IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 5, NO. 3, AUGUST 1990 911 State Estimation Using Augmented Blocked Matrices

, Fernando L. Alvarado, Senior Member William F. Tinney, Fellow The University of Wisconsin - Madison, Wisconsin Consultant - Portland, Oregon Known methods for improving numerical robustness tend to ABSTRACT reduce speed. The method introduced here improves robustness of the most widely used formulation of state Blocked structures of the augmented matrix of power estimation with very little, if any, sacrifice in speed. The system state estimation are studied and it is shown that there method is based on a particular block smcture and the use of are advantages in blocking according to a nodal formulation. block arithmetic in matrix operations. Block matrix The topological and computational properties of three methods and their importance in power system computations blocking alternatives are studied and their numerical were described in 131. The block method used in this paper condition numbers compared. Recommendations are given is similar the method used in [ 11 for contingency ranking. for developing software for the proposed nodal blocking to formulation. Kevwords: Sparsity, State Estimation, Sparse EXISTING STATE ESTIMATION METHODS Matrices, Matrix Conditioning. In this section the main known methods to solve the INTRODUCTION state estimation problem are reviewed. The small 8 node example of Figure 1 is used to illustrate these methods. The static state estimation problem in power systems can be defined as the solution to the following mathematical The H matrix is divided into two parts, a part problem: corresponding to branch measurements and a part corresponding to nodal injection measurements. The Given: topology of G and H for this example is: an unknown vector of n state variables x a known vector of m measurements z g g g g a known function h: Rn +R m G = g 1 a vector of m residuals r defmed as: r = z - h(x) -h h a known function g: Rn +RP h h a vector of p equality constraints on x: g(x) = 0 h h h h Find: h h a value for x that minimizes rt W r, the weighted sum h h of the squares of the residuals. W is a (usually h h diagonal) m by m matrix. . H = h h h h All methods considered rely equally on iterative com- h h putations. Therefore only the linearized version of the state h h estimation problem is considered here. The linearized state h h h estimation problem is: h h h h h h h h. Given: - an unknown vector of n state variables x Following are the main formulations and solution a known vector of m measurements z methods for the state estimation problem: aknownmbynmatrixH a vector of m residuals r defmed as: r = z - H x Traditional formulation without ecpal ie constraints a known p by n matrix G 1 Minimize J = [z-Hxl W [z-Hxl a vector of p equality constraints on x: G x = g -2 Find: This problem is solved from: a value for x that minimizes rt W r. [HfW HI x = H'W z The solution to this problem requires methods that are both computationally efficient and numerically robust. Amroximate handling of equalitv constraints 1 1 Minimize J = - [z-Hxl' W [z-Hxl + 5[ g-Gx]' V [g-Gxl 90 irll 241-0 PWHS A paper recommended and approved 2 by t h e ISE% Power System Engineering Committee of V is a diagonal matrix of artificial weights. Large t h e IZEZ Power Zngineering Society for p r e s e n t a t i o n values of V make the equality constraints more rigid, but a t t h e IEE

0885-8950/90/08OO-0911$01. OO 0 1990 IEEE 912 I Augmented matrix with constraints r122 1 A slight extension of the above method permits the introduction of equality constraints into the formulation: Minimize J = rt W r

__t Subjectto r = W - l p = z - H x and G x = g This problem is solved from: Nodal measurement O G 4 L [ b i t :l[tl=[aI Flow measurement The computational implications of using this 6 L formulation are described in the discussion and closure to 7 0 [14] and in [21]. Passive node (equality constraint) Pseudo-inverses method. no constraints r6.13.162 This method (also known as the method of Peters and Wilkinson) performs least squares errcr minimization on a Figure 1: Sample system with 8 nodes, 11 branches, 11 transformation of the original problem, First all references branch measurements, 3 nodal injection measurements to W are eliminated by re-defining H and z: (nodes 1,3 and 6) and 2 passive nodes (nodes 2’and 5). A = W-112 H b = W-112 z Exact enforcement of equality constraints r S l The problem becomes: 1 Minimize J = [Z-HXW]~ [z-Hx] Minimize ;itY where li = A x - b subject to G x = g The matrix A continues to be an m by n matrix. It can be factored into factors L and U, where L is an m by n This problem is solved by explicit introduction of matrix and U is a nonsingular n by n matrix. This Lagrange multipliers k factorization is not unique. In order to keep U nonsingular renumbering of the rows and columns of A is usually required (off-diagonal pivoting). A most convenient factorization of A is to require that L be lower trapezoidal, and that U be upper triangular. Augmented matrix (Hachtel) method. no constraints r152 A = L U Minimize J = rt W r Define: y = u x where r = W-l p = z - H x The original state estimation problem can be solved This problem is solved from the augmented equations: by first solving the following least squares error problem:

