distribution system state estimation through

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www.ietdl.orgnetwork shows a number of variations at different nodes and cannot be represented by any specicdistribution. The approach presented in this study represents all the load pdfs through the Gaussian mixturemodel (GMM). The expectation maximisation (EM) algorithm is used to obtain the parameters of the mixturecomponents. The standard weighted least squares (WLS) algorithm utilises these load models as pseudo-measurements. The effectiveness of WLS is assessed through some statistical measures such as bias,consistency and quality of the estimates in a 95-bus generic distribution network model.

1 IntroductionThe majority of the power distribution networks wereplanned, designed and built way back in the 1950s and1960s. They served as a passive but reliable link betweenthe bulk power transmission point and the individualconsumer. Usually the cables and lines were built withenough latent capacity to accommodate the projecteddemand growth. In view of this, the system is leftunmonitored and any customer interruption because of thefault or network outage is usually attended through properrestoration service specic to the individual distributionnetwork operator (DNO).

Following the signicant development in businessregulations, technology evolutions and various governmentpolicies towards low carbon generation technology, it hasbecome necessary to operate the distribution systemefciently. One of the important issues is to accommodateas much distributed generation as possible in thedistribution voltage level. This obviously needs monitoringand control of the network for greater asset utilisationthrough modern distribution management system (DMS)in primary substation levels. The state estimation will be atthe heart of the DMS which will act on the network

information to produce network voltage magnitudes andangles. The output of the state estimator will drive theDMS to perform a host of network functions foroperational decisions. The distribution system is typicallycharacterised by the unbalanced, shorter lines with highR/X ratios connecting highly distributed loads. Unlike in thetransmission systems, the scarcity of measured informationoffers formidable challenge to the state estimator to providereasonably meaningful estimate of the system states. Theconsumer loads are not measured. Usually voltage, currentand ows are measured in primary substation with virtuallyno monitoring of secondary substations. The researchefforts have concentrated on few key issues such as the stateestimator algorithms, modelling of pseudo-measurementsand phase unbalance. While the algorithm mainly focusesaround maximum-likelihood estimates of the weighted leastsquare (WLS) type, the pseudo-measurements targetrealistic representations of the loads. The majority of thedistribution networks operate under varying degree ofunbalance and so the phase wise estimation is preferredover the balanced three phase for accuracy.

There are mainly two approaches for estimating the state ofthe system in distribution system state estimation (DSSE)literature:Published in IET Generation, Transmission & DistributionReceived on 30th March 2009Revised on 24th September 2009doi: 10.1049/iet-gtd.2009.0167

Distribution system staGaussian mixture modpseudo-measurementR. Singh1 B.C. Pal1 R.A. Jabr1Department of Electrical and Electronic Engineering, Imper2Department of Electrical and Computer Engineering, AmeriE-mail: [email protected]

Abstract: This study presents an approach to utilisedistribution system state estimation (DSSE). The loahe Institution of Engineering and Technology 2009ISSN 1751-8687

e estimation throughl of the load as

College, London, UKUniversity of Beirut, Beirut, Lebanon

e loads as pseudo-measurements for the purpose ofprobability density function (pdf) in the distributionIET Gener. Transm. Distrib., 2010, Vol. 4, Iss. 1, pp. 5059doi: 10.1049/iet-gtd.2009.0167

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www.ietdl.org1. WLS based [15]

2. Load ow based [68]

Lu et al. [1] have proposed a current-based estimator thatsought to minimise the WLS objective. The advantage ofhaving a constant gain matrix was achieved. This has beenfurther improved for the speed of convergence throughdecoupled formulation [2] with the help of measurementpairing. Baran and Kelley [3] have introduced an algorithmwhich denes branch currents as state variables. Themethod is computationally very efcient and demonstratedto work well in radial and weakly meshed systems. It wasalso shown to be insensitive to parametric variations. Thisconcept has been further rened in [4] through theautomated meter reading assisted load estimation. Theimpacts of measurement locations and measurement typeson the accuracy of estimates have been also investigated. Li[5] has presented a distribution system state estimatorbased on WLS approach and three phase modellingtechniques.

