Indirect Field Oriented Control with rotor time constant adaptation

Transcription

1 4th WSEAS Internatonal Conferene on EECTRONICS, CONTRO and SIGNA PROCESSING, Ma, Florda, USA, Noveber, 005 (pp Indret Feld Orented Control wth rotor te onstant adaptaton PHD. ADO PARDO GARCÍA, MSC. JORGE UIS DÍAZ RODRÍGUEZ. Unversty of Paplona. Cudadela Unverstara. Paplona. Tel: Ext. 41, Fax: Ext NORTE DE SANTANDER. COOMBIA Abstrat - Ths work dls wth the desgn of an Adaptve Referene odel Controller for (MRAC for paraeters adaptaton apply to an ndret feld orented ontroller for nduton otor ontrol. The paraeters behavor was analyzed usng three dfferent referene odels. The onsdered referene odels were: dret (d axs voltage, the quadrature (q axs voltage and the rtve power. For h referene odel the adaptaton error was analyzed. Usng these equatons the paraeters onvergene was analyzed and dsussed under several otor ondtons. The results were presented and the sulaton for all the referene odels onsdered, llustratng the paraeters behavor under several otor operatng ondtons. The analyss and sulaton results show the superorty of the dyna referene odels usng quadrature (q axs voltage and rtve power odel. Keywords: Feld Orented Controller, Feld Orented MRAC Control, Induton otor, Feld Orentaton Drve Systes, Paraeter Adaptaton for the Speed Controlled, Feld-Orented Induton Mahne Drves 1. Introduton The feld orented ontrol s the ost popular ethod to obtan the best dyna benefts n ndustral applatons wth nduton otors. Two bas shees of feld orented ontrollers exst: the dret and the ndret ethod. The an dfferene s that the dret ethod aheves the feld orentaton by ns of dret sureents or alulatons (estaton of the eletr flux of the rotor, whle the ndret ethod aheves t by ns of the sureents of the speed and the slp alulaton wth the referene values. Typally, both shees use soe knd of stator urrent regulaton. The ndret ethod has been the preferred one, anly buse t s not needed to sure or to estate the flux, splfyng the ontroller and then redung the osts. However the slp frequeny alulaton depends on the rotor te onstant whh vares anly wth the rotor teperature, and f ther exat value s not know, the dyna perforane ndators deterorate. It has been proposed several shees usng MRAC ontrollers to adapt suh varatons [7] - [11]. As t s known n ths tehnque (MRAC the requred behavor of the otor t s spefed by a referene odel. To aheve the adaptaton t s use the error sgnal, dfferene between the odel referene output and the otor ternal sureents. Several referene odels have been presented n the lterature, n [7] the referene odel orresponds to an odfed funton of the rtve power. In [9] a referene odel s used, based on the otor torque. In [10] 3 referene odels are based on the voltages. All these referene odels are algebra, and ost of the based on nduton otor stdy state operaton.. Adaptaton Method for MRAC Fgure 1 shows the bas struture of an MRAC shee wth slp frequeny gan adaptaton, slar to generalzed MRAC shee, used n [10] and [11]. The varable Y * s the odel referene output and represents the value of soe otor agntudes, Y s the varable urrent value, the one, obtan by ns of sureents n the ternals of the otor. The Y * Y dfferene produes the adaptaton error (e a, whh s ultpled by the referene of the torque produng urrent, and then s ntegrated to produe the slp frequeny gan ( kˆ s, that s used dretly by the feld orented ontrol, to opute the slp frequeny ( w sl. The goal of ths ethod s to adapt the gan kˆ s to aheve the onvergene to zero of the adaptaton error. The referene odel plays a ldng roll n ths adaptaton struture, sne the behavor of the syste s spefed, n ths ase the behavor of the nduton otor wth feld orented ontrol. The

