A METHOD THAT COMBINES INTERNAL STATE ES
IEEE Transaction on Power Apparatus and Systems, Vol. PAS-104, No. 1, January 1985
91
A METHOD THAT COMBINES INTERNAL STATE ESTIMATION
AND EXTERNAL NETWORK MODELING
A. Monticelli
Felix F. Wu
Department of Electrical Engineering and Computer Sciences
and the Electronics Research Laboratory
University of California, Berkeley, California 94720
Abstract
A method that combines internal state estimation and external
network modeling is developed. The external system is represented by
an unreduced load flow model. One state estimation covering both the
internal system and the external system is used. The external system
operating data on power injections and bus voltages are entered as
pseudo-measurements. At each iteration the set of active pseudomeasurements are selected to conform with the specified variables in a
load flow program. Because such a set of non-redundant measurements
is used, the internal state estimation is not affected by the external system pseudo-measurements. External generation MVAR and controlled
bus voltage limits are enforced. A technique is developed to make a
pseudo-measurement dormant. Using the dormant-measurement technique, it is possible to maintain the same external system state estimation formulation while the PV-PQ switching takes place from iteration
to iteration. The method can easily be implemented by modifying an
existing state estimation program. It has been implemented in a fast
model-decoupled estimation program. Because of the dormant measurement technique, the constant gain matrix evaluated at flat voltage is
used in every iteration. The method has been tested on the IEEE 14bus, 30-bus, and Brazilian 66-bus systems. Excellent results are
obtained. The number of iterations for the method to converge is usually the same as the regular state estimation runs. Thus the method
presented here achieves simultaneously the internal state estimation
and the external network modeling without adversely affecting the
internal state estimation.
I. INTRODUCTION
For control center applications, the interconnected power system
is divided into two parts: the internal system, which is basically one's
own power system, and the external systems, which is the rest of the
interconnected network. Real time data of the internal system are gathered through a data acquisition system: from time to time, system
substations are scanned and measurements reflecting a snapshot of the
present operating condition of the system are obtained. The state estimator utilizes these data to set up a load flow model of the internal system. For the state estimator to estimate the present state of the system, no information about the rest of the interconnected system is
required.
Most control centers today are equipped with the state estimation
software. The weighted least square (WLS) estimation method has
been a popular choice [1]. Recently it has been demonstrated that a
fast model-decoupled version of the WLS state estimation is reliable
and efficient [2,3]. The method utilizes a constant gain matrix
evaluated at flat voltage (V=1,0=0) for each iteration.
On leave from Departamento de Engenharia Eletrica, UNICAMP, Campinas, S.
P., Brazil.
In addition to monitoring the current operating condition of the
system, the control center also performs the task of security analysis.
Security analysis is concerned with the study of line and generation
outages (contingency) and is carried out using an on-line load flow program. In an interconnected system, a contingency in the internal system can perturb the operating conditions of the neighboring systems
(external system); the change in the external system operating point, in
turn, can have a significant effect on the system where the contingency
occurred - we call this effect "external system reaction." So, for such
applications, unlike the state estimation, an external system model is
needed in order to represent, as accurately as possible, the external network reaction when internal system perturbations have to be simulated.
The usual approach to setting up the on-line load flow model of
the interconnected system for security analysis involves three main
steps:
(i) state estimation for the current state of the internal system;
(ii) modeling of external system; and
(iii) attachment of the external system model to the internal sys-
tem for contingency analysis.
The external system has been represented by an unreduced load
flow model or a reduced network equivalent. A comprehensive review
of the external network modeling is available [5-7]. In [7], a unified
procedure for external system modeling is suggested, where the external system is further partitioned into inner external and outer external
systems. The inner external system (or buffer zone) is represented by
an unreduced load flow model and the outer external system is modeled
by a reduced equivalent, which may be a Ward-type or an REI-type
equivalent. An advantage of having an unreduced load flow model for
at least a part of the external system is that certain load flow information (e.g., net interchange, key flows) available on the external system
can easily be incorporated. This paper is concerned solely with the
unreduced load flow model for the external system. The results can be
applied to the whole external system or just to the inner external system.
When the external system is represented by an unreduced load
flow model [8], the operating-point data (loads, generations, and
specified voltages) of the external system is needed. Some external
system data may be available in real time, for example, due to partial
data exchange between control centers. But in general these data are
not all available to the internal system in real time, and so extrapolated
values or off-line data have to be augmented. In order to obtain a set
of operating-point data from the extrapolated values that are compatible
to the current state of the internal system, a load flow program for the
external system and the boundary buses has been used. The input data
for the external system load flow consist of
-
extrapolated real and reactive power injections at the
external load (PQ) buses,
- extrapolated real power injections and specified
voltages at the external generation (PV) buses,
- voltages magnitudes and phase angles from the state
A paper recommended and approved
WM
by the IEEE Power System Engineering Committee of
the IEEE Power Engineering Society for presentation at the IEEE/PES 1984 Winter Meeting, Dallas,
Texas, January 29
February 3, 1984. Manuscript
submitted September 1, 1983; made available for
printing December 13, 1983.
84
034-5
-
estimator for the boundary buses (treated as slack
buses).
A fundamental requirement in the attachment of the external system model to 'the internal system is that it must not affect the values of
the internal state given by the state estimation for the present situation
(called the base case). In other words, the solution of the load flow
model consisting of the internal system and the external model for the
0018-9510/85/0001-0091$01.00©1985 IEEE
92
base case should agree with the solution of the internal system obtained
from the state estimation. This is accomplished by adjusting the external system model via the so-called boundary matching procedure
described below:
(a) From the state estimation, calculate the flows from the boundary buses into the internal system.
(b) Use the external system model together with the voltage magnitudes and angles of the boundary buses from the state estimation to
compute the flows from the external system into the boundary buses.
(c) For each boundary bus, add boundary power injections so that
the flows are balanced.
In summary, the load-flow based external network modeling
requires in real time the coordinated execution of a state estimator and
a load flow. For real-time analysis, simplicity in software and database
is of paramount importance. Any improvement along that line is certainly desirable. Since a state estimator is absolutely essential in control
center applications and it is now generally available, it would be nice to
eliminate the necessity to run a separate6load flow for external network
modeling.
In this paper we develop a method that combines the internal
state estimation and the external network modeling by modifying an
existing state estimation program. The external network model is
obtained at the same time the internal system state is estimated. The
resulting unreduced load flow model of the external system has the properties that the variables lie within their operating limits.
An overview of the proposed method is given in Sec. II. The
theoretical foundation of the method is presented in Sec. III. The solution algorithm of the method implemented in a fast model-decoupled
state estimation program is outlined in Sec. IV. The method has been
tested- on a simple example system, as well as the IEEE 14-bus, 30-bus
system and a Brazilian 66-bus system. The results are shown in Sec. V.
