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Electric PowerSystems Research 104 (2013) 917
Contents lists available at SciVerse ScienceDirect
Electric Power Systems Research
journal homepage: www.elsevier .com/ locate /epsr
Robust WLS estimator using reweighting techniques for electricenergy systems
Eduardo Caroa,, RobertoMnguezb, AntonioJ. Conejoc
a UniversidadPolitcnicade Madrid, Madrid, Spainb Universidadde Cantabria, Cantabria, Spainc Universidad de Castilla-LaMancha, Ciudad Real, Spain
a r t i c l e i n f o
Article history:Received 31 July 2012
Received in revised form 24 April 2013
Accepted 31May2013
Keywords:
Weighted least squares
Power system stateestimation
Outlier detection
DependentGaussian measurements
a b s t r a c t
The state estimator is a key tool in the operation ofany real-world electric energy system. In this paper,
a state estimator based on a weighted least squares model is proposed which is robust against outliers.
This algorithm presents two relevant features: robustness that is achieved by readjusting measurement
weights, and accuracy that is attained by consideringmeasurement dependencies. The proposedmethod
is tested in the IEEE 57-bus and 118-bus systems and the obtained results are analyzed using Design of
Experiments and ANOVA techniques.
2013 Elsevier B.V. All rights reserved.
1. Introduction
1.1. Motivation
In any real-world electric energy system, the Control Center
monitors andcontrols the functioningof the network in real-time,
ensuring operational security. To accomplish this task the Control
Center needs to know accurately the actual state of the system
(node voltages, power flows, etc.) at any time. These values are
estimatedby the state estimator (SE).
The state estimator is a mathematical algorithm which com-
putes the most-likely state of the network, given a redundant set
of measurements captured from the system. From the statistical
point ofview, thestateestimationalgorithm isa nonlinearmultiple
regression problem, whose parameters to be estimated are those
which characterize the network state: node voltage magnitudes
and angles.
This estimated state is generally computed using the Maxi-
mum Likelihood Estimator, minimizing the weighted sum of thesquared residuals(i.e.,WeightedLeastSquaresapproach).Once the
most-likely state is obtained, the Control Center performs a bad
data detection and identification procedure to detect and elim-
inate those measurements whose associated standardized errors
are larger than a pre-established tolerance. The statistical tests
commonlyemployedfor these tasksarethe2-test and the Largest
Corresponding authorat: C/ Jos GutirrezAbascal, 2,28006 Madrid, Spain.
Tel.: +34 913363149.
E-mail address: [email protected] (E. Caro).
Normalized Residual test, and arewell established in the technical
literature [1]. Once outliershavebeenremoved, thenonlinearmul-
tipleregressionproblemissolvedagain,andthe finalstateestimate
is obtained.
Ifoutliers arenot properlydetectedor eliminated, the final esti-
matewillbebiased,andtheControlCenterwillnothavean accurate
knowledge of theactual stateof the system, leading occasionallyto
an insecureoperationof the network.Forthis reason, thedetection
andidentificationof badmeasurementshave a notoriousrelevance
in the estimation process. In fact, an adequate and secure control
is only achieved in the case that the SE procedure is robust enough
to detect andeliminate thepresence of corruptmeasurements.
Traditionally, the outlier elimination problem is solved
iteratively by detecting/removing suspected measurements and
re-estimating the state disregarding the rejected data. These esti-
mators are based on the weighted least squares, which shows a
notable computational efficiency; however the lack of robustness
deteriorates significantly their performance in thepresence of bad
measurements. Specifically, the presence of multiple conformingbadmeasurements in the measurement setmayprovoke a mask-
ingeffect:goodmeasurementsmayberejectedwhereascorrupted
ones may not. This undesirable situation occurs when measure-
ment dependencies are not properlymodeled.
1.2. Aim
The aim of this paper is to present a robust state estimator
based on a weighted least squares regression, which carries out
the estimationand thebad datadetection/identificationprocesses
simultaneouslyby successivelyadjustingtheweightingmatrix and
0378-7796/$ seefrontmatter 2013 Elsevier B.V. All rights reserved.
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10 E. Caro et al./ Electric Power Systems Research 104 (2013) 917
consideringtheeffect ofmeasurement dependencies. Theobtained
estimate does not require further bad measurement processing
algorithms.