2 4th WSEAS Internatonal Conferene on EECTRONICS, CONTRO and SIGNA PROCESSING, Ma, Florda, USA, Noveber, 005 (pp referene odel should be ths way able to predt ths behavor so that the gan kˆ s onverges to ts nonal value, the sae to say that the adaptaton error onverges to zero. The equatons (5 to (8, produe the Y * varable (fg. 1 and Y s obtaned dretly by ns of the sureents of v ds and v of the d and q axs n the voltage referene odels and wth the equaton (7 the rtve power referene odel was obtaned. Fg. 1. Blok dagra of the Feld Orented MRAC Control All these odels are based on nduton otor odel wth lnl agnet rut and onstant referene flux. 3. Algebra referene odel The algebra referene odels are those that only ontan algebra operatons. They don't nlude ntegral or dfferental operatons. The algebra referene odels that we wll onsder n ths work are obtaned only fro stdy state operatons. In ths ase the stator voltage n referene to the synhronous axes are gven fro the followng equatons: v = Rs. λds (1 = Rs. λ ( Where the sybols are defned n the annexes. The stator flux n feld orented ondton are: λ = σ (3 λ ds = s (4 The dret (d axs voltage referene odel was obtaned by ns of the substtutng ( n the feld orented ondton (3 [10]: * v ds = Rs σ (5 The uadrature (q axs voltage referene odel was obtaned by ns of the substtutng (1 n the feld orented ondton (4. [10]: * v = Rs s (6 The rtve power s obtaned by ns of: Q = v (7 The rtve power referene odel was deterned by ns of the substtuton of the equaton (1 and ( n (7, and applyng the feld orented ondtons 3 and 4. [10]: * Q = w + e s σ (8 3.1 Adaptaton error To forulate the adaptaton error, all the otor paraeter who desrbe the referene odels are supposed known and should exst an exat ath between the rl urrents and those of referene n the otor. The saturaton effet were not onsdered. The expressons of the adaptaton error were obtaned takng the dfferene for h referene odel. The stator sured voltage n the otor ternals s gven by equatons (18 and (19. Usng these two equatons and equaton (7 yelds: [11] Dret (d axs voltage referene odel: = ( we λqr p λdr (9 Quadrature (q axs voltage referene odel: v = [ we ( λqr + pλqr ] σ p (10 r Rtve Power referene odel: pr e a = ( λqr + λqr pλqr pλdr σ p r [ ] ( These equatons predt the adaptaton error n both ases, stdy and transent state. (11 3. Stdy state analyss In stdy state, the dervatve ter that apprs n the equatons (9 to (11 are zero. The stdy state adaptaton error s expressed better by ns of the detunng rson: kˆ s x = (1 0 k s Wrtng the adaptaton error n stdy state operaton as a funton of equaton (1, we obtan n ths ase: Dret (d axs voltage referene odel: ( 1 x ds = v0 τ r wr + x (13 1+ Quadrature (q axs voltage referene odel:

3 4th WSEAS Internatonal Conferene on EECTRONICS, CONTRO and SIGNA PROCESSING, Ma, Florda, USA, Noveber, 005 (pp x( x 1 v ds = v0 τ r wr + x 1+ (14 Rtve Power referene odel: ( x 1 pr ds = Q0 τ r wr + x ( x Where: 0 v = Rr and Q0 = v0 (16, 17 The rato / s strateg n the equatons (13 to (15. As ths rato s a torque's funton dnded by the load, ths expressed the load dependene wth the adaptaton errors. Of ourse, sall values of the rato /, ply sall values adaptaton errors. In the farthest ase, the adaptaton error onverge to zero when = 0, that s way, there n not adaptaton wthout load. Ths s not a peular proble of the MRAC ethod, other shees have the sae proble [1]. The adaptaton error also depends on frequeny, as s shown n equatons fro (13 to (15. At rated speed the adaptaton error wll be bgger that n low speed. The adaptaton error dependene s nonlnr. For adaptaton error x = 1 doesn't take plae the adaptaton proess and ths deternes an equlbru pont for gan kˆ s. Ths orresponds to the tuned ondton. 