Decoupling Power System State Estimation by Means of Stochastic Collocation

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 60, NO. 5, MAY 2011 1623

Decoupling Power System State Estimation byMeans of Stochastic Collocation

Andrea Benigni, Graduate Student Member, IEEE, Junqi Liu, Graduate Student Member, IEEE,Ferdinanda Ponci, Senior Member, IEEE, Antonello Monti, Senior Member, IEEE,

Giuditta Pisano, Member, IEEE, and Sara Sulis, Member, IEEE

Abstract—This paper presents a new approach to the problemof system decoupling in a power-system state estimation problem.The complexity of power systems is growing, thus challenging theway measurements for state estimation are traditionally managed.Following a previous experience in defining a decentralized solu-tion for state estimation, the authors here propose a decentralizeddynamic state estimation method for a large-scale power systemin combination with a procedure to automatically identify howand which state information to exchange for reconstructing thestates starting from partial knowledge. In particular, the problemof selecting the variables that each observer has to estimate ispartially solved within the framework of a stochastic approach,i.e., the so-called collocation method. An optimization algorithmbased on dynamic programming is also developed to determinethe optimal set of strongly coupled variables necessary for a suffi-ciently accurate estimation. The developed method is evaluated byapplying to an IEEE test bus system.

Index Terms—Decentralized estimation, decoupling of systems,dynamic programming (DP), power systems, power-system stateestimation.

I. INTRODUCTION

POWER-SYSTEM state estimation is a key tool for theenergy management system. Based on measurement data,

the state estimator provides the network status information topower-system control centers to facilitate reliable monitoringand support decision making of control processes such as faultdetection, contingency analysis, and protection actions. Withsuch processes, a safe and efficient operation of power systemscan be achieved.

The classical power-system state estimation, as a part ofa centralized control system, is based on a nonlinear steady-state system model by measuring slow varying quantities, suchas real and reactive power injections and flows, and voltagemagnitude, for a certain time window to generate snapshots

Manuscript received June 20, 2010; revised November 1, 2010; acceptedNovember 8, 2010. Date of current version April 6, 2011. This work wassupported by the U.S. Office of Naval Research under Grant N00014-07-1-0818. The Associate Editor coordinating the review process for this paper wasDr. Carlo Muscas.

A. Benigni, J. Liu, F. Ponci, and A. Monti are with the Institute for Automa-tion of Complex Power Systems, E.ON Energy Research Center, Rheinisch-Westfälische Technische Hochschule Aachen University, 52074 Aachen,Germany (e-mail: [email protected]; [email protected]; [email protected]; [email protected]).

G. Pisano and S. Sulis are with the Department of Electrical and Elec-tronic Engineering, University of Cagliari, 09123 Cagliari, Italy (e-mail:[email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIM.2011.2124270

of the system conditions. The iterative weighted least square(WLS) algorithm is widely applied to solve the estimationproblem, e.g., in [1]–[3]. This approach requires a sequence ofmeasurement samples in order to conduct the estimation andcan only provide static or quasi-static state information. Thus,the classical state estimation is not able to capture the dynamicbehavior of the power system.

Dynamic state estimation for the power systems using re-cursive WLS or more efficient Kalman filter has been alsoinvestigated [4]–[6]. In [4] and [5], the dynamics of the systemis modeled by the static system model driven by the Gaussiannoise process, while the dynamic states are the node-voltagemagnitudes and the phase angles. In [6], the generator rotorangles and the generator speed are considered as dynamic statesto be estimated. In these works, the developed Kalman filteralso performs in a centralized manner.

The current evolution of the power system toward the so-called future smart grid is introducing new elements of noveltysuch as the integration of widely dispersed renewable energysources mostly through power electronics or the active partic-ipation of the end consumer in power systems through distrib-uted power generation and storage units. All these changes aretransforming the power grid in a system with fast dynamicsand much higher complexity. Thus, the classical static or quasi-static model of the power system is expected to become lessreliable for the reconstruction of dynamic system conditionsvia state estimation. Moreover, under the same circumstances,the centralized state estimation method requires a very longexecution time due to the large size of the system and the largeamount of measurement to be processed. Such a complexity isalso not compatible with the typical communication infrastruc-tures currently adopted in power systems.

