1st Edition

# Symmetrical Components for Power Systems Engineering

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Emphasizing a practical conception of system unbalances, basic circuits, and calculations, this essential reference/text presents the foundations of symmetrical components with a review of per unit (percent), phasors, and polarity--keeping the mathematics as simple as possible throughout. According to IEEE Electrical Insulation Magazine, this book "…provides students and practicing engineers with a fundamental understanding of the method of symmetrical components and its applications in three-phase electrical systems. . .A useful feature of this book. . .is the incorporation of numerous examples in the text and 30 pages of problems."

**Preface**

**Introduction and Historical Background**

Introduction and General Aims

Historical Background

Per Unit and Percent Values

Introduction

Per Unit and Percent Definitions

Advantages of Per Unit and Percent

General Relationships Between Circuit Quantities

Base Quantities

Per Unit and Percent Impedance Relationships II

Per Unit and Percent Impedances of Transformer Units

Changing Per Unit (Percent) Quantities to Different Bases

Phasors, Polarity, and System Harmonics

Introduction

Phasors

Circuit and Phasor Diagrams for a Balanced Three-Phase Power System

Phasor and Phase Rotation

Polarity

Power System Harmonics

Basic Fundamentals and the Sequence Networks

Introduction

Positive-Sequence Set

Nomenclature Convenience

Negative-Sequence Set

Zero-Sequence Set

General Equations

Sequence Independence

Sequence Networks

Positive-Sequence Network

Negative-Sequence Network

Zero-Sequence Network

Impedance and Sequence Connections for Transformer Banks

Sequence Phase Shifts Through Wye-Delta Transformer Banks

Sequence Network Voltages

Sequence Network Reduction

Thevenin Theorem in Network Reduction

Wye-Delta Network Transformations

Short-Circuit MV A and Equivalent Impedance

Equivalent Network from a Previous Fault Study

Example: Determining an Equivalent Network from a Previous Fault Study

Network Reduction by Simultaneous Equations

Other Network Reduction Techniques

Shunt Unbalance Sequence Network Interconnections

Introduction

General Representation of Power Systems and Sequence Networks

Sequence Network Interconnections for Three- Phase Faults

Sequence Network Interconnections for Phaseto- Ground Faults

Sequence Network Interconnections for Phaseto- Phase Faults

Sequence Network Interconnections for Two- Phase-to-Ground Faults

Other Sequence Network Interconnections for Shunt System Conditions

Fault Impedance

Substation and Tower Footing Impedance

Ground Faults on Ungrounded or High Resistance Grounded Systems

Fault Calculation Examples for Shunt-Type Faults

Introduction

Faults on a Loop-Type Power System

Basic Assumptions

Fault Calculation

Summary of Fault Current

Voltages During Faults

Summary of Fault Voltages

Fault Calculations With and Without Load

Solution by Thevenin's Theorem

Solution by Network Reduction

Solution Without Load

Summary

Neutral Inversion

Example: Ground Fault on an Ungrounded System

Example: Ground Fault with High Resistance Across Three Distribution Transformers

Example: Ground Fault with High Resistance in Neutral

Example: Phase-a-to-Ground Fault Currents and Voltages on Both Sides of a Wye-Delta Transformer

Series and Simultaneous Unbalance Sequence Network Interconnections

Introduction

Series Unbalance Sequence Interconnections

One Phase Open: Broken Conductor or Blown Fuse .

Example: Open Phase Calculation

Simultaneous Unbalance Sequence Interconnections

Example: Broken Conductor Falling to Ground on Bus Side

Example: Broken Conductor Falling to Ground on Line Side

Example: Open Conductor on High Side and Ground Fault on Low Side of a Delta-Wye Transformer

Ground Fault on Low Side of a Delta-Wye Transformer

Example: Open Conductor on High Side and Ground Fault on Low Side of a Wye-Groundedl Delta-Wye-Grounded Transformer

