DIFFERENTIAL EVOLUTION OPTIMIZATION COMBINED WITH CHAOTIC SEQUENCES FOR OPTIMAL DESIGN OF SWITCHED RELUCTANCE MACHINE

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1 DIFFERENTIA EVOUTION OPTIMIZATION COMBINED WITH CHAOTIC SEQUENCES FOR OPTIMA DESIGN OF SWITCHED REUCTANCE MACHINE 1 M.BAAJI, 2 V.KAMARAJ 1 Att Pof., Depatment of Electcal Engneeng, SSN College of Engneeng, Inda 2 Pof., Depatment of Electcal Engneeng, SSN College of Engneeng, Inda E-mal: balajm@n.edu.n, kamaajv@n.edu.n ABSTRACT Th pape peent Dffeental Evoluton (DE algothm combned wth chaotc equence fo degn optmzaton of Swtched Reluctance Machne (SRM. Two dffeental evoluton appoache baed on chaotc equence ung logtc equaton ae popoed n th wok. Stato and oto pole ac of SRM condeed a degn vaable wth the objectve of maxmzng toque denty, maxmzng nductance ato and mnmzng coppe lo. The feablty of the popoed method demontated fo an 8/6, fouphae, 5 HP, 1500 pm SRM and compaed wth clacal dffeental evoluton and Genetc Algothm (GA method. The eult how that the popoed method effectve and obut. Keywod: Dffeental Evoluton(DE, Swtched Reluctance Machne(SRM,Chaotc Sequence, Toque Denty, Inductance ato 1. INTRODUCTION Swtched Reluctance Moto (SRM ae wdely ued n vaou applcaton due to the nheent mplcty and ugged contucton[1,2].in [2] the fundamental of SRM degn and pefomance pedcton have been decbed.srm ntended to opeate n deep magnetc atuaton to nceae the output powe denty n contat to tadtonal moto. Due to the effect of atuaton and the vaaton of magnetc eluctance, the flux-lnkage, nductance and toque chaactetc ae hghly nonlnea functon of both oto poton and phae cuent[3]. Thee nonlneate make the degn and analy of SRM dffcult. The conventonal degn method of a SRM to maxmze the oveall tatc aveage toque o mnmze the toque pple by ung optmal machne geomety. Seveal degn paamete, uch a the numbe of phae, pole ac, boe damete, a gap, etc., hould be taloed accodng to the equement of a pecfc applcaton. Fom the lteatue [4], [5] t evdent that toque output a well a the toque pple ae entve manly to tato and oto pole ac. Hence th wok focue on pole hape optmzaton of SRM.In [6], an appoach to detemne optmum pole ac of SRM to maxmze aveage toque and mnmze toque pple ung genealzed egeon neual netwok baed optmzaton dcued. In [7,8] Genetc Algothm (GA have been appled fo optmal degn of SRM. In th pape, the pole hape optmzaton of SRM appoached a a contaned nonlnea optmzaton poblem wth the objectve of maxmzng toque denty, maxmzng nductance ato and mnmzng coppe lo. Th pape ntoduce an optmzaton technque fo SRM motvated by two dffeental evoluton appoache baed on chaotc equence ung logtc equaton [9]. The man advantage of tadtonal DE baed optmzaton algothm ntoduced by [10] ove othe moden heutc fndng the tue global mnmum of a mult modal each pace egadle of the ntal paamete value, fat convegence,veatalty and ue of few contol paamete. To mpove the pefomance of DE, the ue of chaotc equence combned wth DE ha been epoted n lteatue and th technque ha been uccefully appled to vaou optmzaton poblem [11,17].The feablty of the popoed method demontated fo an 8/6, fouphae, 5 HP, 1500 pm SRM and compaed wth clacal dffeental evoluton and Genetc Algothm (GA method. The eult how that the popoed appoach pefom bette n tem of oluton qualty, accuacy and convegence tme. The oganzaton of pape a follow. In ecton 62