This perspective thus leads to the concept of distributedstate estimation by distributing the computational effort toa group of local state estimators in order to enhance theperformance and the reliability of the future power grids. Inaddition, the introduction of synchronized phasor measurementunits (PMUs) [7], [8] also opens up possibilities of collectingand exchanging more accurate and more frequent measure-ments via synchronized data communication from differentmeasurement locations that also facilitates state estimation ina distributed manner. The distributed state estimation problemin power systems has been addressed extensively as in [9]–[12]. In [9], the estimation is based on the classic estimator incombination with a coupling-constraint optimization problem.The decoupling of the system is performed by simply ignoring

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1624 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 60, NO. 5, MAY 2011

the off-diagonal elements of the estimator gain matrix. In [10],it is assumed that the neighboring subsystem only affects theboundary buses, while in [11] and [12], the internal buses ineach subsystem are also affected by the neighboring subsystem.The partitioning of the system is achieved by inspection ongeographical basis or by removing the tie line measurements.The state estimation is obtained in two steps: local estimationin subsystems and aggregation at the coordination level witha centralized estimator. In these mentioned works, only thestatic state estimation is considered, and a large amount ofmeasurement data is required.

As already underlined in [20], even if, on one side, PMUshave the advantage of being a mature technique [13]–[16], theyalso have some limitations: The main limitation is the assump-tion to be in a quasi-steady state with very slow variationsof voltage and frequency. Even if a lot of problems such asfrequency variations and fault detection were widely analyzedin literature, this kind of approach does not seem adaptable forthe estimation of power systems with fast dynamics [17]. Theseconsiderations make the solutions based on PMUs unpracticalfor the future smart grid with high penetration of distributedgeneration and renewable energy sources where the assumptionof a quasi-static alternating-current (ac) condition is becomingincorrect. For example, the growing role of direct-current (dc)transmission, due to high-voltage dc light, is calling for ap-proaches that are able to support ac and dc estimations at thesame time [18], [19].

As consequence of these considerations, in [20], a time-domain dynamic state estimation approach applying a decen-tralized information filter (DIF) is proposed and validated inan experimental environment. The use of a dynamic modelallows for the relaxation of the redundancy requirement on thenumber of measured data, as compared with the classical stateestimator, and increases the reliability of the estimation againstthe failure of some measuring units and/or data communicationchannels. A problem that arose in the experimental validationperformed on a hardware setup, presented in [20], concerns theduration of the execution time of decentralized estimators. It isshown that the computational and communication times neededfor the state information exchange between the local estimatorssignificantly increase with the number of state variables to beestimated. In this perspective, when a large-scale power systemis considered, tradeoffs among the computational efforts, thecommunication time, and the estimation accuracy have to betaken into account.

In this paper, we present a novel approach aiming at re-ducing the complexity and the execution time of the DIFapplied to large-scale power system as a development of [20].In particular, the transmitted state information is limited toselected strongly coupled state variables in order to reduce thecomputational and communication efforts. Those with strongcoupling are transmitted for assimilation required by the DIF,whereas those with weak coupling are not transmitted withoutaffecting the estimation results significantly. The assessmentof strongly coupled state variables is done via a stochasticprocess, i.e., the so-called collocation method (CM), whichevaluates the sensitivity of each state variable with respect tothe other variables. The CM could achieve fast convergence

by using multivariate polynomial interpolations compared withother stochastic methods such as Monte Carlo (MC), and it issimple to implement. With an optimization process based onthe dynamic programming (DP), the minimal set of coupledstate variables are determined while guaranteeing a given es-timation accuracy in order to minimize the execution efforts ofestimation.

Preliminary simulation results on a simple test power systemwith four electric machines modeling four subsystems wereshown in [21]. In this paper, the DIF in combination with theCM and the DP is extended to form a distributed dynamicstate estimator for large-scale power systems. The proposedtechnique is evaluated in simulation on an IEEE 14 bus testsystem. The currents flowing through inductive transmissionlines and the currents drawn by the loads, as well as the voltageover the shunt capacitors in the three-phase power system,are considered as the dynamic states. This results in a large-sized state space model representing the interconnected systemwith all coupled state variables. By inspection on geographicalbasis, the overall system is decomposed into subareas. In eachsubarea, only few (even only one) measurements are chosento enable estimating all the state variables. Then, in order toreduce the complexity of the large-sized system model, theCM and the DP optimization taking into account the givenestimation accuracy are performed to determine the minimumset of strongly coupled state variables of different subareas. Inthis way, the state information transmitted between the localestimators of the subareas is minimized.

