NEW EVOLUTIONARY PARTICLE SWARM ALGORITHM (EPSO) APPLIED TO VOLTAGE/VAR CONTROL Vladmro Mranda vmranda@nescporto.pt Nuno Fonseca nfonseca@power.nescn.pt INESC Insttuto de Engenhara de Sstemas e Computadores do Porto P. Repúblca 93 4050 Porto Portugal Fax 35108417 FEUP Faculdade de Engenhara da Unversdade do Porto, Portugal Abstract Ths paper presents a new optmzaton model EPSO, Evolutonary Partcle Swarm Optmzaton, nspred n both Evolutonary Algorthms and n Partcle Swarm Optmzaton algorthms. The fundamentals of the method are descrbed, and an applcaton to the problem of Loss mnmzaton and Voltage control s presented, wth very good results. Keywords: Evolutonary Algorthms, Partcle Swarm Optmzaton, Voltage/Var control 1. INTRODUCTION Ths paper has the man objectve of ntroducng to the Power System communty a new and powerful metaheurstc hybrd varant called EPSO Evolutonary Partcle Swarm Optmzaton. EPSO s a general-purpose algorthm and t can thus be appled to a dversty of problems n any scentfc area. However, n order to llustrate the technque, we have selected a problem n the Power Systems envronment and wll therefore present n the paper, n the applcaton sectons, a soluton for the Voltage/Var control problem obtaned by EPSO and the comparson of ts performance wth exstng methods The Partcle Swarm Optmzaton s an optmzaton algorthm that was ntroduced n 1995 by Kennedy [1]. We wll refer to t as: Classc PSO. Imagne that we have a populaton of partcles loong around n a gven search space for the global optmum. Ths partcle movement mmcs, n a way, the coordnated movement of flocs of brds, schools of fsh or swarms of nsects: ths s a good mage of a PSO optmzaton algorthm. In Evolutonary Algorthms, there s no coordnaton n the movement of ndvduals wthn the search space. However, the powerful selecton procedure allows solutons wth superor characterstcs to pass these from generaton to generaton, whle the mutaton (and recombnaton) schemes produce dversty n the soluton pool. EPSO jons together the best of two worlds. It s a Partcle Swarm algorthm, because there s exchange of nformaton among solutons, when they are successvely moved around n the search space; and t s an Evolutonary Computaton method, because soluton characterstcs are mutated and passed to the followng generatons by the acton of a selecton mechansm. In a classcal PSO model, partcle movement s condtoned by three strategc parameters: nerta, memory and coordnaton (nformaton exchange). In prevous applcatons of the Classc PSO and other varants le CPSO (Cooperatve PSO) [], the strategc parameters of the algorthms were set to certan values that had already been used wth god results. But there s no vald explanaton to sustan that we should use a partcular value for those parameters f we have a dfferent problem. Also, we can t say that a certan value s the best durng all the process of optmzaton. That had already been realzed for the nerta factor. Ths parameter s usually decreased as the number of teraton ncreases [3]. EPSO [4] defnes these parameters as the genotype of a movng soluton. Therefore, they are subject to mutaton and the partcles holdng them as phenotypes are subject to selecton. Ths scheme turns out to be a successful selftunng mechansm, a self-adaptve evolutonary process actng on strategc parameters, to use the language of the Evoluton Strategy communty. As we wll show, EPSO has a better behavor than Classcal PSO (namely, t s robust, nsenstve n a large degree to ntal values of parameters) and t also has a better behavor than other meta-heurstcs (n ths paper, a comparson wll be made wth smulated annealng). The hybrd characterstcs of Evolutonary and of Partcle Swarm model gve t guaranteed convergence propertes. In terms of effcency, therefore, lower bounds are guaranteed, but experence demonstrates that there s an effectve acceleraton and a better search for the optmum than classcal approaches.
