untitled 2
TRANSCRIPT
Genera
ted o
n 2
01
4-0
3-2
4 0
0:4
1 G
MT /
htt
p:/
/hd
l.hand
le.n
et/
20
27
/mdp.3
90
15
00
29
00
27
5Public
Dom
ain
, G
oog
le-d
igit
ized
/
htt
p:/
/ww
w.h
ath
itru
st.o
rg/a
ccess
_use
#pd-g
oogle
POWER SYSTEM
STABILITY
VOLUME II
Transient Stability
Genera
ted o
n 2
01
4-0
3-2
4 0
0:4
2 G
MT /
htt
p:/
/hd
l.hand
le.n
et/
20
27
/mdp.3
90
15
00
29
00
27
5Public
Dom
ain
, G
oog
le-d
igit
ized
/
htt
p:/
/ww
w.h
ath
itru
st.o
rg/a
ccess
_use
#pd-g
oogle
GENERAL ELECTRIC SERIES
Magnetic Control of Industrial Motors (In
preparation)
By Gerhart W. Heumann
Power System Stability
By Selden B. Crary. Volume I: Steady State
Stability. 291 pages. 152 figures 5}^ by 8%.
Cloth. Volume II: Transient Stability. 329
pages. 191 figures. Sy2 by 8%. Cloth.
Fields and Waves in Modern Radio
By Simon Ramo and John R. Whinnery. S02
pages. 214 figures. 5^ by 8%. Cloth.
Materials and Processes
Edited by J. F. Young. 628 pages. 410 figures.
SV2 by 8Vt. Cloth.
Modern Turbines
By L. E. Newman, A. Keller, J. M. Lyons,
L. B. Wales. Edited by L. E. Newman. 175
pages. 93 figures. 5K by 8%. Cloth.
Circuit Analysis of A-C Power Systems
Volume I by Edith Clarke. 540 pages. 167 fig-
ures. 5Hby8%. Cloth.
Electric Motors in Industry
By D. R. Shoults and C. J. Rife. Edited by
T. C. Johnson. 389 pages. 219 figures. 6 by 9.
Cloth.
A Short Course in Tensor Analysis for
Electrical Engineers
By Gabriel Kron. 250 pages. 146 figures. 6 by
9. Cloth.
Tensor Analysis of Networks
By Gabriel Kron. 635 pages. 330 figures. 6 by
9. Cloth.
Transformer Engineering
By L. F. Blume, G. Camilli, A. Boyajian, and
V. M. Montsinger. Edited by L. F. Blume. 496
pages. 348 figures. 6 by 9. Cloth.
Mathematics of Modern Engineering
Volume I by Robert E. Doherty and Ernest G.
Keller. 314 pages. 83 figures. 6 by 9. Cloth.
Volume II by Ernest G. Keller. 309 pages. 91
figures. 6 by 9. Cloth.
Vibration Prevention in Engineering
By the late Arthur L. Kimball. 145 pages. 82
figures. 6 by 9. Cloth.
Published by JOHN WILEY & SONS, Inc.
Genera
ted o
n 2
01
4-0
3-2
4 0
0:4
3 G
MT /
htt
p:/
/hd
l.hand
le.n
et/
20
27
/mdp.3
90
15
00
29
00
27
5Public
Dom
ain
, G
oog
le-d
igit
ized
/
htt
p:/
/ww
w.h
ath
itru
st.o
rg/a
ccess
_use
#pd-g
oogle
POWER SYSTEM
STABILITY
VOLUME n
Transient Stability
By tr''
SELDEN B. CRARY
CENTRAL STATION ENGINEERING DIVISIONS
GENERAL ELECTRIC COMPANY
SCHENECTADY, NEW YORK
One of a Series Written in the Interest of
the General Electric Advanced
Engineering Program
New York » JOHN WILEY & SONS, INC.
London . CHAPMAN & HALL, LTD.
Genera
ted o
n 2
01
4-0
3-2
4 0
0:4
3 G
MT /
htt
p:/
/hd
l.hand
le.n
et/
20
27
/mdp.3
90
15
00
29
00
27
5Public
Dom
ain
, G
oog
le-d
igit
ized
/
htt
p:/
/ww
w.h
ath
itru
st.o
rg/a
ccess
_use
#pd-g
oogle
Copyright, 1947
By
GENERAL ELECTRIC COMPANY
All Rights Reserved
This book or any part thereof must not
be reproduced in any form without
the written permission of the publisher.
