Mrinal K Pal. Power system stability
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REPRESENTATION OF LOADS IN STABILITY STUDIES
7-11
DQssDQsDQsDQs
r
dt
d
,,
1
,
1
,
)( iΨTWTΨTTe −+=
−−
ω
After carrying out the indicated matrix operations, we obtain
DQssDQsDQsDQs
r
dt
d
,,o,,
iΨWΨe −+=
ω
(7.31)
From (7.21)
DQrrDQr
r
dt
d
,
1
,
1
)( iTΨT0
−−
+=
or
DQrrDQr
r
dt
d
,,
1
)( iΨTT0 +=
−
After carrying out the indicated matrix operation, we obtain
DQrrDQrDQr
r
dt
d
,,,
iΨWΨ0 ++=
θ
&
(7.32)
The power input is
)(
2
3
][
2
3
2
3
,
11
,,, sQsQsDsDDQsDQsdqsdqs
ieieP +=
′′
=
′
=
−−
iTTeie (7.33)
We obtain the electrical torque, as in the case of synchronous machine, as
()
sDsQsQsD
o
e
iiT
ψψ
ω
ω
−=
2
3
(7.34)
Using the same per unit system as used for synchronous machine, we convert equations (7.29) -
(7.34) to per unit form, to obtain
rsss
ML iiΨ
+
−
=
(7.35)
rrsr
LM iiΨ
+
−
=
(7.36)
sssss
r
dt
d
iΨWΨe −+=
o
1
ω
(7.37)
rrrr
rs
dt
d
iΨWΨ0 ++=
o
1
ω
(7.38)
where s is the per unit slip
o
o
o
ω
ωω
ω
θ
−
==
&
s (7.39)
Note that in the above equations we have dropped DQ from the subscript for clarity.
sQsQsDsD
ieieP
+
= (7.40)
sDsQsQsDe
iiT
ψ
ψ
−
=
(7.41)
REPRESENTATION OF LOADS IN STABILITY STUDIES
7-12
Induction Motor Representation in Stability Studies
We now derive the induction motor equations suitable for use in stability studies. The effect of
the
s
dt
d
Ψ terms is small and can be neglected.
From (7.36)
s
r
r
r
r
L
M
L
iΨi +=
1
(7.42)
Substituting (7.42) in (7.35),
s
r
sr
r
s
L
M
L
L
M
iΨΨ
−−=
2
(7.43)
Substituting (7.43) into (7.37), and using the notation
r
s
L
M
Lx
2
−=
′
,
sssr
r
s
xr
L
M
iWiΨWe
′
−−= (7.44)
Defining
rQ
r
D
L
M
e
ψ
−=
′
(7.45)
rD
r
Q
L
M
e
ψ
=
′
(7.46)
we obtain, from (7.44)
DsQDD
irixee
−
′
+
′
=
(7.47)
QsDQQ
irixee
−
′
−
′
= (7.48)
In the above equations we have dropped the subscript s in voltages and currents for clarity.
From (7.42) and (7.38)
s
r
r
r
r
r
rr
L
rM
L
r
s
dt
d
iΨΨWΨ −−−=
o
1
ω
Multiplying both sides by
rr
r
L
M
r
L
, using (7.45), (7.46), and defining
r
r
o
r
L
T
o
ω
=
′
ss
D
Q
D
Q
o
D
Q
o
xL
e
e
e
e
Ts
e
e
dt
d
T
iW )(
o
′
−−
′
−
′
−
′
−
′
′
−=
′
−
′
′
ω
or
′
−−
′
−
′
−
′
′
′
−=
′
−
′
′
Q
D
D
Q
Q
D
o
D
Q
o
i
i
xx
e
e
e
e
Ts
e
e
dt
d
T )(
o
ω
(7.49)
where xL
s
=
REPRESENTATION OF LOADS IN STABILITY STUDIES
7-13
Equation (7.49) can be expressed as separate equations
DoDQ
Q
o
esTixxe
dt
ed
T
′′
−
′
−−
′
−=
′
′
o
)(
ω
(7.50)
QoQD
D
o
esTixxe
dt
ed
T
′′
+
′
−+
′
−=
′
′
o
)(
ω
(7.51)
Using (7.37), (7.47), (7.48), the torque expression (7.41) can be written as
QQDDe
ieieT
′
+
′
=
(7.52)
As in the case of synchronous machine,
QDQD
iiee ,,, , etc. can be treated as phasors, and (7.47),
(7.48), and (7.50), (7.51), and (7.52) can be written as
)()(
QDsQDQD
ijixjrejeeje
+
′
+
−
′
+
′
=+
or
ixjree
s
ˆ
)(
ˆˆ
′
+−
′
=
(7.53)
and
)()()()()(
o QDoQDQDQDo
ejesTjijixxjejeeje
dt
d
T
′
+
′′
−+
′
−−
′
+
′
−=
′
+
′′
ω
or
esTjixxje
dt
ed
T
oo
ˆ
ˆ
)(
ˆ
ˆ
o
′′
−
′
−−
′
−=
′
′
ω
(7.54)
)
ˆ
ˆ
Re(
∗
′
= ieT
e
(7.55)
Figure 7.2 shows the equivalent circuit representing the transient behavior of induction motor.