Minimize ’i t “r Subject to ’i = L y - b Where D = W-l. The augmented method requires that the This transformed problem is solved for y from: computations be properly organized. There are many possi- bilities for doing this. Which is best depends on the [LtL]y=Ltb application. Because at the onset many of the diagonal Solution of these equations requires the use of elements in the augmented matrix are zero or very small, it conventional sparsity-preserving factorization. The solution is not possible to select diagonal pivots arbitrarily. There to the original problem is obtained by solving the upper are three possibilities: triangular system of equations: Perform diagonal pivoGng but defer the selection of u x = y an element as a pivQt until its numeric value becomes sufficiently large. The numerical properties of the pseudo-inverse method can be shown to be better than those of the Perform of€-diagonal pivoting, making the matrix traditional method [16]. unsymmetric. Allow for the automatic recognition of nonsingular 2 Unconstrained Orthoeonal Factorization r19.20.231 by 2 blocks which include a zero component along This method solves the minimization problem by the diagonal. This approach was introduced by first performing an orthogonal factorization of the A matrix. [9,10]. It is effective but does not realize all the The A matrix is factored into the product of a rectangular benefits that can be attained in this type of problem, orthogonal matrix Q and an upper triangular matrix U. as this paper illustrates. Solution to the estimation problem proceeds in two steps: 913 Let y = Q t b The augmented Hachtel matrix for the example is: Solve the triangular equations U x = y for x Branch measurements Node Equality Residuals (1) -1 2 3 4 5 6 7 8 9 1 0 11 1 3 6 2 5 1 2 3 4 5 6 7 8 The matrix U has "second-neighbors'' topology. As 1 d in the pseudo-inverses method, ordering of rows and 2 d columns of A is required to preserve sparsity. Constraints are treated as measurements with large weighting factors. Pseudo-inverses with s u a lit^ constraints This method was introduced by Clements et al. [7]. The equality and measurement matrix equations are organized into a single partitioned matrix problem as:

