A Novel Evolutionary Algorithm for Capacitor Placement in Distribution Systems

Transcription

1 DOI.703/s x STF Journal of Engneerng Technology (JET), Vol. No. 3, Dec 013 A Novel Evolutonary Algorthm for Capactor Placement n Dstrbuton Systems J-Pyng Chou and Chung-Fu Chang Abstract Ths paper uses an effectve method, method, wth nteger programmng for solvng the capactor placement problems n dstrbuton systems. Dfferent from the orgnal dfferental evoluton (DE), the concepts of chaotc search, opposton-based learnng, and quantum mechancs are used n the method to overcome the drawback of selecton of the crossover factor and scalng factor used n the orgnal DE method. One benchmark functon and one 9-bus system from the lterature are used to compare the performance of the method wth the DE, and smulated annealng (SA). Numercal results show that the performance of the method s better than the other methods. Also, the method used n 9-bus system s superor to some other methods n terms of soluton power loss and costs. Index Terms Capactor Placement. Chaotc Search.. Opposton-based Learnng. Quantum Mechancs. I. INTRODUCTION C apactors are wdely nstalled n dstrbuton systems for reactve power compensaton to acheve power and energy reducton, voltage regulaton and system capacty release. And, the nstallaton of shunt capactors n prmary dstrbuton systems can also effectvely reduce peak power and energy losses. The extent of these benefts depends greatly on how the capactors are placed on the system, namely on the locaton and sze of the added capactors [1,]. The objectve n the capactor placement problem s to mnmze the annual cost of the system, subject to operatng constrants under a certan load pattern. ranger et al. [3-] proposed the concept that the sze of capactor banks was consdered as a contnuous varable. Bala et al. [] presented a senstvty-based method to solve the optmal capactor placement problem. Usng genetc algorthm (A) to select capactors for radal dstrbuton systems was proposed n [7]. In the above-mentoned methods, the capactors were often assumed as contnuous varables, n whch cost s proportonate to the capactor sze. However, commercally avalable capactors are dscrete. Selectng nteger capactor szes closest to the optmal values found by the Manuscrpt receved October 8, 013. J. P. Chou s wth the Department of Electrcal Engneerng, Mng Ch Unversty of Technology, New Tape Cty, Tawan (phone: #4804; fax: ; e-mal: jpyng@ mal.mcut.edu.tw). C. F. Chang s wth the Department of Electrcal Engneerng, WuFeng Unversty, Chay County, Tawan (e-mal: cfchang@wfu.edu.tw). contnuous varable approach may not guarantee an optmal soluton [8]. Therefore the optmal capactor placement should be vewed as an nteger-programmng problem, and dscrete capactors wll be consdered n ths paper. The teachng learnng based optmzaton (TLBO) approach to mnmze power loss and energy cost by optmal placement of capactors n radal dstrbuton systems [9]. Muthu Kumar et al. [] proposed an opposton based dfferental evoluton (ODE) method for a dstrbuton system reconfguraton to operate the system at mnmum cost and at the same tme mproves the system relablty and securty. Olamael et al. [11] proposed a new adaptve modfed frefly algorthm to solve optmal capactor placement problem. Dfferental evoluton (DE) as developed by Stron and Prce [8] s one of the best Evolutonary algorthms (EAs), and has proven to be a promsng canddate to solve real valued optmzaton problems [1]. Ths method also turned out to be one of the best genetc algorthms for solvng the real-valued test functon sute of the frst Internatonal Contest on Evolutonary Computaton, whch was held n Nagoya n 199. DE s a stochastc search and optmzaton method. The fttest of an offsprng competes one-to-one wth that of the correspondng parent, whch s dfferent from the other EAs. Ths one-to-one competton gves rse to a faster convergence rate. However, ths faster convergence also leads to a hgher probablty of obtanng a local optmum because the dversty of the populaton descends faster durng the soluton process. To overcome ths drawback, the parameters selecton s very mportant for the DE method. However, the parameters selecton s more senstve wth the problem. For example, a fxed scalng factor s used n DE. Usng a smaller scalng factor, DE becomes ncreasngly robust. However, much computatonal tme should be expanded to evaluate the objectve