Sliding mode flux observer with online rotor parameter estimation for induction motors

Transcription

1 Sliding mod flux obsv with onlin oto aamt stimation fo induction motos Amuliu Bogdan Poca, Mmb, IEEE and Ali Kyhani, Fllow, IEEE Abstact-- Fild ointation tchniqus without flux masumnts dnd on th aamts of th moto, aticulaly on th oto sistanc o oto tim constant(fo oto fild ointation. Sinc ths aamts chang continuously as a function of tmatu it is imotant th valu of oto sistanc b continuously stimatd onlin. A fouth od sliding mod flux obsv is dvlod in this sach. Two sliding sufacs snting combinations of stimatd flux and cunt os a usd to nfoc th flux and cunt stimats to thi al valus. Switching functions a usd to div th sliding sufacs to zo. Th quivalnt valus of th switching functions (low fquncy comonnts a ovn to b th oto sistanc and th invs of th oto tim constant. This oty is usd to simultanously stimat th oto sistanc and invs of th tim constant without io knowldg of ith th oto sistanc o th magntizing inductanc. Simulations and ximntal sults ov th validity of th oosd aoach. Indx Tms-- Fild ointd contol, sliding mod, onlin aamt stimation, induction moto. I. NOMENCLATURE v, v : stato voltag in stationay fnc fam i, i : cunts in stationay fnc fam i i, : cunts in synchonous fnc fam λ d, λ q : oto fluxs in stationay fnc fam λ : oto flux in synchonous fnc fam n : numb of ols ais ω, ω : synchonous fquncy and mchanical sd. L, L l : magntizing and lakag inductanc m R, : stato, oto sistanc s R T : lctomagntic toqu T: stato tmatu ^, *: stimatd, fnc valus f = f f : stimation o, wh f can b any flux, cunt o aamt A.B. Poca is with Solitat Contol Inc., Columbus, Oh A. Kyhani is with th Datmnt of Elctical Engining, Th Ohio Stat Univsity, Columbus, OH 430 (-mail: kyhani.@osu.du II. INTRODUCTION N high fomanc alications, th induction moto is Icontolld though stato and oto fild ointation tchniqus. Fild ointation tchniqus a basd on th knowldg of oto (o stato fluxs, and ths fluxs a not masuabl in actical alications. Th a two common mtho of oto fild ointation: th dict fild ointation uss a flux obsv fo fild ointation whil th indict fild ointation mthod uss th sli quation to calculat fild ointation. Both mtho dnd on th aamts of th moto, aticulaly on th oto sistanc and magntizing inductanc. Whil th magntizing inductanc can b considd constant if th flux lvl is kt constant and can b asily mad to th flux lvl by a satuation function if flux is vaiabl, th oto sistanc vais as a function of skin ffct and oto tmatu. Th lat is unmasuabl in actical conditions. It is thfo imotant fo fild ointation that th valu of oto sistanc is continuously stimatd onlin. Extnsiv sach has bn fomd on th toic by Sul, Tolyiat, Kubota, Maino, Zai [-5],[,[] and oths[6-0]. Dict calculation mtho comut oto sistanc valus algbaically [], [], [6], [8]. In [] th oto sistanc is comutd dictly fom moto quations at stady-stat conditions. In [] th inducd voltag in a has (of a modifid PWM invt is usd to calculat th oto sistanc. In [6] th oto sistanc is calculatd algbaically ovidd that th is a flux and sd tansint. In [8], th authos liminat th flux tms fom th moto quations and us a last squas algoithm to stimat th nw systm aamts. In [3], [- 3] signal injction mtho a usd whil in [4] magntic salincy otis a mloyd to stimat oto