2 2, the poblem fomulaton explaned, whle the DE algothm wth chao befly ntoduced n Secton 3. The pefomance of the algothm peented and compaed n Secton 4. Fnally, concluon ae gven n Secton DESIGN OPTIMIZATION OF SRM- PROBEM FORMUATION 2.1 Pefomance ctea The tuctue of 8/6 SRM hown n fgue 1.The thee cteon ued to evaluate the degn of SRM ae aveage toque pe volume, coppe lo and nductance ato. Fgue1. Stuctue of SRM The computaton of aveage toque gven by T ( W W u N N 4π a ave = (1 I \ W = d (2 W a I a = (3 2 u u whee I epeent the ated phae cuent, a epeent the nductance at the fully algned poton and u epeent the nductance at the complete unalgned poton. A compehenve pogam wtten n Matlab to compute the dffeence of co enege at algned and unalgned poton ung the pocedue decbed by [1]. The moto lamnaton volume calculated a V = V + V (4 whee V epeent the volume of tato lamnaton and V epeent the volume of oto lamnaton. Conequently, the aveage toque pe moto lamnaton volume detemned a Tave TV = (5 V The coppe lo computed a 2 P = I cu R whee R epeent the phae etance. Toque pple expected fom SRM evaluated fom the toque dp n T--θ chaactetc. Toque dp the dffeence between the peak toque of a phae and the toque at an angle whee two ovelappng phae poduce equal toque at equal level of cuent. Th due to the defcency of the ncomng phae n upplyng the neceay toque n thoe oto poton [12]. The effect of pole ac vaaton on mean toque and toque dp can be evaluated fom Inductance ovelap ato K gven by equaton (6.Inductance ovelap ato gve a dect meaue of toque ovelap of adjacent phae. ε K = 1 (6 mn( β, β Fom equaton (12 t evdent that by wdenng the tato and oto pole, toque ovelap can be nceaed. The hghe the K, the lowe wll be the toque dp and the hghe wll be the mean toque a well Objectve functon The thee cteon ae elected a the degn objectve of SRM.The multobjectve poblem fomulaton gven by F opt = ( W tv TV TV whee TV b = max(tv C = mn( C K = max( K W b b tv + Wcu + WKl b = 1 + W cu C C b W kl K K b (7 In the above equatontv denote aveage toque pe volume, C pe coppe lo and K denote nductance ato. W TV, Wcu and W kl epeent the weght facto of the aveage toque pe volume, coppe lo and nductance ato.tv b epeent the bae value of aveage toque pe volume, C b epeent the bae value coppe lo and K b epeent the bae value of nductance ato. Fom equaton t een that the optmzaton wth thee objectve mplfed to an optmzaton functon 63

3 by ung thee weght facto. Vaou weght facto ndcate the hae whch ae taken up by aveage toque pe moto volume, coppe lo and nductance ato n the objectve functon. In th wok a weght facto of 1/3 condeed fo all the objectve. Snce the toque denty and nductance ato of the machne ha to be maxmzed, the ftne functon to mnmze taken equal to mnu the aveage toque pe moto volume and nductance ato Degn contant The followng ae the contant ae mpoed on the degn optmzaton poblem accodng to the ule of feable tangle[4]. β (8 β 2π β f β (9 N β f ε (10 To have a pactcally feable and acceptable fnal degn the followng pefomance contant ae mpoed. ( Aveage toque hould be geate than 21 N-m. ( Cleaance pace between the tp of wndng hould be geate than 5 mm.the contant ae taken nto account by penalzng the ftne popotonally to the contant volaton. 3. OPTIMIZATION USING DIFFERENTIA EVOUTION A DE algothm a tochatc paallel dect each optmzaton method that fat and eaonably obut. DE combne mple athmetcal opeato wth the clacal opeato of ecombnaton, mutaton and electon to evolve fom a andomly geneated tatng populaton to a fnal oluton. The fundamental dea behnd DE a cheme wheeby t geneate the tal paamete vecto. In each tep, the DE mutate vecto by addng weghted, andom vecto dffeental to them. If the cot of the tal vecto bette than that of the taget, the taget vecto eplaced by the tal vecto n the next geneaton. Pce and Ston [13] popoed 10 dffeent vaant fo DE baed on the ndvdual beng petubed, the numbe of ndvdual ued n the mutaton poce and the type of coove ued. Each tategy geneate tal vecto by addng the weghted dffeence between othe andomly elected membe of the populaton. The geneal conventon ued above DE/x/y/z. DE