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Summary
Summary
Featuring extensive calculations and examples, this reference discusses theoretical and practical aspects of short-circuit currents in ac and dc systems, load flow, and harmonic analyses to provide a sound knowledge base for modern computer-based studies that can be utilized in real-world applications. Presenting more than 2300 figures, tables, and equations, the author explores matrix methods for network solutions and includes load flow and optimization techniques. He discusses ac and dc short-circuit systems calculations in accordance with standards set by the American National Standards Institute (ANSI) and the International Electrotechnical Commission (IEC) .
Author Notes
J. C. Das is a Staff Consultant, Electrical Power Systems, for Amec, Inc., Atlanta, Georgia
Table of Contents
Series Introduction | p. iii |
Preface | p. v |
1. Short-Circuit Currents and Symmetrical Components | p. 1 |
1.1 Nature of Short-Circuit Currents | p. 2 |
1.2 Symmetrical Components | p. 5 |
1.3 Eigenvalues and Eigenvectors | p. 8 |
1.4 Symmetrical Component Transformation | p. 9 |
1.5 Clarke Component Transformation | p. 15 |
1.6 Characteristics of Symmetrical Components | p. 16 |
1.7 Sequence Impedance of Network Components | p. 20 |
1.8 Computer Models of Sequence Networks | p. 35 |
2. Unsymmetrical Fault Calculations | p. 39 |
2.1 Line-to-Ground Fault | p. 40 |
2.2 Line-to-Line Fault | p. 42 |
2.3 Double Line-to-Ground Fault | p. 43 |
2.4 Three-Phase Fault | p. 45 |
2.5 Phase Shift in Three-Phase Transformers | p. 46 |
2.6 Unsymmetrical Fault Calculations | p. 53 |
2.7 System Grounding and Sequence Components | p. 61 |
2.8 Open Conductor Faults | p. 64 |
3. Matrix Methods for Network Solutions | p. 72 |
3.1 Network Models | p. 73 |
3.2 Bus Admittance Matrix | p. 73 |
3.3 Bus Impedance Matrix | p. 78 |
3.4 Loop Admittance and Impedance Matrices | p. 81 |
3.5 Graph Theory | p. 82 |
3.6 Bus Admittance and Impedance Matrices by Graph Approach | p. 86 |
3.7 Algorithms for Construction of Bus Impedance Matrix | p. 89 |
3.8 Short-Circuit Calculations with Bus Impedance Matrix | p. 103 |
3.9 Solution of Large Network Equations | p. 113 |
4. Current Interruption in AC Networks | p. 116 |
4.1 Rheostatic Breaker | p. 117 |
4.2 Current-Zero Breaker | p. 118 |
4.3 Transient Recovery Voltage | p. 120 |
4.4 The Terminal Fault | p. 125 |
4.5 The Short-Line Fault | p. 127 |
4.6 Interruption of Low Inductive Currents | p. 127 |
4.7 Interruption of Capacitive Currents | p. 130 |
4.8 Prestrikes in Breakers | p. 133 |
4.9 Overvoltages on Energizing High-Voltage Lines | p. 134 |
4.10 Out-of-Phase Closing | p. 136 |
4.11 Resistance Switching | p. 137 |
4.12 Failure Modes of Circuit Breakers | p. 139 |
5. Application and Ratings of Circuit Breakers and Fuses According to ANSI Standards | p. 145 |
5.1 Total and Symmetrical Current Rating Basis | p. 145 |
5.2 Asymmetrical Ratings | p. 147 |
5.3 Voltage Range Factor K | p. 148 |
5.4 Capabilities for Ground Faults | p. 148 |
5.5 Closing-Latching-Carrying Interrupting Capabilities | p. 149 |
5.6 Short-Time Current Carrying Capability | p. 153 |
5.7 Service Capability Duty Requirements and Reclosing Capability | p. 153 |
5.8 Capacitance Current Switching | p. 155 |
5.9 Line Closing Switching Surge Factor | p. 160 |