This paper is organized as follows: In Section II, the theoryabout the DIF is introduced. The CM is discussed in Section III.In Section IV, the optimization process of variable selectionwith the DP is proposed. The decomposition of a large-scalepower system for estimation purposes is explained with thehelp of a case study in Section V. In Section VI, the re-sults of the proposed technique including collocation indexcalculation, variable selection, and state estimation are shown.Moreover, the achieved accuracy of the state estimation, theelaboration, and communication time are also analyzed throughthe help of two examples.

II. DIF FOR DECOUPLED SYSTEMS

For the DIF, each observer uses a dynamic model of thesystem, as well as for the decentralized Kalman filter. The fore-casts of each observer are corrected based on locally performedmeasurements and on the information exchanged with otherareas.

Consider a power system modeled as a discrete linear systemwith the model in the following:

xi(k) = Aixi(k − 1) + Bi · ui(k − 1) + wi(k − 1) (1)

where index i refers to the ith partition of the considered powersystem. Matrices Ai and Bi describe the power subsystemin the state space. Vector xi(k) is the state vector (the threephase bus voltage and line current instantaneous values) attime k, including all states needed for the estimator of the ithpartition, ui(k) is the driving input vector (generator voltages),

BENIGNI et al.: DECOUPLING POWER SYSTEM STATE ESTIMATION BY MEANS OF STOCHASTIC COLLOCATION 1625

and wi(k) is the input noise modeled as a Gaussian independentidentically distributed (i.i.d.) random process with zero meanand E[wi(k)wi(j)] = Qi(k)δ(k − j).

The power system is observed by measuring few bus voltagesand line currents in zone i, and it is modeled with the followingobservation equation, derived on the basis of the Ohm andKirchhoff laws:

zi(k) = Cixi(k) + Di · ui(k) + vi(k) (2)

where zi(k) is the observation vector at time k and vi(k) is theassociated noise modeled as a Gaussian i.i.d. random sequencewith E[vi(k)vi(j)] = Ri(k)δ(k − j). Matrices Ci and Di areproper constant matrices.

As outlined in [20], given the state vector xi(k) and thecovariance matrix of its estimate, i.e., PA

i (k), the DIF requiresthe introduction of two new variables: the information matrixYi(k) and the information state yi(k). The expression of thetwo new variables is

Yi(k) =[PA

i (k)]−1

(3)

yi(k) =[PA

i (k)]−1 · xi(k). (4)

The DIF is implemented based on the following four funda-mental steps:

1) Initialization: In general, the information filter, i.e., theinformation matrix and state, is initialized by considering somegiven data, which has the covariance Pi(0). The initial statexi(0) can be chosen as xi(0) = 0, i.e.,

YAi (0) = [Pi(0)]−1 (5)

yAi (0) = [Pi(0)]−1 · xi(0) (6)

where superscript A indicates the assimilated values.2) Local Prediction: In this phase, the information matrix

and state are predicted via the power-system model usingtheir estimates YA

i (k − 1) and yA(k − 1) at the previous timestep, i.e.,

YFi (k) =

{Ai ·

[YA

i (k − 1)]−1 · AT

i + Qi(k − 1)}−1

(7)

Li(k) =YFi (k) · Ci ·

[YA

i (k − 1)]−1

(8)

yFi (k) =Li(k) · yA

i (k − 1) + YFi (k) · Bi · ui(k − 1). (9)

Li(k) is the information prediction gain and is the dual of theKalman gain.

3) Communication: In this step, the estimators of each itharea communicate with the others, exchanging matrix Ii(k) andvector ii(k) obtained from the following formulas:

Ii(k) =CTi · [Ri(k)]−1 · Ci (10)

ii(k) =CTi · [Ri(k)]−1 · [zi(k) − Di · ui(k)] (11)

where Ii(k) and ii(k) are the information associated with theobservation needed for each zone and zi(k) represents thevector of measurements in zone i and is related to the statethrough the measurement model (2).

