Mrinal K Pal. Power system stability
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VOLTAGE STABILITY
10-43
The incremental power flow equations can therefore be written as
∆
∆
′
=
∆
∆
V
JJ
JJ
Q
P
QVQ
PVP
θ
θ
θ
(10.72)
where
kJk
X
V
X
BXV
J
X
VV
J
X
V
J
X
VV
J
QV
S
L
QV
LS
Q
S
PV
LS
P
+=+−
−
=
′
=−=−=
θ
θθθ
θθ
cos
)1(2
sinsincos
11
1
111
where J
QV
is the corresponding element of the conventional power flow Jacobian. Therefore, the
effect of the induction motor load is to modify the power flow Jacobian. The modification is
indicated by the addition of the extra term to J
QV
.
The ∆V - ∆Q relationship is now derived by setting ∆P = 0 in (10.72), which yields
[
]
LPVPQQV
VJJJJQ ∆−
′
=∆
−1
θθ
(10.73)
After carrying out the indicated operations, and noting that V
L
(1 – BX
1
) cos α = V
S
cos(θ + α), from
the phasor diagram of Figure 10.25b, and applying some algebraic manipulations, the following is
obtained:
)(2cos
2coscos
1
αθ
αθ
+
=
∆
∆
S
L
V
X
Q
V
(10.74)
In a static analysis, stable operation would be indicated by the above expression being positive. It is
positive when θ + α < 45
o
. At stability limit ∆V
L
/∆Q is infinity, which occurs when θ + α = 45
o
. As
θ + α exceeds 45
o
, ∆V
L
/∆Q becomes negative, indicating instability. So far, the results agree with
those obtained from the dynamic analysis presented earlier. Note, however, that as α exceeds 45
o
,
∆V
L
/∆Q becomes positive again and therefore the static analysis would indicate stability, although
the operation is clearly unstable. Actually, in the analysis presented, this condition is easy to detect.
In large power system studies using a computer, a more complex equivalent circuit than that used
here would probably be employed, and the computation would probably be performed in terms of
the motor slip, rather than the angle α. The possibility of converging on the wrong value of slip
during initialization is there, especially under low voltage conditions. Consider the following
example, where the parameter values are:
X
1
= X
2
= 0.3 r = 0.03 V
S
= 1.0 V
L
= 0.95 and P
L
= 1.5
Assume that the entire reactive requirement of the motor (Q
L
) is supplied by a static capacitor of
susceptance B
2
, so that, in the steady state, the load power factor is unity. Also assume that B
1
supplies the reactive requirement to maintain V
L
at the specified value for the unity power factor
load.
Solving the system equations, θ = 28.3
o
and B
1
= 0.243. From (10.69) and (10.70), the motor slip is
calculated as s = 0.093 or 0.108, corresponding to the two possible operating points. The
corresponding values of B
2
are 1.545 and 1.789.
VOLTAGE STABILITY
10-44
∆V
L
/∆Q for the two operating points calculated from (10.73) are –0.024 and 0.022, respectively.
This would suggest that the second operating point is stable, whereas the dynamic analysis will
show both operating points to be unstable. However, it would be a simple matter to detect this
possibility and guard against it.
Effect of finite speed of voltage control
In the analysis of voltage stability due to induction motor load presented so far, the assumption was
made that the system voltage would be maintained constant, either by virtue of the system being
large compared to the motor load or due to the action of the fast acting voltage control equipment. If
the motor load constitutes a substantial portion of the total load, the assumption of constant source
voltage is not valid. Also, when the actual response characteristic directly influences the stability
performance, such as in a large disturbance situation, when the duration of the disturbance is
appreciable, a detailed motor representation, including the internal flux dynamics, may be desirable,
if the study results are not to be overly pessimistic. It will be shown later that, when the source
voltage is constant, the level of modeling details does not influence small-disturbance stability
results, provided the basic motor dynamics is included in the model.
The effect of the finite speed of response of the voltage control equipment on motor stability will be
demonstrated using a simplified representation of the combined generator-excitation system as
shown in Figure 10.26. This simple approach is followed here since the objective is to identify the
voltage stability issues rather than to calculate the exact stability limit and/or detailed system
response.
Fig. 10.26 Schematic of induction motor load and supply voltage controlled with finite time lag
In the simplified representation, the system voltage is held constant by adjusting an internal voltage,
E
g
, behind an equivalent reactance, x
g
, through a first order time delay. In order to keep the algebra
simple, the shunt support at the load bus has been neglected.
