Fully Distributed State Estimation for Wide-Area Monitoring Systems

1154 IEEE TRANSACTIONS ON SMART GRID, VOL. 3, NO. 3, SEPTEMBER 2012

Fully Distributed State Estimationfor Wide-Area Monitoring Systems

Le Xie, Member, IEEE, Dae-Hyun Choi, Student Member, IEEE, Soummya Kar, Member, IEEE, andH. Vincent Poor, Fellow, IEEE

Abstract—This paper presents a fully distributed state estimationalgorithm for wide-area monitoring in power systems. Through it-erative information exchange with designated neighboring controlareas, all the balancing authorities (control areas) can achieve anunbiased estimate of the entire power system’s state. In comparisonwith existing hierarchical or distributed state estimation methods,the novelty of the proposed approach lies in that: 1) the assumptionof local observability of all the control areas is no longer needed;2) the communication topology can be different than the physicaltopology of the power interconnection; and 3) for DC state esti-mation, no coordinator is required for each local control area toachieve provable convergence of the entire power system’s states tothose of the centralized estimation. The performance of both DCandAC state estimation using the proposed algorithm is illustratedin the IEEE 14-bus and 118-bus systems.

Index Terms—Distributed state estimation, power system com-munication, wide-area monitoring systems.

I. INTRODUCTION

T HE electric power industry is undergoing profoundchanges as our society emphasizes the importance of

a smarter grid in support of sustainable energy utilization.Technically, enabled by advanced control, communication,and computation, wide-area monitoring systems (WAMSs) ofthe future are likely to involve many more fast informationgathering and processing devices (e.g., phasor measurementunits) [1]. Institutionally, power industry deregulation has ledto the creation of multiple regional transmission organizations(RTOs) to operate portions of a large interconnected powersystem [2]. Both technical and institutional changes suggest theneed for more decentralized estimation and control in wide-areapower system operations [3].The main objective of this paper is to propose a fully dis-

tributed approach to state estimation in large multi-area powersystems. State estimation is one of the key functions in control

Manuscript received May 15, 2011; revised November 02, 2011 and March01, 2012; accepted April 21, 2012. Date of publication May 30, 2012; date ofcurrent version August 20, 2012. This work was supported in part by Power Sys-tems Engineering Research Center (PSERC), in part by Texas Engineering Ex-periment Station (TEES), and in part by the DTRA under Grant HDTRA1-07-1-0037. Paper no. TSG-00178-2011.L. Xie and D.-H. Choi are with the Department of Electrical and Computer

Engineering, Texas A&M University, College Station TX 77843 USA (e-mail:[email protected]; [email protected]).S. Kar is with the Department of Electrical and Computer Engi-

neering, Carnegie Mellon University, Pittsburgh, PA 15213 USA (e-mail:[email protected]).H. V. Poor is with the Department of Electrical Engineering, Princeton Uni-

versity, Princeton NJ 08540 USA (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSG.2012.2197764

Fig. 1. Contrast of communication architectures for hierarchical and fully dis-tributed state estimation.

centers’ energy management systems (EMSs) [2], [4]. A stateestimator converts redundant meter readings and other availableinformation obtained from a supervisory control and data acqui-sition (SCADA) system into an estimate of the state of an inter-connected power system [5] and distribution system [6]. Whilelarge power interconnections such as the eastern/western inter-connections are usually operated by several RTOs, advanced ap-plications such as wide-area monitoring and control require thestate of the entire system to be available to all the RTOs [3],[7]. This creates the need for a more decentralized approach toestimating the entire interconnection’s state information via ad-vanced communication. Fig. 1 contrasts the communication ar-chitecture of fully distributed state estimation with that of hier-archical state estimation in a multi-area power system.Several approaches to more decentralized state estimation

(see [8] and [9], for example, for a treatment of decentralizediterative algorithms for system analysis and optimization) havebeen proposed in the literature. In [10] and [11], a star-likehierarchical state estimation method was proposed. More re-cently, two-level state estimation for multi-area power systemhas been studied in [12]–[14] and [15], driven by the capabilityand need to conductWAMS. The local state estimation obtainedat the first level is coordinated at a higher level via synchro-nized phasor measurements. A survey of multi-area state es-timation is given in [16]. Most recently, a multilevel state es-timator (feeder, substation, transmission system organization,and regional levels) is described for the purpose of monitoringlarge-scale interconnected power systems [17]. However, as the

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XIE et al.: FULLY DISTRIBUTED STATE ESTIMATION FOR WIDE-AREA MONITORING SYSTEMS 1155

number of measurements and sampling rate increase, hierar-chical state estimation approaches may suffer from communi-cation bottleneck and computational reliability issues inherentin a system architecture with one single coordination center. Aparallel and distributed state estimation system was envisionedin [18]. By leveraging the naturally decoupled system character-istics of weighted least squares (WLS) estimation, the state esti-mation problem is decomposed into each area’s local estimatorwith a coupling constraint optimization technique to ensure con-vergence of the boundary buses’ estimates. Numerical results il-lustrate that the distributed algorithm can not only speed up thecomputational time, but also yields acceptable accuracy. How-ever, local observability of each control area is always requiredin the aforementioned algorithms. In other words, all the localcontrol areas need to have enough measurement redundancy inorder to compute the locally decoupled weighted least squaresestimates (excluding the boundary bus measurements). This as-sumption may not always hold due to 1) the increasing vul-nerability of measurements subject to potential bad/maliciousdata, and 2) the emergence of smaller control areas such asmicro-grids. A fully distributed algorithm for estimating powersystem dynamic states is proposed in [19] without a requirementof local observability. However no analytical study has beenconducted for provable convergence of the distributed state es-timation algorithms to the centralized estimates.In this paper, a fully distributed static state estimation algo-

rithmwith relaxed local observability is exploited. Starting fromone of the authors’ recent work [20], an iterative distributed stateestimation scheme is proposed, under which the local controlareas begin with their own estimates of the entire system, com-municate their estimates with pre-specified neighboring controlareas, and eventually making all local estimates converge to thecentralized state estimation result. In summary, the main contri-bution of this paper is threefold:• A distributed, fast state estimation algorithm is proposed. Itdoes not require either local observability or a central coor-dinator. As long as the entire interconnection is observableand the communication graph is connected (which does notnecessarily need to overlap with the physical topology ofthe power network), all local areas’ estimates of the entiresystem will converge to the centralized estimates.

