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Power System Analysis


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Table of Contents

Short-Circuit Currents and Symmetrical Components

Nature of Short-Circuit Currents

Symmetrical Components

Eigenvalues and Eigenvectors

Symmetrical Component Transformation

Clarke Component Transformation

Characteristics of Symmetrical Components

Sequence Impedance of Network Components

Computer Models of Sequence Networks

Structure and Nature of Electrical Power Systems

Power System Studies

Unsymmetrical Fault Calculations

Line-to-Ground Fault

Line-to-Line Fault

Double Line-to-Ground Fault

Three-Phase Fault

Phase Shift in Three-Phase Transformers

Unsymmetrical Fault Calculations

System Grounding

Open Conductor Faults

Matrix Methods for Network Solutions

Network Models

Bus Admittance Matrix

Bus Impedance Matrix

Loop Admittance and Impedance Matrices

Graph Theory

Bus Admittance and Impedance Matrices by Graph Approach

Algorithms for Construction of Bus Impedance Matrix

Short-Circuit Calculations with Bus Impedance Matrix

Solution of Large Network Equations

Current Interruption in AC Networks

Rheostatic Breaker

AC Arc Interruption

Current-Zero Breaker

Transient Recovery Voltage

The Terminal Fault

The Short-Line Fault

Interruption of Low Inductive Currents

Interruption of Capacitive Currents

TRV in Capacitive and Inductive Circuits

Prestrikes in Breakers

Overvoltages on Energizing High-Voltage Lines

Out-of-Phase Closing

Resistance Switching

Failure Modes of Circuit Breakers

Operating Mechanisms-SF6 Breakers

Vacuum Interruption

Stresses in Circuit Breakers

Application and Ratings of Circuit Breakers and Fuses according to ANSI Standards

