Algorithms for Reordering Power Systems Matrices to Minimal Band Form

Transcription

1 174 Algorithms for Reordering Power Systems Matrices to Minimal Band Form P.S. Nagendra Rao Pravin Kumar Gupta Abstract - Efficient exploitation of sparsity of matrices is very critical for the successful implementation of many power system computation problems. This paper introduces a non-traditional approach to handling sparsity in power system computations. An attempt is made to provide evidence to suggest that minimum band ordering of matrices also could be considered as an option in this regard. This is done by showing that of the y- bus matrix of power networks is amenable to minimum band ordering. We propose several new algorithms for reordering symmetric matrices to minimal band form demonstrate that these new algorithms are very effective in reducing the band width of not only the y-bus matrices of power network bus also problems. In addition, the performance of the new algorithms are compared with several existing band reduction algorithms. Index Terms Linear system of equations, sparse matrices, optimal ordering, minimal band form, finite element methods, spectral methods. I. INTRODUCTION Minimum degree ordering schemes proposed by Tinney in the late sixties are by and large the only optimal ordering methods in use in power system computations. However, there are many other ordering scheme that have not been explored in this context. In this paper we study an alternate approach to reordering power systems matrices. The scheme considered here is, reordering to minimize the bandwidth of matrices. Here, we propose several algorithms for reducing the bandwidth of sparse symmetric matrices and compare their performance with several popular approaches. Reduced bandwidth ordering is of interest due to several reasons. There are many schemes for solving linear equations for the form Ax=b, where A is in banded form. These schemes are considerably simple to implement a the schemes of solving linear systems where the coefficient matrices have arbitrary sparsity. The execution time for "band solvers" for example, is O(NB 2 ) for large N and B. Thus it is desirable to reduce B as much as possible Even if it is possible to reduce bandwidth by a small margin, one archives a significant reduction in computation. Bandwidth reduction is also of interest in the case of solving large sets of linear equations, having arbitrary sparsity Prof. P.S. Nagendra Rao, Department of Electrical Engineering, Indian Institute of Science, Bangalore , id nagendra@ee.iisc.ernet.in Pravin Kumar Gupta, former student, Indian Institute of Science, Bangalore structure, in parallel. A banded structure facilitates efficient assignment of threads to processors. It helps to reduce the amount of communication between processors and in most cases this can be limited to neighbouring processors connected as linear array. Besides this, banded structure also helps to reduce the complexity of computation in each of the processors and in situations where the processors support vector instruction, leads itself to easy vectorisation. Reordering large matrices for reducing bandwidth is not in wide use in power system applications because the network topology is not regularly structured and it is believed that finding reduced bandwidth ordering is not easy for these systems. However, bandwidth reducing algorithms are widely used in civil and structural engineering applications in the context of finite element analysis where the physical being analyzed are well structure. Most of the existing work in bandwidth reduction is motivated by the needs of these applications. During the past decade, considerable amount of research efforts has gone into evolving efficient methods for minimizing the bandwidth of sparse symmetric matrices. Some of the important algorithms are discussed briefly here: In the following discussion, we use the words row/column number and node as equivalent. The graph terminology used pertaining to the graph which represents the structure of the given symmetric matrix. Each row/column is represented by a node and the nonzero off-diagonal elements are represent the edge of this graph. In Cuthill-McKee(CM) algorithm[5,7], among the nodes of low degree, we select the one as a potential starting node from which we find a tree using BFS approach