IEEE Transactions on
Power
Systems. Vol. 9.
No.
2.
May
1994
757
SVC PLACEMENT USING CRITICAL MODES
OF
VOLTAGE INSTABILITY
Yakout Mansour Wilsun Xu Fernando Alvarado Chhewang Rinzin
B.C.
Hydro University
of
Wisconsin Department
of
Power
Burnaby, B.C., Canada Madison, Wisconsin, USA Bhutan
Abstract
The location of SVC (Static VAR Compensators) and
other types of shunt compensation devices for voltage
support is an important practical question. This paper
considers a tool based on the determination of critical
modes. Critical modes are computed by studying the
system modes in the vicinity of the Point of Collapse.
System participation factors for the critical mode are
used to determine the most suitable sites for system
reinforcement. Because the method does not rely on
base case linearizations, the method is able to properly
consider all system limits and nonlinear effects.
The
paper tests the proposed method by performing an
as-
sessment of the impact of the addition of Static VAR
compensators to a
1380
bus model of the BC Hydro
system.
1
Introduction
The location of SVC (Static VAR Compensators) and
other types of shunt compensation devices for voltage
support is an important practical question. This paper
uses a tool based on critical modes.
The most critical
system mode can be computed by performing a Point
of
Collapse power flow solution, which requires the com-
putation of the system Jacobian matrix at the point of
maximum loadability
[5].
The method amounts to solv-
ing a power flow problem subject to the constraint that
the Jacobian of the power flow itself be singular
[4, 141.
This paper considers an approximation to this ap-
proach that works well in practice: the system is loaded
to near its point of collapse, and modal analysis is per-
formed on this stressed system. One advantage of the
pragmatic approach is that standard software can be
used in the analysis, namely ordinary power flows and
eigen-structure analysis programs
[7,
lo].
The arguably
more accurate notion of using the precisely determined
This
paper
was
presented at the 1993 IEEE
Power
Industry
Computer Application Conference
held
in
Scottsdale,
Arizona,
May
4
-
7,
1993.
Point of Collapse
[4]
requires the use of more special-
ized software
[5],
and may form the basis for further
refinements of the method.
The manner in which system nonlinearities are con-
sidered can have significant impact on the identification
of critical modes. Base-case linearizations are generally
not acceptable. The full nonlinear effect of components
such
as
generators and SVCs must be taken into ac-
count in the identification of critical modes. The be-
havior of system components (and their impact on the
system eigenvalues) can change drastically as limits are
encountered.
Voltage itself is a poor indicator of proximity to sys-
tem collapse conditions. Consider, for example, the
voltage versus reactive power relationship for an ide-
alized generator (Figure
1).
The figure illustrates the
difficulties associated with the linearization of a com-
ponent that exhibits discontinuous behavior.
A
small
change in demand can move the system from point “a”
to point “b.” If this generator is linearized at point “a,”
the slope of the
V
vs. Q is zero, indicating that the ma-
chine is of a PV type and fully capable of regulating its
voltage. If the generator is linearized at “b,” the slope
is
00,
indicating a machine of a PQ type and unable
to regulate its voltage. This drastic slope change can
lead to very different results from modal analysis. Since
components are likely to reach limits in the vicinity of
the maximum loadability point, representation of these
limits is important. In fact, the reaching of limits helps
identify those system components most responsible for
a critical mode. The Point of Collapse method fully
accounts for all system nonlinearities and limits. If one
does not wish to solve a problem exactly at its point of
collapse, a reasonable alternative for the consideration
of limits is to solve the problem near the collapse point
and
to use the notion of a “conversion margin” which
can be either additive
or
multiplicative:
&actual- Qmargin
>
Qlimit
(1)
&actual’ Qmargin
>
Qlimit
(2)
For cases where one does not solve the system
precisely at the collapse point, a conversion margin
Qmargin enables a user to have some control on which
slope to use in the modal analysis.
0885-8950/94/$04.00
Q
1993 IEEE
758
ages
AV
and
modal
injections
Aq:
[
1;;
e
] [
":I
=
[
":I
(5)
0
0
'..
X,
Avn Aqn
In this equation the modal voltages and modal injec-
tions are obtained from
[AV]
=
[T][Avl
and
[Aq]
=
[T][AQ]
where
[TI
is the modal matrix of eigenvectors.
The vectors
[AV]
and
[Aq]
have the same units
as
the
actual vectors. They are linear combinations of the ac-
tual physical voltages and injections.
As
XI
.--)
0,
modal
voltage
1
becomes very sensitive to modal injection
1
while other modes remain unaffected.