• The convergence rate of the proposed distributed estima-tion algorithm is analytically shown for DC weighted leastsquares estimation.

• The proposed algorithm is implemented for both AC andDC state estimation for wide-area power systems.

This paper is organized as follows. In Section II, the problemformulation for distributed state estimation is introduced. Theproposed distributed state estimation algorithm is presented inSection III. Analytical results on the convergence of the pro-posed algorithm are also discussed. Illustrative case studies ofboth distributed DC and AC state estimation using the proposedalgorithm are presented in Section IV. Concluding remarks anda discussion of future work are included in Section V.

II. PROBLEM FORMULATION

A. Preliminaries

In -dimensional Euclidean space , the identity ma-trix is denoted by , while and represent the column vec-tors with all ones and all zeros in , respectively. The operator

applied to vectors and matrices corresponds to the stan-dard Euclidean 2-norm and the induced 2-norm, respectively.The induced 2-norm is equivalent to the matrix spectral radiusfor symmetric matrices.In this paper, we assume that all the random variables are

defined on a common measurable space, . In addition,all inequalities involving random variables are to be considereda.s. (almost surely); see [21].In the spectral graph theory literature, for an undirected graph

is the set of nodes or vertices with, and is the set of edges with , where

denotes the cardinality. The unordered pair belongs to theset if nodes and are connected to each other through anedge. We consider only simple graphs that contain no self-loopsand multiple edges. A graph is connected if there exists a path,1

between each pair of nodes. The neighborhood of node isdefined as

(1)

Node has degree , the number of neighboringedges of node . The structure of the graph can be expressedby the symmetric adjacency matrix , inwhich element , if , and , other-wise. Assuming that the degree matrix is the diagonal matrix

, the graph Laplacian matrix, , is

(2)

Due to a positive semidefinite property of the Laplacian matrix,its eigenvalues can be ordered as

(3)

The smallest eigenvalue is always equal to zero, withbeing the corresponding normalized eigenvector.

The multiplicity of the zero eigenvalue equals the number ofconnected components of the network; for a connected graph,

. This second eigenvalue is the algebraic connectivityor the Fiedler value of the network; see [22], [23], or [24] fordetailed treatment of graphs and their spectral theory. For thecomputation of vectors, Kronecker products will be involved inmost of the matrix manipulations. For example, the Kroneckerproduct2 of the matrix and will be anmatrix, denoted by .

B. Multi-Area Power System State Estimation

An interconnected multi-area power system is assumed tobe partitioned into a total of regions, each region cor-responding to a geographically non-overlapping control area.Each control area is allowed, if necessary, to exchange infor-mation with its neighboring areas. The measurement model formulti-area state estimation is formulated as follows:

(4)

where is measurement vector (including the boundary injec-tion and flow measurements) in control area is the state

1A path between nodes and of length is a sequenceof vertices, such that .

2The Kronecker product of an matrix with an matrixis an matrix denoted by whose th block is the

matrix , where denotes the th (scalar) entry of .


vector of the entire interconnected power system, is anonlinear measurement function for control area , and is ameasurement error vector with zero mean in area .Initially, we study the linearized DC state estimation problem

with 1.0 per unit (p.u.) voltage magnitudes at all buses and j1.0p.u. branch impedance. Then, the state vector is taken as thevoltage phase angle vector for all control areas. Therefore, thenonlinear measurement model for multi-area state estimation(4) is modified to

(5)

Centralized state estimation computes the optimal estimateof by minimizing the weighted least squares of measurementerror:

(6)

(7)

where and. is the positive definite covariance ma-

trix of the noise vector for area . Definingand for each control area , the centralizedweighted least squares estimate of is given by

(8)

where and . The solutionin (8) is obtained under the following assumption:Assumption (E.0)—Global Observability: The matrix

(9)

is full-rank.Remark 1: Under assumptions (E.0), the weighted Gramian

(10)

is also full-rank.It is obvious that the centralized computation for the optimal

estimate requires the knowledge of all the measurement Ja-cobian matrices , the covariances , and observationsat the central control center. In the next section, we study theproposed distributed - (Modified-Coordinated State Es-timation) algorithm, where by inter-control area data exchange,each control area with only its local measurements and the cor-responding Jacobian matrix in the network is able to constructthe estimate .

III. DISTRIBUTED ALGORITHM

In this section, we investigate the distributed estimation al-gorithm - for wide-area monitoring systems. The pri-mary goal of this section is to show that for linearized DC powerflow-based measurement models, it is possible to design totally

distributed iterative schemes where each control area convergesalmost surely3 (a.s.) to the centralized least squares estimatorof the state . In particular, we show that the convergence ofthe - algorithm at each control area holds true underthe assumption of global observability and connectivity of theinter-control area communication network. In [20], [25], and[26], it is shown that the algorithm - can be extended 1)to yield more general centralized estimates at each control area,for example, the maximum likelihood estimate; 2) to deal withnonlinear observation models; 3) to operate in unpredictableenvironments with random inter-control area communicationfailure or transmission noise; and 4) to process observations ar-riving sequentially over time. The second statement mentionedabove implies that the algorithm - can be applicable tononlinear AC state estimation. However, due to including theinversion of the nonlinear measurement functions in adistributed iterative algorithm, it is intractable to implement dis-tributed AC state estimation. Instead, we adopt a new approachof updating a linearized Jacobian matrix without the inversionof the corresponding nonlinear measurement function. In otherwords, each local linearized Jacobian matrix is updated onceevery few iterations to assure the convergence of distributedstate estimation to centralized state estimation. In the next sec-tion, it is shown by simulation results that the performance ofdistributed AC state estimation based on the proposed approachis satisfactory.

A. DC State Estimation

For simplicity, we assume that the vector of initial estimatesof the states, , is deterministic where is the totalnumber of buses. A sequence of estimate vectors, , iscomputed by each control area in a distributed iterative manner.The state estimate vector of the th control area at the

th iteration is a function of its previous estimate vector,the communicated estimate vectors at the th iteration from itsneighboring control areas, and the local measurement vector .Algorithm - : Based on the current state vector ,

the exchanged data , and the measurement vector, we update the estimate of the states at the th control area

via the following distributed iterative algorithm:

(11)

In (11), are appropriately chosen time-varyingweight sequences (here, time implies iteration). Algorithm (11)is distributed because for the th control area, it involves onlythe data from the sensors in its neighborhood .

