-
Published in IET Generation, Transmission &
DistributionReceived on 19th September 2008Revised on 22nd December
2008doi: 10.1049/iet-gtd.2008.0485
ISSN 1751-8687
Choice of estimator for distribution systemstate estimationR.
Singh1 B.C. Pal1 R.A. Jabr21Department of Electrical and Electronic
Engineering, Imperial College, London, UK2Electrical Computer and
Communication Engineering Department, Notre Dame University, Zouk
Mosbeh, LebanonE-mail: [email protected]
Abstract: In this study, a statistical framework is introduced
to assess the suitability of various state estimation(SE)
methodologies for the purpose of distribution system state
estimation (DSSE). The existing algorithmsadopted in the
transmission system SE are recongured for the distribution system.
The performance of threeSE algorithms has been examined and
discussed in standard 12-bus and 95-bus UK-GDS network models.
Nomenclaturem, n number of measurements and state variables
xt, x true state and estimated state vectors, respectively(n
1)
Px, Px numerically computed and estimated errorcovariance
matrices, respectively (n n)
E[.] expectation operator
e normalised state error squared variable
z measurement vector (m 1)h(x) expectation of measurement vector
(m 1)szi standard deviation of the ith measurement
Rz measurement error covariance matrix (m m)ez measurement error
vector (m 1)ri normalised residual of ith measurement
1 IntroductionDeregulation of power system and the introduction
ofdistributed generation (DG) to distribution networks
haschallenged the operational philosophy of the
distributionsystems. The passive nature of the network can
onlyaccommodate restricted amount of DG capacity. This meansthat
signicant network reinforcement will be necessary toaccommodate DG
and load growth in the future. Analternative would be to change the
approach to networkoperation such as introducing control to
distribution network
operation. A range of technology innovations is needed tochange
the way distribution systems operate. The innovationsmust pin down
on new architecture for distribution networkcontrol centre with
performance critical software functionslike state estimation (SE),
optimal power ow (OPF) andnetwork-specic sensor placement and
integration.
In transmission systems, SE is a fairly routine task and a
hostof established methodologies exist [1]. These cannot simply
betransferred to distribution systems because the planning,
designand operation philosophy of distribution networks are
differentfrom those in the transmission networks. The
distributionnetwork topology and characteristics are different and
mostimportantly the amount of available network measurements isvery
limited. The SE methodologies adopted in transmissionsystems start
showing their limitations when exposed to thespecics of
distribution networks [2].
Furthermore, the potential benets of using SE technologiesin
distribution network control have not been explored mainlybecause
of the absence of adequate network measurementsand also the lack of
rigorous methodology and tools thatcould be applied on restricted
measurements. Thedevelopment of new distribution system state
estimation(DSSE) is a challenging task as the tools to evaluate
thequality of SE must consider a number of issues relating
tomeasurement types, locations and numbers.
Methodologies on which such tools could be built are
notavailable at present. However, some interesting research has
666 IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 7, pp.
666678
& The Institution of Engineering and Technology 2009 doi:
10.1049/iet-gtd.2008.0485
www.ietdl.org
emaddoxWATERMARK
-
been done in DSSE [310]. Lu et al. [3] propose a three-phase
DSSE algorithm. The algorithm uses a current-based formulation of
the weighted least-squares (WLS)method in which the power
measurements, currentmeasurements and voltage measurements are
converted totheir equivalent currents, and the Jacobian terms
areconstant and equal to the admittance matrix elements.
Theobservability analysis of the proposed distribution system
isalso discussed. Lin and Teng [4] have proposed a new
fastdecoupled state estimator with equality constraints.
Theproposed method is based on the equivalent currentmeasurement in
rectangular coordinates. Baran and Kelley[5] have introduced a
computationally efcient algorithmbased on branch currents as state
variables. The method isdemonstrated to work well in radial and
weakly meshedsystems. This concept is further rened by Wang
andSchulz [6] and they have presented a revised branchcurrent-based
DSSE algorithm. In this algorithm, the loadestimated at every node
from an automated metre readingsystem is used as a pseudo
measurement. Li [7] haspresented a distribution system state
estimator based onWLS approach and three-phase modelling
techniques. Lihas also demonstrated the impact of the
measurementplacement and measurement accuracy on the
estimatedresults. A rule-based approach for measurement placementis
presented by Baran et al. [8]. Ghosh et al. [9] havepresented an
alternative approach to DSSE using aprobabilistic extension of the
radial load ow algorithmtreating the real measurements as solution
constraints. Thealgorithm that accounts for non-normally
distributed loads,incorporates the concept of load diversity and
can interactwith a load allocation routine. The eld results
arediscussed in [10].
