observability - The University of Texas at Arlington

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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 1, FEBRUARY 2006

Jacobian Matrix-Based Observability Analysis for State Estimation Bei Gou, Member, IEEE

Abstract—In this paper, numerical observability analysis is restudied. Algorithms to determine observable islands and to decide a minimal set of pseudo-measurements to make the unobservable system observable are presented. The algorithms make direct use of the measurement Jacobian matrix. Gaussian elimination, which makes the whole process of observability analysis simple and effective, is the only computation required by the algorithms. Numerical examples are used to illustrate the proposed algorithms. Comparison of computation expense on the Texas system among the proposed algorithm and the existing algorithms is performed. Index Terms—Measurement placement, multiple measurement placements, observability analysis, observable islands, trial matrix.

I. INTRODUCTION

O

BSERVABILITY analysis, as a pre-procedure of state estimation, must be executed prior to state estimation in the energy management system (EMS). Its function is to decide if the given set of measurements is sufficient to solve the whole system. When the given set of measurements is not sufficient, it must identify all the possible observable islands (or subnetworks) that can be independently solved, respectively, and place a minimal set of pseudo-measurements to merge the obtained observable islands. Observability analysis has so far been accomplished by topological and numerical approaches. Topological approaches make use of the graph theory to determine network observability based on the type and location of the measurements, without any floating point arithmetic [1]–[7]. Numerical approaches are based on the decoupled measurement Jacobian matrix and the associated gain matrix. An algorithm is developed to determine all the observable islands if the system is found to be unobservable [8]. A similar approach is employed to place pseudo-measurements to make the whole network observable [9], [19]. Based on the work in [8], [9], and [19], the factorization-based observability analysis and the normalized residual-based bad data processing have been derived for state estimation using the normal equation approach [10]. All the above numerical approaches need to solve the measurement equations. By only investigating the gain matrix instead of the measurement equations, the authors in [11] and [12] developed different numerical observability analysis algorithms to determine observable islands and place pseudo-measurements

Manuscript received November 1, 2004; revised August 11, 2005. Paper no. TPWRS-00575-2004. The author is with the Energy System Research Center, Department of Electrical Engineering, University of Texas at Arlington, Arlington, TX 76019 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TPWRS.2005.860934

when the system is not observable. These algorithms depend only on the processing of the gain matrix, which results in fewer computations. The algorithms proposed in [11] and [12] were utilized and applied in [13]–[16] to develop topological approaches and hybrid algorithms for observability analysis. In this paper, numerical observability analysis is restudied. It will be shown that the information about the observable islands can be extracted from the transformed matrix of measurement Jacobian matrix by Gaussian elimination. A so-called trial matrix is defined based on the transformed matrix. Furthermore, the trial matrix, which is defined differently from the test matrix defined in [11] and [12], is shown to carry the necessary information about the location and type of the pseudo-measurements that will merge the observable islands. The proposed algorithms can be implemented with minimal effort in an existing state estimation, since Gaussian elimination is the only required computation. This paper is organized as follows. Section II provides the theoretical results for numerical observability analysis based on Gaussian elimination. Algorithms to determine observable islands and to place pseudo-measurements to make the whole networks observable are presented in Sections III and IV. Comparison between the existing numerical algorithms and the proposed algorithm is provided in Section V. Section VI provides the results of test examples. Section VII gives the conclusion. II. PRELIMINARY RESULTS Consider the linear decoupled measurement equation given below: (1) where mismatch between the measured and calculated real power measurements; decoupled Jacobian matrix of the real power measurements versus all bus phase angles; incremental change in the bus phase angles at all buses, including the slack bus; measurement error vector. The decoupled gain matrix for real power measurements can be formed as (2) where the measurement error covariance matrix is assumed to be the identity matrix without loss of generality. Note that, since the slack bus is also included in the formulation, the rank of

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GOU: JACOBIAN MATRIX-BASED OBSERVABILITY ANALYSIS FOR STATE ESTIMATION

(and ) will be at most ( is the number of buses), even for a fully observable system. Previous numerical observability analysis methods utilize the gain matrix to identify observable islands and decide the minimal set of pseudo-measurements to make the whole system observable [11], [12]. Even though the triangular factorization is a common calculation procedure in state estimation, its computation is still a big burden. In this paper, observability analysis is solely based on the measurement matrix . Therefore, the construction of the gain matrix from is not necessary. The traditional approach of power system modeling for observability analysis is used in this paper. That is, all branch im, and all weighting factors are equal pedances are equal to to 1.0. Since state estimation uses the real parameters, the triangular factors generated from observability analysis cannot be used by state estimation. Suppose we have measurements and buses. In order to simplify our derivation, we first need to reorder measurement as follows: Jacobian matrix with dimension (3)

