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IEEE Transactions on Power Systems, Vol. 11,No.
3,
August 1996
etection of Power System Small Disturbance Voltage Instability
Liancheng W ang Adly A. Girgis
Student
Member,
IEEE
Fellow,
IEEE
Clemson University Electric Power Research Association
Department of Electrical and Computer Engineering
Clemson University
Clemson,SC 29634-0915
Abstract Small disturbance (SD) voltage stability (or
instability) deals with a system's ability to maintain
satisfactory voltages following a small disturbance. For an
operating condition, a system's
SD
voltage stability depends on
the proximity of the condition
to
the critical point (or voltage
collapse point).
A
Q angle and Q directional derivatives are
proposed for SD voltage instability detection and weak bus
identification, respectively. The
Q
angle index can handle
different kinds of loads, e.g., constant P and Q , constant
impedance, and constant current, or a combination of them,
and
is
effective in dealing with generator Var limits. Moreover,
the computation speed of the Q angle is fast, which makes it
suitable for on-line application. Simulation results using two
power systems are provided.
I. INTRODUCTION
Voltage stability studies have attracted great attention in
recent years [l , 21. According to the study method and problem
formulation, voltage stability is classified into two categories
[2]:
small disturbance (SD) voltage stability and large disturbance
(LD) voltage stability. SD voltage stability is concerned with a
system's ability to control voltages following small disturbances
such as gradual load variations. On-line detection of SD voltage
instability is to determine how far a steady-state operating
condition, which is always SD stable, is from the voltage collapse
point or critical point. Due to its nature, SD voltage stability can
be analyzed by using the power flow Jacobian. Large disturbance
(LD) voltage stability
is
concerned with the system's ability for
voltages to recover at acceptable steady state values following
large disturbances such as system faults, loss of generations, or
circuit contingencies. For LD voltage stability analysis, dynamic
system models are required.
Several algorithms have been developed to detect SD voltage
instabi lity (collapse). The minimum singular value of the system
Jacobian matrix has been proposed as a voltage collapse index [3].
However, calculating the minimum singular value is time-
consuming due to the high dimension of the Jacobian matrix.
To
improve the feasibility of this method, a fast algorithm to compute
the minimum singular value was proposed [4]. Modal analysis was
also reported for voltage instability assessment [5, 61. This method
calculates a set of the smallest eigenvalues
of
the reduced Jacobian
matrix and the associated participation factors. The eigenvalues are
95
SM
525-6 PWRS paper recommended and approved
by the IEEE Power System Engineering Committee of the
IEEE Power Engineering Society for presentation at
the 1995 IEEE/PES Summer Meeting July
23-27,
1995
Portland OR. Manuscript submitted June 28 1994;
made available for printing June
5,
1995.
used
as
voltage instability indicators, and the participation factors
for weak area identifications. Voltage-power (real or reactive)
sensitivity is another index for voltage collapse detection [7-lo].
During normal operating conditions, the voltage-power sensitivity
is a finite value, and it will increase with the system loading. When
voltage collapse
occnrs,
the voltage-power sensitivity will be
infinite. Another developed method includes using the distance in
the load parameter space between a given operating condition and
the critical point, which is the voltage collapse point, as an index
[ l l] . The critical point in the state space was assumed lying in the
arithmetic center of a pair of closely located load flow solutions.
Algorithms for calculating this distance were further investigated
[12, 131. While this index can provide a load power margin for an
operating condi tion, which is particularly valuable to system
operators, the computationalburden is a main concern.
The study of LD voltage stability requires accurate dynamic
load modeling and further developments in the stability of
nonlinear systems [14-171.
The paper begins with a presentation of the voltage collapse
mechanism and the introduction of the Q and P angle concept.
Indices are then proposed for SD voltage instability detection and
weak bus identification. This is followed by simulation results
using
two
IEEE test systems. Finally, conclusions are provided as
to the effect iveness and computational speed of the proposed
indices.
II
Q ANGLES, ANGLES,NDVOLTAGEOLLAPSE
A new concept of Q and
P
angles is introduced based on the
geometric interpretation of the load flow solutions. The
characteristics of these angles with respect to heavy loading is
presented. The voltage collapse mechanism is clarified by this
concept.
