Parameter Identification of a PMSG Using a PSO Algorithm Based on Experimental Tests

Transcription

1 Vol.6 No.1, 1 مجلد 6, العدد 1, 1 Proc. 1st Intrnational Conf. Enrgy, Powr and Control Basrah Univrsity, Basrah, Iraq Nov. to Dc. 1 Paramtr Idntification of a PMSG Using a PSO Algorithm Basd on Exprimntal Tsts A.J.Mahdi,W.H.TangMmbr, IEEE, Q. H. Wu Snior Mmbr, IEEE Abstract An accurat modl for a prmannt magnt synchronous gnrator (PMSG) is important for th dsign of a high-prformanc PMSG control systm. Th prformanc of such control systms is influncd by PMSG paramtr variations undr ral opration conditions. In this papr, th lctrical paramtrs of a PMSG (th phas rsistanc, th phas inductanc and th rotor prmannt magnt (PM) flux linkag) ar idntifid by a particl swarm optimisation (PSO) algorithm basd on xprimntal tsts. Th advantags of adopting th PSO algorithm in this rsarch includ asy implmntation, a high computational fficincy and stabl convrgnc charactristics. For PMSG paramtr idntification, th normalisd root man squar rror (NRMSE) btwn th masurd and simulatd data is calculatd and minimisd using PSO. Indx Trms Paramtr idntification, prmannt magnt synchronous gnrator (PMSG), particl swarm optimisation (PSO), modlling, dspace. I. INTRODUCTION Rcntly, PMSGs hav bn widly usd in variabl-spd powr gnration systms spcially in wind turbin gnrator (WTG) systms [1]. A PMSG has a numbr of attractiv faturs, such as a high fficincy, a high torqu to inrtia ratio, a high torqu to volum ratio, a high air-gap flux dnsity, a high powr factor, high acclration and dclration rats, lowr maintnanc cost and a compact structur []. Gnrally, th prformanc of a PMSG control systm is dpndnt on th good knowldg of gnrator paramtrs that vary with th tmpraturs and th frquncis []. Th accuracy of a PMSG modl is dtrmind by its paramtrs, which ar th phas rsistanc, th phas inductanc and th PM flux linkag. Ths paramtrs should b idntifid as accurat as possibl. Thr ar many analytical mthods which hav bn proposd for calculating paramtrs of lctrical machins [], []. Most of ths mthods ar basd on physical spcifications of a PMSG such as mchanical dimnsions, stator winding spcifications and rotor PM charactristics. As most of ths information is unknown, ths analytical mthods ar not fasibl for calculating optimisd paramtrs. On th othr hand, xprimntal mthods for paramtr idntification, hav also bn dvlopd such as a dc currnt dcay tst [6], a no-load tst, a blockd-rotor tst and a load tst [7]. In th dc currnt tst, th stady-stat paramtrs can b idntifid without considring transint oprating conditions. A. J. Mahdi, W. H. Tang and Q. H. Wu ar with th Dpartmnt of Elctrical Enginring and Elctronics, Th Univrsity of Livrpool, Livrpool, L69 GJ, U.K. Corrsponding author is W. H. Tang, Tl: , Fax: , s: ali.mahdi@liv.ac.uk, whtang@liv.ac.uk and qhwu@liv.ac.uk. Implmnting th blockd-rotor tst givs inaccurat paramtrs, bcaus th voltags and currnts ar not masurd at normal oprating conditions. In th load tst, accurat paramtrs may b idntifid but this tst rquirs complx xprimntal procdurs. In [8], th lctrical paramtrs of a prmannt magnt synchronous motor (PMSM) ar idntifid using an adaptiv idntification algorithm, which is basd on output idntification rrors. Th lctrical paramtrs (th phas rsistanc, th phas inductanc and th iron loss rsistanc) wr idntifid in stady-stat oprating conditions, whil th mchanical paramtrs (th lctromotiv