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Now let us study the gain matrix when a measurement is added into the existing measurement set. The addition of will modify the existing gain matrix as follows:

(7) and . where Since upper triangular matrix is of full rank, the following lemma shows that matrix is also of full rank. Lemma 2.2: If is of full rank, then matrix is also of full rank. Proof:

where square and full rank submatrix of of

is the rank

;

submatrix containing the redundant part of measurements; submatrix containing the redundant part of nodes; remaining submatrix in . is linearly dependent on maSince the remaining matrix trix and , then we have the following lemma. Lemma 2.1: If we partition the measurement Jacobian matrix as (3), then we have . and are linearly dependent of and Proof: Since , respectively, then there exists a nonzero matrix such that and . We thus have and . can be factorized as , where Suppose matrix and are lower and upper triangular matrices, respectively. in (3), we have the following facBased on the partition of torization:

where . must be of full According to [12, Lemma 3.1], matrix rank. Let us partition vector into , where is a vector with dimension , and is a vector with . Since is a symmetrical matrix, then dimension it can be factorized into triangular matrices in the form of , and then matrix can be further transformed as

(8) where (4) where (5) (6)

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If the measured system is not observable, we are interested in can increase the rank that if the newly added measurement of . The following theorem shows their relationship. Theorem 2.1: rank rank if and only if is a nonzero vector. is decided Proof: From (8), we know that the rank of , since and are of full rank. Define by the rank of . According to the above reordering of and , we have

(9) Since is of full rank, and by [12, Lemma 3.1], then the exists. inverse of So, the triangular factorization can reduce the above matrix to the following form:

where

Fig. 1. Example 1: six-bus system.

is nonzero, then must have a rank of 1. This ends the proof. Example 1: A power system of six buses is given in Fig. 1, where branch flows on branches 1–3, 4–5, and 2–3 are measured, and injection at bus 2 is measured. All branch impedances , and all weighting factors are equal to 1.0. The are equal to original Jacobian matrix can be directly obtained. Using the triangular factorization algorithm presented in [11], we can get the diagonal matrix and lower triangular matrix . It is obvious that this measured system is not observable

(10) Let us now consider matrix

In matrix , and indicate the bus number, and , and indicate the measurements. We first reorder so that all linearly indethe nodes and measurements in pendent rows and columns are ordered first. The resulted matrix, which includes the exchange of columns 3 and 4, and the insert of row 4 between row 1 and 2, is shown as follows:

where (11) During the above derivation, the Sherman-Morrison-Woodbury formula [18] is used

By linear column elimination, we can eliminate in , and then it is easy to see that the rank of is decided by the rank of and the rank of . Considering that is a , it is obvious that the rank of is scalar in . The maximum rank of is 1. If decided by the rank of

As in (4), we obtain the following matrices:


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Each column in corresponds to a branch. The columns are listed in the order of branch 1–2, 1–3, 2–3, 3–4, in shows that the 4th and 6th columns 4–5, and 4–6. Matrix are nonzero vectors. Therefore, we know branches 3–4 and 4–6 are not observable. III. ALGORITHM TO DETERMINE OBSERVABLE ISLANDS

The purpose of defining is that through the can instudy of , we could find whether the measurement crease the rank of the gain matrix. In other words, suppose is a branch flow measurement (actually, it is the same logic if is an injection measurement), and then the study of will provide the information whether or not the th branch is obcannot increase servable. If the branch is observable, then the rank of the gain matrix; if the branch is not observable, then is able to increase the rank of the gain matrix. We thereto check the observability of all the fore define branches. The calculation of the branch-node incidence matrix for this example is then necessary. The branch-node incidence matrix is constructed as follows. Assign a direction to each branch, and each row of the matrix corresponds to a branch; each column corresponds to a bus. There are two nonzero elements for each row: 1 at a bus if the at a bus if the branch enters the branch leaves the bus and is used when form matrix bus. The order of the columns in . , and indicate the branches

