TRANSIENTS IN ELECTRICAL SYSTEMS

ANALYSIS, RECOGNITION, AND MITIGATION

J. C. Das

Mc Graw Hill

New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul

Singapore Sydney Toronto

CONTENTS

Preface xiii

CHAPTER 1 INTRODUCTION TO TRANSIENTS

1-1 Classification of Transients 1

1-2 Classification with Respect to Frequency Groups 1

1-3 Frequency-Dependent Modeling 2

1-4 Other Sources of Transients 3

1-5 Study of Transients 3

1-6 TNAs—Analog Computers 3

1-7 Digital Simulations, EMTP/ATP, and Similar Programs 3

References 4

Further Reading 4

CHAPTER 2 TRANSIENTS IN LUMPED CIRCUITS

2-1 Lumped and Distributed Parameters 5

2-2 Time Invariance 5

2-3 Linear and Nonlinear Systems 5

2-4 Property of Decomposition 6

2-5 Time Domain Analysis of Linear Systems 6

2-6 Static and Dynamic Systems 6

2-7 Fundamental Concepts 6

2-8 First-Order Transients 11

2-9 Second-Order Transients 15

2-10 Parallel RLC Circuit 18

2-11 Second-Order Step Response 21

2-12 Resonance in Series and Parallel RLC Circuits 21

2-13 Loop and Nodal Matrix Methods for Transient Analysis 24

2-14 State Variable Representation 25

2-15 Discrete-Time Systems 28

2-16 State Variable Model of a Discrete System 30

2-17 Linear Approximation 30

Problems 31

Reference 32

Further Reading 32

CHAPTER 3 CONTROL SYSTEMS

3-1 Transfer Function 33

3-2 General Feedback Theory 35

3-3 Continuous System Frequency Response 38

3-4 Transfer Function of a Discrete-Time System 38

3-5 Stability 39

3-6 Block Diagrams 41

3-7 Signal-Flow Graphs 41

3-8 Block Diagrams of State Models 44

3-9 State Diagrams of Differential Equations 45

3-10 Steady-State Errors 47

3-11 Frequency-Domain Response Specifications 49

3-12 Time-Domain Response Specifications 49

3-13 Root-Locus Analysis 50

3-14 Bode Plot 55

3-15 Relative Stability 58

3-16 The Nyquist Diagram 60

3-17 TACS in EMTP 61

Problems 61

References 63

Further Reading 63

CHAPTER 4 MODELING OF TRANSMISSION LINES AND CABLES

FOR TRANSIENT STUDIES

4-1 ABCD Parameters 65

4-2 ABCD Parameters of Transmission Line Models 67

4-3 Long Transmission Line Model-Wave Equation 67

4-4 Reflection and Transmission at Transition Points 70

4-5 Lattice Diagrams 71

4-6 Behavior with Unit Step Functions at Transition Points 72

4-7 Infinite Line 74

4-8 Tuned Power Line 74

4-9 Ferranti Effect 74

4-10 Symmetrical Line at No Load 75

4-11 Lossless Line 77

4-12 Generalized Wave Equations 77

4-13 Modal Analysis 77

V

v i CONTENTS

4-14 Damping and Attenuation 79

4-15 Corona 79

4-16 Transmission Line Models for Transient Analysis 81

4-17 Cable Types 85

Problems 89

References 89

Further Reading 89

CHAPTER 5 LIGHTNING STROKES, SHIELDING,

AND BACKFLASHOVERS

5-1 Formation of Clouds 91

5-2 Lightning Discharge Types 92

5-3 The Ground Flash 92

5-4 Lightning Parameters 94

5-5 Ground Flash Density and Keraunic Level 98

5-6 Lightning Strikes on Overhead lines 99

5-7 BIL/CFO of Electrical Equipment 100

5-8 Frequency of Direct Strokes to Transmission Lines 102

5-9 Direct Lightning Strokes 104

5-10 Lightning Strokes to Towers 104

5-11 Lightning Stroke to Ground Wire 107

5-12 Strokes to Ground in Vicinity of Transmission Lines 107

5-13 Shielding 108

5-14 Shielding Designs 110

5-15 Backflashovers 113

Problems 117

References 121

Further Reading 121

CHAPTER 6 TRANSIENTS OF SHUNT CAPACITOR BANKS

6-1 Origin of Switching Transients 123

6-2 Transients on Energizing a Single Capacitor Bank 123

6-3 Application of Power Capacitors with Nonlinear Loads 126

6-4 Back-to-Back Switching 133

6-5 Switching Devices for Capacitor Banks 134

6-6 Inrush Current Limiting Reactors 135

6-7 Discharge Currents Through Parallel Banks 136

6-8 Secondary Resonance 136