The DSSE literature is either based on the probabilisticload ow
or direct adaptation of transmission system SEalgorithms
(particularly WLS). The issue of measurementinadequacy is addressed
through pseudo measurements thatare stochastic in nature. However,
the performance of theSE algorithms under the stochastic behaviour
of pseudomeasurements is not addressed in the DSSE literature.
The work presented in this paper investigates the
existingtransmission system SE techniques and algorithms
andassesses their suitability to the DSSE problem. The
selectedalgorithms are tested on the 12-bus and 95-bus
UK-GDSnetwork models against some statistical measures like
bias,consistency and overall quality of the estimates. Unlike
manyother distribution systems, the UK distribution network
isfairly balanced and that has prompted us to go with a
single-phase approach, although the method is generic.Furthermore,
the statistical measures utilised in this papermainly depend on the
probability distribution of themeasurements and not on the line
model of the network.Following this introduction, a theoretical
framework forthe statistical measures is established in Section 2.
Theconsistency and the quality of the estimates utilise
theasymptotic state error covariance matrix. The various SE
techniques along with the details of their state errorcovariance
matrices are discussed in Section 3. The efcacyof the algorithms is
examined on standard test systems anddiscussed in Section 4.
2 Statistical measuresIn distribution systems, measurements are
predominantlyof pseudo type, which are statistical in nature, so
theperformance of a state estimator should be based on
somestatistical measures. Various statistical measures such asbias,
consistency and quality have been adopted forassessing the
effectiveness of SE in other technology areassuch as target
tracking [11]. We explore these for theDSSE applications. Briey we
describe the statisticalmeasures as follows.
2.1 Bias
A state estimator is said to be unbiased if the expected valueof
error in the state estimate is zero. Mathematically anunbiased
estimator can be dened as
E[(xt x^)] 0 (1)
2.2 Consistency
If the error in an estimate statistically corresponds to
thecorresponding covariance matrix then the estimate (andhence the
technique generating this estimate) is said to beconsistent. One
measure of consistency is the normalisedstate error squared
variable
e (xt x^)TP^1x (xt x^) (2)
where, Px, denotes the estimated state error covariance
matrix.
For the estimator to be consistent e should be within
itscondence bounds, which can be obtained from the
errorstatistics.
2.2.1 Choice of condence regions: In the univariatecase when the
estimation error is represented by a normaldistribution with zero
mean and known variance, one canuse the tables of normal
distribution to compute thecondence intervals. However, in the
multivariate case whenthe estimation error is represented by a
normal distributionwith zero mean vector and known covariance
matrix, suchcondence intervals are difcult to compute because
tablesare available only for the bivariate case. Alternatively,
onecould setup limits for each component on the basis
ofdistribution, but this procedure has the disadvantages thatthe
choice of limit is somewhat arbitrary and in some casesleads to
tests that may be poor against some alternatives.Moreover, such
limits are difcult to compute. Theprocedure given below, which is
based on x2-statistics, canbe easily computed and applied in the
multivariate case.Furthermore, it can be theoretically justied
based on thefollowing lemma. The proof the lemma can be found in
[12].
IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 7, pp. 666678
667doi: 10.1049/iet-gtd.2008.0485 & The Institution of
Engineering and Technology 2009
www.ietdl.org
emaddoxWATERMARK
-
Lemma 1: If an n-component vector v is distributedaccording to N
(0, T ) (non-singular), then vTT1v isdistributed according to
x2-distribution with n degrees offreedom.
2.2.2 x2-statistics: It can be shown that if the errorsin
measurements are normally distributed, the SE errorcorresponding to
these measurements will be normallydistributed with zero mean
vector and covariance matrixgiven by E[(xt x^)(xt x^)T]. By
utilising this fact andLemma 1, the normalised squared error e (2)
should followa x2-distribution with n degrees of freedom for a
consistentestimator, where n is the number of states. In other
wordsfor the estimator to be consistent, e should lie within
itscondence bounds that can be obtained from the standardx2-table
for a chosen condence level a. Lower and upperbounds for this
condence level can be given byx2n((1 a)=2) and x2n((1 a)=2),
respectively. In statistics,a 95% condence interval is considered
to be adequate.
2.2.3 x2-test over Monte Carlo simulations: Inpractice,
statistical tests are performed using a number ofMonte Carlo
simulations. Consider the system has nnumber of states and M is the
number of Monte Carlosimulations, then the normalised squared error
follows ax2-distribution withMn degrees of freedom.
Mathematically
E[e] x2Mn(a)M
(3)
For large number of Monte Carlo runs x2Mn(a) Mn, whichresults
in
E[e] n (4)
Hence the mean of e should approach to the number of stateswith
the increase in the number of simulations.
2.3 Quality
Quality of an estimate is inversely related to its variance.
