Capacitor Placement In Distribution Systems Using Genetic Algorithms and Tabu Search

Transcription

1 Capactor Placement In Dstrbuton Systems Usng Genetc Algorthms and Tabu Search J.Nouar M.Gandomar Saveh Azad Unversty,IRAN Abstract: Ths paper presents a new method for determnng capactor placement n dstrbuton systems. The capactor placement problem consst of fndng places to nstall capactor ban n an electrcal dstrbuton system amng to reduce losses due to the compensaton of the reactve component of power flow. The capactor placement s hard to solve n sense of global optmzaton due to the hgh non lnear and mxed nteger problem. To solve the problem effcently, ths paper focuses on complex Genetc Algorthm (GA) and tabu search (TS) that s two of the effcent optmzaton methods. The proposed method has been tested on a real dstrbuton system. Keyword: capactor placement, dstrbuton systems, genetc algorthm, tabu search,optmzaton 1. Introducton Capactors are often nstalled n dstrbuton system for reactve power compensaton to carry out power and energy loss reducton, voltage regulaton, system securty mprovement and system capacty release. Economc benefts of the capactor depends manly on where and how many capactes of the capactor are nstalled and proper control schemes of the capactors at dfferent load levels n the dstrbuton system [1]. The problem of capactor placement n a dstrbuton networ conssts of fndng szes, locaton and the number of capactors that have to be placed on the networ. Ths s one of the combnatonal optmzaton problems wth the sze n of search space s beng equal ( + 1), where n s the number of buses on dstrbuton networ and s the number of possble capactor szes that can be placed on the networ. Ths problem has tradtonally been solved usng mathematcal non lnear and mxed nteger programmng technques. A capactor placement technques can be found n [2]. Among varous approaches, the metaheurstcs play a relevant role, snce exact optmzaton methods are not sutable for tacng real world nstances. Focusng only on metaheurstc methods, [3-6] propose dfferent methods for capactor placement problem. Ths paper presents a new approach base on a complex Genetc Algorthm (GA) and Tabu search (TS) for solvng the problem. Snce, power flow s one of most mportant tools for solvng the problem, ths paper presents a very fast and smple power flow problem for solvng the capactor placement. The man contrbuton of ths study s combnaton of GA-TS and a proper power flow base on the networ-topology. To demonstrate the effectveness of proposed algorthm, ths method s appled to a real radal dstrbuton feeder. 2. Problem Formulaton The objectve functon of the problem can be expressed as follows to mnmze the capactor nvestment cost and system energy loss: I L j j Mn ( ) (, ) q, q C + ejt j Ploss x q (1) = 1 j= 1 subject to P flow ( z, q ) = (power flow constrants) (2) mn max V V V (voltage constrants) (3) : dscrete varables, for fxed typed capactor: l = =, l, j {1,2,3,,L} (4) for swtched type capactor j q q {1,2,3,,I} (5), where, q s the szng vector whose components are multples of the standard sze of one capactor

2 ban. q s the control scheme vector at load level j whose components are dscrete varables. C ( ) represent the nvestment cost assocated wth the capactor nstalled at locaton. P loss s the power loss at load level j wth at tme duraton T j and K ej s dfferent energy loss cost for each load level. 2 x = [ P, Q, V ], Z = [ P, Q ] represent state varable vectors of real and reactve powers P, Q as well as squared voltage magntude V at branch,l and I denote numbers of load levels and canddate locatons to nstall the capactors.[4] 3. Genetc Algorthm 3.1 Overvew Genetc Algorthm are general-purpose search technques based on prncples nspred from the genetc and evoluton mechansms observed n natural systems and populatons of lvng bengs. Ther basc prncple s the mantenance of a populaton of solutons to a problem (genotypes) as encoded nformaton ndvduals that evolve n tme [7-11]. Generally, GA comprses three dfferent phases of search: phase 1: creatng an ntal populaton; phase 2: evaluatng a ftness functon; phase 3: producng a new populaton. A genetc search starts wth a randomly generated ntal populaton wthn whch each ndvdual s evaluated by means of a ftness functon. Indvdual n ths and subsequent generatons are duplcated or elmnated accordng to ther ftness values. Further generatons are created by applyng GA operators. Ths eventually leads to a generaton of hgh performng ndvduals [7]. 3.2 The genetc algorthm operators There are usually three operators n a typcal genetc algorthm [7]: the frst s the producton operator (eltsm) whch maes one or more copes of any ndvdual that posses a hgh ftness value; otherwse, the ndvdual s elmnated from the soluton pool; the second operator s the recombnaton (also nown as the ' crossover' ) operator. Ths operator selects two ndvduals 2 wthn the generaton and a crossover ste and carres out a swappng operaton of the strng bts to the rght hand sde of the crossover ste of both ndvduals. Crossover operatons synthesze bts of nowledge ganed from both parents exhbtng better than average performance. Thus, the probablty of a better performng offsprng s greatly enhanced; the thrd operator s the 'mutaton' operator. Ths operator acts as a bacground operator and s used to explore some of the nvested ponts n the search space by randomly flppng a 'bt' n a populaton of strngs. Snce frequent applcaton of ths operator would lead to a completely random search, a very low probablty s usually assgned to ts actvaton. 4. Tabu Search 4.1 Overvew Tabu search s characterzed by an ablty to escape local optmal by usng a short-term memory of recent solutons. Ths s acheved by a strategy of forbddng certan moves. The purpose of classfyng a certan move as forbdden (tabu) s bascally to prevent cyclng. Moreover, TS permts bactracng to prevous solutons, whch may ultmately lead, va a dfferent drecton to better solutons [12]. The man two components of TS algorthm are the tabu lst (TL) restrctons and the aspraton level (AV) of the soluton assocated wth the recorded moves. 4.2 Tabu Lst TL s managed by recordng moves (tral solutons) n the order n whch they are made. Each tme a new element s added to the 'bottom' of a lst, the oldest element on the lst s dropped from the 'top'. Emprcally, TL szes whch provde good results often grow wth the sze of the problem and stronger restrctons are generally coupled wth smaller sze. Best szes of TL le n an ntermedate range between these extremes. In some applcatons a smple choce of TL sze n a range centered seem to be qute effectve [13].