6-9 Phase-to-Phase Overvoltages 139

6-10 Capacitor Switching Impact on Drive Systems 140

6-11 Switching of Capacitors with Motors 140

6-12 Interruptions of Capacitance Currents 144

6-13 Control of Switching Transients 147

6-14 Shunt Capacitor Bank Arrangements 150

Problems 152

References 153

Further Reading 153

CHAPTER 7 SWITCHING TRANSIENTS AND

TEMPORARY OVERVOLTAGES

7-1 Classification of Voltage Stresses 155

7-2 Maximum System Voltage 155

7-3 Temporary Overvoltages 156

7-4 Switching Surges 157

7-5 Switching Surges and System Voltage 157

7-6 Closing and Reclosing of Transmission Lines 158

7-7 Overvoltages Due to Resonance 164

7-8 Switching Overvoltages of Overhead Lines and Underground Cables 165

7-9 Cable Models 166

7-10 Overvoltages Due to Load Rejection 168

7-11 Ferroresonance 169

7-12 Compensation of Transmission Lines 169

7-13 Out-of-Phase Closing 173

7-14 Overvoltage Control 173

7-15 Statistical Studies 175

Problems 179

References 180

Further Reading 180

CHAPTER 8 CURRENT INTERRUPTION IN AC CIRCUITS

8-1 Arc Interruption 181

8-2 Arc Interruption Theories 182

8-3 Current-Zero Breaker 182

8-4 Transient Recovery Voltage 183

8-5 Single-Frequency TRV Terminal Fault 186

8-6 Double-Frequency TRV 189

8-7 ANSLTEEE Standards for TRV 191

8-8 IEC TRV Profiles 193

8-9 Short-Line Fault 195

8-10 Interruption of Low Inductive Currents 197

8-11 Interruption of Capacitive Currents 200

8-12 Prestrikes in Circuit Breakers 200

8-13 Breakdown in Gases 200

CONTENTS v i i

8-14 Stresses in Circuit Breakers 204

Problems 205

References 206

Further Reading 206

CHAPTER 9 SYMMETRICAL AND UNSYMMETRICAL

SHORT-CIRCUIT CURRENTS

9-1 Symmetrical and Unsymmetrical Faults 207

9-2 Symmetrical Components 208

9-3 Sequence Impedance of Network Components 210

9-4 Fault Analysis Using Symmetrical Components 211

9-5 Matrix Methods of Short-Circuit Current Calculations 221

9-6 Computer-Based Calculations 224

9-7 Overvoltages Due to Ground Faults 224

Problems 232

References 233

Further Reading 233

CHAPTER 10 TRANSIENT BEHAVIOR OF SYNCHRONOUS

GENERATORS

10-1 Three-Phase Terminal Fault 235

10-2 Reactances of a Synchronous Generator 237

10-3 Saturation of Reactances 238

10-4 Time Constants of Synchronous Generators 238

10-5 Synchronous Generator Behavior on Terminal Short-Circuit 239

10-6 Circuit Equations of Unit Machines 244

10-7 Park's Transformation 246

10-8 Park's Voltage Equation 247

10-9 Circuit Model of Synchronous Generators 248

10-10 Calculation Procedure and Examples 249

10-11 Steady-State Model of Synchronous Generator 252

10-12 Symmetrical Short Circuit of a Generator at No Load 253

10-13 Manufacturer's Data 255

10-14 Interruption of Currents with Delayed Current Zeros 255

10-15 Synchronous Generator on Infinite Bus 257

Problems 263

References 264

Further Reading 264

CHAPTER 11 TRANSIENT BEHAVIOR OF INDUCTION AND

SYNCHRONOUS MOTORS

11-1 Transient and Steady-State Models of Induction Machines 265

11-2 Induction Machine Model with Saturation 270

11-3 Induction Generator 271

11-4 Stability of Induction Motors on Voltage Dips 271

11-5 Short-Circuit Transients of an Induction Motor 274

11-6 Starting Methods 274

11-7 Study of Starting Transients 278

11-8 Synchronous Motors 280

11-9 Stability of Synchronous Motors 284

Problems 288

References 291

Further Reading 291

CHAPTER 12 POWER SYSTEM STABILITY

12-1 Classification of Power System Stability 293

12-2 Equal Area Concept of Stability 295

12-3 Factors Affecting Stability 297

12-4 Swing Equation of a Generator 298

12-5 Classical Stability Model 299

12-6 Data Required to Run a Transient Stability Study 301

12-7 State Equations 302

12-8 Numerical Techniques 302