aamts. Modl fnc adativ systms (MRAS w usd in [9], [5-7] to stimat oto sistanc o th oto tim constant. Th authos of [5], [8-0] us an xtndd Kalman filt fo th stimation of both fluxs and aamts. In [4], [0], [], [] a non-lina adativ obsv is usd to simultanously stimat fluxs and aamts. Basd on th induction moto modl in stationay fnc fam, a fouth od sliding mod obsv is dvlod fo both flux and aamt stimation. Two sliding sufacs snting th stato cunt amlitud and a combination of stimatd flux and cunt os a usd to nfoc th flux and cunt stimats to thi al valus. Th obsv dos

2 not dnd on th valus of th oto sistanc and oto tim constant. Emloying th us of Lyaunov functions, it is ovn that th sliding sufacs can b focd to zo in finit tim by using two switching functions. Futhmo, th quivalnt valus of th switching functions (low fquncy comonnts a ovn to b th oto sistanc and th invs of th oto tim constant. This oty is usd to simultanously stimat th oto sistanc and invs of th tim constant without io knowldg of ith th oto sistanc o th magntizing inductanc. Futhmo, sinc th algoithm is quit obust to th vaiation of lakag inductanc, th only offlin (o slf commissiond stimation ndd is fo stato sistanc. III. INDUCTION MOTOR MODEL In this wok, w will us th induction moto modl in stationay fnc fam as shown in Fig. fo q-axis (d-axis a simila. As notd in [3], th modl is quivalnt (without any loss of infomation to th mo common T modl in which th lakag inductanc is saatd in stato and oto lakag. V Rs Ll Lm ω λ d Fig. Induction moto modl in stationay fnc fam(q-axis Th following quations fo th induction machin can b divd: di di dλ q = β n ω λ + ηλ R i + v R i ( ( d q s = β ( n ω λ + ηλ R i + v R i ( d q = n ω λ ηλ q d + R i s dλd = nωλq ηλd + Ri (4 with constants dfind as follows : R η =, β = Lm L l Th lctomagntic toqu xssd in tms of th stat vaiabls is: T ( λ i λ i d q R (3 = 3 n (5 IV. EFFECT OF VARIATION OF ROTOR RESISTANCE ON FIELD ORIENTATION Fild (stato, oto o aiga ointation mtho nd flux infomation to calculat th ointation angl. Th oto flux can b masud using flux snsos in th d-q axis ositions; howv, ths snsos a costly and duc th ovall liability of th systm. A fd mthod in th absnc of flux masumnts is th us of modl basd fild ointd contolls. In this cas, th vaiation of oto sistanc imacts th fild ointation. Dict fild ointd contolls us th flux quations in stationay fnc fam (qn.3,4 to calculat th ointation angl. Indict fild ointd contolls us th sli quation fo th sam uos: dλ dθ = η λ + R i (6 i = n ω + R (7 λ Both mtho dnd on th oto aamts (. Sinc oto sistanc vais as a function of tmatu and skin ffct, both oto aamts will b affctd, and os in th valus of ths aamts will imly os in fild ointation [-]. An o in fild ointation will sult in an o in both th valus of th tackd flux and of th tackd toqu (assuming that th contolls a wll tund. Th ffcts of th unaccountd fo vaiation of oto sistanc on th valus of flux and toqu w analyzd though simulation, using th atd aamts of th moto mloyd in this sach. Th sults a not sntd to maintain th lngth of th a shot. Although tacking os w obsvd thoughout th nti ang of sd and loading, a high tacking o in toqu and flux will aa at high loa sinc th snsitivity of th stato cunts (and consquntly, toqu and flux to oto sistanc incass with loading. At high load, th o fo both flux and