Scroll Plate Optimization Based on GA-PSO

Transcription

1 Purdu Unvrsty Purdu -Pubs Intrnatonal Comprssor Engnrng Confrnc School of Mchancal Engnrng 2006 Scroll Plat Optmzaton Basd on GA-PSO Bn Png Lanzhou Unvrsty of Tchnology Jun Wang Lanzhou Unvrsty of Tchnology Zhnquan Lu Lanzhou Unvrsty of Tchnology Follow ths and addtonal works at: Png, Bn; Wang, Jun; and Lu, Zhnquan, "Scroll Plat Optmzaton Basd on GA-PSO" (2006). Intrnatonal Comprssor Engnrng Confrnc. Papr Ths documnt has bn mad avalabl through Purdu -Pubs, a srvc of th Purdu Unvrsty Lbrars. Plas contact pubs@purdu.du for addtonal nformaton. Complt procdngs may b acqurd n prnt and on CD-ROM drctly from th Ray W. Hrrck Laborators at Hrrck/Evnts/ordrlt.html

2 C090, Pag 1 Scroll Plat Optmzaton Basd on GA-PSO BIN PENG 1, JUN WANG 2, ZHENQUAN LIU 2 1 Collg of Mchano-Elctronc Engnrng, Lanzhou Unv. of Tch., Lanzhou Phon: +86(931) E-mal: pngb2000@163.com 2 Collg of Ptrochmcal Tch, Lanzhou Unv. of Tch. Lanzhou Phon: +86(931) E-mal: pngb2000@163.com ABSTRACT Th parts optmzaton ar vry mportant for scroll comprssor dsgn. Accordng to xstng problms of currnt optmzaton algorthm and actual optmzaton problms, th mprovd optmzaton algorthm gntc-partcl swarm optmzaton (GA-PSO) s proposd for scroll plat optmzaton. Th optmzaton mthod ntgrats crossovr of gntc algorthm (GA) and volutonary mchansm of partcl swarm optmzaton (PSO), th man structur paramtrs ar bn as control varabl, th optmzaton mathmatcs modl s dvlopd, makng us of crossovr of GA and volutonary mchansm of PSO, GA-PSO ralzs th purpos of mnmzng valu of objctv functon. GA-PSO s appld to scroll plat optmzaton on computr, t s shown that th mprovd approach convrgs to bttr soluton much fastr than th arlr rportd approachs through compard wth othr mthods and tstd of prototyp prformanc. All th rsults supply thory and tchnology support for wd applcaton of GA-PSO n ngnrng. 1. INTRODUCTION Th scroll comprssor s a knd of smpl structur, hgh-ffcnt, low nos, hgh dpndablty, low vbraton nwtyp dsplacmnt comprssor, t has alrady xtnsvly appld to rfrgraton, ar condtonr, varous knds of gas comprsson, and prssurzd pump tc. But at prsnt frcton loss and lakag ar th man barrrs that affct th mchancal ffcncy of scroll comprssor, fnally lad to th low rfrgraton quantty and EER. Th ky componnt - fxd scroll and orbtng scroll form a srs of crscnt-shapd workng pockts, ts basc paramtrs dcd th frcton loss and lakag, also affct th comprssor prformanc, so paramtrs optmzaton dsgn s vry mportant to nhanc th ntr machn prformanc of scroll comprssor. At prsnt thr hav many optmzaton rsarch about scroll comprssor, Fan Lng t al. [1] stablshd th optmzaton mathmatcal modl of scroll comprssor and usd th mprovd Rung-Kutta mthod to solv th modl; Zhang Lqun and Lu Yongbo [2] usd th SWIFT to optmz to th man dsgn paramtrs, t mprovd th machn prformanc; Chn Jn t al. [3] usd th multobjctv gntc algorthm to optmz th scroll shap; Partcl swarm optmzaton (PSO) s an volutonary computaton tchnqu dvlopd by Dr. Ebrhart and Dr. Knndy n 1995, nsprd by socal bhavor of brd flockng or fsh schoolng. PSO s a populaton basd optmzaton tool, th systm s ntalzd wth a populaton of random solutons and sarchs for optma by updatng gnratons. In PSO, th potntal solutons, calld partcls, ar "flown" through th problm spac by followng th currnt optmum partcls. It sks th optmal soluton through ndvdual coopraton and nformaton