POWER
SYSTEM
DYNAMICS
Stability
and
control
Second
Edition
"This page is Intentionally Left Blank"
POWER
SYSTEM
DYNAMICS
Stability
and Control
Second Edition
K.
R.
Padiyar
Indian Institute
of
Science, Bangalore
SSP
BS
Publications
4-4-309, Giriraj Lane, Sultan Bazar,
Hyderabad -
500 095 -
AP.
Phone: 040-23445677,23445688
e-mail:

Publications
Printed
at
:
4-4-309, Giriraj Lane, Sultan Bazal,
Hyderabad - 500 095
AP.
Phone: 040-23445677,23445688
e-mail:

website:
www.bspublications.net
Adithya Art Printers
Hyderabad.
ISBN:
81-7800-186-1
TO
PROF.
H
N.
RAMACHANDRA
RAO
"This page is Intentionally Left Blank"
Contents
1
Basic
Concepts
1
1.1
General

Synchronous Machine
3.1
Introduction

.
3.2
Synchronous
Machine.
3.3
Park's
Transformation
3.4 Analysis of Steady State Performance.
3.5
Per Unit Quantities

.
3.6
Equivalent Circuits of Synchronous Machine
3.7 Determination of Parameters of Equivalent Circuits
3.8
Measurements for Obtaining
Data
. . . . . . .
3.9
Saturation Models . . . . . . . . . . . . . . . .
3.10 Transient Analysis of a Synchronous Machine
9
9
13
16


.
4.2 Excitation System Modelling. . . . . . .
4.3 Excitation
Systems- Standard Block Diagram
4.4 System Representation by
State
Equations
4.5 Prime-Mover
Control
System.
4.6
Examples
. .
5
Transmission
Lines,
SVC
and
Loads
5.1
Transmission Lines

.
5.2 D-Q Transformation using
a -
(3
Variables
5.3 Static Var compensators
5.4 Loads

.
6.6
Consideration
of
other Machine Models .
6.7 Inclusion of
SVC Model

.
7
Analysis
of
Single
Machine
System
177
178
181
188
191
199
211
221
7.1
Small Signal Analysis with Block Diagram Representation
221
7.2
Characteristic Equation (CE)
and
Application

257
257
259
263
Contents
ix
8.4 Structure and tuning of PSS . . . . . . . . . . . 264
8.5
Field implementation and operating experience 275
8.6
Examples of PSS Design and Application. . . . 277
8.7
Stabilization through HVDC converter
and
SVC controllers
291
8:8 Recent developments
and
future trends
9
Analysis
of
Multimachine
System
9.1 A Simplified System
Model.
9.2 Detailed Models: Case I

9.3 Detailed
Model:

10.4
Analysis of the Combined System . . 348
10.5
Computation of Ye(s) : Simplified Machine
Model.
350
10.6 Computation of Ye(s): Detailed Machine Model . . 354
10.7
Analysis of Torsional Interaction - A Physical Reasoning 356
10.8
State
Space Equations and Eigenvalue Analysis 360
10.9 Simulation of SSR . 369
10.10
A Case Study . . . . . . . . . . . . . . . . . . . 369
11
Countermeasures
for
Subsynchronous
Resonance
387
11.1 System Planning Considerations .
11.2 Filtering Schemes . . . .
11.3 Damping Schemes . . . .
11.4 Relaying and Protection
12
Simulation
for
Transient
Stability

Direct
Stability
Evalua-
tion
441
13.1
Introduction
441
13.2 Mathematical Formulation . . . . . . . . . . . . . . . .
442
13.3 Energy Function Analysis of a Single Machine
System.
446
13.4 Structure Preserving Energy Function. . . . . . . . . .
451
13.5 Structure-Preserving Energy Function with Detailed Generator
Models.
. . . . . . . . . . . . . . . . . 457
13.6 Determination of Stability Boundary .
13.7 Extended Equal Area Criterion (EEAC)
13.8 Case
Studies . . . . . .

14 Transient
Stability
Controllers
14.1
System
resign
for Transient

is Voltage
Stability?