II. OVERVIEW OF THE METHOD
We have developed a method which combines the internal state
estimation and the external network modeling. One modified state estimation covering both the internal system and the external system is
used. The external system operating data on power injections and bus
voltages are entered as "pseudo-measurements" in the state estimation
program. The internal state estimation and the external network
modeling are achieved simultaneously.
1. The Requirements
Prior to the development of the method a wish list was drawn
which specifies the requirements or desired attributes of the new
method. They are listed below.
(1) The part on the external network modeling must not affect
the result of the internal . state estimation. In other words, the
estimated internal state variables should agree with the estimated internal state variables that would be obtained from the state esitmation
based only on the same set of internal system measurements.
(2) In the external system, Q-limits at generator buses (PV
buses) and V-limits at load buses (PQ buses) with voltage control
should be enforced.
(3) The state estimation formulation for the external network
modeling should remain the same throughout the iterations for computational efficiency. In particular, the gain matrix with flat-voltage
approximation should remain constant so that the fast model-decoupled
state estimation algorithm can be employed.
-2. The Approach
We now describe the ideas developed in the new method to fulfill
the requirements stated above.
(1) The set of pseudo-measurements for the exiernal system is
selected to conform with the PV- and PQ-designations in a load flow
program. Initially for load buses, extrapolated values of P and Q injec-
tions are used as pseudo-measurements, and for generator buses, extrapolated P generations (say, based on AGC data) and specified voltages
V are used. Such a set of measurements actually forms a minimal set
of measurements which makes the external system observable [14].
It is shown in Sec. 111-2 that if a set of non-redundant pseudomeasurements is used for the external system, the estimated state of
the internal system depends only on the internal system measurements,
independent of the values of the external system pseudomeasurements.
(2) The way the enforcement of Q-limits at PV buses and Vlimits at PQ buses is handled in a standard load flow program is the socalled bus switching scheme, i.e., changing bus types whenever a limit
violation occurs. We propose to do exact the same thing for the external system in our method, for example, when a Q-limit at a PV bus is
violated the pseudo-weasurement.V = VSP is replaced by a pseudomeasurement Q = Q m, where Q im is the enforced limit.
(3) To avoid the change of state estimation formulation whenever
a change of pseudo-masurement occurs, we propose to retain all
relevant variables as pseudo-measurements in the formulation and
activate at each iteration only those pseudo-measurements to make up a
non-redundant set. For example, for an external generator bus we put
P, Q and V all as pseudo-measurements in the formulation. Initially P
and V are active, and the pseudo-measurement Q is kept as a dormant
measurement as long as it remains within the specified limits. Whenever a limit violation,.occurs, the pseudo-measurement Q is then
activated with Q = Q lm and the pseudo-measurement V is made a
dormant measurement.
A measurement is dormant if its presence has no effect on the
result of the estimation. Thus the value of a dormant measurement
should be equal to the calculated value of that variable obtained from
the state estimation using a set of measurements from which that particular measurement is absent. In Sec. III-3, we show that using sensitivity analysis it is possible to calculate an approximate value of a dormant measurement without actually removing that measurement from
the measurement set in the state estimation. At each iteration of the
state estimation, for a pseudo-measurement which is supposed to be
dormant, the value of that pseudo-measurement is replaced by one calculated using sensitivity analysis that will make it dormant.
3. The Implementation
The proposed approach sketched above can easily be implemented
into any existing state estimation program. It seems to be generally
agreed that the fast model-decoupled state estimation is more reliable
and efficient [2,31. We have modified a fast model-decoupled state estimation program to do the combined internal state estimation and external network modeling. The required modification is not extensive.
The modified algorithm is presented in Sec. IV.
The computational efficiency of the fast model-decoupled state
estimation stems from the fact that a constant gain matrix (evaluated at
the flat voltage) is used throughout the iterations. This is still the case
for the modified algorithm.
The limit enforcement (as well as the limit back up) of external
system variables is handled in each iteration, just like a standard load
flow. Tests have shown that the inclusion of limit enforcement does
not seem to affect the number of iterations for the state estimation to
converge.
4. Comments on Related Work
The idea of using a set of non-redundant pseudo-measurements
of the external system so that the internal state estimation will not be
affected was first pointed out in [5, p. 2302], and was recently used in
[9]. The problem that the state estimation may result in external generation MVAR and controlled voltages outside the limits has not been
resolved.
Geisler and Bose [10] have proposed to use state estimation for
the external network solution. In their approach, two separate state
estimation runs are needed, one for the internal system and one for the
external system. For the state estimation to handle the load flow task
93
of enforcing external generation MVAR and voltage limits, they
gested to manipulate the weighting factors of the WLS estimator,
when the calculated value of a pseudo-measurement is outside
range, its corresponding weighting factor is increased in order to for(
back to its limits. Once the weighting factors are changed for enfor
the limits, the state estimation formulation is changed. Consequent
new gain matrix will have to be formed and LU-decomposition
have to be refactorized. Thus computational efficiency suffers.
The covariance matrix of the residual vector r is given by [111
cov (r) = R - W-1-H(x-)G-1 (x)H(x)
The sensitivity matrix a' is given by [41:
OZ
Ox
The fact which prompted Geisler and Bose to replace the I
flow by a state estimation for the external network solution is
observation that large boundary mismatches sometimes occur. T
method, as a matter of fact, may be interpreted as a way of accompl
ing boundary matching by adjusting external operating-point d
rather than simply adding boundary injections [7]. Simulation exp
ments have concluded [6] that the inaccuracy of external operat
point data does not seem to influence very much the results of c
tingency studies as long as the boundary matching is done for the
case. However if boundary mismatchs are really of concern
modification of our proposed method can be used to eliminate bc
dary mismatches at specified boundary buses at the possible expens
external operating-point data mismatches.
ar
Or
=_
I-H (x) G- (£A)H ((x) W c R W
fz,J_
1z2J
H
are:
whereHll =
h(x,)+w
where
z is the (m x 1) measurement vector, h (x ) is the (m x 1) i
tor of non-linear functions, x, is the (n x 1) true state vector, w is
(m x 1) measurement error vector, m is the number of measureme
and n is the number of state variables.
z-
where the Jacobian matrix H(x) is
aJ
ax
=
0
estimate
ix
is
2)]+H22W2[Z2-h2(i2)]
= 0
(14)
state estimation.