Forthe multivariate case, the square root of the determinant ofthe
error covariance matrix measures the volume of 12 sellipsoid and is
used here to quantify the total variance ofan estimate. Hence, the
quality of the estimate can bedened as
Qdet log1
det (Px)p
!(5)
Sometimes, in large networks, it becomes difcult tocompute the
determinant of the error covariance matrixnumerically because of
precision limits of the solver. In thissituation, an alternate way
to dene the quality is to use thetrace of the error covariance
matrix. However, this ignoresthe off-diagonal information. The
quality as function of the
trace of the error covariance matrix can be written as
Qtrace log1
tr(Px)
(6)
3 State estimation techniquesVarious algorithms have been
suggested for transmission systemSE [1]. All these algorithms work
well in transmission systemsbecause there is high redundancy in the
measurements.However, in distribution systems, because of sparsity
ofmeasurements, there is less or no redundancy in themeasurements.
Hence, when these algorithms are exposed todistribution systems
they start showing their limitations. Forexample, in transmission
systems, weighted least absolutevalue estimator (WLAV) eliminates
bad data out of redundantmeasurements, but in distribution systems
it fails to workbecause it treats every pseudo measurement as bad
data andthere is no redundancy to eliminate these
pseudomeasurements.
This section briey explains the most common SEtechniques to
examine their suitability for the DSSE problemunder stochastic
behaviour of the pseudo measurements andlimited or no redundancy.
All these techniques use thefollowing measurement model.
3.1 Measurement model
z h(x) ez (7)
where, ez N (0, Rz) is zero mean Gaussian noise with
errorcovariance matrix Rz( diag{s2z1, s2z2, . . . , s2zm}). We
denethe normalised residual of ith measurement ri as
ri zi hi(x)
szi(8)
where, ri N (0, 1). The class of estimators discussed in
thissection are based on maximum likelihood theory. Theyrely on a
priori knowledge of the distribution of themeasurement error
(Gaussian in this case, with zero meanand known covariance). A
generalised estimation problemseeks to minimise the following
objective
J Xmi1
r(ri) (9)
The different estimators can be characterised based on thechoice
of the r function.
3.2 Weighted least squares estimation
WLS is a quadratic form of the maximum likelihoodestimation
problem. The WLS problem can be stated asthe minimisation of the
following objective function
12[z h(x)]TR1z [z h(x)] (10)
668 IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 7, pp.
666678
& The Institution of Engineering and Technology 2009 doi:
10.1049/iet-gtd.2008.0485
www.ietdl.org
emaddoxWATERMARK
-
The above objective takes the form given in (9) for
r(ri) 12r2i (11)
An estimate of state was obtained iteratively using theNewton
method according to
x^k1 x^k (HT(x^k)R1z H (x^k))1HT(x^k)R1z [z h(x^k)](12)
where
H (x^k) @h(x)@x
xx^k
(13)
3.3 Weighted least absolute valueestimator
WLAV estimator is based on the minimisation of the L1norm of
weighted measurement residual, and can beexpressed as
jjR1=2z [z h(x)]jj1 (14)
which is equivalent to (9) when
r(ri) jrij (15)
Existing techniques use linear programming or interior
pointmethods to solve this problem. In this paper we have used
aprimal dual interior point method [13].
3.4 Schweppe Huber generalised M(SHGM) estimator
This estimator combines both WLS and WLAV estimators.The r
function for SHGM estimator is given by
r(ri) 12r2i if jrij avi
avijrij 12a2v2i otherwise
8>: (16)
The performance of this estimator highly depends upon theweight
factor vi and tuning parameter a. In this paper, thesolution to
this problem was obtained using iteratively re-weighted least
squares (IRLS) method [1]. The parametera 1.5 was used in
simulations.3.4.1 State error covariance matrix: An estimate ofthe
asymptotic covariance matrix at convergence can beexpressed as [14,
15]
P^x a(HT(x^)R1z H (x^))1 (17)
where x^ limk x^k. The value of a depends on the choice of
the estimator. An expression for a is given by [14]
a E[c2(r)]
(E[c0(r)])2(18)
where c(r) @r(r)=@r and c0(r) @c(r)=@r.
The numerical computation of a for various estimatorsis given in
the Appendix. Table 1 summarises thevarious estimators used in this
paper. Table 1 indicates atypical value of a for SHGM considering
avi 1:5. Sincethe IRLS method is used for SHGM, the value ofa
changes during the estimation process depending uponthe weight
vi.
4 Case studyThe algorithms discussed in the previous section
were appliedon a 12-bus radial distribution network model and on a
partof the UK generic distribution system model (95-bus UK-GDS).
Figs. 1 and 2 show the schematic of the testsystems. Network and
load data for these networks can befound in [16] and [17],
respectively.
4.1 State variables
The bus voltage magnitudes and angles were considered asstate
variables except at the reference bus (Bus #1) forwhich the bus
angle was assumed to be zero. Hence, thenumber of states to be
evaluated was 23 and 189 for the12-bus test system and the UK-GDS,
respectively.
