483
IEEE Transactions on Power Systems, Vol. 10, No. 1, February 1995
A BRANCH-CURRENT-BASEDSTATE ESTIMATION METHOD FOR DISTRIBUTION SYSTEMS
Mesut E. Baran
Arthur
W.Kelley
Department of Electrical and Computer Engineering
North Carolina State University
Raleigh, NC 27695
Abstract A branch-current-based three-phase state
estimation (SE) method is proposed for distribution
systems. The method is tailored for distribution feeders
with a few loops. The method is computationally more
efficient than the conventional node-voltage-based SE
methods. To further improve the computational
efficiency it is shown that distribution systems can be
reduced without much loss of accuracy in SE.
Key Words: Distribution Systems, State Estimation,
Real-Time Monitoring.
I. INTRODUClYON
The interest and efforts of the last couple years to
"modernize" distribution systems have resulted in the
development of many new concepts and methods for
distribution system problems, many of which are
challenging [l]. Most of the feeder analyses needed for
real-time monitoring and operation of distribution
systems a r e power flow based, such as v o l t h a r
control, feeder reconfiguration and restoration. A
detailed power flow analysis of a distribution system
is challenging because distribution systems a r e
unbalanced, have very short lines with high r/x
ratios, and the loads a r e highly distributed and
diverse. The biggest challenge, though, i n real time
applications is i n obtaining the necessary data which
includes the data for on-line power flow based feeder
analysis methods.
monitoring of distribution feeders in real-time. A
branch-current-based three-phase SE method is
developed. The method can handle radial and weakly
meshed feeders which may have a few loops created by
closing some normally-open tie or line switches. The
method is computationally more efficient and more
insensitive to line parameters than the conventional
node-voltage-based SE methods. This improvement
mainly comes from the fact that the branch current
formulation decouples the SE problem into three
subproblems, one for each phase. Furthermore, a
simple rule based feeder network reduction method is
proposed in this paper to further improve the
computational efficiency of SE without sacrificing
accuracy.
11. STATE ESTIMATION
The branch-current-based SE method, like
conventional node-voItage-based SE methods, is based
on the weighted least square (WLS) approach. An
excellent review of the state-of-the-art on SE can be
found in [5]. Rather than using the node voltages as the
system state, the proposed method uses the branch
currents and solves the following WLS problem to
obtain an estimate of the system operating point
defined by the system state x:
min J(x) =
Recently, encouraged by the fact t h a t many
utilities are i n the process of installing or planning to
install Supervisory Control and Data Acquisition
(SCADA) systems to automate some of the distribution
system operations, new methods are proposed for
obtaining the consistent and accurate real-time data
needed for monitoring and operation of distribution
systems. One of the approaches is power flow based [21
and the others [3,41 are extensions of the conventional
state estimation (SE) method for three-phase analysis.
In 171 an alternative single-phase SE is developed for
balanced radial feeders. Although SE is preferred over
t h e power flow approach, i t s computational
complexity may prevent i t s use i n practical
applications.
X
$
wi(zi - h,(x))' = [ z
i=l
- h(x)ITw[z - h(x)]
(1)
where W i and hi(x) represent the weight and the
measurement function associated with measurement
zi, respectively. For the solution of this problem the
conventional iterative method is adapted by solving
the following normal equations at each iteration to
compute the correction x k + l = xk + Axk,
[G(xk)IAxk = HT(xt)W[z- h(xk)]
(2)
(3)
is the gain matrix and H is the Jacobian of the
measurement function h(x).
The main focus of this paper is to develop a
computationally efficient SE method that is tailored for
One of the main challenges in implementing this
approach for SE in distribution feeders is incorporating
the unbalanced nature of distribution feeders into the
94 SM 602-3 p m s A paper recommended and approved
by the I E E Power System Engineering Committee of the problem. Other challenges are the lack of enough realtime measurements and the fact that most of the
IEEE Power Engineering Society f o r presentation at
the IEEE/PES 1994 Summer Meeting, San F r ~ c i s c o ,CA, available measurements are branch current magnitude
jUly 24 - 28, 1994. Manuscript submitted January 4, measurements which are not usually included in the
1994; made available f o r printing June 10, 1994.
conventional SE methods. These issues are addressed
below.