toqu is comaabl in magnitud with th oto sistanc o. V. SLIDING MODE OBSERVER DESIGN Th sliding mod flux obsv is dvlod basd on th moto dynamic quations: di = β ( n ωλd + ηλq Rsi + v Ri (8 di = β ( n ωλq + ηλd Rsi + v Ri (9 d λq = n ω λd η λq + R i (0 d λd = n ω λq η λd + R i ( Th fouth od obsv aas simila in stuctu with a full od on loo obsv. Th diffnc consists in th tms R, η (lacing th aamts R, η that a switching functions usd to assu th convgnc of th obsv. Dfin two sliding sufacs as: s = i i + i i ( s i i λq i i = ( + ( λd (3 Lt a Lyaunov function (ositiv dfinit b

3 T s s& = 0, s = 0 ([4], sction.3. Thi divation and a oof V = 0. 5 s s wh s = (4 s that th quivalnt valus a idntical to R, η is sntd in To ov th xistnc of a sliding mod, w a intstd Andix A. Th analytical solution of R, q, η q will contain in finding th condition fo which unknown tms ( R, η, λd, λq and thfo can only b V & = s& s + s& s < 0 (5 usd fo dmonstation uoss. A low ass filt is usd Aft som algbaic divation, th divativ of s is instad to xtact th quivalnt valus ([4]. f R s& = + G (6 R q f η = R, + s τ ( wh: ηq = η ( f = R β ( i + i Rs β ( i + i i i + s τ + ηβ ( λqi + λdi Th valus of R βn ω ( λqi λdi λqi + λdi, q, η q + βv i + βv i will b usd as stimats of R, η in th analysis of th obsv f ( ( = nω λq i λd i βnω λq λ λd λ VI. SIMULATION RESULTS R ( ( ( β λqi + λdi + βη λqλq + λdλd Rsβ λq i + λd i A. Convgnc of th obsv Extnsiv simulations w un to vify th convgnc of g g G = = th obsv. Th modl usd in th simulations has th sam g3 g4 aamts as th on usd in th ximnts; ths β( i + + i β( λdi λqi aamts a summaizd in Tabl A of th Andix. Th = i i + i i + β( λdi + λqi β( λd + λq λd i λq i following tabl summaizs th stimation os fo R and η and th subsqunt tacking of flux and toqu at stady_stat. Lt Sinc os in th obsv yild os in fild ointation, th R K 0 K sign( gs + g3s os in oto flux stimation (in synchonous fnc T = sign( G s = (7 fam and toqu a also shown. As xctd, lag η 0 Kη Kη sign( g s + g 4s aamt stimation os can b xctd in th low loadhigh sd gion. Howv, flux stimation os a small Th Lyaunov function divativ bcoms: V& = fs + f s K ( gs + g3s sign( gs + g3s sinc th obsv is last snsitiv to oto aamts in that (8 Kη ( g s + g 4s sign( g s + g 4s gion. if K, K η a ositiv constants chosn so that: K gs + g3s + Kη gs + g4s > fs + fs (9 thn th Lyaunov function divativ is ngativ and s and s will convg to zo in a finit tim, imlying th cunt stimats ( i, i will convg to thi al valus in a finit tim. Th choic of switching functions is not uniqu. Du to th fact that gnally g s > g3s, g 4s > gs, g > 0 whil g can b mad ngativ by choosing initial valus fo th cunt clos to th al valus, th switching functions can b simlifid to th fom: R K sign( s = (0 η Kη sign( s This choic of switching functions is asi to comut (no nd to comut matix G and it allows th dcouling of th sliding sufacs; it was thfo fd in this wok. Both R, η a switching functions, containing a high od hamonics. Th quivalnt valus of R, η (th smoothd stimats, R, q, η q can b found by solving th quation TABLE CONVERGENCE OF THE PROPOSED OBSERVER Sd (m Load(i in ams R(% η(% λ(% Τ(% B. Robustnss to aamt vaiation Two of th modl aamts, R, η a stimatd on-lin using th sntd obsvs, whas R s, and L l a not stimatd on lin. Th ffct of th vaiation of R s and L l ov th stimation of R, η an thi imact on fild ointation a shown in th nxt two tabls. It can b sn that th ffct of unctaintis of th lakag inductanc on flux and toqu stimation is minimal vn fo lag unctaintis (+/- 75%. Futhmo, th lakag inductanc can b asily mad to th stato cunt amlitud as a satuation function. Th ffct of stato sistanc unctaintis is not ngligibl, scially in th low 3

