TM
Marcel Dekker, Inc. New York
•
Basel
Power System
Analysis
Short-Circuit Load Flow and Harmonics
J. C. Das
Amec, Inc.
Atlanta, Georgia
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
ISBN: 0-8247-0737-0
This book is printed on acid-free paper.
Headquarters
Marcel Dekker, Inc.
270 Madison Avenue, New York, NY 10016
tel: 212-696-9000; fax: 212-685-4540
Eastern Hemisphere Distribution
Marcel Dekker AG
Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland
tel: 41-61-261-8482; fax: 41-61-261-8896
World Wide Web

The publisher offers discounts on this book when ordered in bulk quantities. For more
information, write to Special Sales/Professional Marketing at the headquarters address above.
Copyright # 2002 by Marcel Dekker, Inc. All Rights Reserved.
Neither this book nor any part may be reproduced or transmitted in any form or by any
means, electronic or mechanical, including photocopying, microfilming, and recording, or by
any information storage and retrieval system, without permission in writing from the pub-
lisher.
Current printing (last digit):

13. Restructured Electrical Power Systems: Operation, Trading, and Volatility,
Mohammad Shahidehpour and Muwaffaq Alomoush
14. Electric Power Distribution Reliability, Richard E. Brown
15. Computer-Aided Power System Analysis, Ramasamy Natarajan
16. Power System Analysis: Short-Circuit Load Flow and Harmonics, J. C. Das
17. Power Transformers: Principles and Applications, John J. Winders, Jr.
18. Spatial Electric Load Forecasting: Second Edition, Revised and Expanded, H.
Lee Willis
19. Dielectrics in Electric Fields, Gorur G. Raju
ADDITIONAL VOLUMES IN PREPARATION
Protection Devices and Systems for High-Voltage Applications, Vladimir Gure-
vich
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
Series Introduction
Power engineering is the oldest and most traditional of the various areas within
electrical engineering, yet no other facet of modern technology is currently under-
going a more dramatic revolution in both technology and industry structure. But
none of these changes alter the basic complexity of electric power system behavior,
or reduce the challenge that power system engineers have always faced in designing
an economical system that operates as intended and shuts down in a safe and non-
catastrophic mode when something fails unexpectedly. In fact, many of the ongoing
changes in the power industry—deregulation, reduced budgets and staffing levels,
and increasing public a nd regulatory demand for reliability among them—make
these challenges all the more difficult to overcome.
Therefore, I am particularly delighted to see this late st addition to the Power
Engineering series. J. C. Das’s Power System Analysis: Short-Circuit Load Flow and
Harmonics provides comprehensive coverage of both theory and practice in the
fundamental areas of power system analysis, including power flow, short-circuit
computations, harmonics, machine modeling, equipment ratings, reactive power
control, and optimization. It also includes an excellent review of the standard matrix

Short-circuit analyses are included in chapters on rating structures of breakers,
current interruption in ac circuits, calculations according to the IEC and ANSI/
IEEE methods, and calculations of short-circuit currents in dc systems.
The load flow analyses cover reactive power flow and control, optimization
techniques, and introduction to FACT controllers, three-phase load flow, and opti-
mal power flow.
The effect of harmonics on power systems is a dynamic and evolving field
(harmonic effects can be experienced at a distance from their source). The book
derives and compiles ample data of practical interest, with the emphasis on harmonic
power flow and harmonic filter design. Generation, effects, limits, and mitigation of
harmonics are discussed, including active and passi ve filters and new harmonic
mitigating topologies.
The models of major electrical equipment—i.e., transformers, generators,
motors, transmission lines, and power cables—are described in detail. Matrix tech-
niques and symmetrical component transformation form the basis of the analyses.
There are many examples and problems. The references and bibliographies point to
further reading and analyses. Most of the analyses are in the steady state, but
references to transient behavior are included where appropriate.
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
A basic knowledge of per unit system, electrical circuits and machinery, and
matrices required, although an overview of matrix techniques is provided in
Appendix A. The style of writing is appropriate for the upper-undergraduate level,
and some sections are at graduate-course level.
Power Systems Analysis is a result of my long experience as a practicing power
system engineer in a variety of industries, power plants, and nuclear facilities. Its
unique feature is applications of power system analyses to real-world problems.
I thank ANSI/IEEE for permission to quote from the relevant ANSI/IEEE
standards. The IEEE disclaims any responsibility or liability resulting from the
placement and use in the described manner. I am also grateful to the International
Electrotechnical Commission (IEC) for permission to use material from the interna-