4) Assimilation: In this phase, each observer assimilates thedata calculated with the local prediction and measurementswith the data obtained from the observers of other areas. Theassimilation yields the following new values of YA

i (k) andyA

i (k):

YAi (k) =YF

i (k) +∑j �=i

Ij(k) (12)

yAi (k) =yF

i (k) +∑j �=i

ij(k) (13)

where superscript F indicates values calculated in the predic-tion stage.

III. COLLOCATION METHOD

In this paper, to determine the level of coupling between twovariables, one of the two is perturbed, and the effect on theother is evaluated. A first remark to this approach is that, fora generic nonlinear system, the result of any perturbation isconsistent only for that specific testing condition and cannotbe generalized. To overcome this limitation, we have createda stochastic process, where the amplitude and the frequencyof perturbation are stochastic variables. We then introduce asinusoidal perturbation, with the frequency and the amplitudebeing stochastic variables.

Considering the overall system under test in (14), the objec-tive is to analyze the level of coupling between any two statevariables xi and xj ∀i �= j, i.e.,

x(k + 1) = f (x(k),u(k)) . (14)

After multiple simulations in different working conditions,the input vector became a vector of stochastic variables. Weobtain the sets of all possible values that variable xj can assumein the steady state. This information is used to define the rangeof variation for the stochastic variables.

To synthesize these results for the purpose of the estimator,a sensitivity function Si, to evaluate the level of coupling, isdefined as the following difference:

Si = diff(xA

i , x̂Ai

)(15)

where xAi is the estimation of xi in the nonperturbed system

and x̂Ai is the estimation of xi in the perturbed system. Since

we focus on the state estimation problem in the power system,the sensitivity is expressed in per unit.

A common numerical approach to solve (14), with a randominput, for the purpose of numerical simulation is the MCmethod. With the MC approach, a set of random realizationsfor the random inputs is chosen, and each realization is thensolved as a standard deterministic problem. This process pro-vides a set of solutions from which the statistical informationcan be extracted. The implementation of the MC approach isstraightforward, but the convergence can be slow. Consideringk realizations of the random inputs, the MC method asymp-totically converges at a rate of 1/

√k. This characteristic may

be unacceptable if the solution of the deterministic problem isalready expensive in terms of computation effort [24].


Fig. 1. Schematic flowchart of the DP.

To speed up the convergence, several methods have beendeveloped. The most well-known is the quasi-MC method(QMC). These methods are based on replacing the randominput points of traditional MC with a set of points that coversmore uniformly the hypercube, defined by the inputs. Undercertain conditions, the QMC method converges faster andprovides more accurate results than crude MC. However, onedrawback of the QMC, as compared with MC, is the muchhigher order [25], [26].

Another approach to solve the aforementioned stochasticproblem is the use of nonsampling methods such as perturba-tion method and second-moment analysis. These methods areusually suitable for systems with limited number of the inputsand outputs. Nonsampling methods that offer fast convergenceand are applicable to systems with large sets of inputs andoutputs are the Stocasthic Galerkin methods. If the solutionof the problem is sufficiently smooth in the random space, theStocasthic Galerkin methods provide a fast convergence rate.The main limitation of this approach, however, is the need for adedicated solver.

A more efficient method is the CM that combines, in a way,the crude MC and the Galerkin methods. When the solutionof the problem is sufficiently smooth, the CM achieves fastconvergence like the Garlerkin methods by using multivariatepolynomial interpolations. In addition, the CM is trivial to im-plement since it only requires the solution of the deterministicproblem at each sampling point.

The core issue of the CM is the definition of the interpolationpoints, which also identify the different CMs. For this imple-mentation, a sparse Clenshaw–Curtis grid with dimensionaladaptivity is chosen. A comprehensive review of CMs is givenin [22]. To solve the stochastic problem using collocation, aMATLAB toolbox developed at the Massachusetts Institute ofTechnology is used [22], [23].

IV. OPTIMIZATION OF SYSTEM DECOUPLING

An optimization algorithm based on the DP is developedto determine the optimal set of strongly coupled variables inorder to decouple the system with respect to a given estimation

performance metric. This optimization process is conductedoffline for the all possible configurations of the network.