Consider a power system, and let n be the total number of
buses minus the slack bus. Allow the
n+
1)th bus to be the slack
bus, and assume
rn +1
to be the number of generator buses. The
real and reactive power balance equations for the power system
are expressed by
n+1
C y y ~ J c o s ( 6 , - 6 ,
e I J ) - p z ( y ) = e
( i = I ,
.-.,
n ) (la)
xV;V,x,sin(6,-SJ -OzJ)-qs(y)=Q
( i= l , ...,
n - m ) (lb)
J=1
n+l
J=1
where
(?,a)
s the constant part of the net power entering bus i,
p I y ) , q z y ) )s the voltage dependent part of the net
power entering bus i,
Y L 6 is the ith bus complex voltage, and
LeL, is the (i, j)th element of the network admittance
Equations (la ) and (lb ) may be written compactly in vector
matrix.
form as
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F ( x )= h (2)
where
F = [ f ,
f 2 ...
f2,-,3 represents the real and
x . 6, a . . 5
. .r
epresents the system state
I = [
..
lj
...
Qj o ] r
represents the system
Equation (2) includes 2n
-
n nonlinear algebraic equations
with 2n-rn unknowns. From a geometric point of view, every
equation in (2) represents a space surface, and any (2n-rn)-l
simultaneous equations of (2) represents a space curve. The
intersection points between the space surface and the space curve
are the load flow solutions.
Let the injection power in (2n-rn)--l equations of (2) be
constant; then, the space curve expressed by the (2n
-
m)
-
simultaneous equations is determined in the space. If the injection
power in the remaining equation is increased, the load flow
solutions will vary, and their trajectories will be along the space
curve expressed by the (2n-
n
- 1 simultaneous equations. For
some injection power value, the space surface and the space curve
will be tangent to each other, and this point corresponds to the
voltage collapse point
[
181. Figure 1 illustrates the relative
movement between the space surface and the space curve as the
injection power in the space surface equation is increased from
Q
to Q.
reactive power equation vector,
variable vector, and
parameter vector.
Gradient
Vector
Tangent
Vector
X :Load flow solution candidates
Figure
1:
Relative
movement
between the space surfaceand
the space curve,
Q
< Q
It is noted that as the injection powler in the space surface
equation increases, the angle
a
between the gradient vector of the
space surface and the tangent vector of the space curve at the load
flow solutions will also increase. At the point where the space
surface and the space curve are tangent to each other, the angle is
90 degrees.
The above discussions are for a special change mode of the
parameter vector
h
. That is, only one element of the parameter
vector is varying and all the other elements are fixed. As a matter
of fact, the observations about the angle between the gradient
vector of the space surface and the tangent vector of the space
curve are the same for any arbitrary change mode of
h .
Since (2) includes 2n-rn equations, there exist
2 n - m
angles corresponding to 2n - n different surface-curve
combinations. Let a vector
a
denote these angles, then
where ai is the angle between the gradient vector of the surface
represented by the ith equation of
(2)
and the tangent vector of the
curve defined by the remaining
2n
-
n -
simultaneous
equations.
To facilitate discussions, we make the following definitions.
The angles corresponding to the gradient vector of the space
surfaces expressed by the active power balance equations will be
defined as P angles, the others will be defined as Q angles.
III.
VOLTAGEOLLAPSE
NDEX
In this section, an index for
SD
voltage collapse detection is
proposed, and the computational requirements of the index
are
discussed. Also, the observation that all the Q and P angles are
equal to 90 degrees at the critical point is verified. Finally, a
simple example of a two bus system
i s
provided to explain the
proposed index.
A . Voltage Collapse
Index
Consider the load flow solutions of (2) as the intersections
between the surface 5 represented by the ith equation
of
(2) and
the curve Cvi implicit ly defined by all the equations in
(2)
except
the ith. Therefore,
S
(x)= hi 4)
f Z n - m ( x ) = h 2 n - m
0 0
The gradient vector of
Sf,
at an arbitrary operating point
( x
,h
)
is defined by
r
l t
which is the transpose of the ith row of the power flow Jacobian
matrix of 2). The equation of the tangent line of Cvi at the
operating point ( x
,h
) is
0
0 0
0
(7)
2n-m -
3 n - m
2 -
-
l - x l
- x
- x
-
... =
31 2
2n-m
where si
( l= l , 2,
...,
2n-m),
and t i is the
parameter in the parametric representation of Cvi .