3Note here that all estimates (centralized or distributed) are random objects,being a functional of the random observations . Hence, any meaningful con-vergence of such estimate sequences needs to be interpreted in a probabilisticsense. In this paper, all convergence results are shown to hold in the almost sure(with probability one) sense, which means convergence for a set of sample pathsor instantiations having probability one.


The iterations in (11) can be written in compact form as

(12)

Here, is the vector of sensor states(estimates.) The Laplacian matrix captures the topology ofthe sensor network. We also define the matrices and as

(13)

and

(14)

We refer to the recursive estimation algorithm in (12) as- (Modified-Coordinated State Estimation). We note

that the estimate vector sequence is random, due to thestochasticity of . Hence, all convergence results will be provedin an almost sure (a.s.) sense. Based on (11), the procedure ofthe proposed distributed state estimation can be summarized asfollows.Step 1) Each control area knows only its own local Jaco-

bian matrix , local measurement vector , localcovariance matrix , and time varying weightparameters , and . The aforementioneddata will be kept at every iteration of the proposedalgorithm.

Step 2) A global observability test is conducted in a dis-tributed manner as follows. Starting from an arbi-trary positive semi-definite local weighted Gramian

, each local area participates in the followingupdate:

Then, each area obtains the normalized weightedGramian andcomputes the rank of to check global observ-ability. If the network is not globally observable,observability restoration can be performed usingthe method proposed in [27].

Step 3) At the 0th iteration, each area sets the initial esti-mate vector . We note that this estimate vectorfor each control area is partitioned into controlarea subvectors conformally with the sets of busesassociated with the control areas. For example,in Fig. 2, the initial estimate vector correspondingto area can be expressed as

(15)

Fig. 2. IEEE 14-bus system.

where

and denote area and the initial voltagephase angle at bus in the th area, respectively.Then, each area concurrently sends its estimatevector to the neighboring areas.

Step 4) At the first iteration, two tasks are sequentially con-ducted using (11):a) Computation: each area computes the es-timate vector based on its previousestimate vector and the communicatedestimate vectors together with

, and .b) Communication: each area again sends itsestimate vector to the neighboring areasfor the next iteration process.

Step 5) The distributed iterations in Step 4) are repeated fora finite number of times, say , such that the con-trol area estimates approach the centralized leastsquares estimate to within a desired level of accu-racy. Note that the number depends on severalfactors including the size of the physical network,the sparseness of the communication graph, and thelevel of accuracy desired. In practice, reasonable ap-proximations for may be obtained through offlinetraining or simulations.

The following assumption on the connectivity of the inter-area communication network is assumed:Assumption (E.1)—Connectivity: The inter-area communi-

cation network is connected, i.e., .


Further, the time-varying weight sequences andassociated to the agreement (consensus) and innovation

potentials,4 respectively, are assumed to satisfy:Assumption (E.2)—Time Varying Weights: The sequences

and are of the form

(16)

where are constants and the exponents satisfy

(17)

Under (E.0)–(E.2), the convergence of DC state estimation isasserted in the following theorem:Theorem 2: Consider the - under (E.0)–(E.2). Fur-

ther, if , assume, in addition

(18)

where denotes the largest eigenvalue of . Then, foreach , the estimate sequence converges a.s. to the cen-tralized least squares estimator , i.e.,

(19)

Remark 3: The proof of Theorem 2 is provided inAppendix A for (note that the case can betreated similarly, the proof being omitted to avoid unnecessaryrepetition). We remark on the consequence of Theorem 2 andthe assumptions – . First, we point out the necessityof the time-varying weight sequences and asso-ciated to the innovation and consensus potentials, respectively.In fact, a constant weight version of - , the wasproposed and analyzed in [28]. The state update rule (in vectorform) in such a case reduces to

(20)

where are constants. The following was establishedfor the in [28]:Theorem 4 (Corollary 4 in [28]): Let (E.0)–(E.1) hold and

the constant satisfy

(21)

where denotes the matrix . Then, if inaddition, the centralized least squares satisfies , theestimate sequence for each control area converges to

4Broadly speaking, the agreement potential corresponds to the agreementterm in (12) and quantifies the tendency of the sensor esti-mates to mutually agree, thus highlighting the collaborative nature of the es-timation procedure. Innovation potential here corresponds to the effect of ob-servation data on the estimate update process and is manifested by the quantity

in (12).

at a geometric rate, i.e., there exists constants and, such that

(22)

for each .Thus, without the additional assumption , although

the may be shown to converge, the limiting value, in gen-eral, is not the centralized least squares estimate . As shownin [28], this is primarily due to the fact that, in general

(23)

and hence, is not a fixed point of the update process (20).Only in cases where (for example, is an invertiblesquare matrix), the limiting value is , otherwise it deviatesfrom gracefully.To overcome this, we resort to time-varying weight sequences

in the - scheme, which (as shown by Theorem 2) con-verges to for all observable . The use of time varyingweight sequences makes the convergence proof (Appendix A)much more involved and is beyond the purview of fixed pointbased techniques as used in [28]. Moreover, as shown in theproof, the design of and in accordance with as-sumption (E.2) provides the right interplay between the collab-orative network information flow and the local sensor observa-tions leading to the desired convergence to .Theorem 2 establishes the convergence of the - . As

long as the weight sequences satisfy (E.2), convergence is guar-anteed. However, the convergence rate depends on the partic-ular choices of the algorithm parameters, , and . Con-vergence rate analysis of such mixed time scale procedures israther technical and will be addressed in future work. The ex-tensive numerical simulations presented in the paper show thatthe convergence rate is reasonable and also provide guidelinesin choosing the various algorithm parameters.We comment in this context that under more specific assump-

tions, for example, , the convergence rate can be spedup further by employing the fixed weight version of the- . In fact, as shown by Theorem 4, the control area esti-

mates are guaranteed to converge at a geometric rate to pro-vided the condition holds. As pointed out before, ingeneral, the limiting value, say , of the deviates from. However, as our simulations demonstrate, as long as

is close to (or, in other words, there is not much redundancyin the measurement process), the limiting value of the

does not deviate much from . This, in general, points tothe following interesting trade-off between accuracy and con-vergence rate: for generic measurement models, one may usethe - (with decaying weights) if estimate convergence isdesired to the exact centralized least squares ; on the otherhand, if one is willing to tolerate a little error (small deviationfrom ), one may instead use the which exhibits geo-metric (exponential) convergence.