functon. DE wth a larger scalng factor should result generally falls nto local soluton or msconvergence. Ln et al. [13] used a random number that ts value s between zero and one as a scalng factor. However, a random scalng factor could not guarantee the fast convergence. So, Omran et al. [14] presented an effectve method, method, to overcome the drawback of the parameters selecton problem. Only two parameters, populaton sze and the maxmum teraton, are necessary for the method. In ths study, a method [14-1] wth nteger programmng for solvng the optmal capactor placement of The Author(s) 013. Ths artcle s publshed wth open access by the STF DOI:.17/1-3701_.3.8

2 STF Journal of Engneerng Technology (JET), Vol. No. 3, Dec 013 dstrbuton systems s proposed. Here, the concepts of chaotc search, opposton-based learnng, and quantum mechancs are used n the method to overcome the drawback of selecton of the crossover factor and scalng factor used n the orgnal DE method. Optmal capactor placement s a combnatoral optmzaton problem that s commonly solved by employng mathematcal programmng technques. However, n those methods, the capactors are often assumed as contnuous varables n whch cost s proportonate to the capactor sze. Selectng nteger capactor szes closet to the optmal values found by the contnuous varable approach does not guarantee an optmal soluton. Therefore, the optmal capactor placement should be vewed as an nteger-varable problem. The method can be used to solve the nteger-varable problems effectvely. To llustrate the convergence property of the proposed method, one benchmark functon and one 9-bus system from the lterature are solved respectvely by the proposed method, DE, and SA. From the computatonal results, t s observed that the convergence property of the method s better than that of the other methods. II. PROBLEM FORMULA Subject to () Where V s the voltage magntude of bus, Vmn and Vmax are the mnmum and maxmum voltage lmts, respectvely. The objectve functon COST n (1) s an overall cost relatng to power loss and capactor placement. The voltage magntude at each bus must be mantaned between ts mnmum and maxmum voltage lmts. To avod the complex teraton process for power flow analyss, a set of smplfed feeder-lne flow formulatons s appled. Consderng the sngle-lne dagram depcted n Fg.1, the followng set of recursve equatons s used for power flow computaton [17]. Q ) / V Q Q Q L X, ( P Q ) / V V () The total power loss of the feeder, PT,Loss, may then be determned by summng up the losses of all lne sectons of the feeder. Whch s gven by n 1 PT,Loss P Loss (, 1) (7) 0 Consderng the real-world capactors, there exsts a fnte number of standard szes whch are nteger multples of the smallest sze Q0C. Besdes, the cost per kvar vares from one sze to another. In general, capactors of larger sze have lower unt prces. The avalable capactor sze s usually lmted to C (8) Qmax LQ0C Where L s an nteger. Therefore, for each nstallaton locaton, there are L capactor szes Q0C,Q0C,, LQ0C avalable. ven the annual unt capactor nstallaton cost for each compensated bus, the total cost due to capactor placement and power loss change s wrtten as COST K p PT,Loss K C C Q (9) 1 Where K p s the equvalent annual cost per unt of power loss n $ /(kw year), and here $ s a fctonal monetary unt. Vmn V Vmax P P PL R, ( P P Q n The mathematcal model of the optmal capactor placement of dstrbuton systems can be expressed as follows: (1) mn COST PLoss (, 1) R, (3) V V ( R, P X, Q ) ( R, X, ) (4) ( P Q ) V () Where P and Q are the real and reactve powers flowng out of bus, and PL and QL are the real and reactve load powers at bus. The resstance and reactance of the lne secton between buses and 1 are denoted by R, and X,, respectvely. The power loss of the lne secton connectng buses and 1 may be computed as The constant K s the annual unt capactor nstallaton cost. And, 1,,, n are the ndces of buses selected for compensaton. The bus reactve compensaton power s lmted to QC n Q () L 1 Where QC and Q L are the reactve power compensated at bus and the reactve load power n bus, respectvely. III. METHOD The man dea of the method s to use the concepts of chaotc search, opposton-based learnng, and quantum mechancs nto the orgnal DE method to overcome the drawback of selecton of the crossover factor and scalng factor. The method s brefly descrbed n the followng. Step 1. Intalzaton Input system data and generate the ntal populaton. The ntal populaton s chosen randomly and would attempt to cover the entre parameter space unformly. The unform probablty dstrbuton for all random varables as followng s assumed (11) 0 mn round ( ( max mn )), 1,..., N p Where 0,1 s a random number, and round(b) represented as the nearest nteger for the real number b. The The Author(s) 013. Ths artcle s publshed wth open access by the STF