4 4 TABLE EFFECT OF UNCERTAINTY OF STATOR RESISTANCE ON OBSERVER Sd (m Load(i in ams R(% Rs=-75% η(% λ(% Τ(% R(% Rs=-50% η(% λ(% Τ(% R(% Rs=-5% η(% λ(% Τ(% R(% Rs=5% η(% λ(% Τ(% R(% Rs=50% η(% λ(% Τ(% R(% Rs=75% η(% λ(% Τ(% TABLE 3 EFFECT OF UNCERTAINTY OF LEAKAGE INDUCTANCE ON OBSERVER Sd (m Load(i in ams R(% Ll=-75% η(% λ(% Τ(% R(% Ll=-50% η(% λ(% Τ(% R(% Ll=-5% η(% λ(% Τ(% R(% Ll=5% η(% λ(% Τ(% R(% Ll=50% η(% λ(% Τ(% R(% Ll=75% η(% λ(% Τ(%

5 5 sd ang. In situations in which th stato sistanc vais considably, a stato sistanc obsv is ndd. In this sach, stato tmatu was availabl fom a tmatu snso. Th stato sistanc was mad to stato tmatu using a lina function. VII. EXPERIMENTAL RESULTS A. Eximntal Stu Th ximntal stu usd in this sach is shown in Fig.. Th induction moto (I.M. is 3 has, 4 ol, 5H, 750 m 0 V squil cag. Th synchonous gnato (S.G. is has, ol, 5 H 440 V and is usd as a load. Th 5 kw vaiabl sisto box (R loa th synchonous gnato. A vaiabl DC ow suly contols th xcitation of th synchonous gnato. Th moto is divn by a 400V/30 A ow convt caabl of switching at 0 khz. A dual ocsso (TMS30C3 Mast and TMS30P4 Slav DSP boad usd both fo contol and data acquisition. A 04 uls/volution incmntal otical ncod is usd fo sd masumnt. Th PWM cycl is 40 µs and th data acquisition samling tim is 60 µs. In od to avoid aliasing, th masud voltag is assd though a low ass filt io to bing acquid. Th synchonous gnato can b contolld simultanously with th moto using th DSP boad though th xcitation voltag. Fig. 3 Obsv convgnc fo low sd/no load P.C. Excitation contol S.G. R Fig.4 Obsv convgnc fo high sd/no load DSP boad Pow convt V I Signal conditioning Filt I.M. E Fig. Eximntal stu sd tmatu Fig.5 Obsv convgnc fo low sd/high load B. Obsv convgnc Extnsiv tsting has bn fomd with th oosd obsv. To show th convgnc of th flux obsv tsts w un at low sd (00 m, mdium sd (000 m and high sd (000 m. Fo ach sd lvl, 3 loading conditions w considd: No load, mdium load (8 Nm and high load (6 Nm. Th following lots a som xamls of th sults. Th gahs on th ight sid a th cunt and flux os (sctivly fo th d-comonnt (th q comonnt is simila. Th gahs on th lft sid snt th stimats of oto sistanc and tim constant invs; th dottd lin snts th al valu. Fig.6 Obsv convgnc fo high sd/high load

6 6 C. Load St Anoth st of tsts consistd in a st in th moto load. Th to gahs show th bhavio of th stimats of oto sistanc and invs of tim constant whil th bottom gahs show th toqu cunt and th sd, sctivly. unctaintis in th lakag inductanc valus but lativly snsitiv to unctaintis in th stato sistanc valus. Simulations and ximntal sults ov th validity of th cunt aoach. Fig.7 Load st at 400 m Fig. 9 Obsv bhavio fo a sinusoidal sd fnc IX. APPENDIX A Fig. 8 Load st at 600 m D. Tacking of a vaiabl fnc To snt a continuously dynamic condition, th sd fnc was mad sinusoidal. Th following gahs show th flux convgnc and cunt convgnc togth with th stimats of oto sistanc and invs of tim