B. AC State Estimation

Alternatively, for AC state estimation, the iterative equation(11) is modified to

(24)

Here, is the transformed Jaco-bian matrix of . In addition,

= is the time-varying coefficient whereimplies modulo , and is a fixed number

of iterations. For each intervalis updated as follows:

(25)

where represents the th fixed iteration interval.

C. Bad Data Processing

In the presence of bad data, the proposed distributed frame-work could also detect and identify bad data in a similar manner.Due to space limitations, we do not elaborate on this issue here.Interested readers are referred to [29]–[31] for detailed analysisof this situation.

IV. CASE STUDIES

In this section, we analyze the performance of the proposeddistributed state estimation in the IEEE 14-bus and IEEE118-bus systems. The centralized WLS state estimate pro-vides a performance benchmark for the proposed algorithm.In Section IV-A, we introduce performance indices used forassessing the performance of the proposed distributed state es-timation algorithm. The performance of the proposed algorithmis analyzed in the two subsequent subsections, correspondingto DC state estimation and AC state estimation, respectively.The first part of these two subsections includes the results ofthe observability analysis for the IEEE 14-bus and 118-bussystems. This is followed by detailed numerical simulations toinvestigate the convergence rate of the proposed algorithm.

A. Performance Evaluation

The performance of the proposed distributed state estimationalgorithm is evaluated in terms of the following performanceindices:1) Estimation Accuracy: We choose the bus phase angle and

voltage magnitude difference between distributed and central-ized algorithms as the performance indices to evaluate the con-vergence of the proposed algorithm:

(26)

where subscripts and correspond to buses and , respec-tively. andrepresent the absolute values of bus and bus ’s phase angle

Fig. 3. Multi-area IEEE 118-bus system illustrating two different inter-controlcommunication networks.

differences in distributed and centralized state estimation, re-spectively, and

(27)

where and represent bus ’s estimated voltagemagnitudes in distributed and centralized state estimation,respectively.2) Execution Time Efficiency: The execution time efficiency

of the proposed algorithm is defined as

% (28)

where and represent the system-wide execution time ofthe centralized and distributed state estimation algorithms, re-spectively. Note that the system-wide execution time of the dis-tributed state estimation algorithm is always equal to the totalexecution time of the slowest area among local control areas.

B. Case 1: DC State Estimation

In this case, we assume a DC state estimation model with1.0 p.u. voltage magnitudes in all buses and p.u. branchimpedance. For the IEEE 14-bus system, the measurement con-figuration including the types as well as locations of the mea-surements and network decompositions comes from [12]. Thesystem has four nonoverlapping control areas as shown in Fig. 2.Control areas 1, 2, 3, and 4 contain ,and buses, respectively. On the other hand, the IEEE118-bus system has nine nonoverlapping control areas, as con-sidered in [32] and shown in Fig. 3. The control areas contain

, and buses, respectively. Initially, mea-surement noises are assumed to be Gaussian with zero meanand uniform finite variances . We assume that one or mul-tiple local control areas become locally unobservable for bothsystems, but that the entire system is still globally observable.1) IEEE 14-Bus System: We conduct global and local ob-

servability analysis in the IEEE 14-bus system. We define the


Fig. 4. Multi-area IEEE 14-bus system illustrating two different inter-controlcommunication networks.

local observability based on [12, (24)]. An area is observ-able if and only if

(29)

where is the local Jacobian matrix related to all the internalmeasurements of area (excluding boundary bus measure-ments). This system has a total of 22 measurements, including6 power injection and 16 power flow measurements. Then, therank of the system-wide measurement Jacobian matrix is 13so that the system is still globally observable. The rank test re-sult for each local measurement Jacobian matrix is shownas follows:• ;• ;• ;• .

Therefore, area shaded in Fig. 2 becomes locallyunobservable.Next, we show in this relaxed observability setup that by the

proposed iterative algorithm, all the control areas’ estimates ofthe system-wide state will converge to the centralized WLS so-lution. We assume that all the power flow and injection mea-surements are corrupted by additive Gaussian noises with equalvariances . The IEEE 14-bus system has a totalof 22 measurements, which is comprised of 16 branch flowand 6 nodal power injection measurements. The detailed busnumber and measurement type of each area are shown in Fig. 4.Constants , and are chosen for the distributed itera-tive algorithm with , and ,respectively. In addition, the tolerance of simulation is set to

. For both centralized and distributed state estima-tion, bus 1 is selected as the slack bus.The performance test is conducted for a total of 91 pairs of

phase angle differences . We ran-domly illustrate five pairs , and in thefigure. In Fig. 5, it is observed that after a short period of os-cillation (30 iterations), the distributed estimates exponentially

Fig. 5. Convergence of bus phase angle difference between distributed and cen-tralized algorithms in the IEEE-14 bus system of Case 1.

converge to the centralized estimates. Furthermore, in our re-cent work [28] (based on the algorithm), we investigatedthe sensitivity of the convergence rate of the proposed algorithmwith respect to 1) step size coefficients and ; 2) the measure-ment error covariancematrix ; and 3) different communicationtopologies. The main results for the aforementioned sensitivityanalysis are summarized as follows:• Sensitivity of convergence rate with respect to a and b:Larger values of and [provided they satisfy (21)] lead tohigher convergence rate at the expense of potentially moreoscillations, and vice versa.

• Sensitivity of the convergence rate with respect to valuesin the covariance matrix: The convergence rate seems tobe robust to the measurement error covariance matrix.

• Convergence rate of the algorithm with different communi-cation topologies: The communication topology is elasticfor the convergence of the proposed distributed state esti-mates to the centralized state estimates on condition thatthe inter-area power network is connected, and the wholesystem is globally observable.