h1(A1,A2)
(15)
0
=
H22W2[z2-h2('2)]
ah (x)
ax
by solving the non-linear system
through the iteration process:
HT(xk) W(z-h(xk))
xk+I
(13)
0
It follows that Eq. (14) is equivalent to
=
obtained
G(xk)Axk
=
When the measurements z1 are selected to be a minimal set of
measurements which makes the external system observable, the observability implies that the columns of H1l are independent [14] and the
minimality implies that the dimension of h1 must be equal to the
dimension of xl. Hence H1I is nonsingular. The diagonal matrix WI
is of course nonsingular. Equation (13) is therefore equivalent to
inverses of
HT(x)W[z-h(x)]
The
(12)
H1I H12)
° H22j
where (x1,x2) is the solution of the
The condition for optimality is that the gradient of J vanishe
the optimal solution x, i.e.,
H(x)
J+W2J
h2(X2)
Ox.H12 H22' = Ox2
H12 W1 [z1-h1(i,
[z-h(x)JTW[z-h(x)]
diagonal matrix whose elements are the
measurements variances, that is, W = [cov(w)1-1.
rh,(XI,X2) lWI
|
The optimality condition (3) becomes
HI, W1[z1-hl (XI1, X2)]
The estimate of the unknown state vector xt is designated b
and is obtained by minimizing the least squares function:
a
(11)
In Eq. (12) h, may involve boundary bus state variables x2, where h2
does not involve external state variables xl. The injection measurements at the boundary buses, if any, are treated as part of z1. The
Jacobian matrix (4) is
This section describes the conventional WLS State Estima
equations, in order to introduce basic concepts and notations.
non-linear equations relating the measurements and the state ve
where W is
by [4]:
Let subscript 1 correspond to the external system and subscript 2
correspond to the internal system and the boundary buses of the interconnected system. The state estimation equation (1) can be written as
1. Basic WLS Algorithm
=
is given
(10)
2. Effect of Non-redundant External System Measurements
The basic weighted least square (WLS) state estimation algori
and the sensitivity matrices are briefly reviewed. Two results u
which the proposed method is based are then derived.
J(x)
= G-l(k)HT( )W
and the sensitivity matrix ar
III. THEORETICAL FOUNDATION OF THE METHOD
z =
(9)
=
xk+Axk
for k = 0.1,2,... until appropriate convergence is
gain matrix G is
G = HT(xk)WH(xk)
(5)
(6)
attained. Here the
(7)
The following observations are made
(15)-(16) at the solution point (x1, x2).
(16)
= 0
on
the optimal conditions
(i) Equation (16) is the same optimality condition for the state
estimation involving only the internal system and the boundary buses.
In other words, the estimated internal system state x2 depends only on
the internal system measurements Z2 and is independent of the external
system measurements z1.
(ii) Equation (15) implies that when
a
non-redundant set of
measurements is used the calculated values hl (XA, x2) of the external
system measurement variables z1 are identical to the measurements z1
3. Calculation of the Value of Dormant Measurement
The estimation residuals
are defined as:
r =
z-h(x)
(8)
We are going to show that in the linear case if a measurement j
has zero residual, rj = 0, then the estimated state x is not sensitive to
94
the weighting factor Wj, which means that such a measurement plays
no role in the estimation process (let WJ-O).
Let us consider the linear model
z = Hx + w
(17)
The estimated state x is given by
(18)
(HTWH) =HT Wz
Introducing now a perturbation W in the weighting matrix, we havte
(19)
[HT (W+A W)H] (x+Ai) = HT (W+A W)z
which leads to:
[HT (W+A& W)H]A Ax =HTA Wr
(20)
r = z-H£
(21)
where
is the vector of residuals. Let us make
.0 AW)
(22)
0*
.0
We then have
A Wr =
ejrjA Wj
(23)
where e1 is a m x 1 vector formed by zeros except for the element j,
which is equal to 1.
We say a measurement is dormant if its presence has no effect on
the result of the state estimation. Setting W1 = 0 makes measurement zj dormant. So, from (20) and (23) we can write
Ax
=
S.j
=
[HT (W+A W)H]-HTej.rjA Wj
(24)
which implies that, whenever r = 0 Ax = 0, whatever the value of
A Wj. It follows from Eq. (24) that a measurement z is dormant if
the corresponding residual rj = 0.
Now we consider the problem of how to adjust the value of zj so
as to make rj = 0.
(30)
where W. is the weighting factor of measurement j and Rii is the
(jUth-element of matrix R - which, for the places where true measurements are used, gives the residuals covariances, and otherwise is simply
a sensitivity relation.
Comments: 1. Because relation (29) is obtained from a linear approximation, if the state estimate x is recomputed after substituting zj by
in
nonlinear model, the new residual re, may not b precisely
z1e,
equal tothe
zero. Of course, it is expected that the
better is the linearization the smaller is the magnitude of rnew. For the standard WLS
estimation algorithm with matrix S computed at the solution point x,
this approximation is quite accurate. This would mean extra computation burden and programming complexity. On the other hand,if a fast
decoupled estimation algorithm with constant gain matrices computed
at V = 1 p.u. and 0 = Odegree is used, the sensitivity matrix is also
approximated, and the accuracy of rnew will not be as good as in the
standard WLS method. Specifically, for systems with low X/R ratios
as, for example, the IEEE 30-bus system (a low voltage transmission
line have X/R = .9), recomputations of Zfnew may be necessary in order
to keep rn.ew within a specified tolerance.
The procedure described above can be easily generalized to the
more common case where there are more than one dormant measurement. The direct generalization would require the computation of the
off-diagonal elements of sensitivity matrices. However, we propose to
ignore these off-diagonal elements and use only the diagonal elements
Sfi, which can be easily computed using the sparse inverse matrix technique [12]. This means that each dormant measurement is treated as
being independent of the rest (Eq. (28) is used). The errors introduced
by such approximation, when significant, may be reduced by recomputing (iteratively) zflew, as described in the previous paragraph.
IV. OUTLINE OF THE ALGORITHM
The proposed method has been implemented in a fast modeldecoupled state estimation program. The required modification is not
significant. In this section, we first briefly review the fast modeldecoupled state estimation algorithm (2]. The overall algorithm of the
combined method for internal state estimation and external network
modeling is presented next. Finally the details of the modifications on
the state estimation program (bus type switching and making a measurement dormant) are described.
1. Fast Model-Decoupled Estimator [21
The state vector x is defined as
X =
Let
rT = Zj-hj (x)
(25)
be the residual of measurement j. We want to correct zj,
z new = z
WjRjj
2. A similar approach has been used in [41 to handle bad data,
where the suspected measurement is substituted by a value that makes
it dormant (rj = 0).
0.
