POWER SYSTEM HARMONIC ANALYSIS Jos Arrillaga, Bruce C Smith Neville R Watson, Alan R Wood University of Canterbury, Christchurch, New Zealand JOHN WILEY & SONS Chichester New York Weinheim Brisbane Singapore Toronto Copyright 1997 by John Wiley & Sons Ltd, Baffins Lane Chichester West Sussex PO19 IUD, England Nutiotinl I243 779777 Inkwrationril ( + 44) 1243 779777 e-mail (for orders and customer service enquiries): cs-books(cc wiley.co.uk Visit our Home Page on http://www.wiley.co.uk or http://www.wiley.com Reprinted October 1998, November 2000 All Rights Reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means electronic mechanical photocopying recording scanning or otherwise, except under the terms of the Copyright Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency, 90 Tottenham Court Road London, UK W1P 9HE without the permission in writing of the Publisher Otller Wi1c.r E d i t o r i d Ofliccs John Wiley & Sons Inc 605 Third Avenue, New York NY 10158-0012, USA WILEY-VCH Verlagsgesellschaft GmbH Pappelallee D-69469 Weinheim, Germany Jacaranda Wiley Ltd, 33 Park Road, Milton, Queensland 4064 Australia John Wiley & Sons (Asia) Pte Ltd, Clementi Loop #02-01 Jin Xing Distripark, Singapore 129809 John Wiley & Sons (Canada) Ltd 22 Worcester Road, Rexdale, Ontario M9W I LI Canada Library o$ Congress Cataloguing in Publication Data Power system harmonic analysis i Jos Arrillaga [et al.] p cm Includes bibliographical references and index ISBN 471 97548 I Electric power systems - Mathematical models Harmonics (Electric waves) - Mathematics I Arrillaga J TK3226.P378 1997 97-309 621.319’1 - d ~ CIP British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 471 97548 Cover design by J N Arrillaga Typeset in 10/12pt Times by Dobbie Typesetting Limited PREFACE The subject of Power System Harmonics was first discussed in a book published by J Wiley & Sons in 1985 which collected the state of the art, explaining the presence of voltage and current harmonics with their causes, effects, standards, measurement, penetration and elimination Since then, the increased use of power electronic devices in the generation, transmission and utilisation of systems has been accompanied by a corresponding growth in power system harmonic problems Thus, Power System Harmonic Analysis has become an essential part of system planning and design Many commercial programmes are becoming available, and CIGRE and IEEE committees are actively engaged in producing guidelines to facilitate the task of assessing the levels of harmonic distortion This book describes the analytical techniques, currently used by the power industry for the prediction of harmonic content, and the more advanced algorithms developed in recent years A brief description of the main harmonic modelling philosophies is made in Chapter and a thorough description of the Fourier techniques in Chapter Models of the linear system components, and their incorporation in harmonic flow analysis, are considered in Chapters and Chapters and analyse the harmonic behaviour of the static converter in the frequency domain The remaining chapters describe the modelling of non-linearities in the harmonic domain and their use in advanced harmonic flow studies The authors would like to acknowledge the assistance received directly or indirectly from their present and previous colleagues, in particular from E Acha, G Bathurst, P S Bodger, S Chen, T J Densem, J F Eggleston, B J Harker, M L V Lisboa and A Medina They are also grateful for the advice received from J D Ainsworth, H Dommel, A Semylen and R Yacamini Finally, they wish to thank Mrs G M Arrillaga for her active participation in the preparation of the manuscript CONTENTS Preface xi Introduction 1.1 Power System Harmonics 1.2 The Main Harmonic Sources 1.3 Modelling Philosophies 1.4 Time Domain Simulation 1.5 Frequency Domain Simulation 1.6 Iterative Methods 1.7 References Fourier Analysis 2.1 Introduction 2.2 Fourier