0885-8950/95/$04.000 1994 IEEE
484
Feeder Representation
In general, main feeders are three-phase, however,
some laterals can be two-phase or single-phase. The
lines are usually short and untransposed. Loads can be
three-phase, two-phase or single-phase (like residential
customers). Therefore it is desirable to use a three phase
model as also recommended for power flow analysis of
feeders 161. A three-phase line model takes into account
the magnetic coupling between the phases in lines,
which for a line section I , l=l...b. such as the one
shown in Fig.1, is of the following form
or
VI= v, - ZJ,
where ZI = glZ is the line impedance matrix and g1 is
the line length. Note that (4) is written for the assumed
branch current direction shown in Fig.1, and the phases
are numbered as 9, = 1,2,3 rather than labeled as a,b,c.
complexity of the SE problem, especially for radial
feeders. To introduce the method, first consider a radial
feeder and choose the branch currents
I/,,, = Ir/,+, + j I x / , ,
9, = 1.2.3
1 = l...b
(5)
as state variables,
xq
=[~l,,"'lb,,l=[I~,c.'x,cpl
9,=1*2*3
(6)
Branch currents define the state of a system because if
the branch currents are known, the node voltages and
loads can then be determined. Taking the substation
bus as the reference bus, a forward sweep procedure can
find the node voltages by starting from the substation
and moving down the feeder visiting branches and
updating the bus voltages at their receiving end by
using the line equations of (4), [6]. Load currents can
also be calculated by applying Kirchhoffs Current Law
(KCL) at every phase of every node, for example, for
node t in Fig.1,
4,
= Il,q
- 4+l,
9, = 1,2,3
(7)
To use branch currents as state variables in SE, we
need to determine the measurement functions, h i ( x )
for each measurement zi first. If all the measurements
were of complex branch currents and node injection
currents, then the measurement functions would be
linear as indicated by (5) and (7), and hence the SE
would be simple. This observation is exploited by
converting the measurements into equivalent current
measurements as follows.
Figure 1: A three-phase line section
Node Voltage Based SE Method
Conventional SE methods take the node voltages as
the state variables, i.e., x = [S VI where 0 and V are
vectors containing the node voltage phase angles and
magnitudes, respectively. These methods have been
extended for three-phase analysis [3,4]. However,
coupling between the phases, as indicated by (4),
increases the dimension and decreases the sparsity of
the gain matrix G and hence increases both the
memory requirements and computational complexity
of the method as compared to SE at transmission level
which uses a single-phase model. However, the
method has the advantage that it can handle feeders
with different topologies, radial feeders as well as
feeders with grid topology. But, in practice, feeders are
predominantly radial, except in some cases in which a
grid topology is created by closing some normally open
loops. Therefore, the goal here is to develop a method
that is specially tailored for SE in distribution systems.
Details of the proposed method are elaborated below.
i) Power Measurements
Two types of power measurements are assumed to
be available: actual power flow measurements and
pseudo load measurements obtained form the load
forecast data. These power measurements are
converted into equivalent current measurements
which are calculated at each iteration by using the
available voltage estimates. For example, the
equivalent current measurement for the power flow
measurement (fl".QY) at the sending end of line I in
Fig.1 is
where V," is the available value of the node voltage at
the kth iteration of the solution process. Note that for
notational simplicity, the subscript 9, for phase index
was dropped. The measurement function for the
equivalent current measurement (I$,I,"f) is then a
linear function of the following form,
hriVr) +JMI,)= Ir/ +J{A
(9)
Measurement Functions
It will be shown in this section that choosing branch
currents as state variables simplifies the measurement
equations h(x) and hence reduces the computational
Similarly, pseudo load measurements are converted
into equivalent load current measurements.
485
Using equivalent currents amounts to a n
approximation of the objective function J ( x ) . For
example, the correct term in the objective function
corresponding to the power measurement (fl”,@)
would be
function are for the power a n d the current
measurements.
