A Novel Method for Modeling Magnetic Saturation in the Main Flux of Induction Machine

Transcription

1 Poceeding of the 5th WSEAS Int. Conf. on Syte Science and Siulation in Engineeing, Teneife, Canay Ilan, Spain, Decebe 16-18, A Novel Method fo Modeling Magnetic Satuation in the Main Flux of Induction Machine A. NESBA 1, R. IBTIOUEN 1, S. MEKHTOUB 1, O. TOUHAMI 1 and S. BACHA 2 1 Depatent of Electical Engineeing National Polytechnic School BP 182 El-Haach Algie Algeia ALGERIA achid.ibtiouen@enp.edu.dz 2 Laboatoie d'electotechnique de Genoble- ENSIEG Intitut National Polytechnique de Genoble ENSIEG - BP Saint-Matin-d'Hèe Cedex FRANCE Seddik.Bacha@leg.enieg.inpg.f Abtact: - The phenoenon of atuation of electic achine contitute an ipotant ubject of invetigation in electical engineeing. Thi pape peent a new odeling technique fo atuation in the cae of induction achine. Neceay calculation to intoduce thi technique in the linea flux odel ae peented. Coputed eult and expeiental tet on the tanient of the elf-excited induction geneato ae clealy copaed. Thee expeiental and coputed eult agee cloely, validating the popoed technique a well a the developed odel. Key-Wo: - Induction achine, Geneato, Modeling, Satuation, Expeiental invetigation. 1 Intoduction The induction achine ha taditionally been odeled with the auption of linea agnetic in teady-tate and ot tanient analyi. Howeve, it i well known that the induction achine ae uually deigned to opeate in atuated ange. In fact, fo pope utilization of the agnetic ateial, the achine opeating egion ha to extend above the knee point of the agnetic chaacteitic. That technically foce the achine into atuation. Futheoe, in any vaiable toque application, it i deiable to opeate with the ain flux path in agnetic atuation, allowing an induction achine to poduce highe toque. To ipove the achine odeling and to enable a oe accuate pediction of the pefoance, it i, thu, neceay to conide the agnetic atuation [1]-[8]. Conventional odel geneally epeent thi phenoenon by the technique of atuation facto o by adjuting the agnetizing eactance [8]. Thee odel ae often inaccuate and do not give enough peciion, paticulaly duing tanient condition [8]. In thi wok, we peent a new technique of odeling of atuation in the cae of the induction achine, a well a calculation neceay to the intoduction of thi technique in the linea flux odel. Thi technique i baed piaily on a function called Satuation Degee Function (SDF). 2 Induction Machine Flux Model The induction achine voltage equation in the q-d axi abitay efeence fae ay be witten a [9] v = i + ω + p (1) v = i - ω + p (2) v q = q + (ω - ω ) d + p q (3) v d = d - (ω - ω ) q + p d (4) All oto quantitie and paaete ae efeed to the tato. p = d/dt, denote the diffeential opeato with epect to tie t. ω and ω ae the angula pee of efeence fae and oto epectively. v, v, v q and v d denote the q-d axi coponent of the tato, and oto voltage efeed to the tato epectively.