AW =
with
+A z
The measurement vector z is partitioned as follows
z =
(26)
(31)
[O T;VT]T
[TT;IT;UTKT;E]T
(32)
where the components of the vectors T ,I , U, K and E are, respec-
tively:
such that the new residual
rnew rg+Ar1
=
=
0
(27)
i.e., we want to keep zj as a dormant measurement.
Assuming Azk = 0 for all k .j, the relation between Aq and
Az is given by the (jj)th-element of the sensitivity matrix RW (Eq.
Ti == P1 /V15P1m= = active power flow from bus I
PjVI;.P active power injection into bus 1,
U1 = Qim/ V; QIm = reactive power flow from bus I to m,
K, = Ql/ VI; Q, = reactive power injection into bus 1,
Ei = VI; VI = voltage magnitude at bus 1.
I,
to
m,
(IY)):
AGri
=
S91Az1
To enforce (27), we have to make Arj = -rj, that means that
zfnewj= ziz+Az
+,& =i zj jj
=
(28)
The Jacobian matrix is given by
H(O,V)
(29)
=
[H1 H12
(33)
95
where H1l corresponds to the active measurements (T and I), and H2j
corresponds to the reactive measurements (U , K , and E).
Step 3.
a) If all components of AO(k) are less than the angle tolerance eq, then make kp = 0 and go to step 4.
The gain matrix can be partitioned as follows
G (@ V)s - 11%
GQO
GPV]
GQv
(34)
where the gain matrices G.- and GQV can be written as functions of
the Jacobian submatrices hr.
The following approximations are made in
fast-model-decoupled state estimator:
order to
(a) Flat voltage profile, i.e., V = 1 p.u. and 0
(b) Submatrices H12 and H21 are ignored (decoupling).
(c) Transmission line resistances are neglected in forming submatrix HI,. (This is the same approximation used in obtaining matrix B'
in fast decoupled load-flow [161.)
Implementing the foregoing approximations, matrices H and G
take the decoupled form:
HO
b) Otherwise go to Step 6.
Step 4.
b) Otherwise go to Step 6.
Step 5.
Make kq = 1.
Step 6
If k > 1, update dormnant measurements to keep zero residuals, check Q-limits and V-limits, and change bus types to
enforce limits whenever necessary.
Step 7.
Compute reactive residuals AU(k), AK(k), and AE(k).
Step 8.
Compute voltage magnitude
(39), and then make V(k+l
(k)
correction AV
= Vk&+A y(k)
by solving Eq.
a) If all components of AV(k) are less than the voltage mag-
(35)
=[%Q H102
Check whether kq = 0.
a) If so, make k-k+1/2 and stop.
obtain the
0 radians.
Compute anzle correction AO(k) by solving Eq. (38), and
then makeS~ v+1) - (k) + AO(k).
nitude tolerance eV, then make kq = 0 and go to Step 9.
b) Otherwise go to Step 10.
4[Gpo0
G°
-(36)
]
Step 9.
Check whether kp = 0.
a) If so, make k-k+1 and stop.
where
p = (Hf l ) TW1H
b) Otherwise to to Step 10.
(37)
Step 10. Make kp = l,k-k+1, and go to Step 2.
GO= (H22)T W2H22
Remark: Notice that the only difference of the above algorithm from a
standard state estimator is in Step 6. Thus the proposed new method
can easily be implemented by modifying an existing state estimation
program.
Introducing (35)-(36) into iteration (5)-(6), it follows that:
0-iteration
0Ok
GR
(HO j)TWiAZ
=
ok+l
(38)
ek+Aok
=
where
AZT
V-iteratio
=
[ATT(Gk,Vk),AIT(ok,Vk)]
3. Update Dormant Measrurements to Keep Zero Residuals
In the following we give the details about the computation performed at Step 6:
a)
Reestimate all dormant pseudo-measurements using Eq. (29), and
make kd = 0.
b)
residuals
Compute
kd (kd'-kd+ 1).
AU,AK and AE,
and
increase
In
GQVA V
Vk+l
=
b. 1) If the residuals of all dormant pseudo- measurements are less
than a given tolerance, or kd = 2, then go to Step 7.
(H2) TW2Zr
=
vk+AVk
b.2) Otherwise go to (c).
where
AZT
=
[AUT(Ok+lVk),
c)
AKT(0k+l,
weight'ing
active
measurements,
Wl is the weighting matrix for the reactive measurements
the voltage magnitude).
2. Overall Algorithm
1, kq
1, and k
Step 1.
Make kp
Step 2.
Compute active residuals
-
0.
AT(k) and
Solve Eq. (39) for AV, update V, and then go to (a).
Vk), EIT]
AI(").
(including
Remark: Though the convergence of the algorithm is not affected very
much by the weighting factors of the external system for a wide range
of values of the weighting factors, better results can be achieved when
the weighting factors are appropriately chosen. The effect of the
weighting factors can be seen by considering, say, a PY bus. Notice
that, if WQ is very small, although it is easy to lkeep measurement Q
dormant, when a bus type change occurs (PV-PQ), it may be
difficult to keep V dormant, because in this case the estimate given by
Eq. t29) may not be very accurate. Therefore we have to balance the
weighting factors Wv and WQ, so that the relative importance of
measurements Q and V are approximately the same for the estimation
process.
96
Case B (narrow limits)
V. TEST RESULTS
Both the performance and reliability o the new method have been
tested on four power systems: a 6-bus example-system, the IEEE 14bus and 30-bus systems, and a Brazilian 66-bus system (345/500 kv).
All tests have been performed using tolerances E
.001 both for
A V(k) (p.u.) and AO(K) (rad.).
-
- limits:
.990< V5( 1.010(p..U)
-50 Q6< 5.0(MVAR)
1. Six Bus Example System
The network is shown in Fig. 1. The following set of measurements have been considered:
pseudo-measurements: same as in Case A
- estimated values:
P5
active power injections P1 and P2;
P6
reactive power injections Qi and Q2;
active power flows P1,2,P1,3,P2,3, and
=
-150.OMW;Q5 =-16.4MVAR (dormant);
Vs
.990(enforced)
40.OMW;Q6 = 5.0(enforced);
=
=
V6 = 1.009(dormant)
P2,4;
- number of iterations: 3 1/2.
reactive power flows Ql,2,Ql,2,Q2,3, and Q2,4;
Case C
and the voltage magnitude V1.
- pseudo-measurements: same as in Case A.
These measurements make the internal system (buses 1 and 2) and the
boundary (buses 3 and 4) observable.
- limits:
.900< V5s 1.010(p.u.)