. EPSO DESCRIPTION In EPSO, each partcle (soluton at a gven stage) s defned by the followng characterstcs: poston n the search space ( x ; value of the coordnate poston, for the partcle) velocty ( v ; value of the coordnate velocty for the partcle). At a gven moment, there s at least one partcle that holds the best poston n the search space. The populaton of partcles s aware of such poston, represented as ( best x, value of the coordnate poston, for the best partcle). Each partcle also eeps trac of ts prevous best poston ( mem x,, value of coordnate poston, memorzed as ts prevous best, for the partcle). The partcles wll reproduce and evolve along a number of generatons, accordng to the followng steps: - Replcaton: each partcle s replcated a number r of tmes, gvng place to dentcal partcles (n ths paper we tae r = 1). - Mutaton: the strategc parameters of the replcated partcles undergo mutaton accordng to: * w = w τn (0, ) (1), j, j + σ where τ s a learnng dsperson parameter and N(0,1) s a random number followng a the normalzed Gaussan dstrbuton wth zero mean and varance σ. The strategc parameters are randomly set between 0 and 1 at the begnnng of the algorthm. In each teraton, the strategc parameters of the replcated partcles are mutated accordng to equaton (1). In ths equaton, j can be the nerta, memory or the coordnaton factor. - Reproducton (movement): each partcle generates as offsprng a new partcle accordng to the transformaton process, smlar to the Classc PSO basc equaton:,mem * *v = w v + w (x x ) + w (x best x ) (),nerta,mem * x x + * v,coop = (3) The offsprng s held separately for the orgnal partcles and for the mutated partcles. Furthermore, nstead of defnng a crsp best-so-far pont as a target, the partcles are attracted to a sort of foggy best-so-far regon (another change relatve to Classc PSO). Ths s done by ntroducng random nose n the defnton of the best-so-far pont: best* best x = x + τ' N(0,1) (4) τ s a nose dsperson parameter, usually small, and N(0,1) s a random number followng a the normalzed Gaussan dstrbuton wth zero mean and varance 1. - Evaluaton: each offsprng partcle plus the orgnals are evaluated accordng to ther current poston. - Selecton: among the offsprng of a partcle, wth and wthout mutated parameters, a stochastc tournament s played to select the partcle that wll survve to the next generaton. As n many other meta-heurstcs, EPSO deals wth nequalty constrants through a penalty strategy. In the case of EPSO, the selectve pressure appled helps n elmnatng the ndvduals or partcles wth excursons outsde the feasble doman, whch receve a penalzed ftness value. 3. TESTING EPSO In ths secton we llustrate the superorty of EPSO regardng the Classc PSO algorthm, n solvng classcal dffcult test problems. 3.1 Test functons Schaffer s functon: (sn x + y ) 0.5 f1(x) = 0.5 + (5) (1.0 + 0.001(x + y )) Rosenbroc functon: n f (x) = (100 (x+ 1 x ) + (x 1) ) (6) = 1 Sphere functon: n 3 (x) = x = 1 f (7) Alpne functon: f = (8) 4 (x) sn(x1) sn(x ) x1x The parameters used n these functons are presented n Table 1. The threshold used as the stoppng crteron s lsted n the Stop column. Functon n Doman Stop f 1 [-50, 50] n 1.0E-10 f 30 [0, 30] n 100 f 3 30 [-50, 50] n 0.01 f 4 [0, 100] n 98.967 Table 1: Parameters used n the test functons.
3. Results n the test functons The followng pctures llustrate the typcal convergence n the test functons, for the EPSO and PSO. Ftness 1.E-06 1.E-08 1.E-10 1.E-1 Schaffe r F6 1.E+00 0 50000 100000 150000 00000 1.E-0 EPS O 1.E-04 PS O Num ber of evaluatons Fgure 1 - Typcal convergence n the Schaffer s functon. The results presented n Fgures 1,, 3 and 4 show a clear superorty of the EPSO algorthm. The PSO results could be optmzed f we ve tuned by hand the strategc parameters. EPSO was able to provde better results ndependently of the strategc parameter ntalzaton. If we are tryng to optmze a dfferent problem (ex: Optmal Power Flow), where we don t now whch are the better strategc parameters, then EPSO s certanly better because of the self-tunng mechansm. To demonstrate the superorty of EPSO over the Classc PSO we compare the average number of evaluatons that both algorthms need to reach the stoppng crteron. The maxmum number of evaluatons was fxed n 00000. Table presents the results of ths test. Ftness Rosenbroc EPSO 1.E+09 PSO 1.E+08 1.E+07 1.E+06 1.E+05 1.E+04 1.E+03 1.E+0 1.E+01 1.E+00 0 50000 100000 150000 00000 Num ber of evaluatons Functon EPSO PSO f 1 1186. 