1.3. Literature review
Thetechnical literatureis richin references concerningthestate
estimationproblem, forinstance,[2]or [1]; andthere isa significantnumber of references on outlier detection: [311]. The previous
worksarefocusedmainly ontheareaof least squares linearregres-
sion. Other statistical models and estimation methods, such as
reweighed techniques [1215], non-linearmethods [16], variance-
varying models [17], or some robust estimators [1820] have
received comparatively less attention. Nevertheless, [21] report
successful results fromthe applicationof thereweighted leastdevi-
ationmethoddevelopedby[14], todetectdatarelatedtohurricanes
and typhoon onwave hindcast databases.
However, no so many works address the power system WLS
estimator using adjusted measurement weights. The pioneering
work reported in [22] proposes a method for readjusting the mea-
surementvariancesbasedon theresidualsof previousestimations.
Ref. [23] develops thisapproach, improving thecomputational effi-ciencyandensuringmathematicalconvergence.Work [24] propose
an iterative reweighted least-squares estimator that is based on
GivensRotations and improves the robustness against outliers.
In [25], the weights of theWLS estimator are artificiallymanip-
ulated, leading to a more robust estimator with the properties of
the weighted least absolute value approach. Recently, in [26], the
WLS regression is addressedusing estimatedweights based on the
measurement variances.
All the aforementioned works consider that the measurement
covariancematrix is diagonal. However,recentworks [27,28]show
that this matrix is generally non-diagonal. Thus, the reweighting
techniques previously proposed in the technical literature can be
improved, since such techniques cannot deal with measurement
dependencies.
1.4. Contribution
The contribution of this paper is threefold:
First, it provides a mathematic procedure that allows apply-
ing a reweighting estimation technique (originally designed
for diagonal covariance matrices) to a non-diagonal estimation
problem. Second, an iterative state estimator is proposed, showing both
robustness against outliers and computational efficiency. Specif-
ically, it requires significantly less time that similar methodsproposed in the technical literature [29].
Finally,DesignofExperimentsandANOVAprocedures areusedto
comparetheperformanceof theproposedmethodwithstatistical
rigor.
1.5. Paper organization
Therest of this paper is organized as follows. Section2 develops
and formulates the Reweighted Least Squares Estimator consid-
ering measurement dependencies. Section3 applies the Design of
Experiments and ANOVAprocedures to theconsidered estimation
problem.Section4 providesandanalyzesresults fromfourrealistic
casestudies. Finally, Section5 provides somerelevant conclusions.
2. Dependent state estimationmodel
Any state estimator can be formulated as a nonlinear multiple
regression problem, where theunknown parameters are thenode
voltage magnitude andangle of every node, represented by Vi and
i, respectively. These two sets of variables form the state vector
x = [VT T]T. There are n state variables. The unknown true state
is represented by xtrue.
The unknown parameters are estimated using the informationprovided by observations {z1, . . .,zm}. These observationsare cap-tured from the system using measuring devices, and are related
with x bymeans of a multifunctional vector h( x). Depending onthe measurement type, the functions hi( x) differ. Expressions offunctions hi(x) arewell-established in the technical literature [1].
The error terms used throughout this paper are definedbelow:
Measurement residual: difference between the measurement ziand the function hi(x) evaluated at the optimal state x,
Residuali = zi hi(x)|x=x =zi hi(x) = ri. (1)
Measurement error: difference between the measurementzi and
the function hi(x) evaluated at any state x,
Errori = zi hi(x)|x=x =zi hi(x) = ei. (2)
Metering error: difference between the measured value and the
unknown true value,
Meteringerrori =zi hi(x)|x=xtrue = zi hi(xtrue) = zi z
truei . (3)
Note that the term measurementresidual is solelyused in the
case of comparing measurement value zi with the function hi()
evaluatedat the optimal state x. Similarly, theterm measurementerroris solelyemployedfor comparingthemeasuredvalueziwith
the function hi(), evaluated at any state x.Measurement errors have been traditionally modeled as an
independent unbiased Gaussian-distributed random variable. The
factualmetering infrastructurewithin substations results in signif-icant statistical correlations between measurement errors. Works
[27] and [28] numerically show that these correlations are signif-
icant, and its consideration may improve the quality of the final
estimate. Therefore, hereafter measurement errors are assumed
to be dependentGaussian-distributed unbiased random variables.
Thedependence structureismodeledbymeans ofpositive-definite
non-diagonal variance-covariance matrix Cz, which can be easilycomputed using the Point Estimatemethod [28].
As it is customary in the technical literature, allmeasurements
areconsidered synchronous.Thisis reasonable since in steady state
power systemmagnitudes change very slowly with respect to the
time needed to transfer measurements to the EMS from Remote
Terminal Units (RTUs) or PhasorMeasurement Units (PMUs).