which partition the nodes to several level sets. Nodes in each of the level sets are ordered consequently, by numbering those nodes, whose parent node have a smaller number, first. Considering each of these nodes as a starting node, Consider each potential starting node as starting node, we choose the particular ordering which gives the minimum bandwidth. The Gibbs-Poole- Stockmeyer(GPs)[8] algorithm differs from CM primarily in the selection of starting node. In GPS, we try to find the pseudo-diameter and the starting node is choosen as an end point of pseudo-diameter. Thus, the number of BFS searches are reduced significantly. Puttonen[3] has proposed a renumbering method by systematically changing rows and columns in the connectivity matrix. In phase 1, it condenses the upper triangular remaining nodes below the highest bandwidth row and checks whether it is possible to reduce bandwidth by interchanging the highest bandwidth row with

2 INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR , DECEMBER 27-29, a row below it. Other algorithms are due to : Collins(RC)[2](used a element-dependent search), Rosen(RR)[12] (used a row permutation scheme), and Akhras[10] (used a row ponderation method). The Gibbs- King(GK)[4,11]algorithm, as compared to GPS algorithm, differs only in the last step (reordering the node similar to CM). Here, nodes are reordered according to a different heuristic algorithm. From a careful analysis of these algorithm it is easy to see that all these algorithms are essentially heuristic search algorithms and their success depends on the correctness of heuristic. Hence, no single algorithm performs uniformly well for all type of systems and the performance of a given algorithm for a particular system could be varied depending on the choice of the initial structure/data structure etc. The major aim of this paper is to show that the power system matrices can indeed be ordered to bounded forms and this is established by considering several existing algorithms as well as by proposing new band reducing algorithms. The performance of the set of algorithms propose here is constant over a range of applications. The proposed algorithms are basically different from the existing ones as they are not based on heuristic search. Instead, they are based on numerical computations. We compare their performance with the existing algorithms for matrices obtained from power systems as well as other engineering application areas. II THE PROPOSED ALGORITHMS The approach adopted here to develop the new bandwidth reduction algorithms is to start with a row numbering which is unique for a given matrix. One such numbering is possible for symmetric matrices. This can be done by assigning a unique value to each row/column of the matrix and order the matrix row/column based on this value. Such a unique value is obtained by computing the eigen vector values corresponding to second eigen value of the matrix which captures the structural information of the given matrix. Second eigen value is the lowest non zero eigen value of connectivity matrix. Such numbering has been very profitably used in several matrix structure related applications [1] such as finding bordered band forms and clusters in sparse matrices. The first of the new algorithms considers this ordering itself as the basis for the final banded solution, the other algorithm uses this ordering as an initial ordering and seek to reduce bandwidth further by some additional simple search techniques. A. Algorithm 1 Step 1: From the adjacency list, form the connectivity matrix C, where C ( i, = 1 if node i and node j are connected for all i i, j = 1,2,..., N j N C ( i, = - C( i, j= 1, j i Step 2: Find out eigen values of matrix C. Step 3: Find out eigen vector corresponding to second eigen value of C. Assign the N components of the eigen vector as values to the N rows/columns of the matrix. Step 4: Renumber the rows/columns such that their valuation are in non-decreasing order. Step 5: stop. B. Algorithm 2 Step1-3: Same as that of algorithm 1. Step4 : Find the rows/columns that has smallest eigen vector value. Level of these nodes is 1. Renumber all the nodes in this level by assigning number 1,2,3.. etc. Step5 : Find nodes in the next level by finding the nodes connected to the nodes at the present level which have not been accounted so far. Order