Torque Ripple Reduction of Brushless DC Motor Using Genetic Algorithm

Transcription

1 Poceedngs of the 4 th Intenatonal Confeence of Contol, Dynamc Systems, and Robotcs (CDSR'17) oonto, Canada August 21 23, 217 Pape No. 15 DOI: /cds17.15 oque Rpple Reducton of Bushless DC Moto Usng Genetc Algothm en-ch Chen 1, S. H. Sheh 1, sa-jun Ren 2 1 Depatment of Compute and Communcaton, Kun Shan Unvesty No. 195, Kunda Rd., Yong-Kang Dst., anan Cty, awan 2 Depatment of Infomaton Engneeng, Kun Shan Unvesty No. 195, Kunda Rd., Yong-Kang Dst., anan Cty, awan tchchen@mal.ksu.edu.tw; cyusen@mal.ksu.edu.tw Abstact - Bushless DC moto toque pple educton has been the man ssue n sevo dvng systems n whch the speeds of fluctuaton, vbaton, and acoustc nose should be mnmzed. Most methods fo suppessng toque pples usually eque Foue sees, analyss, fnte element analyss o least-mean-squae mnmzaton. hese methods mght lead to eos dung complex Foue sees analyss and cost much calculaton tme. hs pape pesents a new method to mpove toque pple based on the Genetc Algothm. he poposed method depends on Genetc Algothms to seach fo the Foue coeffcents of thee-phase stato cuents fo the gven back-emf wavefoms. hese Foue coeffcents can then be used to ecompose thee-phase optmum cuent commands fo thee-phase balanced bushless DC moto dvng. he toque pple must theefoe be expected to mpove n ths way f stato cuents ae pefectly acheved. he valdty and pactcal applcatons of the poposed method ae vefed fom expements usng the MS32F2812 DSP. In the expemental stuctue, the thee-phase optmum cuent commands and the measued thee-phase back- EMFs ae set up as the tables. hey ae obtaned accodng to the oto angle and speed nfomaton fom the encode. he expemental esults can pove that the poposed method povdes a smple and effcent way to obtan thee-phase optmum stato cuents fo the gven back-emf wavefoms and the mnmum toque pple wll also be acqued. Keywods: Foue sees analyss, genetc algothm, bushless DC moto, toque pple 1. Intoducton he pemanent magnet synchonous moto (PMSM) o bushless DC moto (BLDC moto) ae popula n many applcatons because they can poduce acceptable amounts of toque pple wthout addng substantal complexty to the moto dvng ccuty [1, 2]. Howeve, the mnmum toque pple s always had to obtan, because the back-emf wavefoms usually have a non-snusodal shape. hat wll cause lage pulsatons o pples n the toque f pue snusodal stato cuent s fed nto them. oque pple can ceate undesable nose and vbaton and contol naccuaces n speed applcatons. Hence, a mnmum amount of toque pple s always equed n many applcatons [3, 4]. Geneally, the toque pple n pemanent magnet motos s poduced by thee physcal phenomena: () Mutual toque s due to the nteactons between the pemanent magnet feld and the stato cuents n the phase wndngs; () Coggng toque s due to the attacton of the pemanent magnet to the salent potons of the stato on; () Reluctance toque s due to wndng self-nductance vaatons that ae a functon of the oto poston. All of these toque pple souces ae geatly affected by the magnetc stuctue of the pemanent magnet moto. Fo example, coggng toque can be geatly educed by skewng the magnets elatve to the stato slottng; by spacng the stato slots o oto magnets by one slot. Mutual toque and eluctance toque ae also functons of the stato cuent chaactestcs. In ths vew the mutual toque chaactestc can be epesented as a poduct of thee-phase back-emfs and the thee-phase stato cuents. Hence, the back-emf shape and stato cuent one wll affect the magntude of the mutual toque pple f the fed stato cuent shape s not sutable fo the back-emf shape. he pupose of ths pape s to exploe the elatonshp between back-emfs and stato cuents n mpovng mutual toque pple [5-7]. he nteacton between the thee-phase back-emf wavefoms and the thee-phase stato cuent wavefoms has been wdely studed and analysed by many authos [5-9]. In most of these studes the elatonshp to the thee-phase stato cuent wavefoms, thee-phase back-emf wavefoms and toque pple ae analysed usng Foue sees and fnte 15-1

