An Improved Measurement Placement Algorithm for Network Observability

Transcription

1 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 4, NOVEMBER An Improved Measurement Placement Algorithm for Network Observability Bei Gou and Ali Abur, Senior Member, IEEE Abstract This paper presents a fast method for multiple measurement placement for systems that are found to be unobservable. Upon placement of multiple measurements, all observable islands will be merged together and the entire system will be rendered fully observable. The method uses a test matrix whose leading dimension is determined by the rank deficiency of the gain matrix. Therefore, even for very large systems, as long as the number of measurements to be placed is relatively low, proposed method will maintain its computational advantage. Compared to the existing iterative approaches, this method directly provides the entire set of additional measurements for placement. The method is developed based on authors previous work [4] where a direct method for observability analysis was presented. Index Terms Measurement placement, multiple measurement placement, observability analysis, observable islands, test matrix, triangular factors updating with multiple rank. I. INTRODUCTION OBSERVABILITY analysis of power systems involves checking the rank of the state estimation gain matrix, which is sparse, symmetric and positive definite for observable systems. This can be accomplished by topological [1] or by numerical [2], [3] methods. If the gain matrix is found singular, then all observable islands need to be identified and injection pseudo-measurements will have to be introduced in order to merge these islands into a single observable island for the entire system. The process of choosing the right set of additional measurements for this purpose is referred to as the measurement placement. Pseudo-measurements are typically generated from load forecasts, scheduled generation data, or some other source with a degree of uncertainty. Hence, pseudo-measurements are usually of injection type. It is assumed that a set of such pseudo-measurements exits. Further, it can be shown that the eligible candidates for placement in a given islanded system will be those taken at the boundary buses of observable islands [1]. The approach taken in the past for multiple measurement placement has been to place one candidate boundary injection at a time each time updating the observable islands and hence the candidate list. This procedure is repeated until enough candidates are placed to merge all observable islands together. In this paper, an alternative and direct procedure which accomplishes Manuscript received February 11, 2000; revised March 2, B. Gou was with the Department of Electrical Engineering, Texas A&M University, College Station, TX He is currently with ABB Energy Information Systems, Santa Clara, CA ( beigou@ee.tamu.edu). A. Abur is with the Department of Electrical Engineering, Texas A&M University, College Station, TX ( abur@ee.tamu.edu). Publisher Item Identifier S (01) this same task, will be presented. The main idea behind the proposed method is the use of the test matrix, which has been introduced in [4] in order to simultaneously process the candidate measurements and make the final selection in a noniterative and fast manner. II. MOTIVATION The measurement placement algorithm presented in this paper is based on the observability analysis method introduced earlier in [4]. This method will be briefly reviewed first in order to show the motivation behind the algorithm developed in this paper. Consider the real power versus phase angle part of the linearized and decoupled measurement equation. This is obtained by using the first order approximation of the decoupled nonlinear measurement equation around an operating point: where mismatch between the measured and calculated real power measurements; decoupled Jacobian 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 the real power measurements can be formed as: where, 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 (and ) will be at most ( being the number of buses), even for a fully observable system. This leads to the triangular factorization of a singular and symmetric gain matrix. The symmetric matrix can be decomposed into its Cholesky factors where the diagonal factor, may have one or more zeros on its diagonal. Taking the inverse of and collecting only those rows of corresponding to the zero diagonals of, we can form a rectangular submatrix which is called the test matrix and denoted by in [4]. Computation of can be done very efficiently via the use of back substitutions. The following algorithm can then be used to determine all the observable islands simultaneously as shown in [4]: Step 1 Form the gain matrix and perform the triangular factorization. (1) (2) /01$ IEEE