this set according to the increasing position of their parents. Ties are broken based on the eigen vector value (the node that has smaller eigen vector value will be numbered first). Step6 : Continue step 6 until all the nodes are not renumbered. C. Algorithm 3 Same as algorithm 2, only difference is that instead of starting with smallest eigen vector rows, start with largest eigen vector rows and in case of tie in step 5, the node that has the larger eigen vector value will be numbered first. D. Algorithm 4 This algorithm is a heuristic algorithm, which tries to improve to bandwidth of any given ordering, using some limited local searches. Let us define for the given matrix A, for row i, let left(i) and right( represents the position of leftmost and rightmost non zero entry columns respectively. Then left( i) i right( i) Hence, the bandwidth can also be defined as B = max [max[( i left ( i )), ( right ( i ) i )]] i Now, let row i be the highest bandwidth row and let [i-left(i)] be the higher of two values for row i then there will also be another row j(j=left(i)) for which the value [right(-j] will be equal to B. so for reducing the bandwidth of matrix A - (a) either row i has to move upwards, or (b) row j(=left(i)) has to move downwards. This can be achieved by the use of the following algorithm. If we move row i upwards by k rows or say we interchange row i and row (i-k) so maximum bandwidth due to modified row i-k will be =max[((i-k)-left(i)), (right(i)-(i-k))] while previously it was = (i-left(i)) so, for sure reduction in bandwidth of row i, the following two conditions must be satisified: 1. i-left(i)>(i-k)-left(i) which is always true for any k > 0 2. i-left(i) > right(i) (i-k)

3 176 or k < 2*i-(left(i) + right(i)) Similarly, the condition to guarantee reduction in bandwidth while moving row j downwards by k rows is : k < 2*j-left(i) + right(i)) For finding a row 1 suitable for interchange with i, where i is to be moved upwards, it should be that the bandwidth of modified row i should not increase. So, i-left(1) < i-left(i) or, left(1) > left(i) also, if i is to moved downwards by interchanging with some row 1 (1-, then right(1) < right(i) so, after the interchange of row i with row 1, we can see that there will be no effect on left(m) and right(m) of row m iff, 1. either both node i and node 1 are not connected to node m, or 2. left(m) <i, 1 <right(m) So, if any of above two conditions are satisfied, there is no need to calculated the new left(m) and right(m) for row m, where m = 1,2, N. Steps of Algorithm 4 Step 1 : Find the set of bottleneck rows(rows for which bandwidth is equal of bandwidth of matrix) Step 2 : Try to interchange any of these rows (say i ) with some suitable row 1. Step 3 : If successful, calculate new values of right(m) and left(m) for only those rows which are connected to either i or 1. Step 4 : If there is success for even one row, goto step 1. Step 5 : stop. III COMPARISON OF PERFORMANCE We have applied our four algorithms on 24 test matrices, obtained from different fields of engineering. The source of the data, the filed to which it belongs and the size of the network(matrix) for all the examples considered is given in Table1. All the results are given in Table2. In Table2 for each of the examples we have provided the minimum based width achieved by eight different methods. The first column gives the results obtained by the most popular and widely used Reverse Cuthill-Mackee (RCM) method. The next three columns give the results obtained by using the first three algorithm of this paper. As our fourth algorithm is basically a row interchange algorithm, we have applied it on the results of our pervious three algorithms and also on the results obtained by the Reverse Cuthill-Mackee[RCM] algorithm, (MATLAB implementation). The next four sets of results correspond to this experiment.. In Table 3, we also give the best results that the others have been able to obtain for some of these examples. These results show that the results of applying fourth algorithm on the results of the first three algorithms give us minimum bandwidth compared to that achievable by other existing algorithms. No. of Nodes Description Graph Power System Source No. of Nodes [5] 72 [3] 37 Description Source [9] [3] [3] 62 Water network [3] [3] 73 [3] 162 [1] 45 [8] 19 [3] 40 Random [3] 100 