5.10 Out-of-Phase Switching Current Rating | p. 162 |
5.11 Transient Recovery Voltage | p. 163 |
5.12 Low-Voltage Circuit Breakers | p. 168 |
5.13 Fuses | p. 173 |
6. Short-Circuit of Synchronous and Induction Machines | p. 179 |
6.1 Reactances of a Synchronous Machine | p. 180 |
6.2 Saturation of Reactances | p. 182 |
6.3 Time Constants of Synchronous Machines | p. 183 |
6.4 Synchronous Machine Behavior on Terminal Short-Circuit | p. 183 |
6.5 Circuit Equations of Unit Machines | p. 194 |
6.6 Park's Transformation | p. 198 |
6.7 Park's Voltage Equation | p. 202 |
6.8 Circuit Model of Synchronous Machines | p. 203 |
6.9 Calculation Procedure and Examples | p. 204 |
6.10 Short-Circuit of an Induction Motor | p. 214 |
7. Short-Circuit Calculations According to ANSI Standards | p. 219 |
7.1 Types of Calculations | p. 219 |
7.2 Impedance Multiplying Factors | p. 220 |
7.3 Rotating Machines Model | p. 222 |
7.4 Types and Severity of System Short-Circuits | p. 222 |
7.5 Calculation Methods | p. 223 |
7.6 Network Reduction | p. 231 |
7.7 Breaker Duty Calculations | p. 233 |
7.8 High X/R Ratios (DC Time Constant Greater than 45ms) | p. 233 |
7.9 Calculation Procedure | p. 235 |
7.10 Examples of Calculations | p. 236 |
7.11 Thirty-Cycle Short-Circuit Currents | p. 261 |
7.12 Dynamic Simulation | p. 262 |
8. Short-Circuit Calculations According to IEC Standards | p. 267 |
8.1 Conceptual and Analytical Differences | p. 267 |
8.2 Prefault Voltage | p. 271 |
8.3 Far-From-Generator Faults | p. 271 |
8.4 Near-to-Generator Faults | p. 275 |
8.5 Influence of Motors | p. 281 |
8.6 Comparison with ANSI Calculation Procedures | p. 283 |
8.7 Examples of Calculations and Comparison with ANSI Methods | p. 285 |
9. Calculations of Short-Circuit Currents in DC Systems | p. 302 |
9.1 DC Short-Circuit Current Sources | p. 303 |
9.2 Calculation Procedures | p. 304 |
9.3 Short-Circuit of a Lead Acid Battery | p. 306 |
9.4 DC Motor and Generators | p. 312 |
9.5 Short-Circuit Current of a Rectifier | p. 318 |
9.6 Short-Circuit of a Charged Capacitor | p. 324 |
9.7 Total Short-Circuit Current | p. 325 |
9.8 DC Circuit Breakers | p. 326 |
10. Load Flow Over Power Transmission Lines | p. 328 |
10.1 Power in AC Circuits | p. 329 |
10.2 Power Flow in a Nodal Branch | p. 331 |
10.3 ABCD Constants | p. 334 |
10.4 Transmission Line Models | p. 336 |
10.5 Tuned Power Line | p. 345 |
10.6 Ferranti Effect | p. 346 |
10.7 Symmetrical Line at No Load | p. 347 |
10.8 Illustrative Examples | p. 349 |
10.9 Circle Diagrams | p. 352 |
10.10 System Variables in Load Flow | p. 356 |
11. Load Flow Methods | p. 360 |
11.1 Modeling a Two-Winding Transformer | p. 361 |
11.2 Load Flow, Bus Types | p. 366 |
11.3 Gauss and Gauss-Seidel Y-Matrix Methods | p. 367 |
11.4 Convergence in Jacobi-Type Methods | p. 377 |
11.5 Gauss-Seidel Z-Matrix Method | p. 383 |
11.6 Conversion of Y to Z Matrix | p. 384 |
12. Load Flow Methods: Part II | p. 391 |
12.1 Function with One Variable | p. 391 |
12.2 Simultaneous Equations | p. 393 |
12.3 Rectangular Form of Newton-Raphson Method of Load Flow | p. 395 |
12.4 Polar Form of Jacobian Matrix | p. 397 |
12.5 Simplifications of Newton-Raphson Method | p. 405 |
12.6 Decoupled Newton-Raphson Method | p. 408 |
12.7 Fast Decoupled Load Flow | p. 408 |
12.8 Model of a Phase-Shifting Transformer | p. 411 |
12.9 DC Models | p. 413 |
12.10 Load Models | p. 415 |