2.1. State estimation
Given thepreviousassumptions,theestimationof thestatevari-
ables are obtained by minimizing the weighted sum of squared
measurement errors of the multiple nonlinear regression model,
leading to a nonlinear optimization problem:
minimizex
J= [z h(x)]TC1z [z h(x)] (4a)
subject to
c(x) = 0 g(x) 0 (4b)
wherethescalarJistheobjectivefunctionand c(x)and g(x)arethe
equalityandinequality constraintsmodelingzero-injectionsnodes
and physical operating limits, respectively. Note that matrix Cz is
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E. Caro et al. / Electric Power Systems Research 104 (2013) 917 11
not diagonal, since the considered model takes into account the
statistical correlations betweenmeasurement errors, but positive-
definite as it is a variance-covariancematrix.
2.2. Reweighted least squares formulation
The Weighted Least Squares model (4a) accounts for the error
measurements with different variances and their dependency.
However, an alternative weighting matrix can be used account-
ing for (i) different variances, (ii) dependency, (iii) and degree of
confidence related to eachmeasurement.
Inordertoderivethisweightingmatrixletusconsiderthevector
of measurement errors e = z h( x) and optimization problem(4). Using the Cholesky decomposition of matrix Cz, the objective
function in (4a) becomes
J= [e]TC1z [e]
= eT(LLT)1e = eT(L1)
TL1e
= (L1e)TL1e = uTu
(5)
where u is a vector of standard independent normal randomvari-ables, with covariance matrix equal to the identity matrix I, and
L is the lower-triangular Cholesky factor of matrix Cz. Note thatthe derivation above isbasedon theassumption that the estimated
state x issufficientlycloseto the true state xtrue,whichis a commonassumption in the state estimation Literature.
The aim of most outlier detection methods is to determine
whether or not a measurement should be considered as an out-
lier, without allowing for intermediate situations. In contrast, the
methodproposed in this paper, originally developed by [14], aims
at empirically determining a diagonal matrix W to be included inmodel (5), i.e.
JR = uTWu (6)
where wii is a weight for every observation ranging continuously
from 0, for observations that arecompletely unreliable, up to 1, for
observations that are completely reliable.
Considering (6), the objective function (5) becomes:
JR = eT(L1)
TWL1e = eTWRe. (7)
From (5) and (7), the followingobservations are in order:
The measurement error vector e (dependent normal randomvariables) is transformed into a vector of independent standard-
ized normal variables u [29].
The objective functionJcan be expressed as the sum of a set ofsquared independent standardized normal random variables.
The objective function JR is computed as the weighted sum of
squared independent standardized normal random variables.
Each factor u2iis multiplied by the weighting factor wii [0,1].
If the ith weighting factor is null (wii = 0), then the componentu2iis not considered in the objective functionJR. If, on the other
hand, wii = 1, the component u2iis fully considered inJR.
Theunderlyingideaof the RWLSmethodis toadjustempirically
the weighting factors, based on the degree of confidence of each
measurement.Thecoefficients for thosemeasurementscompletely
unreliable areadjusted to zero and, similarly, theweighting factors
for thosemeasurements completely reliable are adjusted to one.
The reweighting method is based on an iterative algorithm
which updates the weighting factors in every iteration, according
to theTuckeys biweight formula:
wii =
1
yi6
22if |yi| 6
0 if |yi| > 6(8)
whereyi =ui/* is thestandardizedresidual related touncorrelatedvector u, and * is the scaledmedianabsolute deviationestimator*:
=mediani|ui|
(3/4)
mediani|ui|
0.6745(9)
Reweighting techniques constitute an effective and computation-
ally attractive alternative to solve M-estimators. A significant
numberofM-estimatorshavebeendeveloped inthetechnical liter-
ature, such as least-absolute value, Huber, Cauchy,Welsh, Tuckey,
etc., and each type has an associated updating formula forweights,
such as (9) for Tuckeys case.Work [30] providea usefuldiscussion
about the choice of the appropriate function for practical cases.
Specifically, Tuckeys biweight do not guarantee uniqueness of thesolution, but the influence of gross errors is reduced considerable
or even eliminated [31].
Thus, theRWLS problem formulation is:
minimizex
J= [z h(x)]TWR[z h(x)] (10a)
subject to
c(x) = 0 g(x) 0 (10b)
Thealgorithmof theRWLSestimatorconsideringdependencies
is:
Initial non-dependent estimation. An initial WLS estimation is
performed to estimate the measurement variance-covariance
matrix. Using the initial estimation obtained x0) , matrix C0)z iscomputed via thePoint-Estimatemethod in [28].