In this section, we will derive an algorithm to determine observable islands based on Gaussian elimination, when the measured system is not observable under a given set of measurements. It can be shown that [11, Corollaries 4.1 and 4.2] are also true here, except that the statement is a little different. Corollary 3.1: Consider a measured power system with several observable islands. 1) For a pseudo measurement, incident to nodes belonging to the same observable islands, must be a zero vector. 2) For a pseudo measurement, incident to nodes belonging to must be a nonzero vector. different observable islands, Let us suppose we have a pseudo branch flow measurement. In order to find out if this pseudo measurement is able to increase the rank of gain matrix for a given set of measurements, we need . Since , now we need to to calculate matrix . From (5), we have the following calculate the inverse of result:

Similarly as in [11], we can define the trial matrix follows:

as (12)

The relationship between

and the trial matrix

is given

as (13)

Then

gives the following matrix:

We use and to indicate the corresponding . The last three rows in form branches to the columns in , which is shown as follows: matrix

From the results of Corollary 3.1, the following is true for the trial matrix . Corollary 3.2: For a measured power system with several observable islands, all the elements in , whose columns correspond to the same observable island, will have equal values. Note: The notion of pathological cases in numerical observability analysis was first introduced in [9] and [19]. However, it has not received any detailed study so far in the literature except [11]. From our study, we currently believe that there are two reasons causing pathological cases: 1) the linear combina[11] and ) columns of corresponding tion of columns of to buses in different observable islands being equal. The first cause is avoided by using the test matrix [11] and trial matrix in this paper, instead of using the angle solutions in [8], [9], and [19]. It is observed that there are cases showing that, without iterations, the second cause is removed by moving the irrelevant injection measurements from the set of measurements. The way of moving the irrelevant injection measurements from the set of measurements is explained at the end of this section. The proof of this observation needs to be provided for general cases.

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The results of this investigation will be reported in future papers. However, in this paper, the iteration scheme is still utilized when deciding the observable islands based on the trial matrix. The following theorem shows a way of how to obtain the trial in a simple approach. matrix Theorem 3.1: For a given set of measurements, there exists a full-rank matrix that can transform the measurement Jacobian matrix , which is partitioned as in (3), into the following form: (14) Proof: Define a full-rank matrix Fig. 2. Example 1: six-bus system.

Then

surements at branch 1–2 and 4–6. The measurement Jacobian matrix is

can be transformed into the following form:

(15) We conclude the proof. Theorem 3.1 shows that the measurement Jacobian matrix can be transformed into the form in (15) by Gaussian eliminain (12), we can see tion. Compare (15) with the trial matrix . that the trial matrix can be directly obtained from matrix if they are in the Since the buses have the same values in same island, the product of the branch-node incidence matrix and will show whether or not a branch in matrix is observable. This implies the following algorithm to determine the observable islands under a given set of measurements. ) Execute Gaussian elimination on , and reorder the rows and columns when zero row is encountered so that all the linearly independent rows and columns are ordered first. ) If has nonzero pivots, stop; if not, form from the resulted matrix by the trial matrix Gaussian elimination, and execute the calcula. If at least one entry in a row tion is not zero, then the corresponding branch of will be unobservable. ) Remove all the unobservable branches to this row, and remove all the irrelevant injection measurements, to obtain the observable islands; then go to Step 1). If there are no irrelevant measurements, then stop. The irrelevant injection measurements are defined to be those measurements connecting more than one island. By checking the number of islands connected by an injection measurement, one can decide if this injection measurement is an irrelevant injection measurement. The following example illustrates the way to obtain the trial matrix . Example 2: The same power system of six buses given in Fig. 1 is used to illustrate how we obtain the trial matrix . The measurement configuration is shown in Fig. 2. We have injection measurements at buses 1, 2, and 3 and branch flow mea-

Since the third row is linearly dependent on the first two rows, the rows are reordered as follows so that all the linearly independent rows and buses can be ordered first in :

Note that the rank of is 4, and buses in are not reordered since the first 4 columns are already linearly independent of each other; row 3 is reordered and listed after row 5. Executing Gaussian elimination on , we can obtain the resulted matrix as follows:

(16)

Therefore, according to (12) and (14), the trial matrix be directly obtained from

can

(17)

IV. ALGORITHM OF MEASUREMENT PLACEMENT For a given set of measurements, if it is not able to make the whole system observable, the algorithm described in the previous section provides a way to find all observable islands. In this section, we will propose an algorithm to select a minimal set of pseudo-measurements to make the whole system observable. Although the matrix is different from the matrix in [12, Theorem 3.1], if we apply Theorem 3.1 on the matrix in (8),


we can see that Theorem 3.1 is still true for in this paper. Then the following algorithm of measurement placement is proposed. ) Execute Gaussian elimination on , and reorder the row and columns when zero row is encountered so that all the linearly independent rows and columns are ordered first. has nonzero pivots, stop; if not, ) If form the trial matrix from the resulted matrix by Gaussian elimination, and calculate , where is the Jacobian matrix of candidate measurements. ) Reduce to its Echelon form [17]. The candidates corresponding to the linearly independent columns in are the measurements that are able to merge all the observable islands into one island. ) Stop.