-40.0< Q6K40.0(MVAR)
V;!S V5 IS
m"
- estimated values:
V
P5 =-149.3MW;Q5 -21.8MVAR (dormant)
V5 = .99Op.u. (enforced)'
M
P6 =
V6
ds6< Q6
91
A METHOD THAT COMBINES INTERNAL STATE ESTIMATION
AND EXTERNAL NETWORK MODELING
A. Monticelli
Felix F. Wu
Department of Electrical Engineering and Computer Sciences
and the Electronics Research Laboratory
University of California, Berkeley, California 94720
Abstract
A method that combines internal state estimation and external
network modeling is developed. The external system is represented by
an unreduced load flow model. One state estimation covering both the
internal system and the external system is used. The external system
operating data on power injections and bus voltages are entered as
pseudo-measurements. At each iteration the set of active pseudomeasurements are selected to conform with the specified variables in a
load flow program. Because such a set of non-redundant measurements
is used, the internal state estimation is not affected by the external system pseudo-measurements. External generation MVAR and controlled
bus voltage limits are enforced. A technique is developed to make a
pseudo-measurement dormant. Using the dormant-measurement technique, it is possible to maintain the same external system state estimation formulation while the PV-PQ switching takes place from iteration
to iteration. The method can easily be implemented by modifying an
existing state estimation program. It has been implemented in a fast
model-decoupled estimation program. Because of the dormant measurement technique, the constant gain matrix evaluated at flat voltage is
used in every iteration. The method has been tested on the IEEE 14bus, 30-bus, and Brazilian 66-bus systems. Excellent results are
obtained. The number of iterations for the method to converge is usually the same as the regular state estimation runs. Thus the method
presented here achieves simultaneously the internal state estimation
and the external network modeling without adversely affecting the
internal state estimation.
I. INTRODUCTION
For control center applications, the interconnected power system
is divided into two parts: the internal system, which is basically one's
own power system, and the external systems, which is the rest of the
interconnected network. Real time data of the internal system are gathered through a data acquisition system: from time to time, system
substations are scanned and measurements reflecting a snapshot of the
present operating condition of the system are obtained. The state estimator utilizes these data to set up a load flow model of the internal system. For the state estimator to estimate the present state of the system, no information about the rest of the interconnected system is
required.
Most control centers today are equipped with the state estimation
software. The weighted least square (WLS) estimation method has
been a popular choice [1]. Recently it has been demonstrated that a
fast model-decoupled version of the WLS state estimation is reliable
and efficient [2,3]. The method utilizes a constant gain matrix
evaluated at flat voltage (V=1,0=0) for each iteration.
On leave from Departamento de Engenharia Eletrica, UNICAMP, Campinas, S.
P., Brazil.
In addition to monitoring the current operating condition of the
system, the control center also performs the task of security analysis.
Security analysis is concerned with the study of line and generation
outages (contingency) and is carried out using an on-line load flow program. In an interconnected system, a contingency in the internal system can perturb the operating conditions of the neighboring systems
(external system); the change in the external system operating point, in
turn, can have a significant effect on the system where the contingency
occurred - we call this effect "external system reaction." So, for such
applications, unlike the state estimation, an external system model is
needed in order to represent, as accurately as possible, the external network reaction when internal system perturbations have to be simulated.
The usual approach to setting up the on-line load flow model of
the interconnected system for security analysis involves three main
steps:
(i) state estimation for the current state of the internal system;
(ii) modeling of external system; and
(iii) attachment of the external system model to the internal sys-
tem for contingency analysis.
The external system has been represented by an unreduced load
flow model or a reduced network equivalent. A comprehensive review
of the external network modeling is available [5-7]. In [7], a unified
procedure for external system modeling is suggested, where the external system is further partitioned into inner external and outer external
systems. The inner external system (or buffer zone) is represented by
an unreduced load flow model and the outer external system is modeled
by a reduced equivalent, which may be a Ward-type or an REI-type
equivalent. An advantage of having an unreduced load flow model for
at least a part of the external system is that certain load flow information (e.g., net interchange, key flows) available on the external system
can easily be incorporated. This paper is concerned solely with the
unreduced load flow model for the external system. The results can be
applied to the whole external system or just to the inner external system.
When the external system is represented by an unreduced load
flow model [8], the operating-point data (loads, generations, and
specified voltages) of the external system is needed. Some external
system data may be available in real time, for example, due to partial
data exchange between control centers. But in general these data are
not all available to the internal system in real time, and so extrapolated
values or off-line data have to be augmented. In order to obtain a set
of operating-point data from the extrapolated values that are compatible
to the current state of the internal system, a load flow program for the
external system and the boundary buses has been used. The input data
for the external system load flow consist of
-
extrapolated real and reactive power injections at the
external load (PQ) buses,
- extrapolated real power injections and specified
voltages at the external generation (PV) buses,
- voltages magnitudes and phase angles from the state
A paper recommended and approved
WM
by the IEEE Power System Engineering Committee of
the IEEE Power Engineering Society for presentation at the IEEE/PES 1984 Winter Meeting, Dallas,
Texas, January 29
February 3, 1984. Manuscript
submitted September 1, 1983; made available for
printing December 13, 1983.
84
034-5
-
estimator for the boundary buses (treated as slack
buses).
A fundamental requirement in the attachment of the external system model to 'the internal system is that it must not affect the values of
the internal state given by the state estimation for the present situation
(called the base case). In other words, the solution of the load flow
model consisting of the internal system and the external model for the
0018-9510/85/0001-0091$01.00©1985 IEEE
92
base case should agree with the solution of the internal system obtained
from the state estimation. This is accomplished by adjusting the external system model via the so-called boundary matching procedure
described below:
(a) From the state estimation, calculate the flows from the boundary buses into the internal system.
(b) Use the external system model together with the voltage magnitudes and angles of the boundary buses from the state estimation to
compute the flows from the external system into the boundary buses.
(c) For each boundary bus, add boundary power injections so that
the flows are balanced.
In summary, the load-flow based external network modeling
requires in real time the coordinated execution of a state estimator and
a load flow. For real-time analysis, simplicity in software and database
is of paramount importance. Any improvement along that line is certainly desirable. Since a state estimator is absolutely essential in control
center applications and it is now generally available, it would be nice to
eliminate the necessity to run a separate6load flow for external network
modeling.
In this paper we develop a method that combines the internal
state estimation and the external network modeling by modifying an
existing state estimation program. The external network model is
obtained at the same time the internal system state is estimated. The
resulting unreduced load flow model of the external system has the properties that the variables lie within their operating limits.
An overview of the proposed method is given in Sec. II. The
theoretical foundation of the method is presented in Sec. III. The solution algorithm of the method implemented in a fast model-decoupled
state estimation program is outlined in Sec. IV. The method has been
tested- on a simple example system, as well as the IEEE 14-bus, 30-bus
system and a Brazilian 66-bus system. The results are shown in Sec. V.
