About the Author xvii
Preface xix
1 Transients in Elementary Circuits and the Laplace Transform 1
1.1 Introduction 1
1.2 Laplace Transform 2
1.2.1 Definition 2
1.2.2 Some Transforms and Their Elementary Properties 2
1.2.3 Inversion of the Laplace Transform 5
1.3 The Convolution Integral 7
1.4 RL Circuit 8
1.4.1 RL Circuit with Sinusoidal Voltage Source 9
1.4.2 RL Circuit with DC Voltage Source 13
1.5 Series RLC Circuit 15
1.5.1 RLC Circuit with Sinusoidal Voltage Source 15
1.5.2 LC Circuit 20
1.6 Resonance at the Nominal Frequency 27
1.7 Analysis of Simple Networks with More Than One Loop 28
1.7.1 Inductive and Capacitive Elements with Initial Conditions 29
1.7.2 Network Analysis 30
References 34
2 Traveling Waves in Single-Phase Lines 35
2.1 Introduction 35
2.2 Basic Equations 38
2.2.1 Transmission Line with Losses 38
2.2.2 Lossless Transmission Line 40
2.3 Voltage and Current Relations and Surge Impedance of a Lossless Transmission Line 44
2.4 Traveling Waves in Discontinuities – Reflected and Refracted Waves 45
2.4.1 A Generic Impedance at the Line Terminal 46
2.4.2 Analysis of Discontinuities Using the Thevenin Equivalent 55
2.5 Nonlinear Elements 58
2.6 Lattice Diagram 63
2.7 Sine Voltage Waves 66
References 67
3 Traveling Waves in Multiphase Lines 69
3.1 Introduction 69
3.2 Elements of Matrix Algebra 70
3.2.1 Calculation of the Exponential Matrix e Ax 70
3.2.2 Modal Decomposition 71
3.2.3 Properties of Symmetric and Balanced Matrices 73
3.2.4 Diagonalization of the Product of Symmetrical Matrices 73
3.3 Phase Domain 75
3.3.1 Multiphase Line 75
3.3.2 Relationship Between Voltages and Currents – Matrix of Characteristic Impedances 78
3.3.3 Lossless Transmission Line 79
3.3.4 Traveling Waves in Multiphase Lines with Discontinuities 81
3.3.5 Thevenin Equivalent in Multiphase Circuits 83
3.4 Modal Domain 84
3.4.1 Modal Analysis 84
3.4.2 Analysis of the Propagation Modes 86
3.4.3 Basic Models in the Modal Domain 91
3.4.4 Traveling Waves in Discontinuities 93
References 106
4 Numerical Solution of Electromagnetic Transients 109
4.1 Introduction 109
4.2 Single-Phase Models 110
4.2.1 Inductance Model 110
4.2.2 Capacitance Model 111
4.2.3 Resistance Model 112
4.2.4 RL Circuit 112
4.2.5 Single-Phase Transmission Line Models 113
4.3 Transient Solution Using Nodal Analysis 120
4.4 Nonlinear Elements 128
4.4.1 Resistive Elements 128
4.4.2 Inductive Elements 131
4.4.3 Conversion of the Saturation Curve 134
4.5 Representation of Switches 138
4.6 Multiphase Models 139
4.6.1 Three-Phase Inductive Circuit with Mutual Inductances 139
4.6.2 Three-Phase Circuit with Resistances and Inductances 141
4.6.3 Three-Phase Capacitive Circuit 142
4.6.4 Three-Phase Transmission Lines 143
4.7 Comments on Numerical Errors 147
References 152
5 Electrical Parameters Dependence on Frequency 153
5.1 Introduction 153
5.2 Elements for Mathematical Modeling 154
5.2.1 Fitting of Rational Functions 155
5.2.2 Convolution Integral by the Recursive Method 157
5.3 Modal Domain Approach 160
5.3.1 Convolution with the Propagation Function 162
5.3.2 Convolution with the Characteristic Admittance 166
5.4 Frequency-Dependent Transformation Matrix 168
5.5 Model of the Transmission Line with the Nodal Admittance Matrix 171
5.5.1 Inverse Fourier Transform 171
5.5.2 State-Space Model of the Transmission Line 173
5.5.3 Norton’s Equivalent 174
5.5.4 Calculation of the Nodal Admittance Matrix in Frequency Domain 176
5.5.5 Frequency-Dependent Network Equivalents-FDNEs 176
5.6 Transmission Line Parameters 177
5.6.1 Internal Impedance of the Conductor 177
5.6.2 Matrix of Series Impedance with Carson’s Corrections 178
5.6.3 Matrix of Series Impedance with a Complex Ground Return Plane 179
5.6.4 Matrix of Capacitances 180
References 180
6 Elements of Power Electronics 185
6.1 Introduction 185
6.2 LCC – Line Commutated Converters 186
6.2.1 Rectifier Bridge without Commutation Angle 187