In principle, the DP enables the decomposition of the originalproblem into subproblems and finds the best solution of thesubproblems, beginning from the problem smaller in size. TheDP can be successfully applied when:

1) the problem can be divided into stages and a decision isrequired at each stage;

2) a finite number of states is associated with each stage;3) the decision at one stage transforms one state into a state

in the next stage;4) a recursive relationship, provided that the states at stage

j − 1 are known, identifies the optimal decisions to reachthe states at stage j;

5) the recursion for determining the optimal decisions atstage j only depends on the states at stage j − 1 and noton the way these states have been reached.

The problem of the optimal decoupling may be formalizedaccording to the aforementioned points. Inputs of the optimiza-tion process are the results of the CM, i.e., the sensitivitiesindex Si (see Section III), and the estimation results of the DIFbased on the overall system (i.e., the estimators estimate all thestate variables). The coupling indexes between state variablesdefined in (15), resulting from the CM, are used to drive theDP to choose set of variables strongly coupled. The estimatesof the complete DIF are assumed as the reference condition forthe state variables; they represent the best estimates that the DIFcould provide, when the communication links between one areaand all others are active. Fig. 1 depicts the approach used hereto solve the problem according to the DP paradigm.

The procedure starts assuming that the state variables couldbe all decoupled. According to this hypothesis, at the startingpoint (stage D0), the state variables of each zone are estimatedwithout any information about the others. In other words, eachzone estimator observes only the local state variables.

At each decision level, i.e., stages DI,DII, . . . , Dn, onemore state variable is added to the temporary optimal set. Thesubscript of the stage represents the total number of variablesadded to each estimator with respect to the initial number. Eachstate (α, β, etc.) in each stage identifies the added variable to


Fig. 2. IEEE 14 bus test system.

be estimated by a given estimator and represents the set of thetemporary optimal estimated variables.

The transition from the ith stage to the next one [(i + 1)th]is driven by the evaluation of label L(i+1)th (cost function)for each state in the (i + 1)th stage. The accuracy constraintis checked at each stage.

The optimization stops when the accuracy constraint is com-pliant for at least one set of variables.

In order to clarify the process (a more detailed explanationcan be found in [27]), let us suppose that, for a given estimatorX , state β in stage DII has to be reached from DI. Possiblestates in DI are α, γ, . . . , η, each one labeled with the optimalvalue of the objective function LI. Preliminarily, all the couplesformed by adding β to α, γ, . . . , η in DI are examined, andthe objective functions are assessed (one for each couple).State β at level DII is then labeled with the minimum valueof such objective functions LII(B). If n calculated values ofL, corresponding to n states, are identical, the algorithm willchoose the variables to be added to the current optimal set onthe basis of a merit list ordered with the coupling indexes of thecollocation. By doing so, the optimal policy to reach β at levelDII from DI is univocally determined.

A critical point of the overall procedure is the definitionof both the stopping criterion and the cost function. In theproposed application, the decision to pass from one stage to thenext is based on the accuracy constraint. For instance, if none ofthe solutions at level DII complies with the accuracy constraint,then the procedure is iterated by labeling the candidates atthe next stage until the goal has been reached for all thezones of the network. The accuracy constraint is checked by

Fig. 3. IEEE 14 bus test system partitioned.

TABLE ILOCAL MEASUREMENTS IN EACH ZONE

comparing the current estimates (i.e., the result of the executionof the DIF with the current optimal set of variables) with thereference condition obtained by the complete DIF. In particular,the moving average of the percentage errors in a given timeinterval must be less than the given constraint (e.g., 10% in thispaper). The time interval is opportunely chosen as a tradeoffbetween the dynamic phenomena that we aim to observe andthe elaboration time.

As for the cost function, the following is considered:

L = a ·(3 · N3

i + 2 · M3i

)+ b · Zi (16)

where a and b are suitable coefficients related to elaborationand communication time, Ni is the number of the variablesestimated by the ith estimator, Mi is the number of local mea-surements, and Zi is the number of the zones that communicatewith the ith estimator. The form of (16) has been chosen forthe cost function since representing the total execution timeindicates the dynamic limit of the observer. A bigger value of


Fig. 4. Sensitivity function between (a) I 3–4 and 2–3 and (b) I 3–4 and 12–13.

cost function implies a slower refresh rate and thus limits therange of dynamic solicitations that can be estimated.