The main conclusion from this is that
voltage collapse
is
actually the collapse
of
a modal voltage.
That is, the
power system cannot support a particular combination
of
reactive power loads. If modal voltages are plotted
as
a function of system stress, only one modal voltage ex-
periences collapse. However, since several (and to some
degree, all) physical voltages participate to a greater
or
lesser degree on the mode in question, this causes
multiple physical voltages to collapse.
The participation of each load in the critical mode de-
termines the importance of the load in the collapse
[SI.
The degree of participation is determined from an in-
spection of the entries of the left eigenvector of the crit-
ical mode. Larger eigenvector entries signify locations
most suitable for voltage support
or
load shedding.
tvoitage
PV
to
PQ conversion margin
el
slope=o
nose point
Q
1
100-0.1
MVar
tb
100
MVar
~Qllrnit)
..
Fig.
1:
Steady-state voltage versus reactive power re-
lationship
for
a generator
2
Modal analysis
Modal
or
eigenvalue analysis of the system Jacobian
matrix near the point of voltage collapse can be used to
identify buses vulnerable to voltage collapse and loca-
tions where injections of power benefit the system most.
When modal analysis is used, there is no need to drive
the system precisely to its "nose point" to ensure that a
maximum level of stress
is
reached. The eigenvectors of
the critical eigenvalue give information about the loads
responsibIe for the voltage collapse.
For
a
single load power system, the linearized rela-
tionship between load voltage and reactive power load
can be described by the following equation:
XAV
=
AQ
(3)
where
X
is the
V-Q
sensitivity.
In general, this sen-
sitivity is obtained from the network Jacobian matrix.
When
X
is close to zero,
a
small load change results in
a
large voltage change. The operating point is considered
to be voltage unstable from the load flow point of view.
Consider a system whose Jacobian matrix is diagonal:
If
XI
is close to zero, a small load change at bus
1
would result in
a
large change in the voltage at the bus.
However, all other buses would be unaffected. Clearly,
voltage collapse would happen at bus
1.
Unfortunately,
the Jacobian matrices
of
actual power systems are not
diagonal matrices. However, they can be diagonalized
using modal analysis. With this technique, one obtains
the following matrix relationship between
modal
volt-
3
Modal Transformation of the Reduced
S
ys tern Jacobian
Eigenanalysis
[8,
11,
121
uses a linearization
of
the non-linear steady state power system equation
f(z,X)
=
0:
HN
[:;I=[
M
L]
[,a:]
(6)
This equation is obtained
as
the steady-state of a dy-
namic model that considers also the impact of the dy-
namics of all generators and generator controls.
For
the
time frames of interest in voltage collapse, this model
can effectively be a constant terminal voltage voltage
provided one is assured that faster modes are stable.
As
in power flow solutions, voltage-dependent load mod-
els can be included in the modal analysis. Steady-
state load-voltage characteristics should be used, since
the modal analysis deals with the steady-state condi-
tions of the system. This is valid for modal analysis of
the critical mode near the voltage collapse bifurcation.
The steady-state load-voltage characteristics are differ-
ent from transient load-voltage characteristics adopted
for transient stability simulations.
For
further infor-
mation on load models appropriate to voltage collapse
studies refer to
[16,
91.
Although both
P
and
Q
changes affect system condi-
tions, it is possible to study the approximate effects of
reactive power injections on voltage stability by setting
AP
=
0
(P
constant) and deriving the
Q-V
sensitivi-
ties at different loads. Using arguments similar to those
of
[14],
an approximate equation
6
can be rewritten
as:
AQ
=
[L
-
MH-lN]AV
=
JRAV
(7)
The matrix
JR
in equation
7
is the reduced Jacobian
and it relates voltage and reactive power at each bus.
The smallest eigenvalue, which at the saddle node
bifurcation becomes zero is calculated by performing a
modal transformation of the reduced Jacobian matrix
of the system
as:
JR
=
<AV
(8)
where
<
and
q
are. the left and right eigenmatrices
re-
spectively and
A
represents the system eigenvalues. The
components of the left eigenvector can be interpreted
as
indicating a direction normal to the operational bound-
ary of the system
[6,
131.
Right eigenvector components
indicate the degree to which given variables are involved
in agiven mode. The use of both left and right eigenvec-
tor information leads to the notion of participation fac-
tors. This paper uses bus participation factors, which
are defined for bus
le
and critical mode
i
as
<ki
vik.
Identifying the critical mode eigenvalue,
Xi,
is not
trivial. The magnitude of an eigenvalue is only a rel-
ative measure of proximity to collapse
[SI.