Parameter initialization. Weights wii are set to one:w0)ii= 1,i
{1,m}.
The iteration counter is set to 1, =1. Dependent state estimation. The state estimation problem (10)
is solved considering measurement dependencies and the
reweighted matrix W1)R . The obtained estimates are denoted
as x).
Convergence checking. Once the estimates x)
are available, if
||x) x
1)|| > theestimation process continues in 5).
Otherwise, a solutionwitha tolerance is x)andthealgorithm
concludes.
Updateweightingmatrix WR. Once the estimatesof the statevari-ables are available (x
)), the weighting matrix WR
) is updated
using (8).
Set +1 and goto step(3).
Thecomputational efficiency of this algorithmcan be improved
by using x1)
as the initial values of the estimation in step (3).
The convergence of the iterative process does not only depend
on the reweighted technique employed, but also on the struc-
ture of the considered optimization problem (i.e., convexity or
linearity of the objective function, presence of equality/inequality
constraints, types ofvariables, etc.). Thus,convergence rulesare not
general. Specifically,work[30] discusses thechoice of anappropri-
atefunction,whereas[32] showsconvergence forcertainweighting
functions. For the optimization problem considered in the paper,
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fast convergence is achieved in all cases, regardless the measure-
ment scenario, presence of baddata, or network used.
3. Design of Experiments
Section2 above presents a novel algorithm to estimate the
state of a power system in a robust manner. In this section, the
statistical procedures Design of Experiments and ANOVA are
briefly described as they are used (i) to analyze the performanceof the proposedmethod, and (ii) to compare it with other existing
methodologies.
Themethods considered in this paper are listedbelow:
WLS. The commonWeighted Least Squares estimator is used as
basic benchmark, using the 2-test and the Largest NormalizedResidual (LNR) test to detect and identify bad measurements,
respectively. These algorithms are well-established in the tech-
nical literature [1]. The WLS results are the final estimates once
the 2-test andLNR test have been successfully passed. DWLS. The estimation, detection, and identification algorithms
considering dependencies, proposed in [27] and [29], are
employed to estimate the state and to detectbadmeasurements
including the statistical correlation between measurements.Thus, the DWLS results correspond to the final estimate once the
badmeasurementshave been removed.
3.1. Performance assessment
In order to rigorously determine which is the methodwith best
performance, a Design of Experiments is carried out.Thisstatistical
procedure allows determining if the proposed algorithm is signif-
icantly better than the rest of the approaches with a pre-specified
confidence level, and taking into consideration the dissimilarities
among the considered measurement scenarios.
The performance of each method is assessed by means of the
followingmetrics:
MetricVMET,, definedas the averageabsoluteerrorof thevoltagemagnitude estimate for theth measurement scenario, consid-ering themethodMET, i.e.,
VMET, =
ni=1
|VMETi,
Vtruei,
|
n(11)
Note that the previousmetric ismeasured in p.u. MetricMET,, definedas the averageabsoluteerrorof thevoltageangle estimate for the th measurement scenario, consideringthe method MET, i.e.,
MET, =
ni=2
|METi,
truei,
|
n 1(12)
Thepreviousmetric ismeasured in radians. Note that the con-sidered reference angle is located at node 1, i.e., 1 =0 rad,for allthe considered scenarios.
Metric CPUMET , defined as the required CPU time to obtain the
final estimate considering themethodMETfor thethmeasure-
ment scenario. Note that this metric ismeasured in seconds.
3.2. ANOVAmodel
The model employed in this Design of Experiments proce-
dure comprises the factors Method and Scenario. This model
is described below,
yMET, = + MET + + uMET, (13)
Table 1
ANOVA model:factors and levels.
Names Levels
Factors Method WLS
(MET) DWLS
RWLS
Scenario () 1,. . ., n
Table 2ANOVA table structure.