2 element analyss. hs pape poposes a new method that adopts the Genetc Algothm [1] to desgn thee-phase optmum cuent wavefoms that poduce the mnmum toque pple fo a non-snusodal back-emf. Geneally, thee-phase stato cuent wavefoms can be wtten as Foue sees expessons. he Genetc Algothm, wdely adopted and studed n vaous felds [11-13], s appled to seach fo the most sutable Foue coeffcents fo thee-phase stato cuents afte achevng the evoluton pocess ncluded n the epoducton, cossove and mutaton opeaton. he seached Foue coeffcents fo the thee-phase stato cuents can be used to compose thee-phase optmum cuent wavefoms whch ae the most sutable fo the gven thee-phase back-emf wavefoms and povde the mnmum toque pple. he dffeence between the poposed method and po woks s that the elatonshps between thee-phase stato cuent wavefoms and thee-phase back-emf wavefoms need not be analysed by the complex Foue sees decomposton, fnte element analyss, and so on. Many assumptons ae not consdeed hee. In pactce, some easons such as manufactung mpefectons, the deteoaton of pemanent magnets, o unbalanced stato wndngs such that the assumptons, whch ae defned n many studes, mght lead to eos. he complex Foue sees decomposton mght esult n eos n the analyss o eque much calculaton tme. heefoe, the poposed method povdes a smple and effectve way to obtan the thee-phase optmum cuent wavefoms and the pue snusodal back-emf wavefom s no longe consdeed. he expemental esults pove that the mnmum toque pple can be successfully acheved usng the poposed method. he DSP-based BLDC moto dve scheme s adopted to acheve a speed contol system wth the optmum cuents. he feasblty of the poposed method can be examned though pesented method. 2. BLDC Moto Modellng Fo a balanced system, the voltage equatons of an nvete-fed thee-phase BLDC moto can be expessed as follows: v v v as bs cs Rs R s R s as bs cs Ls L s L s d dt as bs cs e e e as bs cs (1) whee v as, v bs, v cs ae the appled stato voltages, as, bs, cs ae the appled stato cuents, e as, e bs, e cs ae the back-emf voltage, R s s the stato wndng esstance pe phase, and L s s the stato wndng pe phase self-nductance. he electomagnetc toque can be expessed as ( ) + whee k t s toque constant. he toque, velocty and poston can be expessed espectvely as follows: e = kt aseas + bsebs csecs (2) 2 d 2 = J + B + P dt P e L (3) dt (4) = ( 2/ P) (5) m whee P s the numbe of poles, ω m s the mechancal angula velocty of the oto (ad/s), ω s the electcal oto angula velocty (ad/sec), θ s the oto angula dsplacement. 15-2

3 3. Genetc Algothm Appled to Reduce oque Rpple he goal of the poposed method s to desgn thee-phase optmum cuent wavefoms, whch ae to poduce mnmum pple n the mutual toque. he mutual toque ( e ) s a coss poduct of the stato cuent and back-emf, as shown as: P ( ) e = eaa + ebb + ecc (6) ω whee ω s the oto speed, P s the numbe of poles, e a, e b, e c ae the thee-phase back-emfs, and a, b, c ae the thee-phase stato cuents. Geneally speakng, the thee-phase stato cuent ( a, b, c ) shape can be wtten as the Foue sees expessons, whch ae shown as follows: b c n1 I sn nt (7) a n1 n1 I I n n n sn n t 2 3 (8) sn n t 2 3 (9) whee I n s Foue coeffcent of the stato cuent, and n s the hamonc ode. In ode to obtan the thee-phase optmal