12-9 Synchronous Generator Models for Stability 304

12-10 Small-Signal Stability 317

12-11 Eigenvalues and Stability 317

12-12 Voltage Stability 321

12-13 Load Models 324

12-14 Direct Stability Methods 328

Problems 331

References 331

Further Reading 332

CHAPTER 13 EXCITATION SYSTEMS AND POWER

SYSTEM STABILIZERS

13-1 Reactive Capability Curve (Operating Chart) of a Synchronous Generator 333

13-2 Steady-State Stability Curves 336

13-3 Short-Circuit Ratio 336

13-4 Per Unit Systems 337

13-5 Nominal Response of the Excitation System 337

v i i i CONTENTS

13-6 Building Blocks of Excitation Systems 339

13-7 Saturation Characteristics of Exciter 340

13-8 Types of Excitation Systems 343

13-9 Power System Stabilizers 352

13-10 Tuning a PSS 355

13-11 Models of Prime Movers 358

13-12 Automatic Generation Control 358

13-13 On-Line Security Assessments 361

Problems 362

References 362

Further Reading 363

CHAPTER 14 TRANSIENT BEHAVIOR OF TRANSFORMERS

14-1 Frequency-Dependent Models 365

14-2 Model of a Two-Winding Transformer 365

14-3 Equivalent Circuits for Tap Changing 367

14-4 Inrush Current Transients 368

14-5 Transient Voltages Impacts on Transformers 368

14-6 Matrix Representations 371

14-7 Extended Models of Transformers 373

14-8 EMTP Model FDBIT 380

14-9 Sympathetic Inrush 382

14-10 High-Frequency Models 383

14-11 Surge Transference Through Transformers 384

14-12 Surge Voltage Distribution Across Windings 389

14-13 Duality Models 389

14-14 GIC Models 391

14-15 Ferroresonance 391

14-16 Transformer Reliability 394

Problems 395

References 396

Further Reading 396

CHAPTER 15 POWER ELECTRONIC EQUIPMENT

AND FACTS

15-1 The Three-Phase Bridge Circuits 397

15-2 Voltage Source Three-Phase Bridge 401

15-3 Three-Level Converter 402

15-4 Static VAR Compensator (SVC) 405

15-5 Series Capacitors 408

15-6 FACTS 414

15-7 Synchronous Voltage Source 414

15-8 Static Synchronous Compensator 415

15-9 Static Series Synchronous Compensator 416

15-10 Unified Power Flow Controller 419

15-11 NGH-SSR Damper 422

15-12 Displacement Power Factor 423

15-13 Instantaneous Power Theory 424

15-14 Active Filters 425

Problems 425

References 426

Further Reading 426

CHAPTER 16 FLICKER, BUS TRANSFER, TORSIONAL

BYNAMICS, AND OTHER TRANSIENTS

16-1 Flicker 429

16-2 Autotransfer of Loads 432

16-3 Static Transfer Switches and Solid-State Breakers 438

16-4 Cogging and Crawling of Induction Motors 439

16-5 Synchronous Motor-Driven Reciprocating Compressors 441

16-6 Torsional Dynamics 446

16-7 Out-of-Phase Synchronization 449

Problems 451

References 451

Further Reading 452

CHAPTER 17 INSULATION COORDINATION

17-1 Insulating Materials 453

17-2 Atmospheric Effects and Pollution 453

17-3 Dielectrics 455

17-4 Insulation Breakdown 456

17-5 Insulation Characteristics—BIL and BSL 459

17-6 Volt-Time Characteristics 461

17-7 Nonstandard Wave Forms 461

17-8 Probabilistic Concepts 462

17-9 Minimum Time to Breakdown 465

17-10 Weibull Probability Distribution 465

17-11 Air Clearances 465

17-12 Insulation Coordination 466

17-13 Representation of Slow Front Overvoltages (SFOV) 469

17-14 Risk of Failure 470

17-15 Coordination for Fast-Front Surges 472

17-16 Switching Surge Flashover Rate 473

17-17 Open Breaker Position 474

CONTENTS ix

17-18 Monte Carlo Method 474

17-19 Simplified Approach 474

17-20 Summary of Steps in Insulation Coordination 475

Problems 475

References 476

Further Reading 476

CHAPTER 18 GAS-INSULATED SUBSTATIONS—VERY FAST

TRANSIENTS

18-1 Categorization of VFT 477

18-2 Disconnector-Induced Transients 477

18-3 Breakdown in GIS—Free Particles 480

18-4 External Transients 481

18-5 Effect of Lumped Capacitance at Entrance to GIS 482