For the solution of this problem, note that the
measurement functions in the first summation, which
are for power measurements, are linear i.e.,
hr(Ir)= 4
This term is approximated by using the available
voltage V,” rather than the actual voltage Vs as
where IV,12 is dropped by approximating it as 1.0 p.u.
The proposed SE method uses this approximate
objective function a t each iteration rather than the
linearized measurement functions as the conventional
approach does. Approximations used in (11) also
indicate that the weights used for power measurements
can also be used for equivalent current measurements.
ii) Branch Current Magnitude Measurements
Branch current measurements are handled exactly
since the measurement function for a branch current
magnitude measurement I& can be written as
which is a non-linear function of branch currents.
iii) Voltage Measurements
Voltage measurements (if there are any) will be
ignored, except the voltage measurement at the
substation bus which is taken as the reference bus. This
is based on the observation that the voltage
measurements do not have a significant effect on SE
results provided that the system is observable IS]. We
propose to use voltage measurements as a means for
checking the consistency of the feeder model and the
measurements by comparing the SE results with the
voltage measurements.
(14)
hX(1x)= AI,
where A is the constant matrix with non-zero values of
1 or -1. Hence the solution for the special case with
power measurements only can be obtained directly
from the optimality conditions as,
where G = ATWA is the constant gain matrix. Note that
the normal equations (15) for this case are decoupled on
a phase basis as well as on real and imaginaryparp and
their solution yields the estimated state kk =[I: $1 for
phase cp. Note also that this solution is approximate
since the approximate objective function (11)is used in
obtaining (15). To get the exact solution, one should use
the exact objective function (10) with the voltage V, be
written as a function of the branch currents. Then (15)
would include the extra partial terms a l z ( V ‘ ) / d I ,
a12(Vs)/aI, and dlV,I2/JI. However, these terms are
much smaller than the ones included in (15), i.e.,
ahr//a&[ = ahxl/dI,l = 1, and hence they are neglected.
The current magnitude measurements introduce
coupling terms between the real and imaginary parts of
the normal equations (15), since current measurement
functions are non-linear as indicated in (12). For
example, the current measurement I&, introduces the
following non-zero elements into the measurement
Jacobian H
Branch Current Based SE Method
1
Now using current measurements, actual and/or
equivalent, SE problem (1) needs to be solved to
estimate all the branch currents. However, note that
the objective function is separable on a phase basis,
since the measurement functions for measurements on
a given phase are functions of the branch currents of
that phase only. Hence we can decompose (1) into three
subproblems, one for each phase cp = 1,2,3. The current
only SE problem for phase cp is
where 6 1 ,=~Tan- ( I , I , ~l 4 1 , ~ )
is the phase angle of the branch current. The
contribution of this measurement to the gain matrix G
is
where we dropped the phase subscript cp for notational
clarity. The two summation terms in the objective
For the solution of the normal equations, a good
estimate of the phase angles for branch currents are
needed in the construction of the gain matrix G . To
b+1
-
Therefore, normal equations become nonlinear and
coupled, and thus the real and imaginary parts must be
solved together using (2). However, the phase
decoupling still holds and hence the gain matrices are
still much smaller and sparser than that of the nodevoltage-based case.
486
achieve this, the current measurements are excluded in
the first iteration (which corresponds to a flat voltage
start with $2 = 0 ) and then introduced in the successive
iterations. The gain matrix is reconstructed in the first
few iterations (first three in the implemented version)
to guarantee the convergence.
The current only SE problem (13) is solved together
with the voltage update procedure forward sweep to
obtain the SE solution. This iterative process involves
the following steps at each iteration k:
Branch Current based SE Algorithm, SE-br
the loop which can be assigned arbitrarily, and T is the
set containing all the branches in the loop. Eq.(18) can
be written in matrix form as
For the general case, let there be nt loops in a given
feeder. Then, the KVL equations for these loops can be
incorporated into the SE method by adding (19) to the
current only SE problem of (13) a s an equality
constraint, which yields
Step 1: Given the node voltages V k - l , convert power
measurements into equivalent current measurements
using (8).
Step 2 Use current measurements to obtain an estimate
of branch currents ?$=[iF,, i:,,] by solving the
currenf only SE problem (13) for each phase 9 = 1.2.3.