This method performs the same factorization as the method of pseudo-inverses. It yields a lower trapezoidal and a nonsingular upper triangular matrix for this combined set of equations, except that the renumbering of the rows of the matrix is confined within each group. That is, all rows of G are numbered ahead of all rows of A. The result is a lower trapezoidal matrix L of dimension m+p by n and an upper triangular matrix U of dimension n by n. The lower trapezoidal matrix L has the following structure: A partial factorization is performed to eliminate branch and nodal measurements. Equality constraint equa- tions are retained. The result is a reduced matrix problem The dimension of L11 is p by p. The dimension of where the lower right hand comer fills in with a topology L22 is m by (n-p). The original problem can be solved by corresponding to first-neighbors for branch measurements first solving an ordinary set of permuted equations based on and second-neighbors for nodal measurements. The result is: L11 for an intermediate variable w, then solving a permuted modified least squares problem for an intermediate variable y 5 1 2 3 4 5 6 7 8 (based on L d , and finally solving for the original variables l g g g g 1 of interest. The key steps are: 0 1 g g g = la a a g l Solve L11 w g for w by forward substitution l a a 7l Solve & Lz;! y = I&b for y by sparse factorization a a a a a a a 7l a a a a Solve U x = for x by back substitution a a a a 1 7 l a a a a a a a1 THE BLOCKED FACTORS METHOD The h" and h" entries refer to values that result from The computational burden of existing methods of the incomplete factorization of the matrix, i.e., the state estimation is increased when nodal injection elimination of branch and node measurements. Italicized measurements are included because they create "second- entries refer to second-neighbor entries that result from neighbor" fill-in. Numerical difficulties are caused by the elimination of nodal measurements. These equations can be matrix products involved in the formulations and/or a wide written in "blocked form by grouping terms that correspond range of magnitudes in the weighting factors. to the same node. For example, node 2 is referred to in the The key idea of this paper is the observation that if upper rows of the matrix; it is also referred to in the lower the elements of the augmented matrix are grouped in blocks rows of the matrix. These two rows can be grouped. that correspond to nodes, the resulting blocked augmented Similarly, the two columns that refer to node 2 can be matrix has the same topology as the familiar first-neighbor grouped. References to node 5 can also be grouped into matrices of the power flow Jacobian and of Y-bus. The blocks. The result is a blocked matrix: sparsity properties of these matrices are well understood and excellent methods for sparsity preservation exist. Both traditional ordering methods [ 171 and newer ordering algorithms which have some advantages over the traditional ordering schemes [2,4] can be used. 914 1 2 3 4 5 6 7 8 - This matrix is solved using sparsity-preserving U'DU factorization, using block-arithmetic for all operations. This means, among other things, that diagonal blocks are inverted explicitly. Even though the blocks may have zeros on their diagonals, their inverses are well defined. Although there is no second-neighbor fill-in for the (3) blocked arrangement, the zero elements within the blocks have to be counted the same as the nonzero elements, and therefore may offset or actually increase the total number of nonzero floating point operations and storage requirements. State estimation in power systems usually results in two equations for each branch measurement, one for P and one for Q. There are also two variables to be estimated at each node, voltage magnitude and angle. Nodal measurements and constraints also come in pairs. Therefore, each of the "elements" h and g in the above matrices is actually a 2 by 2 matrix. Thus, the dimension of the blocks As an alternative, only the upper left hand portion that are dealt with are 2 by 2, 2 by 4, 4 by 2, or 4 by 4. containing branch measurements can be eliminated. The More detailed internal block topology is not illustrated here nodal measurements part of the matrix is not eliminated. because it does not contribute to understanding of the Parts of the matrix corresponding to nodal measurements and method, but a production-grade implementation would equality constraints are blended into a single group. The require processing of all these block types. proposed new method results in a matrix that has a first- It is often useful to compute certain elements of the neighbors topology pattern: residual co-variance matrix. The traditional formula for the 1 3 6 2 5 residual CO-variancem amx is: 1 d R = W-* - H (HtWH)-' Ht 3 d (6) 6 d Elements of (HtWH)-I can be computed directly from 2 0 the augmented formulation in equation (5), and these values 5 0 used in equation (6). Computation of post-estimate quanti- 1 h h g ties is not directly affected by the blocking scheme. It ap- 2 h h g pears, however, that other forms of blocking may be used to 3 h h g g speed up these computations, but this has not yet been tried. 4 h g g 5 h g EXAMPLES 6 h h This section illustrates topological and numerical 7 h g ll examples of the proposed blocking method for power 8 h systems ranging from 118 to 707 nodes. In all the comparisons the following is assumed: There is a branch measurement at every branch. There is a nodal injection measurement at one out of 1 every three nodes. That is, there are injection measurements at nodes 1,4,7, etc. 2 One out of every three nodes is a passive node (an equality constraint). That is, the injections at nodes 2,5,8, etc. are treated as equality constraints. 3 h O The actual topology of the example systems is retained but their numerical values are replaced with a 4 more consistent set that permits better comparisons of the numerical properties of different size networks. All figures, operation counts and condition number calculations have been performed using the Sparse Matrix Manipulation System, a prototyping sparse matrix manipulation package [21. Sample Topology Diagrams In Figure 2 the topology diagram of the matrix that 8 1 results from the conventional method is compared with that of the blocked matrix from [12] and with the topology of the 915 blocked matrix proposed in this paper. These comparisons I are illustrated for a 118 node system. . . . . Numerical Conditioning ...... :i 1 * e .: Numerical ill-conditioning has been associated with . . : . . . . . iii f state estimation mamces. The degree of ill-conditioning of .. ..."...... , ...... a matrix can be measured using the notion of "condition :: * ::: .... i .i ...... " ...... number," which is the norm of a matrix times the norm of :. :::U its inverse. The norm used in this paper is the one-norm: * .. :.. 'f iff! :. . :-. : . : .... :.: ti:- 1? I! llAll1 = max Claijl ..".. . i.: . ... **" .." ...... ::. ... .::* "U 1: :I : j i -7 * : .:.:.: :I .:. * :! . ..."...... I..d I The condition number: K(A) = llAll IIA-lII ..I: For more information on matrix norms and condition :i i - .*f.*f.* .":. t: *". : "-7 -3 i **if- :.I:.: . .*.&.* * ::: : numbers refer to [8]. The smallest possible condition ... . z e?: " :::: ...... I : ;;?$:: p:. * iiiiii number for a matrix is one. Small values (below 100) . . . . : ..... :mtn ::::::it::-"- ...... -- . . a : &,-.--: I - . *.- .- ...... I generally indicate good condition and few numerical I . . . ; ...... difficulties. Large values (greater than 1000) indicate ill I conditioning. Multiplying a matrix times itself squares its (a) The normal equations method. Dimension is 118. condition number. It is not hard to see, therefore, that I=.. methods that rely on the explicit computation of HtH are likely to be ill-conditioned. When ill-conditioning is a potential problem, certain measures can be taken to minimize its effects. These include QU factorization based on orthogonalization [19,20], and the use of the method of pseudo-inverses, which avoids many (but not all) of the condition number problems. A good way to avoid ill-conditioning is to avoid the operations that increase the condition number in the first place. The proposed method avoids computing the matrix products that lead to large condition numbers; therefore it never lets the condition number deteriorate. Tabular Cornmisons Three methods are compared using three different matrices and two criteria. The methods compared are: The "traditional" method with equality constraints (b) Augmented method. Dimension is 118 blocks or 157 handled by using larger weights. The weights used individual elements. Individual elements shown. are 10 times the unweighted values. (Note: typical values could actually be much greater). The augmented method, where blocking is used to deal with zeroes in the diagonal. The proposed method, where submatrix blocks are used not only to deal with zeroes but with all nodal equations.