PRINTED IN THE UNITED STATES OF AMERICA
Genera
ted o
n 2
01
4-0
3-2
4 0
0:4
3 G
MT /
htt
p:/
/hd
l.hand
le.n
et/
20
27
/mdp.3
90
15
00
29
00
27
5Public
Dom
ain
, G
oog
le-d
igit
ized
/
htt
p:/
/ww
w.h
ath
itru
st.o
rg/a
ccess
_use
#pd-g
oogle
To
MARJORIE
Genera
ted o
n 2
01
4-0
3-2
4 0
0:4
3 G
MT /
htt
p:/
/hd
l.hand
le.n
et/
20
27
/mdp.3
90
15
00
29
00
27
5Public
Dom
ain
, G
oog
le-d
igit
ized
/
htt
p:/
/ww
w.h
ath
itru
st.o
rg/a
ccess
_use
#pd-g
oogle
Genera
ted o
n 2
01
4-0
3-2
4 0
0:4
3 G
MT /
htt
p:/
/hd
l.hand
le.n
et/
20
27
/mdp.3
90
15
00
29
00
27
5Public
Dom
ain
, G
oog
le-d
igit
ized
/
htt
p:/
/ww
w.h
ath
itru
st.o
rg/a
ccess
_use
#pd-g
oogle
PREFACE
Power system engineers have become engaged, since the end of
World War II, in a race to increase system capacities to meet an
unprecedented demand for electrical energy. In some countries elec-
trical energy has had to be rationed, and system engineers have
been forced to resort to emergency operating and design procedures.
During this critical postwar period the system designer has been
acquiring new practical knowledge of the load-carrying ability of his
system.
This experience will help in obtaining future economical operation.
The trend in system design has been accordingly accelerated in the
direction to increase the unit loadings of circuits. It can be ex-
pected that new transmission line designs will have increased circuit
loadings, which will require more complete knowledge of the sta-
bility characteristics. These developments gave me added encour-
agement to complete this book as quickly as possible so that this
volume may be of assistance for the present emergency as well as
for the future task of the system engineer.
The first six chapters of this second volume are devoted to methods
of analysis, the next three to applications. In the last chapter the
various types of stability are discussed and the term "overall" stabil-
ity is proposed to indicate the necessary unity of the complete
phenomena. Also in this chapter indication is given of the future
possibilities of control equipment, particularly voltage regulators.
Included also is a bibliography which, although limited, indicates
the wealth of material on this subject. It may be used as a source
for reference to more detailed and complete treatises of particular
aspects of a problem.
In this endeavor, as in Volume I, I have drawn from material
developed by many engineers, particularly those with whom I have
been associated in the Central Station Engineering Divisions of the
General Electric Company.
These are some of the specific contributions of my associates.
Material for Sections 11 and 12 of Chapter 3 and Section 14 of Chap-
ter 4 was drawn from the published work of R. H. Park, E. H.
Bancker, J. B. McClure, and the late I. H. Summers, who were
among the first contributors to the fundamental theory of transient
Genera
ted o
n 2
01
4-0
3-2
4 0
0:4
4 G
MT /
htt
p:/
/hd
l.hand
le.n
et/
20
27
/mdp.3
90
15
00
29
00
27
5Public
Dom
ain
, G
oog
le-d
igit
ized
/
htt
p:/
/ww
w.h
ath
itru
st.o
rg/a
ccess
_use
#pd-g
oogle
viii PREFACE
stability. Material for Sections 14 and 16 of Chapter 4 was drawn
from publications by H. L. Byrd, S. H. Pritchard, and Edith Clarke,
who made valuable applications of the theory. Section 27 of Chap-
ter 6 was taken from published tests made by F. A. Hamilton, Jr.;
and Figures 6-13 and 617 were taken from unpublished work of
Charles Concordia. Material for Chapters 8 and 10 was taken from
work done in the preparation of an A.I.E.E. paper with my co-
authors L. F. Kennedy and C. A. Woodrow and another paper with
my co-authors C. Concordia and F. J. Maginniss. The staffs of the
General Electric Company a-c network and differential analyzers
contributed to Chapters 5, 7, and 10. Data for Appendix 5 were
obtained largely from a General Electric Company publication writ-
ten by R. N. Slinger and N. H. Meyers. C. E. Kilbourne contributed
suggestions and photographs for Section 34 of Chapter 7.