Ghosh et al. [6] have approached the problem as a series ofprobabilistic radial power ow computations treating realmeasurements as solution constraints. The algorithmexploited the radial nature of the distribution circuit. Thestate variables were treated as the random variables.The eld results are discussed in [7]. Celik and Liu [8]have used a GaussSeidel load ow algorithm for stateestimation in radial distribution networks. In thisalgorithm, the load proles obtained from the historicalload data are scaled according to the actual measurementsso that the total load as seen at the measurement pointagrees with the measurement. A similar criterion is alsoused by Roytelman and Shahidehpour [9], however, theirapproach is based on current balancing.

The quality of the estimate very much depends on realisticmodelling of estimated loads. In the absence of any realmeasurement of loads which are highly distributed anddiverse, they are treated as pseudo-measurements (randomvariables) with appropriate mean and variances. It is naturalto model the pseudo-measurements through the normaldistribution because of its compatibility with WLS basedon maximum-likelihood estimation. However, the normaldistribution assumptions of load proles, adopted in manypapers, do not reect the realistic situation. Seppala [10]has suggested log-normal distribution models and alsoproposed a model of customer load condence interval.The models were veried from hourly load measurementdata obtained from a Finnish load research project. Ghoshet al. [6] have investigated this issue further through theload correlation coefcients using the diversity factor. Theyhave validated various models such as normal, log-normaland beta distribution through chi-square goodness of ttest. Ghosh et al. [6] concluded that an appropriate modelwas system specic with clear preference for the betadistribution because of its exibility to adapt to theGener. Transm. Distrib., 2010, Vol. 4, Iss. 1, pp. 5059i: 10.1049/iet-gtd.2009.0167skewness in the distribution. However, the beta distributioncannot be incorporated into the WLS formulation.

Singh et al. [11] have shown that if the statisticalcharacteristics (mean and variance) of the measurements areknown, then even in the presence of the pseudo-measurements (with large errors) the WLS gives unbiasedand consistent results. However, the consistency of theWLS relies on the assumption that measurements shouldbe normally distributed. Thus in our research, we havemodelled the variability in the load distribution throughGaussian mixture model (GMM) approximation. Theadvantage of the GMM approach is that different types ofload distributions can be fairly represented as a convexcombination of several normal distributions with respectivemeans and variances. The problem of obtaining variousmixture components (weight, mean and variance) isformulated as a parametric estimation problem. Theexpectation maximisation (EM) algorithm [1214] wasutilised to obtain the solution. The EM algorithm isa powerful tool in parameter estimation problems. It is ageneral method of nding the maximum-likelihoodestimate of the parameters of an underlying distributionfrom a given data set when the data are incomplete or havemissing values.

The above formalism was applied to a 95-bus UK GenericDistribution System (UKGDS) model. Unlike many otherdistribution systems, the UK distribution network is fairlybalanced and that has prompted us to go with a singlephase approach although the method is generic.Furthermore, the GMM-based load modelling depends onthe load proles and their statistical description. If thenumber of customers, their types and their annualmaximum demands in each phase are available with theDNOs, the load proles in each phase can be computed.Once the phase load proles are obtained, the GMMtechnique can be directly employed to model the pseudo-measurements in each phase and the three-phase WLSalgorithm given in [5] can be applied for state estimation.The effectiveness of the DSSE is assessed through threestatistical measures of the estimates: bias, consistency andquality, which will be briey described in Section 4.Section 2 introduces the UKGDS network model and loadprole followed by their distributions. The GMM, the EMalgorithm and mixture reduction techniques are discussedin Sections 3. The simulation results are presented anddiscussed in Section 5.

2 Distribution system loadprolesLoad proles were generated for a part of UKGDS modelconsisting of 95 buses and 94 lines at 11 kV voltage level.The network parameters and load data were obtained from[15]. The buses are renumbered while keeping the topologyand the data the same as in [15] with an exception that the51

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Step 1: Computation of scaling parameter: First the scaling

led every halff the jth classthe annualconsumer at

In the secondith bus) wererameters andoad prole at

(2)

led every halfat the ith

of consumer

d prole: Ins, the power

factors of various classes of consumers were utilised. The

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www.ietdl.orgparameter aj for a particular consumer was computed bydividing the annual maximum demand of that consumer bythe maximum value in the corresponding LPI

aj Pji, max

maxt(P jPI(t))

(1)

Figure 1 UKGDS 95-bus test system modelThe Institution of Engineering and Technology 2009reactive power LPI (Q jPI(t)) corresponding to the realpower LPI was computed as follows

Q jPI(t) P jPI(t) tan(fj) (3)

The reactive power LPIs were scaled and summed uptogether according to Step 2 in order to obtain the reactiveshunt branches are ignored. There are 55 load buses in thisnetwork. Fig. 1 shows the schematic of the network. Atypical UKGDS bus consists of four types of consumers.

