Vector Control Using Series Iron Loss Model of Induction, Motors and Power Loss Minimization

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1 World Acadmy of Scinc, Enginring and Tchnology Vctor Control Using Sris Iron Loss Modl of Induction, Motors and Powr Loss Minimization Khldoun Aissa, and Khodja Djalal Eddin Abstract Th iron loss is a sourc of dtuning in vctor controlld induction motor drivs if th classical rotor vctor controllr is usd for dcoupling. In fact, th fild orintation will not b satisfid and th output torqu will not truck th rfrnc torqu mostly usd by Loss Modl Controllrs (LMCs). In addition, this componnt of loss, among othrs, may b xcssiv if th vctor controlld induction motor is driving light loads. In this papr, th sris iron loss modl is usd to dvlop a vctor controllr immun to iron loss ffct and thn an LMC to minimiz th total powr loss using th torqu gnratd by th spd controllr. Kywords Fild Orintd Controllr, Induction Motor, Loss Modl Controllr, Sris Iron Loss. I. INTRODUCTION HE induction motor is dsignd to oprat undr constant T voltag and frquncy [1, 2]. That mans whn it is usd as a variabl spd driv, th motor will oprat far from th optimum oprating point. In fact, if th fild orintd controllr is usd as a controllr that is mostly usd, th orintd flux will b maintaind to its ratd valu which incrass th powr loss whn th motor drivs light loads. Enrgy saving in inductions motor drivs aims at controlling th motor to match th load rquirmnts but with minimum powr loss. So far, two mains mthods hav bn usd to minimiz th powr loss within th induction motor driv whatvr th dcoupling control tchniqu; sarch controllrs [3, 4, 5] (SC) and loss modl controllrs [1,6, 7, 8] (LMC). Th SCs offrs th advantag to b robust against th paramtrs variation but hav a vry sluggish rspons whras th LMCs ar vry fast but paramtrs dpndnt. In othr hand, for th LMCs to b mor accurat, th modl that will b usd to driv th LMC algorithm must b xtndd to includ all th powr loss componnts, such as: stator and rotor iron losss, stray load losss, tc. For this rason, many paprs hav usd th xtndd dq modl that includs th iron loss componnts, which dominats th stray load losss, in ordr th gt mor prcis LMCs. In this modl, th iron loss is modlld by a rsistanc connctd in paralll to th magntizing inductanc, howvr this approachs has addd two xtras diffrntial quations which nds xtras calculations. In th prsnt work, a sris iron loss modl is usd to driv th fild orintd controllr and th LMC algorithm. A program is dvlopd to tstify th ffctivnss of th proposd architctur in which th LMC fds th modifid Khldoun Aissa is with Systms and Signals Laboratory, Dpartmnt of Elctrical nginring and Elctronics, Boumrds Univrsity, Algria (-mail: aissa1973@gmail.com, fax: ). Khodja Djalal Eddin is with M sila Univrsity, Algria. fild orintd controllr by th optimal flux lvl and rcivs th rquird lctrical spd of th orintd flux from it. Th papr is organizd as follows, aftr a gnral introduction; th modlling of th induction motor is prsntd in sction II whr th sris iron modl is mphasizd. Th nxt sction uss this modl to driv to rotor fild orintd controllr quations. In sction IV, th principal of Loss modl controllr is applid. First th total loss is quantifid using th dq modl and th fild orintd control thn th quation that gnrats th optimal flux as function of th oprating point variabls is drivd. Th obtaind simulation rsults using MATLAB/SIMULINK packag ar discussd in sction V. Finally w nd up by a gnral conclusion and som suggstions for th futur. II. DQ MODELLING OF INDUCTION MOTORS A dynamic modlling of th induction motor has bn widly studid in th litratur. Th six diffrntial quations rlating th stator and rotor voltags to th stator and rotor currnts ar rducd to four quations by using th famous park transformation. This transformation offrs, bsids simplification, th advantag to liminat compltly