Power system analysis short circuit load flow and harmonics (TQL)

Power System Analysis Short-Circuit Load Flow and Harmonics Second Edit ion J.. Highlighting the latest directions in the field, Power System Analysis: Short-Circuit Load Flow and Harm

Power System

Analysis Short-Circuit Load Flow and Harmonics

Second Edit ion

J C Das

Second Edit ion

K11101

Fundamental to the planning, design, and operating stages of any electrical engineering

endeavor, power system analysis continues to be shaped by dramatic advances and

improvements that reflect today’s changing energy needs Highlighting the latest

directions in the field, Power System Analysis: Short-Circuit Load Flow and

Harmonics, Second Edition includes investigations into arc flash hazard analysis and

its migration in electrical systems, as well as wind power generation and its integration

into utility systems

Designed to illustrate the practical application of power system analysis to real-world

problems, this book provides detailed descriptions and models of major electrical

equipment, such as transformers, generators, motors, transmission lines, and power

cables With 22 chapters and 7 appendices that feature new figures and mathematical

equations, coverage includes:

• Short-circuit analyses, symmetrical components, unsymmetrical faults, and

matrix methods

• Rating structures of breakers

• Current interruption in AC circuits and short-circuiting of rotating machines

• Calculations according to the new IEC and ANSI/IEEE standards and

methodologies

• Load flow, transmission lines and cables, and reactive power flow and control

• Techniques of optimization, FACT controllers, three-phase load flow, and

optimal power flow

• A step-by-step guide to harmonic generation and related analyses, effects,

limits, and mitigation, as well as new converter topologies and practical harmonic passive filter designs—with examples

• More than 2000 equations and figures, as well as solved examples, cases

studies, problems, and references

Maintaining the structure, organization, and simplified language of the first edition,

longtime power system engineer, J.C Das, seamlessly melds coverage of theory and

practical applications to explore the most commonly required short-circuit, load-flow,

and harmonic analyses This book requires only a beginning knowledge of the per-unit

system, electrical circuits and machinery, and matrices, and it offers significant updates

and additional information, enhancing technical content and presentation of subject

matter As an instructional tool for computer simulation, it uses numerous examples and

problems to present new insights while making readers comfortable with procedure and

methodology.

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1 Power Distribution Planning Reference Book, H Lee Willis

2. Transmission Network Protection: Theory and Practice

Y G Paithankar

3 Electrical Insulation in Power Systems, N H Malik,

A A Al-Arainy, and M I Qureshi

4 Electrical Power Equipment Maintenance and Testing, Paul Gill

5 Protective Relaying: Principles and Applications, Second

Edition, J Lewis Blackburn

6 Understanding Electric Utilities and De-Regulation

Lorrin Philipson and H Lee Willis

7 Electrical Power Cable Engineering, William A Thue

8 Electric Systems, Dynamics, and Stability with Artificial

Intelligence Applications, James A Momoh and

Mohamed E El-Hawary

9 Insulation Coordination for Power Systems

Andrew R Hileman

10 Distributed Power Generation: Planning and Evaluation

H Lee Willis and Walter G Scott

11 Electric Power System Applications of Optimization

James A Momoh

12 Aging Power Delivery Infrastructures, H Lee Willis, Gregory V

Welch, and Randall R Schrieber

13 Restructured Electrical Power Systems: Operation, Trading,

and Volatility, Mohammad Shahidehpour

and Muwaffaq Alomoush

14 Electric Power Distribution Reliability, Richard E Brown

15 Computer-Aided Power System Analysis

Ramasamy Natarajan

16 Power Transformers: Principles and Applications,

John J Winders, Jr.

17 Spatial Electric Load Forecasting: Second Edition, Revised and

Expanded, H Lee Willis

18 Dielectrics in Electric Fields, Gorur G Raju

19 Protection Devices and Systems for High-Voltage

Applications, Vladimir Gurevich

Vehicles, Ali Emadi, Mehrdad Ehsani, and John Miller

22 Power Distribution Planning Reference Book, Second Edition,

25 Power System Capacitors, Ramasamy Natarajan

26 Understanding Electric Utilities and De-regulation: Second

Edition, Lorrin Philipson and H Lee Willis

27 Control and Automation of Electric Power Distribution

Systems, James Northcote-Green and Robert G Wilson

28 Protective Relaying for Power Generation Systems

Donald Reimert

29 Protective Relaying: Principles and Applications, Third Edition

J Lewis Blackburn and Thomas J Domin

30 Electric Power Distribution Reliability, Second Edition

Richard E Brown

31 Electrical Power Equipment Maintenance and Testing,

Second Edition, Paul Gill

32 Electricity Pricing: Engineering Principles and Methodologies

Lawrence J Vogt

33 Power System Analysis: Short-Circuit Load Flow and

Harmonics, Second Edition, J C Das

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© 2012 by Taylor & Francis Group, LLC

CRC Press is an imprint of Taylor & Francis Group, an Informa business

No claim to original U.S Government works

Version Date: 20110517

International Standard Book Number-13: 978-1-4398-2080-3 (eBook - PDF)