cuent wavefoms, the Genetc Algothm s appled to seach fo the Foue coeffcent I n of Eqs. (7), (8) and (9). he thee-phase optmal cuents can be deved takng the acqued Foue coeffcent I n nto Eqs. (7), (8) and (9). he acqued optmal cuent wavefoms must be most suted to the gven back-emf wavefoms. Moeove, the mutual toque pple wll be sue to be educed n opposton Genetc Algothm John Holland s poneeng book povded a geneal famewok fo vewng all adaptve systems (whethe natual o atfcal) and showed how the evoluton pocess can be appled to atfcal systems. Any poblems n adaptaton can geneally be fomulated n genetc tems. Once fomulated n those tems, such poblems can often be solved so-called the Genetc Algothm. he Genetc Algothm s the most effectvely method to seach the optmum soluton. It s a seach ules based on the Dawnsm poposed by Dawnan; theefoe, people also call t popagaton algothm. he Genetc Algothm concept can be appled to seach fo the Foue coeffcents of the stato cuent. en stato cuent hamonc components ae consdeed hee (the stato cuent hamonc components ae fom 2nd to 11th and the fundamental cuent sze s fxed as 1). he Genetc Algothm evoluton pocess ncludes the epoducton, cossove, mutaton and temnaton condtons. he seach pocess wll be ntoduced n detal n the next secton Intal Populaton he ntal paental populatons ae geneated andomly and the geneated amount s equal to ten sets n each stato cuent hamonc component. hat s, the numbe of geneated 3d stato cuent hamonc components wll be equal to ten. he othes ae geneated n the same way. It s vey mpotant to appopately select the populaton sze. If the populaton sze s too small the algothm wll convege too quckly dung the evoluton pocess and the wose solutons mght be acqued, snce the populaton s unable to povde enough nfomaton. On the othe hand, f the populaton sze s so too lage, the algothm wll eque too much tanng tme dung the evoluton pocess. he algothm wll then have a dffcult acqung a bette soluton. heefoe, defnng ten sets of evey stato cuent hamonc component mples that t wll have one hunded data ponts n the evoluton pocess. hese stato cuent hamonc components wll be nceased to twenty sets (two hunded data, whch s the sum of the cossove and mutaton opeaton) afte completng the fst cycle evoluton pocess. 15-3

4 hs advantage of ths method s that the dffeent Foue coeffcents of the stato cuent, whch can obtan bette esults o not, wll be povded n the evoluton pocess. Hence, the ecombnaton phenomenon n the small egon can be avoded. he cuent hamonc components of the snusodal and ectangula cuents ae also ncluded n the ntal populaton. he obtaned stato cuent hamonc components can etan the cuent hamonc components of both ognals afte completng the evoluton pocess. If bette solutons have not yet been found the pogam s temnated. In ths way, the next geneaton can be ensued to not be wose than the last one. he povded data wll also be enough to fnd a bette esult Encodng and Decodng In ode to effectvely seach fo a bette soluton, all stato cuent Foue coeffcents ae encoded nto the bnay stngs. How many bts ae used fo a stng? he defned bt stng length always depends on the desed pecson and seach ange. he pecson of evey bt stng s expessed as 1 bts n ths pape, and the desed seach ange s defned between -2 and 2. he seach ange mples the stato cuent Foue coeffcent sze. he stato cuent Foue coeffcents (A k ) can be encoded as bnay stngs usng the followng encode equaton. A b - a = a + L gk k= 1, 2,... (1) 2-1 k 3 whee b s the maxmum seach ange and a s the mnmum one, L s the bt stng length and g k s the encoded bt stng, k s the numbe of bt stngs. When g k s equal to b, the maxmum stato cuent