th tim varying paramtrs. It s basd on a st of hypothsis assumptions, among othrs symmtrical thr phas machin and nglctd saturation [2, 9]. In this modl th iron loss has not bn considrd. As th prsnt papr is focusing on th powr loss minimization, this powr loss componnt cannot b ignord. According to th litratur, on can find two main approachs of modlling induction machin taking into account th iron loss, namly: paralll and sris iron loss modls. A) Paralll Iron Loss Modling In this modl, th iron loss, including th loss du to ddy currnt and to hystrsis currnt, is rprsntd by a rsistanc Rf. Th classical DQ modl, rprsntd by four diffrntial quations, of th induction motor is modifid by conncting this rsistanc in paralll to th magntizing inductanc. This modification rsults in a nw dq modl with six ordr diffrntial quations, as xplaind in Lvi t al. [1]. B) Sris Iron Loss Modlling Th sris iron loss modl has bn drivd from th paralll modl by introducing som simplifications and assuming that th chang rat of magntizing currnt is ignord in comparison with that of stator currnt and rotor currnt as wll [11], it follows that: v ds = R s L ls di ds /dt- ω L ls i qs L M d(i ds i dr )/dt -ω L M ( i qr )R ms ( i dr ) (1) 142

2 World Acadmy of Scinc, Enginring and Tchnology v qs = R s L ls d /dt ω L ls L M d( i qr )/dt ω L M ( i dr )R ms ( i qr ) (2) PI d v ds σl ω s R ms / L r σl ω s LM LrT T p 1 mr mr L ω / L M r Fig. 1 IM modlling with sris iron loss rsistanc = R r i dr L lr di dr /dt- ω sl L lr i qr L M d( i dr )/dt -ω sl L M ( i qr )R mr (i dr ) (3) = R r i qr L lr di qr /dt- ω sl L lr i dr L M d( i qr )/dt Whr: ω sl L M ( i dr )R mr (i qr ) (4) L m (ω,s) L M (5) R ms (ω,s) ω 2 (s 2 1) L M 2 /R M (6) R mr (ω,s) ω sl ω (s 2 1) L M 2 /R M (7) Th quivalnt circuit for sris iron loss modl of th IM in th rotating rfrnc fram is rprsntd in Fig. 1. III. FIELD ORIENTED CONTROL USING SERIES IRON LOSS MODEL To apply th rotor fild orintd control principal, th stat vctor must contain th rotor flux componnts. Thus th abov quations of th induction motor ar arrangd by introducing th rotor flux dfind by th followings: = L r i dr L M (8) λ qr = L r i qr L M (9) By liminating th rotor currnt componnts using (8) and (9) and rplacing thm in th induction motor modl, w obtain th following: V ds = (R s L lr /L r R ms σl s p) ω σl s R ms /L r L M /L r d /dt - ω L M /L r λ qr V qs = (R s L lr /L r R ms σl s p) ω σl s R ms /L r λ qr (1) PI q ( L / T R Fig. 2 Block diagram of th FOC drivd from th sris iron loss modl = (p (R r R mr )/L r ) - ω sl λ qr [R mr L M /L r (R r R mr )] = (p (R r R mr )/L r ) λ qr ω sl [R mr L M /L r (R r R mr )] mr) iqs (12) (13) Whr: p = d/dt and σ = (1- L M 2 /L r L s ). By ltting λ qr = dλ qr /dt =, = a constant [6, 7] and using th following notations T mr = L r /(R mr R r ), T r = L r /R r, R s =R s R ms.l lr /L r, w obtain th modifid fild orintd controllr quations: pi ds = -(R s /σl s ) i sd ω i qs (R ms /σl s L r ) λ dr V ds /σl s (14) pi qs = -(R s /σl s )i qd - ω i ds (L M /σl s L r ) λ dr V qs /σl s (15) λ dr = [(L M -T mr R mr )/(T mr p1)] (16) ω sl = (L M /T mr R mr ) i qs / (17) Th dcoupling currnt control is achivd by: V ds =(K p K i /p)( - ) ω σl s (R ms /L r ) V qs =(K p K i /p)( - ) ω σl s (L m /L r ) ω M (18) (19) Whr: K p,k i ar th proportional and intgral gains and, dnot th d- and q-phas currnt commands, rspctivly. Th block diagram, shown in Fig. 2 summarizs th constitution of th modifid FOC controllr. mr w sl dr / λ v qs p. w L M /L r dλ qr /dt ω L M /L r (11) 143