2) IEEE 118-Bus System: For the IEEE 118-bus system, weassume that power injection measurements are placed at all gen-erator buses, power flow measurements at a subset of transmis-sion lines. Therefore, this system has a total of 178 measure-ments, including 49 power injection and 129 power flow mea-surements. Based on this measurement configuration, the rankof the system-wide measurement Jacobian matrix is 117 sothat the system is globally observable; however, shaded areas

, and in Fig. 3 are identified to be locally un-observable by the following rank test:• ;• ;• ;• ;• ;• ;• ;• ;• .


TABLE IPERFORMANCE IN THE IEEE 14-BUS SYSTEM OF CASE 1

TABLE IIPERFORMANCE IN THE IEEE 118-BUS SYSTEM OF CASE 1

Fig. 6. Convergence of bus phase angle differences between distributed andcentralized algorithms in the IEEE-118 bus system of Case 1.

In the above observability setup, we again examine the con-vergence rate of the proposed distributed state estimation algo-rithm with centralized WLS state estimation. Constants ,and in the proposed distributed iterative scheme are chosenwith , and , and the tolerance ofthe simulation is set to . Five pairs ,and are illustrated in the figure. Fig. 6 shows that in theIEEE 118-bus system, the distributed estimates also convergewell (in 30 iterations) to the centralized estimates.Tables I and II summarize the performance of the proposed

distributed state estimation algorithm in the DC state estima-tion model, corresponding to the IEEE 14-bus and 118-bus sys-tems, respectively. In these tables, three performance indicesare used: 1) maximum phase angle difference between the truestate values and the distributed state estimates ; 2) averagephase angle difference between the true state values and the dis-tributed state estimates ; and 3) execution time efficiencydefined in (28). We can see from these tables that increasing theparameters and leads to improved efficiency. In addition, it isobserved that the estimation accuracy of the proposed algorithmdegrades if area or boundary measurements have a higher

noise variance . In particular, these tables show the per-formance of the proposed algorithm for two different commu-nication topologies where communication scheme 1 graph isdenser than communication scheme 2 graph. However, commu-nication scheme 1 does not always outperform communicationscheme 2. It can also be observed by comparing the last rowsof Tables I and II, that the efficiency gain is more significantin the IEEE 118-bus system than in the IEEE 14-bus system.Therefore, we conjecture that the proposed algorithm is moresuitable for large-scale power networks to achieve the improvedefficiency.

C. Case 2: AC State Estimation

In this subsection, we study the performance of AC state es-timation using the proposed algorithm. First, we consider theIEEE 14-bus system for the performance analysis of the pro-posed algorithm. This system is assumed to have the same mea-surement configuration as the IEEE 14-bus system illustratedin Case 1 except assuming that power injection and flow mea-surements are always in pairs and one voltage magnitude mea-surement is placed at each local control area. Suppose that apair of real and reactive flow measurements, and inarea are deleted, which leads to a locally unobservable area, but a globally observable network. This system has a total

of 48 measurements, including 6 pairs of power injection, 16pairs of power flow, and 4 voltage magnitude measurements.Similarly, based on the measurement configuration of the IEEE118-bus system illustrated in Case 1, the IEEE 118-bus systemin Case 2 has a total of 365 measurements, including 49 pairs ofpower injection, 129 pairs of power flow, and 9 voltage magni-tude measurements.For distributed nonlinear AC state estimation, we propose

two types of update rules with respect to the local Jacobianmatrix for area , corre-sponding to: 1) wait-and-update (WAU); and 2) no-wait-and-update (NWAU) rules. In the WAU rule-based algorithm, eachlocal control area does not update its own local Jacobian matrixuntil after a fixed number of iterations. For example, supposethat this fixed number of iterations is equal to 40. In the WAUrule-based algorithm, every local Jacobian matrix remains un-changed while the WAU rule-based algorithm counts the itera-tion from 1 to 39. When the number of iterations arrives at 40,the local matrix is finally updated with its local state estimate


Fig. 7. Convergence of bus phase angle differences between distributed andcentralized algorithms in the IEEE 14-bus system of Case 2.

vector computed at the 39th iteration. Then, the count of itera-tions resets to zero and the proposed algorithm restarts from thefirst iteration. This updating process continues until this algo-rithm converges. Here, the fixed iteration period is denoted byFP. On the other hand, in the NWAU rule-based algorithm, eachlocal control area updates its own local Jacobian matrix at everyiteration. The performance of the WAU and NWAU rule-basedalgorithms is analyzed in the next two subsections.1) IEEE 14-Bus System: Considering both bus voltage mag-

nitude and phase angle as state variables, an area is locallyobservable if and only if

(30)

where is the local Jacobian matrixassociated with only all the internal measurements of areaand with flat start.The rank of the system-wide measurement Jacobian matrix

is 27 so that the system is globally ob-servable. All the local Jacobian matrices satisfy condition (30)except area as follows:• ;• ;• ;• .

Therefore, area is locally unobservable.Next, the performance of the WAU and NWAU rule-based

algorithms is evaluated and compared with each other. For theNWAU rule-based algorithm, it is hard to find optimum ini-tial state and parameters and . However, we already knowfrom the results of the previous subsection that the proposeddistributed state estimation algorithm based on a linearized DCpower flow model converges to a centralized WLS state estima-tion solution. This fact is our motivation for adopting the WAUrule-based algorithm rather than the NWAU rule-based algo-rithm. For illustrating the convergence of the WAU rule-basedalgorithm in the figure, five pairs , andare randomly chosen. The parameters of the proposed algorithmare chosen with , and

Fig. 8. Convergence of with varying FP in the IEEE 14-bus system ofCase 2.

Fig. 9. Convergence of and in the IEEE 14-bus system illustratingthe two different communication schemes.

. Fig. 7 shows the convergence of the distributed esti-mation algorithm to the centralized estimate. From Fig. 8, whichshows the effect of varying FP, two noticeable phenomena areobserved. First, the convergence rate of is changing as thevalue of FP is increasing. In other words, the convergence ratebecomes slower as a larger value of FP is chosen. From thisfigure, the best value of FP among those considered for theconvergence rate is 4. Second, the WAU rule-based algorithmoutperforms the NWAU rule-based algorithm in terms of phaseangle accuracy. The NWAU rule-based algorithm (without FP)shows a poor performance as shown in Fig. 8. Therefore, forfaster convergence of the proposed algorithm, the values of pa-rameters and FP must be carefully chosen. Fig. 9 showsthat the WAU rule-based algorithm is robust to change in thecommunication scheme. In Fig. 10, we can observe that voltagemagnitude estimates using the proposed algorithm are rather in-accurate at some buses.2) IEEE 118-Bus System: In the IEEE 118-bus test system,

the rank of the system-wide measurement Jacobian matrix


Fig. 10. Bus voltage magnitude differences between distributed and central-ized algorithms in the IEEE 14-bus system of Case 2.