Fig. 7.2 Equivalent circuit for transient behavior of induction motor
The equation of motion is
em
TT
dt
dH
−=
ω
ω
o
2
where
m
T is the mechanical torque
or
)(2
em
TT
dt
ds
H −−= (7.56)
Note that for motor action both
m
T and
e
T are negative.
REPRESENTATION OF LOADS IN STABILITY STUDIES
7-14
Reference 1 recommends the following expression for the mechanical torque
am
m
m
m
m
m
m
m
m
DCBA
T
T
++
+
=
oo
2
oo
ω
ω
ω
ω
ω
ω
(7.57)
)(1 DBAC
+
+−= (7.58)
In the steady state
0
ˆ
=
′
dt
ed
, therefore, from (7.54),
i
sTj
xxj
e
o
ˆ
1
)(
ˆ
o
ω
′
+
′
−
−=
′
(7.59)
Substituting (7.59) in (7.53), we obtain
1
)(
)(
ˆ
ˆ
o
sTj
xxj
xjr
i
e
o
s
ω
′
+
′
−
−
′
+−= (7.60)
Equation (7.60) leads to the following equivalent circuit
Fig. 7.3 Steady-state equivalent circuit of induction motor
In Figure 7.3, current direction is shown for motor action, for clarity.
The electrical torque
2
2
o
o
o
)(1
)(
ˆˆ
1
)(
Re)
ˆ
ˆ
Re( i
sT
sTxx
ii
sTj
xxj
ieT
o
o
o
e
ω
ω
ω
′
+
′′
−
−=
′
+
′
−
−=
′
=
∗∗
From Figure 7.3
i
xxj
sT
xx
xxj
i
ii
o
ˆ
)(
)(
ˆ
ˆˆ
o
2
1
′
−+
′
′
−
′
−
−=
−=
ω
or
2
o
o
ˆ
1
ˆ
i
sTj
sTj
i
o
o
ω
ω
′
′
+
−=
2
2
2
o
2
o
2
)(
)(1
i
sT
sT
i
o
o
ω
ω
′
′
+
=
∴
REPRESENTATION OF LOADS IN STABILITY STUDIES
7-15
and
s
r
i
sT
xx
iT
o
e
2
2
2
o
2
2
−=
′
′
−
−=
ω
where
o
2
ω
o
T
xx
r
′
′
−
=
As in synchronous machine, we can express x and
x
′
in terms of mutual and leakage reactances
rm
rm
sms
xx
xx
xxxxx
+
+=
′
+= ,
Also
r
rm
o
r
xx
T
o
ω
+
=
′
Therefore, an alternative form of the equivalent circuit is as shown in Figure 7.4. The derivation
is left as an exercise.