tand fo dffeental evoluton, x epeent a tng denotng the vecto to be petubed, y the numbe of dffeence vecto condeed fo petubaton of x, and z tand fo the type of coove beng ued exponental o bnomal.in th wok mutaton tategy DE/bet/1/bn ued. In th cheme the vecto to be petubed the bet vecto of the cuent populaton and the petubaton caued by ngle dffeence vecto.the optmzaton pocedue of DE/bet/1/bn gven by the followng tep and pocedue Step 1: Paamete etup The ue chooe the paamete of populaton ze, the bounday contant of optmzaton vaable, the mutaton facto (F, the coove ate (CR, and the toppng cteon of maxmum numbe of teaton (geneaton, Gmax. Step 2: Intalzaton of the populaton Set geneaton G = 0. Intalze a populaton of NP ndvdual wth andom value geneated accodng to a unfom pobablty dtbuton n the D dmenonal poblem pace. Thee ntal value ae choen andomly wthn ue defned bound. Step 3: Evaluaton of the populaton Evaluate the ftne value of each ndvdual of the populaton. Step 4: Mutaton opeaton (o dffeental opeaton Mutaton an opeaton that add a vecto dffeental to a populaton vecto of ndvdual. Fo each taget vecto a mutant vecto poduced ung the followng fomula v ( t + 1 = xbet + F ( x, 2 x, 3( t (11 In the above equaton, = 1, 2,...,N the ndvdual ndex of populaton, x (t tand fo the poton of the -th ndvdual of populaton of eal-valued n-dmenonal vecto, v (t tand fo poton of the -th ndvdual of a mutant vecto, F a eal paamete, called mutaton facto, whch contol the amplfcaton of the dffeence between two ndvdual o a to avod each tagnaton. The mutaton tategy petub the bet vecto of the cuent populaton by ngle dffeence vecto.the two ndvdual x, 2 and x, 3 ae andomly elected and the dffeence vecto calculated. Step 5: Coove opeaton To nceae the potental dvety of the populaton a coove opeato ued. DE ue two knd of 64

4 co ove cheme namely Exponental and Bnomal. In th wok bnomal coove ued. In th coove cheme, the coove pefomed on each of the D vaable wheneve a andomly pcked numbe between 0 and 1 wthn the CR value. The cheme may be outlned a u, j = v, j f ( and (0,1 < CR = x, j ele. (12 In th way fo each tal vecto X (t an offpng vecto U (t ceated. Step 6: Selecton opeaton Selecton opeato ued to detemne whch one of the taget vecto and the tal vecto wll uvve n the next geneaton. The electon poce may be outlned a X ( t + 1 = U f f ( U f ( X = X (t f f ( X < f ( U.. (13 whee f the functon to be mnmzed. If the new tal vecto yeld a bette value of the ftne functon, t eplace t taget n the next geneaton othewe the taget vecto etaned n the populaton. Once new populaton ntalled, the poce of mutaton, ecombnaton and electon eplaced untl the optmum located, o a pecfed temnaton cteon atfed, e.g., the numbe of geneaton eache a peet maxmum Gmax.At each geneaton, new vecto ae geneated by the combnaton of vecto andomly choen fom the cuent populaton (mutaton. The upcomng vecto ae then mxed wth a pedetemned taget vecto. Th opeaton called ecombnaton and poduce the tal vecto. Fnally, the tal vecto accepted fo the next geneaton f t yeld a educton n the value of the objectve functon. 3.1Dffeental evoluton wth chaotc appoache The thee vtal contol paamete of DE ae the populaton numbe, the mutaton facto and the coove ate. The peed and obutne of the each ae affected wth the vaaton of thee paamete. The dffculty n the ue of DE ae n vew of the fact that the choce of thee manly baed on empcal evdence and pactcal expeence [14,17]. DE paamete uually ae contant thoughout the ente each poce. Howeve, t dffcult to popely et contol paamete n DE. The applcaton of chaotc equence n mutaton facto degn a poweful tategy to dvefy the DE populaton and mpove DE pefomance n peventng pematue convegence to local mnma[17]. The applcaton of chaotc equence can