3 STF Journal of Engneerng Technology (JET), Vol. No. 3, Dec 013 ntal process can produce N p ndvduals of 0 randomly. Step. Mutaton operaton The essental ngredent n the mutaton operaton s the dfference vector. Each ndvdual par n a populaton at the -th generaton defnes a dfference vector D jk as D jk j k (1) The mutaton process at the -th generaton begns by randomly selectng ether two or four populaton ndvduals j, k, l and m for any j, k, l and m. These four ndvduals are then combned to form a dfference vector D jklm as D jklm D jk Dlm ( j k ) ( l m ) (13) A mutate vector s then generated based on the present ndvdual n the mutaton process by ˆ F D (14) p jklm, 1,..., N p However, the scalng factor value s depends on the problem. Dfferent from the orgnal DE algorthm, the concept of the quantum mechancs [14, 18] s used to generate the nosy replca from the parent ndvdual n algorthm as follows: ˆ round ( ) ln( 1 ), 1,..., N, 1 (1) 1 u p Where u 0,1 s a random number. Step 3. Estmaton and selecton ˆ ) arg mn f ( ), f ( b arg mn f ( ) (1) (17) Where arg mn means the argument of the mnmum. Step 4. Exclude operaton f necessary To ncrease the convergence of the algorthm, the exclude operaton s consdered. Frst, a new ndvdual s created as follows: w mn max worst, f 0. (18) best 1 ( c 1), otherwse Where and are randomly generated numbers unformly dstrbuted n the range of (0,1). worst and best are the worst and best ndvdual n the (+1)th generaton. c s the chaotc varable defned as follow: f c 0, p c / p, c (19) (1 c ) /(1 p), f c p,1 Where c 0 and p are ntalzed randomly wthn the nterval (0,1). The worst ndvdual n the -th generaton s replaced by the generated ndvdual, f the ftness of the generated ndvdual s better than that of worst ndvdual n the -th generaton. Step. Repeat step to step 4 untl the maxmum teraton quantty or the desred ftness s accomplshed. IV. APPLICATION OF THE PROPOSED METHOD One benchmark functon and one 9-bus system from the lterature are nvestgated and the results are used to compare the performance of the method wth the DE, and SA methods. The FORTRAN SA [19] algorthm solver accessed from /decoste/tmls /NN/glopt/glopt.html#sa_codes, s used to solve the optmal capactor placement problems. The SA solver recommended some settng factors for a user. For comparson, the SA package s rewrtten usng Matlab software. Example 1: Let us consder the maxmzaton problem s descrbed by (0) max J z1, z 1. z1 sn 4 z1 z sn 0 z z1, z Where 3 z1 1.1 and 4.1 z.8. Ths problem has been solved by the smple genetc algorthm usng the populaton sze of 0. The best result n generaton 39 had the best objectve functon value of , as also shown n Mchalewcez [0]. To verfy the performance of the method, the convergence property of the method and the orgnal DE method are compared va ths example frstly. The populaton sze N p 0, scalng factor F 0.1, crossover factor C R 0., and maxmum teratons of 300, are used n the DE method for solvng ths example. Sx strateges of mutaton operaton are respectvely used to solve ths example. Ths soluton of ths example s repeatedly solved one hundred tmes. The largest and smallest values among the best solutons of the one hundred runs are respectvely expressed n Table I. The average for the best solutons of the one hundred runs and the standard devaton wth respect to the average are also shown n ths table. A smaller standard devaton mples that almost all the best solutons are close the average best soluton. From the Table I, the standard devaton for the method s smaller than the other mutaton strategy. That mples the convergence property of the method s better than the orgnal DE method. Fve parameters ncludng the populaton sze, mutaton operaton, crossover factor, scalng factor, and the maxmum teraton number must be set n the orgnal DE method. But, only two parameters, populaton sze and maxmum teraton number, needs to set n the method. The best solutons for these one hundred runs are compared