As the sys-
tem approaches collapse, eigenvalues that initially have
small real component may not be critical, and other
eigenvalues may become critical. Proximity of an eigen-
value to the imaginary axis is not sufficient. A method-
ology to identify the critical mode is presented next.
4
Stressing the
system
Based on earlier discussions, it is evident that modal
analysis must be done on
a
stressed system. Stressing
a system all the way to its point of collapse was not
considered a practical approach using an available pro-
duction grade power flow type program. Instead, this
paper performs the determination of the critical mode
in the vicinity
of
the maximum loadability point. In
this application, the real and reactive power loads at
the buses in the area of interest are scaled up accord-
ing to predetermined weighting factors. These prede-
termined weighting factors can correspond to a known
expected direction of demand increase. Although dif-
ferent directions can identify different critical modes,
in practice the approach has proven to be robust in the
identification of critical modes irrespective of the exact
direction chosen, particularly when the procedure for
the elimination of local modes described later in this
section is used. Finally, a procedure has been recently
developed for the computation of the worse possible de-
mand increase direction
[2].
759
Fig.
2:
VP
nose curve
for
base case and contingency.
The load increase is done in many steps until the
nose of the
P-V
curve is reached. While the tech-
nique is simple in concept, its implementation involves
many details. In this regard, reference to continuation
power flows is useful
[l,
51.
At the maximum loadabil-
ity point, the system must satisfy both the power flow
equations
and
a condition for the singularity of the Ja-
cobian matrix. Mathematically, this corresponds to the
simultaneous solution of the power flow equations aug-
mented with a matrix equation of the form
JRY
=
0
with
Iyl
#
0.
In practice, exact solution for this point
of collapse is not necessary. The expedient procedure
of iterating toward the maximum loadability point with
ever decreasing step sizes yields results within
2.0
MW
of
the correct solution within
a
few iterations. Like
the continuation power flow, this method is capable of
taking into account all system limits with the added
advantage of using ordinary computational tools.
The main problem with the solution process concerns
the solvability of a stressed case after a disturbance.
One practical way to resolve the problem and compute
accurate MVA margins is depicted in figure
2.
The
base-case is solved up to a certain load level. At this
point, a discrete disturbance such as the outage of a
heavily loaded transmission line is applied. In order to
render this system solvable, load models are converted
momentarily to constant impedance models, enabling
the system to be readily solvable along the constant
impedance curve. After
a
post-disturbance solution is
obtained, the load models are modified once more and
the same stressed-system power flow solver
as
before is
applied. This approach resembles the response of loads
in the period following a disturbance
[16].
The reason
that
a
constant power load is
a
reasonable model for
power loads is due in part to the action of tap chang-
ers that isolate demands from transmission-side voltage
variations. Thus, an impedance load behaves initially
as a constant impedance, and only after tap changer
action does it behave
as
a constant power load.
760
UlOI
US4
--------I-.
-
-
-
-
-
-
-.
-
CUSTER
IEPN
Fig.
3:
Relevant portion
of
1380
bus
BC
Hydro System model
,
SEL
6507
Allocation of power among generators is another is-
sue. Generator outputs are scheduled in proportion to
their remaining output capability. However, other gen-
eration scheduling methods can be explored
[3].
Other features of the stressed power flow method
used during these tests include: power can be scaled
by area, zones and individual buses, both their active
and reactive components.
Once the system has been stressed, the objective is
to identify the critical modes. This is done by per-
forming modal analysis
on
the last solved case.
A
posi-
tive eigenvalue indicates that the maximum loadability
point is beyond the bifurcation point for the system.
The critical mode is the mode associated either with
this eigenvalue
or
with one of the small real eigenvalues.
In general it
is
useful to track three
or
four eigenvalues.
As
a result of linearization of the control equations, it
may be the case that the critical mode may not be the
one that initially appears to be critical based on the
location of the eigenvalue at base case conditions.
Eigenvalues with few participating buses correspond
to local modes. These normally reveal local voltage
problems such
as
problems resulting from
a
long line
feeding
a
remote load. Alleviating these modes by lo-
cal means (adding reactive power) and proceeding with
the stressing of the system eventually will reveal sys-
tem modes. While in an operations context both local
and system modes are quite important, in a planning
context local modes are somewhat easier to remedy by
local means.
Thus
it is of interest, in addition to find-
ing all important local modes, to find the most impor-
tant system modes. Modes with small eigenvalue real
parts and large numbers
of
participating buses reveal
system-wide
or
area-wide problems. This procedure for
the identification of important system modes by explicit
removal of local modes has been found to be more valu-
able than the notion of identifying modes by tracking
multiple modes with small real eigenvalue components.