sharng. Th nw algorthm provds th vry good tool for prsnt optmzaton doman, Zhou Ch t al. [4] talkd about orgn, dvlopmnt and applcaton of PSO, HPSO was dvlopd by Lovbjrg, Rasmussn and Krnk n 2000 [5]. But at prsnt th optmzaton mthods of scroll comprssor xst th shortcomng of low prcson, low ffcncy and optmal soluton s not dal, accordng to structur, prformanc charactrstcs of scroll comprssor and GA, PSO charactrstcs GA-PSO optmzaton mthod whch combns mrts of two algorthms s dvlopd, th prformanc of scroll comprssor s mprovd by usng th optmzd scroll plat, th tst rsults ndcat that th nw optmzaton mthod has th obvous suprorty for scroll plat optmzaton, t also provds th thory and practc bass for th applcaton of ntllgnt algorthm n th projct. 2. GA-PSO Intrnatonal Comprssor Engnrng Confrnc at Purdu, July 17-20, 2006

3 C090, Pag GA GA was dvlopd by amrcan Profssor John H Hoolland, GA s global optmzaton rconnassanc mthod whch smulats crossovr and mutaton of bology, t transforms optmzaton problm from th soluton spac of polcy-makng to th sarch spac of hrdty gn, lmnats nfror gn ndvdual and dvlops th hgh qualty gn ndvdual, gradually nhancs th qualty of ntr populaton, untl th soluton paramtrs rach convrgnc condton and th optmal soluton of polcy-makng varabl s obtand. GA gts rlvant ftnss accordng to optmzaton quston, dos not rqust contnuous ftnss functon tc. so t gts wd applcaton, GA s a procss of sarch charactrstc charactr strng spac, th goal s that gts th charactrstc charactr strng wth hgh ftnss, Th charactrstc of GA s that t works n turn n th cod spac and soluton spac, t carrs on GA opraton n th cod spac for chromosom, apprasal and choc n soluton spac for th soluton, th procss of choc, crossovr and mutaton s uncasng crculatd from an ntal populaton, untl rachs a trmnaton crtron and obtans th optmal soluton. 2.2 PSO PSO s an volutonary computaton tchnqu, PSO smulats th bhavors of brd flockng. In th modl communty's ndvdual has th ablty to control onslf bhavor basd on th crtan ntrnal nformaton and xtror nformaton, t mans that ach ndvdual has a crtan snsaton ablty and t can apprcv ndvdual poston of local bst and glob bst, partcl adjusts ts nxt acton accordng to th currnt condton and obtand nformaton, so th whol communty dsplays a crtan ntllgnc. Whn solvng optmzaton quston ach ndvdual poston s corrspondngly rgardd as a latnt soluton, accordng to th abov rul th glob optmal soluton can b gottn through rpatd adjustng ths latnt solutons. For th nth traton th partcl of PSO changs accordng to th undr two formulas: v = w v x = x + v (1) n+1 n n+1 d d d n + c rand() ( pd xd ) + c2 rand() ( p x n+ 1 n n d d 1 gd d Whr M s th partcl sum; th dth wght of poston vctor for th nth traton partcl ; ) =1,2,, M (2) n v d s th dth wght of flght vlocty vctor for th nth traton partcl ; p d s th dth wght of Pbst for partcl ; n x d s p s th dth wght of Gbst for partcl ; Pbst s th bst of partcl; Gbst s th bst of all partcls; c 1 and c 2 ar larnng factors; rand() s a random numbr btwn (0,1); w s nrta wght functon. Th formula (2) computs nw vlocty of partcl through thr parts: 1Th frst s th sarch ablty; prvous tm vlocty of partcl, t shows th prsnt condton, can balanc glob and local 2Th scond s cognton part, t ndcats th partcl thought, nabls th partcl to hav th nough strong glob sarch ablty and avod local mnmum; 3Th thrd s socty part, t ncarnats nformaton sharng btwn partcls. Undr functon of th thr parts, th partcls adjust poston basd on hstory xprnc and nformaton sharng mchansm, fnally th glob bst soluton can b obtand. 