forc constant, th inrtia and th viscous friction) wr idntifid xprimntally in dynamic conditions. Bcaus th rotor of PMSGs includs PMs and thr ar no opn-circuit and short-circuit stats, th influnc of PM charactristics must b considrd in calculations of PMSG paramtrs. As a rsult, th mthods basd on xprimntal tsts ar also not accurat nough idntify to paramtrs of a machin [9]. A PSO is a stochastic optimisation mthod, which is basd on th principls of natural biological volution [1]. It has bn usd for optimising paramtrs of lctrical machins and tuning PI controllrs [11]. In [1], a nw approach basd on PSO is dmonstratd for paramtr idntification with a nonlinar PMSM modl. Th motor stator rsistanc and th disturbd load torqu ar idntifid for variabl-frquncy driv systm applications. In [1], th paramtrs of a PID controllr ar optimally tund by an adaptiv particl swarm optimisation (APSO). Although PSO is a robust optimisation algorithm, it gnrally taks a long tim to optimis paramtrs of a systm bcaus of its slow convrgnc spd [1]. In this papr, th bounds of paramtrs ar minimisd using rsults of xprimntal tsts. Th aim of this procdurs of th PSO algorithm is to incras th accuracy of paramtr idntification, rduc th sarch spac of paramtrs and dcras th convrgnc tim (th tim that algorithm spnds to find th global optimal solution) of th PSO algorithm. This papr is organisd as follows: a PMSG modl is rprsntd in Sction II. Paramtr masurmnts of a PMSG ar givn in Sction III. Sction IV illustrats th basics of a PSO algorithm. Th rsults and discussion ar dmonstratd in Sction V. Finally in Sction VI, a conclusion is givn. II. PMSG MODEL Th opration principls of PMSGs ar similar to convntional synchronous gnrators xcpt that an xcitation winding is rplacd by prmannt magnts [1]. Figur 1 9

2 Vol.6 No.1, 1 مجلد 6, العدد 1, 1 of a PMSG in th αβ-axis can b rprsntd as [18]: Stator PM rotor v α = R s i α + dλ α dt, (1) v β = R s i β + dλ β dt, () whr v α and v β ar th α-axis and β-axis stator voltags; i α and i β ar th α-axis and β-axis stator currnts; R s is th phas rsistanc of th stator winding; λ α and λ β ar th stator magntic fluxs which can b xprssd (Fig. ) as follows. Fig. 1. Th stator and th PM rotor of a PMSG. q-axis -axis S d-axis -axis Fig.. Phasor diagram of th PM rotor flux linkag and back-mf in th stationary rfrnc fram. shows th stator and th PM rotor of th PMSG usd in this study. Figur shows th phasor diagram of a PMSG in th stator orthogonal coordinat systm (αβ-axis). As can b sn from Fig., λ is th magnitud of th rotor PM flux linkag; λ α and λ β ar th α-axis and β-axis rotor PM flux linkags; α and β ar th α-axis and β-axis back-mfs; ω and θ ar th lctrical angular rotor spd and position rspctivly; and λ ω is th magnitud of th back-mf. Thr ar two typs of PMSGs which ar classifid according to th location of PMs on th rotor, i.. th surfac-mountd prmannt magnt synchronous gnrators (SPMSGs) and th intrior prmannt magnt synchronous gnrators (IPMSGs). Th typ of th PMSG usd in this study is a SPMSG. Th main diffrnc btwn a SPMSG and an IPMSG is that th latr has a salincy on th rotor and th d-axis inductanc is largr than th q-axis inductanc. This diffrnc btwn th d-axis and q-axis inductancs is du to ithr th asymmtric structur of th PMSG or th flux inducd magntic saturation du to PM [16]. Bcaus of th complxity modlling of th nonlinarity bhavior of a PMSG which coms from th magntic saturation