Table 2: Comparison of condition numbers.

(c) The proposed method. Dimension is 118 blocks or 197 individual elements. Individual elements shown. Figure 2: Comparison of topologies for 118 node system. Each dot represents an individual element within a block. 916 to that of the incidence matrix of the network. This leads to The systems on which these comparisons are the normal first-neighbor topological structure of the performed are all based upon actual power system blocked matrix which is processed in the same sparsity- topologies, ranging from 118 to 707 nodes. The preserving order the nodal admittance matrix. Explicit comparison criteria are: as inversion of diagonal submatrix blocks overcomes former Computational effort, measured as the total number difficulties with small or zero diagonals and further enhances of multiplications required for a complete U'D U numerical stability. factorization step. All multiplications are counted, Equality constraints are enforced exactly instead of as including those needed to construct the matrix. No measurements with artificially large weighting factors. advantage is taken of symmetry or other computa- Another feature of the method is that it can accommodate a tional shortcuts. more realistic covariance matrix having nonzero off-diagonal Condition numbers are measured according to the one- weighting factors corresponding to correlations between the norm of the matrix. active and reactive power measurements at a bus. In fact, In an actual implementation, matrix symmetry any covariance terms that fall within the submatrix blocks (in locations that are now zeroes) can be accommodated with normally would reduce the indicated operations counts by almost one half. Since this reduction would be no effect on computational burden. approximately the same for each of the methods compared, This paper has not considered convergence issues in the relative counts, which are what matter, would be state estimation. Convergence of the proposed method unaffected. A more detailed comparison would also require should be nearly identical to that of the other state an estimation of the computational effort of the sparsity estimation methods when ill-conditioning does not play a logic for each method. In general, block operations reduce significant role, and should be significantly better in those the relative cost of sparsity logic, compared to floating point cases where ill-conditioning affects convergence. arithmetic. The larger the blocks, the lower the ratio of The proposed method can be used with coupled or logical to arithmetic operations. decoupled formulations. The only requirement for achieving The matrices tested had one third of all nodes its potential efficiency is skillful programming of sparsity designated as equality constraints. The benefits of the operations. There are no discernible disadvantages. proposed method increase as the number of nodal measurements or constraints increases. However, the ACKNOWLEDGEMENTS method proves advantageous even when a smaller number of nodal measurements is considered. The test for the 118 node One of the authors (Alvarado) wishes to acknowledge system was repeated with 10% equality constraints instead of partial NSF support under contract ECS-8822654. 33% and with weighting factors of lo00 instead of 10. In this test the condition number was greater than lo6. This REFERENCES compares with 227 for the proposed method, which was 1. 0. Alsa~,B . Stott and F. L. Alvarado, "Analytical unaffected. Additional extensive comparisons among all and Computational Improvements in Performance algorithms are described in [22]. These comparisons indicate Index Ranking Algorithms," Int. J. of Elect. Power that the condition number of the Givens method is superior, and Energy Syst., Vol. 7, pp. 154-166, July 1985. but that the proposed method compares well as far as 2. F. L. Alvarado, "The Sparse Matrix Manipulation number of operations. System," SIAM Symposium on Sparse Matrices, Gleneden Beach, OR, May 22-24 1989. Some investigators have contended that convergence is improved by using weighting factors instead of the 3. F. L. Alvarado and M. K. Enns, "Blocked Sparse Lagrange multipliers for equality constraints because the Matrices in Electric Power Systems," IEEEPES weighting factors "soften" the constraints. Any desired Summer Meeting, Portland, OR, July 18-23 1976. amount of "softness" in constraint enforcement can be 4. F. L. Alvarado, D. Yu and R. Betancourt, obtained with the proposed method by adding suitable "Partitioned Sparse A-l Methods," IEEEPES weighting factors to the normally zero diagonals of the exact Summer Meeting, Long Beach, CA July 9-14, 1989. equalities. Such added weighting factors will not worsen the 5. F. C. Aschmoneit, N. M. Peterson and E. Adrian, condition number compared to the case of zero diagonals for "State Estimation with Equality Constraints," Proc. exact equality constraint enforcement. of PICA Conf., pp. 427-430, Toronto, May 1977. 6. A. Bjorck and I. S . Duff, "A direct method for the CONCLUSIONS solution of sparse linear least squares problems," A new method of blocking the Hachtel formulation Linear Alg. and its Applic, Vol. 34, pp. 43-67, 1980. for power system state estimation is proposed. As indicated 7. K. A. Clements, G. W. Woodzell and R. C. Burchett, by its large reduction in the numerical condition number of "A New Method for Solving Equality-Constrained three representative test problems, it should be able to Power System Static-State Estimation," IEEEPES overcome the usual causes of ill-conditioning of power Summer Meeting, July 9-14 1989. system state estimation. As shown by comparisons of its 8. I. S . Duff, A. M. Erisman and J. K. Reid, "Direct computational requirements with those of other methods, Methods for Sparse Matrices," Oxford Science there is no sacrifice in speed to achieve robustness. Publications, Clarendon Press, 1986. The key idea is organization of the Hachtel 9. I. S . Duff and J. K. Reid, "A comparison of some formulation into a submatrix block structure that conforms methods for the solution of overdetermined systems 917 of linear equations," J. Inst. Maths. Applics., Vol. Fernando L. Alvarada was born in Lima, Peru. He received 17, pp. 267-280, 1976. the BEE and PE degrees from the National University of 10. I. S. Duff and J. K. Reid, "Multifrontal Solution of Engineering in Lima, Peru, the MS degree from Clarkson Indefinite Sparse Symmetric Linear Systems," ACM University, and the Ph.D. degree from the University of Trans on Math Sofhuare, Vol. 9, Sept. 1983. Michigan in 1972. He joined the University of Toledo in 11. I. S . Duff, J. K. Reid, N. Munskgaard and H. B. 1972. Since 1975 he has been with the University of Nielsen, "Direct Solution of Sets of Linear Equations Wisconsin in Madison, where he is a Professor of Electrical whose matrix is sparse, symmetric and indefinity," J. and Computer Engineering. Inst. Maths. Applies., Vol. 23, pp. 235-250, 1979. William F. Tinney received his B. S. and M. S. degrees A. Gjelsvik, S. Aam and L. Holten, "Hachtel's 12. from Stanford University in 1948 and 1949. He worked for augmented matrix method - a rapid method improving the Bonneville Power Administration from 1950 until his numerical stability in power system static state retirement in 1979, at which time he was head of System estimation," IEEE Trans. on Power App. and Syst, Analysis. Most of his work has been concerned with power Vol. PAS-104, pp. 2987-2993, NOV.1 985. system computer applications and he is presently a 13. G. Peters and J. H. Wilkinson, "The least squares consultant in the field. problem and pseudo-inverse,'' The Computer Journal, Vol. 13, NO. 4, pp. 1534-1542, 1970. 