3 World Acadmy of Scinc, Enginring and Tchnology IV. POWER LOSS MINIMIZATION Minimisation of th loss in th induction motor is dirctly rlatd to th choic of th flux lvl. Choosing th lvl of flux in th induction motor rmains an opn problm from th prspctiv of maximising motor fficincy and many rsarchrs continu to work on this problm, and numrous opration schms hav bn proposd by many rsarchrs concrning th optimal choic of xcitation currnt or flux lvl for a givn oprating point. In low-frquncy opration, cor loss (hystrsis and ddy currnt loss) is rathr low compard with coppr loss. As th spd gos up, howvr, th contribution of th ddy currnt loss incrass and finally bcoms dominant. Hnc, th optimal combination of d-axis and q-axis currnts varis, dpnding on th rquird torqu and spd. In our work w ar going to invstigat and dscrib a principl allowing fficincy improvmnt for induction motors: it is th so-calld loss-modl-basd approach, also known as Loss Minimization Controllrs (LMCs), which consist of computing losss using th prvious sris modl and slcting a flux lvl that minimiss ths losss [1,3,8,9]. Rd, Rq (Ω) 18 Rq 16 Rd w (rad/sc) Fig. 3 R d (ω ) and R q (ω ) A) Loss Modl Simplification From th sris iron loss modl shown in fig.1, it follows that i dm, i qm can b approximatd as: i dm i dr i qm i qr (2) (21) And from quation (1) and (2), w can asily s that th iron loss sms brought to stator and rotor sids, which in th paralll modl was prsntd as a paralll rsistanc to th magntizing brunch, this maks th magntizing voltag componnts asy to dduc dirctly from quations (1) and (2): v dm = L M d( i dr )/dt - ω L M ( i qr ) (22) v qm = L M d( i qr )/dt ω L M ( i dr ) (23) Fig. 4 Optimal flux vrsus ω and T Rfrring to th flux quations (8) and (9) and applying th fild orintation principl, w dduc th following: i dr = (λ dr - L M i ds ) / L r (24) i qr = - L M i qs / L r (25) - PI T w T (35) (33) λ r LPF Fig.3 v as v bv v cs PWM Invrtr v as v bv v cs Park Transformation w i as i bv i cs -- IM Fig. 5 Configuration systm 144

4 World Acadmy of Scinc, Enginring and Tchnology Phas currnt (Amp) Powr Loss (W) (a) (c) With LMA Without LMA Rotor flux (wb) Rotor spd (rad/s) (b) (d) λ qr Fig. 6 CasN 1: load torqu=18 Nm, = 12 rad/s at starting, = 6 rad/s at t= 2 sc, = -3 rad/s at t= 3.5 sc Substituting into quations (22) and (23): v dm = (L M L lr /L r ) d /dt (L M /L r ) d /dt - ω (L M L lr /L r ) (26) 2 =R s ( i 2 qs )R r (L M /L r ) 2 i 2 qs [ω 2 (L M L lr /L 2 2 r ω 2 L 2 M i 2 ds ]/R m = i 2 ds [R s (ω 2 L 2 2 M / R m )] [R s R r (L M /L r ) 2 2 ω (L M L lr /L r ) 2 /R m ] v qm = (L M L lr /L r ) d /dt ω (L M /L r ) (L lr ) (27) In th stady stat, λ qr =, d /dt =, i dr =, sinc Whr: = R d (ω ) 2 R q (ω ) 2 (3) = L M. Thrfor, w hav: v dm = - ω (L M L lr /L r ) v qm = ω (L M /L r ) (L lr ) = ω L M (28) (29) In normal opration slip is low, i.. s<<1. Thrfor, w disrgard th iron loss of th rotor, hraftr. Thn th iron loss rducs to (v dm 2 v qm 2 ) /R m. along with th coppr loss, th total motor losss is: P loss =R s ( 2 2 )R r (i dr 2 i qr 2 )(v dm 2 v qm 2 ) /R m R d (ω ) = [R s (ω 2 L M 2 / R m )] (31) R q (ω ) = [R s R r (L M /L r ) 2 ω 2 (L M L lr /L r ) 2 /R m ] (32) R d (ω ) and R q (ω ) ar considrd to b th d-q axs quivalnt rsistors rprsnting th total loss and thir graphs shown in fig.3 rprsnt thir variations with rspct to ω. Fig. 3 shows that R d is dominant ovr R q as ω incrass. Thrfor, it motivats us to rduc th d-axis currnt (or flux lvl) for th loss minimisation. Howvr, too much dcras in (or ) lads to xtrmly larg for a dsird torqu production, yilding a larg coppr loss. 145

5 World Acadmy of Scinc, Enginring and Tchnology Phas currnt (Amp) a) b) Rotor spd (rad/s) With LMA Without LMA λ qr Powr Loss (W) Rotor flux (wb) c) d) Fig. 7 CasN 3: load torqu = 1 Nm at starting, Tl = 6 Nm at t= 1.5 s, Tl = 3 Nm at t= 3 s, Ωr = 12 rad/s at starting, Ωr = 6 rad/s at t= 2.2 s, Ωr = -1 rad/s at t= 4 s Hnc, a compromis btwn iron loss and coppr loss nds to b mad for optimal opration. B) Optimal Solution for Loss Minimization Th xprssion of th powr loss must b writtn as function of th dvlopd torqu and th rotor flux. To do so, th stator currnt componnts, and ar rplacd in (3) by xprssions obtaind from th application of th fild orintation: TABLE I