V. CASE STUDY

The proposed algorithm is verified for an IEEE 14 bus testpower system. As shown in Fig. 2, the power system consideredis characterized by two voltage levels of 132 and 33 kV, withtwo generators connected to nodes 1 and 2 and three compen-sators connected to buses 3, 6, and 8. In the considered three-phase power system, the state variables are chosen as thetransmission line currents, the transformer currents, the currentsfeeding the loads, the voltage over the shunt capacitors, and thevoltage at the output terminal of the compensators to create acorrect state space representation. This results in a dynamic sys-tem model with 123 state variables. As system inputs, the phasevoltages of the generators connected to buses 1 and 2 areconsidered.

First, the network has to be divided in zones, and the mea-surement apparatus has to be placed. As shown in Fig. 3, the testsystem is decomposed into four subareas (zones) by inspectionon geographical basis that is implied by the given inductancevalues for the transmission lines of the standard test system.Since, in the start, we do not have any information regardingthe coupling between variables, we assume that geographicallydistant quantities, in general, are more decoupled than theclose ones.

How to choose the position and the amount of measurementsfor each observer is a critical topic that has been alreadyanalyzed in literature as in [27] and [28]. The use of the DIFimposes new requirements in the definition of the number andthe position of measurement devices to be chosen such that astraightforward use of the methods proposed in the mentionedpapers is not appropriate. The first consideration is to select asmall set of measurements in order to reduce as much as pos-sible the economic cost of installation. In a previous work, wedetermined that, if the number of measurements is really scanty,it is not an appropriated choice to measure quantities that

TABLE IICOUPLING INDICES FOR THE CONSIDERED EXAMPLE

TABLE IIIVARIABLE CONSIDERED FOR 10% ACCURACY

TABLE IVNUMBER OF SELECTED VARIABLES

are also state variables of the system. Choosing measurementquantities that are not state variables means that we reduce thesparsity of matrix C so that each measurement impacts on morestate variables and, in general, thus increases the quality ofthe estimation of a greater number of variables. Therefore, thefirst requirement for choosing the measurements’ position is toselect quantities that are not state variables. The second require-ment considered is to acquire measurements only at the loca-tions where the observers are physically placed. Since the maingoal of this paper is to optimize the selection of variables thateach observer estimates but not to find the optimal allocationof measurements, a minimum set of measurements, as listed inTable I, has been manually selected according to the require-ments just exposed. The formalization and the optimizationof the location selection could be investigated in future works.


Fig. 5. Estimation results of case 1 according with Table V. Zones (a) 1, (b) 2, (c) 3, and (d) 4.

VI. RESULTS

A. Collocation

The first step for determining the state variables that can beneglected in the estimation process is to calculate the couplingindex using the collocation approach reported in Section III.Each state variable has been perturbed using the informationfrom previous simulations. Each perturbation is determined bytwo stochastic variables: the amplitude and the frequency. Thestochastic variable determining the amplitude of the pertur-bation has a uniform distribution between zero and 10% ofthe nominal value of the considered variable. The stochasticvariable determining the frequency of the perturbation for eachvariable has a uniform distribution between 10 and 50 Hz. Thisrange of frequency is not related with the system frequency,but it is the speed of variation that we imagined for thevariables of the system as driven by a variation in the loador the generator. It is important to underline that the methodproposed is completely independent of the value chosen for thisparameters. The sensitivity function, as defined in Section III,is then synthesized in a single index that is the obtained max-

imum value applying a moving mean value on the sensitivityfunction.

To better show how a calculated index can give an indicationof the coupling level between two variables, an example isshown in Fig. 4. The mean and the variance, in p.u., of thesensitivity index by observing the current on phase A betweenbuses 3 and 4 and by perturbing the current on phase A betweenbuses 2 and 3 are shown in Fig. 4(a). As comparison, Fig. 4(b)shows the results while still observing the current on phaseA between buses 3 and 4 but the perturbed variable is nowthe current on phase A between buses 12 and 13. In Table II,the coupling indexes calculated using the moving mean areshown.

This example is an extreme case by considering the structureof the power-system results such as the ones shown in Fig. 4.Table II confirms our interpretation that the current on lines 3and 4 is influenced much more by a variation of the current onlines 2 and 3 than that on lines 12 and 13. At the same time, withso-calculated indexes, a numerical value is given to the couplinglevel between each couple of variables. In Section VI-B, thecalculated indexes will be used to identify, for each observer,


Fig. 6. Comparison of the estimation errors (reference-estimates) between the two cases, according with Table V. Zones (a) 1, (b) 2, (c) 3, and (d) 4.

the minimum set of variables that guarantee a defined level ofaccuracy of the estimation.