3.3 Transent analyss The dervatve ters of n the equatons (10 and (11 deonstrates that the adaptaton error depends on transents. Suh a dervatve ter apprs buse the referene odels were based on stdy state onsderatons and t annot predt the transent behavor of the nduton otor n feld orentaton. The equaton (9 doesn t a deonstrate suh dependene buse e depends on the dervatve of and onstant flux ondton s supposed. The dependene of the adaptaton error n transents ntrodues an sgnfant proble. The proble that arses for any transent of an adaptaton error wll take plae ndependently of the tune ondton. Ths an be seen as a wrong adaptaton error and t wll ause an adaptaton of kˆ s to an erroneous value. For ths rson, the syste an abandon the tune ondton due to these transents [11]. As the tunng sensblty s bgger at low flux levels, the proble wll be larger n ths stuaton. Also, the proble wll be ore sgnfant f frequently torque hanges s requred, e.g., Robots Applatons. 4. Dyna odel forulaton To develop a dyna odel of the nduton otor n feld orentaton, t should be onsdered ther behavor n ths stuaton, as follows: The stator voltage of the nduton otor s: v = Rr σ + ( weλdr + pλqr (18 = Rs σ ( weλqr + pλqr (19 Feld orentaton, supposng onstant flux operaton: λ qr = pλqr = 0 (0 λ r = (1 Substtutng (0 and (1 n (18 and (19 yelds: v = Rs s ( v = R w (3 ds s ds e σ The equatons ( and (3 desrbe the behavor of the stator voltage of the nduton otor n both ases, n transent and stdy state. The only dfferene between the and the equatons (5 and (6 s the dervatve that apprs n the equaton (. To onsder suh a dervatve a slar odel an be buld, as s shown n fgure. In the fgure G(p s standard PID ontroller. It has the sae role that the CRPWM nverter, produng the stator voltage v to aheve that ontnues losely to approah., f G(p were desgned to aheve ths Fg.. Dyna odel blok dagra (4

4 4th WSEAS Internatonal Conferene on EECTRONICS, CONTRO and SIGNA PROCESSING, Ma, Florda, USA, Noveber, 005 (pp Ths an be delared as: v = Rs Rs Rs s s σ Rs σ (5 = (6 Now, lke t was establshed before, that = and =, the equaton (5 delare that ds ds stator voltage n the q axs of the dyna odel v orresponds to the urrent otor voltage v. For ths rson, the dyna odel predts the behavor of the stator voltage of the nduton otor n both ases, n transent and stdy state. Thus the dyna odel an be wrtten as: v v ds = R s 1 ( v s ( p ( s eσ (7 = G (8 = R w (9 4.1 Dyna referene odels The dyna odel obtaned n the prevous seton an be use to buld a slar referene odel to the algebra one. The dret and uadrature voltages referene odels should be obtaned dretly as: * v = ds v ds * v (30 v = (31 Rtve power referene odel s obtaned by ns of: * Q = v v (3 ds ds These odels wll be alled as dyna referene odels buse they are based on the nduton otor dyna odel. 4. Adaptaton error The adaptaton error e a = Y * Y for the dyna referene odels s obtaned n the sae way that the algebra ones. The varable Y * s gven by the equatons fro (30 to (3 whle the varable Y s gven by the equatons (18, (19 and (7. Dret (d axs voltage referene odel: = ( weλqr pλdr (33 Quadrature (q axs voltage referene odel: v = [ we ( λ dr + pλqr ] (34 Rtve Power referene odel: [ ( λ + λ ] Pr e a = we ds dr ds qr ( pλqr