of th iron cor, a simplifid PMSG modl is sufficintly considrd for most control systms [17]. In this rsarch, a PMSG is accuratly modlld using a two-axis rprsntation undr th assumption that th saturation of th iron cor and magntic losss ar nglctd. Morovr, symmtrically - phas sinusoidal currnts ar considrd and th rotor dos not contain dampr windings. Th instantanous voltag quations λ α = L α (θ )i α + λ cosθ, () λ β = L β (θ )i β + λ sinθ, () whr L α and L β ar th inductancs in th αβ-axis. In a SPMSG, ths inductancs ar qual to th slf phas inductanc, L s, of a stator winding and indpndnt to rotor positions [17]. Substituting λ α and λ β of (), and (), into (1), and (), rspctivly, rarranging and solving th latr two quations, th stimatd α-axis and β-axis stator currnts (i α,st and i β,st ) ar obtaind as follows. d dt i α,st = 1 (v α R s i α,st α,st ), L s () d dt i β,st = 1 (v β R s i β,st β,st ). L s (6) From Fig., It is sn that th stimatd α-axis and β-axis back-mfs ( α,st and β,st ) dpnd on th rotor position angl which can b masurd by an ncodr. In practic, an ncodr cannot masur an accurat initial position angl, which may lad to wrong calculations of α,st and β,st [19]. In this rsarch, an stimator basd on an adaptiv PI controllr is dvlopd for th stimation of α,st and β,st as follows. t α,st = k p,mf (i α,st i α,m )+k i,mf (i α,st i α,m )dτ (7) t β,st = k p,mf (i β,st i β,m )+k i,mf (i β,st i β,m )dτ, (8) whr k p,mf and k i,mf ar th proportional and intgral factors of th back-mf stimator; i α,m and i β,m ar th masurd α-axis and β-axis stator currnts. Howvr, th proposd back-mf stimator is vry asy to implmnt and includs a fw paramtrs to b tund using a PSO algorithm. III. PMSG PARAMETER MEASUREMENTS In this sction, th particular xprimntal tsts for calculating th PMSG paramtrs ar brifly dscribd. Th masurd paramtr valus can b utilisd for choosing th paramtr boundaris of th proposd PSO algorithm. In this rsarch, th PM flux linkag is calculatd by a no-load tst. Whil, th phas rsistanc and inductanc ar calculatd by a load tst. In th proposd load tst, th only stator winding of a PMSG is connctd to a variabl-voltag rotary rgavolt transformr. Th advantag of using a load tst is its asy implmntation compard with a short-circuit tst.

3 Vol.6 No.1, 1 مجلد 6, العدد 1, 1 A. PM Flux Linkag In a no-load tst, th PMSG is drivn by a -phas induction motor at various spds and th stator winding is opncircuitd. Th trminal -phas voltags and th rotor spd ar masurd. Th back-mf is considrd as th opn-circuit RMS stator phas voltag, v s. Th RMS PM flux linkag, λ, is givn as []: λ = v ab ω, (9) whr v ab is th opn-circuit lin-to-lin maximum voltag. Figur illustrats th masurd PM flux linkag valus undr frquncy variations. B. Phas Rsistanc and Phas Inductanc Thr is a significant variation in th phas rsistanc of a stator winding ovr an oprating tmpratur rang. Masuring of this variation in rsistanc givs an stimat th tmpratur ris of a stator winding. For a 1 C rsolution in th oprating tmpratur rang, th rsistanc must b stimatd to about mω accuracy [1]. As th stator winding of th PMSG usd in this study is connctd in a star connction and thr is no nutral point supplid, th masurd inductanc is th inductanc btwn phas A and phas B which is givn as: L lin = L sa + L sb L mab L mba, (1) whr L sa and L sb ar th slf inductancs of phas A and phas B rspctivly and L mab and L mba ar th mutual inductanc btwn