The basic motor dynamics is given by equation (10.67). From Fig. 10.26, the voltage control
equation can be written as
Srefge
VVET −=
&
(10.75)
Writing δ
1
= δ + α, and θ
1
= θ + α, the power balance equations can be written as
X
E
X
EV
X
E
X
EE
X
EV
X
EE
GE
S
gg
g
S
g
g
2
1
2
1
11
2
coscos0
sinsin
−=−=
==
θδ
θδ
where
VOLTAGE STABILITY
10-45
X
g
= x
g
+ X
1
+ X
2
and X = X
1
+ X
2
.
Linearizing and eliminating the non-state variables,
∆
∆
=
∆
∆
G
E
aa
aa
G
E
g
g
2221
1211
&
&
where
1
2
22
1
1
22
21
1
2
1
2
1
12
1
1
11
2cos
sin
cos2
sin
sin
1
sin
cos
cos
δ
δ
δ
θ
δθ
θ
δ
LL
g
ee
T
E
a
T
XEG
a
T
EX
a
T
a
−=−=
−−=−=
For stability,
0and0)(
211222112211
>
−
>+− aaaaaa
which yield
02cos
cos
cos
and02cos
cos
cos
1
1
1
2
1
2
1
1
>>+
θ
θ
δ
δ
θ
δ
LeLe
TT
E
T
E
T
The necessary condition for stability is therefore θ
1
< 45
o
. This is also sufficient when T
L
>> T
e
.
Note that, as T
L
→ 0, stability condition becomes δ
1
< 45
o
. Since, in the above analysis, a fictitious
internal voltage behind a fictitious reactance was assumed, the need for a detailed analysis in
situations where T
L
is comparable to T
e
is obvious.
Stability analysis using detailed induction motor model
In this analysis the motor model will include only the rotor flux dynamics. Assuming the motor is
being supplied from an infinite source of voltage V
S
, the two-axes motor transient equations, using
V
S
as reference and neglecting stator resistance, are (see Chapter 7)
QoQDDo
esTixxeeT
′
′
+
′
−
−
′
−=
′′
o
)(
ω
&
(10.76)
DoDQQo
esTixxeeT
′
′
−
′
−
+
′
−=
′′
o
)(
ω
&
(10.77)
QQDDe
ieieT
′
+
′
= (10.78)
QDD
ixee
′
+
=
′
(10.79)
DQ
ixe
′
−=
′
(10.80)
where the terms are defined in Chapter 7.
In the above equations positive directions of current and torque correspond to motor action. Note
that, since V
S
is reference, e
D
= V
S
, and e
Q
= 0.
The equation of motion is given by (10.66). For simplicity, the load torque is assumed constant.
Stability of an operating point can be assessed by linearizing the above set of equations and
applying the stability conditions (i.e., the eigenvalues must have negative real parts). The details of
VOLTAGE STABILITY
10-46
the algebra are omitted here, and left as an exercise. It will be seen that, for the assumed constant
load torque, the stability limit occurs when T
e
is maximum, at which point
o
ω
o
Tx
x
s
′′
= (10.81)
xx
xxV
T
S
e
′
′
−
=
2
)(
2
max
(10.82)
x
xxV
ee
S
QD
2
)(
′
−
=
′
−=
′
(10.83)
(i.e.
[
]
22
QD
eeE
′
+
′
=
′
lags V
S
by 45°)
Substituting the expressions for x, x′ and T′
o
(see Chapter 7) in the above equations, it is seen that
there is close agreement with the results obtained earlier using the basic dynamic model. The small
discrepancy is due to the simplified equivalent circuit used in the basic model, where it was
assumed that
rsm
xxx or>> . As in the case of the synchronous machine without excitation control,
the electrical transients have no effect on small-disturbance stability of induction motors when the
source voltage is constant. The performance difference between the basic dynamic model and a
more detailed model will only be noticeable under large disturbance conditions, and when the
duration of the disturbance is appreciable. The effect of the electrical transients is to slow down the
motor slip excursions somewhat, thereby aiding stability and/or buying some time for possible
corrective action. The effect is more pronounced in large machines than in small ones, due to design
characteristics [33].
Voltage Stability Using Type-Two Load Model
The general form of the load model is given by equations (10.44), reproduced here for convenience
LL
LL
VkkVQQ
VkkVPP
&
&
&
&
43
21
)(
)(
++=
++=
θ
θ
The static part represents the load characteristic in the steady state. A constant static part would
represent a constant power load. However, the dynamic behavior of this model would be distinctly
different from that of the static constant power load analyzed earlier. The complete right hand side
constitutes the load model and it is a bona fide dynamic model. However, if the static part reflects a
truly static load, such as constant impedance, the stability behavior of the model would be similar to
that of the static load (i.e., there would be no voltage instability). This will be evident from the
analysis that follows.