The combined use of system-wide stress and modal
analysis can provide reliable information for the study
of
voltage stability problems. The next section illus-
trates an application to the design of system reinforce-
ments.
5
Application
of
Modal Analysis to Planning
The technique described in the previous sections has
been applied to planning of a full-scale version of the
BC
Hydro
(BCH)
System. Figure
3
illustrates the relevant
portion of this system. The main concerns in this study
were the delivery of power to the main load centers:
the Lower Mainland (in the vicinity of bus ING) and
Vancouver Island (in the vicinity of buses GLD and
SAT), whichever was more likely to collapse, and to
determine the most effective location for the placement
of voltage support. This section presents some results
from these studies.
The first study checks if participation factors can pre-
dict the voltage collapse area. For this purpose, bus
ING (key bus in Lower Mainland) and bus
SAT
(key
761
1
0.8
b.
B
2
0'
0.6
3
'D
'f!
0.4
!3
0.2
0
BCH
230
kV
buses
and areas
Fig.
4:
Critical mode eigenvalue participation factors
for
demand increase at ING and
SAT.
bus in Vancouver Island) are stressed with increased
reactive power load respectively. The near-nose points
computed with the stressed power flow are then an-
alyzed using modal analysis. Figure
4
illustrates the
participation factors for all
230
KV buses.
A
large par-
ticipation factor value indicates a high involvement of
the bus in the voltage collapse. It can be seen that the
Lower Mainland buses have large participation factors
when ING is stressed. If SAT is stressed, the Vancouver
Island buses have high participation factors. The use-
fulness of the modal information
is
thus demonstrated.
If the entire BCH system is stressed with a uniform
MVA
load increase
as
specified in the planning criteria,
identifying the critical areas with intuition becomes im-
possible. Modal analysis has to be used. Figure
5
shows
the participation factors near the nose point when the
entire
BCH
system is stressed according to the plan-
ning criteria. The Lower Mainland is found to be the
critical area. This area can be further divided into sev-
eral sub-areas. Among them, the Vancouver-south is
the most critical one and is probably the best location
for effective voltage support.
Six
230
kV buses are then selected
as
candidate SVC
locations based on figure
5.
The SVC size, f150 MVar,
is determined based on the need to continue to meet the
BCH voltage stability planning criteria for some time
after the SVC is put into operation [15]. The effective-
ness of a SVC is evaluated according to the resulting
improvement
on
system margin, both present and fu-
ture. Different SVC locations yield different margin
improvements,
as
illustrated in Figure 6. Observations
from this figure include:
0
The best SVC locations are buses
SPG
or
K12,
which are in the center of the critical area iden-
tified by the participation factors.
0
There is a good correlation between margin im-
provement and participation factors.
3
2.5
0
;
B'
.-
2
1.5
i
-
U.
.-
a1
0.5
ECH
230
kV
buses
and critical areas under
area
S-stress
Fig.
5:
Identification
of
critical areas in
BCH
System.
Modal analysis
was
also performed for cases with
SVC installed at buses WLT and SAT. Participation
factors (both before and after the addition of SVC com-
pensation) for these two cases are shown in figure
7.
The figure indicates that with a SVC at WLT the par-
ticipation factors, and therefore the risks of voltage col-
lapse around the WLT area, are reduced.
A
similar
conclusion applies to the SAT case.
The SVC used in this study represents a general shunt
compensation device. It is intended to demonstrate
the effectiveness of the proposed method. The speed
with which the shunt device should operate is a sepa-
rate issue. This problem involves dynamic aspects of
the voltage collapse phenomenon, and cannot be prop-
erly analyzed using modal analysis methods based on
power flow static mqdels. Detailed studies of the rate
of voltage collapse and the response time requirements
for reactive power support can be found in [16].
A
concern not properly addressed by the steady-state
methodology in this paper is the situation where the
system is safely operable both before and after the out-
age of a component, but where the sudden transition
itself can cause an instability. This mode of failure is
the dominant mode of failure for angular instability. It
has been assumed that this is not expected to be the
case for voltage stability problems.
6
Conclusions
Modal analysis in the vicinity of the point of collapse
of a system can be a very valuable tool that permits
the direct comparison of alternatives. This paper has
shown that, if some care is taken, power flow programs
and eigenvalue analysis packages can be used to study
the system in the vicinity of these points. The paper
describes a comprehensive methodology for the identi-
fication of system weaknesses in a systematic manner
which
is
both theoretically sound and validated with
practical design experience.