15.2 Factors affecting voltage instability
and
collapse
15.3 Comparison of Angle and Voltage Stability

15.4 Analysis of Voltage Instability and Collapse . .
15.5 Integrated Analysis of Voltage and Angle
Stability.
15.6 Control of Voltage Instability

.
462
471
473
489
489
492
493
498
499
501
502
505
513
513
515
518

General
Modern power systems are characterized by extensive system interconnections
and
increasing dependence
on
control for optimum utilization of existing
re-
sources.
The
supply of reliable
and
economic electric energy is a major deter-
minant
of
industrial progress and consequent rise in
the
standard
of living.
The
increasing demand for electric power coupled with resource
and
environmental
constraints pose several challenges to system planners.
The
generation may have
to
be
sited
at
locations far away from load centres (to exploit

day.
Power system dynamics has
an
important bearing on
the
satisfactory
system operation.
It
is
influenced by
the
dynamics of
the
system components
such as generators, transmission lines, loads
and
other control equipment
(HVDe
and
SVC controllers).
The
dynamic behaviour
of
power systems
can
be
quite
complex
and
a good understanding

is
termed as
the
synchronous operation of
a system. Any disturbance small or large can affect
the
synchronous operation.
2
Power System Dynamics - Stability and Control
For example, there can be a sudden increase in the load or loss of generation.
Another type of disturbance is
the
switching out of a transmission line, which
may occur due
to
overloading or a fault.
The
stability of a system determines
whether
the
system can settle down to a new or original steady
state
after the
transients disappear.
The
disturbance can be divided into
two
categories (a) small and (b)
large. A small disturbance
is

system,
for
purposes of the system analysis, it is divided into two broad classes
[8].
1.
Steady-State or Small Signal Stability
A power system
is
steady state stable for a particular steady
state
op-
erating condition
if,
following any small disturbance,
it
reaches a steady
state
operating condition which
is
identical or close to the pre-disturbance
operating condition.
2.
Transient Stability
A power system
is
transiently stable for a particular steady-state oper-
ating condition
and
for a particular (large) disturbance or sequence of
disturbances

are to
be
considered.
Another important point
to
be noted is
that
while
the
system can be
operated even if
it
is transiently unstable, small signal stability is necessary
at
all times.
In
general,
the
stability depends ·upon the system loading. An increaSe
in the load can bring
about
onset of instability. This shows the importance of
maintaining system stability even under high loading conditions.
1.
Basic Concepts
NORMAL
E,I
SECURE
Load
Tracking,

Cut
losseS,
protect
Equipment
.
SYSTEM
NOT
INTACT
E : Equality Contrainl
System
splitting
and/or
load
loss
I
EMERGENCY
I
Heroic
action
E,
I A-SECURE
SYSTEM
!NT
ACT
I :
Inequality
constraints,
-:
Negation
Figure 1.1: System Operating States

load demand.
The
other set' consists of
inequality constraints (I) which express limitations 'of
the
physical equipment
(such as currents
and
voltages must not exceed maximum limits).
The
classifi-
cation of the system states is based on the fulfillment or violation of
one or
both
sets of these constraints.
1.
Normal
Secure
State:
Here all equality (E)
and
inequality (I) con-
straints are satisfied.
In
this state, generation
is
adequate
to
supply the
existing load demand and no equipment is overloaded. Also in this state,

that
in
this
state,
the
security level is below some threshold
of
adequacy. This
implies
that
there is a danger of violating some of
the
inequality (I) con-
straints when subjected to disturbances (stresses).
It
can also
be
said
that
4
Power System Dynamics - Stability and Control
security constraints are not met. Preventive control enables the transition
from
an
alert
state
to a secure state.
3.
Emergency
State:

both
E and I constraints are violated. The
~iolation
of equality constraints implies
that
parts
of
system load are lost.
Emergency control action should be directed
at
avoiding total collapse.
5.
Restorative
State:
This is a transitional
state
in
which I constraints are
met from the emergency control actions taken
but
the E constraints are
yet to
be
satisfied. From this state, the system can transit to either the
alert or the
I1-ormal
state depending on the circumstances.
In
further developments
in

(In Extremis). The
knowledge of system dynamics is important in designing appropriate controllers.
This involves
both
the detection of the problem using dynamic security assess-
ment and initiation
of
the control action.
1.4
System
Dynamic
Problems
-
Current
Sta-
tus
and
-Recent
Trends
In
the early stages of power system development, (over
50
years ago)
both
steady
state and transient
s~ability
problems challenged system 'planners.
The
develop-

system can help in improving the system dynamic performance, particularly the
frequency stability.
Over last
25
years, the problems of
low
frequency power oscillations have
assumed importance.
The
frequency of oscillations
is
in
the
range of 0.2 to
2.0
Hz.
The
lower
the
frequency, the more widespread are
the
oscillations (also
called inter-area oscillations).
The
presence of these oscillations is traced to fast
voltage regulation in generators and can be overcome through supplementary
control employing power system stabilizers
(PSS).
The
design

behaviour of power systems and development of suitable controllers to overcome
the problems.
The
system has not only controllers located
at
generating stations
- such as excitation and speed governor controls
but
also controllers
at
HVDC
converter stations, Static VAR Compensators (SVC). New control devices such
as Thyristor Controlled Series Compensator (TCSC) and other
FACTS con-
trollers are also available. The multiplicity of controllers also present challenges
in their design and coordinated operation. Adaptive control strategies may be
required.
6 Power System Dynamics - Stability and Control
The
tools used
for
the study of system dynamic problems
in
the past
were simplistic. Analog simulation using
AC
network analysers were inadequate
for
considering detailed generator models.
The