The power balance equations can be written as
θ
sin
X
VV
P
LS
L
= (10.84)
X
BXV
X
VV
Q
L
LS
L
)1(
cos
2
−
−=
θ
(10.85)
VOLTAGE STABILITY
10-47
Note that, when both k
1
and k
3
are zero, then either k
2
or k
4
must also be zero to uniquely determine
V
L
. Similarly, when k
2
and k
4
are zero, either k
1
or k
3
must also be zero.
Assume
LL
GVPVP
2
0
)( += , and Q(V) = 0. In the steady state
)1(cos
sin
2
0
BXVV
X
VV
GVP
LS
LS
LL
−=
=+
θ
θ
For simplicity, two special cases will be considered, first with k
3
= 0, and then with k
4
= 0.
Case (i) k
3
= 0
Linearising and eliminating the non-state variables, the system in state space form is
∆
∆
=
∆
∆
L
L
Vaa
aa
V
θ
θ
2221
1211
&
&
where
Xk
V
a
Xk
VV
a
XG
k
k
Xk
V
a
k
k
Xk
VV
a
L
LS
L
L
LS
′
=−=
′
−+
′
=
+=
4
22
4
21
4
2
1
12
4
2
1
11
sin
2
cos
sin
sincos
θ
θ
θ
θθ
and
BX
X
X
−
=
′
1
For stability,
0and0)(
211222112211
>
−
>+− aaaaaa
Provided k
1
is negative and k
4
is positive, the first condition is satisfied for θ up to 90
o
(i.e., for all
normal operating conditions). (Note that in the literature, k
1
is cited as positive.) After carrying out
the indicated algebra, the second condition reduces to
0
2
2cos
P
GV
LL
−>
θ
which is the same as obtained using the load model of equation (10.43) for the constant power part
and presented earlier.
Case (ii) k
4
= 0
The elements of the state matrix are
VOLTAGE STABILITY
10-48
′
−+
′
=
+=
′
−=−=
XG
k
k
Xk
V
a
k
k
Xk
VV
a
Xk
V
a
Xk
VV
a
L
L
LS
L
LS
2
cos
sin
sincos
sin
3
1
2
22
3
2
2
21
3
12
3
11
θ
θ
θθ
θ
For stability,
0and0)(
211222112211
>
−
>+− aaaaaa
The second condition reduces to
0
2
2cos
P
GV
LL
−>
θ
which is identical to that obtained for case (i). However, there seems to be some anomalies in
satisfying the first condition. At a given load level, which may be well below the voltage stability
limit obtained for case (i), the stability strongly depends on the relative values of k
1
, k
2
, and k
3
.
However, by suitable choice of the parameter values, the small disturbance stability performance of
the two dynamic load models described by equations (10.43) and (10.44) can be made identical.
This is not surprising. In the absence of discontinuities (i.e., when θ and V
L
are differentiable), the
two models are equivalent. For example, consider the case of the constant MVA load. Replace P
L
and Q
L
in equations (10.84) and (10.85) by GV
L
2
and BV
L
2
, obtain the time derivatives of G and B
in terms of
θ
&
and
L
V
&
, and substitute into (10.43). After carrying out the algebra and rearranging
LL
LL
VkkVQQ
VkkVPP
&
&
&
&
43
21
)(
)(
++=
++=
θ
θ
where
XV
XBT
k
XV
VT
k
V
GT
k
XV
VT
k
L
LQ
L
SLQ
L
LP
L
SLP
′
′
+
==
=−=
)1(
,sin
,cos
43
21
θ
θ
This explains why identical small disturbance performances were predicted by the two models. Note
the negative sign of the expression for k
1
. Note also that the model parameters are functions of
system states. This may make the use of this model difficult in general simulation studies.
There is, however, a fundamental problem when using the Type-two model in a large disturbance
analysis. In the network model employed in stability studies, the high frequency electrical
transients are neglected and hence load bus voltages and angles are not state variables (i.e., they
can have discontinuities). A model that employs time derivatives of these variables is therefore
inconsistent with system modeling. The use of such a model would require adjusting the model
parameter values during simulations whenever there is a discontinuity (e.g., a sudden
disturbance). Serious computational difficulties can also be encountered.