pλdr The equatons (33 to (35 an be opared wth the equatons (9 to (11 to ondut a oparatve analyss. ( Stable state analyss In stdy state operaton, the dervatve ters of equatons (33 to (35 are zero. Ths way derses the adaptaton error obtaned by ns of the algebra referene odel (equatons (9 to (11 n stdy state, so both referene odels, the dyna and the algebra one have the sae behavor n stdy state and they produe the sae adaptaton error. Ths an see obvous buse both referene odels predt the sae behavor for the nduton otor n stdy state. Also, the adaptaton error n the d axs dyna referene odel and the algebra odel of the sae axs, are the sae one, buse the d axs stator voltage doesn't depend on the transents of, lke t was entoned before. 4.4 Transent analyss The equatons (34 and (35 shows that the adaptaton error, through the q axs voltage and the power rtve dyna referene odels don't depend on the transents of lke t was n the ase of the algebra referene odel. Ths s the an advantage fro the referene odel dyna envelope to the algebra one. They are able to predt the behavor wanted for the nduton otor n both ases, stdy state and transtory, false adaptaton error doesn't take plae ths way durng the transents of. The equaton (6 deonstrates that the d axs voltage dyna referene odel s slar to the algebra one. Ths way, n both ases the sae behavor s obtaned, n transent and stdy state, ths buse the d axs voltage n the transents doesn't depend of lke t was entoned before. It only depends on the ds transents and they were not onsdered supposng onstant flux operaton. 5. Motor paraeters dependene Both dyna and algebra referene odels s dependent of the otor paraeters, the otor paraeters ontans R s, s and σ, and. The stator resstane hanges wth the teperature,

5 4th WSEAS Internatonal Conferene on EECTRONICS, CONTRO and SIGNA PROCESSING, Ma, Florda, USA, Noveber, 005 (pp whle the ndutanes s and σ vares wth the agnet rut saturaton of the otor. [3, 10]. The q axs voltage and the rtve power dyna referene odels ontans these paraeters, whle the algebra odels only ontans two of these paraeters. Thus expressng superor robust behavor of the algebra referene odel that the dyna one. Also, t s observed that the dyna referene odels are uh ore oplex that the algebra ones, buse t ontans ntegral operatons. Fgure 4 shows the behavor of kˆ s for the dyna referene odels. The detunng haratersts ntrodued n ths ase s saller than n the ase of algebra referene odels. Coparng both results t an be notes that the dyna referenes odels have better behavor that ther algebra equvalent. 6. Sulaton results The sulatons results were obtaned n an ndret feld orented ontrol of an nduton otor wth speed losed loop. A speed referene hange at t = 0.1 seg was appled and the k s vs t graph was obtaned. They were arred out for both, q axs voltage and rtve power algebra and dyna odels. The syste ntally s tuned and workng to rated flux wth 0% of the load. Fg. 5. Adaptaton behavor sulaton durng a referene hange. D Axs Voltage Referene Models. Fgure 5 shows the behavor of kˆ s for the d axs voltage referene odel. It s observed that ther behavor s slar to the ase of the dyna referene odels (fgures 4, ths allows to onlude that the detunng ntrodued n ths ase s not onsequene of the transents of, but to other errors that are not nluded. So the dervatve of was exatly predted wth the dyna referene odels. Fg. 3. Adaptaton behavor sulaton durng a referene hange. Q Axs Voltage and Rtvate Power Algebra Referene Models. Fgure 3 shows the behavor of kˆ s for the ase of the algebra referene odels. The fgure also shows that the kˆ s abandons ts tune value durng the transent of speed for both odels. That s way, a referene speed hange produes a transent of whh ntrodues an adaptaton error, although the syste was tuned ntally. Fg. 4. Adaptaton behavor sulaton durng a referene hange. Q Axs Voltage and Rtvate Power Dyna Referene Models. 