Robustness Analysis of Stator Voltage Vector Direct Torque Control for Induction Motor

Transcription

1 IX Symoium Indutial Elctonic INEL, Banja Luka, Novm 3, Routn Analyi of Stato Voltag Vcto ict oqu Contol fo Induction Moto Alkanda Ž. Rakić, Sloodan N. Vukoavić Univity of Blgad, School of Elctical Engining Blgad, Sia Pta R. Matić Univity of Banja Luka, School of Elctical Engining Banja Luka, Rulic of Ska, Bonia and Hzgovina Atact In thi a, outn analyi of Stato Voltag Vcto ict oqu Contol (SVV C fo induction moto (IM againt oto tim-contant unctainty i fomd uing modn lina out thoy. Aoiat tuctud out modl of th lant i divd and fquncy domain tu i ood to tt out taility, a wll a nominal and out fomanc in th n of clod-loo t-oint tacking. Rult of th analyi confim full taility outn. id nominal and out fomanc i confimd in th moto fild-wakning gim, whil fomanc dgadation i found at low moto d. Kywod: Induction Moto, Fild Wakning, oqu Contol, Rout Staility, Rout Pfomanc. I. INROUCION In IM div, toqu and flux a could. Attmt ha n mad [ 6] to achiv and vify high fomanc contol in th fild wakning gim, uing tato voltag angl a th only indndnt contol vaial. Gnalization of thi aoach i ood in [7] and namd Stato Voltag Vcto ict oqu Contol (SVV C. Algoithm imultanouly contol moto toqu and oto flux y uing tato voltag angl and amlitud in th a d gion, whil th voltag angl only i ud in th fild wakning gion. Routn of th SVV C olution againt oto tim-contant unctainty i not ntd in [7] and it i th ujct of thi a. h a i oganizd a follow. In Sction II aoiat out modl of th gulatd lant i divd. Nonlina tat-ac modl of IM i ntd, a wll a linaizd modl, modl aoximation and ntial chaactitic of dcould contol fom [7]. Alo, tuctud out modl of th lant i divd to fit th modn lina out thoy [8-9]. Routn analyi in th fquncy domain i ntd in Sction III. Fquncy tu i ood to tt clod-loo outn oti. Rout taility, nominal fomanc, and out fomanc a analyzd and ult a dicud in aat uction. Concluion i givn in Sction IV. II. ROBUS MOELING A. Nonlina Modl of Induction Moto h nonlina tat-ac modl of th IM, in nomalizd (-unit [.u] valu, i givn a follow: d = A (, [ ] + U, ( dt / k / / k / A (, =, ( k / / k / / t = σ l ( q Q d. (3 wh th tat vcto conit of tato and oto flux: = d q Q. (4 h modl aum: ynchonou fquncy and voltag modulu U a th contol vaial, moto toqu t and oto flux modulu Q = + a th outut vaial, mchanical tanint much low than lctical on, and th fnc fam alignd with tato voltag: ud = U, u =. (5 In th modl ( (4, tand fo lativ moto li, i a d, R and R a tato and oto itanc, L and L a tato and oto lf inductanc, M i mutual inductanc, k = M/L and k = M/L a tato and oto couling cofficint, = σ L / R and = σl / R a tato and oto tanint tim contant in [.u], σ = M / L L i lakag cofficint and l i tato inductanc in [.u]. q 8

2 B. Linaizd Modl Fo th givn oation gim (, = (,, talihd y tationay contol voltag U = U, nonlina modl ( (3 giv tady-tat vcto of flux,, = A U. (6 Auming incmnt of th contol vcto u = U U (7 oduc incmnt of th tat vcto and outut vcto, ctivly: x =, (8 y = t, (9 t linaizd modl i found in [7] a: (, (,, (, x = A x + B u y = C x, ( wh (, (, A i givn in ( fo (, = (,, and q d Q (, B =, ( k k k k Q q d σ l σ σ σ = l l l Q C.( C. Linaizd Modl Aoximation and couling Contol Fom th tat-ac linaizd modl (-(, tanf function matix can found a G = C I A B. (3 Sinc analytic olution fo th tanf function out of tat-ac ntation ( ( wa found [7] to ath comlicatd, in that a aoximatd tanf function matix wa adotd, in which th tatic gain matix and common ai of comlx-conjugat oto-latd ol hav n includd a th function of oation gim (,. It wa alo hown in [7] that nonlina tatic dcoul achiv tatic gain dcouling and ffctiv dcould lant modl wa adotd in th fom G dc = / n + ( ζ / n +, (4 wh natual fquncy and lativ daming atio w adotd fo th unloadd moto and thy w found to, ctivly: ζ = n = ( + ( kk ( + ( + kk ( + ( + ( kk, (5, (6 Sinc oth toqu and flux channl of dcould lant hav th am tanf function, toqu and flux contoll w ood with th am ootional-intgal tuctu and th am aamt: / z + K = Kt = K K = i, (7 wh K i i contoll intgal gain and z i fquncy of th contoll zo, ctivly: K i = =, (8 ( k k z = min, +. (9 id clod loo oti fo oth channl of contol w: andwidth a dcad fo th limit fquncy of oto ol, and ha magin of 9 dg.. Rout Modl In od to valuat outn of th contol ood in [7] to oto tim-contant unctainty, aoiat out modl of ach channl tanf function i to divd. Roto-latd ol oition, ut without th unloadd moto aumtion, w alo found in [7]: ( ( kk +, (, = +. ( / ± kk j + + ( Natual fquncy of oto-ol i givn y wh: (,, = k + k + k, ( k =, ( ( ( k = k k / +, (3 ( k = k k + / +. (4 8