Source SquaredSum Deg. o f freedom Mean-sq F-stat Fcrit
Method SSM 31 s2M s2M/s2error F
Methodcrit
Scenario SSD n 1 s2D s2D/s2error F
Scenariocrit
Residual SSerror (n 1)(31) s2error
where uMET,iid
N(0, 2) and:
MET
MET = 0;
n=1
= 0
where is theglobal effect, i.e., theaveragevalueof the consideredmetric yMET,. Parameter MET is the main effect of the estima-tion method, and measures the increase/decrease of the average
response of the factor Method (MET)with respect to the average
level. Likewise, parameter is the main effect of the factor Sce-
nario (), and it measures the increase/decrease of the averageresponse for all the methods with respect to the average level at
the th measurement scenario. Finally, the random effect uMET,includes the effects of all other causes not modeled. Taking into
consideration that the particularities of each measurement sce-
nariomayhave influence on the methods performance, the effect
Scenario is included inmodel (13).
The factors considered in this ANOVA analysis and the levels
corresponding toeach factorareprovided in Table 1. Parametern
stands for thenumber of measurement scenarios considered.Thebackground hypotheses of this model are: (i) normality, (ii)
constantvariance, and (iii) independence. To ensure this statistical
properties, an appropriate diagnosis procedure is performed after
the residual computation.
Sincetheaimof thisstudyis tofindthemostaccurateestimation
method and to check if it is significantly different from the other
methods, the following tests are performed:H0 : MET = 0,MET
H1 : MET|MET /= 0
H0 : = 0, = 1, . . . , n
H1 : | /= 0
The null hypothesis for the first test corresponds to the no
statistically-significant influenceof themethodontheaverageper-
formance. The alternative hypothesis establishes that it exists atleast one method performing different from the average. The sec-
ond test is analogous but related to the factor Scenario.
To perform the above two statistical hypothesis testing, the
ANOVA table is computed andanalyzed. Table 2 provides the gen-
eral structure for this table, particularized for the problem under
consideration. Thecomputationof theelements of thetable iswell
established in the technical literature [34]. Appendix A provides
further details concerningthe ANOVAtablestructure andthe com-
putation of its terms.
Once the ANOVA table is obtained, the value of the F-statistic
for the factor Method allows deciding whether or not the per-
formances of the different methods are statistically different. If
methods performances are statistically similar, F-stat will be low.
Otherwise, if there are significant dissimilarities among methods
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Table 3
118-Bus system case study: ranges.
Vm m Pm Qm PFm QFm Im Buses Branches
118 [2; 4] [10; 103] [10; 103] [208; 231] [208; 231] [5; 20] 118 179
performances, the value ofF-stat will be high. In order to check if
the resulting F-stat is low or high, it is comparedwith FMethodcrit
:
IfF-statistic> FMethodcrit
, thenthe relatednullhypothesisisrejected
and it is concluded that the response variable is significatively
affected by the method employed. Then, the average values for
eachmethodand itsconfidenceintervalsareplottedtodetermine
which is the method with best performance. IfF-statistic< FMethod
crit, then it is concludedthattheperformances
of all methods are similar.
Withina certain confidence level (1), the parameter FMethodcrit
is definedas themaximum admissible value for a randomvariable
which follows an F-Snedeccor distribution with the corresponding
degrees of freedom. Alternatively, the parameter FMethodcrit
can be
defined as the limit value for the marginal right-side region with
anarea equal to for the F probability density function (see Fig. 6).The value ofFMethod
critis computed considering a confidence level of
95%, leading to false alarmprobability lower than 5%.
4. Case studies
In this section, four case studies are analyzed to check the esti-
mationaccuracyandcomputationalefficiencyof theproposedstate
estimator.
The networks under study are the IEEE 57-bus and the 118-
bus systems.1 To obtain statistically sound conclusions: (i) a set of
one hundred randomly-generatedmeasurement scenarios is con-
sidered, and (ii) an ANOVA procedure is performed to analyze the
obtained results.
Each scenario involves: (i) a randomactive/reactive power con-
sumption level, (ii) randomlocations of voltageandactive/reactive
power meters (ensuring observability of thewhole system), (iii) a
random redundancy level, and (iv) Gaussian-distributed random
errors in all measurements, (standard deviations of 0.01pu and
0.02pu for voltage andpower measurements, respectively).
Note that each scenario involves randomerrors in allmeasure-
ments. Theseerrors aremodeled as Gaussian-distributed unbiased
random variables, taking into consideration the actual metering
infrastructure of each substation. Ref. [27] provides more details
concerning the random measurement generation. The measure-
ment set is generated assuming perfect synchrony throughout all
substations.Futureworkwill considergenerationofmeasurements
exhibiting accidental asynchrony and its impact on the perfor-
mance of theproposedmethod.The allocation of meters is based on a random algorithm. This
algorithmhas been specifically designed for allocatingmeters in a
realisticway,providing scenariosthat satisfy thefollowing require-
ments:
The resulting measurement configuration provides whole
observability of the system. Themeasurement configuration provides a realistic redundancy
ratio, ranged from2.9 to 3.3.