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where , and will form the test matrix , which is defined to be the matrix containing only those rows of corresponding to the zero pivots in the diagonal matrix [11]. Following the procedure of [8], the state can be solved as (18) Solving (18) for

.. .

.. .

.. .

V. REVIEW OF NUMERICAL OBSERVABILITY ALGORITHMS In order to compare the proposed algorithm in this paper with those algorithms based on the gain matrix, we call the first algorithm of observability analysis based on the gain matrix in [8], [9] and [19], algorithm I; the algorithm developed in [11] and [12], algorithm II; and the algorithm proposed in this paper, algorithm III. Before the comparison of those algorithms, we should note that algorithm II has to be improved as follows in order to consider the second cause of pathological cases presented in Section III. ) Form the gain matrix , and perform the triangular factorization. ) Check if has only one zero pivot. If yes, stop. If not, compute the test matrix from . . If at least one ) Compute the matrix entry in a row is not zero, then the corresponding branch will be unobservable. ) Remove all the unobservable branches, to obtain observable islands, and remove all the irrelevant injection measurements. Then go to Step 1); if there are no unobservable branches, go to Step 5). ) Stop.

(19) is the second before last element in , and in the where above calculation, we notice that the product is a lower triangular matrix, and its last three diagonal entries are units. In algorithm I, is used to find the observable islands. For normal cases, the buses having the same values are in the same observable island. However, algorithm II checks the test matrix to obtain the observable islands, which can reduce the possibility of buses in different islands having the same values due to the linear combination of the columns in . Besides fewer computations, this is another advantage of algorithm II compared with algorithm I. B. Comparison of Algorithms II and III According to the partition of given as

in (3), the gain matrix is

A. Comparison of Algorithms I and II The comparison of Algorithms I and II was made in [11] by using a three zero pivot example. Consider a measured system whose gain matrix is decomposed into its triangular factors with having zero pivots at the last three diagonal entries, which are replaced by 1’s per the method of [8]

where

We suppose that , where Then can be factorized as follows: and, let us define the rows of

is of full rank. (20)

as where .. .

(21)

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Therefore, the inverse of


is

Based on the algorithm proposed in [11, Sec. III], the test matrix defined in [11] is given as follows:

The following theorem shows that the test matrix and the trial matrix defined in this paper are identical. Theorem 5.1: Based on the partition of in (3), we have . Proof: Let us define (22) Considering Lemma 2.2, test matrix

can be further written

as Fig. 3.

Fourteen-bus system.

VI. TEST RESULTS

This concludes our proof. From the discussion in the note following Corollary 3.2 in Section III, we know that when the second cause of the pathological cases occurs, observable islands obtained by algorithm II are different from the final observable islands and need further processes. In order to deal with the second cause, an iterative scheme is utilized by algorithm III. Observations from numerical tests show that the second and further iterations in algorithm III might be avoided by the process of removing the irrelevant injection measurements in the first iteration, that is, the second and further iterations might not be necessary since the first iteration in algorithm III can achieve the final observable islands. However, mathematical proof is needed for algorithm III to avoid the iterative scheme. Results of such research will be reported in future papers. Although Theorem 5.1 shows that the trial matrix is equal to the test matrix , the theoretical analysis showing that can be directly obtained from the measurement Jacobian matrix is still necessary, which is one of the contributions of this paper. That is, although matrix and are identical, a totally different but simpler manner to compute them is proposed in this paper, which makes the algorithm to determine observable islands more efficient. Comparing the above matrix with the trial matrix defined in (12) with the test matrix , we can see that algorithm III is simpler and easier than algorithm II because fewer calculations are required in algorithm III, and an iterative scheme allows algorithm III to be able to deal with the second cause of pathological cases.

