinusoidal oscillator, where the amplifier gain at the oscillation frequency in the presence of a reference signal of zero is written as A0 (Figure 14.3). 0

+

u(t)

Linear amplifier

-

F (p) 1

i(t) Output current

Gain A0

Gain -1 stage F (p) 2

RT

Figure 14.3. Oscillation loop

The oscillation frequency can be determined by using well-known tools of analog electronics for sinusoidal oscillators, tools that were built on the work of Heinrich Georg Barkhausen (1881–1956). We replace the error detector with a gain of −1. The transfer functions F1(p) and F2(p) are given by: F1 ( p ) =

I ( p) 1 1 1 = = . U ( p) R + Lp R 1 + τ1 p

1

F2 ( p) = 1+

2ξ p

ω0

p2

[14.1] [14.2]

.

+ 2 ω 0

The transfer function F(p) for the whole feedback loop, from the amplifier input to the output of the error detector, is found to be equal to the following expression, an expression that must respect Barkhausen’s criteria: V R 1 . F ( jω ) = er = − A0 . T . Ve R 1 + τ1 jω

1 1+

2ξ jω

ω0

ω2 − 2 ω0

.

[14.3]

Self-Oscillating Current Controllers

421

Since the numerator of F(jω) is real, its denominator can be used to determine the oscillation frequency. In this case it is sufficient to set the imaginary part of the denominator to zero to satisfy the first Barkhausen criterion. The expression for the denominator is as follows: D( jω ) =

1





2





0





0





ω ω ω 2 − ( 2ξω0τ1 + 1) ω 2 ) + j  ω0τ1 1 − 2  + 2ξ  . 2( 0   ω0  ω ω 

[14.4]

Thus, if the imaginary part is set to zero, we can easily determine the oscillation frequency, denoted as fosc. This is illustrated by the following formula where the product ω0τ1 is much greater than 10 in the applications we are interested in [LEC 99a]: fosc f 2ξ = 1+ = 1 + 2ξ . 1 . ω0τ1 f0 f0

[14.5]

fosc/f 0 damping coefficients: 4 2 1 0.5 0.25

Figure 14.4. Ratio of oscillation frequency to the natural frequency of filter F2 as a function of the product of the pulsation ω0 and the time constant of filter F1 and as a function of the damping coefficient

We can associate a characteristic frequency f1 with the time constant τ1 of the load, which gives another figure (Figure 14.4) based on the ratio of the oscillation frequency fosc to the natural frequency fo of filter F2. The cutoff

422

Power Electronic Converters

frequencies of transfer functions F1 that are likely to be found in machine control are of the order of a few hundred Hz at most, whereas f0 is several kilohertz. For this reason, the oscillation frequency is not very sensitive to the load parameters and the filter F2 is what determines the oscillation frequency. Now, by setting the function F(jω) to one at the oscillation frequency or applying the second Barkhausen criterion, the gain A0 of the amplifier at that same frequency can be determined [LEC 99a] as: A0 = 2ξ .

R 1 R .(ω0τ1 + .ω0τ1 . + 2ξ ) ≈ 2ξ . RT RT ω0τ1

[14.6]

By combining equations [14.3] and [14.6], the transfer function F(jω) for the control loop now satisfies the following expression:   1 −2ξ .  ω0τ1 + + 2ξ  ω0τ1   F ( jω ) = .    ω2  1 ω 2 2 ω0 − ( 2ξω0τ1 + 1) ω + j  ω0τ1 1 − 2  + 2ξ   ω   ω0  ω02 0    

(

)

[14.7]

modulus of F

imag axis

damping coefficients: 4 2 1 0.5 0.25

4 2 1

0.25 0.5

real axis

Figure 14.5. Nyquist plot of the transfer function –F(jω) and the modulus of F(jω) as a function of the normalized frequency for ω0τ1 equal to 100 and as a function of the damping coefficient

Ignoring the error detector for now, the Nyquist plot of −F(jω) shown below shows the transition through the critical point −1 at the oscillation frequency. This is a consequence of the gain A0, which is automatically adapted if the linear amplifier has automatic gain control or if the linear

Self-Oscillating Current Controllers

423

amplifier is replaced with a nonlinear amplifier. The latter provides significant gain for small input signals and a saturated output. The plot of the modulus of F(jω) shows this unit gain at the oscillation frequency, reflecting the second Barkhausen criterion. 14.2.3. Local low-frequency current control loop The device is therefore capable of current control, where the power converter of a relay type has an equivalent gain at low frequencies. The method used to determine this has already been used [LEC 05]. The schematic diagram is modified to shift the “oscillatory component” to a tighter, internal feedback loop and introduce a carrier c(t) (Figure 14.6). V

ref

(t)

low frequency component

+ V

fb

(t)

second-order low-pass filter

ε c(t)

-

+E

u(t)

-E

carrier path

F (p) 2

low frequency negative feedback loop

+

first-order low-pass filter RT

F (p) 1

Figure 14.6. Oscillation loop with separate carrier

Using a control loop with separate carrier, the structure looks like a closed loop system where the error is compared to a carrier. The lowfrequency reference and feedback signals for the error detector are denoted Vref(t) and Vfb(t). Thus the detector generates the low-frequency error that is applied to the output stage. This incorporates a comparator that outputs the square-wave signal that varies between the voltages +E and –E. To simplify things, we will assume that the switching frequency is stable and equal to the natural frequency of filter F2, we will treat the carrier as a sine wave, in light of the presence of the low-pass filters F1 and F2, the carrier is passed directly to the nonlinear stage, and the low-frequency component of the reference signal (a current image) is fed to the error detector.

424

Power Electronic Converters

The gain of the nonlinear relay amplifier is given by the ratio of the mean value of the output signal u(t) to the value of the error ε at the input, a value that is assumed to be constant given the high switching frequency. This gain is a function of the positive and negative output voltages but also of the duty cycle α: G=

< u (t ) >

ε

[14.8]

.

For a switching period T and a duty cycle α, the output voltage u(t) is negative for a time β T, where that time added to αT makes up one full period T. It follows that: < u ( t ) >= − E. ( 2β − 1) = E. ( 2α − 1) .

[14.9]

The first harmonic of the signal u(t) is denoted u1(t). We will make the hypothesis that the carrier signal fed back through the nonlinear stage is a function of this component only because of the filtering performed by the low-pass filters F1 and F2. The transfer function H(jωosc), resulting from F1(jωosc) followed by RT and then by F2(jωosc), thus determines the amplitude C0 of the carrier c(t). Moreover, since the oscillation frequency fosc is very close to the natural frequency f0 of the filter F2, the modulus of H(jωosc) can be replaced by the modulus of H(jω0). It is therefore possible to write two expressions for the signal u1(t) and the modulus of H(jω0): u1 ( t ) =

4E

π

.sin (πα ) .sin (ω0 t + φ )

H ( jω0 ) = F1 ( jω0 ) .RT . F2 ( jω0 ) =

[14.10] RT . L.ω0 .2.ξ

[14.11]

Equations [14.10] and [14.11] can be used to determine the amplitude C0 of the carrier. If we do this, we find the amplitude and functional form of the carrier c(t) to be: C0 =

4E

π

.sin (πα ) .

RT 2.ξ .L.ω0

c(t ) = C0 .cos (ω0 t ) .

[14.12] [14.13]

Self-Oscillating Current Controllers

425

For a duty cycle α, the carrier c(t) intersects the error signal at a time t equal to half β T, where T is the switching period. The error ε is then given by: RT  β T  4E . .sin ( πα ) .cos (πβ ) . ε = c = 2 2. .L.ω0 π ξ  

[14.14]

We can then use equations [14.8], [14.9], and [14.14] to find the gain G(α) of the nonlinear relay amplifier: G (α ) =

(1 − 2α ) .ω0 .L.ξ .π sin ( 2πα ) .RT

.

[14.15]

If we use a Taylor–Young expansion [PIC 98] for the sin(2πα) term for a duty cycle close to 50%, we can also obtain the minimum gain Gmin of G(α): Gmin =

L.ω0 .ξ . RT

[14.16]

Around a duty cycle of 50%, the gain G(α) takes a “bowl” shape. It is a minimum for a duty cycle of 50% and rises on either side. For values close to 0% or 100%, it drops strongly. Over a broad range around a duty cycle of 50%, the form of the “bowl” is maintained and the gain is therefore greater than or equal to Gmin. The minimum value Gmin of G(α) can be used to calculate the gain of the system in closed loop mode. It should be noted that this minimum gain for the nonlinear amplifier in a low-frequency model can also be determined using two further methods. By using the automation tools developed by Cypkin in the context of motor control [HYU 01] and in work relating to the emulation of power devices [OLI 06a], the gain G(α) can be found and then its minimum value determined. In addition, the use of an alternative approach [BOI 99] enables only the minimum equivalent gain to be found but using a very brief approach. In the case where the control system consists of a nonlinear relaytype element without any hysteresis, the minimum equivalent gain can simply be determined using the following expression [BOI 99]: Geq − min =

2

∑ ( −1)

k∈N

1 k

Re  H ( jk 2π . f osc ) 

[14.17]

426

Power Electronic Converters

We will start from the assumption that the switching frequency, and hence the oscillation frequency fosc, is close to (and can therefore be taken to be equal to) the natural frequency f0 of the second-order filter F2. This, combined with the fact that the transfer function H(jω0) is real at the oscillation frequency and the assumption that the resistance of the coil can be ignored, enables us to use the above formula directly to determine an expression for the minimum equivalent gain of the relay at low frequencies: Geq − min ≈

2.

∑ ( −1)

k∈N

1 k

Re  H ( jk 2π . f o ) 

=

Lω0ξ 1 . [14.18] = −2.H ( j.2π . f o ) RT

The gain Geq-min, obtained using this last method confirms the minimum value Gmin of the gain G(α). However, the two earlier methods offer a greater understanding of how the amplifier gain varies as a function of the duty cycle. These various methods that we have discussed lead to the value Gmin for the relay amplifier gain or to its expression G(α) as a function of the duty cycle. The nonlinear amplifier, a relay amplifier without hysteresis, can then be replaced by its equivalent linear model, making it easier to determine the parameters of the correctors that will be introduced in more complex control systems. Thus the equivalent gain can be used to optimize a voltage source built around a kernel consisting of several current sources [LEC 07b]. The maximum power that can be delivered by this voltage source is a function of the number of modules used, each of which uses self-oscillating current control (SOCC). These power modules are combined into a parallel structure. The linear model of the kernel delivering the global current is obtained by considering the equivalent gain of each nonlinear amplifier present in each elementary current source. An external control loop contributes to the regulation of the output voltage of the DC/AC converter. Voltage regulation is improved by using a simple proportional-integralderivative (PID) corrector, the parameters of which can be determined using the linear model.

Self-Oscillating Current Controllers

427

14.2.4. Stability of the modulator In the case of a power converter, which is intrinsically nonlinear, the linear part of the control loop can be expressed as the product of the transfer function of the first-order filter F1, the current sensor RT, and the secondorder filter F2: This product results in the transfer function H(p), which is equal to: H ( p) =

RT . R

1 .   2ξτ1 1  2 τ1 3 2ξ  1 +  τ1 + + 2p + 2 p  p +   ω0  ω0   ω0 ω0 

[14.19]

In their study of relay systems without hysteresis, G. Schmidt and Preusche [GIL 88] proved the stability of such systems subject to a number of conditions. First, it must be possible for the transfer function for the linear part to be written in the following form: H ( p) =

b0 + b1 p a0 + a1 p + a2 p 2 + a3 p3

[14.20]

.

If this is the case, the following conditions must also be met [GIL 88]: a0 ≥ 0

a1 > 0

a1a2 − a0 a3 > 0

a2 > 0

a3 > 0

[14.21]

.

In the case of an SOCC, the coefficients a0, a1, a2, and a3 from equation [14.20], and present in [14.19], are positive and hence the set of inequalities in [14.21] is also satisfied:  2ξ   2ξτ1 1  τ1 + − a1a2 − a0 a3 = τ1 +  ω0   ω0 ω02  ω02  2ξ  2 2ξτ1 1  + a1a2 − a0 a3 = τ + >0 ω0  1 ω0 ω02 

.

[14.22]

Consequently, the converter with a relay-type power stage, with current and switching frequency controlled by the SOCC, is stable.

428

Power Electronic Converters

14.3. Improvements to the SOCC 14.3.1. Reducing the static error In the case where the load inductance is small or a counter-EMF is present or alternatively if the DC bus voltage is very low, a static error will be visible. This can be observed by attempting to generate a sinusoidal current with a peak of 16 A and a frequency of 200 Hz, supplying a load (R, L) of 5 mH and 15 Ω, using a natural frequency f0 of 10 kHz for filter F2 (which is a Butterworth filter), a current sensor with transresistance 1 V/A, and a voltage E of 300 V. To reduce the static error to the point where it can no longer be “seen”, we simply need to increase the low-frequency gain of the forward chain. To this end, a filter F3 is added to the forward chain. The schematic diagram for a SOCC using such a filter is shown in Figure 14.7. converter +E filter i ref (t)

RT

v ref (t) +

Reference current

-

F3 (p)

C

u(t)

C

F (p) 1

-E

Output current i(t)

RT

non-linear relay amplifier

F (p) 2

Figure 14.7. Modulator with static error corrector

The transfer function for this corrector filter is:

F3 ( p ) =

τ 3 1+τ 2.p 1 + 68.10 −6. p . = 5.6 . τ 2 1+τ 3. p 1 + 381.10 −6. p

[14.23]

The time constants are chosen such that the gain of the forward chain should not be much affected at high frequencies (including the switching frequency). At such frequencies the transfer function F3 has a value very close to 1. This results in the Bode diagrams as shown in Figure 14.8.

Self-Oscillating Current Controllers

429

Figure 14.8. Bode diagrams for the static error corrector F3

The phase shift in the forward chain is thus almost unchanged at high frequencies. At low frequencies, the gain of filter F3 is 15 dB. This causes a sufficient increase in the gain of the forward chain such that no static error is seen in real situations. The Bode diagrams show the effect of filter F3 on the linear part of the control loop for the parameters listed earlier, for the case where the natural frequency f0 of filter F2 is equal to 10 kHz (Figure 14.9).

(2) (1)

(1) (2)

Figure 14.9. Influence of filter F3 on the linear part of the control loop: a) without; and b) with F3

430

Power Electronic Converters

For the transfer function −H(jω)F3(jω), representing the entire linear part of the control loop, a phase shift of zero is again found at frequencies close to f0. Using the parameters listed earlier, sinusoidal current control at 16 A and 200 Hz no longer shows any visible static error. The current oscillates on either side of the reference signal (Figure 14.10).

(a)

(b)

Figure 14.10. Regulation: (a) without; and (b) with correcting filter F3

14.3.2. Controlling the switching frequency The first implementation of a SOCC was equipped with a second-order low-pass filter F2 that incorporated a switched capacitor [LEC 99a, LEC 99b, and LEC 99c] (Figure 14.11). vref (t) Reference voltage input

low-pass filter

+

smoothing filter -

+ +

F (p) 3

optional output

F (p) 2 (MF10)

comparator + buffers (2 states)

antialiasing filter

Power converter (H bridge)

u(t)

inductive load F1 (p)

High-pass filter Low-pass filter

Voltage controlled oscillator

(UAF42) BOARD

input voltage control of f 0

Figure 14.11. Modulator using switched capacitance filter

i(t)

current sensor RT

Self-Oscillating Current Controllers

431

It is thus possible to stabilize or even vary the natural frequency f0. The aim is to be able to judge the impact of this frequency on the behavior and performance of the modulator. The schematic diagram of this first “SOCC” shows an input intended to control the frequency f0. The control signal is a steady voltage but could also be a modulated signal. In this initial design for an SOCC modulator, the bandwidth of the input circuit is limited to avoid any aliasing effects. The smoothing filter limits the bandwidth of the low-frequency error to 72 kHz. The anti-aliasing filter associated with the MF10CCN switched capacitor filter, limits the bandwidth of the signal passing through it to 77 kHz. The forward-phase shifting filter balances out the phase delay introduced by the anti-aliasing filter in the chosen range over which the switching frequency is to be varied. The voltage-controlled oscillator enables the clock frequency for the switched capacitor filter F2 to be selected, and hence the natural frequency f0 of the second-order filter controlling the oscillation frequency of the feedback loop. The high-pass and low-pass filters downstream of the current sensor split the components above and below 980 Hz using a universal active filter UAF42. This choice means that the switched capacitor filter MF10CCN does not process the DC component of the current delivered by the converter. This prevents the significant offset voltage of the switched capacitor filters. This first SOCC design was subsequently modified to enable lower cost boards to be manufactured, where f0 cannot be varied. In that case, the switched capacitor filter was replaced with a second-order Butterworth lowpass filter with a Sallen-Key structure built around a single conventional instrumentation operational amplifier. Although complex, the first modulator offered a great deal of flexibility in early experimental investigations and enabled detailed validation of theoretical results [LEC 00 and LEC 02b]. 14.3.3. Variants on the initial design The filter F2, which enables the switching frequency to be controlled, can just as well be placed in the feedback chain as in the forward chain [LEC 97]. SOCCs have been constructed using both design variants. Their performances are similar. The reason for placing the filter F2 in the forward

432

Power Electronic Converters

chain is that it can be constructed with or without an electronic device. In the case of a passive filter, the frequency f0 can be much higher without encountering any of the difficulties associated with operational amplifiers at high frequencies. In the context of motor control, where the currents to be controlled are at relatively low frequencies, the performance of the system is not limited by either of these locations for the filter F2 since the natural frequency f0 for the SOCC is chosen to lie in the range from 10 kHz to 30 kHz. 14.4. Characteristics of the SOCC 14.4.1. Switching frequency As was previously indicated, the SOCC offers control over the instantaneous commutation frequency of the power switches through the oscillations it induces in the feedback loop. By simulating for a given set of parameters the reproduction of a sinusoidal wave at 200 Hz [LEC 02b], observation of the instantaneous evolution of the ripple reveals that its frequency changes. The same is true for the components at all frequencies that are involved with the ripple. Figure 14.12 illustrates the evolution of the frequencies of the various current components output by the converter. If the natural frequency of the filter F2 is tuned to 10 kHz, then the switching frequency is modulated. Nevertheless, although the switching frequency is significantly decreased for the chosen parameter set, its value is still bounded and limited to a higher value close to the natural frequency of filter F2. The ripple can be studied for a constant reference signal, enabling each of the basic expressions for the feedback signal going to error detector to be determined [LEC 99a and LEC 00]. These elementary expressions can then be used to determine the instantaneous switching frequency [LEC 99a]. Thus, it is possible to establish a static characteristic of the switching frequency as a function of the requested current for a given supply voltage to the converter. It is also possible to confirm that the maximum switching frequency can be well controlled through a suitable choice of parameters for the filter F2.