14. F. F. Wu, W. E. Liu and S. Lun, "Observability Analysis and Bad Data Processing for State Estimation with Equality Constraints," IEEE Trans. on Power Syst, Vol. 3, pp. 541-548, May 1988. 15. G. D. Hachtel, "The Sparse Tableau Approach to Finite Element Assembly," Sparse Matrix Computations, pp. 349-363, 1976. 16. J. W. Gu, K. A. Clements, G. R. Krumpholz and P. W. 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DISCUSSION with the resources we have at our disposal (computer + algorithm). And, if this is not the case, what should be done to fix the L. RADU (Consolidated Edison, New York, NY) possible causes of instability. Preferably and A. MONTICELLI (UNICAMP, Campinas, this should be done on the fly during the Brazil) : The authors are to be congratulated factorization of the gain matrices. on an interesting and well written paper. While the basic algorithms and techniques used over the last three decades the general were known, they have been combined in a novel tendency in power system computer analysis has way and resulted in an improved solution for been to do some kind of matrix prearranging both internal and, especially external state (structural or topological analysis) that estimation. For example, the idea of using a would permit the factorization process of the blocked sparse matrix formulation for the N/EC problem matrices to be carried out without method was suggested in a discussion to the worrying about numerical problems (pivot OPF paper [8]. The authors prpvide a checking). For example, the topological practical strategy in the vtpairingv8o f observability analysis should, ideally, lead equations which is important for the to full rank state estimation matrices, which. observable solution where the pairing is not hopefully, will go through the numerical trivial (as is the case of external solution). factorization process without any problem. Also, while the state estimator based external When the standard normal equations approach is solution was introduced in [6], the handling used, the part of the system defined as of all injections (except zero equality observable by the topological observability constraints) in a "Hachtel I s manner" improves algorithm usually leads to a positive definite the robustness of the solution and is better gain matrix. Usually, but not. always: that suited (than a weighting technique) to the will depend on how the gain matrix is implementation of various power flow controls computed (computer word sizes may play an and adjustments. important role here). When it happens that the gain matrix is indeed positive definite (in The authors report that the N/EC method worked the numerical sense) it may be factorized remarkably well for the observable solution in without numerical pivoting. As for the several control center implementations. methods relying on indefinite matrices (such Recently this method was also implemented in as N/EC and Hachtel's), blocking schemes as the Consolidated Edison EMS as a replacement the ones suggested by this and the related to the original, vendor supplied, normal paper [A] may permit a positive definite like equations based program. Note that, due to factorization of the associated indefinite the very high number of very short line and matrices. This is not always guaranteed, long line (i.e. transformer) connections however, as practical experience has shown in having low local measurement redundancy and several different implementations. the large number of passive buses, this network can be a litmus test for any State Consider, for example, a radial string of two Estimator technique. The N/EC method was flow measured branches connected in the implemented with on the fly reference bus, one with very low impedance ordering-factorization (delayed elimination (see example in [B] ) . Also, while it is scheme), the factorization in double precision true, as the authors state, that critical and scaling of the weighting factors. The measurements could be identified method is robust. However, it has to be topological observability, numerical bcYheckz mentioned that, there are only a few measured might provide additional measurements which injections in this system since flow measured are in fact bad data unidentifiable. subtransmission transformers were explicitly modelled. Numerical problems do occur, qualitative methods sure may help, but are not guaranteed With respect to the fact that the authors to work in all cases. This being the case, we method is "linked to a topological suggest that, since usually such problems can observability routine" and "relies exclusively be traced to a few %ad actors", removing them on positive-definite matrix techniques" we and thus making a small part of the network would like to comment that, perhaps, for the temporarily unobservable, would greatly sake of performance, some flexibility is contribute to the reliability of the state sacrificed. It is well known that the State estimator. For this purpose the numerical Estimator is the one network application in an observability algorithm could be used when EMS which should be very reliable. If it numerical difficulties appear during the does fail, other Security functions are factorization process. endangered and more importanttits lltrustworthinesstinv the eyes of the system For a discussion on a closely related problem, operator can suffer since he/she can always see [C] which deals with numerical problems compare the estimated quantities with the more associated to the Newton OPF approach. familiar SCADA measurements. However, in real-time systems, "pathologicalvtc ases tend to appear from time to time. Let us consider REFERENCES the following situation: a state estimation run converged in, say, io iterations showing [A] F. L. Alvarado and W. F. Tinney,"State some abnormal voltages / flows not flagged as Estimation Using Augmented Blocked Matrices", bad data. The typical number of iterations Paper 9OWM 241-0 PWRS, IEEE/PES 1990 Winter for convergence under similar conditions is 4 Meeting, Atlanta, GA. or 5 (rather than 10). The question is: are the abnormal values and the unexpected number [B] A. Monticelli, C. A. F. Murari, and F. F. of iterations due to, say, an emergency Wu, "A Hybrid State Estimator: Solving Normal condition that should be handled by some Equations by Orthogonal Transformationsfl,IEEE rescheduling function or, instead, the Trans. PAS, vol.PAS-104,No.l2, p.3460-3468, abnormal conditions are simply due to some December 1985. hidden numerical problems (say, small pivots were generated during factorization leading to [C] A. Monticelli and W-H. E. Liu, "Adaptive inaccurate triangular factors). Or, put in a Movement Penalty Method for the Newton Optimal different way: we may want to know if the Power Flow18,Paper9 OWM 252-9 PWRS, IEEE/PES problem we are dealing with is really solvable 1990 Winter Meeting, Atlanta, GA. 919