Foue coeffcent (A k ) wll be equal to 1. On the othe hand, when g k s equal to b, the mnmum one (A k ) wll be equal to -1. If the A k ae substtuted nto Eq. (3.3), the followng expesson can be obtaned. a = I n n=1 sn( nωt) = I sn( ωt) + I sn( 2ωt) I sn( 11ωt ) 1 2 A1 A2 A11 = sn( ωt) + sn( 2ωt) sn( 11ωt ) whee A 1 s fxed as 1, the othes wll be seached though the Genetc Algothm. Afte completng the evoluton pocess the seached bt stngs, whch ae expessed as the bnay code, must be decoded to the decmal value agan Ftness Afte completng one cycle of the evoluton pocess the next geneatons must be checked fo ftness to detemne f the next geneaton should be kept o gven up. he mutual toque pple s defned by Eq. (11) as the evoluton pocess ftness. Dung the evoluton pocess f the ftness becomes smalle mutual toque pple wll be acqued and the composed cuent wavefom becomes moe sutable fo the gven back-emf. 11 M k= 1 (k) - oque Rpple = M 1% e avg avg (11) whee aveage mutual toque ( avg ) s equal to M k= 1 (k) / M, M s the sample ponts. e 15-4

5 3.5. Repoducton he epoducton opeaton selects two sets of supeo stato cuent Foue coeffcents as paents fom the ognal populatons to poduce the next geneaton. he epoduced pobablty s popotoned to the ftness. hs mples that the stato cuent Foue coeffcents, whch povde the lowe ftness, have hghe epoduced pobablty n the evoluton pocess. Dependng on ths method, moe supeo stato cuent Foue coeffcents wll be acqued and the geneated mutual toque pple wll also be smalle. hs s what s called Dawnan natual selecton and suvval Cossove he cossove opeaton task s swappng paental bts untl the next geneaton populaton s equal to that of the ognal. In ths way a moe sutable geneaton mght be acqued dung the evoluton pocess. he cossove opeaton s conducted by andomly selectng the cossove ponts fom epoduced bt stngs. Accodng to the cossove ponts the two bt stngs ae sepaated nto many peces and swapped wth the cossove ponts n these peces. Afte ths opeaton two new bt stngs, dffeent fom the ognal ones wll be obtaned. he cossove ponts must be appopately defned; othewse the ognal bt stng chaactestcs would be destoyed. hee cossove ponts ae used n the evoluton pocess. he foegong method s epeated untl the new geneaton populaton s equal to the ognal populaton Mutaton Afte the cossove opeaton, some bts n the bt stngs ae abtaly changed to poduce a new geneaton wth mpoved chaactestcs. hs method s the so-called mutaton opeaton. he man pupose of mutaton s to avod the soluton fom fallng nto local optma (that s seachng fo a bette soluton wthn a small ange) dung the evoluton pocess. he mutaton opeaton nvolves andomly selectng bts fom the bt stngs as mutaton bts and abtaly changng the state of the mutaton bts. hs sgnfes that the ognal state of mutaton bt s 1. Afte the mutaton opeaton t wll become. On the othe hand f the ognal mutaton bt s, t wll become 1. he numbe of mutaton bts must be appopately defned. If too many mutaton bts ae defned, the system wll seach ove a vey wdely space and obtanng the optmum soluton wll become hade. On the othe hand, f the mutaton bt s defned too naow, t wll become meanngless. hee mutaton bts ae defned n the evoluton pocess emnaton Condton Stoppng the genetc algothm s detemned by settng the temnaton condton. One hunded geneatons s defned as the temnaton condton n the evoluton pocess. hs means that the evoluton pocess wll be stopped when 1 executons has been eached. Accodng to the temnaton settng the optmum stato cuent Foue coeffcents wll be obtaned usng the Genetc Algothm epoducton, cossove and mutaton opeatons. A mnmum mutual toque pple can be acqued fom the thee-phase optmum cuent wavefoms, composed of the seached stato cuent Foue coeffcents and then multpled by the gven back-emf wavefom. 