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Series Introduction xxi

Preface to the Second Edition xxiii

Preface to the First Edition xxv

Author xxvii

1 Short-Circuit Currents and Symmetrical Components 1

1.1 Nature of Short-Circuit Currents 2

1.2 Symmetrical Components 6

1.3 Eigenvalues and Eigenvectors 8

1.4 Symmetrical Component Transformation 9

1.4.1 Similarity Transformation 9

1.4.2 Decoupling a Three-Phase Symmetrical System 10

1.4.3 Decoupling a Three-Phase Unsymmetrical System 14

1.4.4 Power Invariance in Symmetrical Component Transformation 14

1.5 Clarke Component Transformation 15

1.6 Characteristics of Symmetrical Components 16

1.7 Sequence Impedance of Network Components 19

1.7.1 Construction of Sequence Networks 20

1.7.2 Transformers 21

1.7.2.1 Delta–Wye or Wye–Delta Transformer 22

1.7.2.2 Wye–Wye Transformer 23

1.7.2.3 Delta–Delta Transformer 24

1.7.2.4 Zigzag Transformer 24

1.7.2.5 Three-Winding Transformers 25

1.7.3 Static Load 29

1.7.4 Synchronous Machines 29

1.8 Computer Models of Sequence Networks 34

1.9 Structure and Nature of Electrical Power Systems 35

1.9.1 Power System Component Models 37

1.9.2 Smart Grids 37

1.9.3 Linear and Nonlinear Systems 37

1.9.3.1 Property of Decomposition 38

1.9.4 Static and Dynamic Systems 39

1.10 Power System Studies 39

Problems 40

Bibliography 42

2 Unsymmetrical Fault Calculations 43

2.1 Line-to-Ground Fault 43

2.2 Line-to-Line Fault 45

2.3 Double Line-to-Ground Fault 47

2.4 Three-Phase Fault 48

2.5 Phase Shift in Three-Phase Transformers 49

2.5.1 Transformer Connections 49

vii

2.5.2 Phase Shifts in Winding Connections 50

2.5.3 Phase Shift for Negative Sequence Components 51

2.6 Unsymmetrical Fault Calculations 56

2.7 System Grounding 63

2.7.1 Solidly Grounded Systems 64

2.7.2 Resistance Grounding 67

2.7.2.1 High-Resistance Grounded Systems 67

2.7.2.2 Coefficient of Grounding 73

2.8 Open Conductor Faults 74

2.8.1 Two-Conductor Open Fault 74

2.8.2 One-Conductor Open Fault 75

Problems 78

References 81

Bibliography 81

3 Matrix Methods for Network Solutions 83

3.1 Network Models 83

3.2 Bus Admittance Matrix 84

3.3 Bus Impedance Matrix 88

3.3.1 Bus Impedance Matrix from Open-Circuit Testing 89

3.4 Loop Admittance and Impedance Matrices 90

3.4.1 Selection of Loop Equations 92

3.5 Graph Theory 92

3.6 Bus Admittance and Impedance Matrices by Graph Approach 95

3.6.1 Primitive Network 95

3.6.2 Incidence Matrix from Graph Concepts 97

3.6.3 Node Elimination in Y-Matrix 102

3.7 Algorithms for Construction of Bus Impedance Matrix 103

3.7.1 Adding a Tree Branch to an Existing Node 103

3.7.2 Adding a Link 105

3.7.3 Removal of an Uncoupled Branch 106

3.7.4 Changing Impedance of an Uncoupled Branch 107

3.7.5 Removal of a Coupled Branch 107

3.8 Short-Circuit Calculations with Bus Impedance Matrix 114

3.8.1 Line-to-Ground Fault 115

3.8.2 Line-to-Line Fault 115

3.8.3 Double Line-to-Ground Fault 115

3.9 Solution of Large Network Equations 125

Problems 125

Bibliography 127

4 Current Interruption in AC Networks 129

4.1 Rheostatic Breaker 129

4.2 AC Arc Interruption 131

4.2.1 Arc Interruption Theories 132

4.2.1.1 Cassie’s Theory 132

4.2.1.2 Mayr’s Theory 132

4.2.1.3 Cassie–Mayr Theory 132

4.3 Current-Zero Breaker 133

4.4 Transient Recovery Voltage 135

4.4.1 First Pole to Clear Factor 136

4.5 Terminal Fault 139

4.5.1 Four-Parameter Method 139

4.5.2 Two-Parameter Representation 140

4.6 Short-Line Fault 141

4.7 Interruption of Low Inductive Currents 142

4.7.1 Virtual Current Chopping 145

4.8 Interruption of Capacitive Currents 145

4.9 TRV in Capacitive and Inductive Circuits 147

4.10 Prestrikes in Breakers 149

4.11 Overvoltages on Energizing High-Voltage Lines 149

4.11.1 Overvoltage Control 151

4.11.2 Synchronous Operation 152

4.11.3 Synchronous Capacitor Switching 152

4.11.4 Shunt Reactors 153

4.12 Out-of-Phase Closing 154

4.13 Resistance Switching 155

4.14 Failure Modes of Circuit Breakers 156

4.15 Operating Mechanisms-SF6Breakers 159

4.16 Vacuum Interruption 160

4.17 Stresses in Circuit Breakers 161

Problems 162

References 163

Bibliography 164

5 Application and Ratings of Circuit Breakers and Fuses according to ANSI Standards 165

5.1 Total and Symmetrical Current Rating Basis 166

5.2 Asymmetrical Ratings 167

5.2.1 Contact Parting Time 167

5.3 Voltage Range Factor K 170

5.4 Circuit Breaker Timing Diagram 170

5.5 Maximum Peak Current 173

5.6 Permissible Tripping Delay 174

5.7 Service Capability Duty Requirements and Reclosing Capability 174

5.7.1 Transient Stability on Fast Reclosing 175

5.8 Capacitance Current Switching 178

5.8.1 Switching of Cables 183

5.8.2 Effects of Capacitor Switching 186

5.9 Line-Closing Switching Surge Factor 187

5.9.1 Switching of Transformers 188

5.10 Out-of-Phase Switching Current Rating 188

5.11 Transient Recovery Voltage 189

5.11.1 Short-Line Faults 193

5.11.2 Oscillatory TRV 195

5.11.3 Initial TRV 196

5.11.4 Adopting IEC TRV Profiles in IEEE Standards 196

5.11.5 Definite Purpose TRV Breakers 198

5.12 Generator Circuit Breakers 198

5.13 Specifications of High-Voltage Circuit Breakers 203

5.14 Low-Voltage Circuit Breakers 203

5.14.1 Molded Case Circuit Breakers 204

5.14.2 Insulated Case Circuit Breakers 204

5.14.3 Low-Voltage Power Circuit Breakers 204

5.14.3.1 Single-Pole Interrupting Capability 206

5.14.3.2 Short-Time Ratings 207

5.14.3.3 Series Connected Ratings 207

5.15 Fuses 208

5.15.1 Current-Limiting Fuses 209

5.15.2 Low-Voltage Fuses 210

5.15.3 High-Voltage Fuses 210

5.15.4 Interrupting Ratings 211

Problems 212

References 213

6 Short Circuit of Synchronous and Induction Machines 215

6.1 Reactances of a Synchronous Machine 216

6.1.1 Leakage Reactance XI 216

6.1.2 Subtransient Reactance X00d 216

6.1.3 Transient Reactance Xd0 216

6.1.4 Synchronous Reactance Xd 216

6.1.5 Quadrature Axis Reactances X00q, Xq0, and Xq 217

6.1.6 Negative Sequence Reactance X2 217

6.1.7 Zero-Sequence Reactance X0 218

6.1.8 Potier Reactance Xp 218

6.2 Saturation of Reactances 218

6.3 Time Constants of Synchronous Machines 219

6.3.1 Open Circuit Time Constant Tdo0 219

6.3.2 Subtransient Short-Circuit Time Constant Td00 219

6.3.3 Transient Short-Circuit Time Constant T0d 219

6.3.4 Armature Time Constant Ta 219

6.4 Synchronous Machine Behavior on Terminal Short Circuit 219

6.4.1 Equivalent Circuits during Fault 223

6.4.2 Fault Decrement Curve 226

6.5 Circuit Equations of Unit Machines 230

6.6 Park’s Transformation 234

6.6.1 Reactance Matrix of a Synchronous Machine 234

6.6.2 Transformation of Reactance Matrix 236

