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

Das
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.

Power System
Analysis

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.

Short-Circuit Load Flow and Harmonics
Se c ond Ed i t ion

Se cond
Ed i t i on

J. C. Das

K11101

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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


20. Electrical Power Cable Engineering, Second Edition
William Thue
21. Vehicular Electric Power Systems: Land, Sea, Air, and Space
Vehicles, Ali Emadi, Mehrdad Ehsani, and John Miller
22. Power Distribution Planning Reference Book, Second Edition,
H. Lee Willis
23. Power System State Estimation: Theory and Implementation,
Ali Abur
24. Transformer Engineering: Design and Practice, S.V. Kulkarni
and S. A. Khaparde
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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CRC Press
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Contents
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


viii

Contents

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


Contents

ix

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-SF6 Breakers .................................................................... 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

x

Contents

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 Xd00 ......................................................................... 216
6.1.3 Transient Reactance Xd0 .............................................................................. 216
6.1.4 Synchronous Reactance Xd ........................................................................ 216
6.1.5 Quadrature Axis Reactances Xq00 , 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
0
.............................................................. 219
6.3.1 Open Circuit Time Constant Tdo
6.3.2 Subtransient Short-Circuit Time Constant Td00 ......................................... 219
6.3.3 Transient Short-Circuit Time Constant Td0 .............................................. 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


Contents

xi

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


xii

Contents

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


Contents

xiii

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
Flow Methods: Part I............................................................................................... 413
Modeling a Two-Winding Transformer ............................................................... 414
Load Flow—Bus Types ........................................................................................... 418
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

11. Load

11.1
11.2
11.3

12. Load
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8

Flow Methods: Part II ............................................................................................. 449
Function with One Variable ................................................................................... 449
Simultaneous Equations .......................................................................................... 451
Rectangular Form of Newton–Raphson Method of Load Flow ....................... 453
Polar Form of Jacobian Matrix ............................................................................... 455
12.4.1 Calculation Procedure of Newton–Raphson Method............................ 457
Simplifications of Newton–Raphson Method ...................................................... 463
Decoupled Newton–Raphson Method.................................................................. 465
Fast Decoupled Load Flow ..................................................................................... 466
Model of a Phase-Shifting Transformer ................................................................ 469


xiv

Contents

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 Z0 Compensation ........................................................................................ 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


Contents

xv

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


xvi

Contents

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


Contents

xvii

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


xviii

Contents

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


Contents

xix

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


xx

Contents

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


Series Introduction
Power engineering is the oldest and most traditional of the various areas within electrical
engineering, yet no other facet of modern technology is currently undergoing a more
dramatic revolution in both technology and industry structure. But none of these changes
alter the basic complexity of electric power system behavior, or reduce the challenge that
power 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 fails
unexpectedly. In fact, many of the ongoing changes in the power industry—deregulation,
reduced budgets and staffing levels, and increasing public and regulatory demand for
reliability among them—make these challenges all the more difficult to overcome.
Therefore, I am particularly delighted to see this latest addition to the Power Engineering
series. J.C. Das’s Power System Analysis: Short-Circuit Load Flow and Harmonics provides
comprehensive coverage of both theory and practice in the fundamental areas of power
system analysis, including power flow, short-circuit computations, harmonics, machine
modeling, equipment ratings, reactive power control, and optimization. It also includes an
excellent review of the standard matrix mathematics and computation methods of power
system 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 vice
president 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 are
slightly different but equally valid approaches to power engineering. There is much to be
learned 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 System
Analysis among this important group of books. Like all the volumes in the Power Engineering 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 advanced
classroom use. The series includes books covering the entire field of power engineering, in
all its specialties and subgenres, all aimed at providing practicing power engineers with the
knowledge and techniques they need to meet the electric industry’s challenges in the
twenty-first century.
H. Lee Willis

xxi



Preface to the Second Edition
In recent times, two new aspects of power system analysis have emerged: (1) the arc
flash hazard analysis and reduction of hazard risk category (HRC) in electrical systems
and (2) the wind power generation and its integration in utility systems. Maintaining
the structure and order of the first edition of the book, two new chapters, Chapters 21
and 22, have been added to address these new technologies. The ANSI=IEEE rating
structures of the high-voltage circuit breakers have undergone many changes in an
attempt to harmonize with IEC standards. Chapters 7 through 9 have been revised to
reflect these changes and current ANSI=IEEE and IEC standards. New material has been
added to practically each chapter, for example, Chapters 12 through 15 on load flow
and Chapters 17 through 20 on harmonic analysis and harmonic filter designs. Errata of
the first edition have been taken care of. New figures and supporting mathematical
equations have been added where required.
This new edition should prove all the more popular with the academia and practicing
power system engineers as it enhances the technical content and the presentation of the
subjects covered in this book.
I would like to thank Nora Konopka of CRC Press for all her help and cooperation in
publishing this second edition.
J.C. Das

xxiii