II. OVERVIEW OF THE METHOD
We have developed a method which combines the internal state
estimation and the external network modeling. One modified state estimation covering both the internal system and the external system is
used. The external system operating data on power injections and bus
voltages are entered as "pseudo-measurements" in the state estimation
program. The internal state estimation and the external network
modeling are achieved simultaneously.
1. The Requirements
Prior to the development of the method a wish list was drawn
which specifies the requirements or desired attributes of the new
method. They are listed below.
(1) The part on the external network modeling must not affect
the result of the internal . state estimation. In other words, the
estimated internal state variables should agree with the estimated internal state variables that would be obtained from the state esitmation
based only on the same set of internal system measurements.
(2) In the external system, Q-limits at generator buses (PV
buses) and V-limits at load buses (PQ buses) with voltage control
should be enforced.
(3) The state estimation formulation for the external network
modeling should remain the same throughout the iterations for computational efficiency. In particular, the gain matrix with flat-voltage
approximation should remain constant so that the fast model-decoupled
state estimation algorithm can be employed.
-2. The Approach
We now describe the ideas developed in the new method to fulfill
the requirements stated above.
(1) The set of pseudo-measurements for the exiernal system is
selected to conform with the PV- and PQ-designations in a load flow
program. Initially for load buses, extrapolated values of P and Q injec-
tions are used as pseudo-measurements, and for generator buses, extrapolated P generations (say, based on AGC data) and specified voltages
V are used. Such a set of measurements actually forms a minimal set
of measurements which makes the external system observable [14].
It is shown in Sec. 111-2 that if a set of non-redundant pseudomeasurements is used for the external system, the estimated state of
the internal system depends only on the internal system measurements,
independent of the values of the external system pseudomeasurements.
(2) The way the enforcement of Q-limits at PV buses and Vlimits at PQ buses is handled in a standard load flow program is the socalled bus switching scheme, i.e., changing bus types whenever a limit
violation occurs. We propose to do exact the same thing for the external system in our method, for example, when a Q-limit at a PV bus is
violated the pseudo-weasurement.V = VSP is replaced by a pseudomeasurement Q = Q m, where Q im is the enforced limit.
(3) To avoid the change of state estimation formulation whenever
a change of pseudo-masurement occurs, we propose to retain all
relevant variables as pseudo-measurements in the formulation and
activate at each iteration only those pseudo-measurements to make up a
non-redundant set. For example, for an external generator bus we put
P, Q and V all as pseudo-measurements in the formulation. Initially P
and V are active, and the pseudo-measurement Q is kept as a dormant
measurement as long as it remains within the specified limits. Whenever a limit violation,.occurs, the pseudo-measurement Q is then
activated with Q = Q lm and the pseudo-measurement V is made a
dormant measurement.
A measurement is dormant if its presence has no effect on the
result of the estimation. Thus the value of a dormant measurement
should be equal to the calculated value of that variable obtained from
the state estimation using a set of measurements from which that particular measurement is absent. In Sec. III-3, we show that using sensitivity analysis it is possible to calculate an approximate value of a dormant measurement without actually removing that measurement from
the measurement set in the state estimation. At each iteration of the
state estimation, for a pseudo-measurement which is supposed to be
dormant, the value of that pseudo-measurement is replaced by one calculated using sensitivity analysis that will make it dormant.
3. The Implementation
The proposed approach sketched above can easily be implemented
into any existing state estimation program. It seems to be generally
agreed that the fast model-decoupled state estimation is more reliable
and efficient [2,31. We have modified a fast model-decoupled state estimation program to do the combined internal state estimation and external network modeling. The required modification is not extensive.
The modified algorithm is presented in Sec. IV.
The computational efficiency of the fast model-decoupled state
estimation stems from the fact that a constant gain matrix (evaluated at
the flat voltage) is used throughout the iterations. This is still the case
for the modified algorithm.
The limit enforcement (as well as the limit back up) of external
system variables is handled in each iteration, just like a standard load
flow. Tests have shown that the inclusion of limit enforcement does
not seem to affect the number of iterations for the state estimation to
converge.
4. Comments on Related Work
The idea of using a set of non-redundant pseudo-measurements
of the external system so that the internal state estimation will not be
affected was first pointed out in [5, p. 2302], and was recently used in
[9]. The problem that the state estimation may result in external generation MVAR and controlled voltages outside the limits has not been
resolved.
Geisler and Bose [10] have proposed to use state estimation for
the external network solution. In their approach, two separate state
estimation runs are needed, one for the internal system and one for the
external system. For the state estimation to handle the load flow task
93
of enforcing external generation MVAR and voltage limits, they
gested to manipulate the weighting factors of the WLS estimator,
when the calculated value of a pseudo-measurement is outside
range, its corresponding weighting factor is increased in order to for(
back to its limits. Once the weighting factors are changed for enfor
the limits, the state estimation formulation is changed. Consequent
new gain matrix will have to be formed and LU-decomposition
have to be refactorized. Thus computational efficiency suffers.
The covariance matrix of the residual vector r is given by [111
cov (r) = R - W-1-H(x-)G-1 (x)H(x)
The sensitivity matrix a' is given by [41:
OZ
Ox
The fact which prompted Geisler and Bose to replace the I
flow by a state estimation for the external network solution is
observation that large boundary mismatches sometimes occur. T
method, as a matter of fact, may be interpreted as a way of accompl
ing boundary matching by adjusting external operating-point d
rather than simply adding boundary injections [7]. Simulation exp
ments have concluded [6] that the inaccuracy of external operat
point data does not seem to influence very much the results of c
tingency studies as long as the boundary matching is done for the
case. However if boundary mismatchs are really of concern
modification of our proposed method can be used to eliminate bc
dary mismatches at specified boundary buses at the possible expens
external operating-point data mismatches.
ar
Or
=_
I-H (x) G- (£A)H ((x) W c R W
fz,J_
1z2J
H
are:
whereHll =
h(x,)+w
where
z is the (m x 1) measurement vector, h (x ) is the (m x 1) i
tor of non-linear functions, x, is the (n x 1) true state vector, w is
(m x 1) measurement error vector, m is the number of measureme
and n is the number of state variables.
z-
where the Jacobian matrix H(x) is
aJ
ax
=
0
estimate
ix
is
2)]+H22W2[Z2-h2(i2)]
= 0
(14)
state estimation.