Fig. 7.4 An alternative form of equivalent circuit of induction motor
Initial condition calculation
Given P, E (and Q) from power flow, we can calculate the initial slip, s, as follows
From (7.60)
e
rsTxjxsTr
sTj
i
soos
o
ˆ
)(
1
ˆ
oo
o
ωω
ω
′
++
′′
−
′
+
−=
2
2
o
2
o
ooo
)()(
)(
)
ˆ
ˆ
Re( E
rsTxxsTr
rsTxsTxsTr
ieP
soos
sooos
ωω
ω
ω
ω
′
++
′′
−
′
+
′
+
′
′
−
−==
∗
(7.61)
The above yields two values of s (if a solution exists) for given values of P and E. Alternatively,
for a given value of E, P can be calculated for assumed values of s, starting with s = 0 (no-load
condition). For s = 0,
2
2
2
E
xr
r
P
s
s
+
−= , the stator resistance loss
2
2
o
2
o
ooo
)()(
)(
)
ˆ
ˆ
Im( E
rsTxxsTr
xsTrsTrsTx
ieQ
soos
ososo
ωω
ω
ω
ω
′
++
′′
−
′
′
−
′
−
′
+
−=−=
∗
(7.62)
REPRESENTATION OF LOADS IN STABILITY STUDIES
7-16
For s = 0,
2
2
2
E
xr
x
Q
s
+
−=
The mismatch between Q calculated above and Q from power flow is adjusted by adding the
required reactive compensation.
Generic Load Model
In conventional rotor angle -- large and small disturbance -- stability analyses, composite loads
are usually represented by a ZIP or exponential model, although the IP part is not totally
appropriate, as has been pointed out earlier. Large induction motor loads are often represented by
detailed induction motor model, if their influence on the stability outcome is judged to be
important. In voltage stability analyses, if the bulk of the load is induction motor, it should be
represented as such, if the study results are going to be realistic. For loads other than motor
loads, the use of the ZIP model, as is common in angle stability analyses, can produce grossly
misleading results. If the load is known to be impedance load, and the impedance stays constant
during the study period, it should be represented as such. In Chapter 10 it is shown that static
constant I and constant P load models are inappropriate for voltage stability studies, and should
therefore be avoided.
Voltage stability problems often evolve over a longer period of time than do the traditional angle
stability problems. Many loads, e.g., thermostatically controlled heating loads, and loads
controlled by under-load-tap-changing transformers and distribution voltage regulators tend to
behave as constant power load over a longer term, even though the actual load itself is constant
impedance. For example, space and water heaters will temporarily draw less power following a
drop in voltage; however, they will stay on longer to generate the same amount of heat as would
be produced at nominal voltage. Over a longer period the total energy drawn from the system
would be, more or less, the same. An impedance load whose voltage is controlled would appear
as constant power load to the system. After an initial drop in power consumption following a
drop in voltage, these loads would tend to recover to their nominal level, i.e., collectively, they
would behave as constant power load over a longer period of time, provided the control limits
have not been reached. At limit, e.g., when the highest tap setting is reached, or the voltage is so
low that the heating loads stay on continuously, these loads would behave as constant Z load.
Various models have been proposed in the literature to reproduce the behavior of such loads in
voltage stability studies. All these models are, however, flawed. Apparently, sufficient effort was
not expended in understanding the underlying physical nature of the problem, as well as in the
mathematical logic used [10 - 14] (see also Chapter 10). Also, in these models, active and
reactive powers are modeled separately. In reality, there is always some coupling between active
and reactive powers. It is not difficult to show that these models, in their stated forms, if valid,
would be valid only in a specific range of parameter values, and when the right set of parameter
values is selected, they are actually unnecessarily complex ways of representing the simplified
first-order induction motor model.
Actually, the induction motor model, with appropriate parameter values, can faithfully represent
any composite dynamic load, for both angle and voltage stability studies. The selection of the
right parameter values may, however, present some challenge.
REPRESENTATION OF LOADS IN STABILITY STUDIES
7-17
References
1. IEEE Task Force on Load Representation for Dynamic Performance, “Standard Load Models for Power Flow
and Dynamic Performance Simulation,” IEEE Trans. PWRS - 10, (3), pp.1301-1313, August 1995.
2. IEEE Task Force on Load Representation for Dynamic Performance, “Load Representation for Dynamic
Performance Analysis,” IEEE Trans. PWRS - 8, (2), pp.472-482, May 1993.