simony' principle- include only those details which are essential.
References
and
Bibliography
1. N.G. Hingorani, 'FACTS - Flexible
AC
Transmission System', Conference
Publication
No.
345,
Fifth Int. Conf. on 'AC and DC Power Transmis-
sion', London Sept. 1991, pp.
1-7
2.
S.B. Crary,
Power
System
Stability,
Vol.
I:
Steady-State
Stability,
New York, Wiley,
1945
3.
S.B. Crary,
Power
System
Stability,
Vol.

6.
V.A. Venikov,
Transient
Phenomenon
in
Electric
Power
Systems,
New York, Pergamon,
1964
7.
R.T. Byerly
and
E.W. Kimbark (Ed.),
Stability
of
Large
Electric
Power
Systems,
New York, IEEE Press,
1974
8.
IEEE Task Force on Terms and Definitions, 'Proposed Terms and Defini-
tions for Power System Stability', IEEE Trans.
vol.
PAS-101, No.7, July
1982, pp. 1894-1898
9.
T.E. DyLiacco, 'Real-time Computer Control of Power Systems', Proc.

will review the classical methods of analysis of system stabil-
ity, incorporated
in
the
treatises of Kimbark
and
Crary. Although the assump-
tions behind
the
classical analysis are no longer valid with
the
introduction of
fast acting controllers and increasing complexity of
the
system,
the
simplified
approach forms a beginning in the study of system dynamics. Thus,
for
the sake
of maintaining the continuity, it
is
instructive
to
outline this approach.
As
the
objective
is
mainly to illustrate

through
an
equivalent impedance ZT. The infinite bus, by definition, represents
a bus with fixed voltage source.
The
magnitude, frequency
and
phase of the
voltage are unaltered by changes in the load (output of
the
generator).
It
is
to
be
noted
that
the
system shown in Fig. 2.1
is
a simplified representation of a
remote generator connected
to
a load centre through transmission line.
~T
HI L-ine lV~_ 1~
W.
Bw
Figure 2.1: Single line diagram of a single machine system
The

of
the
voltage source
is
determined by
the
field current which is constant.
2.
Damper circuits are neglected.
3.
Transient stability is
judged
by
the
first swing, which
is
normally reached
within one or two seconds.
4.
Flux decay
in
the
field circuit is neglected (This
is
valid for short period,
say
a second, following a disturbance, as
the
field time constant
is

for simplicity.
Xe
is
the
total
external reactance viewed from
the
generator
terminals.
The
generator reactance, x
g
,
is equal
to
synchronous reactance
Xd
for steady-state analysis. For transient analysis,
Xg
is equal
to
the
direct axis
transient reactance
x~.
In
this case,
the
magnitude of
the

Classical Methods
11
The
Swing
Equation
The
motion of
the
rotor is described by
the
following second order equa-
tion
(2.1)
where
J is
the
moment of inertia
Om
is
the
angular position of the rotor with respect
to
a stationary axis
Tm
is
the net mechanical input torque
and
Te
is
the

is
the
rotor angle with re-
spect to a synchronously rotating reference frame with velocity
W
m
.
Substituting
Eq. (2.3)
in
Eq. (2.2)
we
get
(2.4)
This is called the swing equation. Note
that
M is strictly not a constant.
However
the
variation in M
is
negligible and M can
be
considered as a constant.
(termed inertia constant).
It
is
convenient
to
express Eq. (2.4) in

m
JW
m
(WB)
(2)
Jl8
Jw!
Jl8
(2H)
Jl8
(2.6)
SB dt
2
= SB
WB
P dt
2
=
SBWB
dt
2
=
WB
dt
2
12
Power System Dynaraics - Stability and Control
where
a
is

MV
A
The
inertia constant H
has
the
dimension of time expressed
in
seconds.
H varies
in
a narrow range
(2-1O)
for most of the machines irrespective of their
ratings.
From Eq. (2.6),
the
per
unit
inertia
is
given by
- M
2H
M=-=-
BB
WB
(2.7)
The
above relation assumes

where
IBis
the
rated
frequency in Hz.
(2.8)
For convenience,
in
what follows, all quantities are expressed
in
per
unit
and no distinction will
be
made in
the
symbols to indicate
per
unit
quantities.
Thus, Eq. (2.4)
is
revised
and
expressed in p.u. quantities as
(2.9)
From Fig. 2.2,
the
expression for P
e