59547.0 f 7005.3 180310.8 f 3 1641.4 16165.0 f 4 78539.8 199190.1 Table : Comparson of EPSO wth the Classcal PSO: average number of evaluatons. Fgure - Typcal convergence n the Rosenbroc functon. Ftness Sphere 1.E+05 1.E+04 EPS O 1.E+03 PS O 1.E+0 1.E+01 1.E+00 1.E-01 0 50000 100000 150000 00000 1.E-0 1.E-03 1.E-04 Num ber of evaluatons Fgure 3 - Typcal convergence n the Sphere functon. We also compare the average results of both algorthms for a fxed number of evaluatons. So, consderng a number of evaluatons of 00000 (0 partcles over 5000 teratons n the EPSO algorthm and 0 partcles over 10000 teratons n the Classcal PSO) we obtaned the followng results: Functon EPSO PSO f 1.15E-13 5.45E-11 f 33.888 114.443 f 3 7.81E-04 1.91E-0 f 4 98.967 86.1071 Ftness 100 95 90 85 80 75 70 65 60 55 Alpne PS O EPS O 50 0 100 00 300 400 Num ber of evaluatons Fgure 4 - Typcal convergence n the Alpne functon. Table 3: Comparson of EPSO wth the Classcal PSO: average results. The results presented n Table and Table 3 were obtaned wth a populaton of 0 partcles, over 500 smulatons. Notce that the poston and the strategc parameters (nerta, coordnaton factor and memory) were always ntalzed randomly for each partcle. The results of the PSO would have perhaps a margn for mprovement f the strategc parameters were even better tuned by hand, but ths would nvolve a tedous wor of expermentaton case
by case. The EPSO algorthm was able to provde mmedately good results ndependently of such ntalzaton. Ths s a very mportant mprovement n the algorthm, because the results of Classc PSO are reported to be very dependent of the strategc parameter ntalzaton [3], and ths has been confrmed by our experence. Ths statement does not mean that one could not fnd solutons wth less good ntal PSO parameters but the performance of the PSO algorthm, n our experments, never reached the qualty of EPSO, dd not dsplay on average the same qualty of results and, most mportant, dd not dsplay the same robustness, whch s vtal for a practcal applcaton the users must trust the algorthm, must beleve t gves relable and consstent results, must be confdent that, f they run t a number of tmes, they wll get the same nd of answer. 4. APPLICATION OF EPSO TO VOLTAGE/VAR CONTROL 4.1 Loss reducton n dstrbuton systems The applcaton of PSO-le algorthms to the Voltage/Var control problem was poneered by authors n Japan and reported n [6][7][8]. Ther models ncluded a form of blendng evolutonary concepts wth the PSO algorthm, wth postve results. However, ther valuable wor remaned one step away from a true self-adaptve approach, whch s what ths paper now presents. We llustrate EPSO n a loss reducton-voltage/var control problem for a ddactc example wth the IEEE 4 nodes/36 branches networ defned n [9]. Ths networ also ncludes 31 transmsson lnes, 5 transformers, 11 capactor bans and 9 synchronous generators. The sze of a problem of ths nature, however, s not related really wth the sze of the networ but wth the number of controllers avalable. For the sae of a comparson wth a competng algorthm, based on Smulated Annealng, we too as control varables only the set pont of transformers and capactor bans. Ths Smulated Annealng [10] [11], algorthm s a well tested applcaton developed by INESC Porto and ncluded n a commercal DMS, used by a number of utltes. The problem of Voltage/Var control can be formulated as follows: Mnmze I ( u, x) (9) Subject ϕ ( u, x, p) = 0 (10) φ ( u, x, p) 0 (11) Equaton (9) s the objectve functon of ths problem and, n general, represents the actve losses. The constrants of ths problem, (10) and (11), are respectvely the power flow equatons and operaton lmts, namely bands of admssble voltage values at nodes. All these equatons, wrtten n a general form, must be understood as representng a full AC model, wth losses evaluated, for nstance, wth a Newton-Raphson algorthm. Because these are well nown equatons, we felt we could tae the lberty of adoptng the above representaton. We may have also other objectves, such as the preference for eepng control margns,.e., searchng for solutons that do not requre the set ponts of controllers to be at ther maxma or mnma. Ths means that one s facng a multcrtera problem, wth two objectves: Mnmze losses Mnmze dstance of control varables from nomnal set ponts (usually, the center on the ntervals defnng ther range of varaton). In fact, ths s acheved n practce by applyng a penalty factor to the ftness functon, such as depcted n Fgure 5. Penalty mn Nomnal Max Control setpont varable Fgure 5 Example of penalty functon to be added to the loss functon (per control varable, scaled by a weghtng factor) to favor solutons that do not push controls to ther lmts The Voltage/Var control problem n dstrbuton systems s usually a problem of mnmzng losses and controllng voltage levels, by actng on transformer taps and on capactor ban taps. It s rare to fnd synchronous generators drectly connected to the networ where one could act on ther exctaton. However, EPSO can deal wth these varables as well, wth excellent results. 4. Results of the loss reducton problem In order to compare EPSO results wth those obtaned wth the Smulated Annealng (SA) applcaton we needed to establsh the same stoppng crteron. As the Smulated Annealng already had ths crteron fxed as a certan number of teratons wthout mprovement n the best-sofar soluton, we used the same crteron. For ths partcular exercse, the maxmum number of teratons allowed wthout mprovement n the soluton was fxed n 70. In ths partcular applcaton, all the varables of control are dscrete (set pont of transformers and capactor bans). There s a verson of Dscrete PSO [1], but as for now, the EPSO only deals wth contnuous varables.