2 Poceeding of the 5th WSEAS Int. Conf. on Syte Science and Siulation in Engineeing, Teneife, Canay Ilan, Spain, Decebe 16-18, i, i, i q and i d, epeent the q-d axi coponent of the tato, and oto cuent efeed to the tato epectively.,, q and d epeent the q-d axi coponent of the tato, and oto fluxe efeed to the tato epectively. and denote tato and oto eitance efeed to the tato. The tato and oto cuent, in te of q-d axi fluxe can be witten - i = (5) - i = (6) q - q = (7) d - d = (8) whee and l denote tato and oto leakage inductance efeed to the tato. and, which ae ueful quantitie when epeenting atuation, denote the q- and d-axi coponent of the agnetizing flux. and ae defined by = M( i + q ) (9) = M(i +i d ) (10) whee M = 3 2 L (11) In which L epeent the tato agnetizing inductance. The ue of (5)-(8) help to eliinate the cuent in (9) and (10) a well a in the voltage equation given by (1)-(4), and if the eulting voltage equation ae olved fo the q-d axi fluxe, the following tate equation can be obtained p =v -ω+ ( -) l (12) p =v -ω+ ( -) l (13) p ( ω ω ) ( q =v q- d+ - q) (14) p ( ω ω ) ( d =v d- q+ - d) (15) The tanient elected fo veification of the popoed atuation odeling technique and the developed odel i the elf-excitation of the induction geneato, whee coheent eult in the iulation ae only achieved if atuation i taken into account by the odel. In thi opeating ode, the eactive powe i povided by a thee-phae capacito bank. The voltage-cuent equation of the excitation capacito ae expeed in the abitay efeence fae a [10] 1 p v = i - ω C v (16) e 1 p v = i - ω C v (17) e C e denote the excitation capacito. If the atuation effect i not accounted fo (the induction geneato i aued to be agnetically linea), the flux odel, in the abitay efeence fae, ay be obtained by the ue of the agnetizing flux expeion (9), (10), the et of conventionatate equation (12)-(15) and the elfexcitation equation (16)-(17). Howeve, the odeling of the elf-excited induction geneato unde the auption of agnetic lineaity lea to unealitic eult. Figue 1 how the influence of neglecting the atuation effect in the iulation and paticulaly in the coputed tato voltage duing the elf-excitation poce. It can be een that the voltage eache thouan of volt in le then half a econd. Thi coputed eult cannot be obtained expeientally, becaue the geneato output voltage i liited by the agnetic atuation phenoenon. The following ection peent the technique popoed in thi pape, which take into account the atuation effect in the odelling of induction achine. Stato voltage (V) Tie () Fig. 1. Coputed tato voltage duing the elfexcitation poce when the atuation effect i neglected.

3 Poceeding of the 5th WSEAS Int. Conf. on Syte Science and Siulation in Engineeing, Teneife, Canay Ilan, Spain, Decebe 16-18, Satuation Model The atuation effect in the ain flux path of the induction achine i accounted fo by ue of the Satuation Degee Technique. A entioned aleady, the pinciple of thi accuate technique i ainly baed on a function called Satuation Degee Function (SDF). Calculation involved fo the elaboation and ue of thi technique into the flux odel of the induction achine ae et foth in thi ection. 3.1 Contuction of the SDF The contuction of the SDF i caied out in the following way: Fit, we pefo the neceay eaueent fo the deteination of the tudied induction achine agnetizing cuve. (lin) F( (lin)) (19) (lin) Fig. 3, how the SDF cuve of the tet achine. Satuation Degee Function 0 Linea agnetizing flux, (lin) (Wb) Fig. 3. Vaiation of the SDF a a function of the linea agnetizing flux. Magnetizing flux (Wb) (at) Magnetizing cuent (A) Fig. 2. Magnetization cuve of the tudied induction achine. Then, we define fo any value of the agnetizing cuent (I ) the vaiation a = (lin)- (at) (18) whee: (at) epeent the actual value (atuated) of agnetizing flux, (lin) epeent the value of the agnetizing flux unde the auption of the lineaity (Fig. 2), epeent the attenuation of the agnetizing flux with egad to the value that it would have unde the auption of lineaity (i.e., by uppoing that the achine doe not atuate whateve the value of the agnetizing cuent i). The quantitie (at), (lin) and a function of the agnetizing cuent (I ) ae epeented on figue 2. Lat, the atuation degee function, noted F, i defined a follow 3.2 Modeling of the SDF The eaued point on the cuve of the SDF ae then ued to geneate a continuou cuve by ue of an analytical odel. In ode to axiize accuacy, a cobination of two exponential function i ued in thi odel. A leat-quae optiization ethod i applied fo the deteination of the odel coefficient. Thi odel wite a follow: F0 = 0 if 0 F = (20) F1 0 if > 0 with 1= C 1 [1 - (- λ 1( (lin) - 0))] F + C 2 [ exp( λ 2( (lin) - 0) / ( (lin) + exp (21) 1)) - 1] 3.3 Modified Expeion of the Magnetizing Fluxe In the linea odel of the induction achine appea two vey ignificant quantitie a fo the odeling of atuation, naely, the coponent of axe q and d of the agnetizing flux, and. The intoduction of atuation into the dynaic odel of the induction achine i eentially baed on the knowledge of thee two coponent in each opeating point. Thee coponent ae expeed in q-d vaiable by (at) = (lin) - (22) (at) = (lin) - (23)