1. DomesticUnrestricted (D/U)

2. DomesticEconomy (D/E)

3. Industrial (I)

4. Commercial (C)

UKGDS [15] also provides the half hourly normalisedactive power load proles [referred as load prole index(LPI) in this paper] of various classes of consumers overthe year along with their annual maximum demand. Thisinformation is utilised for modelling the load pseudo-measurements. In generating the load proles at a bus, theLPIs of various classes of customers were scaled up andadded together. The scaling of each LPI was done with thehelp of annual maximum demand. The process ofcomputing the real and reactive power demand proles atthe ith bus is given below:

where t is the time instant of the year sampan hour, P jPI(t) is the real power LPI value oof consumer at time instant t, P ji, max ismaximum demand in kW of the jth class ofbus i.

Step 2: Computation of real power load prole:step, the real power LPIs at a bus (say themultiplied with the corresponding scaling pathen added together to get the real power lthat bus

Pi(t) XNcj1

a jP jPI(t)

where t is the time instant of the year sampan hour, Pi(t) is the real power load in kWbus at time instant t and Nc is the numberclasses.

Step 3: Computation of reactive power loacomputing the reactive power load proleIET Gener. Transm. Distrib., 2010, Vol. 4, Iss. 1, pp. 5059doi: 10.1049/iet-gtd.2009.0167

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www.ietdl.orgpower load proles

Qi(t) XNcj1

ajQjPI(t) XNcj1

ajPjPI(t) tan(fj) (4)

In the above expression, fj is the angle of average powerfactor of the jth class of consumer and Qi(t) is the reactivepower load in kVAr at the ith bus at time instant t.

In this research, the typical power factors for all four classesof consumers were taken as 0.95, 0.99, 0.98 and 0.90 lagging,respectively.

2.1 Distribution of load proles

The density histogram of a load prole at a bus was generatedby segmenting the range of load prole data into variousdisjoint categories known as bins. The relative frequency ofload data falling in a bin was utilised to compute theprobability density of that bin. The computation ofthe relative frequency and probability density of each bin isfairly standard [16] and given in the Appendix. Once theprobability densities (pi) for all bins are computed using (22)in the Appendix the bar graph of probability density versusthe bin data range gives the probability density histogram.

The load probability density histograms at all the buseswere generated using the above procedure and thedistributions of real power load proles at buses 1 and 26are displayed in Fig. 2. It is evident that the distributionsystem loads do not follow any known distribution functionand hence, distributions met in practical life cannot be

Figure 2 Probability distribution of load at different buses

a at bus 1b at bus 26Gener. Transm. Distrib., 2010, Vol. 4, Iss. 1, pp. 5059i: 10.1049/iet-gtd.2009.0167simply described by the normal distribution function. Theliterature suggests that the probability density function(pdf) of loads with skewness can be modelled as log-normal or beta distribution functions [6, 10]. However, asshown in Fig. 2, this is not possible for all the buses in thedistribution network as the distribution functions atdifferent buses are not skewed. To model these loads aspseudo-measurements and accommodate them in the SEalgorithm is a difcult task.

An efcient way to overcome this difculty is to model allnon-Gaussian pdfs with Gaussian mixture pdfs, so that theresulting mixture pdf is a convex combination of differentGaussian components. The following section brieyexplains the method of approximating a non-Gaussian pdfwith GMM.