Self-Oscillating Current Controllers

433

Current in the load in Amps

Frequency in kHz

Spectral density in dB

Time-frequency plot

Time in ms

Figure 14.12. Evolution of the switching (oscillating) frequency

Therefore, as a function of the various values of the direct currents to be controlled and the voltages applied to the converter we can build up a representation of the normalized switching frequency in the form of a 3D plot. This shows that the maximum switching frequency is limited to a value that is extremely close to that of the natural frequency f0 of filter F2 [LEC 02b]. It therefore follows that the switching frequency is correctly limited as required, but also that the ratio of the oscillation frequency to the natural frequency f0 of filter F2 is close to 1 and satisfies the expression we found for the case of a linear amplifier. Furthermore, the switching frequency may be lower than this. Under these conditions, it is not possible to damage the power converter. To cause significant lowering of this frequency, the reference current must be high and the supply voltage to the converter should be low (Figure 14.13).

434

Power Electronic Converters

fosc f0

E Iref Imax

Emax

Figure 14.13. Evolution of the switching (oscillating) frequency

14.4.2. Linearity In order to judge the linearity of the modulator, the output current from the converter can be compared to the reference current. It can be seen that the coil’s current accurately tracks the reference current. We will take the example of the following conditions: (L = 5 mH, R = 15 Ω, ξ = 0.707, f0 = 10 kHz, E = 300 V, I = 16 A, fs = 200 Hz), for which we find that the low-frequency output current component and the reference signal follow a linear relationship when the self-oscillating effects are not taken into account in the output [LEC 99a] (Figure 14.14).

Figure 14.14. Linearity with the use of the static error corrector filter F3

Self-Oscillating Current Controllers

435

14.4.3. Harmonic distortion For the parameters given above, the current spectrum at the output of the converter can be determined. The spectrum reflects the presence of low-frequency harmonics with very low amplitudes, at least 40 dB lower in amplitude than the fundamental. By comparing the examples with and without the filter F3, the beneficial effect of that filter can be seen (Figure 14.15).

Figure 14.15. Spectrum before and after introduction of the filter F3

The filter F3 raises the low-frequency gain of the forward chain, in this case by 15 dB, and as with a linear system the harmonic spectrum is attenuated by its presence. 14.5. Extensions to the SOCC concept 14.5.1. Self-oscillating voltage control 14.5.1.1. Application of the principles of an SOCC to self-oscillating voltage control with a resistive load The principle of the SOCC concept is to introduce a spontaneous oscillation within the closed loop system. The high-frequency component clearly reflects the presence of an oscillator within the structure. In what follows, we will apply the same principle to voltage regulation within a DC/AC power converter where the output is filtered by an LC cell. A voltage measurement circuit is used at the terminals of the capacitor. A fraction of this voltage is fed back to the self-oscillating voltage controller

436

Power Electronic Converters

(SOVC). Figure 14.16 gives a schematic diagram for the case of a resistive load [LEC 02c]. Output v voltage o

+E v ref

+

u

C -

r, Lf Cf

C

F2 (p)

Rs

-E non-linear amplifier

load LC filter

K

V

voltage sensor

Figure 14.16. Local voltage control loop

The load may also be an inductive load or something even more complex but in order to keep our discussion simple we will restrict ourselves to discussing a resistive load. We will use the transfer function F1R(p) to represent the ratio between the voltage v0(t) at the output of the LC filter and the voltage u(t) delivered at the leg(s) of the voltage converter. Its expression is as follows: V ( p) Rs F1R ( p ) = o = , U ( p ) r + Rs + r.Rs C f + L f p + Rs L f C f p 2

(

)

[14.24]

where: ω0 LC =

1 Lf C f

[14.25]

.

The parameters r, Lf, Cf, and Rs represent the negligible resistance of the coil, its inductance, the capacitance of the smoothing filter, and the load resistance. This expression for the transfer function F1R(p) can be simplified since the resistance of the coil is small. It becomes: 1

F1R ( p ) ≈ 1+

Lf Rs

p2

[14.26]

p+ 2 ω

0 LC

In the case of an SOCC, the transfer function F1(p) is first order. Since the transfer function F1R(p) of the “SOVC” is second order, we require a

Self-Oscillating Current Controllers

437

new transfer function for filter F2(p), which we will write as F2SOVC (p): p

1+

1+

2ξ p

G ω0 HFPB K .ω0 LC = F2 SOVC ( p ) = , 2ξ p p 2 2ξ p p 2 + 2 1+ + 2 1+

ω0

ω0

ω0

[14.27]

ω0

where the numerator can be used to compensate for the excess phase shift within the new transfer function F1R(p), an excess whose amount is known since it depends on the chosen smoothing filter. This transfer function can be achieved using a low-pass filter and a band-pass filter whose output signal is passed through an amplifier of gain GHFBP. In practice, a universal active filter UAF42 is used in order that the denominators of the low-pass and band-pass filters should be the same. If we do this in the case of voltage regulation, the transfer function for the linear chain H(p) of the loop, which we will now write as HSOVC(p), becomes:

H SOVC ( p ) = F1R ( p ) .KV .F2 SOVC ( p ) .

[14.28]

This expression can also be written in the following form, in light of our earlier analysis:

 p  p .1 − Rs .K V .1 +  K .ω0 LC   K .ω0 LC H SOVC ( p ) = Den[H SOVC ( p )]

  ,

[14.29]

where:

 R 2ξ Rs Den  H SOVC ( p )  = ( Rs ) +  s + L f − K .ω0 LC  ω0  R L f 2ξ  Rs 2ξ +  Rs L f C f + s2 + − + Lf ω0 ω0  ω0 

 1   K .ω0 LC

 p 

 2  p 

 2ξ L f  R L f 2ξ  1  3 +  Rs L f C f + 2 −  Rs L f C f + s2 + p  ω0 ω0  ω0 ω0  K .ω0 LC   RL C  1  5 2ξ L f  1  4  Rs L f C f +  s f2 f −  Rs L f C f + 2  p −  p 2 ω0 ω0  K .ω0 LC   ω0 K .ω0 LC    ω0

[14.30]

438

Power Electronic Converters

As with the case of the SOCC, if we set the imaginary part of Den [HSOVC (jω)] to zero, we can determine the oscillation frequency of the SOVC feedback loop. This oscillation frequency satisfies a complex expression [LEC 02c]. Nevertheless, it can be shown to be very close to f0. In order to illustrate this, we have chosen the following parameters: Lf = 3 mH, r = 0 Ω, Cf = 10 µF, f0 = 10 kHz, ξ = 0,707, and Rs = 170 Ω, 54 Ω, or 27 Ω. With these values, the Bode plots show a phase shift for F1R. This is close to −180º at the highest frequencies (Figure 14.17).

Figure 14.17. Bode plots for the transfer function F1R(jω) for various values of the resistive load

The Bode plots for -HSOVC(jω), on the contrary, show that the phase shift is 0º close to the natural f0 of the filter F2SOVC, which here has a value of 10 kHz. The plots also confirm the results for the oscillation frequency of the feedback loop (Figure 14.18). These plots show very small influence of variations in the load resistance. For three very different values of the resistance, the oscillation frequency remains close to f0.

Self-Oscillating Current Controllers

439

Figure 14.18. Bode plot of the transfer function –HSOVC(jω) for various values of the resistive load

14.5.1.2. Application of the principles of SOCC to SOVC with an inductive load In the field of motor control using power converters, loads are generally inductive. Up to now, we have considered a resistive load for simplicity and ease of comprehension. If we now consider an inductive load, the transfer function F1(p) becomes F1L(p) [LEC 02c]:

F1L ( p ) = 1+

L f + Ls Rs

L 1+ s p Rs

[14.31]

L p + L f C f p + s L f C f p3 Rs 2

This is a third-order low-pass filter. However, if the frequency is greater than a few hundred hertz, this can be simplified to give an approximate expression in the form of a second-order low-pass filter, which we will denote F1HF: F1HF ( p ) =

Ls . L f + Ls

1 1+

Rs L f C f L f + Ls

p+

Ls L f C f L f + Ls

. p

2

[14.32]

440

Power Electronic Converters

This approximate transfer function is consistent with the operating principles of the SOCC, transferred to our present case of an SOVC. The self-oscillating frequency of the smoothing filter, which depends slightly on the nature of the inductive load, is now given by the following expression:

ω0 LC .ind .load =

L f + Ls Ls L f C f

= ωOLC .

L f + Ls Ls

.

[14.33]

Figure 14.19, which uses Bode plots to compare the transfer function F1L(p) to its simplified version F1HF(p), shows that the two are very similar:

Frequency in Hz

Frequency in Hz

Figure 14.19. Bode plots for the transfer function F1(jω) and F1HF(jω)

14.5.1.3. Experiments with a SOVC and a resistive load To demonstrate the performance of the SOVC, an arbitrary generator was programmed with a specific reference wave that was to be reproduced by the converter. The voltage applied to the DC bus was equal to 100 V. The value of the resistive load was 41Ω. The following oscilloscope snapshots illustrate the performance of the SOVC in tracking the reference signal [LEC 02c] (Figure 14.20). In this experiment, the peak voltage reached 85 V and the current 4 A. For information, open-circuit tests were also performed. Even in the absence of a load, and hence damping of the filtering cell, the system was stable (Figure 14.20).

Self-Oscillating Current Controllers

441

Figure 14.20. Tracking of an arbitrary signal by the SOVC: reference voltage and converter output, current in the choke coil

14.5.2. Three-phase SOCC The principle of the SOCC can also be applied to current regulation in power converters supplying three-phase loads [LEC 99a]. It can also be used in multiphase converters [LEC 97]. A schematic diagram for the three-phase case is shown in Figure 14.21, where the three modulators are independent. v refA +

v refB

-

C

F2 (p)

C

v

Lf v

C C

+E -E

F (p) 2

a

+E -E

+ -

v refC

Lf

+E

Power converter

C

b Lf

v

c Inductive load Currents sensors

C

+

-E

IL 1

IL 2

IL 3

F (p) 2

RT

RT

Figure 14.21. Three local regulation loops for a three-phase SOCC

RT

442

Power Electronic Converters

In this case, each SOCC controls the current of one phase of the converter. For a floating-neutral load, the two other phases affect the potential of the floating neutral and are treated as sources of disturbances [LEC 99a]. Each SOCC demonstrates its ability to reject disturbances, and even when the voltage of the DC bus supplying the converter is low, the currents are still sinusoidal. The three-phase SOCC is therefore able to regulate threephase currents. The results obtained for this, both in simulation and experiment, illustrate the excellent reproduction of the reference currents (Figure 14.22).

Figure 14.22. Currents for the three simulated and measured phases

14.5.3. Three-phase SOVC The principle of a single-phase SOVC can be extended to a three-phase SOVC. Its schematic diagram consists of three SOVC systems combined in parallel, where again, the load has a floating neutral [LEC 03] (Figure 14.23). For a given phase and its associated regulated output voltage, the two other load voltages can again be considered as disturbances that are efficiently rejected by the three elementary SOVC systems [LEC 03]. Both in open circuit and under load, highly effective regulation occurs. Tests were performed with the following parameters: f0 = 20 kHz, ξ = 0.707, GHFBP = 10, Lf = 2.2 mH, Cf = 4.7 µF, KV = 0.05, fs = 200 Hz, U = 50 V, f0LC = 904 Hz, Rs = 2400 Ω or 18 Ω, E = 62.5 V, and DC voltage (125 V).

Self-Oscillating Current Controllers

443

During each of the half-periods of the output voltages, each SOVC stops the self-oscillation because of the low value of the DC bus voltage supplying the converter. This can be seen reflected in the resultant currents. In spite of this, the reproduction of the voltages is still extremely high quality (Figures 14.24 and 14.25). v refA +

v refB

C

F (p) 2

C

v

+

F2 (p)

a

v

C C

+E -E

Three-phase inverter

v Cf

Lf

+E -E

-

v refC

Lf

+E -

C

v

Cf v

b Lf c

1 v1n

2 v 2n

Cf v

3

LC filter and load

v1n

C

+

-E -

v 3n

Voltage sensors

K

V

v 2n K

V

v 3n K

V

F (p) 2

Figure 14.23. Three local voltage control loops for a three-phase SOVC

These various tests show that a three-phase SOVC offers extremely high performance in terms of control fidelity and robustness. Even in open circuit, SOVCs achieve stable voltage control.

Figure 14.24. Voltage and current for single-phase, simulated, and measured, in open circuit

444

Power Electronic Converters

Figure 14.25. Voltage and current for single-phase, simulated and measured, in conjunction with a load

14.5.4. Emulation of high-power active loads We have seen how the design of the SOCC has been extended to produce the SOVC [LEC 02c]. These two self-oscillating controllers have been introduced in high-power industrial applications to emulate active loads [GIN 05]. For such applications, the initial SOVC design was refined in order to control high-power converters with greater dynamic capabilities. In this case, a bandwidth of several kilohertz with a phase shift at either end of the band restricted to 10º was required and was successfully obtained [OLI 06a, OLI 06b, and OLI 08]. 14.5.5. Analog-to-digital converter for the measurement circuit The SOCC principle was also applied in the construction of the highvoltage measurement circuits present in power converters [LEC 05]. Such measurements, very useful in feedback control, require galvanic isolation. The modulation process that we have described can be used for one-bit analog-to-digital conversion. In this case the filter F1 of the SOCC takes the form of a simple RC filter. The resultant PWM signal is sent through an optical isolator in the case of a measurement circuit. The output signal from the optical isolator is demodulated using a first-order low-pass filter identical to the one introduced in the modulator feedback loop [LEC 05].

Self-Oscillating Current Controllers

445

Measurement circuits were constructed with bandwidths greater than 150 kHz and dynamic ranges more than 80 dB. 14.6. Conclusion In the context of motor control, ever-increasing performances are required. Consequently, the power converter feedback loops and the control devices must be improved. To meet these requirements, a number of PWM and current or voltage regulation techniques have been studied and developed. The exceptional performances required of them (with good reason), including when used as motor emulators, justify the study and development of new control methods. To this end, a new PWM and current control system has been devised and studied. It has given rise to SOCCs and SOVCs, which are now seeing realworld use in industrial applications. The method used in these modulators exploits the spontaneous SOCC- and SOVC-loops. This technique results in remarkable performance in terms of fidelity, bandwidth, and stability of the control systems. Although this technique is mostly used in SOCC and SOVC, it can be extended to other applications. One such requirement is for analog-to-digital conversion of the signals involved. To this end, high-voltage measurement boards with galvanic isolation have been developed. These too are used in power converters. 14.7. Bibliography [BOI 99] BOIKO I., “Input-output analysis of limit relay feedback control system”, American Control Conference, San Diego, California, USA, 1999. [BOS 90] BOSE B.K., “An adaptative hysteresis-band current control technique of voltage-fed PWM inverter for machine drive system”, IEEE Transactions on Industrial Electronics, vol. 37, n° 5, 1990. [ELS 94] EL-SAYED I.F., “A powerful and efficient hysteresis PWM controlled inverter”, EPE Journal, vol. 4, 1994. [GIL 77] GILLE J.C., Introduction aux systèmes non linéaires, Dunod University, Bordas, Paris, 1977.