Thus to obtain the tangent line, it suffices to know the ratios
s ~ : s ~ : . . . : s ~ ~ - ~ .rom 5), the
sl
(Z=l, 2, ..., 2n-rn) terms are
related by
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where
,x, =-afj 0'=1, 2
. - a ,
i-1, i+ l ,
a . . , 2n-rn;
k = l , 2 , ..., 2n-rn)
ax
is the j,k)th element of the Jacobian matrix of (2).
obtained by solving the following linear algebraic equation
If we let s ~ ~ - ~-1, the terms s1 through s ~ ~ - ~ - ~an be
AS= b
where
Matrix
A is
composed of columns
1
through 2n-m-1 of a new
matrix which is obtained by deleting one row from the Jacobian
matrix, while vector
b
is composed of the (2n-m)th column of the
new matrix. The deleted row corresponds to the gradient vector of
the space surface.
Let Tagi enote the tangent vector of Cvi, hen
On the other hand, the ratios of
s ~ : s ~ : . . . : s ~ ~ - ~
an also
be
found by applying Cramer's rule [191 to
(8)
as
2
s1 z:. . :
~ ~ - ~
(-1)l
det(Mil
. )):(-1)
det(Miz(
) ) :
. :(- n-m
det(Mip,-,,(J)) (11)
where
J
is the Jacobian matrix of
Z),
M i j ( J ) is the minor of
(J)G, J ) u represents the entry in the (i, j) position of J . Equation
(11)provides an alternative way to find the tangent vector
Tagi, o
1
Tugi = [(-1) det(Mil(J)) (-1)' det(Miz(J))
... (-1)2"-mdet(Mi(2n_m,(. ))f (12)
Using
Vfi
and
Tagi,
the angle
ai
an be computed
where * represents the inner product of
two
vectors, and 1111
represents the Euclidean norm of a vector.
At the voltage collapse point, the system Jacobian is singular
[3,4, 20-221, i.e.
det(J)
= 0
(14)
Because the determinant of a square matrix can be computed
as the sum of the products of the elements of any row (column) of
the matrix and their cofactors,
(14)
is rewritten as
2n-m
J l
det J)=
(J),J(-l)'+Jdet(M,J(J))=O (i=l ,
2,
..., 2n-rn) 15)
or
212-rn
(J),(-l)'det(M,(J.))=O ( i = l , 2, . + a , 2n-rn)
where (-l) +' et(Mz,(J)) is the cofactor of ( J ) z j .According to (6)
and (12), (16) can
be
written in vector form as
(16)
J = l
Vf,.Tug,=O ( i = l ,
2,
..., 2n-rn)
(17)
From (13) and (17), it can be seen tha$ at voltage collapse
point, every element of the vector
a
is
90 .
Their closeness to
90'
may indicate the proximity of a given operating condition to
the voltage collapse point. Consequently, any element of the
vector a may be chosen as an indicator for the voltage instability
detection. In the following section, it will be shown through
simulations that the Q angle is a better index than the P angle.
In obtaining
Tugi ,
(9)
will be used since it
is
computationally more efficient [23], and (12) is introduced only as
a theoretical device. Therefore, the main computations involved in
calculating the proposed index are the LU-decomposition of the
matrix A and the corresponding forward and backward
substitutions. As a result, the computational speed for the
proposed index is fast.
Based on the system configurations and transmission line
impedances, J can be found by using the voltage phasors, which
may be obtained either by a load flow program or by measurement
units. Since running a load flow program takes a longer time and
the index is intended for on-line application, the latter will
be
used.
The procedure for on-line SD voltage instability detection
is
summarized as follows:
Form the system Jacobian matrix using transmission line
impedances and voltage measurements.
Form matrix A and vector b, and solve
(9).
Calculate a
Q
angle (one element of the vector a
)
using (13).
Judge the system status with respect to v$tage collapse
according to the closeness of the
Q
angle
to
90
.
B. Example
A simple
two
bus system is used as an example to further
interpret the Q and P angle concept.
1 L O
V2L6
Figure
2: One
line diagram of a two bus test system
Figure 2 is a two bus system from [21]. In this system, a
load with real and reactive power, P and
Q ,
is supplied by an
infinite system via a single lossless transmission line. The series
admittance and shunt admittance of the transmission line are -jB
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and &, espectively. Assume that the source voltage
s
is l.OL0,
and the load bus voltage is V 2 L 6 . The system may represent a
Thevenin equivalent of a system as seen by a load.