Step 3: G’ven the branch currents, update the node
voltages V 1, by the forward sweep procedure.
Step 4 Check for convergence; if two successive updates
of branch currents are less than a convergence tolerance
then stop, otherwise go to step 1.
SE-br is computationally efficient since the
problem is decoupled on a phase basis. Furthermore,
since the gain matrix G is independent of the line
impedance parameters, the method eliminates the
problems associated with line parameters in nodevoltage-based methods, such as ill-conditioning of the
gain matrix [9] and the leverage points [lo]. The
leverage points are known to hamper the bad data
detection and identification abilities of the SE method
which are very important in distribution system
applications.
SE for Weakly Meshed Feeders
Some distribution feeders serving high density load
areas operate with loops created by closing normallyopen tie-switches. Branch-current-based SE introduced
above can be extended for this “weakly meshed”
distribution feeders.
Existence of loops in the system does not affect the
measurement functions, and measurement functions
can be obtained the same way as in radial case.
Similarly the forward sweep process can still be applied
to determine the node voltages for a given set of branch
currents. However, for any loop created by the closure
of a tie-switch, KVL must be satisfied around the loop,
which can be written as
where 6, ~ [ l , - l ] depending on the direction of branch
current with respect to the reference direction taken for
Gm1
[
G,,
i:
GT1 AI:
-b:
GF2 AI:
- hi
] . . - -dy[ G
:
(21)
487
There is an additional incentive in reducing the
feeder models for SE because it is very difficult to
estimate the exact load distribution with only a limited
number of real-time measurements. Therefore, a
simple rule-based network reduction method is
developed to reduce the distribution feeders for SE. The
method exploits the mainly radial topology of the
feeders. The main idea is to divide a given feeder into
sections such that the total load in each section is small
and hence the loads in each section can be lumped at
the bordering nodes of the corresponding section. The
main steps of the feeder reduction method are as
follows:
Step 1 Pruning the Feeder LateraIs
A feeder usually has many laterals and sublaterals,
many of which are short and serves only a few small
loads. If the total load served by any of such laterals is
smaller than a specified value (50 kVA for example)
these loads are lumped at the node at which the lateral
is connected to the rest of the feeder and the lateral is
eliminated.
Step 2: Simplify the Feeder Sections
It is important to retain the general topology of a feeder
during reduction, hence first identify the following
points to be retained in the reduced model: branching
nodes, end nodes, nodes with large loads (for example,
loads which are 100 kVA or greater) and nodes which
have meters. Then calculate the total load in each
feeder section between the nodes to be retained and
divide these sections into subsections such that the
total load in each subsection is about the specified value
(50 kVA, for example). Finally, lump the total load in
each subsection at their terminal nodes.
Note that since the loads are grouped together for
network reduction, the weights for the equivalent loads
should be adjusted accordingly. Assuming that each
load S L ~is forecasted with a certain accuracy, ai, the
accuracy of the equivalent lumped load, aeq is the
summation of the accuricies of the loads lumped
together. Assuming also that the weights for the
individual loads are chosen as w i= 1/02 the weight for
the lumped load would be weq =l/aeq2:’
Using the reduced feeder model improves the
performance of the SE for the following reasons:
i) It is easier to forecast the load of a group of
distribution transformers than each individual
transformer load.
ii) The larger the ratio of real-time measurements to
forecasted loads the better the performance of the SE
1 23
44
+ meter
5
6 78
l;j
11
will be in terms of both correcting the errors in
forecasted loads and detecting and identifying the
bad data due to topology errors and device
malfunction, etc.
IV. OBSERVABILITYANALYSIS
For SE to be effective, a minimum amount of realtime data is necessary. Currently, power and voltage
measurements at the substation are the only real-time
measurements available at distribution level. Some
feeders may also have a few branch current or power
measurements. Since this real-time data is not enough
for SE, load forecast data is used a s pseudomeasurements, measurements that are less accurate
than the actual measurements, to supplement for the
real-time data. It is expected that more measurements
will be available as utilities install SCADA systems on
their distribution systems.
