Optimal Reconfiguration of Distribution System by PSO and GA using graph theory

Transcription

1 Proceedngs of the 6th WSEAS Internatonal Conference on Applcatons of Electrcal Engneerng, Istanbul, Turkey, May 27-29, Optmal Reconfguraton of Dstrbuton System by PSO and GA usng graph theory Mehd Assadan, Malhe M. Farsang, Hossen Nezamabad-pour Department of Electrcal Engneerng, Shahd Bahonar Unversty of Kerman, Kerman, IRAN Abstract-- Ths paper nvestgates the ablty of partcle swarm optmzaton (PSO) n cooperatng by graph theory for network reconfguraton to reduce the power loss and voltage profle enhancement of dstrbuton system. The numercal results are presented on a dstrbuton system to llustrate the feasblty of the proposed method by PSO usng graph theory. Furthermore, to valdate the obtaned results by PSO usng graph theory, Genetc Algorthm (GA) usng graph theory s appled and the results are compared. Key-Words -- Dstrbuton systems, reconfguraton, partcle swarm optmzaton, Genetc Algorthm, loss reducton, graph theory. Introducton Network reconfguraton n dstrbuton system s realzed by changng the status of sectonalzng swtches, and s usually done for loss reducton or for load balancng n the system. Dstrbuton systems are radally structured to smplfy overcurrent protecton. Each dstrbuton system ncludes a number of lnes, sectonalzng swtches nstalled on ths lne, and other component such as te swtches connectng other dstrbuton lnes. By changng the open and closed status of the feeder swtches, load currents can be transferred from feeder to feeder. The early work on feeder reconfguraton for loss reducton s presented by Cvanlar et al. []. In [2], Baran et al. defned the problem for loss reducton and load balancng as an nteger programmng problem. Nara et al. [3] presented an mplementaton usng a genetc algorthm (GA) to look for the mnmum loss confguraton. In [4] [0], the authors suggested to employ a power flow method-based heurstc algorthm for determnng the mnmum loss confguraton of radal dstrbuton networks. In [9] [], the authors proposed a soluton procedure employng smulated annealng to search for the soluton. In [2], the authors had consdered tmevaryng load analyss to reduce loss. In [3], fuzzy theory and evolutonary programmng were employed to solve feeder reconfguraton systems. A parallel tabu search based method s proposed to look for the mnmum loss confguraton [4]. The use of dfferental evoluton s reported n [5]. Also Jn et al. [6] used Bnary PSO for load balancng n a dstrbuton system. The work carred out by the authors n [7] used the PSO to look for the mnmum loss confguraton for a small dstrbuted network. In ths paper, the ablty of PSO usng graph theory s nvestgated to fnd the optmal confguraton for larger network wth the objectve of mnmzng real power losses and mprovng of voltage profle. To valdate the results obtaned by PSO usng graph theory, GA usng graph theory s appled. 2 Overvew of Partcle Swarm Optmzaton The Partcle Swarm Optmzer s a populaton based optmzaton method that ntroduced by Kennedy and

