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Chapter 8 VOLTAGE STABILITY
The small signal and transient angle stability was discussed in Chapter 6 and 7.
Another stability issue which is important, other than angle stability, is voltage
stability. Voltage stability can be defined as the ability of the system to retain system
voltages within acceptable limits when subjected to disturbance. If the disturbance is
large then it is called as large-disturbance voltage stability and if the disturbance is
small it is called as small-signal voltage stability [1]-[2]. Voltage stability can be a
local phenomenon where only a particular bus or buses in a particular region have
voltage stability issue and this may not affect the entire system. Voltage stability can
be a global phenomenon where many of the system buses experience voltage stability
problems which can also trigger angle stability problems and hence can affect the
entire system. Some of the voltage stability problems can start as a local problem and
escalate to global stability problem.
Voltage stability of a system can be analysed either by static analysis or
dynamic analysis. In static analysis the system is assumed to be in steady state and
hence instead of taking the DAE of the system only algebraic equations are
considered. This type of analysis is suitable for small-disturbances in the system. For
large disturbances the DAE are solved and the system response over a certain period
of time is observed. It is important that for voltage stability the loads should be
properly modelled as each type of load will affect the system voltage stability in a
different way. Similarly the tap changing transformers, shunt or series reactive power
compensators, generator reactive power limits, line charging capacitance should be
included in the system representation to get an accurate picture of voltage stability.
8.1 Basic Concept of Voltage Stability
To understand the concept of voltage stability let there be a two bus system with
one end being sending end and the other being receiving end as shown in Fig. 8.1. Let
the voltage of sending end be SV and that of receiving end be RV . The transmission
8.1
line is represented by a short line mode with impedance TZ T . If a load of
impedance LZ L is connected at the receiving end.
Fig. 8.1: Single line diagram of two bus system
The magnitude of current in the transmission line is given as, taking the sending end
voltage as reference,
2 2
0
T T LZ Z 2 cosS S
L T L T L T L
V VI
Z Z Z Z
(8.1)
The receiving end voltage magnitude is then defined as
2 2T LZ Z 2 cos
S LR
T L T L
V ZV
Z Z
(8.2)
The real and reactive power consumed by the load is given as
2
2 2
S L
T L
V Z
Z Z
cos
2 cos
LL
T L T L
PZ Z
(8.3)
2
2 2
S L
T L
V Z
Z Z
sin
2 cos
LL
T L T L
QZ Z
(8.4)
SV TZ
RV
LZ
I
8.2
Let the transmission line impedance be taken as 0.25TZ pu , neglecting
resistance as usually the reactance to resistance ratio is quite high in a transmission
network. Let the sending end voltage be 1 pu . The real and reactive power of the load
can be increased by decreasing the load impedance LZ maintaining a constant power
factor say 0.8. The receiving end voltage is plotted in Fig. 8.2 with varying real power
consumed by the load, with the parameters mentioned above. This curve is called as
P-V curve.
Knee Point
Fig. 8.2: P-V curve
It can be observed from P-V curve that as the load increases from no load
condition the receiving voltage starts dropping form initial no load voltage of 1 pu .
The increase in the current I due to decrease in the load impedance LZ is more than
the drop in the receiving end voltage magnitude and hence the real power
consumed by the load will increase. The maximum real power, 1
RV
pu , will be delivered
to the load when L TZ Z . Once operating point reaches the maximum power point or
critical point or knee point of the curve any further decrease in load impedance, to
8.3
increase the load, will result in further drop in the receiving end voltage and
simultaneously the real power consumed by the load will also decrease. This is
because after knee point the drop in the voltage is more than the increase in the line
current leading to a decrease in the real power consumed by the load.
8.1.1 Effect of load type on voltage stability
Fig. 8.3: P-V curve with stable and unstable operating points
Consider the P-V curve shown in Fig. 8.3. For a given real power consumed by
the load there are two possible receiving end voltages. As can be seen from operating
points A and B from Fig. 8.3, for a real power of 0.6 pu there are two receiving end
voltages of 0.85 pu and 0.20 pu , approximately. If the load is a constant MVA load
then the operating point A is stable where as the operating point B is unstable. In fact
for a constant MVA load any operating point below the knee point is unstable. This
phenomenon can be understood from the Fig. 8.3. Suppose due to some disturbance
the operating point moves from point A to C. At the operating point C the real power
is more than the real power at operating point A. Since, the load is constant MVA
8.4
load the current will decrease and therefore the operating point move back from C to
A. The operating point may not stop immediately at operating point A but may
oscillate around it and will settle down with enough damping. Hence, the operating
point A is a stable operating point. Now consider the operating point B. Again let the
operating point move to point D due to disturbance. At operating point D the real
power is less than that at operating point B. Since the load is a constant MVA load at
operating point D the current increases, as compared to operating point B, which leads
to a further voltage drop and further increase in the line current. This phenomenon
continues till the voltage at the receiving end becomes zero and hence called as
voltage collapse. Therefore, the operating point B is unstable operating point. It can
be observed from P-V curve that the curve above knee point has negative
that is the change in real power and voltage are in opposite direction, for a increase in
power voltage decreases and for a decrease in power voltage increases. The curve
below the knee point has positive and at the knee point is zero.