The real and reactive power balance equations are expressed
-BYV2 sin6 =
P
(184
~ v ~ B c o ~ s - ( B - B ~ ) v ~Q
(18b)
Each of these equations represents a plane curve in the state
space, and the load flow solutions are the intersec tion points
between the two plane curves. To obtain these curves, (18a) and
(18b) are rewritten as
by
(194
-1 p
&=-sin
2 B
6
= -cos -1 Q + ( B - B l ) V ;
(
1%)
W 2 B
The constant P curve and the constant Q curve are obtained
using the (V2,6) values calculated from (19a) and (19b),
respectively. Fig. 3 shows the real and reactive power curves in the
state space, where the active load
P
is fixed (1.0 pu) and the
reactive power
Q
takes three different values. Since
P
is fixed, the
load flow solutions vary along the constant P curve as Q is
changed. When the reactive power is increased, the load bus
voltage magnitude decreases and its angle (absolute value)
increases. It is also found from Fig. 3 that when the reactive load
becomes heavy, the angle
ixl
between the g,radient vector V Q and
the tangent vector of the constant
P
curve art the feasible load flow
solution point (higher voltage magnitude:) will increase. When
Q =
1.08 pu, the two curves are tangent to each other and the angle
a1
quals 90 degrees, which implies that voltage collapse occurs.
Fig. 4hows the curves for the case where the reactive load
Q is fured (1.0 pu) and the active power P takes three different
values. The load flow solutions are along the constant
Q
curve.
Similar observations as seen in Fig. 3 can also be obtained fmm
Fig.
4.
According to the previous definition,
a
s the Q angle, and
a2 s the
P
angle.
To
find al.nd a2 he system Jacobian matrix
J is required. From (18a) and (18b),
Applying ( 6 ) , the gradient vector of the constant Q curve is
obtained
Solving (9) and substituting the result into
(110)
yield
Substituting (21) and (22) into (13), the Q angle a1 s computed as
Similarly, the P angle
a2
s calculated as
VP TagQ
a2 c0s-l
~ ~ v ~ [ ~ T a ~ Q [
,
.
c
1307
24)
-80'
'
I
0.2 0.4
0.6 0.8
1
v @U
Figure 3: Real and reactive pow.
urves
in the state
space
for the 2 bus syst.
(degrees)
om
\
-y= 0.75 =Y
-80
0.2 0.4
0.6
0.8
v2 @U
Figure 4 Real and reactive pow. curves in the
state space
for he 2
bus
syst.
IV. WEAK
us
IDENI~FICATIONS
Weak buses are referred to as those which can withstand a
lesser load demand increase than the others, without causing
voltage instability. This section will show that the directional
derivatives of the injection power in the space surface equation, in
the direction of the tangent of the space curve, may be used as
indices for the weak bus identifications. Moreover, the
computational requirements will also be discussed in finding the
directional derivatives.
Recall that the load flow solutions are the intersections
between a space surface and a space curve, and that the solution
trajectory is along the space curve as the injection power in the
surface equation is varied. If the injection power in the surface
equation is increased, the angle between the gradient vector of the
space surface and the tangent of the space curve will also increase.
However, the directional derivative of the injection power in the
space surface equation, in the direction of the tangent of the space
curve, will decrease. The larger the injection power, the smaller
the directional derivative will be. When voltage collapse occurs,
the directional derivative will vanish. That is, there is no increase
(or decrease) of the injection power along the solution trajectory,
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and the injection power in the surface equation has approached its
maximum. Therefore, the directional derivatives of the injection
power in the space surface equation, in the direction of the tangent
of the space curve at an operating condition, may well reflect the
margin of the injection power in the surface equation. For the
power system described by (21, there are 2n-m directional
derivatives at an operating condition, each of which stems from a
different surface-curve combination. Weak buses may be
identified using the directional derivatives.
Let Dd represent the directional derivative vector, then
D d = [ D d , D ... Dd2n-mr (25)
where
Vfi .Tagi
IlTagill
Ddi = I ~ V ' ~ C O S ~ ~- ( i = l , 2 ,
...,
2 n - m ) ( 2 6 )
represents the directional derivative of the injection power in SA
along Tagi. The directional derivatives corresponding to the P
angles will be defined as P directional derivatives, and the olles
corresponding to the Q angles will be defined as Q directional
derivatives.
As for the Q and
P
angles, the major calculations for
Ddi
are
finding the tangent vectors Tagi i = 1,2, . . ' ,2n- m) by solving (9).
However, finding all these vectors only requires one
LU-
decomposition of the Jacobian
J .