446

Power Electronic Converters

[GIL 88] GILLES J.C., DECAULNE P., PELEGRIN M., Systèmes asservis non linéaires, Dunod, Paris, 1988. [GIN 05] GINOT N., LE CLAIRE J.C., LORON J.C., “Active loads for hardware in the loop emulation of electrotechnical bodies”, IECON’05, Raleigh, North Carolina, USA, 2005. [HYU 01] HYUEL J.F., Commande en courant des machines à courant alternatif, PhD thesis, University of Nantes, France, 2001. [LEC 97] LE CLAIRE J.C., SIALA S., SAILLARD J., LE DOEUFF R., Procédé et dispositif de commutateurs pour régulation par modulation d’impulsions à fréquence commandable, French patent n° 9708548, 1997, publication n° 2765746, 1999. [LEC 99a] LE CLAIRE J.C., Circuits spécifiques pour commande de machines à courants alternatifs, PhD thesis, University of Nantes, France, 1999. [LEC 99b] LE CLAIRE J.C., SIALA S., SAILLARD J., LE DOEUFF R., “A new pulse modulation for voltage supply inverter’s current control”, ELECTRIMACS 99, Lisbon, Portugal, 1999. [LEC 99c] LE CLAIRE J.C., SIALA S., SAILLARD J., LE DOEUFF R., “An original pulse modulation for current control”, EPE’99, Lausanne, Switzerland, 1999. [LEC 00] LE CLAIRE J.C., SIALA S., SAILLARD J., LE DOEUFF R., “Novel analog modulator for PWM control of alternative currents”, Revue internationale de génie électrique, vol. 3, n° 1, p. 109–131, Hermès, Paris, 2000. [LEC 02a] LE CLAIRE J.C. et al., Method and device for controlling switches in a control system with variable structure with controllable frequency, US patent n° 6376935B1, 2002. [LEC 02b] LE CLAIRE J.C., SIALA S., SAILLARD J., LE DOEUFF R., “Une nouvelle modulation d’impulsions pour le contrôle en courant d’un convertisseur de puissance”, Revue internationale de génie électrique, vol. 5, n° 1, p. 163–181, Hermes, Paris, 2002. [LEC 02c] LE CLAIRE J.C., “A new resonant voltage controller for fast AC voltage regulation of a single-phase DC/AC power converter”, PCC 2002, Osaka, Japan, 2002. [LEC 03] LE CLAIRE J.C., MOREAU R., GINOT N., “A resonant voltage controller for fast regulation of a three-phase voltage source”, EPE 2003, Toulouse, France, 2003. [LEC 05] LE CLAIRE J.C., MENAGER L., OLIVIER J.C., GINOT N., “Isolation amplifier for high voltage measurement using a resonant control loop”, EPE 2005, Dresden, Germany, 2005.

Self-Oscillating Current Controllers

447

[LEC 07a] LE CLAIRE J.C., SAILLARD J., SIALA S., LE DOEUFF R., Procédé et dispositif de commande de commutateurs dans un système de commande à structure variable/Process and device for a control switch in a variable structure command system and control frequency, CA patent n° 2295846, 2007. [LEC 07b] LE CLAIRE J.C., LEMBROUCK, “A simple feedback for parallel operation of current controlled inverters involved in UPS”, EPE 2007, Aalborg, Denmark, 2007. [MAL 97] MALESANI L., “High-performance hysteresis modulation technique for active filters”, IEEE Transactions on Power Electronics, vol. 12, n° 5, 1997. [OLI 06a] OLIVIER J.C., Modélisation et conception d’un modulateur auto-oscillant adapté à l’émulation d’organe de puissance, PhD thesis, University of Nantes, France, 2006. [OLI 06b] OLIVIER J.C., LE CLAIRE J.C., LORON J.C., GINOT N., “A self oscillating voltage controller for applications with high bandwidth”, IECON’06, Paris, France, 2006. [OLI 08] OLIVIER J.C., LE CLAIRE J.C., LORON J.C., “An efficient switching limitation process applied to high dynamic voltage supply”, IEEE Transaction on Power Electronics, vol. 23, n° 1, p. 153–162, 2008. [PIC 98] PICHON J., Calcul des limites, Ellipses, Paris, 1986. [SEI 88] SEIXAS P., Commande numérique d’une machine synchrone autopilotée, PhD thesis, INP, Toulouse, 1988.

Chapter 15

Current and Voltage Control Strategies Using Resonant Correctors: Examples of Fixed-frequency Applications

15.1. Introduction The challenge of controlling alternating electrical quantities using static converter is a common one in power electronics. A wide range of applications exist (speed controllers, active filtering, uninterruptible power supplies, etc.). The solution to this problem already exists in theory: it is necessary to employ techniques that lead to the control loop having an infinitely large gain at the operating frequency. In practice, two classes of control techniques for electrical quantities can be identified [HAU 99b], where the switching angles of the static converter are defined by direct control (amplitude or instantaneous value control, creation of sliding modes) or by indirect control (time duration or instantaneous mean value control).

Chapter written by Joseph PIERQUIN, Arnaud DAVIGNY and Benoît ROBYNS.

450

Power Electronic Converters

For control techniques of the first category, a nonlinear element (such as hysteresis) directly defines the switching sequence as a function of the instantaneous difference between the reference value for the quantity under control and its actual current value. The problems that are generally encountered in such control strategies are often associated with the absence of control over the switching spectrum of the semiconductors, which generally limits the use of such strategies to low-power devices (such as servo motor control). As for the second category of servo techniques, the switching angles result from modulation of the desired instantaneous value for the output voltage from the converter (in nonlinear automation the term sweep linearization is also used). Mean value voltage or current control over one switching period is thus often ensured in a manner that is totally decoupled from the modulating or linearizing action through the use of a standard linear corrector (often a proportional integral [PI] corrector). The limits for the use of PI correctors for AC control are well known: in this case, the requirement for good dynamic performance to minimize lag often leads to a marked sensitivity to noise and parametric variations. This leads us to the specific application for resonant correctors, which is discussed in this chapter. Based on resonant effects, the resonant corrector results in a control loop with an infinite gain modulus at one specific frequency, such that the effects of any nonlinearities or disturbances acting at that particular frequency are completely eliminated. The general form for this type of corrector is given by the following equation: C (s) =

N (s) D( s ) (ωo2 + s 2 )

.

[15.1]

This transmittance behaves like a resonant circuit that, at a pulsation ωo, has a very high gain. The resonant corrector is, in fact, the result of adapting standard correctors to high frequencies, making use of a double integration. For this reason, the same dynamic properties are observed around the resonant frequency as in the case of a double integral action for a sinusoidal signal at an infinitely small frequency (elimination of any lag error). A range of studies and articles have described the operating principles of this corrector, the associated design techniques, and the resultant tracking

Resonant Correctors

451

and regulation performance [HAU 99a and WUL 00]. A wide range of applications for this corrector, from synchronous motor control [DEG 00] and induction motor control [PIE 02] to the control of power in electrical grids, have subsequently been explored. In this chapter, we will consider in turn the control of alternating currents and voltages using resonant correctors through two examples of energy generation applications on a fixed-frequency power grid and in an isolated network. The power sources involved in these applications have seen considerable development since the turn of the century: variable-speed wind turbines, photovoltaic generators, small variable-speed hydraulic stations, gas microturbines derived from aircraft turbines, etc. The two cases considered here enable us to illustrate two design methods for resonant correctors: Kessler’s symmetric optimum method and the generalized floating-time method. 15.2. Current control with resonant correctors 15.2.1. Control using Kessler’s symmetric optimum 15.2.1.1. Summary The block diagram for AC control in a single-phase inductive load is given in Figure 15.1.

Figure 15.1. Current control in an inductive load

452

Power Electronic Converters

The transmittance of the associated resonant corrector is given by equation [15.2]: C (s) =

K (1 + τ1 s ) (1 + τ 2 s )

ωo2 + s 2

[15.2]

The control strategy proposed here is an extension to a resonant corrector of Kessler’s symmetric optimum method, widely used in current control loops (switched-mode power supplies, motor control, etc.) [KES 58] and [BER 95]. This method of control is well known for its robustness. It aims to obtain an open loop system function of the form: ω 2 ( 2 s + ωc ) 1 H BO ( s ) = c , ωc = , 2 τ 2 s ( s + 2 ωc ) Σ

[15.3]

where τΣ represents the smallest time constant of the system, the one that often limits stability. This choice is motivated by the desire to level out the transfer function at the origin when operating in closed-loop mode. The design therefore involves introduction of a pole with:

( s + ωc ) ( s 2 + ωc s + ωc2 ) as its characteristic polynomial (a Butterworth polynomial). This method is also particularly well suited to systems that involve two very distinct time constants. This is the case in current control loops, where the frequency domains associated with the time constants of an electrical load and with the commutation of the power converter are well separated. Thus, if we consider an RL (resistance and inductance) type of load, the transfer function for a converter/load system can often be written in the following form: F (s) =

i G = , u (1 + τ e s )(1 + τ s s )

[15.4]

where τe represents the electrical time constant of the load, τs the mean delay introduced by the control loop and the switching frequency of the converter [BER 95], and G is the static gain of the system.

Resonant Correctors

453

The choice is made to compensate for the dominant pole (the slow pole, which limits the speed at which the current can change within the load): we therefore set τ1= τe in equation [15.2]. By combining [15.2] and [15.4], the transfer function for the open loop system HOL therefore becomes: H OL ( s ) = C ( s ) F ( s ) =

K G (1 + τ 2 s ) (ωo2 + s 2 )(1 + τ s s )

[15.5]

This expression can be compared to the expression for the open-loop transfer function required to produce the symmetric optimum of [15.3]: the term (ωo2 + s 2 ) is substituted for the double integration and the smallest time constant of the system corresponds to that introduced by the converter (τΣ = τs). By equating [15.5] to [15.3], this control method [BER 95] leads to the following choices for K and τ2: τ2 = 4τs, K =

1 8 G τ s2

.

[15.6]

It is worth noting that the choice of K, τ1, and τ2 is independent of the pulsation ωo (which simplifies the real-time implementation of the corrector) and that the proposed control strategy can take account of the dynamic model of the converter and the limits on stability that this often introduces. 15.2.1.2. Performance Our aim is to control a sinusoidal current at 50 Hz (ωo = 314 rad/s) in a load whose parameters are R = 50 Ω and L = 0.2 H. The load RL is supplied by a single-phase voltage inverter (“H”-bridge), switching at 1500 Hz. The voltage E on the DC supply bus is 150 V. In order to evaluate its regulation performance, we subject the system to a disturbance vp which is intended to represent EMFs. These disturbance quantities are often encountered in a wide range of systems in electrical engineering which have the characteristic of always being at the current controlled frequency, and of amplitudes close to the controlled voltages.

454

Power Electronic Converters

Thus, the regulated behavior for an inductive load is analyzed by considering a disturbance signal vp chosen such that its frequency is equal to that of the reference current ωo and its maximum amplitude is fixed at a value equal to that of the voltage at the terminals of the load. The response of the system is shown in terms of its tracking and regulation in Figure 15.2. The results obtained, confirm the good performance of the corrector in terms of its speed and robustness: the response time in tracking the reference is approximately one quarter of a period (Figure 15.2a) and the effect of the disturbance is barely visible (Figure 15.2b).

→ iref = 0 0 ≤ t < 0.01  0.01 ≤ t < 0.01 → iref = sin( ωo t) Figure 15.2a. Performance of resonant correctors, response and tracking

The tracking transfer function Hp(s) and regulation transfer function Hr(s) associated with this control strategy are given by: H p (s) =

H BO ( s ) i(s) = iref ( s ) 1 + H BO ( s )

Resonant Correctors

H r ( s) =

i( s) F (s) = v p ( s ) 1 + H BO ( s )

455

[15.7]

and their associated Bode diagrams are shown in Figure 15.3.

→ vp = 0 0 ≤ t < 0.07   π 0.07 ≤ t < 0.1 → v p = 150 sin( ωo t − )  4 Figure 15.2b. Performance of resonant correctors, regulation response

By studying the gain curves and phase curve for Hp(s), we can give a priori justification for the quality of the control loop in the corrector: the gain at the resonance pulsation is indeed unitary and the phase shift is zero. Similarly, the gain curve for Hr(s) shows significant attenuation at around ωo : disturbances near this pulsation value are effectively rejected.

456

Power Electronic Converters

10

Gain (dB)

0

-10

-20

a) -30

ωo

Phase (deg)

45 0 -45 -90 -135 -180 1 10

10

2

ωo

10

3

10

4

ω (rad/sec) 0

Gain (dB)

-50 -100 -150 -200

b)

-250

ωo

Phase (deg)

135 90 45 0 -45 -90 10

1

10

2

ωo

10

3

10

4

ω (rad/sec)

Figure 15.3. Performance of resonant correctors, Bode diagrams, load: R = 50 Ω, L = 0.2 H, τe = 4 ms, supply: single-phase voltage inverter, E = 150 V; Fs = 1500 Hz, τs = 33 ms: a) tracking transfer function; b) regulation transfer function

Resonant Correctors

457

15.2.2. Application to power control: example of a wind turbine 15.2.2.1. System under study In this section, we will consider a specific application area of resonant correctors: that of power control of a voltage inverter supplying an electrical grid with the energy generated from a variable-speed wind turbine (Figure 15.4). PWM rectifier

PWM inverter

L 50 Hz A.C. grid

MS

Θm ωm Power control

u i ic

Figure 15.4. High-level schematic diagram of a wind turbine

The system consists of a generator (synchronous or induction-based) whose torque and speed are controlled by a three-phase voltage inverter. On the electrical grid side, a system consisting of a voltage inverter, an inductance, and a transformer delivers active and reactive powers, the control of which is the subject of this study. This involves control of the current on the grid side of the transformer, and therefore requires a transformer model that is adapted for synthesis of correction signals. 15.2.2.2. Dynamic modeling of the inverter–transformer system The transformer model used in the control strategy is given in Figure 15.5. This is a standard model with distributed leakage inductances whose parameters have been precisely identified [ESS 00].

458

Power Electronic Converters

i abc 1

r1

l1f

l 2f

r2

iabc 2

iabc 1 + iabc 2 Rms v abc 1

v abc 2

Lms

Figure 15.5. Equivalent circuit for the grid transformer

Here, r1 and r2 are the resistances of the primary and secondary windings, Rms and Lms are the magnetization resistances and inductances respectively, l1f and l2f are the distributed leakage inductances in the primary and secondary of the transformer. This diagram can be used to deduce a model whose voltage equations can be expressed in a fixed two-phase reference frame (α, β) in the form: di vα 1 = R1eq iα 1 + σ L1 α 1 + eα 2 dt , diβ 1 vβ 1 = R1eq iβ 1 + σ L1 + eβ 2 dt

[15.8]

where R1eq and σ L1 represent the total resistance and inductance associated with the primary (specified in the appendix to this chapter, section 15.5), and eα 2 and eβ 2 are coupling terms between the primary and the secondary, equivalent to EMFs and expressed as: eα 2 = a1 iα 2 + a2 vα 2

eβ 2 = a1 iβ 2 + a2 vβ 2 ,

Resonant Correctors

459

where: a1 =

( Rms L2 − R2 Lms ) L2

a2 =

Lms . L2

[15.9]

It is fairly well known that these coupling quantities require specific treatment during current control. This problem is often solved using a multivariable control approach or, at least by a “dynamic decoupling” type of operation. Moreover, if single-variable control were considered, the frequency of the coupling terms (50 Hz grid) would make it difficult to use a PI type of corrector. This analysis explains why specific transformations (Park transformations) are used, giving an equation for the model in a rotating reference frame. In this way, the quantities of the model become constant in the steady state, making them more easily controllable by a standard corrector. In what follows, we will show that the robustness properties provided to the current loop by the resonant corrector make it possible to avoid the use of a rotating reference frame or calculation of decoupling terms. 15.2.2.3. Control strategy The control strategy aims to control the active power (P) and the reactive power (Q) delivered to the grid (Figure 15.6).

Figure 15.6. Control strategy for active and reactive powers

460

Power Electronic Converters

The voltage is measured at the secondary coil of the transformer in order to avoid the problems associated with measurement on the high voltage (20 kV) side. The transformation from reference powers (Pref, Qref) to reference currents (iα1ref, iβ1ref) takes place in the following manner: iα 1 ref =

Pref vα 1 − Qref vβ 1

iβ 1 ref =

vα 12 + vβ 12 Pref vβ 1 + Qref vα 1 vα 12 + vβ 12

.

[15.10]

According to [15.8], the relationships linking the currents and voltages in the (α, β) reference frame are: iα 1 = F ( s) (vα 1 − eα 2 ) iβ 1 = F ( s) (vβ 1 − eβ 2 ),

[15.11]

where F(s) has exactly the same form as in [15.4], in other words: F (s) =

G , (1 + τ e s )(1 + τ s s )

[15.12]

where: τe =

σ L1 R1eq

and τs represents the latency introduced by the converter. The control method proposed in section 15.2.1 is therefore strictly applicable.

Resonant Correctors

461

15.2.2.4. Results 15.2.2.4.1. Power control performance The control structure was the same as that shown in Figure 15.6 and it was implemented using rapid prototyping techniques (DSP (Digital Signal Processor) board, with sampling period Ts = 0.4 ms). The transformer parameters are given in the appendix to this chapter. The switching frequency of the inverter is 2 kHz ( τ s = 0.25 ms ) and interphase inductances (Li = 3 mH) are introduced between the converter and the transformer in order to reduce current ripple. The effect of these inductances is taken into account in the control of the corrector by considering an equivalent inductance (σ L1 + Li ) . The test consisted of an active power ramp (from 0 to 1 kW) and a reference reactive power of zero. The experimental results are shown in Figure 15.7.

