Tree Structured Symmetrical Systems of Linear Equations and their Graphical Solution

Transcription

1 Proceedings of the World Congress on Engineering nd Computer Science 4 Vol I WCECS 4, -4 October, 4, Sn Frncisco, USA Tree Structured Symmetricl Systems of Liner Equtions nd their Grphicl Solution Jime Cerd nd Mrio Grff In generl ISBN: ISSN: (Print); ISSN: (Online) WCECS 4

2 Proceedings of the World Congress on Engineering nd Computer Science 4 Vol I WCECS 4, -4 October, 4, Sn Frncisco, USA define the topology of the grph which represents the SLE. If A is very sprse then the grph will hve very few interconnections. It turns out tht mny physicl systems solved using liner systems re very sprse. In prticulr, in electricl power systems, the mtrices which represent trnsmission networks re very sprse. This is the min reson why sprsity techniques hve been improved by reserch whose gol is to solve efficiently the ctul stte of the power system; such s the best elimintion ordering [6], [9]. Sprsity techniques hve been round since t lest 97 using technique known s bifctoriztion []. The min principles used in mtri bifctoriztion re strongly directed towrd eploiting the underlying mtri grph. Therefore in order to pproch the solution using its grph representtion, first n pproprite model hs to be derived. This model hs to be ble to represent the complete SSLE elements (i.e. A,, nd b). The model proposed in this document is bsed on per eqution bsis. This representtion is given in figure. b i i i ii j ij ii ij ji +.. = b i Figur. Conversion from liner system of equtions to its grph model In this model n eqution is represented by two components: node nd set of links. The node is well defined component which consists of two subcomponents: circle consisting of two hlf prts nd n rc. The upper prt of the circle represents the vrible relted to this eqution which hs to be solved by the system (i.e. i ) nd the lower prt represents the coefficient relted to this vrible in eqution i (i.e. ii ). The rc represents the i th component in b (i.e. b i ). The second prt depends on the SLE topology nd is represented by links which connect the nodes. These links will be denoted s (i, j) where i nd j represent the row nd the column number respectively. There cn be zero or more links which connect the node with some other nodes in the grph. Ech link hs n ssocited vlue for the coefficient locted in the row i column j (i.e. ij ). Perhps it is little bsurd to consider the cse where there re no eternl links. However, s will be shown lter, this is the bsic configurtion which will lwys be pursued in order to solve the SLE. III. Tree Structured Symmetric Systems of Liner Equtions nd Its Grphicl Representtion. Tree structured symmetricl systems of liner equtions (TSSSLE) re SLE where the grph representing the SLE is tree. Bsed on this structure, efficient lgorithms cn be derived in order to solve this kind of systems. These lgorithms emerge nturlly, just by eploiting the properties of trees. This kind of grphs hve been pplied to solve electricl distribution networks whose min chrcteristic is its rdil shpe (i.e. no loops eists in the network). Therefore tree structure cn be derived for the SLE representing these systems. Algorithms to solve different problems with different degree of compleity hve been proposed for distribution networks bsed on this structure in [5], [4], [], [7]. A tree-shped grph hs to be free of loops. This work does not del with how to identify nd remove loops from grphs. Let us instntite eqution with eqution.,5 3 = () whose solution for the TSSSLE denoted by eqution is given by 3 4/7 3 = /7 /7 (3) 4 5/7 Applying the model defined in figure, leds to the grph shown in figure which which complins with the TSSSLE properties. This emple will be used throughout this chpter. The system will be solved using different strtegies which hve to led to the sme solution. To this end ISBN: ISSN: (Print); ISSN: (Online) WCECS 4

3 Proceedings of the World Congress on Engineering nd Computer Science 4 Vol I WCECS 4, -4 October, 4, Sn Frncisco, USA 4=? 4 3=? 5 =? =?.5 tht the number of links creted re miniml, or in the idel cse no links re creted. There re three specil cses which deserve specil ttention. The first one is shown in figure 3. This is the well known s strdelt trnsformtion in electricl engineering. Here Γ k = 3 nd by eqution 5 t most three new links will be generted. Figur. Grph corresponding to the system defined by eqution () Schemtic representtion 4 44 k= Gussin elimintion will be used s the min tool to solve the system. IV. Gussin Elimintion nd Its grphicl Interprettion ij = ij ik kk kj 4 4 Gussin elimintion is generl method to solve SLE. It consists of the itertive ppliction of elementry row opertions which led the system to n echelon form. This is chieved by modifying ech of the elements which do not belong to the column nd row to the eqution under reduction. These elements re modified using epression 4. ij = ij ik kj kk (4) Gussin elimintion cn be regrded s mtri trnsformtion from R n n R (n ) (n ). The resulting system hs ll the informtion needed to solve the subsystem resulting from the trnsformtion. Let us define Γ k s the set of nodes djcent to node k. Then, the mimum number of elements generted in the elimintion is given by eqution 5. IV-A. N k = Γ k ( Γ k ) Specil Cses for Gussin Elimintion (5) As mentioned previously, Gussin elimintion hs two min costs: updting nd link cretion. The first one is unvoidble but the second one cn be optimised if n elimintion order is chieved such (b) Complete trnsformtion Figur 3. Gussin elimintion for Γ k = 3. The second one is shown in figure 4. This is the well known series reduction in electricl engineering. Here Γ k = nd by eqution 5 t most one new link will be generted () Schemtic representtion k=3 ij = ij ik kk kj (b) Complete trnsformtion Figur 4. Gussin elimintion for Γ k =. The third nd most importnt cse regrding this document, s it will be shown long this section, is shown in figure 5. This configurtion cn be regrded s dngling node which cn be reduced into the node where it is dngling from. Here Γ k = nd by eqution 5 there will be no new links generted!! This bsic fct will be used in order to reduce the ISBN: ISSN: (Print); ISSN: (Online) WCECS 4