2 820 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 4, NOVEMBER 2001 Step 2 Check if has only one zero pivot. If yes, stop. If not, compute the test matrix from. Step 3 Compute the matrix, where is the branch-node incidence matrix. If at least one entry in a row is not zero, then the corresponding branch will be unobservable. Step 4 Remove all the unobservable branches, to obtain the observable islands. Step 5 Stop. The above algorithm is noniterative, which makes it computationally very efficient. Thus, it will be used to initialize the multiple measurement placement algorithm developed in this paper. III. MULTIPLE MEASUREMENT PLACEMENT In this section, a new method which will identify all the necessary measurements to be added to an existing measurement set in order to render an observable system, will be presented. Assume that the symmetric gain matrix is expressed in terms of its factors as, and the rank deficiency of the diagonal matrix is. Further assume that a set of pseudo-measurements, whose Jacobian matrix is, are chosen from the candidate measurements. The new gain matrix will then be given by: Denoting a singleton vector by, let us choose the vector to be: Substituting (5) for in (4) will yield: which concludes the proof. We can extend this proposition to the following one. Proposition 3.2: Let be the diagonal factor of gain matrix and be a row vector. If is of full rank, then will also have full rank. Proof: Suppose is singular, i.e., any row of it can be expressed as a linear combination of the rest. Let and. Then, for, there exist a set of scalars,, such that (5) (6) (7) and for : Let, then we have: can be reordered to make the last diagonal entries zero: (3) Multiplying (7) by and (8) by, the scaler can be solved: (8) (9) Similar ordering of will yield: Equation (7) can then be rewritten as: where is a matrix, is a matrix whose rows are those of corresponding to the zero diagonals in. The following propositions and lemma are introduced in order to present a theorem, which will facilitate the development of the multiple measurement placement algorithm. Proposition 3.1: If is the diagonal factor of gain matrix, then the diagonal elements of will all be nonnegative. Proof: Suppose the gain matrix is factorized as by the proposed singular triangular factorization in [4]. Since is definite, i.e., for any row vector, the following equation is satisfied: (4) (10) Based on the above proposition for all. Thus, the left hand side of (10) will always be positive, while the right hand side will always be negative or zero, implying contradiction. Hence, the following lemma can be introduced. 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. Proof: The proof will be carried out by induction. Lemma is shown to be true for the special case of,by proposition 3.2.

3 GOU AND ABUR: AN IMPROVED MEASUREMENT PLACEMENT ALGORITHM FOR NETWORK OBSERVABILITY 821 Now, let us suppose it holds for let:. Then, for Let now consider matrix. (11) where is a row vector. Forming the matrix product: (12) Since is a real symmetric matrix with full rank, it can be factorized as: (13) where has full rank by the assumption. So, (12) can be rewritten as: (17) where. During the above derivation, the following formula is used: (14) Using proposition 3.2,, will have full rank. This concludes the proof. We now present the following theorem, which follows from the above propositions and the lemma. 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 Jacobian matrix is, will modify the old gain matrix by. Then, the new gain matrix will have full rank if and only if matrix is of full rank. Proof: From (3), we know that the rank of is decided by the rank of, since is of full rank. According to the above reordering of and,wehave: (15) Since is of full rank, and by lemma 3.1, then the inverse of exists. So, the triangular factorization can reduce the above matrix to: where (16) From (17), it is always true that Assume that is of full rank, then we have (18) (19) must be equal to since is a matrix. Then we conclude that the rank of is totally decided by since is of full rank. This ends the proof. The above theorem implies a simple method to determine a minimal set of pseudo-measurements in a single step. This set of pseudo-measurements will be guaranteed by Theorem 3.1 to make the whole system observable. The selection criteria will be that the matrix must have full rank. This matrix can be easily computed as the product of the test matrix and the Jacobian matrix of the pseudo-measurements chosen from the candidate measurements, Since the rank deficiency of a measurement system is generally small, the computation of will be fast. A pseudo-measurement is called a candidate if it is an available injection measurement connecting different observable islands, which can make at least one zero diagonal entry of nonzero by [4, Theorem 3.1]. The definition of candidate pseudo-measurements here is the same as those given in [2], [3]. After forming the product, the matrix can be transformed into its rectangular Echelon form, by Gaussian elimination [5]. Following the columns with nonzero pivots in the Echelon form will yield the corresponding measurements as the minimal set of candidates to be chosen for observability. Note that the number of such measurements will be exactly equal to the rank deficiency of the gain matrix.

