# Power System Harmonics Second Edition By J Arrillaga and N R Watson

Pages 405
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## Contents

Preface xi
1 Subject Definition and Objectives 1
1.1 Introduction 1
1.2 The Mechanism of Harmonic Generation 1
1.3 Definitions and Standards 5
1.3.1 Factors Influencing the Development of Standards 7
1.3.2 Existing Harmonic Standards 8
1.3.3 General Harmonic Indices 11
1.4 Relevance of the Topic 12
1.5 References 15
2 Harmonic Analysis 17
2.1 Introduction 17
2.2 Fourier Series and Coefficients 18
2.3 Simplifications Resulting from Waveform Symmetry 20
2.4 Complex Form of the Fourier Series 23
2.5 Convolution of Harmonic Phasors 25
2.6 The Fourier Transform 27
2.7 Sampled Time Functions 29
2.8 Discrete Fourier Transform (DFT) 30
2.9 The Nyquist Frequency and Aliasing 33
2.10 Fast Fourier Transform (FFT) 35
2.11 Window Functions 38
2.11.1 The Picket Fence 40
2.11.2 Spectral Leakage Reduction 41
2.11.3 Choice of Window Function 41
2.11.4 Main-Lobe Width Reduction 44
2.11.5 Application to Inter-Harmonic Analysis 45
2.12 Efficiency of FFT Algorithms 47
2.12.3 Real-Valued FFTs 49
2.12.4 Partial FFTs 50
2.13 Alternative Transforms 52
2.13.1 The Wavelet Transform 53
2.13.2 Automation of Disturbance Recognition 56
2.14 Discussion 58
2.15 References 58
3 Harmonic Sources 61
3.1 Introduction 61
3.2 Transformer Magnetisation Nonlinearities 62
3.2.1 Normal Excitation Characteristics 62
3.2.2 Determination of the Current Waveshape 62
3.2.3 Symmetrical Overexcitation 63
3.2.4 Inrush Current Harmonics 64
3.2.5 D.C. Magnetisation 65
3.3 Rotating Machine Harmonics 67
3.3.1 M.m.f. Distribution of A.C. Windings 67
3.3.2 Three-Phase Winding 68
3.3.3 Slot Harmonics 69
3.3.4 Voltage Harmonics Produced by Synchronous Machines 70
3.3.5 Rotor Saliency Effects 72
3.3.6 Voltage Harmonics Produced by Induction Motors 73
3.4 Distortion Caused by Arcing Devices 74
3.4.1 Electric Arc Furnaces 74
3.4.2 Discharge-Type Lighting 76
3.5 Single-Phase Rectification 79
3.5.1 D.C. Power Supplies 79
3.5.2 Line-Commutated Railway Rectifiers 82
3.6 Three-Phase Current-Source Conversion 85
3.6.1 Basic (Six-Pulse) Configuration 88
3.6.2 Effect of Transformer Connection 91
3.6.3 Twelve-Pulse Related Harmonics 91
3.6.4 Higher-Pulse Configurations 92
3.6.5 Effect of Transformer and System Impedance 93
3.6.6 Direct Voltage Harmonics 97
3.6.7 Imperfect D.C. Voltage Smoothing 99
3.6.8 Half-Controlled Rectification 104
3.6.9 Uncharacteristic Harmonic and Inter-Harmonic
Generation 104
3.6.10 Frequency Cross-Modulation in Line-Commutated Converter
Systems 112
3.7 Three-Phase Voltage-Source Conversion 116
3.7.1 Multi-Level VSC Configurations 117
3.8 Inverter-Fed A.C. Drives 119
3.9 Thyristor-Controlled Reactors 126
3.9.1 The Static VAR Compensator (SVC) 126
3.9.2 Thyristor-Controlled Series Compensation (TCSC) 129
3.10 Modulated Phase Control 130
3.10.1 The Switching Function Approach 133
3.10.2 Derivation of Input Current Harmonics 135
3.11 A.C. Regulators 137
3.11.1 Single-Phase Full-Wave Controller 137
3.11.2 Integral Cycle Control 138
3.12 Discussion 140
3.13 References 141
4 Effects of Harmonic Distortion 143
4.1 Introduction 143
4.2 Resonances 143
4.2.1 Parallel Resonance 143
4.2.2 Series Resonance 144
4.2.3 Effects of Resonance on System Behaviour 145
4.2.4 Complementary and Composite Resonances 147
4.2.5 Poor Damping 149
4.3 Effects of Harmonics on Rotating Machines 149
4.3.1 Harmonic Losses 149
4.3.2 Harmonic Torques 151
4.3.3 Other Effects 152
4.4 Effect of Harmonics on Static Power Plant 153
4.4.1 Transmission System 153
4.4.2 Transformers 153
4.4.3 Capacitor Banks 155
4.5 Power Assessment with Distorted Waveforms 156
4.5.1 Single-Phase System 156