3 Gaussian mixture modelA GMM is the weighted nite sum of several Gaussiancomponents. A multivariate GMM is characterised by a setof weights, mean vectors and covariance matrices of themixture components and mathematically it can be expressed as

f (zjg) XMci1

wif (zjmi , Si) (5)

whereMc is the number of mixture components and wi is theweight of ith mixture component, subject to wi . 0and

PMci1 wi 1. g is chosen from the set of parameters

G {g:g {wi, mi , Si}Mci1}, a member of which denes aGM. f (zjmi, Si) is a multivariate normal distribution. Theparameters of the GMM components can be obtainedusing the EM algorithm. The EM algorithm worksrecursively and given gs, one step of the recursion yieldingformulae for gs1 is

ws1j 1N

XNi1

f ( jjzi, gs) (6)

ms1j PN

i1 zi f ( jjzi , gs)PNi1 f ( jjzi, gs)

(7)

Ss1j PN

i1 f ( jjzi , gs)(zi ms1j )(zi ms1j )TPNi1 f ( jjzi, gs)

(8)

where N is the number of data samples and

f ( jjzi, gs) wsj f (zijmsj , Ssj)PMck1 w

sk f (zijmsk, Ssk)

wsjN (msj , Ssj)(z)PMc

k1 wskN (msk, Ssk)(z)

(9)

Let {wj , mj , S

j }

Mcj1 denote the point of convergence.

The EM algorithm and the steps to obtain (6)(8) aredetailed in [14].53


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www.ietdl.orgThe EM algorithm to obtain the parameters of the GM,constructed by sampling from a given distribution has beenadopted in various application areas such as target tracking[17], clustering [18] and pattern recognition [19].

In the context of power distribution load modelling in thispaper, we have applied the GMM methodology to theUKGDS model to obtain the mixture components for realand reactive demands in all 55 buses. Univariatedistribution (d 1) of the loads was considered in thisstudy. The GMM components at every bus were obtainedusing the EM algorithm. The EM algorithm wasinitialised using the K-means clustering algorithm availablein MATLAB [20]. The algorithm was terminated whenthe relative difference of log-likelihood values in twoconsecutive iterations is below a threshold (i.e.j(Ln Ln1)=Ln1j thr). A threshold value of 0.001 wastaken as the termination criterion. Fig. 3 shows thedistribution of various mixture components in buses 1 and26.

3.1 Mixture reduction

There can be a situation in which a load can be represented bymore than one Gaussian mixture component. In these casesan equivalent Gaussian pdf representing that load isobtained by merging the relevant components. Themerging is based on the clustering algorithm that combinesmixtures into groups (clusters). The algorithm operates byselecting the component with the largest weight asprincipal component for a cluster, and merging allcomponents that are within a certain distance of the

Figure 3 GMM approximation of the load pdf with

a 3 Gaussian componentsb 5 Gaussian componentsThe Institution of Engineering and Technology 2009principal component. The distance measure is dened bySalmond [21]

Dp,j wpwj

wp wj(mp mj)TS1p (mp mj) (10)

where the subscript p denotes the principal component.All the mixture components satisfying Dp,j , T forj 1, 2, . . . , Mc are merged together. The componentswhich do not satisfy Dp,j , T are ignored. The thresholdT is determined by the x2-test using a 99% condence.The equivalent mean and covariance of the mergedcomponents are given by

wm Xj[I

wj (11)

mm 1wm

Xj[I

wjmj (12)

Sm 1wm

Xj[I

wj[Sj (mj mm)(mj mm)T] (13)

It is to be noted that because of the exclusion of somecomponents based on the above criterion, the summationof the weights of the components within the cluster willnot add to unity. This has been accounted for bynormalising the mean and variances of the cluster with wm.

4 State estimationThe improved approximation of the load described in theprevious sections is utilised for the state estimation. Thestandard WLS [22] problem was setup to minimise thefollowing objective function. It incorporates the standardmeasurement model

J [z h(x)]TR1z [z h(x)] (14)

where z h(x) ez, ez N (0, Rz) is zero mean Gaussiannoise with error covariance matrix Rz diag{s2z1, s2z2,. . . , s2zm}.

The derivative of this objective function is linearised anditeratively solved to obtain the update in the estimate of thestates as follows

Dx^ P^xHTR1z [z h(x)] (15)

where P^x is the state error covariance matrix as given by [23]

P^x (HTR1z H )1 (16)

4.1 Statistical measures of performance[2326]

Because of the statistical nature of pseudo-measurements, theperformance of the DSSE needs to be assessed throughIET Gener. Transm. Distrib., 2010, Vol. 4, Iss. 1, pp. 5059doi: 10.1049/iet-gtd.2009.0167

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www.ietdl.orgstatistical measures. Three such measures are bias,consistency and overall quality.