A numerical observability method is also
necessary to ensure that the gain matrix is non-singular
which is an indication that the data used is sufficient
for SE. This is especially necessary when there are
coupling terms in the gain matrix due to current
measurements and/or loops. Numerical observability
can be checked during the LDU factorization of the gain
mahix G [ll].
V. TEST RESULTS
The proposed three-phase state estimation method
is implemented on a DEC Workstation environment to
test its performance. The test results are summarized
below.
The test feeder is a 34 bus, 23 kV, 3-phase radial IEEE
test feeder [12]. A one-line diagram of the feeder is
given in Fig.2 with the nodes renumbered to make the
illustration of the results easier. The feeder is
predominantly three-phase with some single-phase
laterals and has both spot and distributed loads. For test
purposes, distributed line section loads are lumped
equally at terminal nodes of the line sections. The
nominal load data is taken as the actual load and the
power flow results are used to determine the correct
measurements for this load. The minimum voltage for
this loading is
Vmin = V ~ I , ,= 0.9402/ - 3.057
which indicates a heavy loading condition on the
feeder. The line data used is given in (131 with line r/x
ratios varying between 0.57 and 1.37.
12 14 1516 18 19
22
24
~
25 26
29
21
T
Figure 2: One-line diagram of the test feeder
T
488
To generate measurement data for testing purposes,
a measurement scheme consisting of four meters is
considered. The meters, marked as m0-m3 on Fig.2,
measure the voltage and power flow at the substation,
and currents on branches 18-19, 24-25 and 24-30. The
forecasted load data is created by perturbing the actual
load data by about 30%. The disturbance is done such
that the net load perturbation between the meters is
zero. This simulates the first measurement case Z1 in
which forecasted loads have an error margin of 30%
and are scaled according to measurements. Another
measurement case 2 2 is created by using the same
forecasted load data except that the value of the
capacitor at node 33 is reduced by 50 kVar/phase to
simulate a case with a bad measurement.
The weights for forecasted loads and selected
measurements are chosen by assuming that their
measurement accuracies are 1% and 10% of their
measured values, respectively. The voltage at the
substation is held at the assumed measured value of 1.0
p.u. The test runs are grouped into three cases as
detailed below.
Case 1:SE on a Radial Feeder
Two test runs were made with the proposed method
S E b r to obtain SE solutions for the two measurement
schemes Z1 and 22. The same tests are also repeated
with a fully-coupled node-voltage-based SE method,
SE-nd, presented in [3], to obtain a base case against
which the other results can be compared. These test
results are summarized in Tables 1 and 2. The test
results indicate that:
i) SE-br converges to the same point as SE-nd, as the
results from these two methods are very close to each
other. However, SE-br is computationally more
efficient. The small differences in the results are mainly
due to the different convergence criteria used in the
two methods. One of the reasons for the efficiency of
the S E b r is due to phase decoupling of the problem.
The other is the small number of iterations in SE-br
solutions which indicate the validity of the
approximations used in (11) by using equivalent
current measurements.
ii) These test results illustrate the main characteristics
of a SE solution based on a limited number of
branch
current measurements. SE in this case adjusts the loads
such that the estimated power flows a t the
measurement points match the measurements. This
can be deduced from the maximum residual values as
the maximum residuals are much smaller than the
actual p e r t u r b a t i o n s (actual maximum load
perturbation is about 11 kW). Nevertheless, residuals
from the bad data case 2 2 indicate that current
measurements can be effective in detecting bad data as
the maximum residual for reactive power, rq, is much
greater than the normal data case 21.
To investigate the impact of measurement error in
substation voltage on the results, test run for 2 2 is
repeated with substation voltage measurement of 1.01
p.u. using SE-br. The same test run is repeated using
SE-nd also, this time with two more voltage
measurements from nodes 18 and 24, which are
assumed to be correct. The minimum voltage estimates
from SE-br and SE-nd are 0.94871-3.12840 and 0.9380L
moo,
respectively. Comparing these results with the
actual minimum voltage given before indicates that
SE-nd has a better performance in correcting the errors
in voltage measurements than SE-br. However, in
SE-br, the error in substation voltage measurement
can be detected and corrected since an error in
substation voltage will result in a biased estimation
error on the other node voltages. This bias can be
calculated by comparing the estimated voltages with
measured values and it is about 0.009 for the example
considered.