3. C. Concordia and S. Ihara, “Load Representation for Power System Stability Studies,” IEEE Trans. PAS -
101, pp.969-977, 1982.
4. C. Concordia, “Representation of Loads,” IEEE Publication 75 CHO 970-4-PWR, Symposium on Adequacy
and Philosophy of Modeling: Dynamic System Performance, pp.41-45, 1975.
5. M.H. Kent, W.R. Schmus, F.A. McCrackin, and L.M. Wheeler, “Dynamic Modeling of Loads in Stability
Studies,” IEEE Trans. PAS - 88, (5), pp.756-763, May 1969.
6. M.K. Pal, "Voltage Stability: Analysis Needs, Modelling Requirement, and Modelling Adequacy," IEE Proc.
C, Vol. 140, (4), pp. 279-286, July 1993.
7. M.K. Pal, discussion of “An Investigation of Voltage Instability Problem,” by N. Yorino, et al., IEEE Trans.
PWRS - 7, (2), pp.608-609, May 1992.
8. D.S. Brereton, D.G. Lewis, and C.C. Young, “Representation of Induction Motor Loads During Power
System Stability Studies,” AIEE Trans. Vol. 76, pt. III, pp.451-461, August 1957.
9. L.W. Matsch, Electromagnetic and Electromechanical Machines, Intext Educational Publishers, N.Y. 1972.
10. M.K. Pal, discussion of “Nonlinear Dynamic Load Models with Recovery for Voltage Stability Studies,” by
D.J. Hill, IEEE Trans. PWRS - 8, (1), pp.173-174, February 1993.
11. M.K. Pal, discussion of “Voltage Stability Analysis Using Generic Dynamic Load Models,” by W. Xu and Y.
Mansour, IEEE Trans. PWRS - 9, (1), pp.487-488, February 1994.
12. M.K. Pal, discussion of “Modelling and Identification of Nonlinear Dynamic Loads in Power Systems,” by D.
Karlsson and D.J. Hill, IEEE Trans. PWRS - 9, (1), p.164, February 1994.
13. M.K. Pal, discussion of “Development and Comparative Study of Induction Machine Based Dynamic P,Q
Load Models,” by B.C. Lesieutre, P.W. Sauer, and M.A. Pai, IEEE Trans. PWRS - 10, (1), pp.189-190,
February 1995.
14. M.K. Pal, discussion of “State Variable Description of Dynamic Load Models in Electrical Power System
Analysis,” by Y. Kataoka, IEEE Trans. PWRS - 10, (1), p.156, February 1995.
8-1
CHAPTER 8
SMALL DISTURBANCE STABILITY
Stability programs designed for large-disturbance (transient) stability studies simulate system
response in time domain following a disturbance. The simulations are normally limited to a short
duration, usually a few seconds of real time. If the generator rotors swing back before reaching a
specified angle, the power system is considered large-disturbance stable. In the early days when
electric power networks were relatively confined, and sophisticated control equipment were not
generally utilized on the generators, the above criterion, also called “first swing stability,” was
enough to assure eventual stability of the system against that particular disturbance. Instability in
the initial period following a large disturbance is generally due to insufficient (or lack of)
synchronizing torque between the interconnected generators. Automatic control equipment, e.g.,
fast acting excitation control, help improve the first swing stability by increasing the
synchronizing torque. However, in the process they often reduce the damping torque, sometimes
even rendering the overall damping negative, thereby causing oscillatory instability. With the
growth of interconnection and application of advanced control equipment, consideration of
proper damping of oscillations became more important. In a system capable of withstanding the
initial shock of the disturbance, as evidenced by first swing stability, oscillations could continue
at a reduced amplitude for a while, only to increase a few cycles later and eventually cause
cascading line tripping and possibly system separation. This type of instability can manifest itself
not only following a major disturbance but also following a sudden small change in system
condition not generally classified as major disturbance, e.g., a moderate amount of load tripping,
a sudden addition of a large load, tripping of a minor transmission line, etc.
In order for a power system to be operable, it must have an equilibrium point that is stable. This
means it must be small-disturbance stable (see Chapter 2). In an interconnected system, due to an
improper selection of control parameters, a stable equilibrium point may not exist at all.