KEVIN A. CLEMENTS (Worcester Polytechnic Institute, Worcester, MA)I constrained state estimation”, paper 90 WM 234-5 congratulate the authors on a most interesting paper describing a new algorithm PWRS, IEEE/PES Winter Meeting, Atlanta, GA, for solving the equality-constrained power system state estimation problem. Feb. 4-9, 1990. The indefinite system of equations that arirres in constrained state estimation problems has a very special structure and the algorithm presented by the authors [ e ] F.F. Wu, U-H. E. Liu, L. Holten, A. Gjelsvik, exploits this structure quite nicely. S. Aam, “Observability a n a l y s i s and bad d a t a pro- One issue that was not fully addressed in the paper is that of selecting the cessing f o r s t a t e estimation using Hachtel-s aug- pivot elements. This cannot be done by a purely symbolic method since it is mented matrix method”, I E E E Trans. on Power Sys- poeaible that candidate pivot blocks may have zero determinants. This problem tems, vol. 3, pp. 604-611, 1988. did not arise in the example presented in the paper but is likely to Occur in leas well measured systems. Manuscript received February 27, 1990. It seems that candidate pivot blocks could be chosen using the minimum degree algorithm but that it is necessary to test the numerical value of the determinant of the block before choosing it as a pivot. Do the authors include such a numerical test in their algorithm? P. A. Machado, G. P. Azevedo (CEPEL, Rio de Janeiro/RJ, Brazil): The authors are to be commended for their Manuscript received March 5, 1990. effort to introduce a method to improve robustness of the state estimation problem.