4.5.2 Three-Phase System 161
4.5.3 Power Factor Under Harmonic Distortion 166
4.5.4 Effect of Harmonics on Measuring Instruments 168
4.6 Harmonic Interference with Ripple Control Systems 169
4.7 Harmonic Interference with Power System Protection 170
4.7.1 Harmonic Problems During Fault Conditions 170
4.7.2 Harmonic Problems Outside Fault Conditions 171
4.8 Effect of Harmonics on Consumer Equipment 171
4.9 Interference with Communications 172
4.9.1 Simple Model of a Telephone Circuit 173
4.9.2 Factors Influencing Interference 173
4.9.3 Coupling to Communication Circuits 174
4.9.4 Effect on Communication Circuits (Susceptiveness) 177
4.9.5 Telephone Circuit Balance to Earth 184
4.9.6 Shielding 185
4.9.7 Mitigation Techniques 186
4.10 Audible Noise from Electric Motors 187
4.11 Discussion 187
References 187
5 Harmonic Monitoring 191
5.1 Introduction 191
5.2 Measurement Requirements 191
5.2.1 The IEC 61000 4-7 Document 191
5.2.2 Inter-Harmonics 193
5.2.3 Harmonic Phase-Angle Displacement 194
5.2.4 Harmonic Symmetrical Components 195
5.3 Transducers 195
5.3.1 Current Transformers 195
5.3.2 Voltage Transformers 197
5.4 Harmonic Instrumentation 200
5.4.1 Digital Instrumentation 202
5.4.2 Structure of a Modern Monitoring System 205
5.5 Data Transmission 206
5.6 Presentation of Harmonic Information 207
5.7 Examples of Application 210
5.7.1 Synchronised Tests 210
5.7.2 Group-Connected HVD.C. Converter Test 215
5.8 Discussion 217
5.9 References 217
6 Harmonic Elimination 219
6.1 Introduction 219
6.2 Passive Filter Definitions 219
6.3 Filter Design Criteria 221
6.3.1 Conventional Criteria 221
6.3.2 Advanced Filter Design Criteria 222
6.4 Network Impedance for Performance Calculations 223
6.4.1 Size of System Representation 223
6.4.2 Effect of A.C. Network Resistance at Low Frequencies 224
6.4.3 Impedance Envelope Diagrams 225
6.5 Tuned Filters 228
6.5.1 Graphic Approach 231
6.5.2 Double-Tuned Filters 233
6.5.3 Automatically Tuned Filters 234
6.6 Damped Filters 235
6.6.1 Types of Damped Filters 236
6.6.2 Design of Damped Filters 236
6.7 Conventional Filter Configurations 237
6.7.1 Six-Pulse Design 237
6.7.2 Twelve-Pulse Configuration 242
6.8 Band-Pass Filtering for Twelve-Pulse Converters 242
6.9 Distribution System Filter Planning 245
6.10 Filter Component Properties 246
6.10.1 Capacitors 246
6.10.2 Inductors 247
6.11 Filter Costs 247
6.11.1 Single-Tuned Filter 248
6.11.2 Band-Pass Filter 250
6.12 D.C. Side Filters 253
6.13 Active Filters 255
6.13.1 Series Connection of Active Filters 256
6.13.2 Shunt Connection of Active Filters 257
6.14 Discussion 259
6.15 References 259
7 Computation of Harmonic Flows 261
7.1 Introduction 261
7.2 Direct Harmonic Analysis 261
7.2.1 Frequency Scan Analysis 264
7.2.2 Incorporation of Harmonic Voltage Sources 264
7.3 Derivation of Network Harmonic Impedances
from Field Tests 266
7.3.1 Use of Existing Sources (Online Non-Invasive Tests) 266
7.3.2 Direct Injection (Online Invasive Tests) 268
7.3.3 From Transient Waveforms (Online Non-Invasive Tests) 268
7.4 Transmission Line Models 269
7.4.1 Mutually Coupled Three-Phase Lines 273
7.4.2 Consideration of Terminal Connections 276
7.4.3 Equivalent PI Model 277
7.4.4 Evaluation of Transmission Line Parameters 282
7.5 Underground and Submarine Cables 286
7.6 Three-Phase Transformer Models 290
7.7 Generator Modelling 295
7.8 Shunt Elements 295
7.9 Series Elements 297
7.10 Distribution System Modelling 298
7.11.1 Induction Motor Model 302
7.11.2 Norton Equivalents of Residential Loads 303
7.11.3 Empirical Models Based on Measurements 304
7.12 Computer Implementation 304
7.12.1 Harmonic Penetration Overview 305
7.12.2 An Advanced Program Structure 305
7.12.3 Data Structure 307