4.1.1 Bias: A state estimator is said to be unbiased if theexpected value of the error in the state estimate is zero.Mathematically an unbiased estimator can be dened as

E[(xt x^)] 0 (17)

where xt and x^ are the true and estimated state vectors,respectively.

4.1.2 Consistency: When the error in an estimatestatistically corresponds to the corresponding covariancematrix, the estimate is said to be consistent and so is themethod. One measure of consistency is the normalisedstate error squared variable

e (xt x^)TP^1x (xt x^) (18)

where P^x 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 [26]. For normally distributed errors, these boundsare dened by x2-statistics. Typically, a 95% condenceinterval is taken into consideration.

4.1.3 Quality: The quality of an estimate is inverselyrelated to its variance. It means larger variance implies poorquality. For the multivariate case, the trace of the errorcovariance matrix can be used to quantify the overall qualityof the estimate. The quality as function of the trace of theerror covariance matrix can be dened as

Qtrace ln1

tr(Px)

(19)

where Px( E[(xt x^)(xt x^)T]) is the numericallycomputed state error covariance matrix.

5 Simulation studyThe performance of the state estimator, consideringGaussian mixture algorithm to model pseudo-measurements, was evaluated on the UKGDS model. Thebus voltage magnitudes and angles were considered as thestate variables except at the reference bus (bus 1) for whichthe bus angle was assumed to be zero. The number of statevariables to be estimated was 189.

5.1 Load pseudo-measurements for stateestimation

Since the loads are not measured in the distribution network,they are modelled from the load prole data. The load prolefor a particular bus if not available is constructed fromthe knowledge of consumer behaviour, the number ofGener. Transm. Distrib., 2010, Vol. 4, Iss. 1, pp. 5059i: 10.1049/iet-gtd.2009.0167consumer classes, the maximum demand of each consumerclass and so on. We generated the yearly half hour loadprole at a bus by mapping the normalised behaviour ofvarious UK consumer classes through their annualmaximum demand at that bus using the proceduredescribed in Section 2. The information about maximumdemands are available with the DNOs as they aremonitored through the maximum demand indicators. Forthe UKGDS model, this information is available in [15].As per UKGDS research, the normalised behaviour of theconsumer class in a particular DNO system is generic [15].

The yearly half hour load prole at each bus is used toconstruct the load pdf at that bus using the procedure givenin Appendix and Section 2.1. Then the mixturecomponents of the bus load pdf are computed throughGMM off-line. The load proles and mixture componentsinformation are stored. The state estimator utilises thestored information at any point in time t according to thefollowing procedure:

Step 1: Select a particular load bus (say bus k).

Step 2: Retrieve all the mixture components that wereobtained earlier off-line through the GMM to representthe bus load pdf.

Step 3: At time t, obtain the value of the load zk(t) from thebus load prole information.

Step 4: Identify the mixture component or group ofcomponents from the following set Sk(t)

Sk(t) { j:j [ {1, 2, :::, Mc}; and jzk(t) mj j 3sj} (20)

(a) If Sk(t) has one element, take the variance of the selectedcomponent along with zk(t) in the WLS computation.

(b) In case Sk(t) has more than one element, apply themixture reduction algorithm described in Section 3.1 toproduce a single equivalent Gaussian distribution. Use theequivalent variance along with zk(t) in the WLScomputation.

5.2 Classication of measurement typesin DSSE

The measurements in the DSSE are used in three differentways as follows:

1. The real measurements are available through themeasuring instruments and have the uncertaintycorresponding to the accuracy of the instrument.