VOLTAGE STABILITY
10-49
Motor dynamic performance equations derived from a rigorous machine theory will contain time
derivatives of voltages and angles. However, these are internal voltages and angles. Unlike motor
terminal voltages and angles, these are true state variables.
One of the arguments presented in the literature in favor of the Type-two model is that it will serve
as a generic model to predict large disturbance performance of any dynamic load including
induction motor loads. However, the point is being made here that in many situations, the stability
performance of induction motor loads determined from the basic dynamic model will be identical to
that determined from a more detailed model, as shown earlier. When the basic model is not
adequate, such as in some large disturbance situations, a comprehensive analysis employing detailed
modeling of all pertinent system components would be needed. The use of the Type-two model
would then certainly be questionable for the reason stated above.
Corrective Measures for Voltage Stability
Concerns for voltage instability and collapse have prompted utilities to devise effective, efficient
and economic solutions to the problem. Many utilities have either implemented or are actively
considering the implementation of load shedding, capacitor switching, and under-load tap-changer
(LTC) blocking, among other things, as emergency measures to combat voltage stability problems.
However, confusion regarding the viability and effectiveness of these measures under voltage
collapse conditions exists among utility engineers and system operators. Intuitively, one would not
expect a decrease in the bus voltage magnitude when a capacitor is switched on, or some load is
shed, at that bus, under any operating condition.(1) The source of the confusion is the conventional
steady-state analysis of voltage stability and misinterpretation of results obtained from the
conventional power flow model. For example, sensitivity calculations using the formal power flow
Jacobian will show both dV/dP and dV/dQ (or dV/dB) change sign as the operating point moves to
the lower portion of the system PV curve. It is therefore natural to wonder whether load shedding or
capacitor switching under such operating conditions would be a prudent thing to do.
The sensitivity results obtained from the conventional power flow Jacobian are not valid when
applied to operation in the lower portion of the PV curve. This is due to the static, constant MVA
load model generally used in the formulation of the power flow problem. While the static power
flow solutions obtained from such a formulation would be mathematically correct, careful
interpretation is necessary when using the sensitivity results. The difficulty clears up when one
considers the behavior of constant MVA loads in real life. A constant MVA load is not a static load.
As explained earlier, failure to consider the appropriate dynamics of the load may lead to erroneous
and confusing conclusions. When the pertinent dynamics of the loads are taken into account, it can
be shown that, while operating in the lower voltage region, system behavior following capacitor
(1) There could be situations when shedding load would cause voltage collapse. However, this is related to the well
known angle stability problem, not voltage stability. Consider, for example, the case where power is being
transmitted from a remote generating source to a large system over a long transmission line, with some portion of
the load tapped off near the source. If this load is lost, with the generation remaining constant, the transmission line
would have to carry this additional load. If the line is not strong enough, there will be large angle difference between
the generating source and the receiving system and voltages at intermediate points would be depressed. If there are
loads connected at these points, some of that might be shed by undervoltage load shedding. This will aggravate the
problem and could cause voltage collapse.
VOLTAGE STABILITY
10-50
switching and/or load shedding is the opposite of what would be indicated from a steady-state
analysis.
It is noted here that, depending on the composition and characteristic of the load, operation in the
low voltage solution region can be viable, at least temporarily. It has been shown earlier that, if a
substantial portion of the load is static, which may be the case in some power systems under cold
weather conditions, stable operation in part of this region is possible, and no corrective measures
may be necessary if the voltage level is acceptable. For loads with constant MVA (self restoring)
characteristics, stable operation in this region is possible through the use of special controls, e.g.,
static var compensators (SVCs). Even without such controls, depending on the response speed of
the load dynamics, there may be ample time available for corrective actions to avoid a voltage
collapse.
Load shedding and capacitor switching can be useful and effective corrective measures to combat
voltage stability problems in system operations. Load shedding will not normally be considered as
an option in the planning against voltage collapse. However, no matter how well a power system is
planned, occasionally situations will arise, such as disturbances which exceed planning and
operating criteria, when load shedding may be the only available emergency corrective action that
will avoid a complete system collapse. A compelling argument in favor of using load shedding as an
emergency corrective measure [25] is that, since load will be lost anyway during abnormal voltage
conditions, because of motor contactor dropout, discharge lighting extinction etc., it would be better
to have the load shed under utility control. However, it is important to explore theoretically its
viability under collapse conditions so as to identify any possible pitfall in the application of this last
ditch effort.