Series and Coefficients 2.3 Simplifications Resulting from Waveform Symmetry 2.4 Complex Form of the Fourier Series 2.5 Convolution of Harmonic Phasors 2.6 The Fourier Transform 2.7 Sampled Time Functions 2.8 Discrete Fourier Transform 2.9 Fast Fourier Transform 2.10 Transfer Function Fourier Analysis 2.11 Summary 2.12 References 7 10 13 15 17 19 20 24 26 31 31 Transmission Systems 33 3.1 3.2 3.3 3.4 3.5 Introduction Network Subdivision Frame of Reference used in Three-Phase System Modelling Evaluation of Transmission Line Parameters 3.4.1 Earth Impedance Matrix [&I 3.4.2 Geometrical Impedance Matrix [Z,]and Admittance Matrix [ YJ 3.4.3 Conductor Impedance Matrix [Z,] Single Phase Equivalent of a Transmission Line 3.5.1 Equivalent PI Models 33 33 35 37 37 39 41 46 46 vi CONTENTS 3.6 Multiconductor Transmission Line 3.6.1 Nominal PI Model 3.6.2 Mutually Coupled Three-Phase Lines 3.6.3 Consideration of Terminal Connections 3.6.4 Equivalent PI Model 3.7 Three-Phase Transformer Models 3.8 Line Compensating Plant 3.8.1 Shunt Elements 3.8.2 Series Elements 3.9 Underground and Submarine Cables 3.10 Examples of Application of the Models 3.10.1 Harmonic Flow in a Homogeneous Transmission Line 3.10.2 Harmonic Analysis of Transmission Line with Transpositions 3.10.3 Harmonic Analysis of Transmission Line with Var Compensation 3.10.4 Harmonic Analysis in a Hybrid HVdc Transmission Link 3.11 Summary 3.12 References Direct Harmonic Solutions 4.1 4.2 4.3 4.4 4.5 4.6 Introduction Nodal Harmonic Analysis 4.2.1 Incorporation of Harmonic Voltage Sources Harmonic Impedances 4.3.1 Generator and Transformer Modelling 4.3.2 Distribution and Load System Modelling 4.3.3 Induction Motor Model 4.3.4 Detail of System Representation 4.3.5 System Impedances 4.3.6 Existing Non-linearities Computer Implementation 4.4.1 Structure of the Algorithm 4.4.2 Data Programs 4.4.3 Applications Programs 4.4.4 Post Processing Summary References AC-DC Conversion- Frequency Domain 5.1 5.2 5.3 Introduction Characteristic Converter Harmonics 5.2.1 Effect of Transformer Connection 5.2.2 Twelve-pulse Related Harmonics 5.2.3 Higher Pulse Configurations 5.2.4 Insufficient Smoothing Reactance 5.2.5 Effect of Transformer and System Impedance Frequency Domain Model 5.3.1 Commutation Analysis 5.3.2 Control Transfer Functions 5.3.3 Transfer of Waveform Distortion 5.3.4 Discussion 52 52 56 58 59 61 65 65 67 67 71 71 75 84 87 94 94 97 97 98 100 101 101 102 104 107 109 114 114 114 116 126 127 128 130 133 133 133 137 138 139 140 141 144 147 150 151 156 CONTENTS vii 5.4 The Converter Frequency Dependent Equivalent 5.4.1 Frequency Dependent Impedance 5.4.2 Converter DC Side Impedances 5.4.3 Converter AC Side Positive Sequence Impedances 5.4.4 Converter AC Side Negative Sequence Impedances 5.4.5 Simplified Converter Impedances 5.4.6 Example of Application of the Impedance Models 157 160 164 166 166 167 168 5.5 5.6 Summary 169 References 171 Harmonic Instabilities 6.1 Introduction 6.2 Composite Resonance -A Circuit Approach 6.2.1 The Effect of Firing Angle Control on Converter Impedance 6.2.2 Test Case 6.2.3 Discussion 6.3 Transformer Core Related Harmonic Instability in AC-DC Systems 6.3.1 AC-DC Frequency Interactions 6.3.2 Instability Mechanism 6.3.3 Instability Analysis 6.3.4 Dynamic Verification 6.3.5 Characteristics of the Instability 6.3.6 Control of the Instability 6.4 Summary 6.5 References -Harmonic Domain Machine Non-linearities 7.1 7.2 7.3 7.4 7.5 Introduction Synchronous Machine 7.2.1 The Frequency Conversion Process 7.2.2 Harmonic Model in dq Axes 7.2.3 Two-phase Transformation dq to aj? 7.2.4 Admittance Matrix [Yap] 7.2.5 Admittance Matrix [Yak] 7.2.6 Illustration of Harmonic Impedances 7.2.7 Model Validation 7.2.8 Accounting for Saturation 7.2.9 Norton Equivalent 7.2.10 Case Studies Transformers 7.3.1 Representation of the Magnetisation Characteristics 7.3.2 Norton Equivalent of the Magnetic Non-Linearity 7.3.3 Generalisation of the Norton Equivalent 