Understanding Symmetrical Components for Power System Modeling

An essential guide to studying symmetrical component theory

• Provides concise treatment of symmetrical components
• Describes major sequence models of power system components
• Discusses Electromagnetic Transient Program (EMTP) models
• Includes worked examples to illustrate the complexity of calculations, followed by matrix methods of solution which have been adopted for calculations on digital computers

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).

FOREWORD xi

PREFACE AND ACKNOWLEDGMENTS xiii

CHAPTER 1 SYMMETRICAL COMPONENTS USING MATRIX METHODS 1

1.1 Transformations 2

1.2 Characteristic Roots, Eigenvalues, and Eigenvectors 2

1.2.1 Definitions 2

1.2.1.1 Characteristic Matrix 2

1.2.1.2 Characteristic Polynomial 2

1.2.1.3 Characteristic Equation 2

1.2.1.4 Eigenvalues 2

1.2.1.5 Eigenvectors, Characteristic Vectors 2

1.3 Diagonalization of a Matrix 5

1.4 Similarity Transformation 5

1.5 Decoupling a Three-Phase Symmetrical System 6

1.6 Symmetrical Component Transformation 8

1.7 Decoupling a Three-Phase Unsymmetrical System 10

1.8 Clarke Component Transformation 11

1.9 Significance of Selection of Eigenvectors in Symmetrical Components 12

References 14

CHAPTER 2 FUNDAMENTAL CONCEPTS OF SYMMETRICAL COMPONENTS 15

2.1 Characteristics of Symmetrical Components 16

2.2 Characteristics of Sequence Networks 19

2.3 Sequence Impedance of Network Components 20

2.4 Construction of Sequence Networks 20

2.5 Sequence Components of Transformers 22

2.5.1 Delta-Wye or Wye-Delta Transformer 22

2.5.2 Wye-Wye Transformer 25

2.5.3 Delta-Delta Transformer 25

2.5.4 Zigzag Transformer 25

2.5.5 Three-Winding Transformers 27

2.6 Example of Construction of Sequence Networks 32

References 36

CHAPTER 3 SYMMETRICAL COMPONENTS-TRANSMISSION LINES AND CABLES 39

3.1 Impedance Matrix of Three-Phase Symmetrical Line 40

3.2 Three-Phase Line with Ground Conductors 40

3.3 Bundle Conductors 42

3.4 Carson’s Formula 44

3.4.1 Approximations to Carson’s Equations 46

3.5 Capacitance of Lines 50

3.5.1 Capacitance Matrix 50

3.6 Cable Constants 54

3.6.1 Zero Sequence Impedance of the OH lines and Cables 54

3.6.2 Concentric Neutral Underground Cable 55

3.6.3 Capacitance of Cables 57

3.7 EMTP Models 58

3.7.1 Frequency Dependent Model, FD 60

3.8 Effect of Harmonics on Line Models 62

3.9 Transmission Line Equations with Harmonics 62

References 66

CHAPTER 4 SEQUENCE IMPEDANCES OF ROTATING EQUIPMENT AND STATIC LOAD 69

4.1 Synchronous Generators 69

4.1.1 Positive Sequence Impedance 69

4.1.2 Negative Sequence Impedance 70

4.1.3 Negative Sequence Capability of Generators 71

4.1.3.1 Effect of Harmonics 71

4.1.4 Zero Sequence Impedance 73

4.1.5 Sequence Component Transformation 75

4.1.6 Three-Phase Short-Circuit of a Generator 77

4.1.7 Park’s Transformation 79

4.2 Induction Motors 81

4.2.1 Equivalent Circuit 81

4.2.2 Negative Sequence Impedance 83

4.2.3 Harmonic Impedances 84

4.2.4 Zero Sequence Impedance 86

4.2.5 Terminal Short-Circuit of an Induction Motor 86

4.3 Static Loads 87

4.4 Harmonics and Sequence Components 87

References 88

Further Reading 89

CHAPTER 5 THREE-PHASE MODELS OF TRANSFORMERS AND CONDUCTORS 91

5.1 Three-Phase Models 91

5.2 Three-Phase Transformer Models 91

5.2.1 Symmetrical Components of Three-Phase Transformers 94

5.3 Conductors 99

References 102

CHAPTER 6 UNSYMMETRICAL FAULT CALCULATIONS 103

6.1 Line-to-Ground Fault 104

6.2 Line-to-Line Fault 106

6.3 Double Line-to-Ground Fault 107

6.4 Three-Phase Fault 109

6.5 Phase Shift in Three-Phase Transformer Windings 110

6.5.1 Transformer Connections 110

6.5.2 Phase Shifts in Winding as per Standards 112

6.5.3 Phase Shift for Negative Sequence Components 115

6.6 Unsymmetrical Long Hand Fault Calculations 116

6.7 Open Conductor Faults 126

6.7.1 Two Conductor Open Fault 126

6.7.2 One Conductor Open Fault 127

6.8 Short-Circuit Calculations with Bus Impedance Matrix 131

6.8.1 Line-to-Ground Fault 131

6.8.2 Line-to-Line Fault 131

6.8.3 Double Line-to-Ground Fault 131

6.8.4 Calculation Procedure 133

6.9 System Grounding 138

6.9.1 Solidly Grounded Systems 140

6.9.2 Resistance Grounded Systems 140

6.9.3 High-Resistance Grounded Systems 141

6.9.4 Coefficient of Grounding 143

References 145

Further Reading 145

CHAPTER 7 SOME LIMITATIONS OF SYMMETRICAL COMPONENTS 147

7.1 Phase Coordinate Method 148

7.2 Three-Phase Models 150

7.2.1 Generators 150

7.2.2 Generator Model for Cogeneration 152

7.2.3 Load Models 152

7.3 Multiple Grounded Systems 154

7.3.1 Equivalent Circuit of Multiple Grounded Systems 156

7.3.2 Equivalent Circuit Approach 156

References 158

INDEX 159

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