Figure 15.7. Performance of resonant correctors: experimental results showing (a) the active power; and (b) reactive power

The reactive power diverges significantly from its commanded value (Figure 15.7b). This discrepancy is due to the different phase shifts between the measured voltages/currents (subscripted as mes) and their reference values (subscripted as ref). These discrepancies, shown in Figure 15.8a, are induced by the various delays in the control chain (measurement, sampling, and PWM). These various phase shifts can be expressed as: −

δ1

i ( s ) = iref ( s ) e ω



s δ2

v mes ( s ) = v( s ) e ω

. s

[15.13]

462

Power Electronic Converters

The phase shift δ1 between the reference current iref and its actual value i is given by: δ1 = ω (τ s +

Te ) = 0.14 rad . 2

[15.14]

The phase shift δ2 between the reference voltage vref and its actual value v is given by: δ 2 = ω (Tsensor +

Te ) = 0.08 rad , 2

[15.15]

where Tsensor represents the phase delay introduced by the voltage sensor and the measurement filters (0.05 ms). To take into account these various delays, the reference voltage is corrected using the following formula: vcor = vmes e(δ1 +δ 2 )s .

[15.16]

The effect of this correction is shown in Figure 15.8b, where although the transient effect of variation in the active power is still visible (coupled power control loops), the mean value of the reactive voltage is virtually zero.

Figure 15.8. Correction of reactive power: (a) principle and; (b) results

Resonant Correctors

463

15.2.2.4.2. Current control performance We have introduced the aim of using a resonant corrector in general terms: its ease of implementation (absence of a rotating reference frame), absence of tracking error, robustness, etc. In Figure 15.9a, the current iα1 is shown during the active power ramp: In closed loop operation the corrector enables the command signal to be followed to the required performance level. In order to evaluate its performance in response to a disturbance from the grid, the effect of a frequency excursion of 0.2 Hz is evaluated. The tracking of the active-power reference signal is exactly the same 50 Hz and 49.8 Hz. Only the reactive power is slightly affected by this variation (Figure 15.9b) since its mean value is no longer zero.

Figure 15.9. Current control: (a) currentand; (b) active and reactive powers

15.3. Voltage control strategy 15.3.1. Introduction The major problem associated with wind farms (Figure 15.10) is that they do not in general contribute any anciliary services to the grid (voltage control, frequency control, ability to work in isolated mode). The fact that they do not participate in these ancillary services of course makes them very suitable for behaving as passive generators from an electrical point of view. Voltage and frequency control is then relegated to standard alternator systems. Their penetration level within the grid must therefore be limited in

464

Power Electronic Converters

order that the stability of the network can be guaranteed within normal conditions [ACK 07 and ERI 05]. Certain research studies have shown that it is possible to introduce a connection (one that exists naturally in the case of standard alternators) between the variations required by consumers and the operating frequency of the power electronics interfaces [DAV 07] and [LEC 04b]. By pursuing this analogy with the operation of a standard alternator, we might hope in the future to enable a higher level of penetration of this type of source within the grid and also to enable them to operate in isolated mode. Isolated operation requires that this type of source must be able to operate as a voltage source.

Converter 1

Converter 2

Lf Cf filter

Lr

MS

Grid or isolated load

Figure 15.10. Variable-speed wind turbine generator

In this section, we will discuss voltage control of a structure consisting of a PWM power converter and a LCL (inductance, capacitor and inductance) filter as shown in Figure 15.10. The filter enables the majority of the harmonics present in the output voltage of the converter to be eliminated in order to obtain a near-sinusoidal output voltage. With the help of resonant correctors, the composite voltages at the capacitor terminals coupled in a triangular configuration, will be regulated, and as a result there is no need to make assumptions about the load being balanced. This type of corrector is well suited to control of alternating quantities [GUI 07, HAU 99a, LEC 03, PIE 05, and WUL 00] and offers very good performance in terms of tracking, regulation, and above all robustness. Good tracking is achieved through the rapid control-response speed (in other words when the reference values change). Regulation refers to the ability to maintain the regulated quantity at its reference value in spite of disturbances.

Resonant Correctors

465

Robustness refers to the ability of the corrector to maintain its performance in spite of changes or variations in the parameters describing the process (the model used for the process, from which we obtained the parameters for the correctors, may vary with time but the corrector is ultimately not very sensitive to these changes). The resonant correctors are set up for a nominal frequency of 50 Hz. Their intrinsic qualities make it possible to obtain a near-perfect three-phase sinusoidal voltage source. The voltage quantities to be regulated are their effective value, frequency, and phase. This control method enables a given active power to be injected, the voltage to be controlled at the point where it is connected to the grid, and both the voltage and frequency to be imposed in the case of an isolated grid. 15.3.2. Principle of power control The output filter for the converter in Figure 15.10 can be represented by the circuit diagram in Figure 15.11.

um1

1

Lf

if1

2

Lf

if2

um2

Lf

if3

Cf

Cf

Cf uc2

ich1

Lr

uc1 ich2

Lr

ich3

Lr

3

a

b ubc c

Figure 15.11. LCL filter

The symbols in Figure 15.11 are defined as follows: – um1 and um2 are modulated voltages at the output of the inverter [V]; – uc1 and uc2 are voltages at the capacitor terminals [V]; – i f 1, i f 2, and i f 3 are currents in the choke Lf [A];

uac

466

Power Electronic Converters

– ich1, ich2, and ich3 are currents in the load [A]; – Lf is the inductance of the choke on the converter side [H]; – Cf is the capacitance of the capacitor [F]; – Rf is the resistance of the résistance choke on the converter side [Ω]; – uac and ubc, are composite voltages in the grid or at the load terminals [V]; – Lr is the added inductance, transformer inductance, or grid inductance [H]. This LCL filter can be modeled using an equivalent single-phase circuit (Figure 15.12). 1

Lf

if1

ich1

a

vc1

3Cf

vm1

Lr

va

Figure 15.12. Single-phase equivalent LCL circuit

If the voltage at the capacitor terminals is perfectly regulated, the converter-LC filter combination can be represented as a voltage source as shown in the diagram in Figure 15.13. Figure 15.13 can be used to deduce the PQ diagram in Figure 15.14. Lr

Converter + Lf Cf filter = voltage source

ich1

vc1

Figure 15.13. Power converter structure and LCL filter

va

Resonant Correctors

467

P

Psg Vc LrωIch

δ ϕ

Vsg

Q Qsg

Ich1

Figure 15.14. PQ diagram for the system

The symbols in Figure 15.14 are defined as follows: – Vsg: effective value of the single-phase voltage νa [V]; – Vc: effective value of the single-phase voltage νc1 [V]; – Ich: effective value of the current ich1 [A]; – δ: phase shift between voltages νa and νc1 [rad]; – ω: voltage pulsation [rad.s-1]. The active power (Psg) and reactive power (Qsg) flowing through the inductance can be written as: Psg =

Qsg =

3VcVsg Lr ω 3Vsg

Lr ω

× sin δ

× (Vc cos δ − Vsg ) .

[15.17]

[15.18]

Control of the active delivered power (fixed at a reference value Psg-ref) by controlling the angle “δ” involves controlling the phase shift between the voltages νc and νsg (Figure 15.14). The output voltage (νsg) is maintained at its nominal value through control of the equivalent voltage of the EMF (νc), corrected according to the reactive power delivered by the converter. The independence of these two control elements (active and reactive power) is based on an approximation: the assumption that the angle “δ”

468

Power Electronic Converters

remains small. The validity of this working hypothesis lies in the size of the constant of proportionality linking Psg to δ, in other words the fact that a significant variation in power can occur in response to small variations in angle. This is satisfied when the ratio: 3VcVsg Lr ω

in equation [15.17] is very large compared to the power Psg to be injected. Thus if δ is small, we can make the small angle approximation: sin δ ≈ δ

and

cos δ ≈ 1

By substituting these approximations into equations [15.17] and [15.18], the following expressions are obtained: Psg =

Qsg =

3VcVsg Lr ω 3Vsg Lr ω

×δ

× (Vc − Vsg ) .

[15.19]

[15.20]

The values for the injected power and the voltage at the crossing point depend, respectively, on the angle δ and the effective value Vc that is imposed at the capacitor terminals. 15.3.3. Voltage control at the capacitor terminals The aim of the control system is to control the generator in such a way that it can operate as a voltage source in order to ensure, when operating in an isolated network, that the loads are correctly supplied. For this, the voltages νc1 and νc2 at the capacitor terminals must be regulated. The reference voltages are obtained: – when connected to the grid, from the reference value for the active power to be supplied, the required voltage at the point the system is connected to the grid, and the grid frequency;

Resonant Correctors

469

– in an isolated network, from the specified voltage and frequency to be imposed. Figure 15.15 shows a block diagram representing the main principles of control of the grid interface. C (s) is the transmittance of the resonant corrector. The transfer function connecting the voltages at the capacitor terminals to the voltages at the converter output is as follows (Figure 15.12): u c1 ou 2 =

1 1 + 3R f C f s + 3L f C f s 2

u m1ou 2

[15.21]

Figure 15.15. Principles of control at the grid interface

Since the voltages at the capacitor terminals are alternating, resonant correctors are chosen. These will enable the composite voltages at the terminals of the capacitors of the Lf Cf filter, coupled in a triangular configuration, to be controlled. This control strategy does not therefore make any assumption about the load being balanced. For a second-order transfer function, the corrector has the following form: C ( s) =

C0 + C1s + C2 s 2 + C3 s3 ( D0 + D1s )( s 2 + ω 2p )

[15.22]

470

Power Electronic Converters

– ωp: pulsation of the quantities under control [rad.s-1]; – C0, C1, C2, C3, D0, D1: corrector parameters. The control loop can be expressed in the form of the block diagram shown in Figure 15.16.

Figure 15.16. Control loop

The method for determining the coefficients of the resonant corrector involves imposing the location of the poles of the corrector transfer function by choosing the real part of that function. This method is known as the generalized floating-time method [LEC 04]. The coefficients are determined by treating the denominator of the closed-loop system transfer function as a polynomial ∆p(s). The open-loop transfer function is: FTbo ( s ) =

C0 + C1s + C2 s 2 + C3 s3 2

1

2

( D0 + D1s )( s + ω p ) 1 + 3R f C f s + 3L f C f s 2

.

[15.23]

Therefore, the closed-loop transfer function can be written: FTbf ( s ) =

C0 + C1s + C2 s 2 + C3 s3 C0 + C1s + C2 s 2 + C3 s3 + ( D0 + D1s )( s 2 + ω p 2 )(1 + 3R f C f s + 3L f C f s 2 )

.

[15.24] The characteristic fifth-order polynomial of the closed-loop transfer function is therefore: ∆ ( s ) = C0 + C1s + C2 s 2 + C3 s3 + ( D0 + D1s)( s 2 + ω p 2 )(1 + 3R f C f s + 3L f C f s 2 ) .

[15.25]

Resonant Correctors

471

The resonant correctors are calculated by solving the BEZOUT equation, setting this polynomial for the closed-loop transfer function equal to a polynomial of the same degree in canonical form, thus imposing the required dynamic response in closed-loop mode: ∆ P( s ) = ( s + P + jω p )( s + P − jω p )( s + P )( s + P + jωn )( s + P − jωn ) ,

[15.26]

where P represents the real part of the (negative) poles and ωn, the imaginary part of the complex root of the transmittance of the Rf Lf Cf filter. By expanding and rearranging the polynomials in [15.25] and [15.26], we obtain:

(

) (

)

∆ ( s ) = C0 + D0ωP2 + C1 + D1ω P2 + D0ωP2 3R f C f s

( ) + ( C3 + D1 + D0 3R f C f + D1ω p 2 3L f C f ) s3 + ( D13R f C f + D0 3L f C f ) s 4 + D13L f C f s 5 + C2 + D0 + D1ω P2 3R f C f + D0ωP2 3L f C f s 2

[15.27]

(

)

∆ P( s ) = P5 + P3 (ω 2p + ωn2 ) + Pω 2pωn2 + 5P 4 + 3P 2 (ω 2p + ωn2 ) + ω 2pωn2 s

(

)

(

)

+ 10 P3 + 3P(ω 2p + ωn2 ) s 2 + ωn2 + ω 2p + 10 P 2 s3 + 5P s 4 + s 5 .

[15.28]

By matching coefficients between equations [15.27] and [15.28], we obtain: DO = (5P − D1 =

Rf Lf

) 3L f C f

1 3L f C f

[15.29] [15.30]

C0 = P5 + P3 (ω 2p + ωn2 ) + Pω 2pωn2 − ω 2p D0

[15.31]

C1 = 5P 4 + 3P 2 (ω 2p + ωn2 ) + ω 2pωn2 − 3R f C f ω 2p D0 − ω 2p D1

[15.32]

C2 = 10 P3 + 3P (ω 2p + ωn2 ) − DO − 3L f C f ω 2p D0 − 3R f C f ω 2p D1

[15.33]

C3 = 10 P 2 + (ω 2p + ωn2 ) − D1 − 3L f C f ω 2p D1 − 3R f C f ω 2p D0 .

[15.34]

472

Power Electronic Converters

P is the value of the real part of the poles, which determines the margin of stability. This method of calculating the parameters of the resonant corrector enables the poles of the transfer function to be positioned as required. Consequently, it is possible to avoid areas of instability and to select a margin of stability that is chosen in advance. P must be selected to meet the three limits shown in Figure 15.17 [HAU 97], the noise and system limit, the damping limit, and the limit representing the margin of stability.

Figure 15.17. Boundaries of the region where the corrector poles must lie

15.3.4. Determination of reference voltages The composite reference voltages are determined using the expressions for the single-phase reference voltages [15.35] to [15.37]: vc1− ref = Vref

2 sin(2π f sg − ref t + δ ref )

vc 2− ref = Vref

2 sin(2π f sg − ref t −

2π + δ ref ) 3

[15.36]

vc3− ref = Vref

2 sin(2π f sg − ref t −

4π + δ ref ) , 3

[15.37]

[15.35]

Resonant Correctors

473

where: uc1−ref = vc1− ref − vc3− ref

[15.38]

uc 2− ref = vc 2− ref − vc3− ref

[15.39]

We therefore obtain: uc1− ref = U ref sin(ωsg − ref t −

uc 2− ref = U ref sin(ωsg − ref t −

π 6

π 2

+ δ ref )

[15.40]

+ δ ref ) .

[15.41]

The parameters to be determined are the effective value Vref and the transmission angle δref. The frequency fsg-ref is that of the grid. The reference value Vref enables the voltage to be controlled at the point of connection of the generation system to the network, while the value δref can be used to control the active power injected into the grid by the generator system. 15.3.5. Power control The delivered active power Psg can be varied by temporarily assigning a different pulsation to the vector Vc in order to adjust the angle “δ” to the required value (the angle to be obtained is calculated from the reference power Psg-ref). We can write [BOR 01, COE 02, DEB 07, and TUL 00]: dδ = ωsg − ωnetwork = 2π ( f sg − f network ) = 2π∆ f dt

[15.42]

To establish a relationship between the frequency variation ∆f and the variation in active power ∆Psg, a proportionality relationship between the frequency and active power is introduced that is similar to the droop control of classical power generation units (Figure 15.18).

474

Power Electronic Converters

fsg

fsg1

k

fréseau

Psg1

Psg

Psg-ref

Figure 15.18. Proportional relationship between frequency and active power

From Figure 15.18 we find: ∆ f = k ∆ Psg = k ( Psg − ref − Psg )

[15.43]

and by substituting [15.43] into [15.42] we obtain: dδ = 2π ( f sg − f réseau ) = 2 π k ( Psg − ref − Psg ) dt

[15.44]

Using equation [15.44], we can determine the block diagram for control of the active power supplied by the generation system (Figure 15.19).

Psg-ref

+-

1 1+τ s

k

2π s

Filter

Psg-mes Figure 15.19. Block diagram for active power regulation

δref

Resonant Correctors

475

REMARK.– A filter was placed in the control loop in order to filter the alternating component at 100 Hz obtained in the measurement of the instantaneous active power (non-zero fluctuating power in the case of an unbalanced load) since the control loop requires a mean active power that it can compare to the reference value. By taking δref and δreal to be the same as each other, which is equivalent to ignoring the response time of the resonant correctors and their static error, we can use [15.19] and assume that Vc ≈ Vsg, in order to find the following expression for Psg-mes: Psg − mes =

2 3Vsg

Lr ω0

× δ ref ,

[15.45]

where ω0: nominal grid pulsation [rad.s-1]. In light of equation [15.45], the block diagram of Figure 15.19 becomes (Figure 15.20):

Psg-ref Psg-mes

1 1+τ s

+-

k

2π s

δref

3Vsg2 Lr ω Figure 15.20. Active power control

The closed-loop transfer function for the system, if: Psg Psg −ref

1

= 1+

Lr ω

2 6Vsg πk

s+

Lr ωτ

2 6Vsg πk

s2

.

[15.46]

476

Power Electronic Converters

We must then equate this to the following conventional form, where ξ represents the damping coefficient and ωn the natural pulsation frequency of the system: 1

H (s) = 1+



ωn

s+

1

ωn2

s2

.