4.2 Measurements
It was assumed that the errors associated with themeasurements
are independent identically distributed(i.i.d.). Three types of
measurements were taken intoconsideration. The telemetered
measurements were utilisedas real measurements. Zero injections
with a very low
Table 1 State estimators: summary
Solution for x Asymptotic error covariance Px
WLS Newton (HT(x^)R1z H(x^))1
WLAV PDIP p2(HT(x^)R1z H(x^))
1
SHGM IRLS 1.037(HT(x^)R1z H(x^))1 a
aa 1.5, vi 1
Figure 1 Twelve-bus test system
IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 7, pp. 666678
669doi: 10.1049/iet-gtd.2008.0485 & The Institution of
Engineering and Technology 2009
www.ietdl.org
emaddoxWATERMARK
-
variance (1028) were modelled as virtual measurements.Loads were
modelled as pseudo measurements. Variousscenarios considering the
errors in real measurements as 1and 3%, whereas 20 and 50% in
pseudo measurementswere examined. The range of error in pseudo
measurementswas chosen on the basis of errors in load estimates
ofvarious classes of customers, like industrial, domestic
andcommercial. The loads of the industrial customers can
beestimated more accurately than the domestic andcommercial, thus
they have less error. On the other hand,loads of domestic customers
are difcult to estimate, hencethey have large error. The error in
commercial loadestimates lies between the two. It was also taken
intoconsideration that with this choice of range, the maximumdemand
limits at various buses are not violated and thecondition of linear
approximation is valid. The mean valuefor these measurements was
obtained using distributionsystem load ow. Table 2 summarises the
measurementsand their redundancy level for the two test network
models.
4.3 Measurement variance
A +3s deviation around the mean covers more than 99.7%area of
the Gaussian curve. Hence, for a given % of maximumerror about mean
mzi , the standard deviation of error wascomputed as follows
szi mzi %error3 100 (19)
The square of standard deviation gives the variance of
themeasurement.
4.4 Simulation results
The performance of the estimators was evaluated for thefollowing
cases:
Case 1: Error in real measurement 1% and pseudomeasurement
20%
Figure 2 UK-GDS: 95-bus test system
Table 2 Measurements used in study
Test system Real measurements (mr) Virtual &
pseudomeasurements (mp)
Redundency(mrmp)/n
twelve-bus 3(V1, P12, Q12) 22 (loads only no zeroinjections)
25
23 1:09
UK-GDS (a) limitedredundancy
5(V1, P12, Q12, P185, Q185) 188 (loads and zeroinjections)
193
189 1:02
UK-GDS (b)increasedredundancy
21(V1, V18, V19, V20, V21, V95, P12, Q12,P185, Q185, P1819,
Q1819, P8295, Q8295P1517, Q1517, P3435, Q3435, d19, d20, d21)
188 (loads and zeroinjections)
209
189 1:11
670 IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 7, pp.
666678
& The Institution of Engineering and Technology 2009 doi:
10.1049/iet-gtd.2008.0485
www.ietdl.org
emaddoxWATERMARK
-
Case 2: Error in real measurement 1% and pseudomeasurement
50%
Case 3: Error in real measurement 3% and pseudomeasurement
20%
Case 4: Error in real measurement 3% and pseudomeasurement
50%
4.4.1 Twelve-bus system: In the 12-bus test system, thevoltage
magnitude measurement at bus #1 and power owmeasurement in line #12
were considered as realmeasurement. Fig. 3 shows the variation of
the expected valueof the normalised state error squared with
various MonteCarlo steps for the 12-bus distribution system. It is
clear fromthe gure that as the number of Monte Carlo steps
increases,the expected value of normalised state error square
variableapproaches the number of states, which agrees with (4).
Alsoafter 400 Monte Carlo steps, the error in E[e] is within 1%
ofthe number of states. Hence, we chose 400 Monte Carlo stepsfor
the simulations. A larger number of Monte Carlo stepsgives slightly
better results but it increases the computation time.
Fig. 4 shows the error plots with the number of simulationsfor
the three estimators. The plots shown are for the worstcase
scenario (Case 4), that is, the error associated with
realmeasurements is 3% and that with pseudo measurements is50%. The
estimation errors for all the states are displayedin Fig. 4,
however, they are indistinguishable because of theoverlaps. It is
evident from the gure that the error variesabout zero mean. This
indicates that all the threeestimators are unbiased. It was also
found that for all othercases, the three estimators were
unbiased.
Figs. 58 show the consistency plots for the estimators forcases
14. A 95% condence level was used to dene thecondence bounds. It
was found that WLS shows consistentresults in all test cases. On
the other hand, WLAV is
inconsistent in all the cases. It is interesting to note
thatSHGM is inconsistent for small errors in pseudomeasurements and
consistent for large errors in pseudomeasurements. The reason is
that the measurement setconsidered for study is predominantly
comprised of thepseudo measurements, and large error in
pseudomeasurements increases the measurement variance (19). Alsothe
computation of variance in (19) is based on the maximumerror. This
results in low normalised residual (jrij) for pseudomeasurements.