2.7 System Grounding and Sequence Components
2.8 Open Conductor Faults
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
viii Contents
3. Matrix Methods for Network Solutions
3.1 Network Models
3.2 Bus Admittance Matrix
3.3 Bus Impedance Matrix
3.4 Loop Admittance and Impedance Matrices
3.5 Graph Theory
3.6 Bus Admittance and Impedance Matrices by Graph Approach
3.7 Algorithms for Construction of Bus Impedance Matrix
3.8 Short-Circuit Calculations with Bus Impedance Matrix
3.9 Solution of Large Network Equations
4. Current Interruption in AC Networks
4.1 Rheostatic Brea ker
4.2 Current-Zero Breaker
4.3 Transient Recovery Voltage
4.4 The Terminal Fault
4.5 The Short-Line Fault
4.6 Interruption of Low Inductive Currents
4.7 Interruption of Capacitive Currents
4.8 Prestrikes in Breakers
4.9 Overvoltages on Energizing High-Voltage Lines
4.10 Out-of-Phase Closing
4.11 Resistance Switching
4.12 Failure Modes of Circuit Breakers
5. Application and Ratings of Circuit Breakers and Fuses According
to ANSI Standards
5.1 Total and Symme trical Current Rating Basis

7.4 Types and Severity of System Short-Circuits
7.5 Calculation Methods
7.6 Network Reduction
7.7 Breaker Duty Calculations
7.8 High X/R Rat ios (DC Time Constant Greater than 45ms)
7.9 Calculation Procedure
7.10 Examples of Calculations
7.11 Thirty-Cycle Short-Circuit Currents
7.12 Dynamic Simulation
8. Short-Circuit Calculations According to IEC Standards
8.1 Conceptual and Analytical Differences
8.2 Prefault Voltage
8.3 Far-From-Generator Faults
8.4 Near-to-Generator Faults
8.5 Influence of Motors
8.6 Comparison with ANSI Calculation Procedures
8.7 Examples of Calculations and Com parison with ANSI
Methods
9. Calculations of Sho rt-Circuit Currents in DC Systems
9.1 DC Short-Circu it Current Sources
9.2 Calculation Procedures
9.3 Short-Circuit of a Lead Acid Battery
9.4 DC Motor and Generators
9.5 Short-Circuit Current of a Rectifier
9.6 Short-Circuit of a Charged Capacitor
9.7 Total Short-Circuit Current
9.8 DC Circuit Breakers
10. Load Flow Over Power Transmission Lines
10.1 Power in AC Circuits
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

13. Reactive Power Flow and Control
13.1 Voltage Instability
13.2 Reactive Power Compensation
13.3 Reactive Power Control Devices
13.4 Some Examples of Reactive Power Flow
13.5 FACTS
14. Three-Phase and Distribution System Load Flow
14.1 Phase Co-Ordinate Method
14.2 Three-Phase Models
x Contents
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
14.3 Distribution System Load Flow
15. Optimization Techniques
15.1 Functions of One Var iable
15.2 Concave and Convex Functions
15.3 Taylor’s Theorem
15.4 Lagrangian Method, Constrained Optimization
15.5 Multiple Equality Constraints
15.6 Optimal Load Sharing Between Generators
15.7 Inequality Constraints
15.8 Kuhn–Tucker Theorem
15.9 Search Methods
15.10 Gradient Methods
15.11 Linear Programming—Simplex Method
15.12 Quadratic Programming
15.13 Dynamic Programming
15.14 Integer Programming
16. Optimal Power Flow
16.1 Optimal Power Flow
16.2 Decoupling Real and Reactive OPF