12.11 Impact Loads and Motor Starting | p. 422 |
12.12 Practical Load Flow Studies | p. 424 |
13. Reactive Power Flow and Control | p. 435 |
13.1 Voltage Instability | p. 436 |
13.2 Reactive Power Compensation | p. 442 |
13.3 Reactive Power Control Devices | p. 447 |
13.4 Some Examples of Reactive Power Flow | p. 460 |
13.5 FACTS | p. 467 |
14. Three-Phase and Distribution System Load Flow | p. 478 |
14.1 Phase Co-Ordinate Method | p. 479 |
14.2 Three-Phase Models | p. 481 |
14.3 Distribution System Load Flow | p. 491 |
15. Optimization Techniques | p. 500 |
15.1 Functions of One Variable | p. 501 |
15.2 Concave and Convex Functions | p. 502 |
15.3 Taylor's Theorem | p. 503 |
15.4 Lagrangian Method, Constrained Optimization | p. 505 |
15.5 Multiple Equality Constraints | p. 507 |
15.6 Optimal Load Sharing Between Generators | p. 508 |
15.7 Inequality Constraints | p. 510 |
15.8 Kuhn-Tucker Theorem | p. 511 |
15.9 Search Methods | p. 512 |
15.10 Gradient Methods | p. 514 |
15.11 Linear Programming--Simplex Method | p. 516 |
15.12 Quadratic Programming | p. 521 |
15.13 Dynamic Programming | p. 521 |
15.14 Integer Programming | p. 523 |
16. Optimal Power Flow | p. 525 |
16.1 Optimal Power Flow | p. 525 |
16.2 Decoupling Real and Reactive OPF | p. 527 |
16.3 Solution Methods of OPF | p. 528 |
16.4 Generation Scheduling Considering Transmission Losses | p. 528 |
16.5 Steepest Gradient Method | p. 536 |
16.6 OPF Using Newton's Method | p. 539 |
16.7 Successive Quadratic Programming | p. 545 |
16.8 Successive Linear Programming | p. 545 |
16.9 Interior Point Methods and Variants | p. 547 |
16.10 Security and Environmental Constrained OPF | p. 551 |
17. Harmonics Generation | p. 554 |
17.1 Harmonics and Sequence Components | p. 556 |
17.2 Increase in Nonlinear Loads | p. 557 |
17.3 Harmonic Factor | p. 557 |
17.4 Three-Phase Windings in Electrical Machines | p. 557 |
17.5 Tooth Ripples in Electrical Machines | p. 559 |
17.6 Synchronous Generators | p. 560 |
17.7 Transformers | p. 560 |
17.8 Saturation of Current Transformers | p. 564 |
17.9 Shunt Capacitors | p. 565 |
17.10 Subharmonic Frequencies | p. 565 |
17.11 Static Power Converters | p. 566 |
17.12 Switch-Mode Power (SMP) Supplies | p. 581 |
17.13 Arc Furnaces | p. 582 |
17.14 Cycloconverters | p. 584 |
17.15 Thyristor-Controlled Factor | p. 586 |
17.16 Thyristor-Switched Capacitors | p. 588 |
17.17 Pulse Width Modulation | p. 588 |
17.18 Adjustable Speed Drives | p. 591 |
17.19 Pulse burst Modulation | p. 591 |
17.20 Chopper Circuits and Electric Traction | p. 592 |
17.21 Slip Frequency Recovery Schemes | p. 594 |
17.22 Lighting Ballasts | p. 594 |
17.23 Interharmonics | p. 595 |
18. Effects of Harmonics | p. 597 |
18.1 Rotating Machines | p. 598 |
18.2 Transformers | p. 603 |
18.3 Cables | p. 607 |
18.4 Capacitors | p. 608 |
18.5 Harmonic Resonance | p. 609 |
18.6 Voltage Notching | p. 613 |
18.7 EMI (Electromagnetic Interference) | p. 613 |
18.8 Overloading of Neutral | p. 614 |
18.9 Protective Relays and Meters | p. 615 |
18.10 Circuit Breakers and Fuses | p. 615 |
18.11 Telephone Influence Factor | p. 616 |
19. Harmonic Analysis | p. 619 |
19.1 Harmonic Analysis Methods | p. 620 |
19.2 Harmonic Modeling of System Components | p. 626 |
19.3 Load Models | p. 630 |
19.4 System Impedance | p. 630 |
19.5 Three-Phase Models | p. 631 |
19.6 Modeling of Networks | p. 633 |