4 822 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 4, NOVEMBER 2001 Thus, the steps of the proposed multiple measurement placement algorithm will be given as follows: Step 1 Form the gain matrix and perform the triangular factorization. Step 2 Check if has only one zero pivot. If yes, stop. If not, form the test matrix. Step 3 Using the test matrix to find the observable islands. Step 4 Form the candidate list. Those injection measurements, which connect at least two different observable islands, should be included in the list if those pseudo-measurements are available. Step 5 Form the rectangular matrix, and reduce it to its Echelon form. The candidates will correspond to the linearly independent columns in. A. Comparison With the Method [3] This section presents a comparative evaluation of the proposed method with that of [3] in order to demonstrate their conceptual equivalence, but computational differences. In the sequel, the algorithm of [3] will be referred to as algorithm 1 and the proposed algorithm of this paper will be called algorithm 2. Algorithm 1 decomposes the gain matrix by assigning 1 s at the diagonal entries whenever a zero pivot is encountered, as opposed to algorithm 2 where these zero entries are left as they are, yielding a singular diagonal factor. If the triangular factors of the gain matrix obtained by algorithm 2 and 1 by,, and by,, respectively then, the following equation gives the relation between these two different factorizations: (20) where is the pseudo-measurements corresponding to the zero pivots. In algorithm 1, the residuals for all the angle pseudomeasurements have to be computed in order to find out if the added pseudo-measurement can increase the rank of the gain matrix. After an injection pseudo-measurement is added into the existing measurement set, if at least one of the residuals is nonzero, then the corresponding angle pseudo-measurement will be moved out since it will now become redundant. Now we will briefly show that this process is equivalent to checking the nonzeros in corresponding to the zero pivots given in Theorem 3.1 in [4]. Before the addition of an injection pseudo-measurement, algorithm 1 first needs to solve the following equation: (21) where is the angle solution before the addition of ; is the measurement vector containing all zeros except for the angle pseudo-measurements with arbitrarily assigned integer values, such as, etc. Factorizing and pre-multiplying both sides by : (22) Note that the submatrix within corresponding to the nonzero rows of and, is an identity matrix. Thus, pre-multiplying these arrays by does not change any of the entries in these arrays. Hence, (22) can be rewritten as below: After introducing, the above equation becomes: (23) (24) where is the new angle solution after the introduction of. Pre-multiplying the above equation by, we obtain: (25) Let us denote by, and denote by. Then can be solved by the compensation method: where. We can rewrite (26) as below: (26) (27) Since all the entries in rows of corresponding to the zero pivots are zero except for a 1 on the diagonal, the entries in corresponding to zero pivots will appear as the difference of angles of and corresponding to the zero pivots. Hence, if the corresponding entries in the right hand side vector are not zero, those differences will also not be zero. As for the right hand side vector, is a diagonal matrix, hence we only need to study the rest part of the right hand side. We also know is a scalar, therefore, the values of those differences are totally decided by. If the elements in corresponding to zero pivots are all zero, then differences of angles in and corresponding to zero pivots must be zero; if the elements in corresponding to zero pivots have at least one non zero value, then there exists non zero differences of angles in and corresponding to at least one zero pivot. Therefore, instead of checking the values of angle residuals, we can compute the values of. This naturally requires less computational effort. The above result implies the modification of algorithm 1 as follows: Step 1 Form gain matrix. Perform singular triangular factorization of. If only one zero pivot occurs, stop. Step 2 Determine the set of nodes that do not have any injection measurement and whose adjacent branches have at least one nonzero flow. These nodes are candidates to have injection pseudo-measurements. If there is no candidate nodes, stop. Else

5 GOU AND ABUR: AN IMPROVED MEASUREMENT PLACEMENT ALGORITHM FOR NETWORK OBSERVABILITY 823 Step 3 Introduce an injection pseudo-measurement at one of the candidate nodes and compute. Step 4 Check if has any nonzero elements corresponding to the zero diagonals in. If yes and rank deficiency of the new gain matrix is still bigger than 1, update the singular triangular factors of new gain matrix; if no, select another candidate, and repeat step 3. Step 5 Return to Step 2. The above modified algorithm is now computationally more efficient since it requires one triangular decomposition rather than two in each iteration. However, the computation time can be further reduced if the triangular decomposition step can be avoided altogether. The procedure of triangular decomposition can be expressed as: Fig. 1. Measurement configuration of IEEE 14 bus system... computing its triangular factors, a diagonal factor is obtained with three zero diagonals indicating an unobservable system. (28) Let us suppose that sequential updating of the triangular factors times and simultaneous multiple-rank ( rank) updating of the triangular factors have the same computational complexity. Then the above process can be assumed to have a computation complexity of the Gaussian elimination of the matrix, where. However, Theorem 3.1 implies the Gaussian elimination of the whole matrix is not necessary. We only need to do the Gaussian elimination of the small submatrix of, i.e., matrix. Hence, algorithm 2 will require less computational effort to achieve the same results provided by algorithm 1. Applying the method of [4], the test matrix is computed and the matrix is formed. This yields the unobservable branches as: 5 6, 6 10, 10 11, 9 11, 9 14 and Removing all the unobservable branches one obtains the observable islands as: { }, { }, {10}, {11} and {14}. All the candidates can then be found. They are injections at bus 5, 6, 10, 11, and 13. The Jacobian matrix corresponding to the candidate injections is then formed and multiplied by to obtain the matrix as below: IV. SIMULATION RESULTS The proposed algorithm for the multiple measurement placement problem is tested on the IEEE 14 and 30 bus systems. A variety of measurement configurations have been studied. In this section, some of the simulation results will be shown to illustrate the steps of the algorithm. Upon reducing the matrix into its Echelon form, we obtain: A. IEEE 14 Bus System The IEEE 14 bus system and its measurement configuration, which is taken from the example used in [3], is shown in Fig. 1. Injections are located at bus 1, 2, 3, 7, 9, 12, and 14, and flows at branches 1 2, 1 5, 2 3, 4 7, 4 9, 6 13, 7 8, and 7 9. These are the must use measurements. Forming the gain matrix and Note that Echelon reduction may require row pivoting and since the rows with nonzero pivots indicate to the minimal set of measurements to be chosen, these should be properly identified based on the new row ordering. In the above example, the first two rows of correspond to the injections at bus 5 and 10 and