2. The virtual measurements are zero injections which arecompletely deterministic and they are modelled with verylow variance in order to make them consistent with MLEtheory.55


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www.ietdl.org3. The pseudo-measurements on the other hand are the loadvalues which are derived from the historical data, MV/LVtransformer ratings at substations and customer behaviour.Since the loads are not measured and their derived valuesthemselves are uncertain, they act as random variables.These random variables (loads) may take any value betweenthe minimum and maximum demands at different buses.The probability of random variable, taking a value, ismeasured through the probability distribution function.And if this random variable follows the normaldistribution, the variance of this distribution measures theuncertainty. If the probability distribution function is notnormal it can be classied into several Gaussiancomponents using the GMM. This divides the total loadvariation (between minimum and maximum demand) intodifferent segments (lower load, medium load and higherload) with each segment represented by a Gaussian pdf.The uncertainty of the loads in each segment is measuredthrough the variance of the corresponding Gaussiandistribution.

Now in the load prole, zk(t) represents the randomvariable and its variance obtained from the GMMrepresents the uncertainty. In DSSE this information isutilised along with the real measurements.

5.3 State estimation at different loadlevels

The WLS state estimation was run at various load levels bysampling the load proles at different time steps coveringthe whole year. From the half hour yearly load proles, theload values at every 100 h were sampled resulting in 88 testcases. A load ow corresponding to each time step wasused to generate the true values. Assuming that the realmeasurements are available at the main substation, only onevoltage measurement (bus 1) and two power owmeasurements (lines 12 and 185) were considered as thereal measurements. These measurements were generated byadding the Gaussian noise components corresponding to3% error in true values. Using the fact that in a Gaussiandistribution +3s deviation about the mean covers morethan 99.7% area of the Gaussian curve, the standarddeviation of real measurement was computed as follows

szi zt %error3 100

where zt is the true value of the real measurement. Inpractice, the standard deviation of the real measurement isobtained from sensor characteristics.

Virtual measurements were modelled with very smallvariance (1028).

It should be noted that the variances of the realmeasurements are small; pseudo-measurements are highvariance estimates of the loads and virtual measurementsThe Institution of Engineering and Technology 2009are modelled with very low variance. Such a high variationof the measurement variances may lead to ill conditioningof the gain matrix (HTR1z H ). Thus the WLS algorithmdescribed in Section may result in numerical instability ordivergence problem. In our research, the problem of illconditioning is taken care of by using QR decomposition[27].

The voltages and angles in the UKGDS were estimated atvarious time steps using GMM. The estimation results forthe two buses, one close to the main substation (bus 2) andthe other away from the main substation (bus 51) aredisplayed in Figs. 4 and 5, respectively. The true values ofthe voltages and angles along with the +3s condencebounds in the estimates are also shown in the gures.

It is observed that the estimates are within the bounds. Thestate estimates at bus 2 match closely with the true values as

Figure 4 True and estimated voltages and angles atdifferent time steps

Figure 5 True and estimated voltages and angles atdifferent time stepsIET Gener. Transm. Distrib., 2010, Vol. 4, Iss. 1, pp. 5059doi: 10.1049/iet-gtd.2009.0167

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www.ietdl.orgcompared to bus 51. This is expected because bus 2 is close tothe main substation where all the real measurements areavailable while bus 51 is away from the main substationand predominately inuenced by pseudo-measurements.Average absolute errors at different buses are also shown inTable 1. The estimation errors increase as one moves awayfrom the the main substation. These errors can be reducedby deploying additional real measurements at differentlocations in the network. Obviously the deployment shouldconsider the tradeoff between the cost and the accuracy ofthe estimates. Singh et al. [28] have proposed a techniquefor real measurement placement on the same network inorder to attain the pre-specied accuracies in both voltagemagnitude and angle estimates.

5.4 Statistical performance evaluation

To evaluate the statistical performance of GMM-basedWLSstate estimator 500Monte Carlo simulations were performed.Under the assumption that there is no correlation between thepseudo-measurements, the covariance matrix correspondingto the pseudo-measurements (Sm) becomes diagonal. In thiscase, the matrix Rz was obtained by diagonal augmentationof Sm and the measurement error variances of real andvirtual measurements. In all the simulations, it was foundthat all the states were unbiased, that is the expected value ofstate estimation error was zero.

Fig. 6 shows the consistency plot with 3% error in realmeasurements. It is clear from the gure that with acondence of 95%, the normalised error squared variable elies within the bounds obtained from the x2-statistics forthe normal distribution. Hence, the estimator is found tobe consistent. Since x2 bounds are based on the assumptionof normally distributed errors, the consistency of theestimator also indicates that the estimates are normallydistributed.