6.7 Park’s Voltage Equation 238

6.8 Circuit Model of Synchronous Machines 240

6.9 Calculation Procedure and Examples 242

6.9.1 Manufacturer’s Data 249

6.10 Short Circuit of Synchronous Motors and Condensers 251

6.11 Induction Motors 251

6.12 Practical Short-Circuit Calculations 255

Problems 255

References 256

Bibliography 257

7 Short-Circuit Calculations according to ANSI Standards 259

7.1 Types of Calculations 259

7.1.1 Assumptions—Short-Circuit Calculations 259

7.1.2 Maximum Peak Current 260

7.2 Accounting for Short-Circuit Current Decay 261

7.2.1 Low-Voltage Motors 262

7.3 Rotating Machines Model 263

7.4 Types and Severity of System Short Circuits 264

7.5 Calculation Methods 264

7.5.1 Simplified Method X=R  17 264

7.5.2 Simplified Method X=R > 17 265

7.5.3 E=X Method for AC and DC Decrement Adjustments 265

7.5.4 Fault Fed from Remote Sources 266

7.5.5 Fault Fed from Local Sources 268

7.5.6 Weighted Multiplying Factors 272

7.6 Network Reduction 273

7.6.1 E=X or E=Z Calculation 273

7.7 Breaker Duty Calculations 274

7.8 Generator Source Short-Circuit Current Asymmetry 274

7.9 Calculation Procedure 276

7.9.1 Necessity of Gathering Accurate Data 276

7.9.2 Calculations—Step by Step 277

7.9.3 Analytical Calculation Procedure 278

7.9.3.1 Hand Calculations 278

7.9.3.2 Dynamic Simulation 278

7.9.4 Devices with Sources on Either Side 279

7.9.5 Switching Devices without Short-Circuit Interruption Ratings 280

7.9.6 Capacitor and Static Converter Contributions to Short-Circuit Currents 280

7.10 Examples of Calculations 281

7.10.1 Deriving an Equivalent Impedance 299

7.11 Thirty-Cycle Short-Circuit Currents 303

Problems 304

References 307

8 Short-Circuit Calculations according to IEC Standards 309

8.1 Conceptual and Analytical Differences 309

8.1.1 Breaking Capability 309

8.1.2 Rated Restriking Voltage 310

8.1.3 Rated Making Capacity 310

8.1.4 Rated Opening Time and Break Time 310

8.1.5 Initial Symmetrical Short-Circuit Current 310

8.1.6 Peak Making Current 311

8.1.7 Breaking Current 311

8.1.8 Steady-State Current 311

8.1.9 Highest Short-Circuit Currents 313

8.2 Prefault Voltage 313

8.3 Far-from-Generator Faults 314

8.3.1 Nonmeshed Sources 315

8.3.2 Meshed Networks 317

8.3.2.1 Method A: Uniform Ratio R=X or X=R Ratio Method 317

8.3.2.2 Ratio R=X or X=R at the Short-Circuit Location 318

8.3.2.3 Method C: Equivalent Frequency Method 318

8.4 Near-to-Generator Faults 319

8.4.1 Generators Directly Connected to Systems 319

8.4.2 Generators and Unit Transformers of Power Station Units 320

8.4.3 Motors 320

8.4.4 Short-Circuit Currents Fed from One Generator 321

8.4.4.1 Breaking Current 321

8.4.4.2 Steady-State Current 322

8.4.5 Short-Circuit Currents in Nonmeshed Networks 323

8.4.6 Short-Circuit Currents in Meshed Networks 324

8.5 Influence of Motors 325

8.5.1 Low-Voltage Motor Groups 326

8.5.2 Calculations of Breaking Currents of Asynchronous Motors 326

8.5.3 Static Converter Fed Drives 326

8.6 Comparison with ANSI Calculation Procedures 327

8.7 Examples of Calculations and Comparison with ANSI Methods 329

Problems 345

References 348

9 Calculations of Short-Circuit Currents in DC Systems 349

9.1 DC Short-Circuit Current Sources 349

9.2 Calculation Procedures 351

9.2.1 IEC Calculation Procedure 351

9.2.2 Matrix Methods 353

9.3 Short Circuit of a Lead Acid Battery 353

9.3.1 IEC Method of Short-Circuit of a Lead Acid Battery 356

9.4 Short-Circuit Current of DC Motors and Generators 358

9.4.1 IEC Method of Short-Circuit of DC Machines 362

9.5 Short-Circuit Current of a Rectifier 364

9.5.1 IEC Method of Short-Circuit of a Rectifier 367

9.6 Short Circuit of a Charged Capacitor 370

9.6.1 IEC Method 370

9.7 Total Short-Circuit Current 371

9.8 DC Circuit Breakers 373

9.8.1 High-Voltage DC Circuit Breakers 373

Problems 375

References 376

10 Load Flow over Power Transmission Lines 377

10.1 Power in AC Circuits 377

10.1.1 Complex Power 380

10.1.2 Conservation of Energy 380

10.2 Power Flow in a Nodal Branch 381

10.2.1 Simplifications of Line Power Flow 382

10.2.2 Voltage Regulation 383

10.3 ABCD Constants 383

10.4 Transmission Line Models 386

10.4.1 Medium Long Transmission Lines 386

10.4.2 Long Transmission Line Model 387

10.4.3 Reflection Coefficient 390

10.4.4 Lattice Diagrams 392

10.4.5 Infinite Line 393

10.4.6 Surge Impedance Loading 393

10.4.7 Wavelength 394

10.5 Tuned Power Line 394

10.6 Ferranti Effect 395

10.6.1 Approximate Long Line Parameters 397

10.7 Symmetrical Line at No Load 397

10.8 Illustrative Examples 398

10.9 Circle Diagrams 402

10.10 Modal Analysis 407

10.11 Corona on Transmission Lines 408

10.12 System Variables in Load Flow 410

Problems 410

Bibliography 411

11 Load Flow Methods: Part I 413

11.1 Modeling a Two-Winding Transformer 414

11.2 Load Flow—Bus Types 418

11.3 Gauss and Gauss–Seidel Y-Matrix Methods 419

11.3.1 Gauss Iterative Technique 421

11.3.2 Gauss–Seidel Iteration 423

11.3.3 Convergence 424

11.3.4 Gauss–Seidel Y-Matrix Method 424

11.4 Convergence in Jacobi-Type Methods 430

11.4.1 III-Conditioned Network 430

11.4.2 Negative Impedances 432

11.4.3 Convergence Speed and Acceleration Factor 432

11.5 Gauss–Seidel Z-Matrix Method 435

11.6 Conversion of Y to Z Matrix 438

11.7 Triangular Factorization Method of Load Flow 442

Problems 446

Bibliography 447

12 Load Flow Methods: Part II 449

12.1 Function with One Variable 449

12.2 Simultaneous Equations 451

12.3 Rectangular Form of Newton–Raphson Method of Load Flow 453

12.4 Polar Form of Jacobian Matrix 455

12.4.1 Calculation Procedure of Newton–Raphson Method 457

12.5 Simplifications of Newton–Raphson Method 463

12.6 Decoupled Newton–Raphson Method 465

12.7 Fast Decoupled Load Flow 466

12.8 Model of a Phase-Shifting Transformer 469

12.9 DC Load Flow Models 471

12.9.1 P–u Network 472

12.9.2 Q–V Network 474

12.10 Second Order Load Flow 477

12.11 Load Models 478

12.12 Induction Motor Models 480

12.12.1 Double Cage Rotors 482

12.13 Impact Loads and Motor Starting 485

12.13.1 Motor Starting Voltage Dips 485

12.13.2 Snapshot Study 486

12.13.3 Motor Starting Methods 486

12.13.3.1 Number of Starts and Load Inertia 491

12.13.4 Starting of Synchronous Motors 492

12.14 Practical Load Flow Studies 497

12.14.1 Contingency Operation 505

Problems 505

References 507

13 Reactive Power Flow and Control 509

13.1 Voltage Instability 511

13.1.1 Relation with Real Power Instability 514

13.2 Reactive Power Compensation 515

13.2.1 Z0Compensation 515

13.2.2 Line Length Compensation 516