Ground Fault Calculation for a Mid-Tapped Grounded Delta Secondary Transformer

Summary

Overview of Sequence Currents and Voltages During Faults

Introduction

Voltage and Current Phasors for Shunt Faults

System Voltage Profiles During Shunt Faults

Voltage and Current Phasors for All Combinations of the Four Shunt Faults

Summary

Transformer, Reactor, and Capacitor Characteristics

Transformer Fundamentals

Example: Impedances of Single-Phase Transformers in Three-Phase Power Systems

Polarity. Standard Terminal Marking, and Phase Shifts

Two-Winding Transformer Banks: Sequence Impedance and Connections

Three-Winding Transformer Banks

Three-Winding Transformers: Sequence Impedance and Connections

Example: Three-Winding Transformer Equivalent

Example: Three-Winding Transformer Fault Calculation

Autotransformers

Example: Autotransformer Fault Calculation

Ungrounded Autotransformers with Tertiary and Grounded Autotransformers Without Tertiary

Test Measurements for Transformer Impedance

Determination of the Equivalent Zero-Sequence Impedances for Three-Winding Three-Phase Transformers Where the Tertiary Delta Winding Is Not Available

Distribution Transformers with Tapped Secondary

Zig-Zag Connected Transformers

Reactors

Capacitors

Generator and Motor Characteristics

Introduction

Transient in Resistance-Inductance Series Circuits

Transient Generator Currents

Negative-Sequence Component

Zero-Sequence Component

Total RMS Armature Component

Rotating Machine Reactance Factors for Fault Calculations

Time Constants for Various Faults

Induction Machines

Summary

Appendix: Typical Constants of Three-Phase Synchronous Machines

Overhead Line Characteristics: Inductive Impedance

Introduction

Reactance of Overhead Conductors

GMR and GMD Values

The X" and Xd Line Constants

Positive- and Negative-Sequence Impedance

Example

Lines with Bundled Conductors

Zero-Sequence Impedance

Zero-Sequence Impedances of Various Lines

Summary for Zero-Sequence Impedance Calculations

Overhead Line Characteristics: Mutual Impedance

Introduction

Mutual Coupling Fundamentals

Positive- and Negative-Sequence Mutual Impedance

Zero-Sequence Mutual Impedance

Mutual Impedances Between Lines of Different Voltages

Power System-Induced Voltages in Wire Communication Lines

Summary

Overhead Line Characteristics: Capacitive Reactance

Introduction

Capacitance of Overhead Conductors

Positive- and Negative-Sequence Capacitance

Example: Three-Phase Circuit Capacitive Reactance

Example: Double-Three-Phase-Circuit Capacitive Reactance

Zero-Sequence Capacitance

Zero-Sequence Capacitance: Transposed Three- Phase Line

B Example: Zero-Sequence Capacitance, Transposed Three-Phase Line

Summary

Cable Characteristics

Introduction

Positive- and Negative-Sequence Constants

Three-Conductor Cables

Zero-Sequence Constants of Cables Problems

**Appendix**: Overhead Line Conductor Characteristics

**Table A.I** All-Aluminum Concentric-Lay Class AA and A Stranded Bare Conductors

**Table A.2** All-Aluminum Concentric-Lay Class AA and A Bare Stranded Conductors 1350-H19 ASTM B 231

**Table A.3** All-Aluminum Shaped-Wire Concentric-Lay Compact Conductors AAC/TW

**Table A.4** All-Aluminum Shaped-Wire Concentric-Lay Compact Conductors AAC/TW

**Table A.5** Bare Aluminum Conductors, Steel- Reinforced (ACSR) Electrical Properties of Single-Layer Sizes

**Table A.6** Bare Aluminum Conductors, Steel- Reinforced (ACSR) Electrical Properties of Multilayer Sizes

**Table A.7** Shaped-Wire Concentric-Lay Compact Aluminum Conductors Steel-Reinforced (ACSR/TW)

**Table A.8** Shaped-Wire Concentric-Lay Compact Aluminum Conductors Steel-Reinforced (ACSR/TW) S

**Table A.9** Bare Aluminum Conductors, - Wires Stranded with Aluminum-Clad Steel Wires (Alumoweld) as Reinforcement (A WAC) in Distribution and Neutral-Messenger Sizes

Bibliography

Index

### Biography

Blackburn, J. Lewis

"…provides students and practicing engineers with a fundamental understanding of the method of symmetrical components and its applications in three-phase electrical systems. . .A useful feature of this book. . .is the incorporation of numerous examples in the text and 30 pages of problems. "

---IEEE Electrical Insulation Magazine