Fig. 11. Convergence of bus phase angle differences between distributed andcentralized algorithms in the IEEE 118-bus system of Case 2.

is 235 so that the system is globally observ-able. The rank of every local area Jacobian matrix is as follows:• ;• ;• ;• ;• ;• ;• ;• ;• .Since areas and have rank deficiency,

both areas become locally unobservable. Five pairs, and are randomly chosen for

the convergence test. The parameters of the WAU rule-basedalgorithm are set to , and

. Fig. 11 shows that after some initial oscillation,the distributed estimation algorithm converges exponentiallyto the centralized estimates. Unlike Fig. 6, this figure shows

Fig. 12. Convergence of with varying FP in the IEEE 118-bus system ofCase 2.

Fig. 13. Convergence of and in the IEEE 118-bus system illus-trating the two different communication schemes.

some spikes in the plot after every 20 iterations. This isdue to the fact that each local Jacobian matrix is updated atevery 20 iterations. Fig. 12 shows the impact of FP on theconvergence of the distributed state estimation algorithm. Inthis figure, we conclude that is the best among thoseconsidered as well as the lower bound for the convergenceof the proposed algorithm since the proposed algorithmdiverges when and the increase of FP leads to slowerconvergence rate. Similar to Fig. 9, Fig. 13 shows that theproposed algorithm is robust to changes in the communicationscheme. As shown in Fig. 14, the WAU rule-based algorithmin the IEEE 118-bus system also provides a poor voltagemagnitude estimate at some buses.Tables III and IV summarize the performance of the proposed

distributed state estimation algorithm in the AC state estima-tion model, corresponding to the IEEE 14-bus and 118-bus sys-tems, respectively. Compared to Tables I and II, two extra per-formance indices are added to the last two rows of Tables IIIand IV: 1) maximum voltage magnitude difference between the


TABLE IIIPERFORMANCE IN THE IEEE 14-BUS SYSTEM OF CASE 2

TABLE IVPERFORMANCE IN THE IEEE 118-BUS SYSTEM OF CASE 2

Fig. 14. Bus voltage magnitude differences between distributed and central-ized algorithms in the IEEE 118-bus system of Case 2.

true state values and distributed state estimates ; and 2)average voltage magnitude difference between true state valuesand distributed state estimates . Table III shows that theWAU rule-based algorithm provides a better phase angle esti-mation accuracy than theNWAU rule-based algorithm. Table IVshows that with the same parameters and FP value, the WAUrule-based algorithm converges whereas the NWAU rule-basedalgorithm diverges. This result is one of the advantages for theWAU rule-based algorithm. The last observation from these ta-bles is that we can also see that as the network size increases,

the improvement in efficiency becomes more significant as forDC state estimation.Lastly, we investigate the communication requirements of our

proposed algorithm. We emphasize here that the algorithm isquite efficient in terms of communication, as only estimate vec-tors are exchanged in each iteration. This is in contrast to otherdistributed approaches which require the transmission of thesystem matrices from sensor to sensor asthese are available only locally. Further, the update rule doesnot require matrix inversions, a key computational bottleneckeven in centralized applications. The communication overheadusually consists of four processing times: 1) the data processingtime; 2) the queueing delay; 3) the transmission time; and 4)the propagation time. In this paper, only the transmission timeis assumed to be the communication time. Therefore, the com-munication time is defined as shown in (31) at the bottom of thepage. We assume that the data (elements of estimate vector) areexpressed as 32-bit real numbers and optical fibers (bps) are chosen as communication channels among local controlareas. Table V shows the number of exchanged data values andthe communication time per one iteration in the IEEE 14-busand 118-bus systems for both Case 1 and Case 2. Due to the ex-change of only estimate vectors, the amount of exchanged dataamong local control areas is always equal to the number of statesin the power system. This fact implies that the communicationtime in our proposed algorithm is not affected by the networktopology and measurement configuration, unlike [12] in whichthe communication time depends on the number of boundarymeasurements and decomposed local control areas.Table VI shows the total execution time, the total communi-

cation time, and the total time (i.e., the sum of the total execution

(31)


TABLE VCOMMUNICATION TIME BETWEEN LOCAL CONTROL AREAS

TABLE VITOTAL EXECUTION AND COMMUNICATION TIME IN CASE 1

time and the total communication time) for each DC state esti-mation scheme in the IEEE 14-bus and 118-bus systems. Here,we assume that the IEEE 14-bus and 118-bus systems have atotal of 22 and 178 measurements, respectively. Since measure-ments collected by the SCADA system are transmitted to thecentralized state estimator only once, the total communicationtime in the centralized state estimation is simply com-puted by (31). On the other hand, the total communication timein the proposed distributed state estimation algorithm is calcu-lated by multiplying the communication time required for oneiteration by the total number of iterations required for the pro-posed algorithm’s convergence. The fourth column of Table VIshows these total communication times with a total of 1 and 50iterations, corresponding to the centralized and proposed dis-tributed state estimation, respectively. From this table, we cansee that for both state estimation schemes, the total communi-cation time is much smaller than the total execution time so thatthe total time is dominated by the total execution time. In viewof the total time, the proposed distributed state estimation al-gorithm still appears to be more efficient for large-scale powernetworks.Finally, the novelty of the proposed distributed estimation al-

gorithm can be summarized as follows:• Fully distributed algorithmwith lower computational com-plexity: No central coordinator is required to be present inorder for each local control center to achieve entire systemstate estimation. Furthermore, the proposed algorithm iscomputationally more efficient compared with centralizedestimation, since it does not require extensive matrix in-version (e.g., gain matrix inversion). As the network sizeincreases, the relative computational saving compared withcentralized estimation becomes more significant.