INDUCTION MOTOR DATA Stator rsistanc 4.85 Ω Rotor rsistanc 3.85 Ω Iron loss rsistanc 5 Ω Mutual inductanc.258 H Stator inductanc.274 H Rotor inductanc.274 H Rotor inrtia.31 Kg.m 2 Friction cofficint.8 Nm.s/rd Output powr 1.5 Kw Pols 2x2 Voltag 22/38 V Currnt 3.64/6.31 A Ratd spd 142 tr/min Frquncy 5 Hz = (2/3P) (L r /L M ) T / = (1/L M ) (33) (34) Thrfor th powr loss quation in th rotor flux orintation schm is: P loss =R d (1/L M ) 2 2 R q (2/3P) 2 (L r /L M ) 2 (T / ) 2 (35) Th optimal rotor flux is obtaind by, first, taking th partial drivativ of th powr xprssion (35) with rspct to, scond, maks th drivativ quals zro and finally, solv for th rotor flux variabl: = [R q (2/3P) 2 (L r /L M ) 2 T 2 /( R d (1/L M ) 2 )] ¼ (36) Obsrving th abov quation, on can notic that th optimal flux valu corrsponding to minimum powr loss is xplicitly dpndant of two variabls: lctromagntic torqu T and th dirct and transvrs rsistancs R d and R q. Howvr, th fild rotating spd ω is involvd. In fact, th transvrs rsistanc is constant whatvr th valu of ω whras th dirct on dpnds dirctly on th rotating spd. 146

6 World Acadmy of Scinc, Enginring and Tchnology Phas currnt (Amp) 1-1 Rotor spd (rad/s) a) b) Powr Loss (W) With LMA Without LMA Rotor flux (wb) λ qr c) d) Fig. 8 CasN 2: Ωr = 15 rad/s, load = 1 Nm at starting, load= 6 Nm at t= 1.5 s, load = 4 Nm at t= 3 s, load = 1 Nm at t= 4 s. Fig. 4 rprsnts th variation of th optimal flux function with rspct to ths two variabls (ω, T ) that rflct th oprating point. V. DISCUSSION OF SIMULATION RESULTS A computr program has bn dvlopd in MATLAB/SIMULINK softwar according to th proposd configuration systm, Fig. 5. Th squirrl cag induction motor whos paramtrs ar shown in Tabl I should b fd through a PWM invrtr. As th prsnt work is focusing on th modling and th loss minimization, th invrtr has bn considrd as linar gain. In fact, th invrtr is a sourc of loss du to harmonics but this typ of loss can not b avoidd by flux control. And sinc th invrtr powr losss ar function of stator currnt, thy will b clos to th minimum as th motor is oprating nar th optimum point. Th configuration systm contains th modifid fild orintd controllr, Fig. 2 and a bloc which gnrats th optimal flux using (36) to th FOC block through a low-pass filtr. Th aim of th LPF is to rduc th torqu oscillations du th suddn variation of th optimal rotor flux whn a suddn load torqu variation is obsrvd. Th motor mchanical spd is controlld by a classical PI whos paramtrs K p and K i ar obtaind by using pol placmnt tchniqu. To chck th ffctivnss of th suggstd systm, svral simulations hav bn prformd undr diffrnt oprating conditions, namly: Cas 1: Constant load torqu T l with variabl rotor spd command. Cas 2: Variabl load torqu T l with variabl rotor spd command. Cas 3: Variabl load torqu T l with constant rotor spd command According to th first cas, th obtaind rsults, fig.6, show that for th sam drivn load th rotor flux is incrasd (fig.6.d) if th spd dcrass and this is justifid by th rquirmnt to maintain th torqu capability (output powr is constant mans any dcras in spd must b compnsatd for by incras in torqu that proportional to flux). At th sam tim, th modifid fild orintd controllr kps th rotor flux orintation wll (transvrs componnt is null). Th optimum point is rachd by th fact that th load torqu is maintaind qual th ratd on and th flux lvl is incrasd, whras th powr loss is approximatly th sam as in th cas without LMA. In th scond scnario fig.7, th spd is maintaind constant but th load is dcrasd gradually, th lss is th load torqu th lowr is th flux lvl and hnc th minimum is th powr loss. It asy 147