Random [3] 150 Random [3] 200 Random [3] 300 Random [9] 400 Random [9] 500 Random Table 1 : Test Problems Description There are five examples considered in Table2, which taken from power systems. The 30, 57 and 118 node examples represent the topology IEEE standard test systems. A perusal of Table 2 shown for all these examples the result indicated in column G is consistency better than all the other methods. This approach corresponds to using Algorithm 4 in combination with Algorithm 1 of this paper. It is interesting to note that for all these systems, it is possible to reorder them so that the bandwidth is quite small. This is a significant outcome as it implies that this method can also be considered as a method of exploiting the sparsity in power system networks. It is true that the minimum degree based optimal ordering methods have matured but the enormous simplicity of the banded form for solving linear equations should provide further incentive to investigations in this direction. [3] [9] [2] [2]

4 INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR , DECEMBER 27-29, A B C D E F G H I * 3 * 4 3 * NC NC NC 4 * 5 NC NC 7 * * 10 * 10 * * 15 * 15 * * * 6 * 7 6* NC NC NC 8 NC NC * 6 * * 18 * 18 * * 18 * 19 NC * * * 3 * 3 * 3 * NC * NC * NC * NC * NC * * NC 7 * 13 7 * * * * * 8 5 * 5 * NC * 6 NC * NC * 19 4 * 4 * 4 * 4 * NC * NC * NC * NC * * 18 NC * * * * * 62 NC - No change * smallest obtained A- No. of Nodes B - Results for RCM method C - Results of Algorithm 1 D - Results of Algorithm 2 E - Results of Algorithm 3 F - On applying algorithm 4 in results of RCM G - On applying algorithm 4 on results of algorithm 1 H - On applying algorithm 4 on results of algorithm 2 I - On applying algorithm 4 on results of algorithm 3 Table 2 Results of bandwidth reduction algorithms IV CONCLUSIONS We have proposed a new approach for reducing the bandwidth of sparse symmetric matrices. While all previous algorithms are heuristic methods, we have used the spectral information to reduce the bandwidth, which is a numerical approach. The use of spectral information for reducing the bandwidth seems to be able to capture the global network structure and hence, results in extremely small bandwidths. The application of the new bandwidth reducing algorithm to the power system matrices shows that these matrices could also be reordered to forms with a reasonably small bandwidth. This is a very significant outcome as it implies that we can consider alternate avenues for solving the large system of linear equations encountered in many power system computations. This aspect needs further studies. A B C D E F G * * * * * * * smallest obtained A No. of Nodes E Results of A&U [2] B Results of Puttonen [3] F Results of HG [2] C Results of GK [4] G Results of RC [2] D Results of RR [2] Table 3 Results of existing bandwidth reduction algorithms V REFERENCES [1] Mohan P., "Graph Partitioning Algorithms: with Applications in Power System", M.Sc Thesis, Deptt. of Electrical Engg., I.I.Sc., Bangalore, 1992 [2] Collins R., :Bandwidth Reduction by Automatic Renumbering", Int J. Num. Meth, Engg. 6(1973), pp [3] Puttonen J., "Simple and effective Bandwidth Reduction Algorithm", Int. J. Num. Meth. Engg., 19(1983), pp [4] Lewis J.G., :Implementation of Gibbs-Poole-Stockmeyer and Gibbs - King Algorithms", ACM Trans. Math. Soft., 8,22(June 1982), pp [5] Tewerson R.P., "Sparse Matrices", Academic Press, New York, 1973 [6] Wallach Y, "Calculations and Programs for Power System ", Engelwood, Prentice Hall, 1986 [7] Cuthill E. and McKee J., "Reducing the Bandwidth of Sparse Symmetric Matrices, ACM Nat. Conf, San Francisco, U.S.A., 1969, PP [8] Gibbs N.E., Poole W.G. and Stockmeyer P.K., "An Algorithm for Reducing the Bandwidth and Praofile of a Sparse Matrix ", SIAM J. Nm. Anal., 13,2 (April 1976), PP [9] Everstine G.C., "A Comparision of Three Resequencing Algorithms for Reducing of Matrix Profile and Wavefront, Int. J. Num. Meth. Engg., 14(1979), pp [10] Akhras G. and Dhatt G., "An Automatic Node relabelling Scheme for Minimizing a Matrix or Bandwidth", Int.J.Num. Meth. Engg., 10(1976), pp [11] Gibbs N.E., "A Hybrid Profile Reduction Algorithm", ACM Trans. Math. Soft.,2,4 (Dec 1976),

5 178 [12] Rosen R., "Matrix Bandwidth Minimisation", Proc. 23rd Nat. Cof., Ass. For Computing Machinery,Brandon system press, princeton, New Jersy, 1968.