Owing to this fact the normalised residualbecomes less than the
cutoff value avi (16), and the estimatorbehaves like WLS. However,
this is not always true.Whenever the normalised residual exceeds
the cutoff value,the estimator becomes inconsistent. It will be
shown that forthe 95-bus UK-GDS system, SHGM becomes
inconsistentfor these cases of large errors too.
Table 3 shows the performance summary of the 12-bus testsystem.
Two types of qualities are shown. As expected, the
Figure 3 Variation of E[e] with different Monte Carlo steps
Figure 4 Twelve-bus system estimation error plot for allstate
variables: error in true measurements 3%, error inpseudo
measurements 50%
IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 7, pp. 666678
671doi: 10.1049/iet-gtd.2008.0485 & The Institution of
Engineering and Technology 2009
www.ietdl.org
emaddoxWATERMARK
-
quality of the estimates decreases with the increase in theerror
in measurements. This decrease is signicant with theincrease in
error in the real measurements as compared tothe pseudo
measurements.
4.4.2 Ninety-ve-bus UK-GDS: The performance ofthe estimators was
also evaluated on the 95-bus test systemmodel for all the test
cases analysed in the 12-bus testsystem. It was observed that in
the 95-bus test system also,400 Monte Carlo steps are sufcient to
bring down theerror in E[e] within 1% of the number of states.
Thefollowing two cases were considered:
(a) Limited redundancy
In this case, the real measurements were considered tobe
available at the main substation. Hence, the voltagemagnitude
measurement at bus #1 and power owmeasurements in lines #12 and
#185 were taken as realmeasurements. It was observed that in the
95-bus testsystem all the estimators were unbiased. However,
only
WLS was found to be consistent in all the test cases.Hence, the
consistency plots of WLS in all four test casesare displayed in
Fig. 9. The consistency plot for SHGMisalso shown in Fig. 10 for
the test Case 2. It is clear fromFig. 10 that the SHGM which was
consistent in Case 2 inthe 12-bus system no longer remains
consistent in largersystems.
(a) Increased redundancy
In this case, the redundancy was increased by placingthe
measurements at DG locations rst and thenmeasurements were placed
at optimal locations. Theoptimality criterion and details of the
measurementplacement appear in [18]. Furthermore, the
phasormeasurements were also deployed at optimally selectedbuses.
The real measurement set in this study consists ofthe following
measurements:
1. Voltage measurements at buses #1, #18, #19, #20, #21and
#95
Figure 5 Twelve-bus system consistency plot: error in
truemeasurements 1%, error in pseudo measurements 20%
Figure 6 Twelve-bus system consistency plot: error in
truemeasurements 1%, error in pseudo measurements 50%
672 IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 7, pp.
666678
& The Institution of Engineering and Technology 2009 doi:
10.1049/iet-gtd.2008.0485
www.ietdl.org
emaddoxWATERMARK
-
2. Line ow measurements in lines #12, #185, #8295,#1819, #1517
and #3435
3. Phasor measurements at buses #19, #20 and #21
The consistency plots forWLS and SHGMwith increasedredundancy
are shown in Figs. 11 and 12, respectively. TheWLS shows the
consistent performance whereas theSHGM shows the inconsistency in
all the simulated cases.
Figure 7 Twelve-bus system consistency plot: error in
truemeasurements 3%, error in pseudo measurements 20% Figure 8
Twelve-bus system consistency plot: error in true
measurements 3%, error in pseudo measurements 50%
Table 3 Twelve-bus system performance summary
Estimator Real 1%, pseudo 20% Real 1%, pseudo 50%
Bias Consistent/E[e] Quality tr/det Bias Consistent/E[e] Quality
tr/det
WLS unbiased consistent/23.13 4.08/199.24 unbiased
consistent/23.25 3.7/179.01
WLAV unbiased inconsisten/136.7 3.17/191.46 unbiased
inconsistent/80.67 2.26/173.98
SHGM unbiased inconsistent/1033.1 3.86/193.1 unbiased
consistent/26.37 3.7/178.28
Real 3%, pseudo 20% Real 3%, pseudo 50%
WLS unbiased consistent/23.85 1.89/198.4 unbiased
consistent/22.97 1.81/178.02
WLAV unbiased inconsistent/130.91 1.68/190.53 unbiased
inconsistent/72.81 1.45/173.07
SHGM unbiased inconsistent/953.8 1.86/192.37 unbiased
consistent/24.1 1.80/177.63
IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 7, pp. 666678
673doi: 10.1049/iet-gtd.2008.0485 & The Institution of
Engineering and Technology 2009
www.ietdl.org
emaddoxWATERMARK
-
A very high degree of inconsistency was observed in WLAV,which
is difcult to show graphically.