18-6 Transient Electromagnetic Fields 483

18-7 Breakdown in SF6 483

18-8 Modeling of Transients in GIS 484

18-9 Insulation Coordination 487

18-10 Surge Arresters for GIS 488

Problems 493

References 493

Further Reading 494

CHAPTER 19 TRANSIENTS AND SURGE PROTECTION

IN LOW-VOLTAGE SYSTEMS

19-1 Modes of Protection 495

19-2 Multiple-Grounded Distribution Systems 495

19-3 High-Frequency Cross Interference 498

19-4 Surge Voltages 499

19-5 Exposure Levels 499

19-6 Test Wave Shapes 500

19-7 Location Categories 502

19-8 Surge Protection Devices 505

19-9 SPD Components 508

19-10 Connection of SPD Devices 512

19-11 Power Quality Problems 516

19-12 Surge Protection of Computers 517

19-13 Power Quality for Computers 520

19-14 Typical Application of SPDs 520

Problems 523

References 523

Further Reading 524

CHAPTER 20 SURGE ARRESTERS

20-1 Ideal Surge Arrester 525

20-2 Rod Gaps 525

20-3 Expulsion-Type Arresters 526

20-4 Valve-Type Silicon Carbide Arresters 526

20-5 Metal-Oxide Surge Arresters 529

20-6 Response to Lightning Surges 534

20-7 Switching Surge Durability 537

20-8 Arrester Lead Length and Separation Distance 539

20-9 Application Considerations 541

20-10 Surge Arrester Models 544

20-11 Surge Protection of AC Motors 545

20-12 Surge Protection of Generators 547

20-13 Surge Protection of Capacitor Banks 548

20-14 Current-Limiting Fuses 551

Problems 554

References 555

Further Reading 555

CHAPTER 21 TRANSIENTS IN GROUNDING SYSTEMS

21-1 Solid Grounding 557

21-2 Resistance Grounding 560

21-3 Ungrounded Systems 563

21-4 Reactance Grounding 564

21-5 Grounding of Variable-Speed Drive Systems 567

21-6 Grounding for Electrical Safety 569

21-7 Finite Element Methods 577

21-8 Grounding and Bonding 579

21-9 Fall of Potential Outside the Grid 581

21-10 Influence on Buried Pipelines 583

21-11 Behavior Under Lightning Impulse Current 583

Problems 585

References 585

Further Reading 586

CHAPTER 22 LIGHTNING PROTECTION OF STRUCTURES

22-1 Parameters of Lightning Current 587

22-2 Types of Structures 587

22-3 Risk Assessment According to IEC 588

22-4 Criteria for Protection 589

22-5 Protection Measures 592

22-6 Transient Behavior of Grounding System 594

x CONTENTS

22-7 Internal LPS Systems According to IEC 594

22-8 Lightning Protection According to NFPA Standard 780 594

22-9 Lightning Risk Assessment According to NFPA 780 595

22-10 Protection of Ordinary Structures 596

22-11 NFPA Rolling Sphere Model 597

22-12 Alternate Lightning Protection Technologies 598

22-13 Is EMF Harmful to Humans? 602

Prohlems 602

References 603

Further Reading 603

CHAPTER 23 DC SYSTEMS, SHORT CIRCUITS, DISTRIBUTIONS, AND HVDC

23-1 Short-Circuit Transients 605

23-2 Current Interruption in DC Circuits 615

23-3 DC Industrial and Commercial Distribution Systems 617

23-4 HVDC Transmission 618

Problems 627

References 628

Further Reading 629

CHAPTER 24 SMART GRIDS AND WIND POWER GENERATION

24-1 WAMS and Phasor Measurement Devices 631

24-2 System Integrity Protection Schemes 632

24-3 Adaptive Protection 633

24-4 Wind-Power Stations 634

24-5 Wind-Energy Conversion 635

24-6 The Cube Law 636

24-7 Operation 638

24-8 Wind Generators 639

24-9 Power Electronics 640

24-10 Computer Modeling 642

24-11 Floating Wind Turbines 645

References 645

Further Reading 645

APPENDIX A DIFFERENTIAL EQUATIONS

A-l Homogeneous Differential Equations 647

A-2 Linear Differential Equations 648

A-3 Bernoulli's Equation 648

A-4 Exact Differential Equations 648

A-5 Clairaut's Equation 649

A-6 Complementary Function and Particular Integral 649

A-7 Forced and Free Response 649

A-8 Linear Differential Equations of the Second Order (With Constant Coefficients) 650