Let
L U = J
where
L
is lower triangular and U is upper triangular.
To
get Tag,,
A
is constructed as
A
=
Mi,2n-m J)
where Mi,2n-m(J)
is
the minor of (J)i,2n-m. Based on
L
and U ,
matrix
A
can be factorized as
L,U,
= A
where
Li
= Ml,2n-m(L) is lower triangular,
U, M2n-m,2n-m
( U )
is upper triangular.
v.
SIMULATION
RESULTS
This section tests the proposed indices for SD voltage
instability detection and weak bus identifications. For simplicity, a
load flow program is used to get the voltage phasors instead of
simulations of measurement units. Two power systems are
simulated the modified Ward-Hale
6
bus system
[ l l ]
and the 10
machine 39 bus
IEEE
test system [ 2 4 ] .
A. SD Voltage Instability Detection
To evaluate the performance of the proposed index under
various operating conditions, two scenarios are considered: with
and without generator Var limits. Comparing the results for these
two scenarios shows the effect of generator Var limits on the
voltage collapse. Whenever a generator reaches its Var limits, a
constantQ limit
[8 ]
is adopted for i t and the generator bus changes
from PV
to
PQ. The occurrence of genera tor's Var limits implies
that a system's Var reserve is decreased and that the system will
become more vulnerable to voltage collapse. This vulnerability is
manifested by the "jumps" of the Q angle and the shrinking of the
voltage stability region [ 1 5 ] in SD and
LD
voltage stability,
respectively
In the simulations, the load factor
k,
which represents the
ratio of the load parameter h to the base case parameter h,, is
increased in steps until it approaches its critical value
characterized by the divergence of the load flow program. While P
and Q loads are increased, the specified outputs of all generators
will
increase by the same factor as the loads.
The modified Ward-Hale 6 bus system is shown in Figure 5,
where bus 1 and bus 2 are generator buses, and buses 2 through 6
are load buses. The load parameters indicated in Figure 5 represent
the base case. In the load flow program, generator 1 is selected as
the slack bus, generator 2 as the PV bus before it reaches its
reactive power limits, and all the load buses as PQ buses. When
generator
2
reaches its reactive power limits, bus
2
changes to a
PQ bus.
1
0.5
- j0.05
-
0 3
-
j0.18
P = 0.5
Figure
5:
One line diagram of the Modified Ward-Hale 6 Bus System
oo 1 1 2 1 4
1 6
Load factor
k
Figure 6:
Q
and P angles vs load factor kat selected buses
for
the
6
bus system
0
0 8
0.3 1 1 1
1 2
1 3
1 4 1 5
1 6
Load factor k
Figure
7:
Voltage magnitudes at selected buses for the
6
bus
system
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The angle vector a s computed at each operating point with
the increase of the load factor
k.
Figure 6 shows a
Q
angle and a P
angle vs the load factor
k
curves at selected buses with no
generator Var limits being applied. It is seen that both
Q
angles
and P angles increase with heavy loading. At the voltage collapse
point, which is signified by the divergence of the load f low
program,
all
the elements of the vector
a
qual 90. For the 6 bus
system, it was found that k = 1.65636 is the critical value, beyond
which voltage collapse will occur. Fig. 7 shows the voltage
magnitudes at bus
4
and bus
5
vs the load factor k. The voltage
magnitudes decrease as k increases. The nose points of the
voltage-load curves correspond
to
the volta,ge collapse point.
Figure 8 shows the Q and P angles vs load factor k at
selected buses while applying generator Var limits. In
Fig.
8,
generator 2 reaches its Var limit and bus 2 changes from the PV
bus into a
P Q
bus when
k =
1.38. At this instant, the Var output of
generator 2 is 449 MVar. The generator
PV-PQ
bus switching
causes the Q and P angles' sudden rises in Fig. 8, which, in turn,
manifests the "jumps" in the distance
of
the system operating
condition to the voltage collapse point. ' n e variations brought
about by the generator Var limits in the Q and P angle are 12.169'
starting from 71.085' to 83.254' and 2.262' starting from
79.97O'to 82.232', respectively. By co mpk ng Fig. 8 with Fig. 6,
it is found that the generator Var limits have a big influence upon
the system's loadabili ty: the critical value of the load factor
k
reduces from 1.65636 to 1.46317.