2.3 GA-PSO GA and PSO ar all basd on traton optmzaton tool, th systm s ntalzd wth a populaton of random solutons and sarchs for optma by updatng gnratons. Th gnral charactrs of two algorthms: Two algorthms ar both th hurstc algorthms, stablsh on rsarch foundaton of th complx systm, th low lvl lmnt forms th complx structur through th smpl organzaton n th hgh lvl, thus algorthm dsplays th ntllgnt charactrstc, auto-organzd, auto-adaptd and auto-studd to solv th complx optmzaton problm; Two algorthms ar both basd on th probablty algorthm, ar paralll n ssnc and blong to th communty sarch algorthm; Two algorthms do not rqust non-dffrntabl or othr assstanc knowldg, only nd objctv functon and th ftnss functon of sarch drcton. Th dffrnt charactrstcs of two algorthms: GA nds to ralz from th phnotyp to th gn mag, t s th cod work, ach ndvdual s th chromosom wth charactrstc ntty n fact, aftr ntal gnraton of populaton producs, th bttr approxmat soluton s gd Intrnatonal Comprssor Engnrng Confrnc at Purdu, July 17-20, 2006

4 Intrnatonal Comprssor Engnrng Confrnc at Purdu, July 17-20, 2006 C090, Pag 3 producd through survval of th fttst and supror wn and th nfror wash out prncpl. PSO drctly carrs on procssng on th quston trrtory, ach ndvdual has th corrspondng natur, partcl do not wthr away n th tratv procss; th nformaton transmsson of GA s hddn and complts by crossovr and mutaton. PSO s dsplayd by two fundamntal quatons and s domnant; th fnal soluton of GA s obtand through comptton and th fnal soluton of PSO s obtand through coopraton. Although GA has succssful applcaton n many optmzaton qustons, but t also has som nsuffcncs, GA has many paramtrs to adjust, th local sarch ablty s bad, t has th phnomnon of mmaturly constrngncy and stochastcally roams and so on, thus causs th bad astrngncy and nds long tm to fnd th optmal soluton. Compard wth th gntc algorthm and othr optmzd algorthms, PSO has no voluton oprators such as crossovr and mutaton, n PSO, th potntal solutons, calld partcls, ar "flown" through th problm spac by followng th currnt optmum partcls. th advantags of PSO ar that PSO s asy to mplmnt and thr ar fw paramtrs to adjust. At prsnt PSO has bn succssfully appld n many aras, such as functon optmzaton, artfcal nural ntwork tranng, fuzzy systm control and othr aras whr GA can b appld, but PSO also xsts shortcomng of not hgh constrngncy prcson, s asy to fall nto th local xtrm valu[6][7] Th artcl combns th advantags of GA and PSO, puts forward GA-PSO, th crossovr s mltd nto PSO n mprovd algorthm, whn th nw soluton producs, crossovr s appld to t, producd nw dscndant substtuts for th parnts partcls, t can produc bttr soluton, th dscndant partcls nhrt th parnts partcls mrts through crossovr, t can nhanc th sarch ablty for soluton and rgon btwn partcls n thory, bcaus crossovr can rorganz th xstng solutons, t s vry possbl to dscovr a bttr soluton, smultanously whn PSO fll nto th partal xtrm valu, Partcls may jump out th local xtrm pont through crossovr and vry quckly achv th glob bst poston, th crossovr prncpl s that crossovr partcls ar chosn by a crtan crossovr probablty from all partcls, random two partcls carrs on th crossovr to produc th dscndant partcl, th poston radus vctor of dscndant partcl