[15.47]

By using the following equation, which connects the response time tr to the natural pulsation and the damping coefficient: tr =

3

ξωn

[15.48]

,

we can express the coefficient k and the time constant τ as a function of Lr, V, ξ, and tr as follows: k=

τ=

Lr ω 2 2 ξ tr 4π Vsg

tr 6

[15.49]

COMMENT.– The value of τ that is obtained must be greater than 10 ms in order to be effective at filtering the alternating component at 100 Hz. 15.3.6. Voltage control Within the small angle approximation, there is a simple relationship between the effective voltage Vc, the voltage Vsg, and the reactive power delivered by the converter crossing the inductance Lr, which can be determined from equation [15.20]: Vc = Vsg +

Lr ω × Qsg 3Vsg

[15.50]

At first glance, we can deduce a simple control law from this, which involves controlling the effective value Vref, [15.35]–[15.37], using the

Resonant Correctors

477

reactive power delivered with a fixed voltage command Vsg referred to as Vsg-nom, the nominal voltage required at the connection point of the generation system: Vref = Vsg − nom +

Lr ω × Qsg − mes . 3Vsg − nom

[15.51]

15.3.7. Simulations In this part, we will illustrate the behavior of the interface when connected to a grid, in an isolated network with balanced loads and in an isolated network with unbalanced loads, with the help of numerical simulations performed using the Matlab-Simulink software. In the first simulation, for a grid of infinite power and in an isolated network with a balanced load, we will show the reference power for the generation system (Psg-ref), the power supplied by the generation system (Psg), the frequency for the voltages generated by the system (fsg), the angle delta (δ), the output currents from the converter (if1 and if2), the output currents from the filter (ich1 and ich2), the voltages at the capacitor terminals (uc1 and uc2), and the reference voltages (uc1-ref and uc2-ref). In the second simulation, for an isolated network with an unbalanced load, we will show the power supplied by the generation system (Psg), the output currents from the converter (if1 and if2), the output currents from the filter (ich1 and ich2), the voltages at the capacitor terminals (uc1 and uc2), and the reference voltages (uc1-ref and uc2-ref). 15.3.7.1. For an infinite power grid and in an isolated network with a balanced load The simulation takes place over the course of 60 s: from 0 to 40 s the grid is present, and then from 40 s to 60 s the system operates as an isolated network with a balanced 200 kW load–50 kVAR. The reference power Psg-ref is regulated at 600 kW from 0 to 20 s and then at 500 kW from 20 s to 60 s (Figure 15.21). The reference frequency for the voltages fsg-ref is regulated at 50 Hz. The power Psg supplied by the generation system follows this

478

Power Electronic Converters

reference when connected to the grid and then at t = 40 s when operating as an isolated network, the load determines the power supplied by the system as shown in Figure 15.22. The frequency fsg of the generation system varies around 50 kHz when connected to the grid, in response to changes in the reference power Psg-ref (Figure 15.23).

Figure 15.21. Reference power for the generation system Psg-ref

Figure 15.22. Active power supplied by the generation system Psg

Resonant Correctors

479

Figure 15.23. Frequencies

Conversely, when operating in isolated mode, it is no longer stabilized at 50 Hz since Psg-ref > Psg (property of the droop curve [6.43]) (Figure 15.23). The angle delta is small and also varies with each change in Psg-ref when connected to the grid (Figure 15.24). In the isolated network, the angle delta will drift towards infinity because it can no longer be controlled (the voltage downstream of the inductance Lr is determined by the upstream voltage).

Figure 15.24. Angle Delta

480

Power Electronic Converters

Figures 15.25, 15.26, 15.27, and 15.28 in turn show the output currents if1, if2 from the converter, the currents ich1, ich2 at the output of the Lf Cf filter, the voltages uc1, uc2 at the capacitor terminals, and the reference voltages uc1-ref, uc2 –ref. The voltages uc1, uc2 at the capacitor terminals follow their reference values uc1-ref and uc2 –ref. The system can supply power to loads when operating in an isolated network. This mode of operation is only possible if the generation system incorporates a storage element [LEC 04a].

Figure 15.25. Currents if1, if2

Figure 15.26. Currents ich1, ich2

Resonant Correctors

481

Figure 15.27. Voltages uc1, uc2

Figure 15.28. Voltages uc1-ref, uc2-ref

15.3.7.2. Isolated network with non-balanced 200 kW load The simulations are identical to the previous case in the isolated network with a balanced load (time interval from 40s to 60s) in terms of Psg-ref (Figures 15.21 and 15.29), the frequency fsg (Figure 15.23), and the angle delta (Figure 15.24). On the contrary, here the output currents if1, if2 from the converter (Figure 15.30) and the output currents ich1, ich2 from the LC filter (Figure 15.31) are not balanced. However, the voltages at the capacitor terminals (Figure 15.32) still follow their reference values (Figure 15.33) since it is the composite voltages at the capacitor terminals that are controlled and the control strategy does not make any assumptions about the load being balanced.

482

Power Electronic Converters

Figure 15.29. Active power Psg supplied by the generation system

Figure 15.30. Currents if1, if2

Figure 15.31. Currents ich1, ich2

Resonant Correctors

483

Figure 15.32. Voltages uc1, uc2

Figure 15.33. Voltages uc1-ref, uc2-ref

15.4. Conclusion In this chapter, we have discussed two methods of designing resonant correctors for current and voltage control in energy generation applications at fixed frequency. The performance of these methods when applied to wind turbines have been illustrated through experiments and numerical simulations.

484

Power Electronic Converters

15.5. Appendix: transformer parameters – Primary winding: r1 = 2.45 mΩ l1f = 0.012 mH

– Secondary winding: r2 = 2.45 mΩ l2f = 0.012 mH

– Magnetizing impedance: Rms = 1.69 Ω Lms = 21.8 mH

– Equivalent circuits: R1 = r1 + Rms

L1 = l1 f + Lms L2 = l2 f + Lms

R2 = r2 + Rms R1eq =

( R1L2 − Rms Lms ) L2

σ = 1−

Lms 2 L1 L2

15.6. Bibliography [ACK 07] ACKERMAN T., ABBAD J.R., DUDURYCH I.M., ERLICH I., HOLTTINEN H., RUNGE KRISTOFFERSEN J., SORENSEN P.E., “European balancing Act”, IEEE Power & Energy Magazine, November/December 2007. [BOR 01] BORUP U., BLAABJERG F., ENJETI P.N., “Sharing of nonlinear load in parallel–connected three-phase converters”, IEEE Transactions on Industry Applications, vol. 37, n° 6, p. 1817–1823, 2001. [BER 95] BERGMANN C., LOUIS J.P., Commande numérique des machines, Techniques de l’Ingénieur, Traité Génie électrique, 1995.

Resonant Correctors

485

[COE 02] COELHO E.A.A., CORTIZO P.C., GARCIA P.F.D., “Small-signal stability for parallel-connected inverters in stand alone AC supply systems”, IEEE Transactions on Industrial Applications, vol. 38, n° 2, p. 533–542, 2002. [DAV 07] DAVIGNY A., Participation aux services système de fermes d’éoliennes à vitesse variable intégrant du stockage inertiel d’énergie, Electrical engineering thesis, Lille University of Science and Technology, Villeneuve d’Asq, France, 2007. [DEB 07] DE BRABANDERE K., BOLSENS B., VAN DEN KEYBUS J., WOYTE A., DRIESEN J., BELMANS R., “A voltage and frequency droop control method for parallel inverters”, IEEE Transactions on Power Electronics, vol. 22, n° 4, p. 1107–1115, 2007. [DEG 00] DEGOBERT P., HAUTIER J.P., “Torque control of permanent magnet synchronous motors in Concordia’s reference frame with resonating currents controllers”, Proceedings of ICEM’00, Espoo, Finland, 2000. [ERI 05] BORRE ERIKSEN P., ACKERMANN T., ABILDGAARD H., SMITH P., WINTER W., GARCIA J.R., “System operation with high wind penetration”, IEEE Power & Energy Magazine, November/December 2005. [ESS 00] ESSELIN M., ROBYNS B., BERTHEREAU F., HAUTIER J.P., “Resonant controller based power control of an inverter-transformer association in a wind generator”, Electromotion, vol. 7, p. 185–190, 2000. [GUI 07] GUILLAUD X., DEGOBERT P., TEODORESCU R., “Use of resonant controller for grid-connected converters in case of large frequency fluctuations”, EPE 2007, 12th European Conference On Power Electronics and Applications, Aalborg, Denmark, 2007. [HAU 97] HAUTIER J.P., CARON J.P., Systèmes automatiques, commande des processus, t. 2, Ellipses, Paris, 1997. [HAU 99a] HAUTIER J.P., GUILLAUD X., VANDECASTEELE F., WULVERYCK M., “Contrôle de grandeurs alternatives par des correcteurs résonnants”, Revue internationale de génie électrique, vol. 2, n° 2, 1999. [HAU 99b] HAUTIER J.P., CARON J.P., Convertisseurs statique : méthodologie causale de modélisation et de commande, Technip, Paris, 1999. [KES 58] KESSLER C., Das Symmetrische Optimum, Regelungstechnik, vol. 6, p. 395-400 et 432–436, 1958. [LEC 03] LECLERCQ L., ANSEL A., ROBYNS B., “Autonomous high power variable speed wind generator system”, Proceedings of EPE 2003, Toulouse, France, 2003.

486

Power Electronic Converters

[LEC 04a] LECLERCQ L., DAVIGNY A., ANSEL A., ROBYNS B., “Grid connected or islanded operation of variable speed wind generators associated with flywheel energy storage systems”, Proceedings of the 11th International Power Electronics and Motion Control Conference, EPE-PEMC 2004, Riga, Latvia, 2004. [LEC 04b] LECLERCQ L., Apport du stockage inertiel associé à des éoliennes dans un réseau électrique en vue d’assurer des services systèmes, Electrical engineering thesis, Lille University of Science and Technology, Villeneuve d’Ascq, France, 2004. [PIE 02] PIERQUIN J., BOUSCAYROL A., DEGOBERT P., HAUTIER J.P., ROBYNS B., “Torque control of an induction machine based on resonant current controllers”, ICEM 02, Bruges, Belgium, 2002. [PIE 05] PIERQUIN J., ROBYNS B., “Variable speed wind generator network interface power control based on resonant controller”, Electromotion 2005, Lausanne, Switzerland, 27–29 2005. [TUL 00] TULADHAR A., JIN H., UNGER T., MAUCH K., “Control of parallel inverters in distributed AC power systems with consideration of line impedance effect”, IEEE Transactions on Industrial Applications, vol. 36, n° 1, p. 131–137, 2000. [WUL 00] WULVERYCK M., Contrôle de courants alternatifs par correcteur résonnant multifréquentiel, Electrical engineering thesis, USTL, Lille, France, 2000.

Chapter 16

Current Control Strategies for Multicell Converters

16.1. Introduction Over the last ten years, power electronics has seen tremendous growth due to the ever-increasing demand for energy in modern society. Its applications cover a wide range of very different fields, over a large range of different powers (electrical energies). The energy conversion structures, the main players in this growth, must therefore be able to handle even greater powers, with constantly increasing efficiencies. The increase in the power levels handled by these structures is enabled by increasing the supply voltage, increasing the current they handle, or both. Nevertheless, from the point of view of efficiency, an increase in voltage is often preferred. This increased voltage and/or current imposes significant constraints on the power switches that lie at the heart of these conversion structures. In parallel with these requirements, the evolution of power semiconductors has followed a trend in the same direction. Over the years, new components have appeared with ever-increasing capabilities and performance capabilities. Conversely, while it is reasonably feasible to Chapter written by Guillaume GATEAU and Thierry MEYNARD.

488

Power Electronic Converters

increase the current capacities by combining several chips or components (a greater surface area of silicon), voltage increases, on the contrary, require an increase in thickness of the silicon rather than in its surface area, which inevitably results in a loss of performance for these new components. Thus, for a given type of component, the static and dynamic properties will not be as good for components with a higher operating voltage. The main aim of novel conversion architectures is thus to handle a given level of power throughput, by distributing the voltage and/or current requirements over several components of reduced size but which have better intrinsic properties. Through the use of higher performance components, the hope is thus to achieve an increase in the global efficiency of the conversion system. Such conversion architectures thus use several semiconductor components arranged in series or in parallel, which, in addition to distributing the voltage load between multiple components, offer increased performance. In particular, the waveforms of the output signals become multilevel and the available frequencies increase, which limits the requirements on the output filters. With the increased number of components, the complexity of the control strategy also increases. However, more components also implies more degrees of freedom. Thus, even though this complexity may initially appear to be a problem, detailed analysis of the new possibilities it offers rapidly turns this into an advantage. In this chapter, we will first make some comments on multilevel conversion structures and then we will concentrate on our main aim, the control of multicell converters. We will therefore review the various strategies proposed in past years by presenting the advantages and disadvantages of each proposed solution. 16.2. Multilevel conversion topology Before specifically discussing current control in multicell converters, we should first briefly recall the various topologies that can be used to obtain multilevel waveforms. Without seeking to achieve an exhaustive overview

Multicell Converters

489

(since the structural aspect of the conversion process is not the main topic of this chapter), we will describe the main structures used in the application area known as “medium voltage drives” for speed controllers. In this application area, the specifications involve voltages of the order of tens of kilovolts at powers that can be as high as 10 MW. With contemporary semiconductors (as of 2010), it is not possible to work at such voltage level, without using multiple component’s association working in tandem. After briefly recalling the main types of available topologies, we will discuss in more depth the advantages and disadvantages of multicell structures that use flying capacitors. 16.2.1. Main types of multilevel structure 16.2.1.1. Association of Converter (example: H-bridge) One of the first ways of constructing a multilevel waveform is simply to use a combination of several simple converters [COR 02, HAM 97, HIL 99, OSM 99, TEO 02, and TOL 99]. Here, several single-phase converters are used in series, in a full-bridge configuration as shown in Figure 16.1.

Figure 16.1. Cascade converter

490

Power Electronic Converters

The output voltage is given by the sum of the output voltages of each associated converter. In our example, each elementary converter can give three output voltage levels (E, 0, −E). The combined structure therefore gives access to five voltage levels (2E, E, 0, −E, −2E), bearing in mind the redundancies inherent in the design. Although this topology operates in a simple and robust manner, it suffers from a major disadvantage: the requirement that the voltage sources VDC must be mutually isolated, requiring the use of a relatively expensive transformer. Despite these drawbacks, industrial solutions exist because of the optimal design of the transformer [HIL 99]. 16.2.1.2. Combination of input sources: Neutral point clamped converter An alternative is to directly modify the topology of the conversion circuit instead of considering a combination of elementary converters. The first possibility we will consider involves splitting the DC bus and also introducing a switching component that is able to deliver the required voltage level (Figure 16.2).

Figure 16.2. DC bus division

A structure widely used in industrial applications that matches this description is the neutral point clamped (NPC) converter [NAB 81] along with even more sophisticated versions, such as Active NPC (ANPC) [BRU 01 and BRU 05] or five-level ANPC [MEI 06]. This type of converter, shown in Figure 16.3 in its most commonly used form, can be used to obtain three output voltage levels and reduces the voltage constraint on each individual power switch by a factor of two.

Multicell Converters

491

Figure 16.3. NPC converter

Two diodes are used to ensure that the voltage constraints on each power switch are respected. Recent versions of this topology, referred to as ANPC, make it possible to optimize the distribution of losses throughout the circuit by prioritizing certain sequences over others [BRU 01] and [BRU 05]. It should be noted that this topology is difficult to generalize, partly due to the complexity of the clamp circuit in the case of more than three levels, and partly due to problems in balancing the primary sources in the presence of more than two sources. 16.2.1.3. Combination of floating sources: multicell converters or interleaved cells Rather than splitting up the input voltage as in the case of the NPC converter, the multicell topology makes use of floating sources (sources not ground-referenced) in order to achieve the required range of voltage levels. The only condition is that the current flowing throw the floating voltage sources must be an alternating current, thus resulting in a stable mean value over one switching period. The block diagram for this structure is shown in Figure 16.4. Figure 16.5 shows a multicell three-level converter. The floating source voltage is half the DC bus voltage. The voltage load across each switch is thus reduced by a factor of two and is equal to half the DC bus voltage.

492

Power Electronic Converters

Figure 16.4. Use of floating sources

Figure 16.5. Three-level multicell converter

One of the advantages of this topology is that the intermediate level can be achieved in two different ways. This property implies that if appropriate commands are sent then the apparent output frequency can be doubled. This property is very important since it makes it possible to significantly reduce the size of the filtering elements required at the output of the converter. Another advantage is the simple and direct way in which this topology can be generalized. It is thus very easy to imagine upgrading to four, five, or more levels, simply by adding more switching cells in series. 16.2.2. Advantages and disadvantages of a multicell structure Our brief review of standard multilevel topologies has highlighted a number of properties of each type of converter. Our aim here is not to present the best topology, but simply to give an overview of the various

Multicell Converters

493

advantages and disadvantages of a multicell structure since we will then move on to the description of various control strategies well suited to this type of topology. We will also see that certain disadvantages of this topology will have a significant impact on the control architecture to be constructed. The main advantage of the multicell structure is that it introduces redundancies in terms of the number of states: several different states can result in exactly the same output voltage. This redundancy can be used to produce an apparent frequency ( f a ) at the output of the converter that is greater than the commutation frequency (fdec) of each individual switch and directly proportional to the number of cells ( n ) in series: f a = n × f dec

The price to be paid for this improvement comes in the form of several drawbacks. First, in contrast to NPC topologies, the power switches are working throughout the entire low-frequency modulation period, which results in slightly greater switching losses (but with the losses better shared between the switches). The second drawback is associated with the increase in the number of levels. It can be seen that as the number of levels is increased, the voltage across the capacitor closest to the power supply will increase and reach a value close to the supply voltage: Vc =

(n − 1) E n

For supply voltages above 6 kV, the requirements for this capacitor present a significant design hurdle for the structure. However, the greatest problem is one that involves the floating sources, which are by definition not referenced to any specific voltage level. To ensure correct operation of the converter, it is essential that these voltages should remain close to their initial values, in other words, a particular fraction of the input voltage. Furthermore, these voltages must remain balanced both in static and dynamic operation of the converter. When considering the control of such a structure, it is therefore necessary to take this issue into consideration so that it can be included in the global constraints for the control design.