3 4.3 Aspraton Crtera Another ey ssue of TS arses when the move under consderaton has been found to be tabu. Assocated wth each entry n the tabu lst there s a certan value for the evaluaton functon AV. Roughly speang, AV crtera are desgned to overrde tabu status f a move s 'good enough' wth the compatblty of the goal of preventng the soluton process from cyclng [12]. and Table 1 shows the sngle lne dagram and specfcatons of test networ, respectvely. 5. The proposed algorthm 5.1 GA Implementaton The soluton of the problem s represented by a bnary codng. As n the example, 4 szes of capactors (5,1,15,25 Var at 38 Volts) are used, 3 bts are consdered for codng: one for presentng the capactors ban on bus and 2 bts for sze of capactor. The ftness of each chromosome s the sum of nstallaton cost and costs of power losses. Selecton of chromosome for applyng varous GA operators s based on ther scaled ftness functon accordance to the roulette wheel selecton rule. The roulette wheel slots are szed accordng to the accumulated probabltes of reproducng each chromosome. Crossover and mutaton operators are carred wth the prespecfed probabltes. 5.2 TS Implementaton In ths paper we ntroduce a proper approach for creatng the TL for capactor allocaton problem. TL s created as a matrx of dmenson Z M, where Z and M are the TL sze and the total number of capactor locaton canddates n the dstrbuton networ, respectvely. Each vector n the matrx represents the TL for one capactor allocaton canddate. Dfferent forms of aspraton crtera are used n lterature. The one we used n ths paper s to overrde the tabu status has better objectve functon than the one obtaned before for the same move. 6. Numercal Examples In order to test the proposed algorthm, a real dstrbuton networ has been consdered, Fgure 1 Fgure1. sngle lne dagram of test networ of Power Impedance Customer of Secton W Var Sngle Three j j j j j j j j j j j j j j j j j j j j j Table1. Specfcaton of test networ

4 Total current and total power losses at networ whtout capactor and the deal networ (reactve power at the loads = ) are A, watt and A, 1169 watt respectvely. After runnng the program, Table 2 shows the placement of capactor at the buses, total current and total power losses n GA and GA-TS algorthms. Capactor (Var) Capactor (Var) Total Current= (A) Power Losses=11169 (watt) Table 2. Capactor placement results Total Current= (A) Power Losses=11127(watt) 7. Cost Analyss For economc evaluatng of the proposed algorthm, the followng equaton were consdered for the economc gan: Annual Gan = 875 wh.cost Ploss (6) Where: Annual Gan: the annual economc gan wth usng the capactors regard to losses reducton for one year. 875: the converson factor of power losses to energy losses. Kwh.cost: the cost of energy. P loss : power losses reducton regard to use of capactors. Annual Cost: ( (. CapCost) /(1 (1/(1 + ) )) (7) Where: Annual Cost: the total cost of capactors and ther accessores for one year. : Investment perod : Interest rate. Accordng to above relatons, the ftness functon can be formulated as: Ftness = Annual Gan Annual Cost (8) In ths study for the example networ and wth consderng of the cost n IRAN, for the plannng study 1 years long and nterest rate 15%, nflaton rate 15%, Wh.cost = 12 Rals (.15 $), the reducton cost of nvestment cost of losses for 3 years was equal to cost of nvestments and the ftness wll be 1379 Rals (1723 $) for 1 years perod. 8. Concluson In ths paper, mplementaton of GA and GA-TS to the optmal placement of capactor ban has been llustrated. The approach can be extended to other networs also. The results showed the GA-TS s a one proper optmzaton method for optmal placement of capactors ban n dstrbuton networ. The economc study showed the nvestments costs wll be compensated n a few 3 years by reducton costs of losses. References 1. Turan Gonen (1986) Electrc power Dstrbuton systems Engneerng, Mc.Graw-Hll Internatonal Edton. 2. Ng, H.N., Salama, M.M.A., and Author (2) Classfcaton of capactor allocaton technque, IEEE Trans on Power Delvery, Vol.15, 1: Chang H.D., and Author (199) Optmal capactor placement n dstrbuton systems, IEEE Trans on Power Delvery, Vol.5, 2: Huang Y.C., and Author (1996) Solvng the capactor placement problem n a radal dstrbuton system usng tabu search approach, IEEE Trans on Power Systems, Vol.11, 4: Gallego R.A., and Author (21) Optmal capactor placement n radal dstrbuton networs, IEEE Trans on Power Systems, Vol.16, 4: Sundhararajan S, and Author (1994) Optmal selecton of capactors for radal dstrbuton systems usng a genetc algorthm, IEEE Trans on Power Systems, Vol.9, 3:

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