Case 2: SE on a Reduced Feeder
The 34 node test feeder is reduced to a 17 node
feeder shown in Fig. 3 by using the proposed feeder
reduction method. First a power flow is run with the
nominal data to make sure that the power flow results
closely matches with that of the original system. Then
two SE-br test runs with data Z1 and 2 2 are repeated
with the reduced feeder and the results are
summarized in Tb1.3. Comparing Tbl.1 and Tb1.2 with
Tb1.3 indicates that:
i) The execution time is reduced considerably due to
network reduction. Network reduction reduces the
execution time of the S E p d as well, to 3.6 and 3.8 sec.
for Z1 and 22, respectively.
w
Table 1: Test Results based on 21
,
Mthd
SE n d
SE br
itr
9
4
Mthd
SE-nd
SE-br
itr
T(sec)
15.
2.7
J(x)
69.98
73.34
r r ( k W ) rOm(kVar)
2.25
2.13
I
I
2.19
2.27
meter
T
,
Table 3: Test Results from Reduced Feeder
I
10
4
J(x)
2064
2108
r,"x(kW) rOm(kVw)
5.48
5.39
I
I
30.05
30.30
T
Figure 3: Reduced Feeder
Table 2: Test Results based on 2 2
T(sec)
15.9
2.7
m
I
Data
Z1
22
I
I
I
itr
4
4
I T(sec) I
I
I
0.7
0.7
J(x)
I rDm(kW)
6380.0
3.64
7.04
I 185.0 I
I
I
I
rom(kVar)
4.9
36.9
489
ii) The performance of the SE with the reduced feeder is
improved in terms of noise filtering and bad data
detection, as the residuals in both normal load data, 21,
and bad data, 22, are higher than that of previous case.
This is mainly because more of the forecasted loads are
lumped together in this case and hence error in them
can be detected better than in previous case where the
loads were more distributed.
iii) The estimated voltage profile also closely matches
the results obtained from the full feeder model as the
minimum voltage estimates given in Tb1.4 indicate.
These results illustrate the effectiveness of the
proposed feeder reduction method as a means of
improving computational efficiency without sacrificing
accuracy.
Table 4: Minimum Estimated Volta es
Data full network
reduced network
0.9365/ -3.2777
0.9365/ -3.2605
Case 3 SE on a Meshed Feeder
To test the performance of SE-br on a weakly
meshed feeder, two loops are created on the test feeder
by adding two new branches between nodes 21-31 and
29-33, each 90000 feet long. The two test runs in Case 1
are repeated here for this modified feeder and the
results are summarized in Tb1.5 for the bad data case 2 2
only. Test results for the same test using SE-nd is given
in Tb1.5 also for comparison. Closeness of the results
from the two methods again indicate that the SE-br
converges to the same point as SE-nd. Note that SE-br
is still computationally more efficient than SE-nd
although its run-time increased compared to that of the
radial case.
Table 5: Test Results for the Meshed Feeder with 2 2
V. CONCLUSIONS
A branch-current-based three-phase SE method is
developed for distribution systems. Test results indicate
that the method has superior performance compared to
the conventional node-voltage-based methods both in
terms of computation speed and memory requirements. The method is specially tailored for weakly
meshed distribution systems which are radial or have a
few loops. Another advantage of the method is its
insensitivity to line parameters, which improves both
its convergence and bad data handling performance.
Finally, it is shown that the proposed feeder reduction
method is very effective in improving the computation
speed and filtering properties of the method without
sacrificing accuracy. These features of the proposed
method makes it very suitable for practical applications.
Acknowledgments
This research has been supported by EPRC, NCSU
and Pacific Gas and Electric Company.
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[lo] Milli L., Phaniraj V., and Rousseeuw P.J. Least
Median of Squares Estimation in Power Systems,
IEEE Trans. on PWRS, May 1991, pp. 511-523.
Ill] Clements, K.A. Observability Methods and
Optimal Meter Placement, Int. Journal of Electrical
Power and Energy Systems, April 1990, pp. 88-93.
