Understanding Symmetrical Components for Power System Modeling

J.C. Das is President, Power System Studies, Inc. Snellville, Georgia. He is an independent consultant, currently with AMEC Foster Wheeler, Inc., a leading supplier of high-value consultancy, engineering, and project management services to the world's energy, power, and process industries. He is the author of IEEE Press titles Power System Harmonics and Passive Filter Designs (2015) and Arc Flash Hazard Analysis and Mitigation (2012).

"This book provides good technical depth, yet also provides a practical treatment of symmetrical components"..."This would be a very good book for power system engineers" IEEE, Oct 2017

• About the Author ixForeword xiPreface and Acknowledgments xiiiChapter 1 Symmetrical Components Using Matrix Methods 11.1 Transformations 21.2 Characteristic Roots, Eigenvalues, and Eigenvectors 21.2.1 Definitions 21.2.1.1 Characteristic Matrix 21.2.1.2 Characteristic Polynomial 21.2.1.3 Characteristic Equation 21.2.1.4 Eigenvalues 21.2.1.5 Eigenvectors, Characteristic Vectors 21.3 Diagonalization of a Matrix 51.4 Similarity Transformation 51.5 Decoupling a Three-Phase Symmetrical System 61.6 Symmetrical Component Transformation 81.7 Decoupling a Three-Phase Unsymmetrical System 101.8 Clarke Component Transformation 111.9 Significance of Selection of Eigenvectors in Symmetrical Components 12References 14Chapter 2 Fundamental Concepts of Symmetrical Components 152.1 Characteristics of Symmetrical Components 162.2 Characteristics of Sequence Networks 192.3 Sequence Impedance of Network Components 202.4 Construction of Sequence Networks 202.5 Sequence Components of Transformers 222.5.1 Delta-Wye or Wye-Delta Transformer 222.5.2 Wye-Wye Transformer 252.5.3 Delta-Delta Transformer 252.5.4 Zigzag Transformer 252.5.5 Three-Winding Transformers 272.6 Example of Construction of Sequence Networks 32References 36Chapter 3 Symmetrical Components-transmission Lines And Cables 393.1 Impedance Matrix of Three-Phase Symmetrical Line 403.2 Three-Phase Line with Ground Conductors 403.3 Bundle Conductors 423.4 Carson’s Formula 443.4.1 Approximations to Carson’s Equations 463.5 Capacitance of Lines 503.5.1 Capacitance Matrix 503.6 Cable Constants 543.6.1 Zero Sequence Impedance of the OH lines and Cables 543.6.2 Concentric Neutral Underground Cable 553.6.3 Capacitance of Cables 573.7 EMTP Models 583.7.1 Frequency Dependent Model, FD 603.8 Effect of Harmonics on Line Models 623.9 Transmission Line Equations with Harmonics 62References 66Chapter 4 Sequence Impedances of Rotating Equipment And Static Load 694.1 Synchronous Generators 694.1.1 Positive Sequence Impedance 694.1.2 Negative Sequence Impedance 704.1.3 Negative Sequence Capability of Generators 714.1.3.1 Effect of Harmonics 714.1.4 Zero Sequence Impedance 734.1.5 Sequence Component Transformation 754.1.6 Three-Phase Short-Circuit of a Generator 774.1.7 Park’s Transformation 794.2 Induction Motors 814.2.1 Equivalent Circuit 814.2.2 Negative Sequence Impedance 834.2.3 Harmonic Impedances 844.2.4 Zero Sequence Impedance 864.2.5 Terminal Short-Circuit of an Induction Motor 864.3 Static Loads 874.4 Harmonics and Sequence Components 87References 88Further Reading 89Chapter 5 Three-phase Models of Transformers and Conductors 915.1 Three-Phase Models 915.2 Three-Phase Transformer Models 915.2.1 Symmetrical Components of Three-Phase Transformers 945.3 Conductors 99References 102Chapter 6 Unsymmetrical Fault Calculations 1036.1 Line-to-Ground Fault 1046.2 Line-to-Line Fault 1066.3 Double Line-to-Ground Fault 1076.4 Three-Phase Fault 1096.5 Phase Shift in Three-Phase Transformer Windings 1106.5.1 Transformer Connections 1106.5.2 Phase Shifts in Winding as per Standards 1126.5.3 Phase Shift for Negative Sequence Components 1156.6 Unsymmetrical Long Hand Fault Calculations 1166.7 Open Conductor Faults 1266.7.1 Two Conductor Open Fault 1266.7.2 One Conductor Open Fault 1276.8 Short-Circuit Calculations with Bus Impedance Matrix 1316.8.1 Line-to-Ground Fault 1316.8.2 Line-to-Line Fault 1316.8.3 Double Line-to-Ground Fault 1316.8.4 Calculation Procedure 1336.9 System Grounding 1386.9.1 Solidly Grounded Systems 1406.9.2 Resistance Grounded Systems 1406.9.3 High-Resistance Grounded Systems 1416.9.4 Coefficient of Grounding 143References 145Further Reading 145Chapter 7 Some Limitations of Symmetrical Components 1477.1 Phase Coordinate Method 1487.2 Three-Phase Models 1507.2.1 Generators 1507.2.2 Generator Model for Cogeneration 1527.2.3 Load Models 1527.3 Multiple Grounded Systems 1547.3.1 Equivalent Circuit of Multiple Grounded Systems 1567.3.2 Equivalent Circuit Approach 156References 158Index 159