494

Power Electronic Converters

In certain cases it is possible to introduce an auxiliary circuit in order to improve the balancing of these floating voltages. A number of studies have determined the conditions in which such a circuit can be used along with the specifications for the design of the circuit elements. The use of this auxiliary circuit is often limited to open-loop control of floating voltages and when highly dynamic control is required, such a solution remains difficult to apply. For this reason, and for the rest of this chapter, we will focus on closed-loop control strategies for these floating voltages (often referred to as flying-capacitor voltages) 16.2.3. Evolution of high-power multicell topologies: stacked multicell converters Our discussion on multilevel topologies, and in particular multicell topologies, would not be complete without discussing a newer topological innovation that has emerged in recent years under the name of stacked multicell converter (SMC). Thus the structure combines the desirable properties of each family, namely, reduced use of semiconductors (like in the NPC topology, using components for a fraction of the low-frequency modulation period) and an increase in the apparent output frequency through the use of redundancies in the topology (flying capacitor principle). Figure 16.6 shows a five-level two-stack SMC topology. Each of the two stacks involves an association of two cells and a floating voltage. Two switching cells (TA/TB and T2A/T2B) are added and these are driven at low frequency in order to activate each half-period of the modulation.

Figure 16.6. Stacked five-level multicell converter

Multicell Converters

495

The main advantage of using a structure like the SMC instead of a standard multicell converter is in terms of the energy stored in the flying capacitors. For a given application, we can consider a multicell structure with N cells and a two-stack SMC with N three-branch cells, with each converter needing to meet the same performance specification. If the switching frequency and the input voltage are both the same, the value for the capacitors must be the same in each case; however, there are twice as many capacitors in the SMC but with only half the voltage across themselves, which means that they store only half the amount of energy. Furthermore, in the SMC each cell is actively switching only half the time and so the switching frequency can be theoretically doubled without engendering any additional losses compared to the multicell structure, which enables the values of the capacitors to be halved. These two factors combined, imply that the actual energy stored in the case of the SMC structure is one fourth than in the multicell structure. Given that for a constant voltage the stored energy is an increasing function of the number of levels, the advantage of SMC is particularly significant when a high number of levels are to be used. Finally, due to the limited voltage capabilities of semiconductors, the highest voltages can only be accessed using high number of cells; in this situation, SMC structures will give better performance. In terms of its control, the SMC topology requires a control very similar to a multicell structure, resulting in a relatively simple control system. Consequently, all available control strategies that can be used for standard multicell converters, can also easily be extended to the SMC topology. 16.3. Modeling and analysis of degrees of freedom for control 16.3.1. Instantaneous modeling The mean value approach is widely used in power electronics for modeling and control of static converters. Multicell converters (whether simple multicell or SMC) are of course no exception to this rule. It is first necessary to establish the instantaneous model of the converter from the relevant circuit equations (Kirchoff’s laws). The state variables are

496

Power Electronic Converters

generally (in the absence of an input filter), the current in the output inductance of the converter and the voltages across the flying capacitors in the chosen topology. To illustrate our model, we will take the example of a simple multicell converter with three interleaved cells as shown in Figure 16.7.

Figure 16.7. Four-level multicell converter

Starting from the relevant circuit equations, it is easy to obtain a state space representation of the working state of this structure as given by equation [16.1]. The state vector consists of two floating voltages ( vc1 , vc 2 ) and the output current ( il ). Each command SCi represents the state of the power switch for each switching cell. 1  x&1 = v&c1 = − × x3 × ( SC1 − SC2 )  C1   1 × x3 × ( SC2 − SC3 ) x&2 = v&c 2 = −  C 2   1 r  x&3 = i&l = − × x3 + × ( x1 × SC1 + x2 × SC2 + E × SC3 ) L L 

[16.1]

The nonlinearities in these equations appear immediately. In the first two equations, the state x3 (output current) is multiplied by a reference value. Furthermore, a strong interdependency can be observed between the state variables: the variation in the voltages depends directly on the output current, and this same output current is affected by changes in the voltages.

Multicell Converters

497

Moreover, it can be seen that the variations in the flying capacitor voltages are directly connected to the difference between two control signals. 16.3.2. Mean value model Based on this instantaneous representation of the structure, it is then possible to replace each state variable by its mean value over one switching period. The resultant model will then give us the mean change of the state variables [GAT 98]. We will use ul , Il , VC1 , and VC 2 to represent the mean values of the quantities SC1 , il , vc1 , and vc 2 taken over one switching period. This then leads us to a new state model as given by equation [16.2]:  1 x&1 = V&C1 = − × x3 × (u1 − u2 )  C 1   1 . x&2 = V&C 2 = − × x3 × (u2 − u3 )  C2   1 r  x&3 = I&l = − × x3 + × ( x1 × u1 + x2 × u2 + E × u3 ) L1 L 

[16.2]

It can be seen that this model differs very little from the previous one, except for the fact that in this case, effects taking place within one switching period cannot be represented. Note that if the duty cycles for each cell (u1, u2, u3) are all equal, then there will be no change in the mean values of the flying capacitor voltages. The coupling between the voltage and current equations is therefore exactly the same as in the instantaneous model. 16.4. Analysis of degrees of freedom available to the control algorithm 16.4.1. Open-loop PWM modulation It is possible to define a simple PWM modulation strategy for this structure. Recall the diagram in Figure 16.7 for some open-loop waveforms. The aim is to drive these three cells with equal duty cycles, using sequences that are shifted by 2π/3 relative to one another [GAT 98]. Under these

498

Power Electronic Converters

conditions, the following waveforms are obtained, consisting of four distinct voltage levels and an apparent frequency that is three times the switching frequency (Figure 16.8).

Figure 16.8. Four-level flying capacitor waveforms

16.4.2. Degrees of freedom in the topology As we have already observed, our converter model involves three control signals, which correspond to the states of the three switching cells in this particular topology. The control signals must be sent within a switching period with a sufficient time resolution to achieve an acceptable level of precision.

Figure 16.9. Direct open-loop control

Given these three instantaneous control signals, we can average these signals over one switching period and thus control the three duty cycles (Figure 16.9). In this case, it is necessary to use a PWM modulator (Chapter 9).

Multicell Converters

499

Figure 16.10. Duty cycle control

The timing constraints are considerably reduced for the generation of the three duty cycles. An update twice per switching period is sufficient. As for the PWM generation block, it suffers from the same constraints as before: it must generate control signals with a sufficient timing precision within the switching period. Moreover, as we have observed in the previous section, there are two additional inputs: the phase shifts between each control signal. Although initially set to: ∆φ12 = ∆φ12 =

2π 3

to achieve the optimal waveforms, these inputs may be controlled in order to change the internal state of the converter. 16.4.3. Objective of command rules It is important to clearly state the objectives of the command rules that we will develop later. The first quantity that must be controlled is the output current that will supply the load. This quantity must follow a fixed or sinusoidal reference according to the type of structure that is being used (chopper or inverter). The second objective (in no particular order, and therefore not necessarily a less important one) is to constrain the floating voltages to a particular fraction of the input voltage. If this voltage is stable, these references will consequently also be constant. In the case of a four-level multicell inverter, these two voltages will be required to be equal to

E 2E and , respectively. 3 3

500

Power Electronic Converters

In line with the title of this chapter, we must also consider the specific issues of such topologies: simultaneous control of the flying capacitor voltages in order to ensure correct operation of the converter. We recall that it is important to regulate these voltages in order to ensure correct distribution of the input voltage between the power switches in the circuit. 16.5. Classification of control strategies We will use a standardized classification throughout this chapter. We will first distinguish between direct and indirect control. By definition, direct control directly commands the power switches in the switching cell. These commands may be the result of a comparison, as is the case with hysteresis control. A schematic diagram of such a control strategy is shown in Figure 16.11. We will also use an indirect control strategy as shown in Figure 16.12. The principle here is even more well established – using a PWM block for generating the control commands.

Figure 16.11. Principle of direct control

Figure 16.12. Principle of indirect control

Multicell Converters

501

The advantage of such a strategy compared to direct control is that it naturally ensures a fixed operating frequency for the static converter. The main drawback of this approach will be a slight degradation of the dynamic performance because the control block samples at the switching frequency. The control laws can then be classified in terms of whether or not they are applied to a three-phase system. It is of course possible to develop a control law for a single-phase application and then to extend it to the threephase case. The alternative involves directly treating the system as a threephase system. The advantage of such an approach is that it directly integrates into the design the various degrees of freedom and redundancies involved in a three-phase system. 16.6. Indirect control strategy for a single-phase leg 16.6.1. Principle of decoupled control The control principle we will discuss here involves first decoupling the initial system before performing the regulation. Recall that when we defined our model, we noted a strong coupling between the state variables. Returning to the example of a four-level flying capacitor, there are three state variables to be controlled: the two floating voltages VC1 , VC 2 and the load current Il . There are five degrees of freedom available for the implementation of this control using a PWM block (Figure 16.10): the three duty cycles and the two phase shifts. To achieve regular sequences, we will 2π choose constant phase shifts equal to . This then leaves us three degrees 3

of freedom to control the three state variables of the system. The next objective is to arrange things so that each control input variable acts on one and only one state variable, which is not the case in the model we proposed earlier. It can be seen from the model in equation [16.2] that a single-input variable (duty cycle) will affect at least two state variables. We thus propose to introduce a suitable decoupling that will enable us to determine new inputs v with the property that they only act on a single state variable. The block diagram for this is shown in Figure 16.13.

502

Power Electronic Converters

Figure 16.13. Principle of decoupled control

Each input v thus acts on one and only one state variable, because of the introduction of a decoupling block. We must then ensure that each state variable is suitably controlled, in other words we must introduce three independent control loops. Due to the nonlinearity of the model, we will suggest two different approaches for developing this linearization block. The first involves linearizing the model around a given operating point. Having obtained a linear model we will be able to apply the principles of non iteractive control. The alternative is to use the principle of exact input/output linearization that can be applied to nonlinear systems, which can be described in a related form. The advantage of this approach will be that it does not require prior linearization. 16.6.2. Linearization and non-interacting control Using the model expressed in equation [16.2], we will consider the following input vector: U = [α1 α 2

u3 ] ,

where: α k = uk +1 − uk .

t

Multicell Converters

503

This gives the following set of equations representing the state: X& = A1 × X + B1 ( X ) × U ,

[16.3]

where:   0 0 0    A1 =  0 0 0   r 0 0 −   L

and:  x3 C  1  B1 ( X ) =  0   − x1   L

0 x3 C2 − x2 L

 0   0  E  L

As we already know, we obtain a nonlinear state model since the matrix B1 depends on the state X . We will therefore need to linearize the system

around an operating point given by: X 0 = [VC10 VC 20

I L 0 ] = [ x10 t

x20

x30 ] . t

This gives us a small-signal model in the following form: δ X& = A2 × δ X + B2 × δ U

[16.4]

After expansion, the matrix A2 is equal to the matrix A1 . The matrix B2 can then be obtained by substituting into B1 ( X ) , the state vector at the linearization point X 0 .

504

Power Electronic Converters

We therefore have:   0 0 0    A2 = 0 0 0   r 0 0 −   L

and:  x30  C  1  B2 =  0   − x10   L

0 x30 C2 − x20 L

 0   0.  E  L

Using the linear system given in equation [16.4], we combine state feedback (matrix R1) and precompensation (matrix L1) as shown in Figure 16.14. This gives us a new linear equation system representing the state, given by: X& = A × X + B × U

Figure 16.14. Principle of non-interacting control

[16.5]

Multicell Converters

505

L1 and R1 are calculated [TAC 98], resulting in a decoupled system that is shown in Figure 16.15. Each variable of state is the result of the integration of one and only one input.

Figure 16.15. Decoupled system for non-interacting control

Using this decoupled system, we simply need to add a standard linear state feedback in order to determine the regulation dynamics. We then obtain the structure shown in Figure 16.16.

Figure 16.16. Imposed dynamics for a decoupled system for non-interacting control

The results obtained from this approach are extremely satisfactory (Figure 16.17) and enable the control of the output current as well as the floating voltages without there being any interaction between these two objectives. However, the figure also reveals that the current does not behave as a first-order system as would be expected. This problem is a result of the linearization. The problem is that when the converter is first turned on, the linearization that was performed around the nominal operating point is not appropriate and this results in a situation where the decoupling does not work correctly. It should be noted however that startup behavior like this is only theoretical (variation in the input voltage is, in practice, limited by the input filter). The interested reader is referred to [TAC98], where a number of possible improvements are discussed to reduce this current overshoot.

506

Power Electronic Converters

Figure 16.17. Simulation of converter startup behavior

16.6.3. Decoupling using exact input/output linearization The main drawback of non-interacting control as described in the previous section is that it requires linearization around an operating point. When operating as a chopper, this operating point varies only slightly. Conversely, when operating as an inverter, the current varies sinusoidally and this makes it very difficult to achieve good decoupling over the entire operating range. Consideration of the model given in [16.2] leads us to rewrite it in a form known as a nonlinear system [16.6]: X& = f ( X ) + g ( X )U ,

[16.6]

where:    0    f (X ) =  0   r   − x3   L 

and:  x3 − C  1  g( X ) =  0   x1   L

x3 C1 x − 3 C2 x2 − x1 L

   x3  . C2  E − x2   L  0

Multicell Converters

507

This transformation into an equation of this form enables us to apply the method of exact input/output linearization, which involves defining a nonlinear state feedback (matrices α ( X ) and β ( X ) ) that result in exact decoupling between each input and output. These two matrices are calculated using the Lie derivative [GAT 98]. This results in the decoupling scheme shown in Figure 16.18.

Figure 16.18. Decoupling using exact input/output linearization

An integral relationship is found between the new inputs v and the outputs y, as shown earlier in Figure 16.15. It is then necessary to complement the system using a linear control loop for each of the state variables (Figure 16.19). Since the equation system involves an integration, a simple proportional corrector can be used. Nevertheless, as its parameters can vary, the decoupled system will behave more like a first-order system and an IP or PI corrector might be used. The IP structure is however the preferred solution since it enables the dynamics to be fixed independently from the integral action.

Figure 16.19. Decoupling using exact input/output linearization and linear PI or IP corrector

508

Power Electronic Converters

Figure 16.20. Experimental results for exact input/output linearization

The experimental results obtained for this control strategy applied to a chopper converter are shown in Figure 16.20 for the case where the converter is turned on with the bus voltage already in place. The upper plot shows a very good dynamic response in the flying capacitor voltages and confirms an appropriate distribution of the voltage load between the power switches. In response to strong variations in the load (lower plot in Figure 16.20), the current follows its reference value perfectly. It should be noted that during these step changes in the load, the input bus voltage varies slightly and these variations are reflected exactly as such in the flying capacitor voltages. If however the device acts as an inverter, a singularity problem appears when this decoupling is performed. If the matrix β ( X ) is considered in detail, it can be seen that the load current appears in the denominator. This means that it is not possible to calculate the decoupling for a very small current, and because this leads to saturation of the reference. This singularity arises from the fact that when the current is too small there is no way to control the flying capacitor voltages

Multicell Converters

509

since their variations are directly proportional to the current load. This singularity forces us to supplement the control with a strategy to detect the zero-current crossing and to perform a special treatment of this situation. When this treatment is applied, the results are again very positive and do not involve any significant differences from the results already shown [GAT 98]. The handling of this singularity problem has however led to proposal of alternative regulation structures that are variants of the one we have described here. In [GAT 98] or [PIN 99], partial decoupling is used to avoid this singularity and this has produced satisfactory results. 16.6.4. Control exploiting the phase shifts between the command signals When we considered the degrees of freedom of the multicell structure, we saw that the phases of the command signals could be exploited as input variables. If we return to the case of the four-level converter, three variables of state must be driven, and therefore three inputs must be used. A control strategy has therefore been proposed that uses a common duty cycle for all three switching cells, along with the two phase shifts. This approach results in the following block diagram (Figure 16.21) for our openloop system.

Figure 16.21. Block diagram for control exploiting the phase shifts

The duty cycle will be used to control the output current. Standard linear regulation based on the model presented earlier can be used. The two remaining degrees of freedom (the two phase shifts) will then be used to control the two flying capacitor voltages.

510

Power Electronic Converters

In order to describe the behavior of the system in response to these two inputs, we will use a harmonic model of the converter [DAV 97], which we will not describe here. This model leads to a nonlinear state space representation, whichwe can express in the following form: & = A(u, ∆φ ) Xv + B(u, ∆φ ) E , Xv

[16.7]

where: V  Xv =  c1  Vc 2 

is the state vector, u is the duty cycle applied to the three cells, and:  ∆φ  ∆φ =  12   ∆φ23 

is the phase shift vector between the three cells. To control the state vector using this highly nonlinear model, an inversion-based control strategy was proposed. The principle of this is to perform the inversion using a fuzzy-logic based system to represent the inverse model of the converter. We first consider the variations X& v as a function of the possible phase shifts between the inputs. This results in the two plots shown in Figure 16.22.