3 Intoducing th fouth aamt { } k k + ( ( 4 = R, = k channl tanf function Gc ( + = = R{, } + +,, can dcid y th lock diagam givn in Fig.., (5 Figu. Stuctual lock diagam of th channl tanf function. (6 Howv, th actual valu of th oto tim-contant i highly dndnt, mainly ut not only, on th IM woking tmatu. It can dviat ignificantly fom it umd nominal valu vn duing nomal oation of th moto. hfo, out IM modl hould incooat adquat infomation on th oto tim-contant unctainty. Auming th oto tim-contant i unctain, ut li within th cntag ang R of th nominal valu, th aoiat dcition of th unctainty i ( δ = + R, (7 wh δ i th unknown al valu in th nomalizd unity ang δ. (8 Block diagam of th channl tanf function with incooatd unctainty dcition (7 i givn in Fig.. It i quivalnt to th gnic lock diagam of th tuctud out modl in Fig. 3, whn th known at P of th lant i givn y th tat-ac matic, ctivly: A =, (9 k ( k + k / / k4 / B =, (3 R / R k + ( k + k / / k / / / k + k C =, (3 k4 / R k / = R / R ( k + k / /. (3 Figu. Stuctual lock diagam of th channl tanf function with th unctainty dcition of th oto tim contant δ = δ w = w w z z = z Figu 3. Gnic modl with tuctud unctainty 83

4 III. CLOSE-LOOP ROBUSNESS ANALYSIS A. Fquncy omain Stu h outn analyi tu i ntd in Fig. 4. Figu 4. Block diagam of out fnc tacking contol tu. Following ignal a dnotd: th fnc (t oint ignal, = y i th o in fnc tacking, - th fomanc wightd o, u th contol ignal, w unctainty outut, and z unctainty inut. Rlvant tanf function a: K( th contoll (7 (9, P th known at (9 (3 of th unctain lant, an unknown unctainty of modling, and W ( th fomanc wighting function. h fomanc wighting function i adotd in th fom * * + M S W =, (33 M * S with th choic of aamt nuing adquat fnc-tacking fomanc valuation: did clodloo andwidth (a tnth of oto ol limit fquncy * / =, (35 and tight ound fo maximum nitivity [8 9] M =.. (34 If known at P of th lant i atitiond a z y * S P P = P P w u, (36 lvant clod-loo tanf function matic a: N = P P K I + P K P (tanf w z, (37 N = W I + P K P (tanf w, (38 N = P K I + P K (tanf z, (39 N = W I + P K (tanf, (4 and indx of clod-loo out taility, nominal fomanc, and out fomanc a, ctivly: RS = max µ N ( j, (4 ( NP = max σ N ( j, (4 = max µ N( j N ( j N ( j N ( j, (43 wh: σ ( i u ingula valu, and µ ( i tuctud ingula valu. h clod-loo ytm ha out taility (it i tal fo vy tuation of unctain aamt if RS <, (44 whil nominal fomanc and out fomanc a atifid if, ctivly: NP <, (45 <. (46 Condition (44 i mandatoy cau RS > man th i at lat on tud valu that mak clod-loo ytm untal. Violation of condition (45 man th clod-loo ytm with nominal lant (nominal do not fulfill did andwidth (35 o/and maximum nitivity (34. Violation of condition (46 man th i at lat on tud valu that mak clod-loo ytm not fulfilling did andwidth (35 o/and maximum nitivity (34. B. Nominal Pfomanc Rult Nominal fomanc indx NP (4 with th tt moto (aamt givn in Andix fo incaing ynchonou d at diffnt tady tat li i lottd in Fig. 5. NP [.u] Figu 5. Nominal fomanc indx NP. = =.5 =. =.5 84