/RdV dP
/R LdV dP
L
L/RdV dP
Fig. 8.4: P-V curve for different power factor loads
8.5
Figure 8.4 show the P-V curves for loads with different power factors. It can be
observed from the figure that as the load power factor moves from lagging to leading
the knee point is shifted towards higher real power and higher voltage. This shows
that the voltage stability improves as the power factor moves from lagging to leading
loads.
Fig. 8.5: Q-V curve for different load real powers
Just as the load real power was plotted with respect to the receiving end voltage
for varying load impedances, the load reactive power variation with respect to the
receiving end voltage, with load real power being constant, can also be plotted.
Figure 8.5 shows the variation of receiving end voltage with variation in load reactive
power for three different real power loads. Again the locus of knee point is marked.
On the curve left of the locus of the knee point the reactive power and the voltage
move in opposite directions that is is negative and this region is unstable.
The curve on the right of the locus of the knee point the reactive power and the
voltage move in the same direction and hence is positive and this region is
stable. In the unstable region even if reactive power compensation is done through
shunt capacitance at the receiving end bus, the voltage of the receiving end bus will
/ RdQ dV
/ RdQ dV
8.6
not improve. One way of finding the voltage stability is to check the sensitivity of
each bus voltage with respect to the reactive power injected at that bus and if the
sensitivity is positive then it means the operating point is stable and is on the right
side of the locus of the knee points in Fig. 8.4 and if it is negative then it is on the left
side. This stable region and unstable region are only applicable to constant MVA
loads. In case of constant impedance and constant current loads the loads interact with
the system and settle at a new operating point as there is no requirement of constant
MVA.
8.2 Static Analysis
In static analysis [3] of voltage stability the snapshots of the entire system at
different instants is considered and at each instant the system is assumed to be in
steady state that the rates of changes of the dynamic variables are zero. Hence, instead
of considering all the differential algebraic equations only algebraic power balance
equations are considering assuming that the system is in steady state. At each instant
whether the system is voltage stable or not and also how far the system is from
instability can be assessed. There are two methods for assessing whether system is
voltage stable or not. They are sensitivity analysis [4] and modal analysis [5].
8.2.1 Sensitivity analysis
The power balance equations are taken assuming that the system is in steady
state. For, , assuming bus number 1 is a slack bus, the real power balance
equation is given as
2,......,i n
1
( ) cos 0n
Gi Di i i j ij i j ijj
P P V VV Y
(8.5)
The reactive power balance equations are taken at the load buses, for
1, 2,......,g gi n n n
8.7
(8.6) 1
( ) sin 0n
Di i i j ij i j ijj
Q V VV Y
The linearized form of equations (8.5) and (8.6) is given as
2 2
1 1g g
n nP PV
n nQ QV
n n
P
P J J
Q VJ J
Q V
or
PJ
Q V
(8.7)
The matrix is the Jacobian matrix used in Newton-Raphson load flow analysis. For
constant real power is zero hence from equation (8.7)
J
P
1
P PVJ J V
(8.8)
Substituting (8.8) in (8.7) an simplifying leads to
(8.9) 1
QV P PV QQ J J J J V
Let, 1
R QV P PV QJ J J J J , then
1RV J Q (8.10)
The diagonal elements of the matrix 1RJ represents the sensitivity of the voltage
with respect to the reactive power injected at that bus. This is nothing but the slope of
8.8
Q-V curve given in Fig. 8.4. Hence, if the V-Q sensitivity of an ith bus is positive then
the system is voltage stable and if it is negative then the system is voltage unstable. A
small positive value of sensitivity indicates that the system is more voltage stable and
if the sensitivity is small with negative value then the system is highly voltage
unstable.
8.2.2 Modal Analysis
Voltage stability can also be estimated by the eigen values and eigen vectors of
the matrix . The matrix can be expressed in terms of right and left eigen vector
matrices as
1RJ
,
RJ
RJ (8.11)
1RJ 1 (8.12)
Where, is a diagonal matrix with eigen values along its diagonal. Substituting
(8.12) in (8.10) lead to, noting that
I
1 1RV J Q Q (8.13)
1V Q
Q
(8.14)
Let, v V and , then or q 1v q
1i
i
v iq
(8.15)
Where, is the ith bus modal voltage and is the modal reactive power. If the
eigen value is positive then the system is voltage stable. The larger the eigen value
with positive sign the better the stability and if the eigen value has very small positive
value then it is very close to
iv iq
0i , that is the critical point or knee point, and hence
8.9
very close to instability. If the eigen value is negative then the system is voltage
unstable.