7.3.4 Full Harmonic Electromagnetic Representation 7.3.5 Case Study Summary References AC-DC Conversion -Harmonic Domain 8.1 Introduction 173 173 174 175 176 179 180 180 182 183 187 188 189 190 191 193 193 193 194 195 196 198 199 200 202 202 205 206 207 208 209 21 216 216 22 22 223 223 CONTENTS viii 8.2 8.3 8.4 8.5 8.6 8.7 8.8 The Commutation Process 8.2.1 Star Connection Analysis 8.2.2 Delta Connection Analysis The Valve Firing Process DC-Side Voltage 8.4.1 Star Connection Voltage Samples 8.4.2 Delta Connection Voltage Samples 8.4.3 Convolution of the Samples Phase Currents on the Converter Side Phase Currents on the System Side Summary References Iterative Harmonic Analysis 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 Introduction Fixed Point Iteration Techniques The Method of Norton Equivalents ABCD Parameters Model Newton's Method 9.5.1 Functional Description of the Twelve Pulse Converter 9.5.2 Composition of Mismatch Functions 9.5.3 Solution Algorithm 9.5.4 Computer Implementation 9.5.5 Validation and Performance Diagonalizing Transforms Integrated Converter and Load Flow Solution Summary References 10 Converter Harmonic Impedances 10.1 Introduction 10.2 Calculation of the Converter Impedance 10.2.1 Perturbation Analysis 10.2.2 The Lattice Tensor 10.2.3 Derivation of the Converter Impedance by Kron Reduction 10.2.4 Sparse Implementation of the Kron Reduction 10.3 Variation of the Converter Impedance 10.4 Summary 10.5 References Appendix I 234 234 240 240 241 24 24 242 246 246 248 250 253 259 265 27 278 279 28 283 283 284 284 288 294 300 304 307 309 Efficient Derivation of Impedance Loci 311 Adaptive Sampling Scheme Winding Angle Criterion 31 I 1.2 Appendix I1 224 224 226 227 229 229 230 232 Pulse Position Modulation Analysis 11.1 11.2 11.3 11.4 11.5 The PPM Spectrum Contribution of Commutation Duration to DC Voltage Contribution of Commutation Duration to AC Current Contribution of Commutation Period Variation to AC Current Reference 31 I 317 317 318 320 322 325 CONTENTS Appendix I11 Pulse Duration Modulation Analysis Appendix IV 329 330 Derivation of the Jacobian 331 IV.2 IV.3 IV.4 IV.5 Voltage Mismatch Partial Derivatives IV.1.I With Respect to AC Phase Voltage Variation IV.1.2 With Respect to D C Ripple Current Variation IV I With Respect to End of Commutation Variation IV 1.4 With Respect to Firing Angle Variation Direct Current Partial Derivatives IV.2.1 With Respect to AC Phase Voltage Variation IV.2.2 With Respect to Direct Current Ripple Variation IV.2.3 With Respect to End of Commutation Variation IV.2.4 With Respect to Firing Angle Variation End of Commutation Mismatch Partial Derivatives IV.3.1 With Respect to AC Phase Voltage Variation IV.3.2 With Respect to Direct Current Ripple Variation IV.3.3 With Respect to End of Commutation Variation IV.3.4 With Respect to Firing Instant Variation Firing Instant Mismatch Equation Partial Derivatives Average Delay Angle Partial Derivatives IV.5.1 With Respect to AC Phase Voltage Variation IV.5.2 With Respect to D C Ripple Current Variation IV.5.3 With Respect to End of Commutation Variation IV.5.4 With Respect to Firing Angle Variation 321 33 332 335 337 339 340 340 342 344 345 345 346 341 341 348 348 349 349 350 350 35 The Impedance Tensor 353 V V.2 353 356 Impedance Derivation Phase Dependent Impedance Appendix VI Test Systems VI CIGRE Benchmark Index 327 111.1 The PDM spectrum 111.2 Firing Angle Modulation Applied to the Ideal Transfer Function 111.3 Reference IV I Appendix V ix 361 36 365 INTRODUCTION 1.1 Power System Harmonics The presence of voltage and current waveform distortion is generally expressed in terms of harmonic frequencies which are integer multiples of the generated frequency [ 13 Power system harmonics were first described in book form in 1985 (Arrillaga) [2] The book collected together the experience of previous decades, explaining the reasons for the presence of voltage and current harmonics as well as