A-9 Calculation of Complementary Function 650

A-10 Higher-Order Equations 651

A-ll Calculations of Particular Integrals 651

A-12 Solved Examples 653

A-13 Homogeneous Linear Differential Equations 654

A-14 Simultaneous Differential Equations 655

A-15 Partial Differential Equations 655

Further Reading 658

APPENDIX D LAPLACE TRANSFORM

B-l Method of Partial Fractions 659

B-2 Laplace Transform of a Derivative of/(t) 661

B-3 Laplace Transform of an Integral 661

B-4 Laplace Transform of t/(t) 662

B-5 Laplace Transform of {lit) f(t) 662

B-6 Initial-Value Theorem 662

B-7 Final-Value Theorem 662

B-8 Solution of Differential Equations 662

B-9 Solution of Simultaneous Differential Equations 662

B-10 Unit-Step Function 663

B-ll Impulse Function 663

B-12 Gate Function 663

B-13 Second Shifting Theorem 663

B-14 Periodic Functions 665

B-15 Convolution Theorem 666

B-l 6 Inverse Laplace Transform by Residue Method 666

B-l 7 Correspondence with Fourier Transform 667

Further Reading 667

APPENDIX С /-TRANSFORM

C-l Properties of z-Transform 670

C-2 Initial-Value Theorem 671

C-3 Final-Value Theorem 672

C-4 Partial Sum 672

C-5 Convolution 672

C-6 Inverse z-Transform 672

C-7 Inversion by Partial Fractions 674

C-8 Inversion by Residue Method 674

CONTENTS x i

C-9 Solution of Difference Equations 675

C-10 State Variable Form 676

Further Reading 676

APPENDIX D SEQUENCE IMPEDANCES OF TRANSMISSION

LINES AND CABLES

D-l AC Resistance of Conductors 677

D-2 Inductance of Transmission Lines 678

D-3 Transposed Line 678

D-4 Composite Conductors 679

D-5 Impedance Matrix 680

D-6 Three-Phase Line with Ground Conductors 680

D-7 Bundle Conductors 681

D-8 Carson's Formula 682

D-9 Capacitance of Lines 684

D-10 Cable Constants 685

D-ll Frequency-Dependent Transmission Line Models 688

References 688

APPENDIX E ENERGY FUNCTIONS AND STABILITY

E-l Dynamic Elements 691

E-2 Passivity 691

E-3 Equilibrium Points 691

E-4 State Equations 692

E-5 Stability of Equilibrium Points 692

E-6 Hartman-Grobman Linearization Theorem 692

E-7 Lyapunov Function 692

E-8 LaSalle's Invariant Principle 692

E-9 Asymptotic Behavior 692

E-10 Periodic Inputs 693

References 693

Further Reading 693

APPENDIX F STATISTICS AND PROBABILITY

F-l Mean, Mode, and Median 695

F-2 Mean and Standard Deviation 695

F-3 Skewness and Kurtosis 696

F-4 Curve Fitting and Regression 696

F-5 Probability 698

F-6 Binomial Distribution 699

F-7 Poisson Distribution 699

F-8 Normal or Gaussian Distribution 699

F-9 Weibull Distribution 701

Reference 702

Further Reading 702

APPENDIX G NUMERICAL TECHNIQUES

G-l Network Equations 703

G-2 Compensation Methods 703

G-3 Nonlinear Inductance 704

G-4 Piecewise Linear Inductance 704

G-5 Newton-Raphson Method 704

G-6 Numerical Solution of Linear Differential Equations

G-7 Laplace Transform 706

G-8 Taylor Series 706

G-9 Trapezoidal Rule of Integration 706

G-10 Runge-Kutta Methods 707

G-ll Predictor-Corrector Methods 708

G-12 Richardson Extrapolation and Romberg Integration 708

References 709

Further Reading 709

Index 711