h1(A1,A2)
(15)
0
=
H22W2[z2-h2('2)]
ah (x)
ax
by solving the non-linear system
through the iteration process:
HT(xk) W(z-h(xk))
xk+I
(13)
0
It follows that Eq. (14) is equivalent to
=
obtained
G(xk)Axk
=
When the measurements z1 are selected to be a minimal set of
measurements which makes the external system observable, the observability implies that the columns of H1l are independent [14] and the
minimality implies that the dimension of h1 must be equal to the
dimension of xl. Hence H1I is nonsingular. The diagonal matrix WI
is of course nonsingular. Equation (13) is therefore equivalent to
inverses of
HT(x)W[z-h(x)]
The
(12)
H1I H12)
° H22j
where (x1,x2) is the solution of the
The condition for optimality is that the gradient of J vanishe
the optimal solution x, i.e.,
H(x)
J+W2J
h2(X2)
Ox.H12 H22' = Ox2
H12 W1 [z1-h1(i,
[z-h(x)JTW[z-h(x)]
diagonal matrix whose elements are the
measurements variances, that is, W = [cov(w)1-1.
rh,(XI,X2) lWI
|
The optimality condition (3) becomes
HI, W1[z1-hl (XI1, X2)]
The estimate of the unknown state vector xt is designated b
and is obtained by minimizing the least squares function:
a
(11)
In Eq. (12) h, may involve boundary bus state variables x2, where h2
does not involve external state variables xl. The injection measurements at the boundary buses, if any, are treated as part of z1. The
Jacobian matrix (4) is
This section describes the conventional WLS State Estima
equations, in order to introduce basic concepts and notations.
non-linear equations relating the measurements and the state ve
where W is
by [4]:
Let subscript 1 correspond to the external system and subscript 2
correspond to the internal system and the boundary buses of the interconnected system. The state estimation equation (1) can be written as
1. Basic WLS Algorithm
=
is given
(10)
2. Effect of Non-redundant External System Measurements
The basic weighted least square (WLS) state estimation algori
and the sensitivity matrices are briefly reviewed. Two results u
which the proposed method is based are then derived.
J(x)
= G-l(k)HT( )W
and the sensitivity matrix ar
III. THEORETICAL FOUNDATION OF THE METHOD
z =
(9)
=
xk+Axk
for k = 0.1,2,... until appropriate convergence is
gain matrix G is
G = HT(xk)WH(xk)
(5)
(6)
attained. Here the
(7)
The following observations are made
(15)-(16) at the solution point (x1, x2).
(16)
= 0
on
the optimal conditions
(i) Equation (16) is the same optimality condition for the state
estimation involving only the internal system and the boundary buses.
In other words, the estimated internal system state x2 depends only on
the internal system measurements Z2 and is independent of the external
system measurements z1.
(ii) Equation (15) implies that when
a
non-redundant set of
measurements is used the calculated values hl (XA, x2) of the external
system measurement variables z1 are identical to the measurements z1
3. Calculation of the Value of Dormant Measurement
The estimation residuals
are defined as:
r =
z-h(x)
(8)
We are going to show that in the linear case if a measurement j
has zero residual, rj = 0, then the estimated state x is not sensitive to
94
the weighting factor Wj, which means that such a measurement plays
no role in the estimation process (let WJ-O).
Let us consider the linear model
z = Hx + w
(17)
The estimated state x is given by
(18)
(HTWH) =HT Wz
Introducing now a perturbation W in the weighting matrix, we havte
(19)
[HT (W+A W)H] (x+Ai) = HT (W+A W)z
which leads to:
[HT (W+A& W)H]A Ax =HTA Wr
(20)
r = z-H£
(21)
where
is the vector of residuals. Let us make
.0 AW)
(22)
0*
.0
We then have
A Wr =
ejrjA Wj
(23)
where e1 is a m x 1 vector formed by zeros except for the element j,
which is equal to 1.
We say a measurement is dormant if its presence has no effect on
the result of the state estimation. Setting W1 = 0 makes measurement zj dormant. So, from (20) and (23) we can write
Ax
=
S.j
=
[HT (W+A W)H]-HTej.rjA Wj
(24)
which implies that, whenever r = 0 Ax = 0, whatever the value of
A Wj. It follows from Eq. (24) that a measurement z is dormant if
the corresponding residual rj = 0.
Now we consider the problem of how to adjust the value of zj so
as to make rj = 0.
(30)
where W. is the weighting factor of measurement j and Rii is the
(jUth-element of matrix R - which, for the places where true measurements are used, gives the residuals covariances, and otherwise is simply
a sensitivity relation.
Comments: 1. Because relation (29) is obtained from a linear approximation, if the state estimate x is recomputed after substituting zj by
in
nonlinear model, the new residual re, may not b precisely
z1e,
equal tothe
zero. Of course, it is expected that the
better is the linearization the smaller is the magnitude of rnew. For the standard WLS
estimation algorithm with matrix S computed at the solution point x,
this approximation is quite accurate. This would mean extra computation burden and programming complexity. On the other hand,if a fast
decoupled estimation algorithm with constant gain matrices computed
at V = 1 p.u. and 0 = Odegree is used, the sensitivity matrix is also
approximated, and the accuracy of rnew will not be as good as in the
standard WLS method. Specifically, for systems with low X/R ratios
as, for example, the IEEE 30-bus system (a low voltage transmission
line have X/R = .9), recomputations of Zfnew may be necessary in order
to keep rn.ew within a specified tolerance.
The procedure described above can be easily generalized to the
more common case where there are more than one dormant measurement. The direct generalization would require the computation of the
off-diagonal elements of sensitivity matrices. However, we propose to
ignore these off-diagonal elements and use only the diagonal elements
Sfi, which can be easily computed using the sparse inverse matrix technique [12]. This means that each dormant measurement is treated as
being independent of the rest (Eq. (28) is used). The errors introduced
by such approximation, when significant, may be reduced by recomputing (iteratively) zflew, as described in the previous paragraph.
IV. OUTLINE OF THE ALGORITHM
The proposed method has been implemented in a fast modeldecoupled state estimation program. The required modification is not
significant. In this section, we first briefly review the fast modeldecoupled state estimation algorithm (2]. The overall algorithm of the
combined method for internal state estimation and external network
modeling is presented next. Finally the details of the modifications on
the state estimation program (bus type switching and making a measurement dormant) are described.
1. Fast Model-Decoupled Estimator [21
The state vector x is defined as
X =
Let
rT = Zj-hj (x)
(25)
be the residual of measurement j. We want to correct zj,
z new = z
WjRjj
2. A similar approach has been used in [41 to handle bad data,
where the suspected measurement is substituted by a value that makes
it dormant (rj = 0).
0.