ANDERS GJELSVIK and SVERRE AM4 (Norwegian Electric Our comments regarding the paper are the following: Power Research I n s t i t u t e , Trondheim, Norway) and LARS before the factorization process the method establishes blocks HOLTEN (ABB Network Control, Vasteras, Sweden). I n correspondin to zero injection measurements. It may happen t h i s paper t h e authors have modified t h e Hachtel-s that an earfy arrangement of the blocks may lead to an augmented matrix method, and by a block matrix unsolvable system (zero pivot),even when the system has a t r i a n g u l a r i z a t i o n scheme they avoid t h e d i f f i c u l t i e s solution (if the Tinney I1 criterion of ordering is respected). Let t h a t arise s i n c e t h e augmented gain matrix is not us consider the following example. This network is observable p o s i t i v e d e f i n i t e . W e congratulate t h e authors f o r with respect to the Q-V criterion (decoupled version of state t h i s achievement. What seems t o be similar ideas, have estimation). appeared simultaneously i n [A].

We would l i k e t o make t h e following comments: I n our own work with Hachtel-s method, t h e Harwell MA27 r o u t i n e was used. It performs numerical tests during f a c t o r i z a t i o n and uses 2 x 2 p i v o t s when it is found -4 necessary. For t h i s a dynamic d a t a s t r u c t u r e is where: needed. With t h e present method one seems t o be a b l e t o use a f i x e d d a t a s t r u c t u r e , determined from spar- ZI - zero injection equality constraint s i t y alone. This is a big advantage i n a computer i m - 0 - voltage measurement plementation. One then has t o assume t h a t s i n g u l a r i - x - reactive power flow measurement ties w i l l not occur. We would l i k e t o know whether t h e authors have encountered problems where t h e p i v o t After voltage and flow measurements elimination, as diagonal block matrices have become c l o s e t o s i n g u l a r suggested in the paper, we obtain the following matrix: during f a c t o r i z a t i o n . Z l 1 2 3 4 5 The formula ( 6 ) f o r t h e r e s i d u a l covariance is t h e one f o r t h e unconstrained case. For t h e constrained 21 case a s l i g h t l y d i f f e r e n t formulation has t o be used ( r e f . (141 of paper). I n [B] (Section I V and Appendix 1 B ) it is described how t o obtain r e s i d u a l covariances from Hachtel-s augmented matrix i n a very e f f i c i e n t 2 way. With s l i g h t modifications a similar approach should be useful a l s o with t h e present formulation. 3

Table 2 i n t h e paper shows s i g n i f i c a n t l y lower 4 condition numbers f o r t h e proposed method than f o r t h e augmented matrix method. We wonder whether t h i s can be 5 due t o t h e p a r t i c u l a r system d a t a used. The d i f f e r e n c e i n t h e two approaches lies i n t h e elimination of t h e where: x - original elements upper l e f t hand portion containing branch measurements - fill-in elements i n t-- h e augmented matrix (1). This operation l e a d s t o c - diagonal contributions t h e h elements i n ( 4 ) . One might think t h a t t h i s elimination would i n general be more l i k e l y t o Using the blocked arrangement we get: increase t h e condition number of t h e matrix than t o reduce it, s i n c e it is similar t o t h e formation of t h e normal equations. This would depend on t h e condi- t i o n i n g of t h e eliminated p a r t of H . I n general t h e r e 1 may be sources of ill-conditioning a l s o i n t h i s p a r t , e.g. mixture of s h o r t and long l i n e s . Taking t h e 21 square root of t h e “ t r a d i t i o n a l “ condition numbers i n Table 2 i n d i c a t e s t h a t t h e systems d e a l t with are w e l l 2 conditioned. We would appreciate t h e authors’ comments on t h e observed reduction i n condition numbers. 3 REFERENCES 4

[A] R.R. Nucera, Michel L. G i l l e s , “A blocked sparse 5 matrix formulation f o r t h e s o l u t i o n of equality- 920

Accordin to the Tinney I1 criterion the next pivot will that require such modification. Pseudo-measurements be row 1 (one off4iagonal element) which has zero diagonal. introduced to prevent singularity (option (c)) does not cause fill-ins. However, their effect on the estimate must We would appreciate the authors' comments on how the subsequently be nullified by compensation [A,B] in the proposed approach can handle situations as the one discussed same manner as identified bad measurements are nullified, above. and this adds to the computational burden. The burden is small, unless there are many such pseudo-measurements.