13.2.3 Compensation by Sectionalization of Line 516

13.2.4 Effect on Maximum Power Transfer 518

13.2.5 Compensation with Lumped Elements 520

13.3 Reactive Power Control Devices 522

13.3.1 Synchronous Generators 523

13.3.2 Synchronous Condensers 524

13.3.3 Synchronous Motors 525

13.3.4 Shunt Power Capacitors 526

13.3.5 Static Var Controllers 528

13.3.6 Series Capacitors 530

13.4 Some Examples of Reactive Power Flow 533

13.5 Flexible AC Transmission Systems 538

13.5.1 Synchronous Voltage Source 540

13.5.2 Static Synchronous Compensator 542

13.5.3 Static Series Synchronous Compensator 544

13.5.4 Unified Power Flow Controller 547

Problems 549

References 550

14 Three-Phase and Distribution System Load Flow 551

14.1 Phase Coordinate Method 552

14.2 Three-Phase Models 554

14.2.1 Conductors 554

14.2.2 Generators 555

14.2.3 Generator Model for Cogeneration 557

14.2.4 Three-Phase Transformer Models 559

14.2.4.1 Symmetrical Components of Three-Phase Transformers 562

14.2.5 Load Models 566

14.3 Distribution System Load Flow 567

14.3.1 Methodology 569

14.3.2 Distribution System as a Ladder Network 570

14.4 Optimal Capacitor Locations 572

References 575

15 Optimization Techniques 577

15.1 Functions of One Variable 578

15.2 Concave and Convex Functions 579

15.3 Taylor’s Theorem 580

15.4 Lagrangian Method: Constrained Optimization 582

15.5 Multiple Equality Constraints 584

15.6 Optimal Load Sharing between Generators 585

15.7 Inequality Constraints 587

15.8 Kuhn–Tucker Theorem 589

15.9 Search Methods 590

15.9.1 Univariate Search Method 591

15.9.2 Powell’s Method of Conjugate Directions 592

15.10 Gradient Methods 592

15.10.1 Method of Optimal Gradient 593

15.11 Linear Programming—Simplex Method 595

15.12 Quadratic Programming 599

15.13 Dynamic Programming 601

15.13.1 Optimality 602

15.14 Integer Programming 605

Problems 606

References 607

16 Optimal Power Flow 609

16.1 Optimal Power Flow 609

16.1.1 Handling Constraints 610

16.2 Decoupling Real and Reactive OPF 611

16.3 Solution Methods of OPF 612

16.4 Generation Scheduling Considering Transmission Losses 614

16.4.1 General Loss Formula 615

16.4.2 Solution of Coordination Equation 617

16.5 Steepest Gradient Method 621

16.5.1 Adding Inequality Constraints on Control Variables 623

16.5.2 Inequality Constraints on Dependent Variables 623

16.6 OPF Using the Newton Method 624

16.6.1 Functional Constraints 625

16.6.2 Lagrangian Function 626

16.6.3 Hessian Matrix 627

16.6.4 Active Set 628

16.6.5 Penalty Techniques 629

16.6.6 Selecting Active Set 629

16.6.7 Algorithm for the Coupled Newton OPF 629

16.6.8 Decoupled Formation 630

16.7 Sequential Quadratic Programming 631

16.8 Successive Linear Programming 632

16.9 Interior Point Methods and Variants 634

16.9.1 Karmarkar Interior Point Algorithm 635

16.9.1.1 Check for Infeasibility 636

16.9.1.2 Check for Optimality 636

16.9.2 Barrier Methods 636

16.9.3 Primal–Dual IP Method 637

16.10 Security and Environmental Constrained OPF 638

References 640

17 Harmonics Generation 643

17.1 Harmonics and Sequence Components 645

17.2 Increase in Nonlinear Loads 646

17.3 Harmonic Factor 646

17.4 Three-Phase Windings in Electrical Machines 646

17.4.1 Cogging and Crawling of Induction Motors 648

17.5 Tooth Ripples in Electrical Machines 649

17.6 Synchronous Generators 650

17.7 Transformers 651

17.8 Saturation of Current Transformers 654

17.9 Shunt Capacitors 655

17.10 Sub-Harmonic Frequencies 656

17.11 Static Power Converters 656

17.11.1 Single-Phase Bridge Circuit 657

17.11.1.1 Phase Control 658

17.11.1.2 Power Factor, Distortion Factor, and Total Power Factor 660

17.11.1.3 Harmonics on Output Side 662

17.11.2 Three-Phase Bridge Circuit 663

17.11.2.1 Cancellation of Harmonics Due to Phase Multiplication 668

17.11.2.2 Effect of Source Impedance 669

17.11.2.3 Effect of Output Reactor 671

17.11.2.4 Effect of Load with Back EMF 671

17.11.2.5 Inverter Operation 671

17.11.3 Diode Bridge Converter 672

17.12 Switch-Mode Power (SMP) Supplies 673

17.13 Arc Furnaces 675

17.14 Cycloconverters 676

17.15 Thyristor-Controlled Reactor 678

17.16 Thyristor-Switched Capacitors 679

17.17 Pulse-Width Modulation 679

17.18 Adjustable Speed Drives 681

17.19 Pulse Burst Modulation 682

17.20 Chopper Circuits and Electric Traction 683

17.21 Slip Frequency Recovery Schemes 684

17.22 Lighting Ballasts 685

17.23 Voltage Source Converters 686

17.23.1 Three-Level Converter 686

17.24 Inter-Harmonics 688

Problems 690

References 691

18 Effects of Harmonics 693

18.1 Rotating Machines 694

18.1.1 Pulsating Fields and Torsional Vibrations 694

18.1.2 Sub-Harmonic Frequencies and Sub-Synchronous Resonance 695

18.1.3 Increase of Losses 695

18.1.4 Effect of Negative Sequence Currents 695

18.1.5 Insulation Stresses 697

18.1.6 Bearing Currents and Shaft Voltages 698

18.1.7 Effect of Cable Type and Length 698

18.2 Transformers 699

18.2.1 Calculations from Transformer Test Data 701

18.2.2 Liquid-Filled Transformers 703

18.2.3 UL K-Factor of Transformers 705

18.3 Cables 706

18.4 Capacitors 708

18.5 Harmonic Resonance 710

18.5.1 Parallel Resonance 710

18.5.2 Series Resonance 714

18.5.3 Part-Winding Resonance 715

18.6 Voltage Notching 715

18.7 Electromagnetic Interference 716

18.8 Overloading of Neutral 717

18.9 Protective Relays and Meters 718

18.10 Circuit Breakers and Fuses 719

18.11 Telephone Influence Factor 719

Problems 721

References 722

19 Harmonic Analysis 725

19.1 Harmonic Analysis Methods 725

19.1.1 Frequency-Domain Analysis 726

19.1.2 Frequency Scan 727

19.1.3 Voltage Scan 728

19.1.4 Phase Angle of Harmonics 728

19.1.5 Newton–Raphson Method 729

19.1.6 Time-Domain Analysis 730

19.1.7 Switching Function 731

19.2 Harmonic Modeling of System Components 732

19.2.1 Transmission Lines 732

19.2.2 Underground Cables 735

19.2.3 Filter Reactors 736

19.2.4 Transformers 736

19.2.5 Induction Motors 738

19.2.6 Generators 739

19.3 Load Models 739

19.4 System Impedance 740

19.5 Three-Phase Models 741

19.5.1 Uncharacteristic Harmonics 742

19.5.2 Converters 743

19.6 Modeling of Networks 745

19.6.1 Industrial Systems 745

19.6.2 Distribution Systems 746

19.6.3 Transmission Systems 746

19.6.4 Sensitivity Methods 747

19.7 Power Factor and Reactive Power 749

19.8 Shunt Capacitor Bank Arrangements 751

19.9 Unbalance Detection 756

19.10 Study Cases 757

Problems 775

References 776

20 Harmonic Mitigation and Filters 779

20.1 Mitigation of Harmonics 779

20.2 Band-Pass Filters 780

20.2.1 Tuning Frequency 782

20.3 Practical Filter Design 783

20.4 Relations in an ST Filter 792