A third measure often advocated under voltage collapse conditions is locking of taps or even
tapping down of LTCs [27, 29], frequently utilized to maintain load voltages under normal
operating conditions. While for many load types this would be effective in halting an impending
collapse, indiscriminate LTC blocking, or tapping down may be counterproductive. A detailed and
rigorous analysis is presented supporting this view.
Load shedding
There is some confusion as to what would actually happen if load shedding were attempted while
operating in the lower portion of the system PV curve. It has been shown that, in the absence of
special controls, stable operation in this region is possible only when at least a part of the load is
static. The operating points in the lower portion of PV curves for such loads are illustrated by the
intersections of the steady-state load characteristic and the system PV curve in Figure 10.27. A is the
stable operating point, whereas point A′ is unstable. It is clear that, while operating at the stable
operating point A, shedding some load will simply move the operating point upward along the
system PV curve, as illustrated by the intersection of the load characteristic, in dashed line, and the
PV curve.
VOLTAGE STABILITY
10-51
Fig. 10.27 Illustration of the effect of load shedding when the load is a combination of constant
power and resistive load.
For the purpose of illustration, consider a case when the load has constant power characteristic and
the system is operating in the lower portion of the PV curve. The operating point is shown at A, for
a power level P
o
in Figure 10.28. Note that without special controls, the operating point cannot get
into this region for constant power load, except temporarily. In other words, the system cannot be
operated in the steady state in this region, for constant power load, unless special controls, such as
SVCs, are utilized. This discussion is therefore mainly of academic interest.
Fig. 10.28 Illustration of the effect of load shedding when the load has constant power characteristic
and is operating in the lower portion of the system PV curve.
Suppose now, an amount of load ∆P is shed. As the instantaneous load characteristic is constant
impedance, the operating point will temporarily move to B′ following the load shedding. Since this
is within the region of attraction of the stable equilibrium point B, corresponding to the new load P,
the operating point will move to and settle at B. Note that the above scenario would be true
irrespective of the magnitude of ∆P.
In the numerical simulation of load shedding using the load model (10.43), it is only necessary to
change the power set-point from P
o
to P (= P
o
– ∆P). From (10.43) and from a consideration of the
physical behavior of the system, it is clear that the system will settle at B (Figure 10.28). Note,
however, that along with the change in the power set-point, the system states (G, V
L
) could also be
reinitialized, if desired, to reflect the instantaneous effect of load shedding as illustrated in Figure
10.28.
VOLTAGE STABILITY
10-52
We can use the same argument for the case shown in Figure 10.27, if the system is initially
operating at A′. The operating point immediately following the load shedding, given by the
intersection of the instantaneous load characteristic (not shown in Figure 10.27, but similar to that
shown in Figure 10.28) and the PV curve, will fall within the region of attraction of the stable
equilibrium point corresponding to the new load. The operating point will therefore move and settle
there.
Consider the case shown in Figure 10.29, where the system is initially operating at load level P
o
,
which is greater than the maximum load capability of the post-disturbance system, P
max
. It is clear
that, for constant power load, the minimum load needed to be shed in order for the post-disturbance
system to have an equilibrium point is (P
o
–
P
max
). However, in order to keep the load shedding
requirement to a minimum, load must be shed promptly. The reason is as follows:
Immediately following the disturbance the system state moves from A to A′. If the load tries to
maintain constant power characteristic, the system state would start moving down the post-
disturbance system PV curve on the way to voltage collapse. As the system state immediately
following the load shedding must be within the region of attraction of the final stable equilibrium
point, the need for a prompt load shedding is clear. A delayed shedding would require shedding of
load in amount much greater than (P
o
–
P
max
), in order to bring the system state within the region of
attraction. The delay that can be tolerated would depend on the speed of the mechanism (the value
of T
L
in (10.43)) that restores the load to constant power.
Fig. 10.29 Illustration of the minimum load needed to be shed as an emergency control measure
when the pre-disturbance (constant MVA) load is greater than the maximum load supplying
capability of the post-disturbance system.
For loads with fast response characteristics, e.g., motor loads, the voltage will drop at a much faster
rate than would be the case, for example, if the load were static and restored to constant MVA by
LTC action. The rate of drop of voltage would, therefore, serve as an indicator for an immediate
need for load shedding.
It is also clear that the load shedding requirement is less stringent when a part of the load is static. If
a large portion of the load is static with overall load characteristic similar to that shown in Figure
10.27, as opposed to a vertical line for a constant power load, load shedding may not be necessary,
except to bring back the voltage to acceptable level, if it has gone down too low. This is because a
stable equilibrium point may already exist in the post-disturbance system, although the resulting