5 It can ovd that nominal fomanc i not dndant th tationay moto li, i.. th moto load. Nominal fomanc i not fulfilld only at vy low ynchonou d inc (45 not atifid at <..u. C. Rout Staility Rult Fo th tt moto modl with th unctain oto timcontant within th lu/minu cntag ang R = 99% (47 of th nominal valu, out taility indx RS (4 i lottd in Fig. 6 fo incaing ynchonou d at diffnt tady tat li = =.5 =. = x = =.5 =. = [.u] Figu 7. Rout fomanc indx fo unctainty ound R = 99%..4 RS 6.35 = = =. = [.u].5. Figu 6. Rout taility indx RS fo unctainty ound R = 99%.. It can ovd that out taility condition (44 i fulfilld fo th full ang of moto d and load. hfo, out taility i achivd ov th whol t of tud lant and in all oating gim.. Rout Pfomanc Rult Rout fomanc indx (43 i lottd in Fig. 7 fo th tt moto modl with th unctain oto timcontant within th lu/minu cntag ang (47. It can ovd that out fomanc condition (46 i not fulfilld fo th unloadd moto ( = on th full ang of moto d. Howv, thi chaactitic xcd ounday valu of fo l than 5% in th fild-wakning gim, wh th contol chma fomanc i ntial. Moov, all loadd chaactitic ( > achiv l than 5% dgadation tating fom th half of th nominal moto d, and th fully achiv out fomanc in th whol fild-wakning aa. Rout fomanc dgadation i futh tudid fo tight unctainty ound. In Fig. 8-9, out fomanc indx i lottd fo unctainty cntag ang, ctivly: R = 8%, (48 R = 5%. ( [.u] Figu 8. Rout fomanc indx fo unctainty ound R = 8% [.u] = =.5 =. =.5 Figu 9. Rout fomanc indx fo unctainty ound R = 5%. 85

6 On oth lot, out fomanc i fully achivd in th whol fild-wakning aa, and ach of out fomanc ound a ignificant only u to th half of th nominal moto d. ight unctainty ound (48 nal full out fomanc achivmnt fo all loadd chaactitic tating fom =.75.u, whil th unloadd chaactitic xcd unity ound on th whol d ang. Evn tight unctainty ound (49 mak all th chaactitic fully achiv out fomanc tating fom th d =.75.u. IV. CONCLUSION In th a out contol thoy wa utilizd to vify outn oti of th SVV C olution fo IM contol. Conductd outn analyi ovid vidnc that clod-loo ytm will tal oth in a-d and fild-wakning gim, vn with high unctainty of timatd oto tim-contant. Nominal and out fomanc i ovn to fully achivd only in th fild-wakning gim, whil fomanc dgadation i dtctd in th low-d gion. Howv, SVV C olution i intndd fo high-d low-cot IM div woking in d fild wakning ang, o a-d fomanc i not ntial. Pood analyi i conductd on th tuctud out modl of th dcould lant channl tanf function. Futh wok could dictd towad mo comlx analyi, ad on th oiginal multivaial lina IM modl, which could giv d inight on achival out oti and otntially tt contol dign. APPENIX Moto data: 75W, 95V, 7Hz, R =.8Ω, R = 5.673Ω, L = L =.55 H, L m =.58 H. REFERENCES [] P. Matić, A. Rakić, and S. Vukoavić, "Induction Moto oqu Contol in Fild Wakning y Voltag Angl Contol," in Poc. 4th Int. Pow Elctonic and Motion Contol Conf. (EPE/PEMC, Ohid, Macdonia, S. 6 8,, [] P. Matić, A. Rakić, and S. Vukoavić, "ict oqu Contol of Induction Moto in Fild Wakning Bad on Gain-Schduling Aoach," in Poc. 6th Int. Sym. Pow Elctonic (E, Novi Sad, Sia, Oct. 6 8,. [3] A. Rakić, P. Matić, "Rout Modling and Rfnc acking Contol of Voltag Angl Contolld Induction Moto in Fild Wakning Rgim," in Poc. INEL Conf., Banja Luka, Rulic of Ska - Bonia and Hzgovina, Nov. 4 6,, [4] P. Matić, I. Kčma, A. Rakić, S. N. Vukoavić, "Imovd Induction Moto Pfomanc in th Fild Wakning Bad on th PI Contoll Gain Scduling (in Sian," in Poc. th Infoth- Jahoina Conf., Jahoina, Rulic of Ska - Bonia and Hzgovina, Ma. 6 8,, Vol., Rf. A-7, [5] A. Rakić, P. Matić,. Ptović, "Rout Modling and Gain- Schduling Contol of th Induction Moto in th Fild Wakning Rgim," in Poc. LV ERAN Conf., Banja Vućica-lić, Rulic of Ska - Bonia and Hzgovina, Jun. 6-9,,. AU [6] P. Matić, A. Rakić, S. N. Vukoavić, "Sac Vcto Induction Moto Modl in Fild Wakning (in Sian," in Poc. LV ERAN Conf., Banja Vućica-lić, Rulic of Ska - Bonia and Hzgovina, Jun. 6-9,,. EE [7] P. R. Matić, A. Ž. Rakić, and S. N. Vukoavić, "Stato Voltag Vcto ict oqu Contol of Induction Machin," in Poc. 5th Int. Pow Elctonic and Motion Contol Conf. (EPE/PEMC, Novi Sad, Sia, S. 4 6,. [8] S. Skogtad, I. Potlthwait, Multivaial Fdack Contol. England: John Willy & Son, 996. [9] K. Zhou, J. C. oyl, K. Glov, Rout and Otimal Contol. Nw Jy: Pntic-Hall,