can b xprssd as follows: whr 1( 1 2 2( x) = p parnt2( x) + (1 p) parnt1 ( x chld x) = p parnt ( x) + (1 p) parnt ( x) (3) chld ) (4) x s th poston vctor of d dmnsons, chld ( x k ) and parnt ( x k ) ( k=1, 2) ar poston of chld partcl and parnts partcl; p s d dmnsons random numbr wght btwn [0, 1]. Th vlocty vctor of dscndant partcls s obtand by th undr formulas: parnt1 ( v) + parnt2 ( v) chld 1( v) = parnt1 ( v) (5) parnt ( v) + parnt ( v) 1 2 parnt1( v) + parnt2( v) chld 2( v) = parnt2( v) (6) parnt ( v) + parnt ( v) 1 2 whr v s th vlocty vctor of d dmnsons, chld ( v k ) and parnt ( v k ) (k =1, 2) ar vlocty of chld partcl and parnts partcl, ach gnraton ndvdual xchangs ts part gns accordng to a crtan probablty and producs th nw gn combnaton, ach soluton has th opportunty to xchang ts outstandng gn, th bttr soluton structur can b obtand, smultanously th nrta wght w uss th lnar functon n tranng, frstly th grat nrta wght s usd, thn th smallr nrta wght s gradually usd, th phnomnon of slow tranng spd n th bgnnng and stochastcally roams n th nd s solvd. Th mthod ovrcoms th orgnal algorthm shortcomng and s appld to scroll plat optmzaton, th satsfactory rsults obtand[8]. 3. OPTIMIZATION DESIGN OF SCROLL PLATE Th scroll comprssor s rgardd as th nw gnraton dsplacmnt comprssor, ts applcaton prospct s mor and mor xtnsv. Th man structur of scroll comprssor shows n Fgur1, t maks up of fxd scroll, orbtng

5 C090, Pag 4 scroll, crank shaft, fram, Oldham rng, man balanc wght and assstant balanc wgh tc. Th fxd scroll and orbtng scroll ar assmbld at a rlatv angl of 180 0, so that thy touch at svral ponts and form a srs of crscnt-shapd pockts. On of th scroll plat s fxd and th othr orbts around th cntr of th fxd scroll wrap. Th orbtng scroll s drvn by a smpl short-throw crank mchansm. Th par of contact ponts btwn th two spral walls ar shftd along th spral curvs. Th rlatv angl of th two scroll plats ar mantand by mans of an ant-rotaton couplng mchansm locatd btwn th back of th orbtng scroll plat and th fram. Th fxd scroll and orbtng scroll ar th most ky parts, ts dsgn and manufactur nfluncs th workng prformanc n a grat xtnt, th sucton-comprss-xhaust cours of scroll comprssor gos on at th sam tm, th whol cours s rpatd contnuously and stadly, so th vbraton and nos of th scroll comprssor ar low compard wth othr comprssors [9]. In ordr to mprov th prformanc of th scroll comprssor, th scroll plats should guarant own hgh prcson and scroll plat optmzaton s vry mportant, n th artcl th mathmatcal modl of scroll plat s stablshd accordng to th optmzaton dsgn, thn GA-PSO s appld to solv th modl accordng to rstrant condton, th optmzd dsgn paramtrs ar gottn, t gud dsgn of scroll comprssor. 3.1 Dsgn Varabl Th basc paramtrs of fxd scroll and orbtng scroll ar th ky factors for comprssor prformanc and th optmzaton man varabls, th basc varabls hav: (1) Ptch scroll: P; (2) Radus of basc crcl: a; (3) Scroll wdth: t ; (4) Scroll hght: H; (5) Scroll turns:n; Th four paramtrs ar bn as dsgn varabls: X = x, x, x, x ] = [ a, t, H, ] (7) [ N Th chang of fv paramtrs affcts prformanc ndx of dlvry, comprsson rato, mchancal loss and ffcncy tc. 3.2 Objctv Functon Th optmzaton objctv functon of scroll comprssor can b dtrmnd accordng to dsgn rqust, such as manufactur cost, ntr machn lf and rlablty, thrmal charactrstc, rato powr, EER or mult- goals and so on. Rato powr s chosn as th objctv functon of scroll comprssor, th rato powr P of scroll comprssor s gvn by: P P S = (8) V Whr P S s shaft powr; V S s actual dlvry. Fgur 1: Structur of scroll comprssor S P = 2πa (9) Intrnatonal Comprssor Engnrng Confrnc at Purdu, July 17-20, 2006