20.4.1 Number of Series Parallel Groups 795

20.5 Filters for a Furnace Installation 797

20.6 Filters for an Industrial Distribution System 799

20.7 Secondary Resonance 801

20.8 Filter Reactors 802

20.8.1 Q Factor 803

20.9 Double-Tuned Filter 805

20.10 Damped Filters 806

20.10.1 Second-Order High-Pass Filter 808

20.11 Design of a Second-Order High-Pass Filter 810

20.12 Zero-Sequence Traps 812

20.13 Limitations of Passive Filters 813

20.14 Active Filters 814

20.14.1 Shunt Connection 814

20.14.2 Series Connection 815

20.14.3 Hybrid Connection 815

20.14.4 Combination of Active Filters 816

20.15 Corrections in Time Domain 818

20.16 Corrections in the Frequency Domain 819

20.17 Instantaneous Reactive Power 819

20.18 Harmonic Mitigation at Source 821

20.18.1 Phase Multiplication 821

20.18.2 Parallel Connected 12 pu Converters, with Interphase Reactor 822

20.18.3 Active Current Shaping 824

20.19 Multilevel Converters 824

References 828

21 Arc Flash Hazard Analysis 831

21.1 Relating Short-Circuit Currents with Arc Flash and Personal Safety 831

21.1.1 Arc Blast 832

21.1.2 Fire Hazard and Electrical Shock 832

21.1.3 Time Motion Studies 832

21.2 Arc Flash Hazard Analysis 833

21.2.1 Ralph Lee’s Equations 834

21.2.2 IEEE 1584 Equations 835

21.3 Hazard=Risk Categories 838

21.3.1 Hazard Boundaries 838

21.4 System Grounding: Impact on Incident Energy 839

21.5 Duration of an Arc Flash Event and Arc Flash Boundary 841

21.5.1 Equipment Labeling 842

21.6 Protective Relaying and Coordination 843

21.6.1 Unit Protection Systems, Differential Relaying 844

21.6.2 Arc Flash Detection Relays 846

21.7 Short-Circuit Currents 847

21.7.1 Reducing Short-Circuit Currents 847

21.8 Arc Flash Calculations in Medium-Voltage Systems 848

21.8.1 Reduction of HRC through a Maintenance Mode Switch 850

21.8.2 Arc Resistant Switchgear 853

21.9 Arc Flash Calculations in Low-Voltage Systems 854

21.10 Accounting for Decaying Short-Circuit Currents 864

Problems 872

References 873

22 Wind Power 875

22.1 AEP 765 kV Transmission Grid Initiative in the United States 877

22.1.1 Maximum Transfer Capability 879

22.1.2 Power Reserves and Regulation 882

22.1.3 Congestion Management 882

22.2 Wind Energy Conversion 883

22.2.1 Drive Train 883

22.2.2 Towers 885

22.2.3 Rotor Blades 885

22.3 Cube Law 885

22.4 Operation 888

22.4.1 Speed Control 889

22.4.2 Behavior under Faults and Low-Voltage Ride Through 889

22.5 Wind Generators 890

22.5.1 Induction Generators 890

22.5.2 Direct Coupled Induction Generator 892

22.5.3 Induction Generator Connected to Grid through Full Size Converter 892

22.5.4 Doubly Fed Induction Generator 892

22.6 Power Electronics 894

22.6.1 ZS Inverters 894

22.7 Reactive Power Control 895

22.8 Harmonics 897

22.9 Computer Modeling 898

22.9.1 Wind Turbine Controller 898

References 900

Appendix A: Matrix Methods 903

Appendix B: Calculation of Line and Cable Constants 933

Appendix C: Transformers and Reactors 955

Appendix D: Sparsity and Optimal Ordering 983

Appendix E: Fourier Analysis 991

Appendix F: Limitation of Harmonics 1009

Appendix G: Estimating Line Harmonics 1023

Index 1033

Power engineering is the oldest and most traditional of the various areas within electricalengineering, yet no other facet of modern technology is currently undergoing a moredramatic revolution in both technology and industry structure But none of these changesalter the basic complexity of electric power system behavior, or reduce the challenge thatpower system engineers have always faced in designing an economical system that operates

as intended and shuts down in a safe and noncatastrophic mode when something failsunexpectedly In fact, many of the ongoing changes in the power industry—deregulation,reduced budgets and staffing levels, and increasing public and regulatory demand forreliability among them—make these challenges all the more difficult to overcome

Therefore, I am particularly delighted to see this latest addition to the Power Engineeringseries J.C Das’s Power System Analysis: Short-Circuit Load Flow and Harmonics providescomprehensive coverage of both theory and practice in the fundamental areas of powersystem analysis, including power flow, short-circuit computations, harmonics, machinemodeling, equipment ratings, reactive power control, and optimization It also includes anexcellent review of the standard matrix mathematics and computation methods of powersystem analysis in a readily usable format

Of particular note, this book discusses both ANSI=IEEE and IEC methods, guidelines,and procedures for applications and ratings Over the past few years, my work as vicepresident of technology and strategy for ABB’s global consulting organization has given

me an appreciation that the IEC and ANSI standards are not so much in conflict as they areslightly different but equally valid approaches to power engineering There is much to belearned from each, and from the study of the differences between them

As the editor of the Power Engineering series, I am proud to include Power SystemAnalysis among this important group of books Like all the volumes in the Power Engin-eering series, this book provides modern power technology in a context of proven,practical application It is useful as a reference book as well as for self-study and advancedclassroom use The series includes books covering the entirefield of power engineering, inall its specialties and subgenres, all aimed at providing practicing power engineers with theknowledge and techniques they need to meet the electric industry’s challenges in thetwenty-first century

H Lee Willis

xxi

In recent times, two new aspects of power system analysis have emerged: (1) the arcflash hazard analysis and reduction of hazard risk category (HRC) in electrical systemsand (2) the wind power generation and its integration in utility systems Maintainingthe structure and order of thefirst edition of the book, two new chapters, Chapters 21and 22, have been added to address these new technologies The ANSI=IEEE ratingstructures of the high-voltage circuit breakers have undergone many changes in anattempt to harmonize with IEC standards Chapters 7 through 9 have been revised toreflect these changes and current ANSI=IEEE and IEC standards New material has beenadded to practically each chapter, for example, Chapters 12 through 15 on load flowand Chapters 17 through 20 on harmonic analysis and harmonicfilter designs Errata ofthe first edition have been taken care of New figures and supporting mathematicalequations have been added where required.

