Power system state estimation

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

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& The Institution of Engineering and Technology 2009 doi: 10.1049/iet-gtd.2008.0485

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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].

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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)



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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


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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



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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%


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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%



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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


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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%



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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


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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/



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


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/



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



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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.

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[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)


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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)



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