19.7 Power Factor and Reactive Power | p. 637 |
19.8 Shunt Capacitor Bank Arrangements | p. 640 |
19.9 Study Cases | p. 644 |
20. Harmonic Mitigation and Filters | p. 664 |
20.1 Mitigation of Harmonics | p. 664 |
20.2 Band Pass Filters | p. 665 |
20.3 Practical Filter Design | p. 668 |
20.4 Relations in a ST Filter | p. 678 |
20.5 Filters for a Furnace Installation | p. 681 |
20.6 Filters for an Industrial Distribution System | p. 683 |
20.7 Secondary Resonance | p. 684 |
20.8 Filter Reactors | p. 686 |
20.9 Double-Tuned Filter | p. 687 |
20.10 Damped Filters | p. 689 |
20.11 Design of a Second-Order High-Pass Filter | p. 693 |
20.12 Zero Sequence Traps | p. 694 |
20.13 Limitations of Passive Filters | p. 696 |
20.14 Active Filters | p. 698 |
20.15 Corrections in Time Domain | p. 701 |
20.16 Corrections in the Frequency Domain | p. 702 |
20.17 Instantaneous Reactive Power | p. 704 |
20.18 Harmonic Mitigation at Source | p. 706 |
Appendix A Matrix Methods | p. 712 |
A.1 Review Summary | p. 712 |
A.2 Characteristics Roots, Eigenvalues, and Eigenvectors | p. 716 |
A.3 Diagonalization of a Matrix | p. 718 |
A.4 Linear Independence or Dependence of Vectors | p. 719 |
A.5 Quadratic Form Expressed as a Product of Matrices | p. 719 |
A.6 Derivatives of Scalar and Vector Functions | p. 720 |
A.7 Inverse of a Matrix | p. 721 |
A.8 Solution of Large Simultaneous Equations | p. 725 |
A.9 Crout's Transformation | p. 727 |
A.10 Gaussian Elimination | p. 729 |
A.11 Forward-Backward Substitution Method | p. 730 |
A.12 LDU (Product Form, Cascade, or Choleski Form) | p. 733 |
Appendix B Calculation of Line and Cable Constants | p. 736 |
B.1 AC Resistance | p. 736 |
B.2 Inductance | p. 736 |
B.3 Impedance Matrix | p. 739 |
B.4 Three-Phase Line with Ground Conductors | p. 739 |
B.5 Bundle Conductors | p. 741 |
B.6 Carson's Formula | p. 742 |
B.7 Capacitance of Lines | p. 748 |
B.8 Cable Constants | p. 751 |
Appendix C Transformers and Reactors | p. 756 |
C.1 Model of a Two-Winding Transformer | p. 756 |
C.2 Transformer Polarity and Terminal Connections | p. 761 |
C.3 Parallel Operation of Transformers | p. 763 |
C.4 Autotransformers | p. 765 |
C.5 Step-Voltage Regulators | p. 770 |
C.6 Extended Models of Transformers | p. 770 |
C.7 High-Frequency Models | p. 776 |
C.8 Duality Models | p. 776 |
C.9 GIC Models | p. 779 |
C.10 Reactors | p. 780 |
Appendix D Sparsity and Optimal Ordering | p. 784 |
D.1 Optimal Ordering | p. 784 |
D.2 Flow Graphs | p. 785 |
D.3 Optimal Ordering Schemes | p. 788 |
Appendix E Fourier Analysis | p. 792 |
E.1 Periodic Functions | p. 792 |
E.2 Orthogonal Functions | p. 792 |
E.3 Fourier Series and Coefficients | p. 792 |
E.4 Odd Symmetry | p. 795 |
E.5 Even Symmetry | p. 795 |
E.6 Half-Wave Symmetry | p. 796 |
E.7 Harmonic Spectrum | p. 797 |
E.8 Complex Form of Fourier Series | p. 799 |
E.9 Fourier Transform | p. 800 |
E.10 Sampled Waveform: Discrete Fourier Transform | p. 803 |
E.11 Fast Fourier Transform | p. 807 |
Appendix F Limitation of Harmonics | p. 809 |
F.1 Harmonic Current Limits | p. 809 |
F.2 Voltage Quality | p. 811 |
F.3 Commutation Notches | p. 813 |
F.4 Interharmonics | p. 816 |
F.5 Flicker | p. 817 |
Appendix G Estimating Line Harmonics | p. 819 |
G.1 Waveform without Ripple Content | p. 819 |
G.2 Waveform with Ripple Content | p. 821 |
G.3 Phase Angle of Harmonics | p. 827 |
Index | p. 831 |