4 Poceeding of the 5th WSEAS Int. Conf. on Syte Science and Siulation in Engineeing, Teneife, Canay Ilan, Spain, Decebe 16-18, The eulting agnetizing flux (lin) i elated to it coponent (lin) and (lin) by 2 2 (lin) = (lin) - (lin ) (24) The vaiation and ae given by (lin) = (lin) (25) (lin) = (lin) (26) Accoding to the definition of the SDF, thee expeion of and can be witten a: (lin)f( (lin)) (27) = = (lin)f( (lin)) (28) Theefoe, fo given value of (lin) and (lin) it i poible to deteine and fo (24), (27) and (28). The two coponent of the atuated agnetizing flux ae obtained fo (22 (24), (27) and (28), and can be witten a (at) = (lin)[1- F( (lin))] (29) (at) = (lin)[1 - F( (lin))] (30) To deteine the linea and atuated coponent of the agnetizing flux, we wite thei expeion in te of q-d axi cuent a: (lin) = M( i + q ) (31) (lin) = M( i + d ) (32) The expeion of the q-d axi cuent in te of atuated fluxe ae witten a follow (at) - (at) i= (33) (at) - (at) i = (34) q(at) - (at) q = (35) d(at) - (at) d = (36) By ubtituting thee equation fo cuent (33)- (36) in the expeion of the linea coponent of agnetizing fluxe (31) and (32), we obtain (lin) 1 1 (at) q (at) + (at) + = + (37) M l l Eliinating (at) in (20), by uing of (5) and (6), we aive at 1 1 q )+( + ) (lin)= Lq ( + (38) l l In a iila anne a fo (lin), the expeion of (lin) can be witten a follow 1 1 q (lin) = L d ( + )+ ( + ) (39) whee Lq = Ld = + + (40) M l To deduce the expeion of (at), we ue again (22) ((23) in the cae of (at) ) 1 1 q (at) = ( L ( + ) - 1) + L ( + ) (41) q q Uing the definition of L q a in (40), it can be een that L q L q( + ) - 1 = L ( + - )= - (42) q l l L q M Hence, (41) becoe q L q (at) = L q ( + ) - (43) l M In a iila anne a fo (at), the expeion of (at) can be witten a d d (at) = L Ld ( + ) - (44) l M 3.4 Equation of the Satuated Model Finally, the flux odel of atuated induction achine in the abitay efeence fae i obtained by ue of the expeion of linea and atuated coponent of agnetizing fluxe (38), (39), (43), (44) jointly with the et of conventionatate equation (12)-(15) given above. Fo elf excited induction geneato opeating ode, ued in the peent pape fo the veification of the developed odel, the voltage-cuent equation of the excitation capacito (16) and (17) ut alo be added to the developed odel. 1

5 Poceeding of the 5th WSEAS Int. Conf. on Syte Science and Siulation in Engineeing, Teneife, Canay Ilan, Spain, Decebe 16-18, Expeiental Veification The expeiental and coputed eult peented in thi ection (Fig. 4-9) decibe the dynaic epone of the induction achine duing the elfexcitation poce. The achine ued fo the tet i a wound-oto induction achine ated: 3.5kW, 220/380V, 14/8A, 50Hz, 4 pole. The elf-excited induction geneato opeating ode i obtained by diving the achine at ynchonou peed, and then a thee-phae capacito bank i witched onto the tato teinal of the achine. The thee-phae capacito bank ued in tet i 90µF. Figue 4 and 5 illutate the coputed and eaued tato voltage duing the elf-excitation poce, the achine being diven at 1500 p. The coputed and eaued tato cuent ae peented on figue 6 and 7. Figue 8 and 9 how the tanient of the oto cuent duing the elfexcitation poce. The copaion of thee eult how a vey good ageeent between the expeientation and iulation. The coheence between coputed and eaued eult i vey good, a well fo dynaic condition a fo teady tate. The aplitude of the ignal, thei hape a well a thei duation peent pactically the ae value fo both iulation and expeientation. The exaination of all thee eult how a vey good ageeent between the expeientation and iulation and thu confi the validity of the developed odel. Stato cuent (A) Stato voltage (V) Tie () Fig. 5. Meaued tato voltage duing the elfexcitation poce at 1500 p and 90µF. Tie () Fig. 6. Coputed tato cuent duing elf excitation poce at 1500 p and 90µF. Stato voltage (V) Stato cuent (A) Ti e () Fig. 4. Coputed tato voltage duing the elfexcitation poce at 1500 p and 90µF. Tie () Fig. 7. Meaued tato cuent duing elf excitation poce at 1500 p and 90µF.