Total and Symmetrical Current Rating Basis

Asymmetrical Ratings

Voltage Range Factor K

Circuit Breaker Timing Diagram

Maximum Peak Current

Permissible Tripping Delay

Service Capability Duty Requirements and Reclosing Capability

Capacitance Current Switching

Line-Closing Switching Surge Factor

Out-of-Phase Switching Current Rating

Transient Recovery Voltage

Generator Circuit Breakers

Specifications of High-Voltage Circuit Breakers

Low-Voltage Circuit Breakers


Short Circuit of Synchronous and Induction Machines

Reactances of a Synchronous Machine

Saturation of Reactances

Time Constants of Synchronous Machines

Synchronous Machine Behavior on Terminal Short Circuit

Circuit Equations of Unit Machines

Park's Transformation

Park's Voltage Equation

Circuit Model of Synchronous Machines

Calculation Procedure and Examples

Short Circuit of Synchronous Motors and Condensers

Induction Motors

Short-Circuit Calculations according to ANSI Standards

Types of Calculations

Accounting for Short-Circuit Current Decay

Rotating Machines Model

Types and Severity of System Short Circuits

Calculation Methods

Network Reduction

Breaker Duty Calculations

Generator Source Short-Circuit Current Asymmetry

Calculation Procedure

Examples of Calculations

Thirty-Cycle Short-Circuit Currents

Short-Circuit Calculations According to IEC Standards

Conceptual and Analytical Differences

Prefault Voltage

Far-from-Generator Faults

Near-to-Generator Faults

Influence of Motors

Comparison with ANSI Calculation Procedures

Examples of Calculations and Comparison with ANSI Methods

Calculations of Short-Circuit Currents in DC Systems

DC Short-Circuit Current Sources

Calculation Procedures

Short Circuit of a Lead Acid Battery

Short-Circuit Current of DC Motors and Generators

Short-Circuit Current of a Rectifier

Short Circuit of a Charged Capacitor

Total Short-Circuit Current

DC Circuit Breakers

Load Flow over Power Transmission Lines

Power in AC Circuits

Power Flow in a Nodal Branch

ABCD Constants

Transmission Line Models

Tuned Power Line

Ferranti Effect

Symmetrical Line at No Load

Illustrative Examples

Circle Diagrams

Modal Analysis

Corona on Transmission Lines

System Variables in Load Flow

Load Flow Methods: Part I

Modeling a Two-Winding Transformer

Load Flow-Bus Types

Gauss and Gauss-Seidel Y-Matrix Methods

Convergence in Jacobi-Type Methods

Gauss-Seidel Z-Matrix Method

Conversion of Y to Z Matrix

Triangular Factorization Method of Load Flow

Load Flow Methods: Part II

Function with One Variable

Simultaneous Equations

Rectangular Form of Newton-Raphson Method of Load Flow

Polar Form of Jacobian Matrix

Simplifications of Newton-Raphson Method

Decoupled Newton-Raphson Method

Fast Decoupled Load Flow

Model of a Phase-Shifting Transformer

DC Load Flow Models

Second Order Load Flow

Load Models

Induction Motor Models

Impact Loads and Motor Starting

Practical Load Flow Studies

Reactive Power Flow and Control

Voltage Instability

Reactive Power Compensation

Reactive Power Control Devices

Some Examples of Reactive Power Flow

Flexible AC Transmission Systems

Three-Phase and Distribution System Load Flow

Phase Coordinate Method

Three-Phase Models

Distribution System Load Flow

Optimal Capacitor Locations

Optimization Techniques

Functions of One Variable

Concave and Convex Functions

Taylor's Theorem

Lagrangian Method: Constrained Optimization

Multiple Equality Constraints

Optimal Load Sharing between Generators

Inequality Constraints

Kuhn-Tucker Theorem

Search Methods

Gradient Methods

Linear Programming-Simplex Method

Quadratic Programming

Dynamic Programming

Integer Programming

Optimal Power Flow

Optimal Power Flow

Decoupling Real and Reactive OPF

Solution Methods of OPF

Generation Scheduling Considering Transmission Losses

Steepest Gradient Method

OPF Using the Newton Method

Sequential Quadratic Programming

Successive Linear Programming

Interior Point Methods and Variants

Security and Environmental Constrained OPF

Harmonics Generation

Harmonics and Sequence Components

Increase in Nonlinear Loads

Harmonic Factor

Three-Phase Windings in Electrical Machines

Tooth Ripples in Electrical Machines

Synchronous Generators


Saturation of Current Transformers

Shunt Capacitors

Sub-Harmonic Frequencies

Static Power Converters

Switch-Mode Power (SMP) Supplies

Arc Furnaces


Thyristor-Controlled Reactor

Thyristor-Switched Capacitors

Pulse-Width Modulation

Adjustable Speed Drives

Pulse Burst Modulation

Chopper Circuits and Electric Traction

Slip Frequency Recovery Schemes

Lighting Ballasts

Voltage Source Converters


Effects of Harmonics

Rotating Machines




Harmonic Resonance

Voltage Notching

Electromagnetic Interference

Overloading of Neutral

Protective Relays and Meters

Circuit Breakers and Fuses

Telephone Influence Factor

Harmonic Analysis

Harmonic Analysis Methods

Harmonic Modeling of System Components

Load Models

System Impedance

Three-Phase Models

Modeling of Networks

Power Factor and Reactive Power

Shunt Capacitor Bank Arrangements

Unbalance Detection

Study Cases

Harmonic Mitigation and Filters

Mitigation of Harmonics

Band-Pass Filters

Practical Filter Design

Relations in an ST Filter

Filters for a Furnace Installation

Filters for an Industrial Distribution System

Secondary Resonance

Filter Reactors

Double-Tuned Filter

Damped Filters

Design of a Second-Order High-Pass Filter

Zero-Sequence Traps

Limitations of Passive Filters

Active Filters

Corrections in Time Domain

Corrections in the Frequency Domain

Instantaneous Reactive Power

Harmonic Mitigation AT Source

Multilevel Converters

Arc Flash Hazard Analysis

Relating Short-Circuit Currents with Arc Flash and Personal Safety

Arc Flash Hazard Analysis

Hazard=Risk Categories

System Grounding: Impact on Incident Energy

Duration of an Arc Flash Event and Arc Flash Boundary

Protective Relaying and Coordination

Short-Circuit Currents

Arc Flash Calculations in Medium-Voltage Systems

Arc Flash Calculations in Low-Voltage Systems

Accounting for Decaying Short-Circuit Currents

Wind Power

AEP 765 kV Transmission Grid Initiative in the United States

Wind Energy Conversion

The Cube Law


Wind Generators

Power Electronics

Reactive Power Control


Computer Modeling

Appendix A: Matrix Methods

Appendix B: Calculation of Line and Cable Constants

Appendix C: Transformers and Reactors

Appendix D: Sparsity and Optimal Ordering

Appendix E: Fourier Analysis

Appendix F: Limitation of Harmonics

Appendix G: Estimating Line Harmonics

About the Author

J.C. Das is a consultant with Electrical Power Systems in Atlanta, Georgia.


"The usefulness of this text comes from showing how to calculate and analyze many aspects of the power system without relying on a computer analysis. ... [A] very useful text for an undergraduate or even graduate course in power engineering. ... It would also make for a good reference book for practicing power system engineers."
- IEEE Electrical Insulation Magazine, Vol. 20, No. 1, Jan/Feb 2004

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