6 C090, Pag 5 t = 2aα (10) r = P / 2 t (11) Whr a s radus of basc crcl; α s orgnaton angl of nvoluts of th crcl; r s basc crcl cntr dstanc of fxd scroll and orbtng scroll. Comprsson ntrnal work W for comprsson 1m 3 gas s xprssd as follows: m m k m m 1 W = ( ) + m2 1 + m2 m ps τ τ pd p ps τ 1 1 (12) m2 1 k 1 m1 1 Whr W s comprsson ntrnal work; τ s nnr prssur rato; m1 s mut-stag comprsson ndx wthout coolng; m 2 s cours mut-stag ndx; p s nnr prssur; p d s dscharg prssur; p s s sucton prssur. (consdrng hat xchang of comprsson gas wth outsd, affxaton loss of dffrnt nnr and outr prssur rato) Th shaft powr of scroll comprssor s : Whr W s s mchancal loss of comprssor; W = W + W + W (13) d W s motor loss. 3.3 Rstrant Condton Th rstrant condtons of scroll comprssor manly nclud ntnsty, rgdty condton, procssng condton, th thrmal prformanc and dynamc prformanc tc. Th boundary rstrant condtons ar stablshd as follows for scroll plat optmzaton dsgn: (1) Bg scroll crcl numbr N wll produc ovr comprsson loss and small N wll produc undr comprsson loss, crcl numbr nfluncs outln sz of scroll plat, accordng xprnc N satsfs rstrcton 2.75 N 4.5; (2) For a fxd dsgn scroll comprssor, comprsson rato ε should guarant fxdnss ε = ε c, th rstrcton of ε s: ε c ε 1.02ε c ; (3) For a fxd dsgn scroll comprssor, dlvry V should guarant fxdnss V = Vc, th rstrcton of V s: Vc V 1.02V c ; (4) In ordr to guarant rgdty of procss cuttr and consdr prcson of scroll plat, th rstrcton s: H 2 P t 6; (5) Th sz of scroll tooth thck t nfluncs ntnson, rgdty, radal lakag, volum, wght of whol machn and sal prformanc n work, so th t satsfs rstrcton: 2.5 t 5.5; (6) Bg scroll hght H wll rduc lakag, but produc bg ovrturn momnt of orbtng scroll, bg frcton loss, s nstablty movmnt and hard manufactur; small H can rduc ovrturn momnt, but add scroll plat sz undr a crtan dlvry, H satsfs rstrcton: H H H max ; mn 3.3 Optmzaton Rsults and Analyss Bcaus ach dsgn varabl has th dffrnt physcs sgnfcanc, th varabl magntud and th varaton rang ar nconsstnt, zro dmnson scal transformaton s appld to varabl, t maks thr chang scop undr th clos magntud, dos not produc th srous rror. Through th scal transformaton t can also mprov th condton of objctv functon n a crtan dgr, th convrgnc rat of optmzd computaton and th valu stablty, th scal transformaton s gvn as follows: ' x = x x =1, 2, 3, 4 (14) / 0 d Intrnatonal Comprssor Engnrng Confrnc at Purdu, July 17-20, 2006

7 C090, Pag 6 Whr x 0 s orgnal valu of dsgn varabl; Scal transformaton of rstrcton condton can gt sam magntud rstrcton functon and satsfy rstrcton condton, GA-PSO s appld to solv objctv varabl X = x, x, x, x ] = [ a, t, H, ] through [ N mathmatcal modl undr rstrcton condton, t maks P rach mnmum, fgur 2 s th algorthm procss,th program of mprovd arthmtc s ralzd by MATLAB,GA, PSO and GA-PAO ar compard through objctv functon, th basc optmzaton paramtrs of scroll comprssor s X = [ a, t, H, N ] = [3, 4, 38, 3.2]; thr algorthms ar appld to optmz th scroll plat mathmatcal modl, ach algorthm runs 80 tms, n th optmzaton procss, GA falls nto local xtrm valu 10 tms, PSO falls nto local xtrm valu 7 tms, GA- PSO falls nto