Figure 16.22. Variations in X& v1 = V&C1 and X& v 2 = V&C 2 as a function of the input phase shifts (normalized)

Multicell Converters

511

These two figures can be combined into a single plot parameterized as a function of the respective phase shifts to be used. The result of this is shown in Figure 16.23, which shows the variations in the state vector as a function of these two coordinates.

& = V& Figure 16.23. Variations in Xv  C1

t V&C 2 

A fuzzy-logic system is then adapted to represent the inverse model and enable the voltages to be regulated. The result of this learning process by the fuzzy-logic system is the SF nonlinear function given by equation [16.8], which gives the phases to be applied to the system in order to obtain the desired variations:  ∆φ 

[ ∆φ ] =  ∆φ12  = SF ( X& vref ) . 

23 

[16.8]

The diagram of the complete regulation system is shown in Figure 16.24, where the first loop enables the output current to be regulated by using the common duty cycle and the second loop uses this inversion principle to control the two flying capacitor voltages. Simulation results are shown in Figure 16.25, which confirm that the system is correctly regulated. In particular, the floating voltages exactly follow variations in the input voltages. In this case these variations are due to the presence of an LC type input filter.

512

Power Electronic Converters

Figure 16.24. Full diagram for inversion-based control

Figure 16.25. Simulation results for inversion-based regulation

Multicell Converters

513

This type of control therefore operates perfectly in spite of a number of problems involved in its implementation. Inversion of the model is performed for a specific load and thus is only valid for this type of load. The plot shown in Figure 16.23 may therefore be modified by variations in the load, causing problems for the inversion system. A number of solutions have been proposed in [GAT 97], in particular those involving the introduction of a selective output filter. However, discussion of this type of control remains hypothetical due to the challenges of implementing it in a digital environment. 16.7. Direct control strategy for a single-phase leg In this section, we will discuss direct control strategies, that is to say, a control that interacts directly with the power switch state without going through an intermediary PWM generation block. 16.7.1. Sliding mode control Regulation of the state variables for a multicell converter highlights the requirement for multidimensional control. As we have already seen earlier, there is a need for regulation of both the output current and the voltages across the internal capacitors within the structure. This dual objective does however prioritize the voltages. These specified voltages guarantee limits on the voltage stresses acting on the power switches, stresses that have a deleterious effect on the lifetime of the converter. We will describe a control law that uses the sliding modes developed by D. Pinon [PIN 97]. The example we will discuss is based on a three-level converter in order to minimize the complexity of our discussion. The diagram for the converter we will consider is shown in Figure 16.26. The model of the converter is stated in equation [16.2]. The state vector thus v 

consists of two quantities X =  1  . i Sliding mode control is also involved in “amplitude-based” control and generally makes use of the instantaneous model of the converter.

514

Power Electronic Converters

Figure 16.26. Circuit diagram for a three-level converter

For this type of control, we will assume that the control input can only take either of the two states, positive or negative. This principle of variable structure systems, was established in the 1970s by V. Utkin [UTK 78], in particular. We will first define a switching function s( X ) that can be used to determine, via a sign function, the value of the command to be applied to the system. The values for which the switching function is zero form the switching surface (Chapter 12). In the plane representing the regulation error (Figure 16.27), we thus obtain the simple operating principle whereby the system possesses two distinct operational modes. The first, the “attractive mode”, enables it to reach the switching surface without any commutation. The second mode known as the “sliding mode” then enables the system to be maintained around the sliding surface where it ultimately reaches the point of equilibrium.

Figure 16.27. Diagram for a three-level converter

Multicell Converters

515

Thus the sliding surface divides the state space into two regions representing two different structures for the system. At the end of its trajectory, the switching frequency is infinitely high and the system slides towards equilibrium. In order to apply this principle to control a multicell converter, we must first define a switching function. In order to derive this, the Lyapunov criterion is often used, which guarantees the stability of the system. The proposed function is often chosen to be:

(

V ( x) = X − X ref

) Q ( X − X ref ) , T

[16.9]

where Q is a positive definite matrix. The system is stable in closed-loop mode if the derivative of the Lyapunov function is negative. This calculation reveals a control law governing the converter commands that will ensure that this value remains negative. This results in the following switching functions being chosen [PIN 97]: 2 I ref  ( v1 − E ) − i − iref  s1 ( X ) =  E .   s ( X ) = − 2 I ref v − E − i − i (1 ) ref  2 E

(

)

(

)

[16.10]

The results obtained using this type of control are shown in Figure 16.28. The operating point is given for E = 800V and i = 30 A . The reference current value is changed at t = 4 ms , t = 8 ms , and t = 10 ms . The input voltage changes from 800V to 600V at t = 15 ms . Throughout these disturbances, the simulation displays very good performance with, in particular an almost perfect tracking of the reference values. Nevertheless, we must emphasize a number of negative points to counterbalance these excellent initial results. The first negative issue appear on the steady state of the converter. Although the mean reference values are followed exactly, detailed inspection of the results reveals that the phase shift between the switching cells is not correct. This results in output waveforms that are sub-optimal compared to the possibilities inherent in such a structure.

516

Power Electronic Converters

Figure 16.28. Simulation results for sliding mode control

The second negative point involves the operating frequency of the power switches. Nothing in these initial results guarantees that the switching frequency will remain constant. On the contrary, it varies extremely and is in part determined by the operating point. In order to avoid these major problems, it is possible to add a PI corrector associated with a classical PWM modulator, as shown in Figure 16.29.

Figure 16.29. Modification of the sliding mode control

In this case, a system is obtained that operates at fixed frequency and with good symmetry for the output voltage. This modification comes at the cost of a slight degradation of the dynamic performance of the converter, partly due to the addition of a PI corrector whose dynamic performance is slower than the switching frequency.

Multicell Converters

517

This solution does however remain an interesting one, and one that is relatively simple to implement experimentally. 16.7.2. Current mode control This type of control is currently used is switched-mode power supplies, and its main advantage is that it ensures a fixed commutation frequency for the power switches. However, this type of control does introduce a static error (Chapter 13). Figure 16.30 recalls the operating principles for a twolevel converter.

Figure 16.30. Operating principle for two-level current mode operation

The switching frequency is fixed by the clock signal. Each rising edge of the clock triggers transistor T to conduct. The voltage VL at the terminals of the inductance then becomes positive, which leads to an increase in iL . When iL reaches its reference value, denoted iLref , the control signal for transistor T returns to zero. Once transistor T is turned off, VS becomes zero again and iL decreases until the next rising edge of the clock signal. We have two a priori degrees of freedom for the voltage VS : we can choose the times at which the transistor begins to conduct as well as the moments at which it is turned off. The first of these two degrees of freedom is used to determine the switching frequency of the power switches. The second can be used to control the peak magnitude of the current iL .

518

Power Electronic Converters

Studies of this type of control show that it is necessary to add a compensation ramp to the reference signal in order to avoid the appearance of a double cycle for the output current (Chapter 13). The dynamic performance obtained with this type of control is very good, close to those obtained with hysteresis control.

Figure 16.31. Block diagram for multi-level current mode control

It is then possible to generalize this approach to multilevel converters, and in particular multicell converters. For this, it has been shown that not just one ramp but two are required. The result of the comparisons with the reference no longer directly controls the power switches but selects the required (discrete) output level. We then end up with the system as shown in Figure 16.31 for the case of an inverter with three switching cells. Block 1 is

Multicell Converters

519

used first of all to determine the output level. As in the two-level case, here the first switching operation takes place at fixed frequency. The second is then the result of a comparison with one of the two ramps added to the reference signal. The second stage, represented in the diagram by block number 2, has the task of selecting the best power switch configuration, based on the input desired, voltage level, and making use of any redundancies within the topology. These redundancies are then exploited in order to achieve dynamic balancing of the flying-capacitor voltages. The output from this block 2 will then directly control the power switches. Management of the numerous possible behaviors for these two blocks [AIM 03] makes the system relatively complex to specify but this does lead to very encouraging results. Figure 16.32 shows simulation results obtained for the case of a sinusoidal reference current. The current is perfectly regulated and a study of the system’s sensitivity to the various converter parameters reveals that it has a high level of robustness.

Figure 16.32. Simulation results for current mode

520

Power Electronic Converters

Figure 16.33 shows a step change in the commanded reference current, from −20 A to +20 A and back again. The dynamics are excellent and only require a couple of switching periods before the controlled value is correctly tracked.

Figure 16.33. Dynamic tests for current mode control

This type of current mode control is thus ideally suited to the operating mode of multicell inverters. It can be implemented in a small FPGA (Field Programmable Gate Array) without any difficulties [AIM 03]. All the control strategies discussed so far have been developed for singlephase systems. Their extension to three-phase inverters is often relatively easy to anticipate. Nevertheless, in this case the particularities of three-phase systems are not taken into account when the command strategy is formulated and consequently certain degrees of freedom that become available with the use of a three-phase system are not exploited. In order to address this shortcoming, the following section discusses a number of control strategies that have been realized based directly on a three-phase approach. Although this makes the design more complex, the benefits are often far from negligible.

Multicell Converters

521

16.8. Command strategy, three-phase approach In this section we will discuss several control strategies for multicell converters in the case of three-phase systems. There are many types of three-phase loads (motors, power grids, etc.) and it is often difficult to discuss control of the inverter without discussing its load. For this reason we will not discuss control of an induction motor as our example, but rather we will discuss the various degrees of freedom made available by this three-phase approach along with the gains that can be expected from this. The interested reader can refer to the various cited references to obtain further details. 16.8.1. Features of two-level inverters for three-phase systems Briefly recalling three-phase systems [MON 95], Figure 16.34 shows a three-phase load supplied by a two-level voltage inverter. It is assumed that the neutral point of the load, which is often not accessible, is not connected to the midpoint of the DC bus.

Figure 16.34. Three-phase load fed by a two-level voltage inverter

This circuit can be used to obtain the equations governing the time evolution of the single-phase voltages at the terminals of the load. A Concordia type three-phase/two-phase transformation can be used to obtain

522

Power Electronic Converters

the various voltage vectors accessible by the system shown in Figure 16.35 (Chapter 2). These vectors define a hexagon whose summit represents the possible states of the inverter. An arbitrary vector lying within this hexagon is obtained by using the two states that define the sectors that this vector belongs to. These projections must then be used to calculate the times for which each of these two vectors should be applied. In the case of a two-level inverter, seven different vectors are obtained. The vertices of the hexagon can only be obtained using a single combination for the inverter. Conversely, the central point can be obtained using two different combinations of the inverter.

Figure 16.35. Accessible voltage vectors for a two-level inverter, in the (α , β ) frame

16.8.2. Features of a three-phase N-level system In the case of a three-phase system using an N-level inverter [BEN 03], [MAR 00], the number of possible combinations increases very rapidly. Figure 16.36, for example, can be used to represent a three-phase multicell inverter with N voltage levels and (N-1) interleaved cells.

Multicell Converters

Figure 16.36. Three-phase N-level multicell inverter

523

524

Power Electronic Converters

We will define N nsp as the number of phase sequences representing the number of points accessible in the hexagon. We also define N vt as the number of possible voltage vectors. For a thee-phase two-level system, we find N nsp = 8 phase sequences and N vt = 7 different pairs of voltage vectors. Generally speaking, we can use the following formulae:  N snp = N 3   Nvt = 3 × N × ( N − 1) + 1

[16.11]

Figure 16.37 clearly shows the rapid increase in the number of sequences and the number of voltage vectors as the number of levels increases from 2 to 7.

Figure 16.37. Number of phase sequences and number of distinct voltage vectors as a function of the number of levels

It can be observed that the number of phase sequences increases faster than the number of voltage vectors. This can easily be explained by the fact that there is redundancy within the phase sequences. This observation can be understood by the fact that several different sequences within the inverter will ultimately give exactly the same output voltage vector. In what follows, we will see that this redundancy can be exploited to optimize the behavior of the system.

Multicell Converters

525

The representation in the (α , β ) plane then gives us, for N = 3 and N = 4, the diagrams shown in Figure 16.38. Here again, the redundancies mentioned earlier are encountered. Thus, the points in the outer hexagon (in red) can only be obtained using one single configuration.

Figure 16.38. Voltage vectors and phase level sequences for N = 3 and N = 4

The points of the first inner hexagon can be reached using two different phase sequences. In the case where N = 4, the points lying in the final inner hexagon can be reached using three different phase sequences. The central

526

Power Electronic Converters

point can be reached in three or four different ways, depending on whether three or four levels are used. 16.8.3. Analysis of degrees of freedom made available by the use of multilevel inverters The additional degrees of freedom can be classified into three different types [BEN 03]. 16.8.3.1. Number of available vectors (Type I) This degree of freedom is common to all multilevel inverter topologies. The number of available voltage vectors increases with the number of inverter levels according to the quadratic relationship stated earlier and as shown in Figure 16.37. This enables us to reach additional points in the (α, β) plane. This degree of freedom offers the option of choosing a voltage vector from the set of available voltage vectors and enables better control of the inverter load. 16.8.3.2. Redundancy in phase sequences (Type II) A voltage vector can be obtained using several sequences of phase levels. This degree of freedom is also common to all topologies of multilevel inverters. For an N-level inverter, and depending on the position of a voltage vector in the (α, β) plane, this vector may be achieved using one or more sequences of phase levels. Thus, and as shown in Figure 16.38 for an N-level inverter, if the voltage vectors lie within the outer hexagon in the (α, β) plane then each can be achieved by only one sequence of phase levels. If it belongs to the second hexagon, it can be obtained using two sequences of phase levels, etc. The null vector, at the origin of the (α, β) plane, can be achieved using N-1 sequences of phase levels. It is for this reason that multilevel inverters offer redundancies that can be a benefit for the control. Thus, just as the control of variables associated with the inverter load has determined the choice of point in the (α, β) plane, the choice of sequence of phase levels can be made in various ways. In certain control strategies, for example, this choice is made so as to have a fixed mean switching frequency for the inverter. Other criteria that make use of these degrees of freedom to different objectives are possible: it is also

Multicell Converters

527

possible to choose the sequence of phase levels to address constraints associated with the common-mode voltage. 16.8.3.3. Phase redundancy (Type III) This degree of freedom is very specific to multicell inverters and SMC inverters. These multilevel inverter topologies make it possible to generate a given phase voltage level using several different leg configurations. Thus, at the output of one phase of a multicell inverter with p cells (and thus N levels with p = N–1), we can have the voltage level Ec/p with p possible phase configurations. This degree of freedom defines the direction of circulation of the currents in the flying capacitors, which can for example be used to achieve active balancing of the flying capacitors. 16.8.4. Examples of use of the degrees of freedom made available by using multilevel inverters As we have just described, there are numerous possibilities for exploiting the degrees of freedom available in multilevel converters depending on the application (machine control, power grid control, etc.). Next we will give just one possible example application involving control of an induction motor. This first direct control strategy that we will describe is also based on hysteresis-based regulation of the torque and stator flux [MAR 00], [MAR 02]. As usual, the torque and stator flux are the two variables to be regulated and they have equal priorities for regulation. The regulation actions must meet the following requirements and criteria: – ensure minimum impulse-response time for the torque; – guarantee the stability of torque and flux regulation; – minimize the changes in torque and stator flux in the steady state regime so that for given hysteresis bandwidths, the switching frequency of the inverter will be minimized. This direct control strategy (Figure 16.39) will thus be particularly well suited to high-power applications, where the use of multi-level inverters is also indicated.

Figure 16.39. Summary of a DTC strategy for multi-cell inverters, based on hysteresis regulation

528 Power Electronic Converters

Multicell Converters

529

The first stage (Figure 16.40) corresponds to a selection of the inverter voltage vector and enables the degrees of freedom of Type I to be exploited in terms of the instantaneous control of the torque and stator flux. The chosen voltage vector is identified by the variable Qk+1. This vector is then passed as the reference to the “phase-level sequence selection” block, which forms stage two of the algorithm, where the Type II degrees of freedom are exploited to balance the number of switching operations between the three inverter phases. Finally, the three-phase voltage level references are passed to the three “leg-configuration selection” blocks that implement the actual semiconductor control signals. This third stage involves exploiting the Type III degrees of freedom and will focus on stabilizing the flying-capacitor voltages. Stage

Stage

Stage

Figure 16.40. The three stages in the CoDiFI strategy, applied to direct torque control for an induction motor fed using a multicell inverter

530

Power Electronic Converters

16.9. Features of multicell converters: need for an observer Although it is not the primary aim of this chapter, we should however mention the fact that in given industrial applications, multicell converters (standard or SMC) require a measurement of the internal voltages, which are referred to as “flying” since they are not referenced to a fixed potential. The picture in this case is not a promising one since the number of sensors required increases directly with the number of levels in the topology. Moreover, the measurements become rapidly more complicated due to the presence of very high-voltage levels (several kilo-volts). The problem of measuring all these values and isolation of the various voltages then becomes restricting and other techniques must be implemented in order to perform indirect measurements of these quantities. To perform active control of these voltages without adding additional voltage measurements, it is necessary to develop observers or reconstructors that can act as virtual sensors. The measured inputs are then very often the ones already used for global regulation of the system, the phase currents, and the voltage(s) of the DC bus. A wide range of studies, which we will not detail here, have been undertaken in order to develop these voltage observers or reconstructors [BEN 01], [GAT 98], and [LIE 06]. Without going into detail, we can imagine the possibility of reconstructing the various voltages internal to the converter using current measurements at the output of each leg and the command orders [GAT 98] and [LIE 06]. In more general terms and without adding any voltage sensor, we can implement the observation strategy shown in Figure 16.41. In this case, an unbalancing of the internal voltages has a direct effect on the output current of each leg. This effect is very weak compared with the main currents and has a frequency equal to the switching frequency. The observer must therefore be capable of extracting this information in order to cause its outputs to converge to their estimated quantities.