8.3 Dynamic Analysis
The voltage stability of a system can also be assessed by the transient simulation
over a period of time [3]. The differential algebraic equations, given in (8.16), of the
system should be solved through numerical methods and the transient simulation
should be carried out for few minutes to completely observer the interaction of
generators, static loads, dynamic loads, tap changing transformer etc.
( , , , )
( , )
0 ( , , )
d q
d q
d q
x f x I V u
I h x V
g x I V
-
-
-
=
=
=
(8.16)
The transient simulation should be done for different fault scenarios and the
system behaviour should be observed. If the system voltages are restored to
acceptable values after fault clearing then the system is voltage stable if not the
system is voltage unstable. The simulations needs to be done for few minutes because
the time constant involved can be very small like generator exciter or very large like
induction motors or tap changing transformers. For voltage stability assessment load
modelling is important hence both static and dynamic loads should be modelled. The
reactive power compensating devices like series / shunt capacitor and static VAr
compensator should also be included in the system model.
8.3.1 Small-disturbance analysis
Small-disturbance (signal) analysis can also be used for voltage stability
analysis. The DAE given in equation (8.16) can be represented as
( , ( , ), , )
0 ( , ( , ), , )
x f x h x V V u
g x h x V V l
=
= (8.17)
8.10
Where, in (8.17) is a constant representing the loading of the system, as given in
(8.18)
0
0
1
1
D D
D D
P P
Q Q
(8.18)
0 , 0D DP Q are the real and reactive power loads at all the load buses for base case.
The idea is to increase the loading factor in steps and at each step find the eigen
values of the linearized system given in (8.17). The linearized form of the system
given in (8.17) is given as
0
Vf f f fx x u
x V u
Vg g gx
x V
é ùDé ù¶ ¶ ¶ ¶ê úê úD = D + + Dê úê ú D¶ ¶ ¶ ¶ë û ë ûé ùDé ù¶ ¶ ¶ ê úê ú= D + ê úê ú D¶ ¶ ¶ë û ë û
(8.19)
Here, 1 1 1
T T
nV V V V V n n hence the linearized form shown in
(8.19). On further simplification of (8.19) the following expression can be derived
1
sysJ
sys
f f f g g g fx x u
x V V x u
fJ x u
u
q q
-é ùé ù é ù é ù é ù é ù¶ ¶ ¶ ¶ ¶ ¶ ¶ê úê ú ê ú ê ú ê ú ê úD = - D + Dê úê ú ê ú ê ú ê ú ê ú¶ ¶ ¶ ¶ ¶ ¶ ¶ë û ë û ë û ë û ë ûê úë ûé ù¶ê ú= D + Dê ú¶ë û
(8.20)
The matrix sysJ is called as the system Jacobian and is different from the load
flow Jacobian. The eigen value of the system Jacobian sysJ for different loading
conditions, obtained by varying the loading factor , are computed. If a one of the
eigen value becomes zero or a pair of complex conjugate eigen values are on the
imaginary axis for a particular loading factor then the load corresponding to the
8.11
loading factor is a critical load at which the system becomes voltage unstable. If a
complex conjugate pair of eigen values are on imaginary axis then that operating
point is called as Hopf-bifurcation point. If one of the eigen value is on the origin then
that operating point is called as saddle node point. The saddle node bifurcation point
represents the knee point. Hopf-bifurcation point occurs at a load which is less than
the load at which the saddle node bifurcation point occurs. Hence, due to the
dynamics involved the system can become voltage unstable even before the operating
point reaches the knee point.
8.12
8.13
References
1. IEEE, Special publication 90TH0358-2-PWR, Voltage stability of power
systems: concepts, analytical tools, and industry experience, 1990.
2. CIGE task force 38-02-10, Modelling of voltage collapse including dynamic
phenomena, 1993.
3. G. K. Morison, B. Gao, and P. Kundur, “Voltage stability evaluation using
static and dynamic approaches,” IEEE PES summer meeting, July 2-16, 1992,
Seattle, Washington.
4. N. Flatabo, R. Ognedal and T. Carlsen, “Voltage stability conditions in a
power transmission system calculated by sensitivity methods,” IEEE Trans.,
Vol. PWRS-5, No. 4, pp. 1286-1293, November.
5. 1990G. K. Morison, B. Gao, and P. Kundur, “Voltage stability evaluation
using modal analysis,” IEEE Trans., Vol. PWRS-7, No. 4, pp. 1529-1542,
November 1992.