• Flexible communication topology: The proposed dis-tributed estimation algorithm is applicable for anycommunication topology provided that the inter-areacommunication graph is connected, and the whole systemis globally observable.

• Local observability assumption relaxed: Unlike manyexisting distributed or hierarchical state estimation algo-rithms, in our proposed algorithm, all the control areasare not required to be locally observable. Therefore, thepotential performance degradation due to pseudo-mea-surement placement can be prevented.

V. CONCLUDING REMARKS

In this paper, a fully distributed algorithm has been pro-posed for multi-area state estimation in interconnected powersystems. Compared with existing hierarchical or distributedstate estimation algorithms, our proposed algorithm is imple-mentable under the more relaxed assumption that not all thecontrol areas must be simultaneously locally observable. In thecase of DC state estimation, we have proven the convergenceof the proposed algorithm to the centralized state estimationresult. In the case of AC state estimation, we have proposed animplementable distributed WAU rule-based algorithm whichshows satisfactory convergence behavior compared with cen-tralized state estimation results. Illustrative examples in bothIEEE 14-bus and 118-bus systems confirm the effectiveness ofthe proposed algorithm.Future work should address the question of designing the

most efficient communication topology for fast and robust con-vergence of the distributed state estimation algorithm. Also, thepractical implementation of the proposed algorithm should betested in large-scale realistic AC state estimation. Last but notleast, we plan to integrate bad data processing with the state es-timation algorithm in a distributed framework (preliminary re-sults along these lines are reported in [29], [30], and [31]).

APPENDIX AANALYSIS AND PROOFS

This section is devoted to the proof of Theorem 2. Due to themixed time-scale behavior, the - does not fall under thepurview of standard techniques (as used for the in [28])for establishing convergence of iterative schemes. The proof islengthy and requires a thorough understanding of the two dif-ferent potentials, agreement and innovations, and their interac-tion over different time scales. The proof is accordingly accom-plished in steps.We provide a brief outline of the major arguments required to

establish the convergence of the - . The first step consistsof showing that the sensors reach agreement in the long run, i.e.,as , the sensors follow an averaged or mean behavior,the estimate trajectories over different sensors merging towardseach other. This part is accomplished in Section A-2, where it isshown that under the assumption (E.2), the agreement potentialeventually dominates the local innovation potentials, leading theestimates to consensus. Once consensus is achieved, the nextstep is to show the convergence of the averaged system to thedesired least squares estimate. This is further accomplished (seeSection A-3) in two steps: the recursion satisfied by the averagedestimate is quite different from that of a recursive centralizedestimator. In the first step, we construct a fictitious recursivecentralized estimator, which is shown to converge to . Theconvergence of the averaged estimate (and hence, the individualsensor estimates) to is established by a pathwise comparisonwith the aforementioned centralized estimate sequence.1) Some Intermediate Results: We state two results from [20]

to be used for establishing the convergence of - .The first result provides convergence conditions of general

time-varying scalar recursions.Lemma 5 [20, Lemma 4]: Let the sequences and

be given by

(32)


where and . Then, if ,there exists , such that, for sufficiently large non-negativeintegers,

(33)

Moreover, the constant can be chosen independently of .Also, if , then, for arbitrary fixed

(34)

(We use the convention that , for .)The next result presents estimates of ellipticity of a sequence

of relevant time-varying matrices, to be used in the sequel.Lemma 6 [20, Lemma 6]: Under (E.0)–(E.2), there exists

sufficiently large and a constant , such that, for

(35)

2) Estimate Consensus: We state the key result of this sub-section, Lemma 7. To this end, denote by , the se-quence of network-averaged estimates, i.e.,

(36)

Lemma 7: Under (E.0)–(E.2), the sensor estimates achieveconsensus, i.e.,

(37)

Remark 8: Lemma 7 presents the interesting fact, that, irre-spective of different initial estimates and observations, all thesensors in the network eventually reach agreement (consensus)in terms of their estimates of the parameter .The rest of the subsection is devoted to the proof of Lemma

7. We first establish the boundedness of the estimate sequence:

Lemma 9: Under (E.0)–(E.2), the sensor estimates are path-wise bounded, i.e., there exists a random variable , such that

(38)

Proof: Since, the observation noise is finite a.s., there ex-ists a random variable , such that

(39)

Note that (12) may be written as

(40)

We then have from (39)

(41)

By Lemma 6, there exists sufficiently large, such that, for all

(42)

where is a constant.Also, note that, under (E.2), the weight sequence falls

under the purview of Lemma 5 and, hence, there exists suffi-ciently large, such that, for every

(43)

where is a constant and independent of .Choose and note that the recursion (41) may

be upper-bounded as

(44)

for . The above leads to (for )

(45)

The second term in (45) is bounded by (43) and the finitenessof . For the first term, we note that

(46)

where the first step follows from the identity forsmall positive [note that the -s go to zero by (E.2)]. Thelast step is a consequence of the persistence condition, namely,

for all .Hence, from (43), (45), and (46), we finally obtain

(47)

Since the above holds pathwise (for all sample paths), we con-clude that

(48)

which establishes the claim in (38).We now complete the proof of Lemma 7, which shows even-

tual agreement between the sensor estimates.


Proof of Lemma 7: Define the sequence by

(49)

where is defined in (36). Recall the matrix

(50)

and note that

(51)

and

(52)

for all .From (12), the sequence satisfies

(53)

where we use the identity

(54)

and corresponds to the zero matrix of appropriate dimension.Using (51) and (53), the sequence may be shown to

satisfy the following recursion:

(55)

where the residual is of the form

(56)

By Lemma 9, the random objects and are dominatedin norm by a.s. finite random variables and , respectively.Hence, there exists another a.s. finite random variable , suchthat

(57)

From (55) and (57), we conclude

(58)

For sufficiently large, using the properties of the matricesand and the Kronecker product manipulations demon-

strated in Section II-A, we have

(59)

and note that , by the connectivity of the communica-tion network [(E.1)]. It then follows from (58), that, for

(60)

The above recursion leads to

(61)

Note that the last step follows from the fact that the term

goes to zero as , which is due topositivity of (due to the network connectivity) and thepersistence condition [(E.2)], namely, .The remaining term on the right-hand side of (61) falls under

the purview of Lemma 5 (choose , and notethat ) and hence goes to zero also. We thus concludefrom (61), that