The difficulty with singular diagonal blocks is not F. ALVARADO and W. F. TINNEY: We thank the discussers necessarily disposed of even after a satisfactory initial for their excellent discussions. They add important ideas factorization. If the set of measurements is changed and point out a significant omission in our paper. The during the estimation process, the condition can occur and main concern in all four discussions is that a sparsity- must be guarded against. Removal of bad measurements, oriented ordering/blocking scheme based strictly on except by compensation, can create singular diagonal topological properties is vulnerable to numerical blocks the same as a missing original measurement. breakdown under certain conditions even though the Identification and/or modification of operations for network is topologically observable. The discussion of critical conditions must therefore be applied again. The Machado and Azevedo illustrates this problem quite welf. tradeoffs may lead to different choices for these operations Two of the discussions refer to a concurrent paper by in the initial and any following factorizations. If Nucera and Gilles describing the same general blocking compensation (which we recommend) is used to effect the scheme which also discusses this condition. Unless the removal of bad measurements, new singularities cannot be given scheme is modified in some way when this condition introduced and the initial blocking/ordering need not be exists, it will cause a diagonal block to be singular and changed. Compensation can be performed with high thereby cause numerical breakdown. Suggestions and efficiency, particularly if approximations are permitted. insights provided by the discussers are included in our response to this concern. Combinations of alternatives (1) and (2) with options (a), (b) or (c) and certain hybrid combinations lead to a There are two aspects to coping with a condition that can variety of modification schemes for overcoming the cause a diagonal block to be singular: jdentific- of the difficulty. The choice depends to a considerable extent on condition and m o d. i. f i .c w of the ordering/blocking the schemes used for observability and bad data analysis. scheme to prevent numerical breakdown from the Whether the formulation is coupled or decoupled may also condition. Identification can be accomplished in either of affect the choice. two basic ways: (1) topologically or (2) numerically. A combination of both is a third possibility. Modifications to Gjelvik, Aam and Holten express a concern about cope with an identified singular block can be numerical difficulties caused by the reduction of the line accomplished in at least three different ways: (a) flows. It is possible to come up with a system where widely witholding the block from the sparsity-directed ordering differing values in the line lengths could lead to a less until is has acquired a fill-in from an adjacent elimination, than ideal condition number. However, the "squaring" of (b) expansion of the block to include another node, or (c) the condition number observed in the normal equations introduction (and subsequent removal) of an innocuous method does not occur. The ill-conditioning, if any, is not pseudo-measurement into the block. artificially added by the numerical method of solution (as it is in the normal equations method), but is inherent in Identification of conditions requiring some modification of the problem. We have experimented with a wide range of the ordering/blocking scheme can be performed prior to values for line impedances, with the result that the or concurrently with factorization. No matter what condition numbers deteriorates somewhat for all methods, precautions are taken prior to factorization, they may not but the relative merit of the proposed approach remains. be sufficient to overcome all possible numerical Another suggested technique for improving numerical difficulties. This was pointed out by Monticelli and Radu. conditioning with or without blocking is the direct A network may be observable topologically and evaluation of the algebraic expressions for branch flows unobservable numerically. As a safeguard against this in the gain matrix, as suggested by Monticelli and Radu. possibility, a test to detect near singularity of diagonal We were aware of this technique but did not program it blocks must be performed during factorization no matter and test it. We are not certain whether true or only what is done beforehand. The identification of singularity apparent benefits are derived from this approach. concurrently with the recognition of blocks has the advantage of not only detecting this condition, but also the Gjelvic, Aam and Holten also bring up the subject of ability to correct it. The most general version of this covariance matrix computation. In our opinion, only approach is that taken by the Harwell MA28 routine. In selected elements of the co-variance matrix R are actually this approach all blocking is done strictly on numerical required. These elements can be computed "on the fly" as grounds. For some applications this may be the preferred the need for them arises in identifying and compensating approach. However, for many other applications we think for the removal of bad measurements. This computation that detection of the singularity followed by introduction can be done by using sparse vector methods to find selected residual covariance elements. Typically, this is of a pseudo-measurement by compensation may be a better alternative. only a fraction of the effort required to compute a sparse inverse. The paper by Wu et al. mentioned by these Another alternative to prevent singularities on observable discussers gives a formula for the computation of the covariance matrix. The following version of the formula is systems is described in the Nucera and Gilles paper. In this paper only pairings are permitted (no larger blocks). The more suitable to the use of blocked matrix methods: observability tree approach of reference [9] in the Nucera and Gilles paper is used to determine when pairings are to occur. That is, a bus may have to be paired with a remote measurement one bus away to resolve its singularity.

Any modification of the sparsity-directed ordering (option (a) above) or block sizes (option (b) above) of the ordering ]=[,'I scheme is likely to increase the fill-ins in the matrix factors and thereby increase the computational burden. In this formula, the upper left hand side has a known left The increase is small unless there are many conditions hand side and an unknown right hand side. Selected 921

elements of R are computed by selective solutions and Hypothesis Testing Identification in Power System partial matrix-vector products. State Estimation," IEEE Transaction on Power Systems, Vol. 3, No. 3, pp. 887-983, August 1988. To summarize, we think that for most problems and under [B] 1. W. Slutsker, "Bad Data Identification in Power most circumstances a compensation-based modification of System State Estimation Based on Measurement our topological approach to blocking is sufficient to Compensation and Linear Residual Calculation," IEEE eliminate numerical solution concerns, but we recognize Transaction on Power Systems, Vol. 4, No. 1, pp. 53- that many variants are possible. 60, February 1989.

[A] L. Mill and T. Van Cutsem, "Implementation of the Manuscript received April 9 , 1990.