constant. Th scond lot show th fnc and masud sd. VIII. CONCLUSIONS A sliding mod flux obsv was dvlod in this sach. Th obsv allows not only flux obsvation but also calculats th valus of oto sistanc and th invs of th oto tim constant. Du to th low snsitivity to oto sistanc of th stato cunts, high os in th R and η a obtaind whn th moto oats at low lvls of toqu. Howv, du to th sam snsitivity, th os in flux and toqu stimation a small. Futhmo, thoughout th nti sd and load ang, ovidd that th lakag inductanc and stato sistanc a known, th flux and toqu stimation os a small (blow %. Th obsv is obust to Rs Ll Lm R TABLE A INDUCTION MOTOR RATED PARAMETERS 0.39 Ω Hny Hny 0. Ω A. Equivalnt Valus Th quivalnt valus of R, η (th smoothd stimats, R, q, η q can b found by solving th quation s& = 0, s = 0. This yil: i ( s R + v i + v i + n ω ηq = η (A ( R βis λis + Kis [ ( β + i i + i i + β is λ ] R β ( λis + K R, q = (A ( β + i i + i i βis λ wh K = [ n ω ( λd i + λq i βn ω ( λqλd λdqλq ( ] + βη λ λ + λ λ + βλ [ η + v i + v i + n ω ( ] d d q q

7 7 = λ i + λ i = λ i + λ i, = λdi λqi d q, d q = λ i d λ i q λ = +, i = i + i λd λq s, If th flux and cunt stimats convg to thi al valus, thn =, = K = 0 and th quivalnt valus of R, η will convg to thi al valus: R, q = R and η q = η To ov th cunt and flux convgnc on n to substitut th quations of R, q, η q in th obsv quations and ov that th stimats convg to thi al valus. At th tim of witing this a, th authos w not abl to ov th convgnc analytically. Howv, simulation and sults fo vaious sd and loading conditions show th validity of th conct. X. ACKNOWLEDGMENT Th authos would lik to thank Pofsso V.I. Utkin of th Ohio Stat Univsity fo his hl in foming this sach. XI. REFERENCES [] S.K. Sul, A novl tchniqu of oto sistanc stimation considing vaiation of mutual inductanc, IEEE Tans. on Industy Alications, v. 5,no. 4, 989, [] H.A.Toliyat, M.S. Afn, K.M. Rahman, D. 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Bnchaib, Dtction of bokn bas in induction motos using an xtndd Kalman filt fo oto sistanc snsolss stimation, IEEE Tans. on Engy Convsion, v. 5, no., 000, [] R. Maino, S. Psada, P. Tomi, Global adativ outut fdback contol of induction motos with unctain oto sistanc, IEEE Tans. on Automatic Contol, v. 44, no. 5, 999, [] R. Maino, S. Psada, P. Tomi, Exonntially convgnt oto sistanc stimation fo induction motos, IEEE Tans. on Industial lctonics, v.4, no. 5, 995, [3] G.R. Slmon, Modling of induction machins fo lctic divs, IEEE Tans. on Industy Alications, v. 5, no. 6, [4] V. Utkin, J. Guldn, J. Shi, Sliding mod contol in lctomchanical systms, Taylo & Fancis, 999 XII. BIOGRAPHIES Amuliu Bogdan Poca (S 96- M 0 civd his MSEE and PhD in Elctical Engining fom th Ohio Stat Univsity, Columbus, Ohio in 997 and 00 sctivly. H is cuntly woking fo Solitat Contols Inc, Columbus, Ohio. M. Poca's sach intsts a in th aas of induction machin contol, modling, aamt stimation and dsign. Ali Kyhani (S 7 M 76 SM 89 F 98 civd th Ph.D. dg fom Pudu Univsity, Wst Lafaytt, Indiana in 975. Fom 967 to 969, h wokd fo Hwltt-Packad Co. on th comut-aidd dsign of lctonic tansfoms. Cuntly, h is a Pofsso of Elctical Engining at th Ohio Stat Univsity, Columbus, Ohio. His sach intst is in contol and modling, aamt stimation, failu dtction of lctic machins, tansfoms, and div systms.