85
8 8
0 9
1 1 1 1 2 13 1 4 15
1.6
Load factor k
Figure
8: Q
and
P
angles vs load factor k at .selectedbuses for the
6 bus system considering generatorVar limits
The one line diagram of the 10machine test system is shown
in Figure 9. In the load flow program, generator 10 was chosen as
the slack bus; all other generators were chosen either as
PV
buses
or as P Q buses depending on their operating status; the load buses
were chosen as
P Q
buses.
Fig. 10shows a Q angle and a P angle at selected buses
without applying generator Var limits. For this system, even at t?
base case, most of the P angles are very large (larger than 85
),
and they change very little when increas,ing the load factor
k.
However, in the vicinity of the critical point, they all increase to
90" abruptly. Accordingly, the
P
angles can not reflect the
tendency toward voltage collapse with the increase of load, and are
not good indices for voltage instability. Contrary to the P angles,
the Q angles are much better in measuring the proximity of an
operating condition to the voltage collapse: point. I t is found that
all the Q angles possess similar properties; hence, any one of them
can be used as a voltage instability indicator. The critical va lue of
the load factor k for the 10 machine system was 2.22905 with
k
=
1
representing the base case.
9
7
17 24
1
L
6
15
14
39
9
Figure
9:
One line diagram of the 10 machine
39
bus system
Load factor
k
Figure 1 0 Q and
P
angles vs load factor
k at
selected buses
for the
39
bus system
With generator Var limits being applied, the
Q
and P angles
are shown in Figure 11.The generator Var limits reduce the
critical value of the load factor
k
to 1.49084. In Fig. 11, the fis t
discontinuity in the
Q
angle is caused by generator
3
hitting its Var
limits when k = 1.20, and the second discontinuity is caused by
generator 2 hitting its Var limits when k = 1.39. As mentioned
earlier, these discontinuities signify the "jumps" in the distance of
the system operating condition to the voltage collapse.
P angle
atbus
29
(degrees) ________________________________________
a
:
Q angle at bus
4
75u
0
1
1.1
1.2 1.3 1 4
5
Load
factor k
Figure 11:Q and P angles vs load factor
k
at selected buses for the
39
bus system considering generator reactive power limits
For the
10
machine system, the CPU time required to
calculate the Q angle is 0.05 s, considering that the voltage
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magnitudes and phase angles are already available via
measurement units, on a Sun SPARCstation 2. On the same
computer, the computation of the V-Q sensitivities at all the buses
required 0.06 s.
B. Weak Bus Identifications
To identify weak buses for an operating condition, the Q and
P
directional derivatives
are
calculated. Then, from this operating
point, increasing the reactive (or active) load demand one load bus
at a time until the voltage collapse point, the reactive (or active)
load increment is the
Q
margin (or
P
margin) of th e stressed bus.
Table
1
shows the results for load factor
k =
1.0 in the 6 bus
system. The system buses are ordered according to the magnitudes
of
their Q directional derivatives. It is seen that, the order of the
Q
directional derivatives coincides with that of the Q margins, and
the order of the
P
directional derivatives coincides with that of the
P margins. Thus, for this system, the Q weak buses and P weak
buses can
be
identified according to the
Q
directional derivatives
and the
P
directional derivatives, respectively. Here, the
Q
weak
buses ( P weak bu ses ) represent the buses which can withstand the
least amount of reactive load (active load) increase without
causing voltage collapse.
A
smaller
Q
directional derivative
implies a smaller Q margin, and similarly, a smaller
P
directional
derivative implies a smaller
P
margin.
Table 1: Real and reactivepow. marg. for the 6
bus syst., k
=
1.0
Table 2 provides the results for the load factor k = 2.20 in
the
10
machine 39 bus system. As for the 6 bus system, the system
buses are ordered according to their Q directional derivatives. It is
noted that the
Q
weak buses can still be determined by comparing
the Q directional derivatives at all the load buses although the
order of the Q directional derivatives does not exactly reflect the
order of the Q margins. However, there is no linear relationship
between the P directional derivatives and the
P
margins. Thus, the
P weak buses can not be identified using the P derivatives for this
system.