The algorithms proposed in this paper have been tested on IEEE 14-bus, IEEE 118-bus, and the actual Texas systems. A variety of measurement systems, including observable and unobservable cases, has been studied. The test on the Texas system is to compare the computational efficiency among the previous algorithms and the proposed algorithm. The results of the IEEE 14-bus system are shown in Tests 1 and 2. ) The measurement configuration is given in Fig. 3. is a row vector with all values The trial matrix equal to 1.0. The whole network is observable. ) Injections at buses 6 and 9 are removed from the above measurement configuration. The trial matrix is obtained as follows:

From the trial matrix , we can obtain the observable islands: [1]–[9], [10]–[13], [14], and [19].


Removing all the irrelevant injections and redundant measurements, we have the Jacobian measurement matrix shown in

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TABLE I CPU TIME (IN SECOND) USED BY THREE ALGORITHMS

Matrix

is obtained as follows:

(23)

Removing the redundant measurements is just for simpler calculation in the next step. After the execution of Gaussian elimination in the first step, redundant measurements are then decided. They are those measurements that are linearly dependent on the previous measurements. Execute Gaussian elimination on , and then we can obtain the trial matrix as shown in

We obtain the same observable islands as in the previous step. Now we need to decide a minimal set of pseudomeasurements to make the whole system observable. The candidates are the injections at buses 6, 9, 10, 11, 12, 13, and 14, which are denoted by , and . The candidate measurement Jacobian matrix is shown in

Executing Gaussian elimination on Echelon from of

, we get the

Candidate Injection Measurements

(24)

From matrix , we can decide that the candidate injections at buses 6 and 9 form the minimal set of pseudo-measurements, which can make the whole system observable. ) In this test, we compare the computation complex among these three algorithms. The Texas system with 4694 buses provided by ERCOT is used for this comparison. Two measurement configurations are tested: all branch flows and all node injections. A Dell/Inspiron 1150 laptop computer is utilized for the test. The test is performed in a MATLab environment. All three algorithms finished the calculations within the first iteration and achieved the same solution for these two measurement configurations. Their CPU time is provided in Table I. Algorithm III proposed in this paper has an obvious superiority of computation expense. The main reason is that the proinposed algorithm uses the measurement Jacobian matrix stead of the gain matrix. Tables II and III show the number and and their triangular factors of nonzero elements in . is the lower triangular matrix, and is the upper triangular matrix. A much larger number of fill-ins are introduced to the gain matrix and its triangular factors than those of the matrix. One should note that because of computational accumulated errors, the numbers of nonzero elements in and are not equal.

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TABLE II NUMBER OF NONZERO ENTRIES IN AND AND THEIR TRIANGULAR FACTORS FOR THE MEASUREMENT CONFIGURATION OF BRANCH FLOWS

H

G

TABLE III AND AND THEIR TRIANGULAR NUMBER OF NONZERO ENTRIES IN FACTORS FOR THE MEASUREMENT CONFIGURATION OF BUS INJECTIONS

H

G

VII. CONCLUSION This paper presents numerical algorithms to determine observable islands and to decide pseudo-measurements to merge all islands of a measured network. It is shown that transforming the measurement Jacobian matrix by Gaussian elimination is possible to obtain all the necessary information for observability analysis. This constitutes the main contribution of this paper. Results of test examples are provided to illustrate the proposed algorithms. Implementation of the algorithms is easy since Gaussian elimination is the only computation required. Comparison of the computation expense on the Texas system shows that the proposed algorithm has the obvious superiority and is ready for the practical applications. APPENDIX THEOREMS AND LEMMAS IN [12] For a good understanding of this paper, theorems and lemmas in [12] used in this papers are recited in the following. Lemma 3.1: Let be the diagonal factor of gain matrix and be a row vector. If has full rank, then will have full rank. Theorem 3.1: Let the triangular factors of the gain matrix be . Assume the rank deficiency of diagonal matrix is , and a set of pseudo-measurements are chosen from the candidate measurements. The pseudo-measurements, whose , will modify the old gain matrix by Jacobian matrix is . Then, the new gain matrix will have full is of full rank. rank if and only if matrix Their proofs are provided in [12]. REFERENCES [1] G. R. Krumpholz, K. A. Clements, and P. W. Davis, “Power system observability: A practical algorithm using network topology,” IEEE Trans. Power App. Syst., vol. PAS-99, no. 4, pp. 1534–1542, Jul. 1980.