6.2.2 Rectifier Bridge with Commutation Angle 189
6.2.3 Inverter Bridge 192
6.2.4 Fourier Analysis of Current in Six-Pulse Bridges 194
6.3 Thyristor Controlled Reactors and Switched Capacitors 198
6.4 Power Electronics – with VSC 202
6.4.1 Voltage Source Converters – VSC in Transmission Systems 202
6.4.2 Application of VSC in Renewable Generation 207
6.5 VSC Elements 208
6.5.1 Converter Bridges 208
6.5.2 Gate Drivers 210
6.6 MMC – Modular Multilevel Converter 212
6.7 Converter Control 217
6.7.1 Transformation abc/αβ and αβ/dq 217
6.7.2 PLL – Phase-Locked Loop 222
6.7.3 Elementary Control 226
6.8 VSC Models 227
6.8.1 Switching Models 228
6.8.2 Averaged Switch Models 228
6.8.3 Simple Source Models 232
References 233
7 Phasor Domain Analysis and Temporary Overvoltages 235
7.1 Introduction 235
7.2 Line Energization and Load Rejection 235
7.2.1 Line Energization 236
7.2.2 Load Rejection 245
7.3 Faults 251
7.4 Open Phases in Transmission Lines 257
7.4.1 Introduction 257
7.4.2 Network Modeling 259
7.4.3 Model for Single-Phase Autoreclosure 271
7.4.4 Model for Stuck Breaker Analysis 277
7.4.5 Single-Phase Autoreclosure 277
7.5 Voltages Induced in Parallel Circuits 278
7.5.1 General Considerations 278
7.5.2 Model for the Capacitive Coupling Between Circuits 278
7.5.3 Circuits with Reactive Compensation 281
7.5.4 Comments on Resonance Analysis in Parallel Circuits 286
7.6 Frequency Response Analysis 290
7.6.1 Introduction 290
7.6.2 Modeling the Network Elements 290
7.6.3 Harmonic Flow 292
7.6.4 Harmonics of Transformers 293
7.6.5 Harmonics of Converters and Filtering 294
7.7 Temporary Overvoltages with Transformers 301
7.7.1 Transformer Energization and Load Rejection 301
7.7.2 Ferroresonance 302
References 314
8 Switching Surges 317
8.1 Introduction 317
8.2 General Considerations 318
8.3 Line Energization and Line Autoreclosure 320
8.3.1 Energization 320
8.3.2 Autoreclosure 325
8.3.3 Residual Voltage for Tripolar Opening 328
8.3.4 Preinsertion Resistor 334
8.4 Faults 342
8.4.1 AC Systems 342
8.4.2 dc Transmission Line 344
8.5 Fault Clearing 346
8.6 Load Rejection 347
8.7 Transformer Energization 348
8.8 Controlled Switching 353
8.8.1 Opening and Closing Switching 354
8.8.2 Switching of Reactive Compensation and Transmission Lines 357
8.9 VFTO – Very Fast Transient Overvoltages 360
8.9.1 Disconnector Operation in Gas-Insulated Substations 360
8.9.2 GIS Components Modeling 362
References 364
9 Lightning Surges 367
9.1 Introduction 367
9.2 Data to Calculate Lightning Surges 369
9.2.1 Lightning Current 369
9.2.2 Wavefront and Tail Time 371
9.2.3 Ground Flash Density 373
9.2.4 Topography and Soil Resistivity 373
9.3 Models for Overvoltage Calculations 374
9.3.1 Lines and Cables 374
9.3.2 Towers 374
9.3.3 Tower Grounding 377
9.3.4 Substation Equipment 380
9.3.5 Lightning Stroke Attachment 380
9.3.6 Dielectric Strength of the Insulation 382
9.4 Transmission Line Analysis 382
9.4.1 Lightning Strokes 383
9.4.2 Direct Stroke 383
9.4.3 Back-Flashover 383
9.4.4 Line-Arrester Application 393
9.4.5 Induced Overvoltages in Transmission Lines 402
9.4.6 Underground Cables 410
9.4.7 Corona 411
9.5 Substations Studies 413
9.5.1 Air Insulated Substations 415
9.5.2 Gas Insulated Substations-GIS 419
References 422
10 Transients in Systems with Shunt Capacitors 427
10.1 Introduction 427
10.2 High-Frequency Current and Voltage Transients 427
10.2.1 Energization of Shunt-Capacitor Banks 428
10.2.2 Restrike and Trapped Charge 431
10.2.3 Overvoltages and Arresters 433
10.2.4 Voltage Amplification 437
10.2.5 Lightning Surges 437
10.3 Back-to-Back Shunt Capacitor 439
10.3.1 Transient Inrush Currents 439
10.3.2 Back-to-Back Energization 440
10.3.3 Restrike 442
10.3.4 Faults 442
10.4 Three-Phase Circuits 456
10.4.1 Stored Charges in Ungrounded Shunt Capacitors 456
10.4.2 Trapped Charges in Grounded Shunt Capacitors 460
10.4.3 Reclosing and Restrike in Three-phase Circuits 460