1 Power Systems Test Case Archive. Available at: http://www.ee.washington.
edu/research/pstca/.
The numbers of measurements are well-balanced across mea-
surement types. Measurements are geographically allocated in an uniform man-
ner throughout the considered system.
The random allocation of metering devices is realistic since the
quantity and proportion of the allocated measurements are based
on reasonable ranges. Table 3 provides the ranges used for this
random allocation of measurements. Columns Vm, m, Pm, Qm,
PFm, QFm, and Im correspond to the voltage meters, PMU devices,
active/reactive power injectionmeters, active/reactive power flow
meters and current meters, respectively. PMUmeasurements are
modeled as in [35,36] The number of buses and branches of the
considered systemareprovided in the last twocolumns.
The computational analyses have been performed using a
Windows-basedpersonal computerwitha 64-bits four-core third-
generation i5 processor at 1.73GHz and 4Gb of RAM. Sections4.1
to 4.4 consider the IEEE 118-bus system, whereas Section4.5 con-
siders the IEEE 57-bus system.
4.1. First case study: no gross errors
In this case, the measurement vector z is free of gross errors.For each measurement scenario, the estimates for methods WLS,
DWLS, and RWLS are computed and an ANOVA analysis is per-
formed.
Table4provides the ANOVAanalysis formetrics VMET, MET, and
CPUMET, and Fig. 1 depicts the average value for these metrics and
theconfidence intervals for a 95%confidence level.
FromTable 4 and Fig. 1, the following observations arein order:
From theANOVAanalysis ofmetricVMET in Table 4, it is observedthat F-stat= 39.2 > FMethod
crit= 3.03. This indicates that the accu-
racy of the methods for estimating voltage magnitudes are not
the same. Thus, in order to determine graphically which is the
methodwith higher accuracy, the mean and confidence interval
of metric VMET are plotted in Fig. 1. Similarly, the F-statistics formetricsMET and CPUMET are higherthan the critical values. Then, the mean and confidence interval
of these metric are plotted in Fig. 1 to allow determining which
is themethod with better accuracy estimating the voltage angle,
andwhich is themethod computationally more efficient. Regarding estimation accuracy (left and center subplots), the
WLS estimator is the less accurate, whereas theDWLS andRWLS
Table 4
ANOVA table for thefirst case study.
Source Squared Sum DoF Mean-sq F-stat Fcrit
Metric VMET
Method 2.4106 2 1.2106 39.232 3.03
Scenario 4.1105 99 4.1107 13.490 1.30
Residual 6.0106 198 3.0108
Metric MET
Method 2.1105 2 1.1105 111.094 3.03
Scenario 4.0105 99 4.0107 4.196 1.30
Residual 1.9105 198 9.6108
Metric CPUMETMethod 6.0101 2 30 11.975 3.03
Scenario 4.3102 99 4.3 1.721 1.30
Residual 5.0102 198 2.5
http://www.ee.washington.edu/research/pstca/http://www.ee.washington.edu/research/pstca/http://www.ee.washington.edu/research/pstca/http://www.ee.washington.edu/research/pstca/http://www.ee.washington.edu/research/pstca/8/22/2019 Caro 2013
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14 E. Caro et al./ Electric Power Systems Research 104 (2013) 917
Fig. 1. Results for thecase studywith no gross errors: performance comparison.
procedures provide the most accurate results for a confidence
level of 95%. There is no significant difference between theaccuracy provided
by the DWLS and RWLSmethods. Regarding the required CPU time, methods WLS and RWLS are
the most efficient.
4.2. Second case study: three gross errors
In this case, the measurement vector contains bad data. The
corresponding scenarios are generated as follows [27]:
1. Vector ztrue isderived fromapowerflowsolution. Then, random
Gaussian-distributednoiseis added toallmeasurements in ztrue
to derive measurement vector z.2. Three substations are randomly selected, and for eachof them:
A metering transformer is randomly selected. Both phase and
transformer type (current or voltage) are also randomly cho-
sen. Then, a gross error is included in theselected transformer
signal.All measurements from this substation are computed takinginto account the erroneous transformer signal. Thus, two or
moreprocessedmeasurements in thissubstation aredistorted
as a result of an interacting and conforming error [29].