B. Variables Selection

The variable selection has been performed, imposing a maxi-mum accepted estimation error equal to 10% for a restricted setof variables for each zone. A set of variables has been selectedfor each zone, assuming that it is not necessary to estimate allthe state variables with the same level of accuracy and that theaccuracy needed for each estimated variable is a function of thecontrol strategy adopted.

Assuming a generic control strategy, we selected the statevariables reported in Table III.

To show the relations among the accuracy level, the numberof variables, and the cost function, we choose here two setsof variables that both satisfy the condition of a maximumaccepted estimation error equal to 10% but with very differentvalues of cost function. The optimization algorithm has tobe able to choose among the possible sets of variables the

TABLE VVARIABLES CHOSEN TO EVALUATE THE RESULTS OF THE

STATE ESTIMATION

TABLE VICOMPARISON OF THE ESTIMATION ACCURACY

BETWEEN THE TWO CASES

set of variables that minimize the cost function while stillsatisfying the accuracy constraint. In Table IV, the number ofvariables selected in the two cases is shown. In the succeeding


TABLE VIICOMPARISON OF THE COMPUTATIONAL AND COMMUNICATION TIMES BETWEEN THE REFERENCE CASE AND THE TWO CASES

subsections, results about the accuracy and the cost function arepresented.

C. Estimation

To evaluate the results of the selection, four variables arechosen as reference, i.e., one for each zone (see Table V).

Each variable represents the worst case behavior (maximumerror) among the set variables selected for each zone.

As an example, Fig. 5 shows the estimation results of case 1in comparison with the reference case, where the reference caseis the estimation performed using all the variables of the powersystem.

As shown in Fig. 6 and Table VI, the estimation errorobtained with the two sets of variables is very different. In case1, we have a maximum error observed on the current throughphase A between nodes 12 and 13 in zone 4 with a value closeto 7%, whereas in case 2, the maximum error is found on thecurrent through phase A between nodes 3 and 4 in zone 2 witha value of 1.4%.

D. Cost Function and Execution Time

The cost function as defined in (16) is the sum of thecomputational and communication times.

Table VII shows the values of cost functions for both con-sidered cases. These values have been computed, assuminga = 2 ns and b = 2 ms.

As shown in Table VII, the second case has a larger computa-tional cost, and thus, the set of variables associated with case 1has to be preferred by the selection algorithm. It is obvious thatthe hypothesis behind the selection of an error tolerance is thatany estimation that defers from the real value within the fixedtolerance interval is equally valuable. Thus, the better accuracyprovided by the second case cannot be taken in account in theselection of the variables set.

Moreover, since the communication among all the observersis still required, the communication time remains equal to 6 ms.If we look at the computational time, we can appreciate how thereduction in the number of locally estimated variables resultsin a significant reduction of the computational time. With theselection of case 1, the elaboration time is 82% smaller than theoriginal one, whereas the total execution time is 66% smaller.It is important to underline that, since the observers have tosynchronously work, the refreshment time of all observers hasto be synchronized with the execution time of the slowestobserver, in this case, the one in zone 3.

VII. CONCLUSION

In this paper, an approach to system decoupling for thedistributed power-system state estimation, based on the DIF,the CMs, and the DP, has been proposed. The goal is toreduce the execution time of the DIF, reducing elaboration andcommunication effort. To do this, each observer has limited itsestimation only to a reduced set of state variables. Variableshave been selected according with their level of coupling. Thecoupling level among the state variables has been evaluated bymeans of a CM. The proposed optimization algorithm, based onthe DP, has been able to find the optimal set of strongly coupledvariables necessary to provide a state estimation of the systemthat complies with a predetermined accuracy level. This newapproach has been applied to a realistic test case, i.e., the IEEE14 bus power system.

Results and considerations about the variables’ selection, ac-curacy, and elaboration time have been presented and discussed.As shown in Section VI, the communication among all fourzones is still necessary. However, due to the reduction of theinformation transmitted and to the reduction of the number ofvariables estimated by each observer, a significant reduction ofthe elaboration time has been obtained.