I wish to take this opportunity to thank Charles Concordia for
advice during the preparation of the manuscript, Robert Treat for
suggestions during his review of the galley proof, and the American
Institute of Electrical Engineers for permission to use text material
and figures of published papers. I also wish to thank H. A. Peterson
and A. W. Rankin for reading the manuscript and suggesting many
valuable improvements in arrangement and material; F. J. Maginniss,
who checked the galley proof and gave critical assistance; Dorothy
Rogers for her skill in preparing the drawings; and Doris Stupp for
her careful calculations. Finally, I wish to express appreciation to
Rose Pileggi for her excellent help in typing the manuscript, super-
vising the drawing of figures and proofreading, and corresponding
with the publishers.
May, 1947 Selden B. Crary
Genera
ted o
n 2
01
4-0
3-2
4 0
0:4
4 G
MT /
htt
p:/
/hd
l.hand
le.n
et/
20
27
/mdp.3
90
15
00
29
00
27
5Public
Dom
ain
, G
oog
le-d
igit
ized
/
htt
p:/
/ww
w.h
ath
itru
st.o
rg/a
ccess
_use
#pd-g
oogle
NOMENCLATURE
The following nomenclature defines the symbols used generally throughout the
text. There are instances where the same symbol is used locally for a different
quantity and is so defined.
Page
Where Defined
or First Used
E m voltage 6
e = voltage 9
/ = frequency 45
H = inertia constant 45
/ = current 6
< = current 9
j = vector operator 150
k = acceleration constant 48
L = inductance 6
M = iirfH 245
P = real power 7
Q = reactive power 122
R = resistance 118
r = resistance 10
5 ■ speed 6
T = torque 6
V = voltage 34
X = reactance 6
x = reactance 12
y = admittance 203
Z = impedance 7
a = 90 — 8 = complement of impedance angle 7
6 = electrical angular displacement between voltages ... 7
t = flux linkages 6
6 = power factor angle 13
Subscripts
a = armature or accelerating
av = average
amort =' amortisseur
c — condenser
d = direct axis
e = external
/ = fault or field
g = generator
Genera
ted o
n 2
01
4-0
3-2
4 0
0:4
5 G
MT /
htt
p:/
/hd
l.hand
le.n
et/
20
27
/mdp.3
90
15
00
29
00
27
5Public
Dom
ain
, G
oog
le-d
igit
ized
/
htt
p:/
/ww
w.h
ath
itru
st.o
rg/a
ccess
_use
#pd-g
oogle
x NOMENCLATURE
I = field excitation
L — line, load
m = mechanical or motor
max = maximum
min = minimum
o = open circuit or initial
p = Potier
Q = quadrature
q = quadrature
r = rated receiving end
J = saturation, sending end, switching
5 = sending end, synchronous
t = terminal
T — transformer
Prime indicates transient.
Arabic numerals designate synchronous machines or synchronous machine groups.
Letters designate points in the system or loads.
Genera
ted o
n 2
01
4-0
3-2
4 0
0:4
6 G
MT /
htt
p:/
/hd
l.hand
le.n
et/
20
27
/mdp.3
90
15
00
29
00
27
5Public
Dom
ain
, G
oog
le-d
igit
ized
/
htt
p:/
/ww
w.h
ath
itru
st.o
rg/a
ccess
_use
#pd-g
oogle
CONTENTS
Introduction
CHAPTER
1 Fundamental Concepts For Transient Analysis .... 5
1. Electrical torque. 2. Constant flux linkage theorem. 3. Illus-
tration of use of constant flux linkage theorem. 4. Application of
equal area criterion. 5. Illustrative problems.
2 Synchronous Machine Torque Angle Characteristics ... 22
6. Salient pole torque angle characteristics. 7. Cylindrical rotor
torque angle characteristics. 8. Subtransient effects and flux decay.
9. Approximate method.
3 System Torque Angle Characteristics 33
10. Method of analyzing faults. 11. Switching. 12. Derivation of
the step-by-step method.