Table2:Real and reactivepow. marg. for the 39 bus syst.,k =2.20
VI.CONCLUSIONS
This paper has proposed a Q angle index for small
disturbance (SD) voltage instability detection. The Q angle is
defined as the angle between two vectors. One vector is the
gradient of
a
space surface defined by the reactive power balance
equation at a bus, and the other is the tangent of a space curve
defined by the simultaneous load flow equations except the one for
the space surface. In a similar manner, the
P
angle has also
been
defined. It has been proved that all the Q and P angles
are
equal to
90' at the voltage collapse point. Simulations have shown that the
Q angle can reflect the proximity of an operating condition to
voltage collapse point and
is
efficient in handling generator
reactive power limits. Specifically, the Q angle increases gradually
with the increment of system loads, and it rises suddenly, which
indicates a "jump" in the distance
to
voltage collapse, whenever a
generator reaches its Var limits. The paper has also shown that the
Q
weak buses can
be
identified by the
Q
directional derivatives.
The computational speed is fast in finding the
Q
angle. The
main arithmetic operations are the LU-decomposition of a matrix,
which is obtained by deleting one row and one column from the
load flow Jacobian, and the corresponding forward and backward
substitutions.
W . ACKNOWLEDGMENT
The authors acknowledge the support of the NSF and
Clemson University Electric Power Research Association for this
research.
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Liancheng
Wang (S'92) obtained his B.S. and M.S. degrees in
Electrical Engineering from Shandong Institute of Technology, Jinan,
China, in 1983 and 1986, respectively. From 1986 to 1991, he was a
lecturer in the Electrical Engineering Department, Shandong Instituteof
Technology. He joined Clemson University in January 1992. He is
currently working toward his Ph.D. degree. His research interests are
power system protection, control and stability.
Adly A. Girgis
(S'80-SM81-F92)received the B.S. (with distinction
first class honors) and M.S. egrees from Assuit University, Egypt, and
the Ph.D. degree from Iowa State University, all in Electrical
Engineering. He taught at Assuit University, Egypt, Iowa State
University, and North Carolina State University.
Dr. Girgis joined Clemson University in 1985.He is currently the
Duke Power Distinguished Professor of Power Engineering in the
Electrical and Computer Engineering Department and the director of
Clemson University Electric Power Research Association. Dr. Girgis
has published more than 90 technical papers and holds four U.S.
patents. He is the recipient of the 1989 McQueen Quattlebaum-Faculty
outstanding achievement award, the
1990
Edison Electric Institute
Power Engineering Education Award and the
1991
Iowa State
Professional Achievement Citation in Engineering Award.
His
present
research interests are real-time computer applications in power system
control, instrumentation and protection, signal processing, and Kalman
filtering applications. Dr. Girgis is a member of Phi Kappa Phi, Sigma
Xi, and is a registered Professional Engineer.
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Discussion
Claudio A. Caiiizares (University of Waterloo): The authors
present an interes ting paper t ha t proposes a new index for on-
line detection of proximity to static voltage collapse points,
also known as maximumloading points, singular points,
or
saddle-
node bifurcation points.
Gradient and tangent vectors to the
bifurcation manifold
or
system states profiles are used to define
an angular voltage stability index, obtaining results compara-
ble, both in terms of resulting shapes and computa tional costs,
to those obtained with test functions in
[A]
and reduce deter-
minants in [B]. It is interesting to observe that the proposed
angular index has also a quadratic or quartic shape that allows
to predict t he proximity of the system to a maximum loading
point; furthermore, generator limits are shown to produce in-
stantaneous changes on this angular index, similar to those re-
ported on
[B]
for test functions and reduce determinants. The
latter makes all these indices somewhat unreliable to predict
proximity to collapse, although that can be partially overcome
by continuos calculations of these indices, as these are relatively
inexpensive to compute.
One issue, however, that is not discussed in the paper, and
tha t may be related to the size of the sys tem used for the compu-
tation of the proposed index, is the different shape that this an-
gular index may have for different system buses. This can be em-
pirically justified by observing that in large systems, many buses
present a relatively flat profile as th e system load changes, re-
sulting in
a
highly nonlinear variation of the angle between the
tangent and gradient vectors, similar to what is depicted on
Figs. 10 and 11 for the P angle at bus 29 This same problem
was also observed and reported in [B] for the te st function and
reduced determinant indices. In this case, the index behaves
similarly to eigenvalue or singular value indices, which are cer-
tainly not adequate to predict proximity to collapse. Hence, it
becomes important to detect
a
critical bus or area where the
variables of interest change in a way so that an adequate angular
index can be obtained. The problem then lies on detecting these
critical buses at any system loading (a possible solution based
on tangent vector and clustering techniques is proposed in [C]
for the te st function and reduced determinant indices).
The authors comments regarding these issues would be ap-
preciated.