The performance summaries for both cases are shown inTables 4(a)
and (b). In both the cases, the quality denedin (5) gives numerical
instability in computations, hence itdoes not appear in the tables.
Furthermore, the quality forWLAV estimator is inconsistent and
shows negative values.This is because of very high variance of
state estimates thatare unacceptable for SE. In WLS and SHGM, as
expectedthe qualities decrease with increase in errors in real
and
pseudo measurements. The value of E[e] in the case ofSHGM does
not converge to the number of states (i.e.189), which numerically
veries its inconsistency.
It is also important to note that with limited redundancy
thetrace qualities dened in (6) are close for both WLS andSHGM in
Case 2 and Case 4. This gives the impressionthat SHGM should be
consistent for these cases. Since tracecaptures the diagonal
information of the error covariancematrix, it can be attributed
that inconsistency in SHGM ismainly due to off-diagonal elements.
In case of increasedredundancy there is signicant difference in the
qualities ofWLS and SHGM in all the test cases. The quality of
WLSis better than the quality of SHGM.
In all the simulated cases only WLS satises the threestatistical
criteria (bias, consistency and quality) under theassumption of
normal distribution of measurement errors.It can be concluded that
the WLS is a suitable solver forthe DSSE problem.
4.5 Comments on error distribution andchoice of solver
The statistical criteria discussed in this paper depend on
thecharacteristics of the distribution of measurement errors.
Theresults presented are based on the assumption that
themeasurement errors are normally distributed. Under
thisassumption, the WLS satises the statistical criteria andhence
was found to be the suitable solver for the SE.However, this may
not be true if the measurement errorsare not normally distributed.
For instance if the errorsfollow the Laplace distribution [19], the
WLAV estimatorgives better performance than WLS and SHGM. Thereason
for this is that the WLAV is consistent with theLaplace
distribution and maximisation of log-likelihood ofthe Laplace
density function results in the WLAVformulation. Hence, depending
on the distribution of theerrors, the corresponding statistical
criterion discussed inSubsection 2.2 can be modied in order to
identify theconsistent solver for that distribution.
Figure 9 Ninety-ve-bus system consistency plot withlimited
redundancy: WLS shows consistency in all test cases
Figure 10 Ninety-ve-bus system consistency plot withlimited
redundancy: error in true measurements 1%,error in pseudo
measurements 50%
674 IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 7, pp.
666678
& The Institution of Engineering and Technology 2009 doi:
10.1049/iet-gtd.2008.0485
www.ietdl.org
emaddoxWATERMARK
-
In reality, different probabilistic load distributions exist in
thedistribution networks and no standard distribution can t all
ofthem. Furthermore, the large size of the distribution
networkhaving various probability distributions at different
busesmakes accommodating them in a single state estimator
impractical. A more practical approach is to model the
actualprobability distributions as a mixture of several
Gaussiandistributions (Fig. 13) and apply the WLS state
estimatorwhich is consistent with the normal distribution.
Thisrequires the modelling of the distribution of errors
through
Figure 11 Ninety-ve-bus system consistency plot withincreased
redundancy: WLS shows consistency in all thetest cases
Figure 12 Ninety-ve-bus system consistency plot withincreased
redundancy: SHGM shows inconsistency in allthe test cases
IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 7, pp. 666678
675doi: 10.1049/iet-gtd.2008.0485 & The Institution of
Engineering and Technology 2009
www.ietdl.org
emaddoxWATERMARK
-
Gaussian mixture model (GMM) [2022]. As shown inFig. 13, the GMM
represents an arbitrary distribution as aweighted combination of
several Gaussian components.Mathematically, a GMM having Mc mixture
componentswith mean and variance of kth component as mk and s
2k can
be written as
f (x) XMck1
wkN (mk, s2k )(x) andXMck1
wk 1 (20)
The expectation maximisation algorithm [2022] is used toobtain
the parameters (wk, mk, s
2k ) of the GMM.
In transmission systems, all the estimators work wellbecause of
very high redundancy and thus the statisticalmeasures to evaluate
the performance are not required. Forexample, a highly erroneous
measurement is treated as abad data by the WLAV estimator and a
redundantmeasurement is always available to replace this. But
indistribution systems, the measurements are mainly thepseudo
measurements with very limited redundancy. Sincepseudo measurements
are derived from the historical loadproles and customer behaviour,
they are highly erroneous.This is why the statistical framework is
required to identifythe suitable solver for the DSSE.