4 Two-Machine Stability 51
13. Two-machine stability problem. 14. Two-machines — resistance
neglected. 15. Results obtained from two-machine stability analyses.
16. Two machines — with resistance. 17. Suddenly applied loads.
5 Multi-Machine Problem 88
18. Multi-machines — procedure. 19. Multi-machine methods. 20.
Special considerations. 21. Calculation of system voltages and cur-
rents during transient swing. 22. Relay operation during swing.
6 Generator Characteristics — Methods of Analysis . . . 129
23. Additional characteristics to be analyzed. 24. Saliency effects.
25. Excitation system response. 26. Effect of saturation. 27. Damp-
ing. 28. Negative phase sequence braking torque. 29. Summary.
Methods of analysis.
7 Generator Stability Characteristics 167
30. Importance of generator characteristics. 31. Transient reactance.
32. Inertia. 33. Excitation response. 34. Amortisseur windings.
35. Prime mover damping. 36. Generator characteristics. Conclu-
sions. 37. Additional methods for improving generator stability.
38. Generator braking resistors. 39. Watt-responsive governor. 40.
Steam topping units.
8 High-Speed Reclosing Circuit Breakers 201
41. High-speed reclosing. General considerations. 42. Conditions
for study of system interconnections. 43. Three-phase reclosing tie
lines and system interconnections. 44. Single-phase reclosing tie lines
Genera
ted o
n 2
01
4-0
3-2
4 0
0:4
6 G
MT /
htt
p:/
/hd
l.hand
le.n
et/
20
27
/mdp.3
90
15
00
29
00
27
5Public
Dom
ain
, G
oog
le-d
igit
ized
/
htt
p:/
/ww
w.h
ath
itru
st.o
rg/a
ccess
_use
#pd-g
oogle
xii CONTENTS
and system interconnections. 45. Conditions for study of hydro
systems. 46. High-speed reclosing for hydroelectric system. 47.
Summary. Three-phase reclosing on system ties. Single-phase re-
closing on system ties. Single-phase switching for a hydro station
delivering power over a single-circuit to a large system. 48. Addi-
tional considerations. 49. General conclusions.
9 System Design 225
50. System design for stability. 51. Transmission. 52. Generators.
53. Synchronous motors. 54. Synchronous condensers. 55. Shunt
capacitors. 56. Frequency changers. 57. Concluding remarks.
10 Overall Stability 241
58. Types of stability. 59. Automatic voltage regulation. 60. Effect
of excitation systems. 61. Oscillations. 62. Overall stability.
Appendix
I. Natural Frequency Of Oscillation Between Two Synchronous
Machines 283
II. Equivalent Circuits 285
III. Synchronous Machine Relations 287
IV. Excitation System Response 289
V. Stability Data 291
Bibliography 297
Index 325
Genera
ted o
n 2
01
4-0
3-2
4 0
0:4
6 G
MT /
htt
p:/
/hd
l.hand
le.n
et/
20
27
/mdp.3
90
15
00
29
00
27
5Public
Dom
ain
, G
oog
le-d
igit
ized
/
htt
p:/
/ww
w.h
ath
itru
st.o
rg/a
ccess
_use
#pd-g
oogle
INTRODUCTION
Power system stability is naturally divided into two parts, steady
state and transient. Steady state is concerned with small or gradual
changes, whereas the transient state deals with large or sudden
changes. This major division of subject matter provides a necessary
first step in the classification of the phenomena.
This natural division was taken as the basis for including the
steady state theory with its applications in the first book of this
series (Volume I); the transient analysis was left for inclusion in
this book (Volume II).
The advisability of treating steady state stability first as com-
pared to a general treatment with steady state as a special case may
be at first glance open to question. However, as becomes quickly
evident, idealized cases need to be used as stepping stones to a more
complete solution.
The practical solution of a system stability problem comes from
a physical conception of a knowledge built from such idealized cases,
using reasonable assumptions tempered wherever possible by oper-
ating experience. This puts the solution of system stability prob-
lems in the province of the engineer, who must know above all what
assumptions and simplifications to make and also how to test the
system analytically for stability. An actual power system has too
many machines and too many devices affecting its operation to allow
for a direct, rigorous mathematical solution. The solution comes
from simplified analytical tests which in effect approach the actual
answer without ever quite attaining it. Although such is the method
of attack for most important practical engineering problems, the
student should not be deluded into thinking that power system stabil-
ity is different.