This new edition should prove all the more popular with the academia and practicingpower system engineers as it enhances the technical content and the presentation of thesubjects covered in this book

I would like to thank Nora Konopka of CRC Press for all her help and cooperation inpublishing this second edition

J.C Das

xxiii

Power system analysis is fundamental in the planning, design, and operating stages, andits importance cannot be overstated This book covers the commonly required short-circuit,loadflow, and harmonic analyses Practical and theoretical aspects have been harmoni-ously combined Although there is the inevitable computer simulation, a feel for theprocedures and methodology is also provided, through examples and problems PowerSystem Analysis: Short-Circuit Load Flow and Harmonics should be a valuable addition to thepower system literature for practicing engineers, those in continuing education, andcollege students.

Short-circuit analyses are included in chapters on rating structures of breakers, currentinterruption in ac circuits, calculations according to the IEC and ANSI=IEEE methods, andcalculations of short-circuit currents in dc systems

The loadflow analyses cover reactive power flow and control, optimization techniques,and introduction to FACT controllers, three-phase loadflow, and optimal power flow.The effect of harmonics on power systems is a dynamic and evolvingfield (harmoniceffects can be experienced at a distance from their source) The book derives and compilesample data of practical interest, with the emphasis on harmonic powerflow and harmonicfilter design Generation, effects, limits, and mitigation of harmonics are discussed, includ-ing active and passivefilters and new harmonic mitigating topologies

The models of major electrical equipment—i.e., transformers, generators, motors, mission lines, and power cables—are described in detail Matrix techniques and symmet-rical component transformation form the basis of the analyses There are many examplesand problems The references and bibliographies point to further reading and analyses.Most of the analyses are in the steady state, but references to transient behavior areincluded where appropriate

trans-A basic knowledge of per unit system, electrical circuits and machinery, and matrices isrequired, although an overview of matrix techniques is provided in Appendix A The style

of writing is appropriate for the upper-undergraduate level, and some sections are atgraduate-course level

Power Systems Analysis is a result of my long experience as a practicing power systemengineer in a variety of industries, power plants, and nuclear facilities Its unique feature isapplications of power system analyses to real-world problems

I thank ANSI=IEEE for permission to quote from the relevant ANSI=IEEE standards TheIEEE disclaims any responsibility or liability resulting from the placement and use in thedescribed manner I am also grateful to the International Electrotechnical Commission(IEC) for permission to use material from the international standards IEC 60660-1 (1997)and IEC 60909 (1988) All extracts are copyright IEC Geneva, Switzerland All rightsreserved Further information on the IEC, its international standards, and its role isavailable at www.iec.ch IEC takes no responsibility for and will not assume liabilityfrom the reader’s misinterpretation of the referenced material due to its placement andcontext in this publication The material is reproduced or rewritten with their permission.Finally, I thank the staff of Marcel Dekker, Inc., and special thanks to Ann Pulido for herhelp in the production of this book

J.C Das

xxv

J.C Dasis currently staff consultant, Electrical Power Systems, AMEC Inc., Tucker, Georgia.

He has varied experience in the utility industry, industrial establishments, hydroelectricgeneration, and atomic energy He is responsible for power system studies, including shortcircuit, loadflow, harmonics, stability, arc-flash hazard, grounding, switching transients,and protective relaying He conducts courses for continuing education in power systemsand has authored or coauthored about 60 technical publications He is the author of the bookTransients in Electrical Systems—Analysis Recognition and Mitigation, McGraw-Hill, NewYork, 2010 His interests include power system transients, EMTP simulations, harmonics,power quality, protection, and relaying He has also published 190 electrical power systemstudy reports for his clients

Das is a life fellow of the Institute of Electrical and Electronics Engineers, IEEE (UnitedStates), a member of the IEEE Industry Applications and IEEE Power Engineering societies,

a fellow of the Institution of Engineering Technology (United Kingdom), a life fellow of theInstitution of Engineers (India), a member of the Federation of European Engineers(France), and a member of CIGRE (France) He is a registered professional engineer inthe states of Georgia and Oklahoma, a chartered engineer (CEng) in the United Kingdom,and a European engineer (Eur Ing.)

He received his MSEE from Tulsa University, Tulsa, Oklahoma, and his BA (advancedmathematics) and BEE from Punjab University, India

xxvii

Short-Circuit Currents and Symmetrical

Components

Short circuits occur in well-designed power systems and cause large decaying transientcurrents, generally much above the system load currents These result in disruptiveelectrodynamic and thermal stresses that are potentially damaging Fire risks and explo-sions are inherent One tries to limit short circuits to the faulty section of the electricalsystem by appropriate switching devices capable of operating under short-circuit condi-tions without damage and isolating only the faulty section, so that a fault is not escalated.The faster the operation of sensing and switching devices, the lower is the fault damage,and the better is the chance of systems holding together without loss of synchronism.Short circuits can be studied from the following angles:

1 Calculation of short-circuit currents

2 Interruption of short-circuit currents and rating structure of switching devices

3 Effects of short-circuit currents

4 Limitation of short-circuit currents, that is, with current-limiting fuses and faultcurrent limiters

5 Short-circuit withstand ratings of electrical equipment like transformers, reactors,cables, and conductors

6 Transient stability of interconnected systems to remain in synchronism until thefaulty section of the power system is isolated

We will confine our discussions to the calculations of short-circuit currents, and the basis ofshort-circuit ratings of switching devices, that is, power circuit breakers and fuses As themain purpose of short-circuit calculations is to select and apply these devices properly, it ismeaningful for the calculations to be related to current interruption phenomena and therating structures of interrupting devices The objectives of short-circuit calculations, there-fore, can be summarized as follows:

Determination of short-circuit duties on switching devices, that is, high-, medium-,and low-voltage circuit breakers and fuses

Calculation of short-circuit currents required for protective relaying and ation of protective devices

coordin- Evaluations of adequacy of short-circuit withstand ratings of static equipment likecables, conductors, bus bars, reactors, and transformers

Calculations of fault voltage dips and their time-dependent recovery profilesThe type of short-circuit currents required for each of these objectives may not beimmediately clear but will unfold in the chapters to follow

1

In a three-phase system, a fault may equally involve all three phases A bolted faultmeans that the three phases are connected together with links of zero impedance prior tothe fault, that is, the fault impedance itself is zero and the fault is limited by the system andmachine impedances only Such a fault is called a symmetrical three-phase bolted fault or asolid fault Bolted three-phase faults are rather uncommon Generally, such faults give themaximum short-circuit currents and form the basis of calculations of short-circuit duties onswitching devices.