10.5 High-Frequency Requirements for Substation Equipment 465
10.5.1 Circuit Breakers 466
10.5.2 Current Transformers 468
10.5.3 Shunt Capacitors 470
10.5.4 Surge Arrester 470
References 470
11 Transients in Systems with Series Capacitors 473
11.1 Introduction 473
11.2 Protection Schemes for Series Capacitor Banks 474
11.2.1 Protection by Spark Gaps 475
11.2.2 Protection by Metal Oxide Varistor 476
11.3 Protection Schemes Performance 477
11.3.1 Triggering Levels for Spark Gaps 477
11.3.2 Reinsertion Overvoltages 478
11.3.3 Protection Schemes with MOV 483
11.4 Complementary Studies 490
References 493
12 Transient Recovery Voltage 495
12.1 Introduction 495
12.1.1 Fault Currents 495
12.1.2 Extinction of the Fault Current 496
12.2 Transient Recovery Voltage 497
12.2.1 Steady-State Component and Transient Component 497
12.2.2 Opening Sequence for the Circuit Breaker Poles 498
12.3 Calculation of the Transient Recovery Voltage 499
12.3.1 Current Injection Method and Principle of Superposition 499
12.3.2 Calculation with Electromagnetic Transient Programs 501
12.4 TRV in Single Phase Inductive Circuits 502
12.4.1 Current Interruption in Inductances 502
12.4.2 Inductance and Capacitance 504
12.4.3 Transient Recovery Voltage with Transmission Lines 509
12.5 Calculation of the TRV in Three-Phase Circuits 512
12.5.1 Three-phase Ungrounded Fault in the Transmission Line 513
12.5.2 Three-Phase Ungrounded Fault in the Substation Bus 516
12.5.3 Rate of Rise of the Recovery Voltage – RRRV 517
12.5.4 Analysis with Symmetrical Components 520
12.5.5 Traveling Waves 525
12.5.6 TRV Analysis in the Frequency Domain 530
12.6 Short Line Fault 534
12.6.1 Time Domain Analysis 534
12.6.2 Analysis with Two-Port Network 540
12.7 TRV in Systems with Series Capacitors 541
12.8 Electric Arc 543
12.8.1 Cassie’s Model 545
12.8.2 Mayr’s Model 546
12.8.3 Stability of the Electric Arc for Small Currents 547
12.9 Comments on Asymmetrical Faults and ITRV 547
12.9.1 Asymmetrical Current 547
12.9.2 Initial Transient Recovery Voltage 548
12.10 Standards for Transient Recovery Voltage 549
References 551
13 Surge Arrester 553
13.1 Introduction 553
13.2 Overvoltage Control – Basic Concepts 554
13.2.1 Analysis Using the Thevenin Equivalent Circuit 554
13.2.2 Three-Phase Transmission Line 557
13.3 Types and Characteristics of Surge Arresters 558
13.3.1 Silicon–Carbide Surge Arrester 558
13.3.2 Metal Oxide Surge Arrester (MOSA) 559
13.4 Surge Arrester Application 563
13.4.1 Rating Selection 564
13.4.2 Protection Levels and Insulation Coordination 565
13.5 Performance of Surge Arresters 567
13.5.1 Simplified Model of the Surge Arrester 567
13.5.2 Arrester Energy Dissipation 568
13.5.3 Arrester and Switching Surges 578
13.5.4 Surge Arrester and Fast-Front Overvoltages 580
References 592
14 Insulation Coordination of Transmission Lines and Substations 593
14.1 Introduction 593
14.2 Basic Probabilistic Concepts 594
14.2.1 Elementary Concepts 594
14.2.2 Probability Density Function and Distribution Function 595
14.2.3 Function of Random Variable 600
14.2.4 Joint Probability Density Function and Distribution with Two Random Variables 601
14.3 Insulation Strength 602
14.3.1 Impulse Tests for Lightning and Switching Surges 603
14.3.2 Self-Restoring and Non-Self-Restoring Insulation 603
14.3.3 Withstand Levels for Self-Restoring Insulation 606
14.4 Insulation Coordination Methods 610
14.4.1 Deterministic Method 612
14.4.2 Statistical Method 612
14.4.3 Simplified Statistical Method 616
14.4.4 Further Comments on Slow-Front and Fast-Front Overvoltages 616
14.5 Insulation Coordination of Substations 617
14.5.1 Power-Frequency Voltage 618
14.5.2 Fast-Front Overvoltages 618
14.5.3 Slow-Front Overvoltages 620
14.6 Insulation Coordination of Transmission Lines 625
14.6.1 Insulation Coordination for Lightning Surges 627
14.6.2 Insulation Coordination for Switching Surges 645
References 650
Index 653