7. Conluson In ths work t were analyzed the effets of the transents n the perforane of the adaptaton by ns of voltages and rtve power algebra referene odels. Also, a dyna odel of the ndret feld orented nduton otor that onsders ts transent behavor, dyna referene odels based on the voltages and n the rtve power were obtaned. Fnally, a oparatve analyss of the behavor of algebra and dyna referene odels was presented, nludng the sulaton results, whh show an perforane nrse through these last ones. Referenes: [1].- F. Blashke, The Prnple of Feld Orentaton ae Appled to New the Transvetor Closed-oop Control Syste for Rotatng-Feld Mahnes, Seens Revew, Vol.XXXIX, pp.17-0,197. [].- D. W. Novonty and P.C.Sen, Introduton to Feld Orentaton and Hgh Perforane AC Drves, Tutoral ourse presented at IEEE Ind. Appl. So. Annual Meetng, Sep

6 4th WSEAS Internatonal Conferene on EECTRONICS, CONTRO and SIGNA PROCESSING, Ma, Florda, USA, Noveber, 005 (pp [3].- R. Krshnan, F. C. They gld, Study of Paraeter Senstvty n Hgh-Perforane Inverter-Fed Induton Motor Drve Systes, IEEE Trans. Ind. Appl, Vol. IA-3, pp , Jul/Aug [4].- K. B. Nordn, D. W. Novonty and D. S. Znger, The Influene of Motor Paraeter Devatons n Feed-forward Feld Orentaton Drve Systes, IEEE Trans. Ind. Appl., Vol. IA-1, pp , July/Aug [5].- R. Krshnan, A.S. Bharadwaj, A Revew of Paraeter and Adaptaton n Vetor Indret Controlled Induton Motor Drve Systes, IEEE Transaton or Power Eletrons, Vol.6, pp , Ot [6].- R. D. orenz and D. W. Novonty, Saturaton Effets n Feld-Orented Induton Mahnes, IEEE Trans. Ind. Appl., Vol. IA- 6, pp , Mar/Apr [7].-. J. Garès, Paraeter Adaptaton for the Speed Controlled Stat AC Drve wth to Squrrel-Cage Induton Motor, IEEE Trans. Ind. Appl., Vol. IA-16, pp , Mar/Apr [8].- K. Ohnsh, Y. Ueda and K. Myah, Model Referene Adaptve Syste Aganst Rotor Resstane Varaton n Induton Motor Drve, IEEE Trans. Ind. Elet. Appl., Vol. IE- 33, pp. 17-3, Aug [9].- R. D. orenz and D.B awson, TO Splfed Approah to On-lne Contnuous Tunng of Feld-Orented Induton Mahne Drves, IEEE Trans. Ind. Appl., Vol. 6, pp , May/June [10].- T. M. Rowan, R. J. Kerkan and D. eggate, A Sple On-lne Adaptaton for Indret Feld-Orentaton of Induton Mahne, IEEE Trans. Ind. Appl., Vol. 7, pp , July/Aug [11].- R. Regnatto, Control por Capo Orentado Indreto do Motor of Indução o Adaptação of Parâetros Va MRAC, M. S. Dssertaton, CMI, UFSC. [1].- H. Sugoto and S. Taa, Seondary Resstane Identfaton of Induton - Motor Appled Model Referene Adaptve Syste and Its Charatersts, IEEE Trans. Ind. Appl., Vol. IA-3, pp , Marh/Aprl Annexes Induton otor paraeters: Three-phase nduton otor, star onneted, 0 Volt, 4 poles, 60 Hz, 1710 rp. X = Ω, X s = Ω, X r = Ω Sybols and Notatons: v v d and q axs Stator Voltages ds, ds dr λ ds λ dr, d and q axs Stator Currents., qr d and q axs Rotor Currents., λ d and q axs Stator Fluxes., λ qr d and q axs Rotor Fluxes. Supra ndex of the referene varables. * Supra ndex of output of the ref odels. Supra ndex of the varables of the dyn. odels. w e Synhronous frequeny. w r Rotor eletr angular speed. w sl Slp frequeny. k Slp frequeny gan. s 0 k s Tunng value of the slp frequeny. kˆ s Adaptaton value of the slp frequeny. x Detunng rato. e Adaptaton error. a