4 Proceedings of the World Congress on Engineering nd Computer Science 4 Vol I WCECS 4, -4 October, 4, Sn Frncisco, USA TSSSLE s the lefs cn be regrded s dngling nodes. Once they re reduced, then the nodes where they were dngling from cn be reduced, nd so on. () Schemtic representtion k= ij = ij ik kk (b) Complete trnsformtion Figur 5. Gussin elimintion for Γ k =. V. Grph-bsed Solution for Tree Structured Symmetric Systems of Liner Equtions A tree is specil kind of grph whose min chrcteristic is the bsence of loops. A stndrd representtion for tree is presented in figure 6. A tree hs n lyers, n, numbered from to n. The number of lyers represents its depth. A function lyer() cn be defined to epress the lyer where node is locted, for instnce lyer(e) =. Another useful function is prent() which returns the node where node is hnging from or nil if it hs no prent, for instnce prent(e) = B nd prent(a) = nil. D B A E F G H I J K L M N O Figur 6. An emple of tree. Let us denote Υ i s the set of indices j which corresponds to vribles j which re hnging from kj C Lyer 3 i s denoted by eqution 6 (i.e. the children of i ). Υ i = {j : (i, j), prent(i) j} (6) For instnce Υ A = {B, C}, nd Υ G = {N, O} in figure 6. Bsed on these definitions, the nodes in tree cn be clssified in three types root node: There is just one of them in the grph such tht lyer(root) = nd prent(root) = nil. For instnce the root node in figure 6 is A (in blck), internl node: re those nodes such tht Υ nd root. The set of internl nodes, I, in figure 6 is {B, C, D, E, F, G} (in gry), terminl node: lso clled lef node, re those nodes where Υ =. Therefore the set L representing the lef nodes in figure 6 is {H, I, J, K, L, M, N, N, O} (in white). This kind of grphs re one of the most useful structures in computer science. Their ppliction spns from contet-free grmmr nlysers in compiler theory, XML representtion in internet technologies, nd so on. A common chrcteristic bout those pplictions is tht only the lef nodes hve rel informtion. However, when pplied to TSSSLEs, the internl nodes will hve some informtion which need to be processed in order to solve the SLE. It is very importnt to preserve the tree structure of the system in the reduction process. The best wy to reduce these grphs is by finding n order which does not generte new elements t ll. From figures 3, 4 nd 5 bsic fct cn be sserted. The only reductions which do not generte new elements re those which re pplied to nodes k where Γ k =. Therefore those re the idel cndidtes to pply the elimintion process. Gussin elimintion when pplied to ll nodes j where j Υ i, is epressed in equtions 7 nd 8 for the coefficient corresponding to vrible i nd the independent term respectively. ii = ii j Υ i ij jj (7) b i = b i b j ij (8) jj j Υ i The solution for the SLE once the reduction process hs been pplied is strightforwrd. To this ISBN: ISSN: (Print); ISSN: (Online) WCECS 4

5 Proceedings of the World Congress on Engineering nd Computer Science 4 Vol I WCECS 4, -4 October, 4, Sn Frncisco, USA References ISBN: ISSN: (Print); ISSN: (Online) WCECS 4

6 Proceedings of the World Congress on Engineering nd Computer Science 4 Vol I WCECS 4, -4 October, 4, Sn Frncisco, USA ces in Power System, Control, Opertion nd Mngement, APSCOM, pges , October. [3] Thoms H. Cormen, Chrles E. Leiserson, Ronld L. Rivest, nd Cliff Stein. Introduction to Algorithms. MIT Press,. [4] D. Ds, H. S. Ngi, nd D. P. Kothri. Novel method for solving rdil distribution networks. IEE Proceedings on Genertion, Trnsmission nd Distribution, 4(4):9 98, July 994. [5] S. K. Goswmi nd S.K. Bsu. Direct solution of distribution systems. IEE Proceedings on Genertion, Trnsmission nd Distribution, 38:78 85, 99. [6] Hrry M. Mrkowitz. The elimintion form of the inverse nd its ppliction to liner progrmming. Mngement Science, 3(3):55 69, April 957. [7] S.F. Mekhmer, S.A. Solimn, M.A. Moustf, nd M.E. El- Hwry. Lod flow solution of rdil distribution feeders: A new contribution. Electricl power nd Energy Systems, 4:7 77,. [8] Gilbert Strng. Liner Algebr nd Its Applictions. Brooks Cole, 4 edition, July 5. [9] W. F. Tinney nd J. W. Wlker. Direct solutions of sprse network equtions by optimlly ordered tringulr fctoriztion. Proceedings of the IEEE, 55():8 89, 967. [] K. Zollenkopf. Bifctoriztion: Bsic computtionl lgorithm nd progrmming techniques. In Oford, editor, Conference on Lrge Sets of Sprse Liner Equtions, pges 76 96, 97. ISBN: ISSN: (Print); ISSN: (Online) WCECS 4