their causes, effects, standards, measurement, simulation and elimination Since then the projected increase in the use and rating of solid state devices for the control of power apparatus and systems has exceeded expectations and accentuated the harmonic problems within and outside the power system Corrective action is always an expensive and unpopular solution, and more thought and investment are devoted at the design stage on the basis that prevention is better than cure However, preventative measures are also costly and their minimisation is becoming an important part of power system design, relying heavily on theoretical predictions Good harmonic prediction requires clear understanding of two different but closely related topics One is the non-linear voltage/current characteristics of some power system components and its related effect, the presense of harmonic sources The main problem in this respect is the difficulty in specifying these sources accurately The second topic is the derivation of suitable harmonic models of the predominantly linear network components, and of the harmonic flows resulting from their interconnection This task is made difficult by insufficient information on the composition of the system loads and their damping to harmonic frequencies Further impediments to accurate prediction are the existence of many distributed non-linearities, phase diversity, the varying nature of the load, etc 1.2 The Main Harmonic Sources For simulation purposes the harmonic sources can be divided into three categories: (1) Large numbers of distributed non-linear components of small rating (2) Large and continuously randomly varying non-linear loads INTRODUCTION (3) Large static power converters and transmission system level power electronic devices The first category consists mainly of single-phase diode bridge rectifiers, the power supply of most low voltage appliances (e.g personal computers, TV sets, etc.) Gas discharge lamps are also included in this category Although the individual ratings are insignificant, their accumulated effect can be important, considering their large numbers and lack of phase diversity However, given the lack of controllability, these appliances present no special simulation problem, provided there is statistical information of their content in the load mix The second category refers to the arc furnace, with power ratings in tens of megawatts, connected directly to the high voltage transmission network and normally without adequate filtering The furnace arc impedance is randomly variable and extremely asymmetrical The difficulty, therefore, is not in the simulation technique but in the variability of the current harmonic injections to be used in each particular study, which should be based on a stochastic analysis of extensive experimental information obtained from measurements in similar existing installations As far as simulation is concerned, it is the third category that causes considerable difficulty This is partly due to the large size of the converter plant in many applications, and partly to their sophisticated point on wave switching control systems The operation of the converter is highly dependent on the quality of the power supply, which is itself heavily influenced by the converter plant Thus the process of static power conversion needs to be given special attention in power system harmonic simulation 1.3 Modelling Philosophies A rigorous analysis of the electromagnetic behaviour of power components and systems requires the use of field theory However, the direct applicability of Maxwell’s equations to the solution of practical problems is extremely limited Instead, the use of simplified circuit equivalents for the main power system components generally leads to acceptable solutions to most practical electromagnetic problems Considering the (ideally) single frequency nature of the conventional power system, much of the analytical development in the