7.13 Examples of Application of the Models 311
7.13.1 Harmonic Flow in a Homogeneous Transmission
Line 311
7.13.2 Impedance Loci 318
7.13.3 Harmonic Analysis of Transmission Line with
Transpositions 326
7.13.4 Harmonic Analysis of Transmission Line with VAR
Compensation 334
7.13.5 Harmonic Analysis of an HVD.C. Transmission Line 335
7.14 Simulation Backed by Field Tests 344
7.14.1 Post-Processing of Transmission Line Harmonics for Test Result
Comparisons 346
7.15 Discussion 347
7.16 References 348
8.1 Introduction 351
8.2 Transfer Function Model 351
8.3 Iterative Harmonic Analysis (IHA) 353
8.3.1 Fixed-Point Iterative Method 353
8.3.2 The Method of Norton Equivalents 354
8.3.3 Hybrid Time/Frequency Domain Solution 354
8.3.4 The Harmonic Domain 356
8.4 Harmonic Power Flow 359
8.4.1 Components of a Three-Phase Newton HPF Solution 360
8.5 Harmonic State Estimation 365
8.5.1 Load and Harmonic Source Identification 368
8.6 The Electromagnetic Transients Solution 369
8.6.1 Time Step Selection 370
8.6.2 A.C. System Representation 370
8.6.3 Frequency-Dependent Network Equivalents 370
8.6.4 Case Study 371
8.7 Discussion on Advanced Harmonic Modelling 387
8.8 References 388
Index 391

## Preface

Following the first international conference on Power System Harmonics held in
Manchester in 1982, J. Arrillaga was commissioned by John Wiley & Sons to prepare
a book on the subject. The book, co-authored by D.A. Bradley and P.S. Bodger
and published in 1985, has provided the basis for a variety of courses and workshops
on power quality issues. It has also been of considerable assistance to power
system designers.
In the past two decades other books and an innumerable number of publications
have appeared in the technical literature on the general topic of harmonics. Wiley has
probably been the main contributor, with three further books, Power System Harmonic
Analysis and Power System Quality Assessment (both by J. Arrillaga and his colleagues)
and Power System Harmonics Computer Modelling and Analysis (by E. Acha
and M. Madrigal). All these, however, have mostly included material coming out of
academic research and on computer simulation techniques. In North America the subject
is currently offered in the form of an IEEE CD-ROM tutorial course (Modelling
and Simulation of Power System Harmonics) and an IEEE (5-hour) videotape on Power
System Harmonics.
In recent years there have been numerous requests for an update of our original
text, maintaining the practical approach to the subject. Therefore the scope of this
new edition is not particularly different from the original, namely to provide a general
understanding of power system harmonics generation, their effects, monitoring, analysis
and elimination, but taking into account the main developments (particularly in power
electronics) accepted by the power industry in the past two decades.
It is impractical for most users to develop their own harmonic assessment programs.
Thus the analysis sections of the book provide basic understanding of the techniques
involved in harmonic assessment and rely on existing available software, with special
emphasis on generally available programs such as EMTP and MATLAB. The
only exception is an advanced and complex frequency-domain program developed by
the authors, called The Harmonic Domain, which is provided in CD-ROM form for
demonstration purposes.
We would like to acknowledge the contributions made to the development
of this book by many of our colleagues, and in particular by P.S. Bodger,
D.A. Bradley, G. Bathurst, S. Chen, A.R. Wood, B.C. Smith, E. Acha, J.F. Eggleston,
G. Heydt, A. Medina, M.L. Viana Lisboa, S. Round, A. Semlyen, R. Yacamini and
J.D. Ainsworth.