The quality of the estimates was evaluated for differentaccuracies of the real measurements. The results aredisplayed in Fig. 7. It is as anticipated that the qualityimproves with the increasing accuracy of the realmeasurements.

The voltage magnitude estimates (by considering the 3%accuracy in the real measurements) for all 500 simulationsat all the buses are shown in Fig. 8. The voltagemagnitudes corresponding to all the cases except one are

Table 1 Average absolute errors in voltages and angles

%Average absolute error (AAE)

Bus#2

Bus#9

Bus#51

Bus#95

100=NP

i jV^ i Vit=Vitj 0.03 0.068 0.16 0.38100=N

Pi jd^

i dit=ditj 2.27 3.22 8.28 11.16T Gener. Transm. Distrib., 2010, Vol. 4, Iss. 1, pp. 5059i: 10.1049/iet-gtd.2009.0167within the bounds dened by Vt + 3s. The values of swere obtained from the diagonal elements corresponding tovoltage magnitude components of the error covariancematrix P^x, averaged out over the Monte Carlo simulations.The true value of the voltage magnitudes and mean valueof the estimated voltage magnitudes for all the simulationsare shown in the middle of the plot, and represented by

Figure 6 95-bus system consistency plot-error in realmeasurements 3%

Figure 7 Overall quality of the estimates with differentaccuracy in real measurements

Figure 8 Voltage variations at all 95 buses with differentsimulations57


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www.ietdl.orgblack and white lines, respectively. It is clear that the mean ofthe voltage estimates over all the simulations is close to thetrue value. It is also clear from Fig. 8 that for some casesthe estimated voltages violate the +5% criterion. Thisproblem can be eliminated by installing additional realmeasurements [28].

6 ConclusionAn efcient approach for DSSE based on the GMM of loadsas pseudo-measurements is presented and theoreticallyjustied. The advantage of the approach is that all the loadpdfs irrespective of their distributions are represented byGMM approximation followed by the appropriatereduction. This approach is of particular value becausemethods for generating pseudo-measurements intransmission system state estimation cannot generally beextended to distribution networks mainly due to theabsence of adequate load measurements. The WLS iseffectively applied for the DSSE. The performance of theestimator is tested for different correlated loads at varioustime steps. However, the correlation information is notcaptured in the measurement error covariance matrix. As afuture extension of the work, a multivariate GMM isplanned to be developed to capture the correlation amongthe pseudo-measurements in the measurement errorcovariance matrix. The approach presented in this paper issimple to implement in a practical DMS environment.

7 AcknowledgmentThe authors thank Peter D. Lang of EDF Energy Networksfor his valuable suggestions and discussions.

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[27] JABR R.A., PAL B.C.: Iteratively reweighted least-squares

9 Appendix9.1 Probability density histogram

Let k be the total number of bins and ni be the number of thedata points that fall into the ith bin. The relative frequency ofthe observations ni can be written as

fi niN

(21)

whereN (Pki1 ni) is the total number of observations. Theprobability density of the observations ni was computed bydividing the above frequency by its bin width h

pi niNh

(22)

The bin width was computed using the FreedmanDiaconisrule [29] as follows

h 2 IQR(z)N 1=3

(23)

IETdoi

www.ietdl.orgimplementation of the WLAV state-estimation method,IEE Proc.-Gener. Transm. Distrib., 2004, 151, (1),pp. 103108

[28] SINGH R., PAL B.C., VINTER R.B.: Measurement placement indistribution system state estimation, IEEE Trans. PowerSyst., 2009, 24, (2), pp. 668675

[29] FREEDMAN D., DIACONIS P.: On the histogram as a densityestimator: L2 theory, Z. Wahrscheirdichkeitstheor.Verwandte Geb., 1981, 57, pp. 453476Gener. Transm. Distrib., 2010, Vol. 4, Iss. 1, pp. 5059: 10.1049/iet-gtd.2009.0167where z is the load data and IQR(z) is the interquartile rangeof z, which was obtained using the IQR command inMATLAB [20]. The number of bins was obtained usingthe equal width criterion

k max{z}min{z}h

(24)

where dxe is the smallest integer greater than or equal to x.59


distribution system state estimation through

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