Faults involving one, or more than one, phase and ground are called unsymmetricalfaults Under certain conditions, the line-to-ground fault or double line-to-ground faultcurrents may exceed three-phase symmetrical fault currents, discussed in the chapters tofollow Unsymmetrical faults are more common as compared to three-phase faults, that is,

a support insulator on one of the phases on a transmission line may start flashing toground, ultimately resulting in a single line-to-ground fault

Short-circuit calculations are, thus, the primary study whenever a new power system isdesigned or an expansion and upgrade of an existing system are planned

1.1 Nature of Short-Circuit Currents

The transient analysis of the short circuit of a passive impedance connected to an ing current (ac) source gives an initial insight into the nature of the short-circuit currents.Consider a sinusoidal time-invariant single-phase 60 Hz source of power, Em sin vt,connected to a single-phase short distribution line, Z ¼ (R þ jvL), where Z is the compleximpedance, R and L are the resistance and inductance, Emis the peak source voltage, and v

alternat-is the angular frequency ¼ 2pf, f being the frequency of the ac source For a balanced phase system, a single-phase model is adequate, as we will discuss further Let a shortcircuit occur at the far end of the line terminals As an ideal voltage source is considered,that is, zero Thévenin impedance, the short-circuit current is limited only by Z, and itssteady-state value is vectorially given by Em=Z This assumes that the impedance Z doesnot change withflow of the large short-circuit current For simplification of empirical short-circuit calculations, the impedances of static components like transmission lines, cables,reactors, and transformers are assumed to be time invariant Practically, this is not true,that is, theflux densities and saturation characteristics of core materials in a transformermay entirely change its leakage reactance Driven to saturation under high currentflow,distorted waveforms and harmonics may be produced

three-Ignoring these effects and assuming that Z is time invariant during a short circuit, thetransient and steady-state currents are given by the differential equation of the R–L circuitwith an applied sinusoidal voltage:

Ldi

dtþ Ri ¼ Emsin(vt þ u) (1:1)where u is the angle on the voltage wave, at which the fault occurs The solution of thisdifferential equation is given by

i ¼ Imsin(vt þ u  f)  Imsin(u  f)eRt=L (1:2)

where Imis the maximum steady-state current, given by Em=Z, and the angle

f¼ tan1(vL)

R .

In power systems vL>> R A 100 MVA, 0.85 power factor synchronous generator mayhave an X=R of 110, and a transformer of the same rating, an X=R of 45 The X=R ratios inlow-voltage systems are on the order of 2–8 For present discussions, assume a high X=Rratio, that is, f  908

If a short circuit occurs at an instant t ¼ 0, u ¼ 0 (i.e., when the voltage wave is crossingthrough zero amplitude on the X-axis), the instantaneous value of the short-circuit current,from Equation 1.2, is 2Im This is sometimes called the doubling effect

If a short circuit occurs at an instant when the voltage wave peaks, t ¼ 0, ¼ p=2, thesecond term in Equation 1.2 is zero and there is no transient component

These two situations are shown in Figure 1.1a and b The voltage at the point of boltedfault will be zero The voltage E shown in Figure 1.1a and b signifies that prior to fault andafter the fault is cleared, the voltage remains constant

A simple explanation of the origin of the transient component is that in power systems theinductive component of the impedance is high The current in such a circuit is at zero valuewhen the voltage is at peak, and for a fault at this instant no direct current (dc) component isrequired to satisfy the physical law that the current in an inductive circuit cannot change

Upper envelope

dc component

nondecaying) Fault at

suddenly When the fault occurs at an instant when u ¼ 0, there has to be a transient currentwhose initial value is equal and opposite to the instantaneous value of the ac short-circuitcurrent This transient current, the second term of Equation 1.2 can be called a dc componentand it decays at an exponential rate Equation 1.2 can be simply written as

i ¼ Imsin vt þ IdceRt=L (1:3)where the initial value of Idc¼ Im (1:4)The following inferences can be drawn from the above discussions:

1 There are two distinct components of a short-circuit current: (1) a non-decaying accomponent or the steady-state component, and (2) a decaying dc component at anexponential rate, the initial magnitude of which is a maximum of the ac componentand it depends on the time on the voltage wave at which the fault occurs

2 The decrement factor of a decaying exponential current can be defined as its valueany time after a short circuit, expressed as a function of its initial magnitude perunit Factor L=R can be termed the time constant The exponential then becomes

Idcet=t0, where t0¼ L=R In this equation, making t ¼ t0¼ time constant will result in

a decay of approximately 62.3% from its initial magnitude, that is, the transitorycurrent is reduced to a value of 0.368 per unit after an elapsed time equal to thetime constant, as shown in Figure 1.2

3 The presence of a dc component makes the fault current wave-shape envelopeasymmetrical about the zero line and axis of the wave Figure 1.1a clearly showsthe profile of an asymmetrical waveform The dc component always decays to zero

in a short time Consider a modest X=R ratio of 15, say, for a medium-voltage 13.8

kV system The dc component decays to 88% of its initial value infive cycles Thehigher the X=R ratio, the slower is the decay and the longer is the time for whichthe asymmetry in the total current will be sustained The stored energy can bethought to be expanded in I2R losses After the decay of the dc component, only thesymmetrical component of the short-circuit current remains

4 Impedance is considered as time invariant in the above scenario Synchronousgenerators and dynamic loads, that is, synchronous and induction motors are themajor sources of short-circuit currents The trappedflux in these rotating machines

at the instant of short circuit cannot change suddenly and decays, depending onmachine time constants Thus, the assumption of constant L is not valid forrotating machines and decay in the ac component of the short-circuit currentmust also be considered

5 In a three-phase system, the phases are time displaced from each other by

120 electrical degrees If a fault occurs when the unidirectional component inphase a is zero, the phase b component is positive and the phase c component isequal in magnitude and negative Figure 1.3 shows a three-phase fault currentwaveform As the fault is symmetrical, Iaþ Ibþ Icis zero at any instant, where Ia, Ib,and Icare the short-circuit currents in phases a, b, and c, respectively For a faultclose to a synchronous generator, there is a 120 Hz current also, which rapidlydecays to zero This gives rise to the characteristic non-sinusoidal shape of three-phase short-circuit currents observed in test oscillograms The effect is insignifi-cant, and ignored in the short-circuit calculations This is further discussed inChapter 6

6 The load current has been ignored Generally, this is true for empirical short-circuitcalculations, as the short-circuit current is much higher than the load current.Sometimes the load current is a considerable percentage of the short-circuit cur-rent The load currents determine the effective voltages of the short-circuit sources,prior to fault

FIGURE 1.3 Asymmetries in phase currents in a three-phase short circuit.