2 Proceedngs of the 6th WSEAS Internatonal Conference on Applcatons of Electrcal Engneerng, Istanbul, Turkey, May 27-29, Eberhart [8]. In PSO, each partcle moves n the search space wth a velocty accordng to ts own prevous best soluton and ts group s prevous best soluton [8]. The dmenson of the search space can be any postve nteger. Each partcle updates ts poston and velocty wth the followng two equatons: X ( t + ) = X ( t) + V ( t + ) () where X (t) and V (t) are vectors representng the current poston and velocty respectvely. V, j( t + ) = wv, j ( t) + cr, j ( pb, j X, j ( t)) + (2) c2r2, j ( gbj X, j ( t)) where j,2,..., d represents partcle dmenson. 0 w < s an nerta weght determnng how much of partcle's prevous velocty s preserved, c and c2 are two postve acceleraton constants; and r, j, r2, j are two unform random sequences sampled from U(0, l), pb s the personal best poston found by the th partcle and gb s the best poston found by the entre swarm so far. 3 Overvew of GENETIC ALGORITHM Genetc Algorthm (GA) has desrable characterstcs as an optmzaton tool and offer sgnfcant advantages over tradtonal methods. They are nherently robust and have been shown to effcently search the large soluton space contanng dscrete or dscontnuous parameters and non-lnear constrants, wthout beng trapped n local mnma [9]. GA may be used to solve a combnatoral optmzaton problem. The GA searches for a soluton nsde a subspace of the total search space. Thus they are able to gve a good soluton of a certan problem n a reasonable computaton tme. The optmal soluton s sought from a populaton of solutons usng random process. Applyng to the current populaton the followng three operators create a new generaton: reproducton, crossover and mutaton. The reproducton s a process dependant on an objectve functon to maxmze or mnmze, whch depends on the problem. In graph theory, a graph s made up of dots connected by lnes where two dots can only be connected by one lne [20]. In graph theory, a dot s known as vertex and a lne known as edge. The degree of a vertex n a graph s the number of edges that touch t and t s defned by the followng equaton: deg( V ) = E (3) The sze of a graph s the number of vertces that t has. A path s a route that can be traveled along edges and through vertces n a graph. All of the vertces and edges n a path are connected to one another. Therefore, a graph s a par G = ( V, E) of sets where the elements of V are the vertces or nodes or ponts of the graph G and the elements of E are ts edges or lnes. A cycle s a path whch begns and ends on the same vertex. A cycle s sometmes called a crcut. The number of edges n a path or a cycle s called the length of the path. A vertex v s ncdent wth an edge e f v e, then e s an edge at v. A matrx s defned as ncdence matrx wth the sze of V E as follows: e s connected to v m j = 0 else (4) A graph that s not contanng any cycles s called a tree. In a tree graph wth V vertces and E edges, the followng equaton s satsfed V = E (5) The number of the cycles n graph tree s gven by the followng equaton: cycle = ( E + ) V (6) 5 study system and problem formulaton The study systems s adopted from the lterature whch s a dstrbuton system conssts 28 sectonalzng swtches and 5 te swtches as shown n Fg.. The system data can be found n [2]-[22]. The objectve functon of the network reconfguraton s to mnmze the total power losses as well as mprovng voltage profle. The objectve functon can be expressed as follows: 4 Overvew of graph theory

3 Proceedngs of the 6th WSEAS Internatonal Conference on Applcatons of Electrcal Engneerng, Istanbul, Turkey, May 27-29, For solvng the problem, frst of all, the power flow must be calculated. Snce the dstrbuton systems are radally structured and after reconfguraton the radal characterstcs of network s stll kept, a smplfed power flow equaton used n [7] are appled n ths paper. Fg. : 33 Bus dstrbuton system N mn f = mn( PT, Loss ) + V + C (7) = where, s the total real power loss of the P T, Loss system, and V s the voltage of N buses. After reconfguraton of the network, the radal characterstcs of the network should be kept and all loads have to be n servce. In vew of ths, C s defned as the penalty factor for exstng mesh or n the case of solatng some loads by followng equaton C = Anumber( mesh) + B number( Isolated) (8) where A and B are equal to one. In addton to Equ. (7), other objectve functon s consdered as follows: mn f = mn ( PT, Loss ) + C) (9) where the system s reconfgured based on the mnmzng the reducton of loss. The voltage magntude at each bus must mantan wthn ts lmts. The current on each branch has to le wthn ts capacty ratng. These constrants are expressed as follows: Vmn V V max (0) I I,max () where, V Vmax and V mn are voltage magntude of th the bus, mnmum and maxmum voltage lmts respectvely. I and I, max are current magntude and maxmum th current lmt of branch. 6 PSO & GA usng graph theory for feedder reconfgraton PSO usng graph theory s appled for network reconfguraton n order to have mnmum losses n the network as well as mprovng voltage profle. To use the graph theory, the buses of the network consdered as nodes (V ) and the number of swtches known as edges ( E ). Also, the number of meshes n the network s known as the number of cycles n the graph theory. To retan a radal network structure, only one swtch s opened n each mesh where the total number of open swtches s equal to the total number of meshes so that no feeder secton can be left out of servce. To have a radal network structure, a tree graph should be reached. To fnd the best confguraton for the network, the algorthm started by checkng the degree of the nodes. Those nodes wth one degree wll be deleted followed by deletng of the connected edges. Then the algorthm checks whether the network s radal and all loads are n servce otherwse the penalty factor s ncreased n Equ. (7) or (9). In the PSO algorthm, n partcles are generated randomly where n s selected to be 50. In the study system, there exst 5 meshes. Therefore, each partcle s a d -dmensonal vector n whch d = 5. The ntalzaton s made on the poston randomly for each partcle. The number of teraton s consdered to be 00, whch s the stoppng crtera. The parameter n (2) must be tuned. These parameters control the mpact of the prevous veloctes on the current velocty where, n ths paper, c = c2 =. 4 and w s decreasng lnearly from 0.9 to 0.. Each partcle n the populaton s evaluated usng the objectve functon defned by Equ. (7) or (9) searchng for the partcle assocated wth f best.the best prevous poston of the th partcle s recorded and represented as: pb = ( pb,, pb,2, pb,3, pb,4, pb, 5) and the ndex of the best partcle among all of the