The study of idealized cases therefore is necessary for the student
or the engineer who approaches the problem for the first time. Later,
after a certain vantage point of understanding is gained, it may be
possible to undertake a generalization of the theory. At least these
reasons seem to justify the manner in which these first two volumes
were written. Volume I treats circuit fundamentals and steady
1
Genera
ted o
n 2
01
4-0
3-2
4 0
0:4
6 G
MT /
htt
p:/
/hd
l.hand
le.n
et/
20
27
/mdp.3
90
15
00
29
00
27
5Public
Dom
ain
, G
oog
le-d
igit
ized
/
htt
p:/
/ww
w.h
ath
itru
st.o
rg/a
ccess
_use
#pd-g
oogle
2 INTRODUCTION
state stability; Volume II treats transient stability. The last chapter
of Volume II is the only attempt at a generalization of the theory.
As mentioned, the procedure for analysis is to test the system by
analytical means. These tests for both small and large changes are
usually for conditions close to the boundary between stability and
instability. When conditions are critical, the stability of many less
severe conditions can be determined directly from physical reason-
ing. In this way, a few well-selected test cases can provide a good
evaluation of the system's overall stability.
If the system successfully passes a test for the first swing transient
stability and a test for steady state following the disturbance, it can
in most cases be fairly safely concluded that the system is stable.
This testing of the system at the end points of the phenomena, the
beginning and the end of a disturbance, provides a simple analytical
test procedure which greatly simplifies the analysis and makes it
unnecessary to solve the intermediate phenomena. The last chapter
of this book, Chapter 10, deals briefly with the intermediate phenom-
ena and lends support to this simplified analytical test procedure.
It is important to keep in mind that such a division or classifica-
tion into steady state or transient phenomena is necessary primarily
for purposes of analysis and should not conceal the basic oneness of
the phenomena or a complete physical conception of the problem.
The most important general question and the one toward which the
work of analysis is directed is: Will the system operate with stability
and will it carry the load?
Power systems may be designed and operated with a fair degree
of success without a conscious knowledge of power system stability.
As a result, oftentimes one hears the comment that an understand-
ing of this subject is no longer necessary because of greatly improved
equipment which makes system interruptions relatively rare. How-
ever, this may be an indication that additional loads could be carried
or that full use of the stability factors have not been made to make
possible a further reduction in operating cost and system investment
per unit of load.
Knowledge of system stability has become more, rather than less,
important for the following reasons:
1. Many of the initial misconceptions and some of the distrust
of the theory by practical operating people have been overcome by
the evolution of a sound theory and methods of analysis based on
both theory and operating results. This increased confidence has
resulted in a greater reliance upon analytical methods for assisting
in operating procedures.
Genera
ted o
n 2
01
4-0
3-2
4 0
0:4
7 G
MT /
htt
p:/
/hd
l.hand
le.n
et/
20
27
/mdp.3
90
15
00
29
00
27
5Public
Dom
ain
, G
oog
le-d
igit
ized
/
htt
p:/
/ww
w.h
ath
itru
st.o
rg/a
ccess
_use
#pd-g
oogle
INTRODUCTION 3
2. Interest in long-distance power transmission has been renewed
with the ending of World War II. It has been brought about be-
cause of an increased necessity for conserving natural fuels as well
as increased industrialization in new areas which have undeveloped
hydropower. Furthermore, the development of a sound theory for
long-distance power transmission has increased the confidence of
engineers in the technical practicability of long-distance transmis-
sion for furthering these economic needs.
3. The war experience has shown that reliable operation could
be obtained closer to the stability limits than was previously con-
sidered feasible. This knowledge, although furthered by the war,
has come with the general experience with new lines after their
installation and subsequent operation through emergencies. Thus
new confidence has been obtained for increasing system loadings
more closely to those values determined by the stability limits, with
a resulting change in emphasis in system design. This has resulted
in a trend in the use of larger conductors and more controlled re-
active kva to obtain greater circuit loadings.