past has concentrated on the fundamental (or power) frequency Although the operation of a power system is by nature dynamic, it is normally subdivided into well-defined quasi steady state regions for simulation purposes For each of these steady-state regions, the differential equations representing the system and the dynamics are transformed into algebraic ones by means of the factor (jo), circuit is solved in terms of voltage and current phasors at fundamental frequency (0= 2zj-) By definition, harmonics result from periodic steady state operating conditions and therefore their prediction should also be formulated in terms of (harmonic) phasors, i.e in the frequency domain Index Terms Links Commutation duration contribution to ac current 320 contribution to dc voltage 318 322 Commutation period average 148 modulation 287 sensitivity 148 149 variation 147 156 Commutation process 224 delta connection analysis 226 overlap 143 star connection analysis 224 Compensated line, matrix model Complementary resonance Complex penetration concept Composite resonance Conductor impedance matrix 84 173 43 173 174 41 Connection: Star–Delta 218 Connection: Star–Star 218 Control transfer functions 150 Convergence factor 268 Convergence tolerance 264 Converter characteristic harmonics 133 harmonic model 223 p-pulse 133 transformer core saturation instability 182 see also Twelve-pulse converter Converter frequency dependent equivalent 157 This page has been reformatted by Knovel to provide easier navigation Index Terms Links Converter impedance ac side negative sequence 166 ac side positive sequence 166 dc side 164 effect of firing angle control 175 frequency dependent 160 harmonic 283 Kron reduction method 294 simplified 167 variation 304 Convolution 15 Core saturation instability 182 Cross modulation 173 Current mismatch 252 Cyclo-converters 28 284 232 D Damping 175 Data programs 116 Dc ripple current variation 335 Dc-side voltage 229 delta connection samples 230 samples convolution 232 star connection samples 229 Delay (firing) angle 143 initialization 259 modulation 329 variation 148 342 350 150 175 339 345 This page has been reformatted by Knovel to provide easier navigation Index Terms Links Delay (firing) angle partial derivatives ac phase voltage variation 349 dc ripple current variation 350 end of commutation variation 350 variation 351 Delta connection analysis commutation process 226 voltage samples 230 DFT see Discrete Fourier Transform Diagonalizing transforms 271 Direct current partial derivatives 340 ac phase voltage variation 340 dc ripple variation 342 delay angle variation 345 end of commutation variation 344 Direct frequency domain analysis 183 184 20 Discrete Fourier Transform Discrete polygon concept 111 Distribution system modelling 102 feeder equivalents Double circuits, mutual coupling 22 102 74 dq axes machine behaviour 195 two-phase transformation 196 Dubanton’s formulae 38 E Earth currents 46 This page has been reformatted by Knovel to provide easier navigation Index Terms Links Earth impedance coefficients 40 matrix 37 Earth return 46 Electromagnetic coupling 56 ElectroMagnetic Transients Program Electrostatic (capacitive) coupling EMTDC program simulation results 56 17 187 EMTP see ElectroMagnetic Transients Program End of commutation mismatch partial derivatives 345 ac phase voltage variation 346 dc ripple variation 347 firing instant variation 348 variation 347 350 46 59 Equivalent PI model Euler coefficient 232 F FACTS devices Faraday's law Fast Fourier Transform 215 24 FFT see Fast Fourier Transform Firing angle see Delay angle Firing instant see Delay angle Fix point iteration techniques 241 FORM table (pop-up windows) 119 Forward Transform 17 This page has been reformatted by Knovel to provide easier navigation Index Terms Fourier analysis Links transfer function 26 Fourier coefficients simplification 10 Fourier series 17 10 14 complex form 13 14 harmonic phasor form 15 trigonometric