(62)

Since (62) holds on sample paths (except on a set of measurezero), the claim in (37) follows.3) Averaged Estimate Behavior: Convergence of - :

We start by introducing a fictitious centralized estimator,and establish its convergence to . This estimator will be usedsubsequently to prove convergence of the averaged estimator,

through some comparison techniques.To this end, define the estimator sequence, , by

(63)

where is the Gramian introduced in (E.0).The following holds:Lemma 10: Under (E.0)–(E.2), the sequence con-

verges a.s. to , i.e.,

(64)

Proof: By definition of , the least squares estimate of ,we have

(65)

Define the process by

(66)

Using (63) and (65), we have the following recursion for

(67)


Since is positive definite (full rank) by (E.0), there existssufficiently large, such that, for

(68)

where is a constant. From (67), it then follows

(69)

Due to the persistence condition on the weights , i.e., thatthey sum to , we can take limits on both sides of (69) andconclude

(70)

Since the above holds a.s., the claim in (64) follows.We now show that the averaged estimate sequence merges

with the fictitious centralized estimator constructed in (63).Lemma 11: Under (E.0)–(E.2), we have the following:

(71)

Proof: We note that

(72)

Define the sequence by

(73)

From (63) and (73), we have

(74)

Note that the a.s. boundedness of the sequences(Lemma 9) and (Lemma 10) imply the a.s. boundednessof , i.e., there exists a random variable , such that

(75)

Since is positive definite (full rank) by (E.0), there existssufficiently large, such that, for

(76)

where is a constant.Now consider , arbitrarily small. The eventual agree-

ment between the sensor estimates (Lemma 7) shows the ex-istence of (depending on the sample path), sufficiently large(greater than ), such that, for

(77)

Also, by Lemma 5, there exists sufficiently large (greater than), such that

(78)

for and is a constant independent of . Let. Then, by (74) and (77), we have for

(79)

The above recursion leads to

(80)

As , the first term on the right-hand side of (80) goes tozero [see similar arguments in (69) in Lemma 10]. Hence, wehave

(81)

Since the above holds for all , by taking to 0, we have

(82)

This establishes the claim.We now complete the proof of Theorem 2, which establishes

the convergence of the - estimates to at every sensor.Proof of Theorem 2: Note that, by Lemma 7 and Lemma

11, it follows that

(83)

The result is then a consequence of Lemma 10 and (83).

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LeXie (S’05–M’10) received the B.E. degree in elec-trical engineering from Tsinghua University, Beijing,China, in 2004, the M.Sc. degree in engineering sci-ences from Harvard University, Cambridge, MA, inJune 2005, and the Ph.D. degree from the ElectricEnergy Systems Group (EESG) in the Departmentof Electrical and Computer Engineering at CarnegieMellon University, Pittsburgh, PA, in 2009.He is an Assistant Professor in the Department of

Electrical and Computer Engineering at Texas A&MUniversity, College Station, where he is affiliated

with the Electric Power and Power Electronics Group. His industry experienceincludes an internship in 2006 at ISO-New England and an internship at EdisonMission Energy Marketing and Trading in 2007. His research interests includemodeling, estimation and control of large-scale power systems, and electricitymarkets.

Dae-Hyun Choi (S’10) received the B.S. degree inelectrical engineering from Korea University, Seoul,Korea, in 2002 and the M.Sc. degree in electrical andcomputer engineering from Texas A&M University,College Station, in 2008. He is pursuing the Ph.D.degree in the Department of Electrical and ComputerEngineering at Texas A&M University.From 2002 to 2006, he was a researcher with Korea

Telecom (KT), Seoul, Korea, where he worked ondesigning and implementing home network systems.His research interests include power system state es-

timation, electricity markets, cyber-physical security of smart grid, and theoryand application of cyber-physical energy systems.

Soummya Kar (S’05–M’10) received the B.Tech.degree in electronics and electrical communicationengineering from the Indian Institute of Technology,Kharagpur, India, in May 2005 and the Ph.D. degreein electrical and computer engineering fromCarnegieMellon University, Pittsburgh, PA, in June 2010.From June 2010 to May 2011, he was with the

Electrical Engineering Department at PrincetonUniversity, Princeton, NJ, as a Postdoctoral Re-search Associate. He is currently an AssistantResearch Professor of the Electrical and Computer

Engineering Department at Carnegie Mellon University. His research interestsinclude performance analysis and inference in large-scale networked systems,adaptive stochastic systems, stochastic approximation, and large deviations.

H. Vincent Poor (S’72–M’77–SM’82–F’87) re-ceived the Ph.D. degree in electrical engineeringand computer science from Princeton University,Princeton, NJ, in 1977.From 1977 until 1990, he was on the faculty of the

University of Illinois at Urbana-Champaign. Since1990, he has been on the faculty at Princeton, wherehe is the Michael Henry Strater University Professorof Electrical Engineering and Dean of the School ofEngineering and Applied Science. His research inter-ests are in the areas of stochastic analysis, statistical

signal processing, and information theory, and their applications in wireless net-works and related fields such as social networks and smart grid. Among hispublications in these areas are the recent books Classical, Semi-classical andQuantum Noise (New York: Springer, 2012) and Smart Grid Communicationsand Networking (Cambridge, U.K.: Cambridge Univ. Press, 2012).Dr. Poor is a member of the National Academy of Engineering and the Na-

tional Academy of Sciences, a Fellow of the American Academy of Arts andSciences, and an International Fellow of the Royal Academy of Engineering(U.K.). He is also a Fellow of the Institute ofMathematical Statistics, the Acous-tical Society of America, and other organizations. In 1990, he served as Presi-dent of the IEEE Information Theory Society, and in 2004–2007, he served asthe Editor-in-Chief of the IEEE TRANSACTIONS ON INFORMATION THEORY. Hereceived a Guggenheim Fellowship in 2002 and the IEEE Education Medal in2005. Recent recognition of his work includes the 2010 IET Ambrose FlemingMedal, the 2011 IEEE Eric E. Sumner Awards, the 2011 Society Award of theIEEE Signal Processing Society, and honorary doctorates from the Universityof Edinburgh and Aalborg University, conferred in 2011 and 2012, respectively.

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Fully Distributed State Estimation for Wide-Area Monitoring Systems

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