Again, Table 5 provides the ANOVA analysis for metrics VMET,
MET, and CPUMET, and Fig. 2 depicts the average value for these
metrics and the confidence intervals for a 95% confidence level.
FromTable 5 and Fig. 2, the followingobservations are in order:
From Table 5, note that the three F-stat values correspond-
ing to the factor Method are higher than 3.03, denoting that
there are statistically significant differences regarding the per-
formanceof the consideredmethods, fornumericalaccuracyand
Table 5
ANOVA table forthe secondcase study.
Source Squared Sum DoF Mean-sq F-stat Fcrit
Metric VMET
Method 5.2106 2 2.6106 26.339 3.03
Scenario 6.3105 99 6.3107 6.406 1.30
Residual 2.0105 198 9.9108
Metric MET
Method 3.4105 2 1.7105 95.202 3.03
Scenario 4.6105 99 4.6107 2.597 1.30
Residual 3.5105 198 1.8107
Metric CPUMETMethod 4.7102 2 2.3102 54.899 3.03
Scenario 8.5102 99 8.5 2.010 1.30
Residual 8.4102 198 4.3
computational efficiency. Thus, the confidence interval plots
(Fig.2) arestudied to determinewhichmethod provides the best
performance. Regardingestimationaccuracy (leftandcentersubplots), theWLS
estimator is the less accurate, whereas the DWLSand RWLSpro-
cedures provide the most accurate results for a confidence level
of 95%. There is no significant difference between theaccuracy provided
by the DWLS and RWLSmethods. Regardingthe requiredCPUtime,theRWLS estimatoris themost
efficient. The efficiency provided by the WLS and DWLS algo-
rithms are statistically similar.
4.3. Third case study: six gross errors
In this case, six substations are randomly chosen, and a set of
multiple gross errors is located in each substation. Thus, the mea-
surement vector is corrupted by six sets ofmultiple bad data.
Table6 provides the ANOVAanalysis formetrics VMET, MET, and
CPUMET, and Fig. 3 depicts the average value for these metrics and
the confidence intervals for a 95% confidence level.FromTable 6 and Fig. 3, the followingobservations arein order:
Results fromTable 6 allowwithdrawing the same conclusions as
in the previous section: there are significant differences among
methods accuracy and efficiency. Again, theWLSestimatoris the least accurate,whereas theDWLS
andRWLS procedures provide the most accurate results. RegardingtherequiredCPUtime, theRWLSestimatoris againthe
most efficient, and the DWLS algorithm is the less efficient one. The required CPU time for the RWLS estimator is approximately
75% and 72% smaller than the CPU times required by the DWLS
andWLS methods, respectively.
Table 6
ANOVA table for thethird case study.
Source Squared Sum DoF Mean-sq F-stat Fcrit
Metric VMET
Method 2.6105 2 1.3105 57.645 3.03
Scenario 9.1105 99 9.1107 4.035 1.30
Residual 4.5105 198 2.3107
Metric MET
Method 3.0105 2 1.5105 100.764 3.03
Scenario 6.6105 99 6.7107 4.475 1.30
Residual 3.0105 198 1.5107
Metric CPUMETMethod 2.2103 2 1.1103 133.953 3.03
Scenario 2.1103 99 2.1101 2.573 1.30
Residual 1.6103 198 8.1100
8/22/2019 Caro 2013
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E. Caro et al. / Electric Power Systems Research 104 (2013) 917 15
Fig. 2. Results for thecase studywith three gross errors: performance comparison.
Fig. 3. Results forthe case studywith six gross errors: performance comparison.
4.4. IEEE 118-bus system: results comparison
Analyzing jointly the results obtained for the three case studies
above, the following conclusions can bewithdrawn:
1. Theestimation accuracy of the proposed RWLS algorithm is sig-
nificantly better than the one provided by the traditional WLS
procedure, for a 95% confidence level and considering scenarios
with zero, three, and six sets ofmultiple gross errors.
2. For a 95% confidence level, there is no significant difference
between the estimation accuracy degree of methods RWLS and
DWLS. That is, theobtained estimate for both methods have the
sameaccuracy.
0 3 60
2
4
6
8
Number of errors
AveragerequiredCPUtime(sec.)
WLS
DWLS
RWLS
Fig. 4. Required CPU time comparison.
3. The computational efficiency of the RWLS remains unalteredwith independence of the number of multiple gross errors
corrupting the measurement set. This consideration can be
graphically illustratedbyplottingthe averagerequiredCPUtime
for each method as a function of the number of multiple bad
measurement sets (see Fig. 4).