to the best objectve functon value obtaned by the smple genetc algorthm. The number of tmes that these best solutons were greater than s shown n Table I. From the Table I, the number of the successful runs that the best solutons were greater than s,,,, 70, and 14 for sx dfferent strateges of mutaton operaton. The number of the successful runs that the best solutons were greater than s n the method. Only fourth and ffth mutaton strategy s better than the method. Table II lsts the standard devaton values when the populaton sze s reassgned to and to solve ths example one hundred tmes, respectvely. From the above dscusson, the convergence property of the method s better than the orgnal DE The Author(s) 013. Ths artcle s publshed wth open access by the STF

4 STF Journal of Engneerng Technology (JET), Vol. No. 3, Dec 013 method. Example : The applcaton example [17] s a 3-kV, 9-secton feeder system. Detals of the feeder and the load characterstcs are gven by [17]. The equvalent annual cost per unt of power loss, K p, s selected to be $ 18 /( kw year) and the lmts on the bus voltages are Vmn 0.90 p.u. TABLE I COMPUTATION RESULTS FOR ONE HUNDRED RUNS OF EXAMPLE 1 Mutaton Strategy Largest Smallest Count (1) () Vmax 1. p.u. Two cases are nvestgated. Case 1: It s restrcted that only 3 locatons (buses 4,, and 9) are avalable for placement of capactors. The settng-factors used n the to solve ths case are as follows. The populaton sze, N p, s set to. The maxmum generaton s Mutaton Strategy Largest Smallest Count 00. These ntal-settng factors for the DE method are the same as that for the, except that the DE uses a scalng factor fxed to 0., a crossover factor fxed to 0. and second mutaton operator. The SA s also appled to solve ths problem. To verfy the performance of the proposed algorthm, ths case was repeatedly solved one hundred tmes. The best and worst values among the best solutons of these one hundred runs are lsted n Table. III. The average value and standard devaton () for the best solutons of these one hundred runs s shown n ths table. The best solutons for these one hundred runs are compared wth the best objectve functon value obtaned by the smulated annealng. The Exhaustve Search method s also repeatedly used to solve ths problem one hundred tmes. The best functon value obtaned by the Exhaustve Search s $ 118,38.3 / year. The best functon value obtaned by the Exhaustve Search s the same as that obtaned by the. Case : All buses are avalable for placement of capactors. Parameters for the applcaton are selected as those of case 1, except that the maxmum generaton s set to 000. The DE and SA are also appled to solve ths problem. Table IV expresses the best and the worst values among the best solutons of one hundred runs. The average value and standard devaton for the best solutons of those one hundred runs are also lsted n ths table. From the computatonal results, t s observed that the SA method cannot fnd the global soluton n one hundred runs. The worst objectve functon value of these one hundred runs obtaned by the method s smaller than those of the DE and SA. Comparng the results of the DE and, reveals that they are almost the same. However, the performance of DE s dependent on the choce of the mutaton operator. The best and average results of are relatvely better than those of the other three methods. The best objectve functon value of these one hundred runs s $ 11, / year TABLE II STANDARD DEVIATION FOR VARIOUS POPULATION SIE Populaton sze Populaton sze TABLE III THE COMPUTATION RESULTS OF CASE 1 FOR EXAMPLE Method Best Worst 118, , , DE SA 118, , , , ,73.4 1, , ,999.1 overcome the drawback of selecton of the crossover factor, scalng factor, and mutaton operator used n the orgnal dfferental evoluton (DE) method. From example 1, the convergence property of the method s outperformng than the orgnal DE method. From example, the computaton results shows that the soluton obtaned by the method s better than those obtaned by the DE and SA methods. ACKNOWLEDMENT Fnancal research support from the Natonal Scence Councl of the R. O. C. under grant NSC 0-3-E MY3 3/3 s greatly apprecated. REFERENCES V. CONCLUSION [1] Several heurstc methods ncludng the, DE, and SA used to solve one benchmark functon and one capactor placement problem had been descrbed n ths work. The concepts of chaotc search, opposton-based learnng, and quantum mechancs are used n the method to [] H. D. Chang, J. C. Wang, O. Cockngs, and H. D. Shn, Optmal capactor placements n dstrbuton systems: Part1: A new formulaton and the overall problem, IEEE Trans. on Power Delvery, vol., pp. 34-4, H. D. Chang, J. C. Wang, O. Cockngs, and H. D. Shn, Optmal capactor placements n dstrbuton systems: Part: Soluton algorthms The Author(s) 013. Ths artcle s publshed wth open access by the STF