5 ConclusionThe performance evaluation of SE techniques shows
thatthe existing solution methodology of WLAV and SHGMcannot be
applied to the distribution systems. In order toobtain the
consistent and good quality estimate, signicantmodications are
required in these algorithms. WLS givesconsistent and better
quality performance when applied todistribution systems. Hence, WLS
is found to be a suitablesolver for the DSSE problem.
Table 4 Ninety-ve-bus UK-GDS performance summary
Estimator Real 1%, pseudo 20% Real 1%, pseudo 50%
Bias Consistent/E[e] Quality tr/det Bias Consistent/E[e] Quality
tr/det
(a) Limited redundancy
WLS unbiased consistent/190.02 6.63/ unbiased consistent/188.16
6.24/
WLAV unbiased inconsistent/1 252/ unbiased inconsistent/1
244.42/SHGM unbiased inconsistent/3.06 104 6.46/ unbiased
inconsistent/2.53 105 6.16/
Real 3%, pseudo 20% Real 3%, pseudo 50%
WLS unbiased consistent/189.84 4.61/ unbiased consistent/190.23
4.41/
WLAV unbiased Inconsistent/1 245.18/ unbiased inconsistent/1
241.12/SHGM unbiased Inconsistent/2.89 104 4.75/2 unbiased
inConsistent/2.27 105 4.4/(b) Increased redundancy
Real 1%, pseudo 20% Real 1%, pseudo 50%
WLS unbiased consistent/190 8.86/ unbiased consistent/188.3
8.75/
WLAV unbiased inconsistent/1 255.65/ unbiased inconsistent/1
263.74/SHGM unbiased inconsistent/1.65 109 6.70/ unbiased
inconsistent/1.82 109 6.35/
Real 3%, pseudo 20% Real 3%, pseudo 50%
WLS unbiased consistent/188.65 6.85/ unbiased consistent/189.23
6.75/
WLAV unbiased inconsistent/1 243.73/ unbiased inconsistent/1
249.88/SHGM unbiased inconsistent/6.58 109 4.32/ unbiased
inconsistent/1.0 1010 4.28/
Figure 13 Gaussian mixture approximation of the density
676 IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 7, pp.
666678
& The Institution of Engineering and Technology 2009 doi:
10.1049/iet-gtd.2008.0485
www.ietdl.org
emaddoxWATERMARK
-
The WLS works well if the noise characteristics areknown. In the
absence of this knowledge either the WLSneeds to be modied or a new
class of algorithms need tobe introduced. Furthermore, with growing
interest in thedistribution automation, new DSSE techniques
areexpected to be introduced in the future. However, anymodication
in existing techniques or introduction of newalgorithms should
qualify some statistical criteria because oflimited number of
measurements. This paper highlightssome important statistical
criteria against which a SEalgorithm should be tested to assess its
suitability to DSSE.
6 AcknowledgmentThe authors would like to thank PeterD. Lang of
EDFEnergyNetworks for his valuable suggestions and discussions.
7 References
[1] ABUR A., EXPOSITO A.G.: Power system state estimation:theory
and implementation (Marcel Dekker, Inc., 2004)
[2] SHAFIU A., JENKINS T.V., STRBAC G.: Control of
activenetworks, CIRED, 2005
[3] LU C.N., TENG J.H., LIU W.-H.E.: Distribution system
stateestimation, IEEE Trans. Power Syst., 1995, 10, (1),pp.
229240
[4] LIN W.-M., TENG J.-H.: Distribution fast decoupled
stateestimation by measurement pairing, IEE Proc.-Gener.Transm.
Distrib., 1996, 143, (1), pp. 4348
[5] BARAN M.E., KELLEY A.W.: A branch current based
stateestimation method for distribution systems, IEEE Trans.Power
Syst., 1995, 10, (1), pp. 483491
[6] WANG H., SCHULZ N.N.: A revised branch current
baseddistribution system state estimation algorithm and
meterplacement impact, IEEE Trans. Power Syst., 2004, 19, (1),pp.
207213
[7] LI K.: State estimation for power distribution systemand
measurement impacts, IEEE Trans. Power Syst., 1996,11, (2), pp.
911916
[8] BARAN M.E., ZHU J.X., KELLEY A.W.: Meter placement for
realtime monitoring of distribution feeders, IEEE Trans.
PowerSyst., 1996, 11, (1), pp. 332337
[9] GHOSH A.K., LUBKEMAN D.L., DOWNEY M.J., JONES
R.H.:Distribution circuit state estimation using a
probabilisticapproach, IEEE Trans. Power Syst., 1997, 12, (1), pp.
4551
[10] LUBKEMAN D.L., ZHANG J., GHOSH A.K., JONES R.H.: Field
resultsfor a distribution circuit state estimator
implementation,IEEE Trans. Power Deliv., 2000, 15, (1), pp.