6 Poceeding of the 5th WSEAS Int. Conf. on Syte Science and Siulation in Engineeing, Teneife, Canay Ilan, Spain, Decebe 16-18, Roto cuent (A) Ti e () Fig. 8. Coputed oto cuent duing elf excitation poce at 1500 p and 90µF. Roto cuent (A) Ti e () Fig. 9. Meaued oto cuent duing elf excitation poce at 1500 p and 90µF. 5 Concluion The SDF technique fo the odeling of atuation and the neceay calculation fo it intoduction in the induction achine dynaic odel ae peented. Expeiental and coputed eult that decibe the tanient of the voltage buildup poce of the elf excited induction geneato ae alo peented. The vey good ageeent between the expeiental and coputed eult how the peciion that the SDF technique povide and thu confi the validity of the developed atuated odel. [2] S. D. Sudhoff, D. C. Alipanti, B. T. Kuhn and P. L. Chapan, An Induction Machine Model fo Pedicting Invete Machine Inteaction, IEEE Tan. on Enegy Conveion, Vol.17, No.2, June 2002, pp [3] M. A. de Feita, D. A. de Andade, H. de Paula, J. L. Doingo, Effect of Magnetic Satuation on Induction Machine Diven by Static Convete, Poc. IEEE Intenational Electic Machine and Dive Confeence, 2003, pp [4] T. J. Sobczyk, K. Weineb, T. Wegiel, M. Suiowicz, A. Wazecha, Effect in Stato Cuent of Cage Moto Due to Satuation of Main Magnetic Cicuit, Sypoiu on Diagnotic fo Electic Machine Powe Electonic and Dive, Atlanta, GA USA, Augut 2003, pp [5] T. Kaieh, Y. Lefeve, and J.C. Hapiot, Modeling and expeiental chaacteization of atuation effect of an induction achine, Eu. Phy. J. AP 10, 2000, pp [6] G. Pugley, C. Chillet, A. Foneca, A-L. Bui- Van, New Modeling Methodology Fo Induction Machine Efficiency Mapping Fo Hybid Vehicle, Poc. IEEE Intenational Electic Machine and Dive Confeence, 2003, pp [7] E. Levi, M. Sokola, and S. N. Vukoavic, A Method fo Magnetizing Cuve Identification in Roto Flux Oiented Induction Machine, IEEE Tan. on Enegy Conveion, Vol.15, No.2, June 2000, pp [8] O.J. Ojo, Alfio Conoli and Thoa A. Lipo, An ipoved odel of atuated induction achine, IEEE Tan. on Ind. Appl., vol. 26, no. 2, pp , Mach /Apil [9] P.C. Kaue, Analyi of electic achiney, McGaw-Hill, [10] A. Neba, R. Ibtiouen, S. Mekhtoub, O. Touhai, and N. Takoabet, Rectified Self- Excited Induction Geneato a Regulated DC Powe Supply fo Hybid Renewable Enegy Syte, WSEAS Tan. on Cicuit and Syte, Iue 11, Vol. 4, pp , Novebe Refeence: [1] D. Bipo, L. M. Neto, J. Tacíio de Reende, and D. Alve de Andade, A New Stategy fo Induction Machine Modeling Taking Into Account the Magnetic Satuation, IEEE Tan. on Induty Application, Vol. 37, No. 6, Nov./Dec. 2001, pp