phas A and phas B rspctivly. Th slf and mutual inductancs can b xprssd as: L sa = L ls + L o L ms cos(θ ), (11) L mab = L ba = 1 L o L ms cos(θ π ), (1) whr L ls is th lakag inductanc of th stator winding du to its lakag flux, L o is th avrag inductanc du to th air-gap flux and L ms is th salincy inductanc du to th lctrical rotor position θ. L o and L ms ar givn as: Rotor PM Flux Linkag (pu) Hz 9 Hz 1 Hz. 1 Fig.. Rsistanc and Ractanc (pu) Exprimntal PM flux linkag valus in a no-load tst.. R s X s 1 1 Phas Currnt (A) Fig.. Exprimntal stator phas rsistanc and ractanc in a load tst. wid rang of currnt variations. Th ida of masuring phas rsistanc and ractanc is basd on calculating th magnitud of th phas impdanc and th phas-shift btwn th phas voltag and currnt. It can b sn from Fig. that th phas rsistanc and ractanc of th stator winding incras with currnt incrasing du to th winding tmpratur changs and magntic saturation. L o = 1 (L d + L q ), (1) L ms = 1 (L d L q ). (1) In addition, L sa and L mab dpnd on rotor positions, so that th d-axis should b alignd with th a-axis in ordr to mak θ qual to zro and consquntly th slf inductanc bcoms indpndnt to rotor positions. This condition is only important for a salint rotor machin, IPMSG. In this rsarch, th proposd PMSG is a non-salint typ, SPMSG, thrfor it is not rquird to align th d-axis with th a-axis. As mntiond abov that in a SPMSG, th d-axis inductanc quals to th q-axis inductanc, this considration maks L ms quals to zro. If L ls is nglctd, thn L sa is qual to L o. In practic, L sa can b masurd by svral mthods such as a short-circuit tst, a load tst and a blockd-rotor tst. In this rsarch, th phas stator rsistanc and ractanc ar calculatd and dmonstratd (Fig. ) using a load tst undr a IV. PSO ALGORITHM A PSO algorithm is an volutionary computation mthod inspird by social bhaviors of bird flocking during sarching food. Each bird may b calld a particl in a population, that is a swarm moving ovr a sarch spac to achiv an objctiv. In a PSO algorithm, th position of an particl illustrats th solution of an optimisation problm. Each particl movs in th sarch spac with a vlocity according to th prvious optimum individual solution and th prvious optimum global solution []. It uss a population of N particls, which is th dimnsion of th sarch spac. Th stat of th ith particl is rprsntd as x i (t) =[x i1 (t),x i (t),..., x in (t)]. Th prvious bst stat, pbst, is writtn as p i (t) = [p i1 (t),p i (t),..., p in (t)]. Th indx of th bst stat in th global st, gbst, is rprsntd as p g (t) =[p g1 (t),p g (t),..., p gn (t)]. Th moving vlocity, v i (t), is rprsntd as v i (t) =[v i1 (t),v i (t),..., v in (t)]. A PSO algorithm can b implmntd using (1), and (16), 1

4 Vol.6 No.1, 1 مجلد 6, العدد 1, 1 [1]. v i (k +1) = ωv i (k)+c 1 rand 1 (p i (k) x i (k)) +c rand (p g (k) x i (k)), (1) x i (k +1)=x i (k)+v i (k +1), (16) whr k is an itration numbr, ω is an inrtia wight factor, c 1 and c ar constants which rprsnt th control paramtrs of th PSO algorithm. If c 1 and c ar slctd as small valus, th individual may mov far from th objctiv rgions bfor bing tuggd back. But if c 1 and c ar slctd as larg valus, th individual may mov in suddn towards objctiv rgions. Gnrally, c 1 and c ar slctd around. Th paramtrs rand 1 and rand ar random valus, which ar uniformly distributd random numbrs in [, 1]. It is important to know that ths valus ar randomly