form 14 Fourier Transform 17 17 see also Fast Fourier Transform; Discrete Fourier Transform Frequency conversion process 194 Frequency domain simulation Fundamental (power) frequency 19 G Gauss-Seidel iteration Generator modelling 101 Geometrical impedance matrix 39 Geometrical line asymmetry 80 Gibbs phenomena 267 GIPS (data gathering system) 116 Ground see Earth Grounded Star configuration 204 H Half-wave symmetry 11 HRM_AC (application program) 126 127 HARM_Z (application program) 126 311 This page has been reformatted by Knovel to provide easier navigation 144 Index Terms Links Harmonic currents excitation 85 114 Harmonic distortion, effect of synchronous machines Harmonic domain modelling 206 202 Harmonic electromagnetic representation, full Harmonic flow Harmonic impedances 216 71 101 Harmonic phasors 15 Harmonic sequences, coupling 72 Harmonic solution, Newton’s method Harmonic sources Harmonic voltage sources excitation 263 100 85 High Voltage direct current 87 back to back interties 189 converter device power rating hybrid transmission link High-pulse configurations 87 139 HVdc see High Voltage direct current I Ideal transfer function 329 Impedance asymmetry 75 Impedance circle 110 Impedance contour concept 316 Impedance loci 109 derivation 311 This page has been reformatted by Knovel to provide easier navigation Index Terms Links Impedance matrix, lumped series 37 Impedance plots 93 Impedance tensor 353 Impedances application of models 168 converter see Converter impedance cross-coupling of generator ground/earth 200 38 modulation theory 287 motor 104 non-linear 114 phase dependent 354 sea return system Induction motor model 41 42 69 109 104 Instabilities analysis 183 characteristics 188 control 189 dynamic verification 187 mechanism 182 resonance 173 transformer-core related 180 INTER (data entry system) 124 Interference, telephone systems 72 Inverse Fourier Transform 17 Inverter 165 Iterative frequency domain analysis 183 Iterative methods 174 75 189 224 see also Fixed point iteration techniques This page has been reformatted by Knovel to provide easier navigation 43 Index Terms Links J Jacobian matrix analytical calculation 255 derivation 331 Newton–Raphson solution 243 Newton’s method 253 for non-linear systems 283 sparsity 272 switching 261 255 264 294 297 K Kron reduction method 293 sparse implementation 300 Lattice equivalent circuits 217 Lattice tensor 288 Load flow studies 278 Load system modelling 102 Loaded line behaviour 81 L M Magnetic circuit laws 211 Magnetic non-linearity, Norton equivalent 209 MATLAB (post-processing program) 127 Mismatch functions converter 250 current 252 This page has been reformatted by Knovel to provide easier navigation Index Terms Links Mismatch functions (Cont.) load flow 278 solution algorithm 253 voltage 252 Modal analysis Modelling philosophies 59 Modulation theory commutation period 287 impedances 287 Motive loads Mutual coupling 104 56 N Negative frequencies 22 Negative resistance 176 Negative sequence dc 181 Network subdivisions 33 Newton–Raphson solution 182 243 Newton’s method (for steady-state interaction) computer implementation Nodal analysis Nominal PI model 246 265 259 98 355 147 181 52 Non-characteristic frequencies 144 Non-linearities, effect 114 Norton admittance 242 Norton equivalents 204 205 fixed point iteration 241 generalization 211 158 This page has been reformatted by Knovel to provide easier navigation 160 Index Terms Links Norton equivalents (Cont.) magnetic non-linearity Nyquist frequency 209 14 22 O Open-ended line behaviour Overlap angle 78 143 P Park’s two-reaction theory 194 Passive loads 103 PCC see Point of Common Coupling PDM see Pulse Duration Modulation Perturbation analysis 284 Phase Locked Oscillator 173 227 equivalent 46 59 nominal 52 PI control see Proportional Integral control/ler PI model PLO see Phase Locked Oscillator Point of Common Coupling 101 Post-processing 127 Power electronic loads 104 114 Power