5 STF Journal of Engneerng Technology (JET), Vol. No. 3, Dec 013 [3] [4] [] [] [7] [8] [9] [] [11] [1] [13] [14] [1] [1] [17] [18] [19] [0] and numercal results, IEEE Trans. on Power Delvery, vol., pp , J. J. ranger and S. H. Lee, Optmum sze and locaton of shunt capactors for reducton of losses on dstrbuton feeders, IEEE Trans. on Power Apparatus and Systems, vol.0, no. 1, pp. 1-18, S. H. Lee, and J. J. ranger, Optmum placement of fxed and swtched capactors on prmary dstrbuton feeders, IEEE Trans. on Power Apparatus and Systems, vol. 0, pp. 34-3, J. J. ranger, and S. H. Lee, Capacty release by shunt capactor placement on dstrbuton feeders: A new voltage-dependent model, IEEE Trans. on Power Apparatus and Systems, vol. 1, pp , 198. J. L. Bala, P. A. Kuntz, and M. Tayor, Senstvty-based optmal capactor placement on a radal dstrbuton feeder, n Proc. Northcon 9 IEEE Techncal Applcaton Conf., pp. -30, 199. S. Sundhararajan and A. Pahwa, Optmal selecton of capactors for radal dstrbuton systems usng a genetc algorthm, IEEE Trans. on Power Systems, vol. 9, pp , R. Storn and K. V. Prce, Mnmzng The Real Functons of the ICEC 9 Contest by Dfferental Evoluton, IEEE Conference on Evolutonary Computaton, pp , 199. S. Sultana and P. K. Roy, Optmal capactor placement n radal dstrbuton systems usng teachng learnng based optmzaton, Internatonal Journal of Electrcal Power and Energy Systems, vol.4, pp , 014. R. Muthu Kumar and K. Thanushkod, Reconfguraton and capactor placement usng opposton based dfferental evoluton algorthm npower dstrbuton system, Internatonal Revew on Modellng and Smulatons, vol., no. 4, pp , 013. J. Olamae, M. Morad, and T. Kabood, A new adaptve modfed frefly algorthm to solve optmal capactor placement problem, 18th Electrc Power Dstebuton Network conference, art. No. 9, 013. K. V. Prce, Dfferental Evoluton vs. Functons of the nd ICEC, IEEE Conference on Evolutonary Computaton, pp.13-17, Y. C. Ln, K. S. Hwang, and F. S. Wang, Plant schedulng and plannng usng mxed-nteger hybrd dfferental evoluton wth multpler updatng, Congress on Evolutonary Computaton, San Dego, CA., pp.93-00, 000 M.. H. Omran and A. Salman, Constraned Optmzaton Usng, Chaos, Soltons and Fractals, vol. 4, pp.-8, 009. M..H. Omran and A. Salman, Improvng the performance of usng quadratc nterpolaton, nd Internatonal Conference on Agents and Artdcal Intellgence Proceedngs, vol. 1, pp. -70, 0. M..H. Omran and A. Salman, Optmzaton of dscrete values usng recent varants of dfferental evoluton, Proceedngs of IASTED Internatonal Conference on Computatonal Intellgence, pp , 009. C. T. Su, C. S. Lee, and C. S. Ho, Optmal selecton of capactors n dstrbuton systems, n Proc. IEEE Power Tech. Conf., BPT , J. Sun, B. Feng, and W. Xu, Partcle swarm optmzaton wth partcles havng quantum behavor, IEEE Congress on Evolutonary Computaton, pp.3-331, 004. B. offe, lobal optmzaton of statstcal functons wth smulated annealng, Journal of Economcs, vol. 0, pp. -0, Mchalewcz, enetc algorthms + data structures = evoluton programs. New York: Sprnger-Verlag, 01. Ths artcle s dstrbuted under the terms of the Creatve Commons Attrbuton Lcense whch permts any use, dstrbuton, and reproducton n any medum, provded the orgnal author(s) and the source are credted. J-Pyng Chou, (19) receved hs Ph. D. degree n Electrcal Engneerng from Natonal Chung Cheng Unversty, Chay, Tawan. He s currently a Professor of the Department of Electrcal Engneerng at Mng Ch Unversty of Technology, New Tape Cty, Tawan. Hs research nterest s n the applcaton of evolutonary algorthms and optmzaton technques. Chung-Fu Chang, (194) receved hs Ph. D. degree n Electrcal Engneerng from Natonal Chung Cheng Unversty, Chay, Tawan. He s currently an Assocate Professor of the Department of Electrcal Engneerng at WuFeng Unversty, Chay Cty, Tawan. Hs feld of nterest ncludes dstrbuton system plannng, relablty engneerng, optmzaton technques, and AI applcaton. The Author(s) 013. Ths artcle s publshed wth open access by the STF