be a good altenatve to povde the each dvety n tochatc optmzaton pocedue. Due to the egodcty popety, chao can be ued to ench the eachng behavo and to avod beng tapped nto local optmum n optmzaton poblem. In th pape, to ench the eachng behavo and to avod beng tapped nto local optmum, chaotc dynamc ncopoated nto the DE. In th context, two chaotc DE appoache ae popoed. Popoed dffeent chaotc DE appoache have ued the well-known logtc equaton, whch exhbt the entve dependence on ntal condton, fo detemnng the mutaton facto. The logtc equaton defned a follow y ( k = µ * y( k 1 *(1 y( k 1 (14 Whee k the ample and µ the contol paamete, 0 p µ 4.The behavo of the ytem epeented by equaton (14 geatly changed wth the vaaton of µ. The value of µ detemne whethe F tablze at a contant ze, ocllate between a lmted equence of ze, o behave chaotcally n an unpedctable patten. And alo the behavo of the ytem entve to ntal value of F [10]. Equaton (14 detemntc, dplayng chaotc dynamc when µ =4 and { 0,0.25,0.5,0.75,1} y ( 1. The two chaotc DE (CDE appoache n combnaton of chaotc equence ae decbed a follow CDE1 appoach: The paamete F of (11 modfed by the fomula (15 though the followng equaton: v ( t + 1 = xbet + y( k ( x, 2 x, 3( t (15 CDE2 appoach: The paamete F of (11 modfed by the fomula (16 though the followng equaton: G F = y( k exp( (16 4. RESUTS G max The pefomance of the popoed method teted on a 5HP moto. The pecfcaton of the ample moto ae gven n Appendx 1. The algothm coded n Matlab and executed ung a Pentum IV baed PC a the tet platfom. Dung the poce the followng paamete ettng ued fo 65

5 tadtonal DE: Populaton ze=30, Coove contant =0.7, Scalng Facto fo Mutaton=0.8, maxmum teaton Itemax = 100Upon executon of the algothm, an optmal tuctue wth the confguaton β =21.91and β =24.08 obtaned. The pefomance paamete of the optmal moto degn ae gven n table1. Fom the table t clea that thee gnfcant mpovement n toque denty and nductance ato. Table 1 Reult of Optmal Degn Intal Degn Optmal Degn Stato Pole 18 deg 21.91deg ac Roto Pole 22 deg deg ac Aveage Nm Toque Nm Toque Denty 1252 Nm/m Nm/m 3 Inductance ato Coppe o 183 W 190 W Toque dp 8.87 Nm 4.46 Nm Table 2 Convegence eult of 20 un ung GA, DE and CDE appoache GA DE CDE1 CDE2 Bet Soluton Wot Soluton Mean Standad Devaton Fgue2. Convegence chaactetc of dffeent optmzaton mehod 4.1Compaatve tude In ode to vefy the obutne of the algothm, mulaton wee caed out fo 20 ndependent un. The eult of tattcal compaon ae ummazed n table2.it een that n tem of mean and tandad devaton CDE1 appoach pefomed gnfcantly bette. The pefomance of the optmzaton technque n tem of convegence hown n fg2.fom the fgue t evdent that the convegence chaactetc of chaotc DE appoache ae bette. 5. CONCUSION In th pape, two DE appoache baed on chaotc equence ung logtc equaton to adapt the mutaton facto fo degn optmzaton of SRM ae popoed. The objectve of the popoed chaotc DE appoache to maxmze toque denty, maxmze nductance ato and mnmze coppe lo. The optmzed geomety wa expoed to fnte-element calculaton. The optmal machne poduced an aveage toque of 28.96Nm wth a toque dp of 4.46 Nm. The eult of fnte-element calculaton confm the applcaton of optmzaton pocedue fo SRM degn. Both the tadtonal DE and the CDE appoache wee uccefully appled to degn optmzaton of SRM.The CDE appoache can be ued a pomng optmzaton method fo olvng SRM degn poblem. APPENDIX 1 Degn Data of the machne Machne confguaton 8/6 Powe output Stato pole ac Roto pole ac A gap length Oute tato damete Boe damete Stack length Shaft damete Speed Heght of tato pole Heght of oto pole 5 hp 18 degee 22 degee 0.5 mm 190 mm mm 200 mm 28 mm 1500 pm 32.7 mm 19.8 mm Tun pe phae 154 Rated cuent 13 A 66

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