1121 IEEE W.G., Radial Distribution Test Feeders. IEEE
Trans. on Power Systems, Aug. 1991, pp. 975-985.
1131 Goswami, S.K. and S.K. Basu, Direct Solution of
Distribufion Systems. IEE Proceedings-C, Jan. 1991.
pp. 78-88,
__-_
Mesut E. Baran received his B.S. and M.S. degrees in
Electrical Engineering from Middle East Technical
University, Turkey and his Ph.D. from the University
of California, Berkeley. He is currently an Assistant
Professor at North Carolina State University.
A r t h u r W. K e l l e y received his B.S.E. from Duke
University, Durham, North Carolina. He continued at
Duke as a James B. Duke Fellow and received his M.S.,
and Ph.D. degrees in 1981, and 1984, respectively. He is
currently an Associate Professor at North Carolina State
University.
490
Discussion
Fan Zhang (AB6 Automated Distribution Division 1021 Main Campus
Drive Raleigh NC 27606-5202) The authors are to be congratulated for
combining the weekly meshed topology feature of distribution systems into
the state estimation of distribution systems This was achieved by using
branch current as state variables instead of conventional node voltage
The authors’ opinion on the following three issues are appreciated
a) For a radial system even though the gain matrix G is independent of the
line impedance parameters the leverage points can not be eliminated
Their existence depends on the measurement schemes and system
topology
b) For a radial system without injection measurements, can the gain matrix
be decoupled based on lines? For a weekly meshed system with limited
injection measurements, can the state variables be grouped into several
islands such that the gain matrix can be decoupled based on these
islands?
c) For a weekly meshed system the number of spanning tree (or the tree
with measiirement assignment) is very limited It seems to me that the
topological melhods are better candidates than the numerical methods
in performing observability analysts
Manusenpl received August 15, 1994
R. L. Lugtu, I. Roytelman (Siemens Energy & Automation, Inc., Empros Power System Control, Plymouth, MN):
The authors propose a new and interesting approach to
distribution system state estimation (SE). The method is
not a simple extension of the conventional SE for transmission systems as proposed the same authors’ previous
paper [ I ] and other recently published papers [2, 31. We
support the understanding that distribution systems have
fundamental characteristics which require exploration of
methods other than conventional SE techniques. The
number of recent papers devoted to the distribution system SE shows the timeliness of this topic.
Choosing the branch currents as the state variables is a
novel approach to achieve separable model of the various
phases. Although the authors present a test case, more
extensive tests, however, need to be performed to get to
the same level of confidence and robustness characteristic
of today’s estimators based on node voltage formulation.
However. the results presented by the authors look very
promising.
Another idea used in the paper is network reduction.
Without real time measurements, it is more reliable to
forecast the load of a group of distribution transformers
than for each individual transformer. The proposed pruning of feeder laterals looks extremely attractive especially
for American type distribution systems where most of the
transformers have power less than 50 kVA.
We would appreciate the authors’ comments on the
following points
1. BRANCH
CURRENT
MAGNITUDE
MEASUREMENTS
We would like to ask the authors what their reasons for
not including the branch current magnitude measure-
ments in the first iteration. It seems that with a large
number of current magnitude measurements, including a
preliminary current angle estimation (unity power factor)
may improve performance. Another issue is the weight
coefficient values that are used. The current magnitude is
measured and therefore is more reliable. The same accuracy or weight cannot be assigned to the current angle or
the real and imaginary pseudo current measurements.
How are the weights of the latter chosen such that the
accuracy of the other real telemetry are not compromised
and yet be able to take advantage of the reliability of the
current magnitude telemetry itself that is being replaced?
2. VOLTAGE
MEASUREMENTS
Voltage measurements are widely used in transmission
state estimators. Although the distribution system does
not have an overwhelming number of these measurements, they are more accurate than the other available
real or pseudo measurements. The significant impact of
voltage measurements appear to be corroborated by one
of the case results presented. We feel that the incorporation of voltage measurements is a necessary feature and
the idea of using voltage measurements as a post-solution
check of the consistency of the results does not look very
attractive. Do the authors have any ideas on how to
incorporate the effects of voltage measurements and still
retain the benefits of the current branch formulation?