We ve used probablstc roundng to solve ths problem. Instead of usng smple roundng,.e., consder the nearest value, we ve consdered that the probablty of roundng to the nearest dscrete value ncreases as the dstance decreases. On average, the value of the varable s probablstcally rounded to the nearest dscrete value, but there s always the possblty that t s not, at any gven pont. Ths scheme avods trappng n local dscrete values, and has all the flavor of the technques used n evolutonary computng. In terms of convergence comparson between both algorthms, we can mmedately reveal that: - EPSO fnds ts best soluton n less teratons. - the ntal soluton s better for the EPSO, because t has a populaton of partcles, whle the Smulated Annealng only starts wth one ntal soluton. - there s an extra computng effort n applyng EPSO, when compared to the Smulated Annealng opton (measured n the number of load flows run); - EPSO consstently dscovers better solutons than the Smulated Annealng algorthm. A typcal convergence pattern observed for both algorthms can be observed n Fgure 5, where EPSO (as usual) found a better soluton than SA. We tested EPSO wth dfferent populaton szes. The objectve of ths test was to fnd out the nfluence of populaton sze n the results. As t can be seen n Table 4, the qualty of the soluton mproves f we ncrease the number of partcles. Of course, there s a prce to pay n terms of computng effort. As for now, we were not able to establsh a secure rule for defnng the optmal number of partcles. Our assumpton s that ths wll depend on the complexty of the problem, and wth ncreasng complexty t wll be necessary to ncrease the number of partcles. Average losses (MW) Std. Devaton (W) to the best soluton EPSO partcles 61.7947 3.10334 EPSO 5 partcles 61.791.64314 EPSO 10 partcles 61.7889 1.96607 EPSO 0 partcles 61.7880 1.48473 Smulated Annealng 61.791 9.81175 Table 4: Comparson of EPSO wth Smulated Annealng As we can see n Table 4, the EPSO reveals superorty n terms of the soluton found (both n the best soluton dscovered and n the average optmum obtaned n 1000 runs, as seen n the Table) and n terms of ts robustness (evaluated as the root of the mean square error, or standard devaton, relatve to the best soluton found). In partcular, EPSO gves consstently a near-optmum result, whle the Smulated Annealng model faled many tmes to reach a soluton as good (and that s why the dsperson of results n ths case s much larger than wth EPSO). Therefore, EPSO s a much more relable algorthm for practcal applcatons. 4.3 Voltage control For ths test we ve ncrease the reactve load n bus 8 of the same IEEE 4-bus system. The voltage at ths bus became very low and we run the EPSO algorthm to redspatch the reactve power n order to set the voltage bac nsde the lmts. As t can be seen n Fgure 6, the algorthm was able to fnd a new set pont, to both transformers and capactor bans, whch forced the voltage nto the acceptable lmt (0.9 1.1 p.u.). The Smulated Annealng algorthm faled to obtan a feasble soluton for ths case. Smulated Annelng vs Evolutonary PSO Voltage control 74000 Actve Losses (W) 7000 70000 68000 66000 64000 6000 Smulated Annelng EPSO 0 partcles Voltage n bar 8 (p.u.) 0,904 0,900 0,896 0,89 0,888 60000 0 500 1000 1500 000 Number of teratons 0,884 1 4 7 10 13 16 19 5 8 31 Number of teratons 34 37 40 43 46 Fgure 5 - Comparson n convergence between EPSO and Smulated Annealng Fgure 6 - Voltage Control wth EPSO evoluton of the controlled voltage along the teratons of an EPSO algorthm.