local xtrm valu 1 tm, smultanously two mthods s appld to compar thr algorthms. Th frst knd mthod uss fxd traton tms, tabl 1 s th optmzd rsults of thr knds of algorthms whn th traton tms ar 400; Th scond knd mthod uss fxd objctv functon prcson, whn th objctv functon rachd 5.5, GA nds traton 250 gnratons, PSO nds traton 190 tms and mprovd algorthm nd traton 200 tms, th rsults ndcat that th mprovd algorthm can obtan th qut dal optmal soluton. Th traton tms of mprovd algorthm ar mor than PSO, t s bcaus that th mprovd algorthm usd crossovr opraton, but th mprovd algorthm falls nto th local xtrm valu just 1 tm, smultanously obtand bttr soluton compard wth two algorthms. Th author uss th optmzd rsult to gud th scroll comprssor dsgn, th prototyp s carrd on 400 hours lf tsts, rato powr rducs from 5.7 KW/m 3 mn -1 to KW/m 3 mn -1, th prformanc tst rsults ndcatd that th mprovd algorthm had hgh qualty soluton, good rstrand charactrstc and quck spd for scroll plat optmzaton. Th mprovd mthod solvd th paramtrs optmzaton quston, rducd rato powr, obtand th satsfyng rsults, provdd a nw thought and mthod for th dvlopmnt hgh prformanc scroll ar comprssor [10]. Now w ar ngagd n th mprovd mathmatc modl and bttr algorthm, f th problm can b dally solvd, I thnk that P can b rducd undr 5.2. Tabl 1: Optmzaton rsults (traton tms=400) Dsgn paramtr Intal valu GA PSO GA-PSO a t H N P REFERENCES [1] Fan Lng,Cao Jujang,H W,Png Guoxu. Study of Th Modl of Scroll Comprssor Dmnsond Optmzaton [J]. JOURNALOFNORTHWESTINSTITUTEOFLIGHTINDUSTRY, 1997, 15(4): 1-6. [2] Zhang Lqun, Lu Yongbo. Study of Ar Condton Scroll Comprssor optmzaton [J]. FLUID MACHINERY,2000, 28(1): [3] Chn Jn, Zhang Yongdong, tc. Profl Optmzaton of Scrolls Basd on Multobjctv Gntc Algorthms[J]. CHINESE JOURNAL OF MECHANICAL ENGINEERING, 2005,41(1) : [4] ZHOU Ch, GAO Habng, GAO Lang, ZHANG Wanguo. Partcl Swarm Optmzaton (PSO) Algorthm[J]. APPLICATION RESEARCH OF COMPUTERS, 2003, 12: [5] Cao Chunhong, Zhang Yongjan tc. Th Applcaton of Crossbrdng Partcl Swarm Optmzr n th Engnrng Gomtrc Constrant Solvng [J]. Chns Journal of Scntfc Instrumnt, 2004, 25(4): [6] Ray T, Lw KM. A Swarm Mtaphor for Mult-objctv Dsgn Optmzaton[J]. Engnrng Optmzaton, Intrnatonal Comprssor Engnrng Confrnc at Purdu, July 17-20, 2006

8 C090, Pag , 34(2): [7] Parsopoulos K E, VrahatsM N. Partcl Swarm Optmzaton Mthod n Multobjctv Problms[A]. Procdngs ACM Symposum on Appld Computng [C]. 2002: [8] Collo Collo C A, Puldo G T,LchugaM S. Handlng Multpl Objctvs wth Partcl Swarm Optmzaton[J]. IEEE Transactons on Evolutonary Computaton, 2004, 8 (3): [9] L Lanshng. Scroll comprssor[m]. Bjng: Chna Machn Prss, [10] Ast K. Dutta, Tadash Yanagsawa, Mtsuhro Fukuta. An nvstgaton of th prformanc of a scroll comprssor undr lqud rfrgrant njcton[j]. Intrnatonal Journal of Rfrgraton, 2001, 24(6): ACKNOWLEDGEMENT Ths rsarch s supportd by Natural Scnc foundaton of GANSU provnc (grant No. 3ZS051-A25-036) and Spcalzd Rsarch Fund for th Doctoral Program of Hghr Educaton(grant No ) Start Poston and vlocty of ntalzaton vry partcl Evaluaton ftnss for partcl swarm Gttng Pbst of vry partcl Gttng Gbst of whol partcls Adjustng partcl poston accordng to formula (1) Adjustng partcl vlocty accordng to formula (2) Crossovr and choc Satsfacton paus condton? Y N Export rsults and nd Fgur 2 Algorthm flow chart Intrnatonal Comprssor Engnrng Confrnc at Purdu, July 17-20, 2006