6 824 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 4, NOVEMBER 2001 Reducing into its Echelon form results in only two nonzero rows, namely 1st and 4th rows. These correspond to the injection pseudo-measurements at buses 22 and 27. Therefore, adding these two measurements to the existing measurement configuration, will make the entire system observable. Fig. 2. IEEE 30 bus system. hence they are chosen as the minimal set of measurements to make the whole system observable. B. IEEE 30 Bus System The IEEE 30 bus system and measurement configuration are shown in Fig. 2. The same procedure as above, is repeated for the given measurement set of the 30 bus system. Upon forming the test matrix and the product of the branch-node incidence matrix and lead to the identification of the following set of unobservable branches: and Removing these unobservable branches, four observable islands are obtained. These islands are composed of buses {24}, {29, 30}, {25, 26, 27} and {all other nodes}. This suggests the bus injections at nodes 22, 23, 25, 27, 28, 29, and 30 are candidates for measurement placement. Forming the corresponding sub-jacobian for these candidate measurements and calculating the product yields the following matrix: V. CONCLUSION This paper investigates the problem of measurement placement in power system state estimation. A new and improved algorithm is developed for multiple measurement placement. This algorithm avoids iterative addition of measurements and instead allows simultaneous placement of a minimal set of pseudomeasurements that will render the system observable. It is based on a previously developed method for observability analysis and makes use of small dimension test matrix in deciding on the placement of measurements. The dimension of the test matrix is equal to the rank deficiency of the existing gain matrix, and therefore is typically only a small fraction of the total number of buses in the system. This makes the method computationally very attractive, yet simple to implement in existing state estimators. Examples are included to illustrate the steps of the proposed algorithm. REFERENCES [1] G. R. Krumpholz, K. A. Clements, and P. W. Davis, Power system observability: A practical algorithm using network topology, IEEE Trans.Power Apparatus and Systems, vol. PAS-99, no. 4, pp , July [2] A. Monticelli and F. F. Wu, Network observability: Theory, IEEE Trans. Power Apparatus and Systems, vol. PAS-104, no. 5, pp , May [3], Network observability: Identification of observable islands and measurement placement, IEEE Trans. Power Apparatus and Systems, vol. PAS-104, no. 5, pp , May [4] B. Gou and A. Abur, A direct numerical method for observability analysis, IEEE Trans. Power Systems, to be published. [5] L. O. Chua and P.-M. Lin, Computer-Aided Analysis of Electronic Circuits: Algorithms and Computational Techniques: Prentice-Hall, Bei Gou is from Sichuan, China. He received the B.S. degree in electrical engineering from North China University of Electric Power, China, in 1990, and the M.E. (Electrical) degree from Shanghai JiaoTong University, China, From 1993 to 1996, he taught at the department of electric power engineering in Shanghai JiaoTong University. He worked as a research assistant at Texas A&M University since 1997 and received the Ph.D. degree in He currently works at ABB Energy Information Systems, Santa Clara, CA. Ali Abur (SM 90) received the B.S. degree from METU, Turkey in 1979, the M.S. and Ph.D. degrees from The Ohio State University, Columbus, OH, in 1981 and 1985, respectively. Since late 1985, he has been with the Department of Electrical Engineering at Texas A&M University, College Station, TX, where he is currently a Professor. His research interests are in computational methods for the solution of power system monitoring, operation and control problems.