7 World Acadmy of Scinc, Enginring and Tchnology to notic that th dcoupling is satisfid sinc th transvrs flux componnt is not altrd by th dirct componnt variation. Th third cas is a combination of th two prvious cass, th motor driv changs its spd and drivs variabl load torqu. Whatvr th oprating point, th powr loss with LMA is lss than that obtaind without LMA (constant flux oprating). In this scnario it s noticd that th ovrshot in th flux rspons is important vn though th LMA output is dlayd by a LPF. On on hand, th flux oscillations com from th fact th two quantitis, load torqu and spd, ar simultanously varid within a tight priod. On th othr hand, th fild orintd controllr gnrats th optimal flux on th basis of th knowldg of th torqu rfrnc rathr than th ral load torqu. That mans to minimiz th flux oscillations, a mor advancd spd controllr may b usd, such as th non-linar controllr, th sliding mod controllr, tc. VI. CONCLUSION Two aspcts hav bn discussd in th papr; th first concrns th rotor fild orintation by using th sris iron loss modlling, whras th scond is dvotd to powr loss minimization using th motor modl. Th advantag coms from using th sris modl is th limination of two diffrntial quations dscribing th magntizing currnt in th paralll modl. Th obtaind rsults show that in difficult situations such that variabl flux-variabl spd opration, th rotor fild orintation is maintaind. Th association to th modifid fild orintd controllr a mchanism to slct th optimal flux lading to minimum powr loss (LMA) has not disturbd th dcoupling hnc th induction motor driv. Furthrmor, th LMA nds th valu of th lctromagntic torqu which its imag is gnratd by th spd controllr. As th modifid fild orintd controllr illuminats th dtuning du to iron loss btwn th output lctromagntic torqu and th rfrnc torqu, thrfor th systm will b simplifid by using th rfrnc torqu rathr than a torqu snsor. As rgards th flux and th torqu oscillations, th low pass filtr sms not nough to smooth th flux rspons whn a succssion of variation in spd rfrnc and load conditions th motor is xposd to. Consquntly, to liminat compltly th flux and torqu oscillations, it s suitabl to us a Dirct Fild Orintation Schm or a non-linar controllr with a robust obsrvr to furnish th flux fdback. Th obsrvr can b xtndd to stimat too th stator and rotor rsistanc as th LMC quation contains thir valus. REFERENCES [1] Abrahamsn F., Blaabjrg F., Pdrson J. K., Grabowski P. Z. and Thogrsn P., On th nrgy optimizd control of standard and high fficincy induction motors in CT and HVAC applications, IEEE trans. on industry applications, vol. 34, No 4, July-August 1998 [2] D.W.Novotny and T.A.Lipo, Vctor Control and Dynamics of AC Drivs, Clarndon Prss, Oxford, UK, [3] S. Vaz-Zadh and F. Hndi, A continuous fficincy optimization controllr for induction motor drivs, Enrgy Convrsion and Managmnt 46 (25) [4] Wang J.B., Liaw C.M., Indirct fild orintd induction driv with fuzzy dtuning corrction and fficincy optimization controls, IEE Proc. Elctr. Powr. Appl., Vol.144 n.1, January 1997, pp [5] Sousa G.C.D., Bos B.K., Fuzzy logic basd on-lin fficincy optimization control of an indirct vctor controlld induction motor driv, IEEE confrnc rcord IAS, pp , /Nov, 1993, Hawai, USA. [6] Lim S. and Nam K., Loss-minimising control schm for induction motors, IEE Proc.-Elctr. Powr Appl., Vol. 151, No. 4, pp , July 24 [7] Nam S. W. and Nasir Uddin M., Modl-Basd Loss Minimization Control of an Induction Motor Driv IEEE ISIE 26, July 9-12, 26, Montral, Qubc, Canada [8] Matsus K., Yoshizumi T., Katsuta S., Tanigushi S., High Rspons Flux Control of Dirct Fild Orintd Induction Motor with High Efficincy Tacking Cor Loss into Account, IEEE Trans. on Indus. Appl., Vol.35, No. 1, JAN./FEB. 1999, pp [9] Dal Y. Ohm, Dynamic Modl of Induction Motors for Vctor Control, Drivtch, Inc., Blacksburg, Virginia, 2. [1] Lvi E., Boglitti A., and Lazzar M. Prformanc Dtrioration in Indirct Vctor Controlld Induction Motor Drivs du to Iron Losss, IEEE trans. on Indus. Appl., pp , [11] Jinh w. J., Nam K., A Vctor Control Schm for EV Induction Motors with Sris Iron Loss Modl, IEEE trans. Ind. Elctron. Vol. 45, No. 4, pp , August