5. CONCLUSIONS Ths paper reports two mportant results: A new optmzaton technque, wth roots both n Evolutonary Computng and n Partcle Swarm algorthms. A new model for loss mnmzaton and voltage control Frst of all, there s a new successful meta-heurstc tool, avalable for optmzaton of complex problems wth multple local optma EPSO, the Evolutonary Partcle Swarm Optmzaton method. EPSO jons together the characterstcs of Evolutonary Algorthms and of Partcle Swarm Algorthms. From an Evolutonary Computng pont of vew, there s another operator ntroduced, sde by sde wth recombnaton and mutaton, whch generates new (and promsng) solutons n the search space t s Reproducton n the form of Partcle Movement. From a Partcle Swarm pont of vew, there s a selfadaptve tunng of the algorthm by evolutonary adjustment of the parameters controllng partcle movement. Both ponts of vew are legtmate and justfy the remarable convergence characterstcs of the method. The second mportant result s that EPSO proves very successful n solvng a Power System optmzaton problem mnmzng losses n a transmsson system. In fact, EPSO performed better than a Smulated Annealng model that has been used by utltes, both n the qualty of the soluton dscovered and n the robustness of the result (dsperson around the best result, found n a number of repeated runs). In the tests done, the Smulated Annealng algorthm demanded a somewhat smaller computer effort (measured n the number of load flows requred) but faled completely to dscover the best solutons, whle EPSO was able to converge to them n all cases. Furthermore, EPSO was agan successful n a Voltage Control problem, easly dscoverng a soluton for a dffcult problem where other technques experment dffcultes n convergng. One expects that EPSO may be appled wth equal success to other problems n Power Systems. REFERENCES [1] Kennedy, J., R.C. Eberhart, Partcle Swarm Optmzaton, IEEE Internatonal Conference on Neural Networs, Pert, Australa, IEEE Servce Center, Pscataway, NJ., 1995 [] F. van den Bergh, A.P. Engelbrecht, Tranng Product Unt Networs usng Cooperatve Partcle Swarm Optmzaton, Internatonal Jont Conference on Neural Networs (IJCNN), Washngton D.C., 001. [3] Yuhu Sh, Russell C. Eberhart, Parameter Selecton n Partcle Swarm Optmzaton, Proceedngs of the Seventh Annual Conference on Evolutonary Programmng, 1998. [4] Vladmro Mranda, Nuno Fonseca, EPSO- Evolutonary self-adaptng Partcle Swarm optmzaton, nternal report INESC Porto, July 001 (obtanable from the authors by request). [5] F. van den Bergh, A.P. Engelbrecht, Effects of Swarm Sze on Cooperatve Partcle Swarm Optmzers, Proceedngs of the Genetc Evolutonary Computaton Conference (GECCO),, 001. [6] Yoshda, H., Fuuyama, Y., Taayama, S. and Naansh, Y., A partcle swarm optmzaton for reactve power and voltage control n electrc power systems consderng voltage securty assessment, IEEE Proc. of SMC '99, Vol. 6, pp.497-50, 1999. [7] H. Yoshda, K. Kawata, Y. Fuuyama, S. Taayama, and Y. Naansh, A Partcle Swarm Optmzaton for Reactve Power and Voltage Control Consderng Voltage Securty Assessment, IEEE Trans. on Power Systems, vol. 15, no. 4, pp.13-139, Nov. 000. [8] Fuuyama, Y., Yoshda, H., A partcle swarm optmzaton for reactve power and voltage control n electrc power systems, IEEE Proc. of Evolutonary Computaton 001, Vol.1, pp. 87-93, 001. [9] Relablty Test System Tas Force of the Applcaton of Probablty Methods Subcommttee, IEEE Relablty Test System, IEEE Trans. On Power Apparatus and Systems, vol. PAS-98, no. 6, Nov./Dec. 1979. [10]Jorge Perera, J. Tomé Sarava, Mara Teresa Ponce de Leão, "Identfcaton of Operaton Strateges of Dstrbuton Networs Usng a Smulated Annealng Approach", Proceedngs of IEEE Budapest Power Tech'99, paper BPT99-357-17, August 1999. [11]Manuel Matos, Mara Teresa Ponce de Leão, J. Tomé Sarava, J. N. Fdalgo, et al., "Meta-heurstcs Appled to Power Systems", Proceedngs of MIC 001-4th Metaheurstcs Internatonal Conference, Porto, Portugal, vol., pp.483-488, July, 001. [1]Kennedy, J. and Eberhart, R. C., A dscrete bnary verson of the partcle swarm algorthm, Proc. Conf. on Systems, Man, and Cybernetcs, 4104 4109. Pscataway, NJ: IEEE Servce Center, 1997.