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[7] A. G. Phadke, J. S. Thorp, and K. J. Karimi, “State estimation with phasormeasurements,” IEEE Power Eng. Rev., vol. 6, no. 2, p. 48, Feb. 1986.

[8] J. S. Thorp, A. G. Phadke, and K. J. Karimi, “Real time voltage-phasormeasurement for static state estimation,” IEEE Trans. Power App. Syst.,vol. PAS-104, no. 11, pp. 3098–3106, Nov. 1985.

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Andrea Benigni (S’09) received the M.S. degreefrom the Politecnico di Milano, Milano, Italy,in 2008. He is currently working toward thePh.D. degree in the Automation of ComplexPower Systems, E.ON Energy Research Center,Rheinisch-Westfälische Technische Hochschule(RWTH) Aachen University, Aachen, Germany.

He was with the University of South Carolina,Columbia, as a Visiting Scholar and, from August2008 to December 2008, a Ph.D. Student. SinceJanuary 2009, he has been a Research Associate

with the Institute for Automation of Complex Power Systems, E.ON EnergyResearch Center, RWTH Aachen University. His research activities mainlyconcern state estimation and real-time simulation.

Junqi Liu (S’10) received the Dipl.-Ing. (FH)degree from the University of Applied SciencesMunich, Munich, Germany, in 2007 and the M.Sc.degree from the Technische Universität München,Munich, in 2009. He is working toward the Ph.D.degree in the Institute for Automation of ComplexPower Systems, E.ON Energy Research Center,Rheinisch-Westfälische Technische Hochschule(RWTH) Aachen University, Aachen, Germany.

He is currently a Research Associate in the In-stitute for Automation of Complex Power Systems,

E.ON Energy Research Center, RWTH Aachen University. His researchinterests include advanced monitoring, control, and optimization techniques fornext-generation power systems.

Ferdinanda Ponci (M’00–SM’08) received thePh.D. degree in electrical engineering from thePolitecnico di Milano, Milano, Italy, in 2002.

In 2003, she was an Assistant Professor with theDepartment of Electrical Engineering, University ofSouth Carolina, Columbia, where she was tenuredand promoted to Associate Professor in 2008. Since2009, she has been with the Institute for Automa-tion of Complex Power Systems, E.ON Energy Re-search Center, Rheinisch-Westfälische TechnischeHochschule Aachen University, Aachen, Germany,

where she is currently an Apl. Professor. Her research activity is focused onagent-based applications in monitoring and energy management, distributedstate estimation, and modeling and simulation of uncertain systems.

Antonello Monti (M’94–SM’02) received the M.S.and Ph.D. degrees in electrical engineering from thePolitecnico di Milano, Milano, Italy, in 1989 and1994, respectively.

From 1990 to 1994, he was with the researchlaboratory of Ansaldo Industria of Milan, designingthe digital control of large power drives. In 1995, hewas an Assistant Professor with the Department ofElectrical Engineering, Politecnico di Milano. From2000 to 2008, he was an Associate Professor andthen a Full Professor with the Department of Elec-

trical Engineering, University of South Carolina, Columbia. He is currentlythe Director of the Institute for Automation of Complex Power Systems,E.ON Energy Research Center, Rheinisch-Westfälische Technische HochschuleAachen University, Aachen, Germany.

Giuditta Pisano (M’03) received the M.S. degreein electrical engineering and the Ph.D. degree inindustrial engineering in 2001 and 2006, respec-tively, from the University of Cagliari, Cagliari, Italy.

She is currently a Postdoctoral Researcher withthe University of Cagliari. Her research interests arein the field of power quality, distribution networkplanning and optimization, distributed generation,state estimation, active distribution networks, andmicrogrids.

Dr. Pisano is a member of the Federazione Italianadi Elettrotecnica, Elettronica, Automazione, Informatica e Telecomunicazioniand participates in the CIGRE working group, study committee 6-11.

Sara Sulis (S’04–M’06) received the M.S. degreein electrical engineering and the Ph.D. degree in in-dustrial engineering from the University of Cagliari,Cagliari, Italy, in 2002 and 2006, respectively.

Since March 2006, she is Assistant Professor ofelectrical and electronic measurements with the De-partment of Electrical and Electronic Engineering,University of Cagliari. Her main research activitiesare in the field of power quality measurements andstate estimation.

Decoupling Power System State Estimation by Means of Stochastic Collocation

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