[2] K. A. Clements, G. R. Krumpholz, and P. W. Davis, “Power system state estimation with measurement deficiency: An algorithm that determines the maximal observable subnetworks,” IEEE Trans. Power App. Syst., vol. PAS-101, no. 9, pp. 3044–3052, Sep. 1982. [3] K. A. Clements, G. R. Krumpholz, and P. W. Davis, “Power system state estimation with measurement deficiency: An observability/measurement placement algorithm,” IEEE Trans. Power App. Syst., vol. PAS-102, no. 4, pp. 2012–2020, Jul. 1983. [4] V. H. Quintana, A. Simoes-Costa, and A. Mondel, “Power system topological observability using a direct graph-theoretic approach,” IEEE Trans. Power App. Syst., vol. PAS-101, no. 3, pp. 617–626, Mar. 1982. [5] R. R. Nucera and M. L. Gilles, “Observability analysis: A new topological algorithm,” IEEE Trans. Power Syst., vol. 6, no. 2, pp. 466–475, May 1991. [6] H. Mori and S. Tsuzuki, “A fast method for topological observability analysis using a minimum spanning tree technique,” IEEE Trans. Power Syst., vol. 6, no. 2, pp. 491–498, May 1991. [7] A. S. Costa, E. M. Lourenco, and K. A. Clements, “Power system topological observability analysis including switching branches,” IEEE Trans. Power Syst., vol. 17, no. 2, pp. 250–256, May 2002. [8] A. Monticelli and F. F. Wu, “Network observability: Theory,” IEEE Trans. Power App. Syst., vol. PAS-104, no. 5, pp. 1042–1048, May 1985. , “Network observability: Identification of observable Islands and [9] measurement placement,” IEEE Trans. Power App. Syst., vol. PAS-104, no. 5, pp. 1035–1041, May 1985. [10] F. F. Wu, W.-H. E. Liu, and S.-M. Lun, “Observability analysis and bad data processing for state estimation with equality constraints,” IEEE Trans. Power Syst., vol. 3, no. 2, pp. 541–548, May 1988. [11] B. Gou and A. Abur, “A direct numerical method for observability analysis,” IEEE Trans. Power Syst., vol. 15, no. 2, pp. 625–630, May 2000. , “An improved measurement placement algorithm for network ob[12] servability,” IEEE Trans. Power Syst., vol. 16, no. 4, pp. 819–824, Nov. 2001. [13] G. N. Korres, P. J. Katsikas, and G. C. Contaxis, “Transformer tap setting observability in state estimation,” IEEE Trans. Power Syst., vol. 19, no. 2, pp. 699–706, May 2004. [14] G. N. Korres and P. J. Katsikas, “A hybrid method for observability analysis using a reduced network graph theory,” IEEE Trans. Power Syst., vol. 18, no. 1, pp. 295–304, Feb. 2003. [15] P. J. Katsikas and G. N. Korres, “Unified observability analysis and measurement placement in generalized state estimation,” IEEE Trans. Power Syst., vol. 18, no. 1, pp. 324–333, Feb. 2003. [16] G. N. Korres, P. J. Katsikas, K. A. Clements, and P. W. Davis, “Numerical observability analysis based on network graph theory,” IEEE Trans. Power Syst., vol. 18, no. 3, pp. 1035–1045, Aug. 2003. [17] L. O. Chua and P.-M. Lin, Computer-Aided Analysis of Electronic Circuits: Algorithms and Computational Techniques. Englewood Cliffs, NJ: Prentice-Hall, 1975. [18] G. H. Golub and C. F. Van Loan, Matrix Computations. Baltimore, MD: The Johns Hopkins Univ. Press, 1996. [19] A. Monticelli and F. F. Wu, “Network observability: Identification of observable Islands and measurement placement,” IEEE Trans. Power Syst., vol. 17, no. 2, pp. 250–256, May 2002.

Bei Gou (M’00) received the B.S. degree in electrical engineering from North China University of Electric Power, Baoding, China, in 1990, and the M.S. degree from Shanghai JiaoTong University, Shanghai, China, in 1993, and the Ph.D. degree from Texas A&M University, College Station, in 2000. From 1993 to 1996, he taught in the Department of Electric Power Engineering, Shanghai JiaoTong University. The, he was a Research Assistant at Texas A & M University. He worked at ABB Energy Information Systems, Santa Clara, CA, for two years and at ISO New England for one year as a Senior Analyst. He is currently an Assistant Professor with the Energy Systems Research Center, University of Texas at Arlington. His main interests are power system state estimation, power market operations, power quality, power system reliability, and distributed generators.