4 Proceedngs of the 6th WSEAS Internatonal Conference on Applcatons of Electrcal Engneerng, Istanbul, Turkey, May 27-29, partcles n the group s for the gb. Usng the gb and pb, partcle velocty and poston s updated accordng to () and (2). To fnd the best reconfguraton for the two networks, sutable lnes to be opened are selected based on 20 ndependent runs, under dfferent random seeds. The obtaned results are gven n Table for two dfferent objectve functons (Equ. (7) or (9)). Fg. 2. Convergence characterstcs of PSO on the average cost functon n fndng the soluton based on Equ. (7). TABLE. THE OBTAINED RESULTS BY PSO FOR THE STUDY NETWORK Network confguraton Orgnal confguraton After reconfgurat on Equ. 9 Equ. 7 Te swtches 33, 34, 35, 36, 37 6, 9, 4, 32, 37 7, 9, 4, 27, 32 Powe r loss (KW) Mn.Volta ge magntude (Pu) Table, shows that consderng the swtches 7, 9, 4, 27 and 32 as te swtches, the dstrbuton system reconfgures to a network wth a mnmum losses and a better bus voltage magntude profle. The average best-so-far and the mean cost functon of each run are recorded and averaged over 20 ndependent runs. To have a better clarty, the convergence characterstcs n fndng the best confguraton for the network are gven n Fgs Also, bus voltage magntude profle of the orgnal system and reconfgured system based on Equ. (7) are shown n Fgs Fg. 3. Convergence characterstcs of PSO on the average best-so-far n fndng the soluton based on Equ. (7). Voltage bus n PU Number bus Fg. 4. Bus voltage magntude profle of the orgnal system. Voltage bus n PU Number bus Fg.5 Bus voltage magntude profle of the system after reconfguraton by PSO based on Equ. (7). To valdate the results obtaned, GA s appled to solve the problem. A confguraton s consdered wth fve genes. The number of chromosomes for a populaton s set to be 50. In ths paper, one pont crossover s appled wth the crossover probablty pc = 0.9 and the mutaton probablty s selected to be changed lnearly from pm = to p m = Also, a weghted roulette wheel s used. As n the PSO, the number of teraton s consdered to be 00. The obtaned results by GA gven n Table 2 shows that the swtches 7, 9, 4, 28 and 32 should be opened to get the best confguraton based on Equ.

5 Proceedngs of the 6th WSEAS Internatonal Conference on Applcatons of Electrcal Engneerng, Istanbul, Turkey, May 27-29, (7). Tables and 2 show that the obtaned results by PSO and GA have dfference n one swtch. PSO fnds the swtch 27 to be opened whle GA fnds the swtch 28 to be opened. Fg. shows that these two swtches are next to each other. Comparng the obtaned results by PSO and GA show that GA performs better than PSO. The convergence characterstcs of GA to fnd the soluton over 20 ndependent runs are shown n Fgs. 6-7, where, these fgures show that GA has a better convergence rate. Table 2. The obtaned results by GA for the frst study network Network confguraton Te swtches Orgnal confguraton 33, 34, 35, 36, 37 After Equ. 7, 9, 4, reconfguraton 9 32, 37 Equ. 7 7, 9, 4, 28, 32 Power loss (KW) Mn.Voltage magntude (Pu) Fg. 6. Convergence characterstcs of GA on the average cost functon n fndng the soluton based on Equ. (7). Fg. 7. Convergence characterstcs of GA on the average best-so-far n fndng the soluton based on Equ. (7). 7 Conclusons In ths paper, PSO and GA usng graph theory are appled to confgure two dstrbuton networks to mnmze losses and voltage profle enhancement. Graph theory helps the GA and PSO algorthms to fnd the radal confguraton for dstrbuton network easly. The obtaned results by two algorthms are compared. Although the obtaned results by PSO n [7] for a small dstrbuton network was promsng but the obtaned results n ths paper show that GA s performng better than PSO. The PSO has been proven to be very effectve for statc and dynamc optmzaton problems. But n some cases as shown n ths paper when the problem becomes complex, t converges prematurely wthout fndng a local optmum. Thus; t s possble for the standard PSO to converge prematurely wthout fndng even a local extermum. To overcome of the premature converge, other verson of PSO known as Guaranteed Convergence Partcle Swarm Optmzaton (GCPSO) should be used where the authors are workng on t. References [] S. Cvanlar, J. J. Granger, H. Yn, and S. S. H. Lee, Dstrbuton feeder reconfguraton for loss reducton, IEEE Trans. Power Del., vol. 3, no. 3, pp , Jul [2] M. E. Baran and F. F. Wu, Network reconfguraton n dstrbuton systems for loss reducton and load balancng, IEEE Trans. Power Del., vol. 4, no. 2, pp , Apr [3] K. Nara, A. Shose, M. Ktagawoa, and T. Ishhara, Implementaton of genetc algorthm for dstrbuton systems loss mnmum reconfguraton, IEEE Trans. Power Syst., vol. 7, no. 3, pp , Aug [4] S. K. Goswam and S. K. Basu, A new algorthm for the reconfguraton of dstrbuton feeders for loss mnmzaton, IEEE Trans. Power Del., vol. 7, no. 3, pp , Jul [5] D. Shrmohammad and H. W. Hong, Reconfguraton of electrc dstrbuton networks for resstve lne loss reducton, IEEE Trans. Power Del., vol. 4, no. 2, pp , Apr. 989.