[A] H.
D
Chiang and R. Jean-Jum eau, Toward a practical per-
formance index for predicting voltage collapse in electric
power systems, IEEE
Trans.
Power Systems, vol. 10, no.
2,
May 1995, pp. 584-592.
[B] C. A. Caiiizares, A.
Z.
de Souza, and V.
H.
Quintana, Com-
parison of performance indices for detection of proximity to
voltage collapse, IEEE/PES
95
SM 583-5 PWRS, Portland,
OR, July 1995.
[C]
A.
C.
Z.
de Souza, C. A. Caiiizares, and
V. H.
Quintana,
Improving continuation power flows using system partition-
ing and reduction techniques, submitted for review for the
IEEE/PES Winter Meeting, Baltimore, MA, July 1995.
Manuscript received August 15, 1995
Liancheng Wang
and
Adly A. Girgis:
We would like to
thank Dr. Canizares for his interest on the paper.
We agree with Dr. Canizares that the jumps, which
were caused by generators hitting their Var limits, on the
Q
angle and test function indices may prohibit an accurate
prediction of the proximity of an operating condition to
voltage collapse. However, we consider these instantaneous
changes as inevitable because it is these changes that reflect
the discontinuity in the system characteristics. Normally,
voltage collapse indices variations are affected by system
structure changes, such as generator Var limits.
To address other issues raised by Dr. Canizares, we
offer the following explanations on the relationship between
the Q angles method and test functions method.
Compared with the test functions method
[D],
the Q
angles method was derived from an engineering point of
view. However, the Q angles method and the test functions
method are closely related. Both
of them
were based on
two
vectors: a
Q
(or P ) angle
was
defined as the angle between
two vectors, while
a
test function was defined as the dot
product of these two vectors.
In add ition, the sam e premise
that the system Jacobian matrix is singular at the bifurcation
point was used in their development.
A
test function was defined
as
trk
=
e f J h
D1)
where
h = Jliel
J represents the system Jacobian matrix: e, is the Eth un it
vector, i.e., a vector with
all
elements being zero except the
Zth which is
1; Jlk is
obtained from J by replacing its Eth row
with a row unit vector e,.
Because the term
e,
J results in
the Eth row vector of J , (Dl) can be seen as the multiplication
of the Ith row vector
of
J
by vector
h.
Figure D1 shows
vectors e: J and
h
and the corresponding Q (or
P)
ngle.
T
T
Figure D l : Q (or P)
ngle
and est function
T
In our approach, the vector el J was interpreted
as
the
gradient vector of the space surface represented by the Ith
power balance equation in (1); vector h was interpreted
as
the
tangent vector of the space curve represented by all the
power balance equations in (1) except the Eth.
The angle
between these two vectors is the Q
(or
P ) angle, and the dot
product of the vectors is the test function. At bifurcation or
voltage collapse point, the angle is equal to 90, nd the test
function is equal to 0.
Once
1
in vector
e r J
is specified, the
Q
or P ) angle is
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Another advantage of the Q angles method is that it
provides a more direct means than the test functions method
in predicting voltage instability.
The occurrence of voltage
collapse is caused by a system's reaching its maximum
loading point, which is chara cterized by the Q (or P ) angle's
equality to 90'. A test function's being equal to
0
is simply a
manifestation of the corresponding Q (or P ) angle's being
90. In the computation of Q (or P ) angles, the effects of
irrelevant variables--vector engths--were eliminated.
Reference
[D]
R. Seydel, From Equilibrium
to
Chaos: Practical
Bifurcation and Stability Analysis, New York
Elsevier, 1988.
Manuscript received October 24, 1995.
unique, but vector h and test functions may have different
forms, which depend on the selection of
k
n (D2). For a load
flow Jacobian of ( 2 n - m ) x ( 2 n x m ) , there exist ( 2 n - r ~ t ) ~
test functions. Every test function has a different variation
pattern when the system load
is
increased. The test functions
which have the desired characteristics need to be identified
from the ( 2 n - n ~ ) ~est functions. In the
Q
angles method,
however, there only exist n-m Q angle s itnd n P angles,
so
the
number of poten tial indices are greatly reduced. Furth er, it
has been found that the Q angles' variations are smoother
with respect to system loading and that all the
n-m
Q angles
have a similar property. Any Q angle may be chosen as a
voltage instability index. The refore , the identification of the
best index is more straightforward in the Q angles method
than in the test functions method.