I a

The ac short-circuit current sources are synchronous machines, that is, turbogenerators andsalient pole generators, asynchronous generators, and synchronous and asynchronousmotors Converter motor drives may contribute to short-circuit currents when operating

in the inverter or regenerative mode For extended duration of short-circuit currents, thecontrol and excitation systems, generator voltage regulators, and turbine governor char-acteristics affect the transient short-circuit process

The duration of a short-circuit current depends mainly on the speed of operation ofprotective devices and on the interrupting time of the switching devices

1.2 Symmetrical Components

The method of symmetrical components has been widely used in the analysis of anced three-phase systems, unsymmetrical short-circuit currents, and rotating electro-dynamic machinery The method was originally presented by Fortescue in 1918 and hasbeen popular ever since

unbal-Unbalance occurs in three-phase power systems due to faults, single-phase loads,untransposed transmission lines, or nonequilateral conductor spacings In a three-phasebalanced system, it is sufficient to determine the currents and voltages in one phase, andthe currents and voltages in the other two phases are simply phase displaced In anunbalanced system, the simplicity of modeling a three-phase system as a single-phasesystem is not valid A convenient way of analyzing unbalanced operation is throughsymmetrical components The three-phase voltages and currents, which may be unbal-anced, are transformed into three sets of balanced voltages and currents, called symmet-rical components The impedances presented by various power system components, that

is, transformers, generators, and transmission lines, to symmetrical components aredecoupled from each other, resulting in independent networks for each component Theseform a balanced set This simplifies the calculations

Familiarity with electrical circuits and machine theory, per unit system, and matrixtechniques is required before proceeding with this book A review of the matrix techniques

in power systems is included in Appendix A The notations described in this appendix forvectors and matrices are followed throughout the book

The basic theory of symmetrical components can be stated as a mathematical concept

A system of three coplanar vectors is completely defined by six parameters, and thesystem can be said to possess six degrees of freedom A point in a straight line beingconstrained to lie on the line possesses but one degree of freedom, and by the sameanalogy, a point in space has three degrees of freedom A coplanar vector is defined byits terminal and length and therefore possesses two degrees of freedom A system ofcoplanar vectors having six degrees of freedom, that is, a three-phase unbalanced current

or voltage vectors, can be represented by three symmetrical systems of vectors eachhaving two degrees of freedom In general, a system of n numbers can be resolved into

n sets of component numbers each having n components, that is, a total of n2components.Fortescue demonstrated that an unbalanced set on n phasors can be resolved into n  1balanced phase systems of different phase sequence and one zero sequence system, inwhich all phasors are of equal magnitude and cophasial:

Va,Vb, , Vnare original n unbalanced voltage phasors

Va1, Vb1, , Vn1are thefirst set of n balanced phasors, at an angle of 2p=n between them

Va2, Vb2, , Vn2are the second set of n balanced phasors at an angle 4p=n

Van, Vbn, , Vnnis the zero sequence set, all phasors at n(2p=n) ¼ 2p, that is, cophasial

In a symmetrical three-phase balanced system, the generators produce balanced voltages,which are displaced from each other by 2p=3 ¼ 1208 These voltages can be called positivesequence voltages If a vector operator a is defined, which rotates a unit vector through 1208

in a counterclockwise direction, then a ¼ 0.5 þ j0.866, a2¼ 0.5 – j0.866, a3¼ 1, 1 þ a2þ

a ¼ 0 Considering a three-phase system, Equation 1.5 reduces to

Va¼ Va0þ Va1þ Va2

Vb¼ Vb0þ Vb1þ Vb2

Vc¼ Vc0þ Vc1þ Vc2

(1:6)

We can define the set consisting of Va0, Vb0, and Vc0as the zero sequence set, the set Va1,

Vb1, and Vc1 as the positive sequence set, and the set Va2, Vb2, and Vc2 as the negativesequence set of voltages The three original unbalanced voltage vectors give rise to ninevoltage vectors, which must have constraints of freedom and are not totally independent

By definition of positive sequence, Va1, Vb1, and Vc1 should be related as follows, as in anormal balanced system:

Vb1¼ a2Va1, Vc1¼ aVa1Note that Va1phasor is taken as the reference vector

The negative sequence set can be similarly defined, but of opposite phase sequence:

Vb2¼ aVa2, Vc2¼ a2Va2Also, Va0¼ Vb0¼ Vc0 With these relations defined, Equation 1.6 can be written as

While this simple explanation may be adequate, a better insight into the symmetricalcomponent theory can be gained through matrix concepts of similarity transformation,diagonalization, eigenvalues, and eigenvectors.

The discussions to follow show that

Eigenvectors giving rise to symmetrical component transformation are the samethough the eigenvalues differ Thus, these vectors are not unique

The Clarke component transformation is based on the same eigenvectors butdifferent eigenvalues

The symmetrical component transformation does not uncouple an initially anced three-phase system Prima facie this is a contradiction of what we said earlier,that the main advantage of symmetrical components lies in decoupling unbalancedsystems, which could then be represented much akin to three-phase balancedsystems We will explain what is meant by this statement as we proceed

unbal-1.3 Eigenvalues and Eigenvectors

The concept of eigenvalues and eigenvectors is related to the derivation of symmetricalcomponent transformation It can be briefly stated as follows

Consider an arbitrary square matrix A If a relation exists so that

where

lis a scalar called an eigenvalue, characteristic value, or root of the matrix A

x is a vector called the eigenvector or characteristic vector of A

Then, there are n eigenvalues and corresponding n sets of eigenvectors associated with anarbitrary matrix A of dimensions n  n The eigenvalues are not necessarily distinct, andmultiple roots occur

Equation 1.9 can be written as

[A  lI] [x] ¼ 0 (1:10)where I is the identity matrix Expanding:

  0

This can be expanded to yield an nth order algebraic equation:

anlnþ an Iln 1 þ    þ a1lþ a0¼ 0, that is,(l1 a1)(l2 a2) (ln an) ¼ 0 (1:13)

Equations 1.12 and 1.13 are called the characteristic equations of the matrix A The roots

l1, l2, l3, , lnare the eigenvalues of matrix A The eigenvector xjcorresponding to ljisfound from Equation 1.10 See Appendix A for details and an example

1.4 Symmetrical Component Transformation

Application of eigenvalues and eigenvectors to the decoupling of three-phase systems isuseful when we define similarity transformation This forms a diagonalization techniqueand decoupling through symmetrical components

Anxn¼ yn

An¼ C1A C (1:15)

Anxn¼ yn is distinct from Ax ¼ y The only restriction on choosing C is that it should benonsingular Equation 1.15 is a set of linear equations, derived from the original equations(Equation 1.14) and yet distinct from them

If C is a nodal matrix M, corresponding to the coefficients of A, then

C ¼ M ¼ [x1, x2   xn] (1:16)

where xiare the eigenvectors of the matrix A, then

A The matrices A and Anhave the same eigenvalues and are called similar matrices Thetransformation matrix C is nonsingular

1.4.2 Decoupling a Three-Phase Symmetrical System

Let us decouple a three-phase transmission line section, where each phase has a mutualcoupling with respect to ground This is shown in Figure 1.4a An impedance matrix of thethree-phase transmission line can be written as

Zaa, Zbb, and Zccare the self-impedances of the phases a, b, and c

Zabis the mutual impedance between phases a and b

Zbais the mutual impedance between phases b and a

Assume that the line is perfectly symmetrical This means all the mutual impedances, that is,

Zab¼ Zba¼ M and all the self-impedances, that is, Zaa¼ Zbb¼ Zcc¼ Z are equal This reducesthe impedance matrix to

It is required to decouple this system using symmetrical components First find theeigenvalues:

Zero sequence