6 Proceedngs of the 6th WSEAS Internatonal Conference on Applcatons of Electrcal Engneerng, Istanbul, Turkey, May 27-29, [6] T. P. Wagner, A. Y. Chkhan, and R. Hackam, Feeder reconfguraton for loss reducton: an applcaton of dstrbuton automaton, IEEE Trans. Power Del., vol. 6, no. 4, pp , Oct. 99. [7] E Ramos.; Exposto, A.G.; Santos, J.R.; Iborra, F.L., Path-based dstrbuton network modelng: applcaton to reconfguraton for loss reducton, IEEE Trans.Power Syst, Vol. 20, No. 2,pp ,May 2005 [8]Y. Yu; J. Wu;, Loads combnaton method based core schema genetc shortest-path algorthm for dstrbuton network reconfguraton, IEEE Trans.Power Syst, Vol. 3,pp , Oct [9] H. C. Cheng and C. C. Kou, Network reconfguraton n dstrbuton systems usng smulated annealng, Elect. Power Syst. Res., vol. 29, pp , May 994. [0] H. D. Chang and J. J. Rene, Optmal network reconfguraton n dstrbuton systems: part : a new formulaton and a soluton methodology, IEEE Trans. Power Del., vol. 5, no. 4, pp , Oct [] H. C. Cheng and C. C. Kou, Optmal network reconfguraton n dstrbuton systems: part 2: soluton algorthms and numercal results, IEEE Trans. Power Del., vol. 5, no. 3, pp , Jul ton and Utrecht, [2] R. P. Broadwater, A. H. Khan, H. E. Shaalan, and R. E. Lee, Tme varyng load analyss to reduce dstrbuton losses through reconfguraton, IEEE Trans. Power Del., vol. 8, no., pp , Jan [3] Y. H. Song, G. S. Wang, A. T. Johns, and P. Y. Wang, Dstrbuton network reconfguraton for loss reducton usng fuzzy controlled evolutonary programmng, Proc. Inst. Elect. Eng., Gen., Transm., Dstrb., vol. 44, no. 4, pp , Jul [4] H. Mor, Y. Ogta, A parallel tabu search method for reconfguratons of dstrbuton systems, IEEE Power Engneerng Socety Summer Meetng, Vol., pp , July [5] J. P. Chou, C. f. Chang and C. T. Su, Varable scalng hybrd dfferental evoluton for solvng network reconfguraton of dstrbuton system, IEEE Trans. Power Sys., vol. 20, no. 2, pp , May [6] X. Jn, J. Zhao, Y. Sun, K. L and B. Zhang, Dstrbuton network reconfguraton for load balancng usng bnary partcle swarm optmzaton Internatonal conf. power system technology, Sngapore, pp , November [7] Assadan M, Farsang MM and Nezamabadpour H, Dstrbuton network reconfguraton for loss reducton usng Partcle swarm optmzaton Internatonal conference on physcal and techncal problems n power system (TPE), Ankara, Turkey, [8] J. Kennedy and R. Eberhart, Partcle swarm optmzaton, n Proc. 995 IEEE Int. Conf. Neural Networks (ICNN 95), vol. IV, pp [9] D. E. Goldberg, Genetc algorthms n search, optmzaton, and machne learnng. New York : Addson-Wesley, 989. [20]R. Destel, Graph theory, Sprnger-Verlag Hedelberg, New York, [2]Venkatesh,B.;Ranjan,R.;Goo,H.B., Optmal reconfgu- raton of radal dstrbuton systems to maxmze loadablty, IEEE Trans. Power Systems, Vol 9, no pp.: Feb [22] F. V. Gomes, S. Carnero, J.L. R. Perera, M.P. Vnagre, P. A. N. Garca, and L. R. Araujo, A new heurstc reconfguraton algorthm for large dstrbuton systems, IEEE Trans. Power Sys., vol. 20, no. 3, pp , August