gnratd and thy may chang during ach itration []. As mntiond prviously, th purpos of using th proposd PSO algorithm is to sarch th optimal paramtrs of th PMSG modl by minimising th normalizd root man squar rrors (NRMSEs) of th currnt, i, back-mf, f,androtor spd, ω. Figur illustrats how th PSO algorithm is usd with th PMSG modl to idntify paramtrs of a ral PMSG by masuring th -phas voltags, v abc, and - phas currnts, i abc. Th stimatd α-axis and β-axis currnts, i αβ,st, th stimatd α-axis and β-axis back-mfs, αβ,st, and th stimatd rotor spd, ω m,st, ar compard with th masurd α-axis and β-axis currnts, i αβ,m, th masurd α-axis and β-axis back-mfs, αβ,m, and th masurd rotor spd, ω m,m, rspctivly to calculat NRMSEs. Ths NRMSEs ar thn minimisd by th proposd PSO algorithm to driv th optimal paramtrs of th PMSG modl. In this rsarch, th rotor spd ω m and position θ ar masurd for monitoring purpos. Thus, for ach particl of a population in PSO, its total fitnss valu is givn as: F = min( i + f + ω ), (17) A NRMSE can b calculatd (in prcnt) according to: = 1 n (A i E i ) n 1 M, (18) i=1 whr A i and E i ar th actual and stimatd valus for th PMSG stat variabls rspctivly, n is th numbr of th sampls involvd for PSO implmntation and M is th man of th stimatd valu. In th proposd PSO algorithm, ach particl contains two paramtrs which ar k p,mf and k i,mf. On th othr hand, th sarch spac of th PSO has two dimnsions and th particls must mov in a thr dimnsional spac. As mntiond bfor that th objctiv of th proposd PSO is to minimis f, i and ω, thus for ach itration, it is rquird to calculat ths rrors from th Simulink modl of th PMSG and updat thir valus insid th PSO algorithm. Finally, a maximum numbr of itrations ar gnrally prdfind in ordr to trminat th PSO algorithm. In this work, th PSO algorithm is trminatd whn th fitnss valu of solutions hav small changs aftr m PMSG v abc i abc abc to Transformation v i Minimising, and using a PSO Algorithm i -ph IM m Currnt Estimator m,st,st AC Drivr i st i Error Calculation Rotor Spd and Position Estimator Idntifid paramtrs k and k PI Controllr a st Fig.. Schm of th PMSG paramtrs idntification using a PSO algorithm. 1 Fig. 6. Parts of th dvlopd tst bnch: (1) a variabl-spd ac drivr, () a -phas induction motor, () a PMSG, () an ncodr, () voltag and currnt snsor boards, (6) a dspace controllr. multipl itrations in th sarching procss. On th othr hand, th trmination is tru whn F i (x) F i 1 (x) <ε,whrε is a prdfind rror limit. V. RESULTS AND DISCUSSION Th xprimntal tsts hav bn implmntd with an xprimntal PMSG (typ GL-PMG-A). Th tst bnch is shown in Fig. 6. Th proposd tst bnch consists of a PMSG which is coupld to a -phas induction motor (IM) that is controlld by a variabl-spd ac drivr. Th masurmnts ar sampld by a dspace DS11 controllr. Th tst bnch is quippd with voltag transducrs (typ LV-p), currnt transducrs (typ LA-p) and an ncodr to masur th rotor spd and position. It is worth noting that all th componnts usd in th proposd tst bnch ar commrcially availabl. Th paramtrs of a PMSG modl (which includs a backmf stimator) ar optimisd (using th proposd PSO) and listd in Tabl II. It is apparnt that, ths paramtrs vary with oprating conditions du to th variation of th masurd physical paramtrs. Tabl I prsnts th comparison rsults of th prcntag NRMSEs undr thr tst cass. In cas 1, th back-mf is calculatd using th masurd rotor spd and position. In cass and, th back-mf is stimatd from a back-mf 6