Multicell Converters

531

Figure 16.41. Schematic diagram for the internal voltage observer

Several studies [BEN 01] and [LIE 06] have shown the feasibility of a sliding mode observer. The results are very convincing and it is not too difficult to visualize using these estimated quantities for active regulation of the internal voltages. 16.10. Conclusions and outlook In this chapter, we have presented various multilevel structures and particularly multicell topologies with interleaved cells. These structures offer improved handling of electrical energy for high-voltage applications (several kilovolts) and high-power applications (several megawatts). These improvements stem largely from division of the input voltage, enabling smaller – and hence higher performance – components to be used. They are also partly due to the intrinsic properties of these structures such as multiplication of the apparent frequency at the output of each phase. Clearly, compared to a traditional two-level converter, the topology is of course more complex but the control aspect of such structures is also more challenging. Since in the case of interleaved multicell converters this voltage division is based on the use of floating sources, it is then necessary to ensure

532

Power Electronic Converters

the stability of these sources in order to ensure a long lifetime for the converter. In addition to the standard control loops (management of output current and voltage), further control criteria must be added, such as control of internal voltages to a specified level and implementation of optimized modulation. These new objectives therefore increase the order of the system to be controlled, bearing in mind that this system is by nature nonlinear, involving both continuous and discrete quantities and also including certain couplings between the various state variables. The last ten years of research in this field has resulted in a number of solutions being proposed for controlling these structures: the main types have been discussed in this chapter. These various solutions give acceptable results and there is no need to perform strict comparisons between their performances, which are often in fact very similar. The decisions to be made, lie rather in the suitability of the chosen method to the intended industrial application. Since each specification is different, it is difficult to provide a concrete list of universal solutions. However, in the works cited here, various selection criteria can be found depending on the intended applications. An important point to be noted in the final part of this chapter: this type of structure demonstrates the need for an observer, which may be a key element in the regulation. The choice of an observer appropriate to the law of associated control may prove challenging. We should also state that these structures have demonstrated the need for high-performance nested digital control, which can often be complex. Without this joint development of topology and nested digital control, many such structures would not be usable. To conclude this chapter on control of multicell inverters and to give some thought to future prospects, we can consider the possibility of adapting existing control strategies to apply them to this type of structure. Contrary to what may sometimes be claimed, there is still work remaining in this area. To give a few examples, we may ask the question of how an SOCC strategy (Chapter 14) may be applied to a multicell converter.

Multicell Converters

533

What would be the benefit? Could we ensure the generation of optimal waveforms? Still on the subject of this aspect of control, we could imagine adapting new fixed-frequency hysteresis control strategies (Chapter 13). How can this type of control be adapted to a multicell structure? Can we ensure that all the state quantities will be appropriately regulated using such an approach? These are all unanswered questions to which we hope answers will soon be found. 16.11. Bibliography [AIM 03] AIMÉ M., Evaluation et optimisation de la bande passante des convertisseurs statiques : application aux convertisseurs multicellulaires, Thesis, Institut national polytechnique de Toulouse, France, 2003. [BEN 01] BANSAID R., Observateurs des tensions aux bornes des capacités flottantes pour les convertisseurs multicellulaires séries, PhD thesis, Institut national polytechnique de Toulouse, France, 2001. [BEN 03] BENANI A., Minimisation des courants de mode commun dans les variateurs de vitesse asynchrones alimentés par onduleurs de tension multicellulaire, PhD thesis, Institut national polytechnique de Toulouse, France, 2003. [BRU 01] BRUCKNER T., BEMET S., “Loss balancing in three-level voltage source inverters applying active NPC switches”, Power Electronics Specialists Conference, 2001, PESC. 2001 IEEE 32nd Annual, vol. 2, n° 1, p. 1135–1140, 2001. [BRU 03] BRUCKNER T., HOLMES D.G., “Optimal pulse width modulation for threelevel inverters”, Power Electronics Specialist Conference, 2003, PESC ‘03. 2003 IEEE 34th Annual, vol. 1, n° 2, p. 165–170, 2003. [BRU 05] BRUCKNER T., BERNET S., GULDNER H., “The active NPC converter and its loss-balancing control”, Industrial Electronics, IEEE Transactions on, vol. 52, n° 3, p. 855–868, 2005. [CAR 94] CARPITA M., “Sliding mode controlled inverter with switching optimisation techniques”, EPE Journal, vol. 4, n° 3, p. 30–35, 1994. [COR 02] CORZINE K., FAMILIANT Y., “A new cascaded multilevel H-bridge drive”, Power, IEEE Transactions on Power Electronics, vol. 17, n° 1, p. 125–131, 2002. [CYP 62] CYPKIN J.Z., Théorie des asservissements par plus-ou-moins, Dunod, Paris, 1962.

534

Power Electronic Converters

[DAV 97] DAVANCENS P., MEYNARD T., “Etude des convertisseurs multicellulaires parallèles : I. Modélisation”, Journal de physique III, 1997. [DON 00] DONZEL A., Analyse géométrique et commande active sous observateur d’un onduleur triphasé à structure multicellulaire série, PhD thesis, Institut national polytechnique de Grenoble, France, 2000. [FOS 93] FOSSARD A., NORMAND-CYROT D., Systèmes non linéaires, Masson, Paris, 1993. [GAT 98] GATEAU G., Contribution à la commande des convertisseurs multicellulaires, commande non linéaire et commande floue, PhD thesis, Institut national polytechnique de Toulouse, France, 1998. [HAM 97] HAMMOND P., “A new approach to enhance power quality for medium voltage ac drives”, IEEE Trans. Ind. Applicat., vol. 33, p. 202–208, 1997. [HIL 99] HILL W.A., HARBOURT C.D., “Performance of medium voltage multi-level inverters”, Industry Applications Conference, 1999. Thirty-Fourth IAS Annual Meeting. Conference Record of the 1999 IEEE, vol. 2, n° 1, p. 1186–1192, 1999. [LIE 06] LIENHARDT A.M., Etude de la Commande et de l’observation d’une nouvelle structure de conversion d’énergie de type SMC (convertisseur multicellulaire superposé), PhD thesis, Institut national polytechnique de Toulouse, France, 2006. [LIE 07] LIENHARDT A.M., GATEAU G., MEYNARD T.A., “Digital sliding-mode observer implementation using FPGA”, IEEE Transactions on Industrial Electronics, vol. 54, n° 4, p. 1865–1875, 2007. [MAR 00] MARTINS C.A., Contrôle direct du couple d’une machine asynchrone alimentée par un convertisseur multiniveau à fréquence imposée, PhD thesis, Institut national polytechnique de Toulouse, France, 2000. [MAR 02] MARTINS C.A., ROBOAM X., MEYNARD T.A., CARVALHO A.S., “Switching frequency imposition and ripple reduction in DTC drives by using a multilevel converter”, IEEE Transactions on, Power Electronics, vol. 17, n° 2, p. 286–297, 2002. [MEI 06] MEILI J., PONNALURI S., SERPA L., STEIMER P., KOLAR K., JOHANN W., “Optimized pulse patterns for the 5-level ANPC converter for high speed high power applications”, IEEE Industrial Electronics, IECON 2006 - 32nd Annual Conference on Industrial Electronics, vol. 1, n° 2, p. 2587–2592, 2006. [NAB 81] NABAE A., TAKAHASHI I., AKAGI H., “A new neutral-point-clamped PWM inverter”, IEEE Trans. Ind. Appl., vol. 17, n° 5, p. 518–523, 1981. [NIJ 91] NIJMEIJER H., VAN DER SCHAFT A., Nonlinear Dynamical Control Systems, Springer-Verlag, London, 1991.

Multicell Converters

535

[OSM 99] OSMAN R.H., “A medium-voltage drive utilizing series-cell multilevel topology for outstanding power quality”, Industry Applications Conference, 1999. Thirty-Fourth IAS Annual Meeting, Conference Record of the 1999 IEEE, vol. 4, n° 3, p. 2662–2669, 1999. [OUK 94] OUKAOUR A., BARBOT J., PIOUFLE B., “Nonlinear control of a variable frequency DC-DC converter”, Proc IEEE Conf on Control Applications, vol. 1, p. 499–500, Glasgow, Scotland, 1994. [PIN 99] PINON D., FADEL M., MEYNARD T., “Sliding Mode controls for a two-cell chopper”, Proceedings of EPE, Toulouse, France, 1999. [ROD 02] RODRIGUEZ J., JIH-SHENG L., FANG ZHENG P., “Multilevel inverters: a survey of topologies, controls, and applications”, Industrial Electronics, IEEE Transactions on, vol. 49, n° 4, p. 724–738, 2002. [SIR 89] SIRA-RAMIREZ H., ILIC-SPONG M., “Exact linearization in switched-mode DC to DC power converters”, Int. J. Control., 50(2), 511–524, 1989. [SLO 89] SLOTINE J., LI W., Applied Nonlinear Control, Prentice-Hall International, 1989. [TAC 98] TACHON O., Commande découplante linéaire des convertisseurs multicellulaires série, modélisation, synthèse et expérimentation, PhD thesis, Institut national polytechnique de Toulouse, France, 1998. [TEO 02] TEODORESCU R., BLAABJERG F., PEDERSEN J.K., CENGELCI E., ENJETI P.N., “Multilevel inverter by cascading industrial VSI”, IEEE Transactions on Industrial Electronics, vol. 49, n° 4, p. 832–838, 2002. [TOL 99] TOLBERT L.M., PENG F.Z., HABETLER T.G., “Multilevel converters for large electric drives”, IEEE Transactions on Industry Applications, vol. 35, n° 1, p. 36–44, 1999. [UTK 78] UTKIN V.I., Sliding Modes in Control and Optimisation, Springer-Verlag, Berlin, Germany, 1978. [WES 94] WESTERHOLT E., Commande non linéaire d’une machine asynchrone. Filtrage étendu du vecteur d’état, contrôle de la vitesse sans capteur mécanique, PhD thesis, Institut national polytechnique de Toulouse, France, 1994.

List of Authors

Arnaud DAVIGNY L2EP HEI Lille France Daniel DEPERNET FEMTO ST UTBM Belfort France Guillaume GATEAU LAPLACE ENSEEIHT Toulouse France Xavier KESTELYN L2EP ENSAM Lille France

538

Power Electronic Converters

Francis LABRIQUE LEI UCL Louvain-la-Neuve Belgium Vincent LANFRANCHI LEC UTC Compiègne France Jean-Claude LE CLAIRE IREENA Polytech Nantes Saint-Nazaire France Christophe LESBROUSSART SEIBO Passel France Jean-Paul LOUIS SATIE ENS Cachan France Jean-Philippe MARTIN GREEN ENSEM Nancy France Farid MEIBODY-TABAR GREEN ENSEM Nancy France

List of Authors

Thierry MEYNARD LAPLACE ENSEEIHT Toulouse France Eric MONMASSON SATIE University of Cergy-Pontoise France Ahmad Ammar NAASSANI SATIE University of Alep Syria Mohamed Wissem NAOUAR LSE ENIT Tunis Tunisia Nicolas PATIN LEC UTC Compiègne France Serge PIERFEDERICI GREEN ENSEM Nancy France Joseph PIERQUIN Millipore Strasbourg France

539

540

Power Electronic Converters

Bertrand REVOL SATIE ENS Cachan France Benoît ROBYNS L2EP HEI Lille France Ilhem SLAMA-BELKHODJA LSE ENIT Tunis Tunisia Eric SEMAIL L2EP ENSAM Lille France Jean-Paul VILAIN LEC UTC Compiègne France

Index

A, C

D, F

asynchronous PWM, 114 carrier-based PWM, 91, 240 closed loop, 122, 309, 419, 423, 425, 435, 463 common mode, 117, 142, 161, 169, 178, 199, 200 Concordia transform, 38, 194, 294 conducted interference, 164, 165 control strategy, 68, 119, 156, 161, 172, 179, 190, 196, 197, 225, 231, 235, 237, 251, 278, 281, 309, 311, 321, 322, 326, 331, 333, 354, 452–454, 457, 459, 463, 469, 481, 488, 500, 501, 508–510, 513, 527 criterion, 25, 94, 100, 101, 102, 105, 111, 237, 251, 372, 421– 423, 515 current control, 120, 237, 287, 288, 303, 309–316, 319, 320, 322, 329, 332, 335, 336, 339, 340, 342, 343, 346, 348, 350, 363, 368, 370, 371, 373–375, 379, 383, 384, 402, 414, 415, 417–419, 423, 426, 430, 445, 446, 450, 452, 453, 459, 488

dead time, 105, 106, 109, 111, 188–190, 196 degrees of freedom, 4, 6, 10, 48, 63, 94, 146, 199, 203, 204, 223, 225, 227, 235, 237, 282, 488, 495, 497, 501, 509, 517, 520, 521, 526, 527, 529 Delta-Sigma, 119, 120, 124–126, 128, 134, 138, 140 DSP, 58–60, 63, 69, 90, 113, 146, 147, 153, 239, 461 five-phase, 231–233, 234–236, 239 FPGA, 4, 34, 58, 59, 90, 113, 203, 334, 520, 534 frequency domain modeling, 169, 200 full wave, 54, 72–74, 76–78, 83, 87, 88, 91, 231 H, I, M harmonic levels, 125 implementation, 40, 42, 46, 58–60, 63, 67, 78, 87–90, 115, 144, 146, 147, 163, 239, 288, 379, 430, 453, 463, 501, 513, 532 induction machine, 156, 242

542

Power Electronic Converters

modulation strategy, 74, 82, 83, 90, 94, 128, 146, 183, 204, 263, 264, 281, 301, 497 modulator, 75, 78–80, 82, 126, 128, 145, 170, 251, 256, 257, 260, 261, 271, 278–280, 427, 431, 434, 444, 446, 498, 516 N, O, P noise, 90, 144, 151, 152, 156, 186, 450, 472 open loop, 41, 122, 237, 279, 452, 453 optimized PWM, 94, 107, 108, 110, 113–115 overmodulation, 26, 30, 71–75, 77, 79, 81–90, 192, 230 Park transform, 227, 295, 297, 315, 316, 332, 459 R, S real time, 78, 88, 94, 99, 105, 106 sampling, 16, 18, 22, 29, 124, 125, 129, 130, 139, 156, 168, 184, 186, 187, 189, 321–323, 325– 327, 329, 333, 368, 397, 401, 461 saturation, 28, 72, 74, 78–80, 83, 87, 91, 147, 192, 204, 221, 228– 230, 237, 238, 252, 268, 271, 288, 314, 508 space vector PWM, 33, 35, 40–42, 45, 46, 55, 58, 60–63 65–69, 78– 84, 87–89, 91, 196–199, 303, 366 space vector, 4, 23, 29, 33, 35, 40– 43, 45–47, 49, 50, 55, 58, 60–63 65–69, 77–84, 87–89, 91, 115, 128, 146, 193, 196–199, 204, 212–214, 216, 221, 229, 237, 238, 240–242, 255, 261, 265, 271, 272, 294, 295, 303, 366

spectrum, 56–58, 69, 81, 87, 94, 123, 124, 139, 141–144, 146, 147, 150, 151, 155, 156, 166– 169, 171–173, 176, 180, 182, 183, 186, 189–193, 196, 197, 199, 200, 254, 257, 258, 268, 271, 274, 435, 450 static converter, 46, 141, 143, 154, 159–163, 169, 204, 338, 342, 350, 375, 397, 449, 495, 501 stochastic PWM, 142, 145, 146, 151, 155 switching losses, 30, 33, 63, 72, 93, 94, 155, 215, 221, 234, 251, 268, 272, 273, 275, 281, 375, 493 synchronous machine, 131, 136, 153 synchronous PWM, 180 T, V, W three-level inverter, 116, 126, 127, 135, 138, 225, 274, 279, 281, 282, 283 torque, 89, 90, 101–103, 135, 141, 174, 199, 226, 227, 231, 299, 300, 305, 306, 369, 457, 527, 529 triangular carrier, 13, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 128, 145, 182, 252, 391 variable speed drive, 160, 178, 199 vibration, 90, 114, 144, 151–154 voltage inverter, 1, 29, 32, 36, 71, 93, 97, 102, 108, 109, 138, 160, 174, 301, 307, 320, 322, 323, 331, 333, 353, 366, 368, 369, 453, 456, 457, 521 waveform, 55–57, 69, 72–75, 77, 87, 88, 95, 97, 100, 103, 104, 114, 127, 146, 147, 181, 188, 251, 257, 281, 372, 374, 381, 382, 385, 389, 402, 408, 489