Oscillations once started can build up gradually although they may not be apparent during the
first few cycles. On the other hand a well designed control system can extend the stability limit
considerably.
Conventional stability programs designed for large-disturbance stability studies can, in principle,
be used to study the above phenomenon by running the program for a much longer period of real
time than is normally required to establish large-disturbance, first swing stability. This approach
can, however, be impractical and the result may not be fully conclusive.
Immediately following a small disturbance, or following a large disturbance after the system has
survived the initial shock (i.e., it’s first swing stable) and entered a state of oscillation, the system
nonlinearities do not play a major role. The power system can therefore be linearized about the
equilibrium point and useful information on the system small-disturbance performance can be
obtained from the linearized model. This would permit efficient design of the control, allowing
necessary fixes to combat instability. Certain aspects of the system stability problem are more
readily detectable, and hence correctable, by studying the linearized system.
Although there are several methods of obtaining stability information from a linearized system, a
state-space approach is desirable since in addition to providing all the information as obtainable
SMALL DISTURBANCE STABILITY
8-2
from other approaches this method provides certain additional advantages and can handle, with
sufficient ease, complex systems.
After linearization of the equations representing the power system dynamics around an operating
point, which will retain the behavior of the system under small disturbances, they can be reduced
to the state-space form
Axx =
&
, where x is an n-vector of the variables representing the state of
the system, and
A is a matrix of real constant coefficients, called the coefficient (or system)
matrix. The solution and stability condition of this system has been discussed in detail in Chapter
2. Briefly, for asymptotic stability, the eigenvalues of
A must have negative real parts. If there is
an eigenvalue with positive real part the equilibrium is unstable. Eigenvalues of A will be either
real or complex. If complex, they will occur in conjugate pairs. By studying these eigenvalues
much of the information on the system performance can be obtained. For a stable system the
larger the magnitude of the real part the faster the transient corresponding to that eigenvalue will
disappear. Any increase in the magnitude of the real part would mean a corresponding increase
in damping. The magnitude of the imaginary part indicates the frequency of oscillation of the
transient.
Interconnected generators, due to the inertias of the rotors, act as spring-mass systems, and as a
result a power system is inherently oscillatory (see Chapter 3). Two types of oscillations are
generally noticeable. When there is a disturbance in an area with closely coupled generators, the
machine rotors swing against each other and with respect to the rest of the system. This is the
local mode of oscillation. The special case is a single generating unit transmitting power over a
long transmission line to a load center, and the generator rotor swings against the rest of the
system. The characteristic local mode frequency is, in general, in the 0.5 - 2 Hz range, depending
on the machine and system parameters and loading conditions.
In large interconnected systems closely coupled generators in one area are connected through tie-
lines to other areas with closely coupled generators. Following a disturbance in the tie-line flow,
generators in one area (swinging more or less in unison) swing against generators in other areas
(again, swinging more or less in unison). This is the inter-area mode. This mode of oscillation
can also appear following a major disturbance in any of the areas separated by tie-lines. The
characteristic frequency of the inter-area mode of oscillation is generally in the range of 0.1 - 0.5
Hz. The range is lower than the corresponding range for local mode because of higher combined
inertias and relatively weak tie-lines Both local and inter-area modes tend to be coupled to some
degree. Although it is possible to identify these modes in time domain simulation, it can often be
done more conveniently from an analysis of the linearized system, and any coupling between
modes can also be revealed.
Due to the excessive time involved in obtaining complete eigensolutions of large matrices,
special methods have been developed which aim to compute a selected number of eigenvalues
and the associated eigenvectors in studies of very large systems. In one method a limited number
of eigenvalues associated with rotor angle modes are computed. In several other methods
eigenvalues associated with a small number of selected modes of interest are computed. These
methods can handle very large systems by taking advantage of network sparsity.
All these methods, however, have major drawbacks. Frequently they fail to identify and compute
the critical mode(s). Note that instability can be caused by any mode, not necessarily a particular
rotor angle mode or any other mode that might appear to be a likely candidate. In a complex
system it is not a simple matter to identify such modes a priori. As can be seen from the