flow see Load flow solution PPM see Pulse Position Modulation Primitive matrices 37 Proportional Integral control/ler 227 PSCADZ2/EMTDC program 202 PSCAD/EMTDC program 265 This page has been reformatted by Knovel to provide easier navigation Index Terms Links Pulse Duration Modulation, analysis 327 Pulse Position Modulation, analysis 317 Q Quality (Q) factor 174 R Reactance, smoothing 140 Resonance instability 173 Resonance terms 255 Reverse Transform 174 17 S Sampled time function 19 Saturation see Transformer core saturation Saturation stability factor 187 Schwarz PDM analysis 327 Schwarz PPM analysis 317 SCR see Short circuit ratio Series elements Short circuit ratio 67 173 Shunt elements (reactors/capacitors) 65 Sinc function 18 Single-phase analysis Six-pulse bridge 133 229 Six-pulse converter 27 Skin effect 41 101 46 71 correction factors Slip 105 This page has been reformatted by Knovel to provide easier navigation Index Terms Links Smoothing reactance, insufficient 140 Sparse bifactorisation 263 Sparse symmetric bifactorization method 263 Spectral density function 17 Square wave function 11 Star connection analysis commutation process 224 voltage samples 229 State variable solutions Stator-rotor harmonic interaction 207 Steinmetz equivalent circuit 216 Submarine cable 67 Subsystem, network 33 Switching system 259 Switching terms 255 Synchronous machines 193 effect on harmonic distortion 88 261 206 System loads representation 103 System representation 107 T Tap change controller 235 Telephone interference 75 Terminal connections 58 Thevenin equivalent impedances 112 Three port terms 255 158 160 Three-phase lines mutually coupled Three-phase static converter 56 26 This page has been reformatted by Knovel to provide easier navigation Index Terms Links Three-phase system modelling 35 Three phase transformer models 61 Time domain simulation TL (data entry system) 124 Toeplitz structure 243 Transfer function concept 144 184 202 245 Transformer core saturation accounting for effects 202 instability 180 182 Transformers effect of connection 137 impedance models 207 magnetisation characteristics 208 magnetisation flux 185 modelling 101 multi-limb 211 216 star-g/delta connection 235 236 237 76 85 three-phase models 61 Transmission lines ABCD parameters/matrix 34 attenuation 79 double circuit 74 equivalent PI 46 homogeneous 71 hybrid HVdc link 87 line loaded 81 mutually coupled 74 nominal PI 52 open-ended 78 parameter evaluation 37 59 This page has been reformatted by Knovel to provide easier navigation 90 Index Terms Links Transmission lines ABCD parameters/matrix (Cont.) transposition 75 VAR compensation 84 Transmission towers 88 Transpositions 75 with current excitation 82 with voltage excitation 77 77 82 89 Twelve-pulse converter configurations 138 functional description 248 U Underground cables 67 V Valve firing process 227 Voltage mismatch 251 Voltage mismatch partial derivatives 331 ac phase voltage variation 332 dc ripple current variation 335 end of commutation variation 337 firing angle variation 339 252 W Waveform distortion 151 square 11 symmetry 10 156 194 This page has been reformatted by Knovel to provide easier navigation Index Terms Windows facilities, GIPS Links 118 Z Zero sequence current Zollenkopf method 73 116 263 This page has been reformatted by Knovel to provide easier navigation ... Applications Programs 4.4.4 Post Processing Summary References AC-DC Conversion- Frequency Domain 5.1 5.2 5.3 Introduction Characteristic Converter Harmonics 5.2.1 Effect of Transformer Connection... utilities of harmonic symmetry The harmonic currents produced by non-linear power plant are either specified in advance, or calculated more accurately for a base operating condition derived from a load... be reproduced, stored in a retrieval system, or transmitted in any form or by any means electronic mechanical photocopying recording scanning or otherwise, except under the terms of the Copyright