4. Note thatthemostaccurateestimatesdoesnotcorrespond tothe
WLS results. This is sobecauseWLSestimationdoes notconsider
measurement dependencies.
4.5. Fourth case study: IEEE 57-bus system
Finally, methods WLS, DWLS and RWLS are applied to another
system: the IEEE 57-bus network.2 Results are graphically shown
in Fig. 5, using the same format as in Figs. 13.From Fig. 5, the followingobservationsare in order:
1. The estimation accuracy provided by algorithms DWLS and
RWLS are similar.
2. The accuracylevel ofmethodWLS is significantlylowerthanthe
levels provided by approaches RWLS andDWLS.
3. TherequiredCPU timeof themethodRWLSis significantly lower
than theCPUtimes required by approachesWLSandDWLS. The
saving is up to 80%.
2 Power Systems Test Case Archive. Available at: http://www.ee.washington.
edu/research/pstca/.
http://www.ee.washington.edu/research/pstca/http://www.ee.washington.edu/research/pstca/http://www.ee.washington.edu/research/pstca/http://www.ee.washington.edu/research/pstca/8/22/2019 Caro 2013
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16 E. Caro et al./ Electric Power Systems Research 104 (2013) 917
Fig. 5. Performance of themethods:IEEE 57-bus system.
5. Conclusion
The method proposed in this paper outperforms alternative
algorithms reported in the technical literature in the consid-
ered cases, especially if the number of gross errors or outliers
in the measurement set is large. If the proposed RWLS approach
is implemented in real-world systems, a similar behavior is
expected, in termsof both computational efficiency andnumerical
accuracy.
The proposed RWLS estimator takes into consideration mea-
surement dependencies to improve accuracy and its weights are
automatically readjusted to increase robustness. In summary, it
provides accurate estimates, is robust against outliers, and is com-
putationally efficient.
The methods performance is tested using several realisticcase study with single and multiple gross errors, and consider-
ing a high number of scenarios. These results are compared in
detail using ANOVA techniques, which allows proving the out-
performance of the proposed method from a statistical point of
view.
Since most of the state estimation algorithms used in practice
arebasedonWLStechniques,suchalgorithmscanbeeasilyadapted
ormodified to include RWLS features.
Acknowledgements
Eduardo Caro would like to thank partial financial support by
Project DPI2011-23500, Ministerio de Ciencia e Innovacin, Spain.
Appendix A. ANOVA table description
In this Appendix, the ANOVA table structure presented in Sec-
tion3.2 is described in detail.
The first column provides the sources of variability accord-
ingto model (13): factor Method, factor Scenario andresiduals.
The corresponding variabilities are quantified by the Squared Sum
terms, provided in the secondcolumn.
These terms are computed using the followingexpressions:
SSM = n
3M=1
(yM y)2 (A.1)
SSD = 3
n=1
(y y)2 (A.2)
SSerror =
n=1
3M=1
(yM yM y + y)2 (A.3)
where yM is the average response for the METth method consid-eringall scenarios; y is theaverage response for theth scenario
consideringallmethods; y is theaverageresponseconsideringall
scenarios andmethods.
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E. Caro et al. / Electric Power Systems Research 104 (2013) 917 17
Fig. 6. F-Snedeccor distribution.
These Squared Sum terms are divided by their corresponding
degrees of freedom3 (third column), obtaining the variances for
each factor, denoted as SSM, SSD, and SSerror.
The aim of the ANOVA analysis is to determine whether or not
the factors affect the response variable. This can be checked by
comparing the above variances. The way of comparing them is by
computing the ratios s2M/s2error and s
2D/s
2error.
If the performance is not influenced by the method employed,
the ratio s2M/s2error is low. On the other hand, if the performance
varies depending on the methodused, this ratio is high.Inordertodetermineit a ratiois lowor high,theF-Snedeccor
distribution is used (see Fig. 6). Theratio s2M/s2error is computed and
located in the X-axis. If it is lower than FMethodcrit
, then the perform-
ances of the methods are similar (the hypothesis H0 is accepted).
On the contrary, if the ratio s2M/s2error is higher than F
Methodcrit
, then
there is statistical evidence that the methods perform different.
Note that thevalue Fcrit depends on the degrees of freedomused
and on the confidence level employed. These values are provided
in the last column of the ANOVA table, for a confidence level of
95%. Note also that Fig. 6 provides an example of value s2M/s2error. In
this case (Fig. 6), thehypothesisH0 is accepted, concluding that the
methods exhibit the same performance.
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