17.20 Chopper Circuits a nd Electric Traction
17.21 Slip Frequency Recovery Schemes
17.22 Lighting Ballasts
17.23 Interharmonics
18. Effects of Harm onics
18.1 Rotating Machines
18.2 Transformers
18.3 Cables
18.4 Capacitors
18.5 Harmonic Resonance
18.6 Voltage Notching
18.7 EMI (Electromagnetic Interference)
18.8 Overloading of Neutr al
18.9 Protective Relays and Meters
18.10 Circuit Breakers and Fuses
18.11 Telephone Influence Factor
19. Harmonic Analysis
19.1 Harmonic Analysis Methods
19.2 Harmonic Modeling of System Components
19.3 Load Models
19.4 System Impedance
19.5 Three-Phase Models
19.6 Modeling of Networks
19.7 Power Factor and Reactive Power
19.8 Shunt Capacitor Bank Arrangements
19.9 Study Cases
20. Harmonic Mitigat ion and Filters
20.1 Mitigation of Harmonics
20.2 Band Pass Filters
20.3 Practical Filter Design

Appendix B Calculation of Line and Cable Constants
B.1 AC Resistance
B.2 Inductance
B.3 Impedance Matrix
B.4 Three-Phase Line with Ground Conductors
B.5 Bundle Conductors
B.6 Carson’s Formula
B.7 Capacitance of Lines
B.8 Cable Constants
Appendix C Transformers and Reactors
C.1 Model of a Two-Winding Transformer
C.2 Transformer Polarity and Terminal Connections
C.3 Parallel Operation of Transformers
C.4 Autotransformers
C.5 Step-Voltage Regulators
C.6 Extended Models of Transformers
C.7 High-Frequency Models
C.8 Duality Models
C.9 GIC Models
C.10 Reactors
Contents xiii
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
Appendix D Sparsity and Optimal Ordering
D.1 Optimal Ordering
D.2 Flow Graphs
D.3 Optimal Ordering Schemes
Appendix E Fourier Analysis
E.1 Periodic Functions
E.2 Orthogonal Functions
E.3 Fourier Series and Coefficients

lower is the fault damage, and the better is the chance of systems holding together
without loss of synchronism.
Short-circuits can be studied from the following angles:
1. Calculation of short-circuit currents.
2. Interruption of short-circuit currents and rating structure of switching
devices.
3. Effects of short-circuit currents.
4. Limitation of short-circuit currents, i.e., with current-limiting fuses and
fault current limit ers.
5. Short-circuit withstand ratings of electrical equipment like transformers,
reactors, cables, and conductors.
6. Transient stability of interconnected systems to remain in synchronism
until the faulty section of the power system is isolated.
We will confine our discussions to the calculations of short-circuit currents, and the
basis of short-circuit ratings of switching devices, i.e., power circuit breakers and
fuses. As the main purpose of short-circuit calculations is to select and apply these
devices properly, it is meaningful for the calculations to be related to current inter-
ruption phenomena and the rating structures of interrupting devices. The objectives
of short-circuit calculations, therefore, can be summarized as follows:
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
. Determination of short-circuit duties on switching devices, i.e., high-, med-
ium- and low-voltage circuit breakers and fuses.
. Calculation of short-circuit currents required for protective relaying and co-
ordination of protective devices.
. Evaluations of adequacy of short-circuit withstand ratings of static equip-
ment like cables, conductors, bus bars, reactors, and trans formers.
. Calculations of fault voltage dips and their time-dependent recovery profiles.
The type of short-circuit currents required for each of these objectives may not be
immediately clear, but will unfold in the chapters to follow.
In a three-phase system, a fault may equally involve all three phases. A bolted

the short-circuit cu rrent is limited only by Z, and its steady-state value is vectorially
given by E
m
=Z. This assumes that the impedance Z does not change with flow of the
large short-circuit current. For simplifica tion of empirical short-circuit calculations,
the impedances of static components like transmission lines, cables, reactors, and
transformers are assumed to be time invariant. Practically, this is not true, i.e., the
flux densities and saturation characteristics of core materials in a transformer may
entirely change its leakage reactance. Driven to saturation under high current flow,
distorted waveforms and harmonics may be produced.
Ignoring these effects and assuming that Z is time invariant during a short-
circuit, the transient and steady-state currents are given by the differential equation
of the R–L circuit with an applied sinusoidal voltage:
2 Chapter 1
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
L
di
dt
þ Ri ¼ E
m
sinð!t þÞð1:1Þ
where  is the angle on the voltage wave, at which the fault occurs. The solution of
this differential equation is given by
i ¼ I
m
sinð!t þ À ÞÀI
m
sinð ÀÞe
ÀRt=L
ð1:2Þ