4. Expement Results he expemental esults nclude the snusodal cuent and optmum cuent fo the speed contol scheme when a speed command s 2 pm and 1 pm, espectvely. he speed contol scheme s shown n Fg. 1. he back-emf data ae measued befoehand and set up as tables n the MS32F2812 DSP ntenal memoy. hs pape assumed that the back-emf magntude s lnealy popotonal to the speed, hence, the back-emf facto can be expessed as 6.5 /1 ( s speed command). he thee-phase optmum cuent commands and the thee-phase back-emf wavefoms ae acqued fom the thee-phase back-emf tables accodng to the oto poston. he speed nfomaton s acqued fom the shaft encode. he measued thee-phase stato cuents ae compaed wth the thee-phase cuent commands though the adaptve cuent contolle to detemne the tun-on tme fo the sx PWM swtchng sgnals wthn the pedetemned bandwdth. he sx PWM swtchng sgnals wll then be delveed to the nvete. he powe tanssto swtchng state wll be contolled by these PWM sgnals such that the stato cuents successfully tack the desed cuent commands. 15-5

6 DC 24 V BLDC Moto PI contolle Optmal thee-phase cuent table as as bs bs cs cs Locked-out ccut Locked-out ccut Locked-out ccut P/2 Implemented by MS32F2812 DSP Fg. 1: he speed contol scheme. wo cases bellow wll explan the expemental esults. Case I: Speed Command of 2 pm he steady-state stato cuent, mutual toque tansent esponse and speed esponse ae shown n Fgues 2(a)-2(c), espectvely, when the speed command s 2 pm and the snusodal cuent command s fed. he steady-state stato cuent, mutual toque tansent esponse and speed esponse ae shown n Fgues 3(a)-3(c), espectvely, when the speed command s 2 pm and the optmum cuent command s appled. Accodng to Fgues 2 and 3 the mutual toque pple fo the optmum cuent scheme s obvously smalle than the snusodal cuent scheme fo a low speed command of 2 pm. he mutual toque pple fo the snusodal cuent scheme and optmum cuent scheme ae 13.61% and 8.76%, espectvely. he speed esponse fo the optmum cuent scheme s also faste than the snusodal cuent scheme. Case II: Speed Command of 1 pm he steady-state stato cuent, mutual toque tansent esponse and speed esponse ae shown n Fgues 4(a)-4(c), espectvely, when the speed command s 1 pm and the snusodal cuent command s fed. he steady-state stato cuent, mutual toque tansent esponse and speed esponse ae shown n Fgues 5(a)-5(c), espectvely, when the speed command s 1 pm and the optmum cuent command s appled. Fom Fgues 4 and 5 the mutual toque pple fo the optmum cuent scheme s obvously smalle than the snusodal cuent scheme fo a hgh-speed command of 1 pm. he mutual toque pple fo the snusodal cuent scheme and optmum cuent scheme s 22.87% and 12.28%, espectvely. he speed esponse fo the optmum cuent scheme s also faste than the snusodal cuent scheme. (a) (b) (c) Fg. 2: he expement wth snusodal cuent command and the speed command of 2 pm. (a) he steady-state stato cuent, (b) the mutual toque tansent esponse, (c) the speed esponse. 15-6

7 (a) (b) (c) Fg. 3: he expement wth optmum cuent command and the speed command of 2 pm. (a) he steady-state stato cuent, (b) the mutual toque tansent esponse, (c) the speed esponse. (a) (b) (c) Fg. 4: he expement wth snusodal cuent command and the speed command of 1 pm. (a) he steady-state stato cuent, (b) the mutual toque tansent esponse, (c) the speed esponse. (a) (b) (c) Fg. 5: he expement wth optmum cuent command and the speed command of 1 pm. (a) he steady-state stato cuent, (b) the mutual toque tansent esponse, (c) the speed esponse. 