3. Wmiiirs OF PSEUDO
MEASURMENTS
When real and pseudo measurements are mixed within
the same estimation, the question arises as to how large
the ratio of the measurement weights should be between
real and pseudo such that the effect of the real measurements is not compromised and yet would not cause convergence problems. Did the authors do experimental runs
to test how the method performs with various ratios?
Another point is that with a great majority of the measurements being pseudo measurements, was there any
attempt to classify some pseudo measurements as more
“reliable” than others? For example, voltage and power
factor are known to almost lie within a certain range. The
same cannot be said of the kw or kva historical values.
4. TESTRESULTS
Did the authors test the method using a subnetwork
that includes the substation transformer and all the connected feeders? The reason is that there are usually
measurements either at the high or low voltage side of the
substation transformer(s) that can add substantially to the
redundancy of the estimation model. It would also be very
interesting to see how the method performs for a network
(parallel) which is supplied from different two different
substation transformers each with a voltage measurement.
As a conclusion, we would like to congratulate the
authors for their novel contribution to a very timely
function which is needed as Distrihution Management
Systems’ evolve.
49 1
References
M. E. Baran, A. W. Kelley, “State Estimation for
Real-Time Monitoring of Distribution Systems,” presented at IEEE PES 1994 Winter Meeting, paper No
94 WM 235-2 PWRS.
C . N. Lu, J. H. Teng, W.-H. E. Liu, “Distribution
System State Estimation,” presented at IEEE PES
1994 Winter Meeting, paper No 94 WM 098-4 PWRS.
C . W. Hansen, A. S. Debs, “Power System State
Estimation Using Three-phase Models,” presented at
IEEE 1994 Summer Meeting, paper 94 SM 601-5
PWRS.
Manuscript received August 22, 1994.
M.E. Baran and A.W. Kelley: We thank the discussors
very much for their interest in and for their very valuable
comments on the paper. We will respond to each
discussor separately.
chosen by assuming that the loads are forecasted with a
certain accuracy. With these weights the loads seem to be
adjusted such that the estimated measurements would
match the actual measurements. The “reliability” or what
we call the accuracy information about the other pseudo
measurements can also be reflected in the weights
associated with these measurements.
(4) We plan to d o the tests the reviewers propose and
publish the results, as we agree that they will contribute
more towards illustrating the robustness of the proposed
method.
Fan Zhang: Our response to Dr. Zhang’s questions are as
follows.
(a) In the proposed method the measurement Jacobian, H,
is independent of line parameters (non-zero elements are
+I, Sin$, Cos$) for radial case. Furthermore, the number
of branches connected to a node is low. These conditions
imp1 that there won’t be an levera e points [Cl].
(b) $he gain matrix wiE be Jecoupled if all the
measurements are of line flow type. When there are
measurement islands in a system, the gain matrix can be
partitioned according to these islands and there won’t be
any coupling between the partitions. However, the part of
the gain matrix corresponding to a measurement island
will be phase coupled if the island contains a looping
branch.
(c) As the discussor suggests, topological observability will
be easy to apply in determining the “topological
observability” of the system. However, this observability
will not guarantee that the system will be numerically
observable (i.e., the gain matrix will be nonsingular),
when there are current measurements and/or the system
contains loops.
We thank again the discussors for their very
valuable comments.
R.L. Lugtu and I. Roytelman: The discussors’questions are
very insightful and our response to them are as follows:
(1)The main reason for not including the branch currents
in the first iteration is the difficulty in initial estimation of
their phase angles. The discussors’ suggestion may indeed
improve the convergence. The selection of weights for the
equivalent real and imaginary current measurements
should be the same as that of power measurements, since
the equivalent currents do not change the solution
appreciably.
[Cl]Milli L., Phaniraj V., and Rousseeuw P.J. Least
(2) The justification for not including voltage measureMedian of Squares E s t i i i i n t i o n in Power Systems,
ments is based on the observation, as made by others, that
IEEE Trans. on PWRS, May 1991, pp. 511-523.
the impact of voltage measurements on the state
estimation results are negligible.
(3) The weights for the pseudo load measurements are Manuscript received October 26, 1994.