in an inductive circuit cannot change suddenly. Wh en the fault occurs at an instant
when  ¼ 0, there has to be a transient current whose initial value is equal and
opposite to the instantaneous value of the ac short-circuit current. This transient
current, the second term of Eq. (1.2) can be called a dc component and it decays at
an exponential rate. Equation (1.2) can be simply written as
i ¼ I
m
sin !t þI
dc
e
ÀRt=L
ð1:3Þ
Where the initial value of I
dc
¼ I
m
ð1:4Þ
The following inferences can be drawn from the above discussions:
1. There are two distinct components of a short-circuit current: (1) a non-
decaying ac component or the steady-state component, and (2) a decaying
dc component at an exponential rate, the initial magnitude of which is a
maximum of the ac component and it depends on the time on the voltage
wave at which the fault occurs.
2. The decrement factor of a decaying exponential current can be defined as
its value any time after a short-circuit, expressed as a function of its initial
magnitude per unit. Factor L=R can be termed the time constant. The
exponential then becomes I
dc
e
t=t

change suddenly and decays, depending on machine time constants.
Thus, the assumption of constant L is not valid for rotating machines
and decay in the ac component of the short-cir cuit current must also be
considered.
5. In a three-phase system, the phases are time displaced from each other by
120 electrical degrees. If a fault occurs when the unidirectional compo-
nent in phase a is zero, the phase b component is positive a nd the phase c
component is equal in magnitude and negative. Figure 1-3 shows a three-
phase fault current waveform. As the fault is symmetrical, I
a
þ I
b
þ I
c
is
zero at any instant, where I
a
, I
b
, and I
c
are the short-circuit currents in
phases a, b, and c, respectively. For a fault close to a synchronous gen-
erator, there is a 120-Hz current also, which rapidly decays to zero. This
gives rise to the characteristic nonsinusoidal shape of three-phase short-
circuit currents observed in test oscillograms. The effect is insignificant,
and ignored in the short-circuit calculations. This is further discussed in
Chapter 6.
6. The load current has been ignored. Generally, this is true for empirical
short-circuit calculations, as the short-circuit current is much higher than

impedances presented by various power system components, i.e., transformers,
generators, and transmission lines, to symmetrical components are decoupled
from each other, resulting in independent networks for each component. These
form a balanced set. This simplifies the calculations.
Familiarity with electrical circuits and machine theory, per unit system, and
matrix techniques is required before proceeding with this book. A review of the
matrix techniques in power systems is included in Appendix A. The notations
described in this appendix for vectors and matrices are followed throughout the
book.
The basic theory of symmetrical components can be stated as a mathematical
concept. A system of three coplanar vectors is completely defined by six parameters,
and the system can be said to possess six degrees of freedom. A point in a straight
line being constrained to lie on the line possesses but one degree of freedom, and by
the same analogy, a point in space has three degrees of freedom. A coplanar vector is
defined by its terminal and length and therefore possesses two degrees of freedom. A
system of coplanar vectors having six degrees of freedom, i.e., a three-phase unba-
lanced current or voltage vectors, can be represented by three symmetrical systems of
vectors each having two degrees of freedom. In general, a system of n numbers can
be resolved into n sets of component numbers each having n components, i.e., a total
of n
2
components. Fortescue demonstrated that an unbalanced set on n phasors can
be resolved into n À 1 balanced phase systems of different phase sequence and one
zero sequence system, in which all phasors are of equal magnitude and cophasial:
V
a
¼ V
a1
þ V
a2

n
, are original n unbalanced voltage phasors. V
a1
, V
b1
; ; V
n1
are the first set of n balanced phasors, at an angle of 2=n between them, V
a2
,
V
b2
; ; V
n2
, are the second set of n balanced phasors at an angle 4=n, and the
final set V
an
; V
bn
; ; V
nn
is the zero sequence set, all phasors at nð2=nÞ¼2, i.e.,
cophasial.
In a symmetrical three-phase balanced system, the generators produce
balanced voltages which are displaced from each other by 2=3 ¼ 120