AW =
with
+A z
The measurement vector z is partitioned as follows
z =
(26)
(31)
[O T;VT]T
[TT;IT;UTKT;E]T
(32)
where the components of the vectors T ,I , U, K and E are, respec-
tively:
such that the new residual
rnew rg+Ar1
=
=
0
(27)
i.e., we want to keep zj as a dormant measurement.
Assuming Azk = 0 for all k .j, the relation between Aq and
Az is given by the (jj)th-element of the sensitivity matrix RW (Eq.
Ti == P1 /V15P1m= = active power flow from bus I
PjVI;.P active power injection into bus 1,
U1 = Qim/ V; QIm = reactive power flow from bus I to m,
K, = Ql/ VI; Q, = reactive power injection into bus 1,
Ei = VI; VI = voltage magnitude at bus 1.
I,
to
m,
(IY)):
AGri
=
S91Az1
To enforce (27), we have to make Arj = -rj, that means that
zfnewj= ziz+Az
+,& =i zj jj
=
(28)
The Jacobian matrix is given by
H(O,V)
(29)
=
[H1 H12
(33)
95
where H1l corresponds to the active measurements (T and I), and H2j
corresponds to the reactive measurements (U , K , and E).
Step 3.
a) If all components of AO(k) are less than the angle tolerance eq, then make kp = 0 and go to step 4.
The gain matrix can be partitioned as follows
G (@ V)s - 11%
GQO
GPV]
GQv
(34)
where the gain matrices G.- and GQV can be written as functions of
the Jacobian submatrices hr.
The following approximations are made in
fast-model-decoupled state estimator:
order to
(a) Flat voltage profile, i.e., V = 1 p.u. and 0
(b) Submatrices H12 and H21 are ignored (decoupling).
(c) Transmission line resistances are neglected in forming submatrix HI,. (This is the same approximation used in obtaining matrix B'
in fast decoupled load-flow [161.)
Implementing the foregoing approximations, matrices H and G
take the decoupled form:
HO
b) Otherwise go to Step 6.
Step 4.
b) Otherwise go to Step 6.
Step 5.
Make kq = 1.
Step 6
If k > 1, update dormnant measurements to keep zero residuals, check Q-limits and V-limits, and change bus types to
enforce limits whenever necessary.
Step 7.
Compute reactive residuals AU(k), AK(k), and AE(k).
Step 8.
Compute voltage magnitude
(39), and then make V(k+l
(k)
correction AV
= Vk&+A y(k)
by solving Eq.
a) If all components of AV(k) are less than the voltage mag-
(35)
=[%Q H102
Check whether kq = 0.
a) If so, make k-k+1/2 and stop.
obtain the
0 radians.
Compute anzle correction AO(k) by solving Eq. (38), and
then makeS~ v+1) - (k) + AO(k).
nitude tolerance eV, then make kq = 0 and go to Step 9.
b) Otherwise go to Step 10.
4[Gpo0
G°
-(36)
]
Step 9.
Check whether kp = 0.
a) If so, make k-k+1 and stop.
where
p = (Hf l ) TW1H
b) Otherwise to to Step 10.
(37)
Step 10. Make kp = l,k-k+1, and go to Step 2.
GO= (H22)T W2H22
Remark: Notice that the only difference of the above algorithm from a
standard state estimator is in Step 6. Thus the proposed new method
can easily be implemented by modifying an existing state estimation
program.
Introducing (35)-(36) into iteration (5)-(6), it follows that:
0-iteration
0Ok
GR
(HO j)TWiAZ
=
ok+l
(38)
ek+Aok
=
where
AZT
V-iteratio
=
[ATT(Gk,Vk),AIT(ok,Vk)]
3. Update Dormant Measrurements to Keep Zero Residuals
In the following we give the details about the computation performed at Step 6:
a)
Reestimate all dormant pseudo-measurements using Eq. (29), and
make kd = 0.
b)
residuals
Compute
kd (kd'-kd+ 1).
AU,AK and AE,
and
increase
In
GQVA V
Vk+l
=
b. 1) If the residuals of all dormant pseudo- measurements are less
than a given tolerance, or kd = 2, then go to Step 7.
(H2) TW2Zr
=
vk+AVk
b.2) Otherwise go to (c).
where
AZT
=
[AUT(Ok+lVk),
c)
AKT(0k+l,
weight'ing
active
measurements,
Wl is the weighting matrix for the reactive measurements
the voltage magnitude).
2. Overall Algorithm
1, kq
1, and k
Step 1.
Make kp
Step 2.
Compute active residuals
-
0.
AT(k) and
Solve Eq. (39) for AV, update V, and then go to (a).
Vk), EIT]
AI(").
(including
Remark: Though the convergence of the algorithm is not affected very
much by the weighting factors of the external system for a wide range
of values of the weighting factors, better results can be achieved when
the weighting factors are appropriately chosen. The effect of the
weighting factors can be seen by considering, say, a PY bus. Notice
that, if WQ is very small, although it is easy to lkeep measurement Q
dormant, when a bus type change occurs (PV-PQ), it may be
difficult to keep V dormant, because in this case the estimate given by
Eq. t29) may not be very accurate. Therefore we have to balance the
weighting factors Wv and WQ, so that the relative importance of
measurements Q and V are approximately the same for the estimation
process.
96
Case B (narrow limits)
V. TEST RESULTS
Both the performance and reliability o the new method have been
tested on four power systems: a 6-bus example-system, the IEEE 14bus and 30-bus systems, and a Brazilian 66-bus system (345/500 kv).
All tests have been performed using tolerances E
.001 both for
A V(k) (p.u.) and AO(K) (rad.).
-
- limits:
.990< V5( 1.010(p..U)
-50 Q6< 5.0(MVAR)
1. Six Bus Example System
The network is shown in Fig. 1. The following set of measurements have been considered:
pseudo-measurements: same as in Case A
- estimated values:
P5
active power injections P1 and P2;
P6
reactive power injections Qi and Q2;
active power flows P1,2,P1,3,P2,3, and
=
-150.OMW;Q5 =-16.4MVAR (dormant);
Vs
.990(enforced)
40.OMW;Q6 = 5.0(enforced);
=
=
V6 = 1.009(dormant)
P2,4;
- number of iterations: 3 1/2.
reactive power flows Ql,2,Ql,2,Q2,3, and Q2,4;
Case C
and the voltage magnitude V1.
- pseudo-measurements: same as in Case A.
These measurements make the internal system (buses 1 and 2) and the
boundary (buses 3 and 4) observable.
- limits:
.900< V5s 1.010(p.u.)
-40.0< Q6K40.0(MVAR)
V;!S V5 IS
m"
- estimated values:
V
P5 =-149.3MW;Q5 -21.8MVAR (dormant)
V5 = .99Op.u. (enforced)'
M
P6 =
V6
ds6< Q6