4. The increased immunity of transmission lines to lightning
brought about by modern transmission line design with ground wires
and low tower footing resistances has also placed emphasis on loading
transmission lines more nearly to limits determined by the stability
limitations. This has resulted in a greater emphasis on control
devices to increase the limits of transmission systems and will re-
quire a greatly increased knowledge of the effect of control, rewarded
by further increases in possible power system loadings and further
reductions in transmission costs.
5. With the passing of the first stage in the development of power
systems, which might be characterized as pioneering to furnish cus-
tomers with sufficient power at a high level of reliability, power
systems are now entering a second stage in which capital investment
and operating costs will be more important factors, requiring all
available knowledge to bring about an increasingly more economical
system.
6. Further comparisons will be made between the d-c and a-c
transmission requiring accurate evaluations of stability limits both
for a-c and combined a-c, d-c transmission systems. These com-
parisons will be made for transmission lines of substantial length
and for large blocks of power, and because of the investment in-
volved it will be important that modern methods of analysis be
used.
7. Methods of compensating long lines to increase the stability
Genera
ted o
n 2
01
4-0
3-2
4 0
0:4
7 G
MT /
htt
p:/
/hd
l.hand
le.n
et/
20
27
/mdp.3
90
15
00
29
00
27
5Public
Dom
ain
, G
oog
le-d
igit
ized
/
htt
p:/
/ww
w.h
ath
itru
st.o
rg/a
ccess
_use
#pd-g
oogle
4 INTRODUCTION
limits will be further studied and used in the next ten or twenty
years for future long-distance transmission lines which can be ex-
pected to exceed 300 miles in length.
Thus the future offers a real challenge to those engineers who seek
a full solution of their overall system problems to obtain a good
understanding of power system stability. This book and Volume I
were written to assist in meeting this challenge.
Genera
ted o
n 2
01
4-0
3-2
4 0
0:4
7 G
MT /
htt
p:/
/hd
l.hand
le.n
et/
20
27
/mdp.3
90
15
00
29
00
27
5Public
Dom
ain
, G
oog
le-d
igit
ized
/
htt
p:/
/ww
w.h
ath
itru
st.o
rg/a
ccess
_use
#pd-g
oogle
CHAPTER 1
FUNDAMENTAL CONCEPTS FOR TRANSIENT ANALYSIS
1. Electrical Torque. Power system stability, both theory and
application, has been necessarily classified in order that an under-
standing of the different phenomena and their relation to each other
will be more readily visualized. The study of stability, for purposes
of analysis, has been divided into two major divisions:
I. Steady State: Stability under slow or gradual load changes.
II. Transient: Stability under transient or sudden load changes.
This grouping is primarily an aid to analysis; actually stability is
one phenomenon. The primary question is always: "Is the system
stable?" To answer such a question accurately is to understand the
whole problem completely. In some respects the above division,
although helpful, has tended to obscure the fact that it is all one
phenomenon. For example, a system will be tested for stability
following a short circuit to determine if it is stable during the first
swing. Also, it can be tested for steady state stability after the fault
has been removed and the oscillations have died out. There exists
an intermediate period, following the first swing and until the new
steady state condition has been reached, that is difficult to analyze
because of the necessity of considering the effect of control devices,
damping torques, etc. This subsequent swing period is usually
neglected because of the difficulty of analysis. The system is usually!
considered stable if it passes the analytical tests of stability during
the first swing apart and then during the final steady state period. ,
Although this is not rigorously correct as stability may be lost during
a subsequent swing, it is practically so. For, as shown in Chapter 10,
if stability is maintained during the first swing, the system will more
than likely be stable during the subsequent swings, particularly if it
is stable for the subsequent steady state condition. This allows for
a great simplification in the analysis.
Volume I was devoted to a study of steady state stability; in this
book the analysis of system performance will be extended to include
the transient stability characteristics. The transient performance is
determined by many factors, some of which have but little influence
on the steady state performance, discussed in Volume I, and others
S
Genera
ted o
n 2
01
4-0
3-2
4 0
0:4
7 G
MT /
htt
p:/
/hd
l.hand
le.n
et/
20
27
/mdp.3
90
15
00
29
00
27
5Public
Dom
ain
, G
oog
le-d
igit
ized
/
htt
p:/
/ww
w.h
ath
itru
st.o
rg/a
ccess
_use
#pd-g
oogle