. These vol-
tages can be called positive sequence voltages. If a vector operator a is defined which
rotates a unit vector through 120


c1
þ V
c2
ð1:6Þ
We can define the set consisting of V
a0
, V
b0
, and V
c0
as the zero sequence set, the set
V
a1
, V
b1
,andV
c1
, as the positive sequence set, and the set V
a2
, V
b2
, and V
c2
as the
negative sequence set of voltages. The three origin al unbalanced voltage vectors give
rise to nine voltage vectors, which must have constraints of freedom and are not
Short-Circuit Currents and Symmetrical Components 7
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
totally independent. By definition of positive sequence, V
a1

a2
Also, V
a0
¼ V
b0
¼ V
c0
. With these relations defined, Eq. (1.6) can be written as:
V
a
V
b
V
c
































ð1:7Þ
or in the abbreviated form:
"
VV
abc
¼
"
TT
s
"
VV
012
ð1:8Þ
where
"

where  is a scalar called an eigenvalue, characteristic value, or root of the matrix
"
AA,
and
"
xx is a vector called the eigenvector or characteristic vector of
"
AA.
Then, there are n eigenvalues and corresponding n sets of eigenvectors asso-
ciated with an arbitrary matrix
"
AA of dimensions n  n. The eigenvalues are not
necessarily distinct, and multiple roots occur.
8 Chapter 1
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
Equation (1.9) can be written as
"
AA À I
ÂÃ
"
xx½¼0 ð1:10Þ
where I the is identity matrix. Expanding:
a
11
À  a
12
a
13
a
1













x
1
x
2

x
n




















ð1:11Þ
This represents a set of homogeneous linear equations. Determinant jA À Ij must
be zero as
"
xx 6¼ 0.
"
AA À  I




¼ 0 ð1:12Þ
This can be expanded to yield an nth order algebraic equation:
a
n

n
þ a
n
À I
n
À 1 þ þ a

"
AA. The eigenvector
"
xx
j
corresponding to
"

j
is found from Eq. (1.10). See Appendix A for details and an
example.
1.4 SYMMETRICAL COMPONENT TRANSFORMATION
Application of eigenva lues and eigenvectors to the decoupling of three-phase systems
is useful when we define similarity transformation. This forms a diagonalization
technique and decoupling through symmetrical components.
1.4.1 Similarity Transformation
Consider a system of linear equations:
"
AA
"
xx ¼
"
yy ð1:14Þ
A transformation matrix
"
CC can be introduced to relate the original vectors
"
xx and
"
yy to

¼
"
CC
"
yy
n
"
CC
À1
"
AA
"
CC
"
xx
n
¼
"
CC
À1
"
CC
"
yy
n
"
CC
À1
"
AA

AA
"
CC
ð1:15Þ
"
AA
n
"
xx
n
¼
"
yy
n
is distinct from
"
AA
"
xx ¼
"
yy. The only restriction on choosing
"
CC is that it
should be nonsingular. Equation (1.15) is a set of linear equations, derived from
the original equations (1.14) and yet distinct from them.
If
"
CC is a nodal matrix
"
MM, corresponding to the coefficients of

AAx
1
; x
2
; ; x
n
½
"
CC
À1
"
AAx
1
;
"
AAx
2
; ;
"
AAx
n
ÂÃ
¼
"
CC
À1

1
x
1


















¼
"
CC
À1
"
CC

1

2
:

n

"
AA
"
CC is reduced to a diagonal matrix
"
, called a spectral matrix. Its diagonal
elements are the eigenvalues of the original matrix
"
AA. The new system of equations is
an uncoupled system. Equations (1.14) and (1.15) constitute a similarity transforma-
tion of matrix
"
AA. The matrices
"
AA and
"
AA
n
have the same eigenvalues and are called
similar matrices . The transformation matrix
"
CC is nonsingular.
1.4.2 Decoupling a Three-Phase Symmetrical System
Let us decouple a three-phase transmission line section, where each phase has a
mutual coupling with respect to ground. This is shown in Fig. 1-4(a). An impedance
matrix of the three-phase transmis sion line can be written as
Z
aa
Z
ab

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.