5. Concluson hs eseach developed a new method fo desgnng thee-phase optmum stato cuents fo mpovng the mutual toque pple n a thee-phase balanced, 6-pole bushless DC moto. Fo any gven back-emf shape, the optmal stato cuent Foue coeffcents can be apdly found usng the Genetc Algothm to compose the thee-phase optmum cuents. he optmum cuents ae most sutable fo the gven back-emfs and mnmum mutual toque pple wll be acqued when these cuents ae mplemented. hese expements wee caed out usng MS32F2812 DSP and the mutual toque pple was evaluated accodng to the thee-phase back-emfs and measued thee-phase stato cuents poduct fom the constucted tables accodng to the oto angle and speed nfomaton. Fom expemental esults, the stato cuents tack the desed cuent commands satsfactoly. he speed esponse s also faste than the othes whch feed the snusodal stato cuents dung low o hgh speeds. he mpoved mutual toque pples ae expected to become smalle than the othes by compang the mutual toque pples obtaned by feedng the snusodal and optmum cuent commands, espectvely. he advantage of the poposed method s that t can be appled to any back-emfs wthout the complex Foue sees decomposton o othe dffcult analyss. Moeove, f the thee-phase 15-7

8 back-emfs have slghtly dffeent shapes fom each othe, the optmum cuents ae also found usng the poposed method. hs poves that the poposed method s a smple and effectve way to mpove the mutual toque pple n the bushless DC moto fo any gven back-emf shape. Acknowledgements he authos would lke to expess the appecaton to Mnsty of Scence and echnology fo ts suppot unde contact MOS E Refeences [1] D. C. Hanselman, Bushless Pemanent-Magnet Moto Desgn. McGaw-Hll Intenatonal Edtons, [2] B. K. Bose, Moden Powe Electoncs and AC Dves. Pentce Hall PR, 22. [3] G. Ranjthkuma and K. N. V. Pasad, Mnmzaton of oque Rpple Content fo BLDC Moto by Cuent Contolle usng MLI, Poceda Engneeng, vol. 38, pp , 212. [4] H. M. Hasanen, oque pple mnmzaton of pemanent magnet synchonous moto usng dgtal obseve contolle, Enegy Conveson and Management, vol. 51, no. 1, pp , 21. [5] W. A. Salah, D. Ishak and K. J. Hammad, PWM Swtchng Stategy fo oque Rpple Mnmzaton n BLDC Moto, Jounal of Electcal Engneeng, vol. 62, no. 3, pp. 1-6, 211. [6] S. J. Pak and H. W. Pak, A New Appoach fo Mnmum-oque-Rpple Maxmum-Effcency Contol of BLDC Moto, IEEE ans. on Industal Electoncs, vol. 47, no. 1, pp , 2. [7] W. Handn, R. Setabudy and R. Gunaw, Mnmzaton of oque Rpple n 24-Slot 16-Pole Inset Pemanent Magnet Geneato by Edgeounded Magnet Poles and Stato eeth Notch echnques, Jounal of heoetcal and Appled Infomaton echnology, vol. 93. no.1, pp. 1-16, 216. [8] B. Jaganathan, R. Bndha, C. Anuadha and S. Gunasekaan, A Smple Novel Method of oque Rpple Mnmzaton n Fuel Cell Based PMSM Dves, ACEEE Intenatonal Jounal on Electcal and Powe Engneeng, vol. 1, no. 2, pp , 21. [9] J. Y. Hung and Z. Dng, Desgn of Cuent to Reduce oque Rpple n Bushless Pemanent Magnet Motos, IEE Poceedngs Electonc, vol. 14, no. 4, pp , [1] J. R. Koza, Genetc Pogammng: on the Pogammng of Computes by Means of Natual Selecton. A Bodfod Book, he MI Pess Cambdge, Massachusetts London, England, [11] D. S. Wele and E. Mchelssen, he Contol of Adaptve Antenna Aays wth Genetc Algothms Usng Domnance and Dplody, IEEE ans. on Antennas and Popagaton, vol. 49, no. 1, pp , 21. [12] J. A. Vasconcelos, J. A. Ramez, R.H.C. akahash and R.R Saldanha, Impovements n Genetc Algothms, IEEE ans. on Industal Electoncs, vol. 37, no. 5, pp , 21. [13] M. A. S. Masoum, M. Ladjevad, A. Jafaan; E. F. Fuchs, Optmal Placement, Replacement and Szng of Capacto Banks n Dstoted Dstbuton Netwoks by Genetc Algothms, IEEE ans. on Powe Delvey, vol. 19, no. 4, pp ,