399406
[11] BLACKMAN S.S.: Multiple target tracking with
radarapplications (Canton St. Norwood: Artech House, Inc.,1986)
[12] ANDERSON T.W.: An introduction to multivariatestatistical
analysis (Wiely, Seventh printing, 1966)
[13] JABR R.A.: Primal dual interior point approach to
computethe L1 solution of the state estimation problem, IEE
Proc.-Gener. Transm. Distrib., 2005, 152, (3), pp. 313320
[14] HUBER P.J.: Robust statistics (Wiely, 1981)
[15] CELIK M.K., LIU W.-H.E.: An incremental
measurementplacement algorithm for state estimation, IEEE
Trans.Power Syst., 1995, 10, (3), pp. 16981703
[16] DAS D., NAGI H.S., KOTHARI D.P.: Novel method for
solvingradial distribution networks, IEE Proc.-Gener. Transm.
andDistrib., 1994, 141, (4), pp. 291298
[17] United Kingdom Generic Distribution Network (UK-GDS)
[Online], available at: http://monaco.eee.strath.ac.uk/ukgds/,
accessed June 2008
[18] SINGH R., PAL B.C., VINTER R.B.: Measurement placement
indistribution system state estimation, IEEE Trans. PowerSyst.,
2009, 24, (2), pp. 668675
[19] KOTZ S., KOZUBOWSKI T.J., PODGORSKI K.: The
Laplacedistribution and generalizations (Birkhauser Boston,
2001)
[20] DEMPSTER A.P., LAIRD N.M., RUBIN D.B.:
Maximum-likelihoodfrom incomplete data via the EM algorithm, J. R.
Statist.Soc. Ser. B, 1977, 39, (1), pp. 138
[21] REDNER R.A., WALKER H.F.: Mixture densities,
maximumlikelihood and the EM algorithm, SIAM Rev., 1984, 26,(2),
pp. 195239
[22] BILMES J.A.: A gentle tutorial on the EM algorithm and
itsapplication to parameter estimation for Gaussian mixtureand
hidden Markov models. Technical Report, InternationalComputer
Science Institute, ICSI-TR-97-021, 1998
8 Appendix8.1 Computation of a for variousestimators
The fact that normalised measurement residual r is
normallydistributed with zero mean and unit variance can be used
tocompute a for the state estimators discussed in Section 3.
8.1.1 Weighted least-squares:
c(r) r, c0(r) 1 (21)
IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 7, pp. 666678
677doi: 10.1049/iet-gtd.2008.0485 & The Institution of
Engineering and Technology 2009
www.ietdl.org
emaddoxWATERMARK
-
E[c2(r)] 12p
p11
r2e(1=2)r2dr Var(r) 1 (22)
E[c0(r)] 12p
p11
e(1=2)r2dr 1 (23)
a E[c2(r)]
(E[c0(r)])2 1 (24)
8.1.2 Weighted least absolute value estimator:
c(r) sgn(r), c0(r) 2d(r) (25)
E[c2(r)] 12p
p11
(sgn(r))2e(1=2)r2dr 1 (26)
We use the fact that11
d(t t0)f (t)dt f (t0) (27)
in the following expression
E[c0(r)] 12p
p11
2d(r)e(1=2)r2dr
2p
r(28)
a E[c2(r)]
(E[c0(r)])2 p
2(29)
8.1.3 Schweppe huber generalised M:
c(r) r if jrj avav sgn(r) otherwise
(30)
c0(r) 1 if jrj av2avd(r) 0 otherwise
(31)
E[c2(r)] 12p
pav1
(av sgn(r))2e(1=2)r2dr
12p
pavav
r2e(1=2)r2dr
12p
p1av(av sgn(r))2e(1=2)r
2dr
(32)
By symmetry of the distribution the above equation can
beexpressed as
22p
pav1
(av sgn(r))2e(1=2)r2dr 2
2pp
av0
r2e(1=2)r2dr
(33)
(2a2v2F(av) 22p
pav0
r2e(1=2)r2dr) (34)
where F is the cumulative probability function. The integralterm
in the above equation is given by
22p
pav0
r2e(1=2)r2dr
2p
rave(a
2v2=2) (2F(av) 1)
(35)
Using the relation F(av) 1F(av) and substituting(35) in to (34),
we obtain
E[c2(r)] 12p
rave(a
2v2=2) 2(a2v2 1)(1F(av))
(36)
E[c0(r)] 12p
pavav
e(1=2)r2dr 2F(av) 1 (37)
In this case a depends on parameters a and v, that is, ifa 1.5
and v 1, the value of a is
a E[c2(r)]
(E[c0(r)])2 0:7785
(0:8664)2 1:0371 (38)
678 IET Gener. Transm. Distrib., 2009, Vol. 3, Iss. 7, pp.
666678
& The Institution of Engineering and Technology 2009 doi:
10.1049/iet-gtd.2008.0485
www.ietdl.org
emaddoxWATERMARK