5 Vol.6 No.1, 1 مجلد 6, العدد 1, 1 TABLE I PERCENTAGE NRMSES FOR A COMPARISON UNDER ROTOR SPEED VARIATIONS Cas 1 Cas Cas Using Masurd Paramtrs Using Fixd Paramtrs Using Optimisd Paramtrs Rotor Spd, (r) i (%) f (%) ω(%) i (%) f (%) ω(%) i (%) f (%) ω(%) Low, Ratd, High, Currnt (A) i α,m i β,m i α,st i β,st Currnt (A) 6 1 Exprimntal Estimatd Fig. 7. Exprimntal and stimatd α and β currnts in th transint. Fig. 8. Exprimntal and stimatd phas currnts. stimator. Th main diffrnc btwn cas and cas is that in cas, th PI paramtrs of a back-mf stimator ar considrd fixd valus, whil in cas, th PI paramtrs ar optimally varid with oprating conditions. It is sn that, i of cas 1 ar vry high (spcially at a low spd) bcaus of nglcting th influnc of th initial rotor position. In cass and, th NRMSEs ar significantly rducd bcaus of considring th influnc of th initial rotor position in th stimatd back-mf. It is clar that th maximal NRMSE in cas 1 is 8.8%. Whil in cass and, th maximal NRMSEs ar rducd to 7.11% and 6.% rspctivly. Ths rsults show th ffctivnss of th proposd PSO algorithm for paramtr idntification of th PMSG compard with th numrical mthods that mntiond prviously. Figur 7 shows a comparison btwn th αβ-axis masurd and th stimatd phas currnts in th transint using th optimisd paramtrs. Figur 8 illustrats th magnitud of th masurd and stimatd phas currnt. It is sn that, th stimatd currnts using th accurat PMSG modl (with using th optimisd paramtrs) is vry clos to th masurd currnts. Figur 9 shows th magnitud of th masurd and stimatd back-mfs at th ratd spd. It is sn that th transint rspons of a back-mf stimator has an ffctiv tim-constant of about 1. s. To furthr validat th PMSG modl, a comparison btwn th masurd rotor spd and position with th stimatd rotor spd and position ar dmonstratd in Figurs 1 and 11. It is sn that th stimatd rotor spd is vry clos to th masurd rotor spd spcially during stady-stat opration. Finally, it is worth to not that in this papr, th prformanc Back-mf (V) 1 1 Fig. 9. Exprimntal Estimatd Exprimntal and stimatd back-mfs. TABLE II OPTIMAL PARAMETER VALUES OF THE PMSG MODEL USING THE PROPOSED PSO ALGORITHM Rotor Spd, (r) R s (pu) X s (pu) k p,mf k i,mf Low, Ratd, High, Fixd paramtr of a PMSG modl is improvd by minimising th NRMSE (which rprsnts th ovrall rror prformanc) whil in [1], th stady-stat rror has bn considrd for masuring th

6 Vol.6 No.1, 1 مجلد 6, العدد 1, 1 Rotor Spd (rad/s) 1 Exprimntal Estimatd Fig. 1. Rotor Position (rad) 6 - Exprimntal and stimatd rotor spd. - Exprimntal Estimatd Fig. 11. Exprimntal and stimatd lctrical rotor position. prformanc of a PSO-basd paramtr idntifir. In addition, in [1] th maximum stady-stat spd rror is around 1.8 rad/s whil in this work, th corrsponding rror is minimisd to. rad/s (at ratd rotor spd) by using th optimisd paramtrs. VI. CONCLUSION A PSO algorithm basd paramtr idntification of a PMSG modl has bn implmntd and tstd with xprimntal data. Th proposd PSO algorithm usd in this rsarch allows rsults to b obtaind with a good accuracy. A clos agrmnt btwn th xprimntal masurd data and th corrsponding simulatd data is obtaind, which vrifis th ffctivnss of th proposd PSO algorithm for paramtr idntification. Furthr work will b focusd on idntifications of mchanical paramtrs of a PMSG. ACKNOWLEDGMENT Th authors thank Miss. 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