Journal of Advanced Reearch in Science and Technology ISSN: 2352-9989 Direct Torque Control of Saturated Induction Machine with and without peed enor Tahar Djellouli,2, Samir Moulahoum, Med Seghir Boucherit 3 Reearch Laboratory of Electrical Engineering & Automation Univerity Dr Yahia Fare, Ain D heb, 26, Medea, ALGERIA 2 Département of Science and Technology, Univerity of Ghardaia, 47, Algeria 3 Laboratoire de commande de proceu LCP, ENP-Alger,, Avenue Pateur El Harrach, B.P 82, 62, Alger, Algeria Abtract. In thi paper, a modified KALMAN filter i propoed to etimate peed. At firt, the influence of the magnetic aturation i taken into account in the modelling. In the econd part, the direct torque control (DTC) i elaborated; the control of the peed loop i enured by an IP controller, the flux and the torque are etimated from ource voltage and meaured current. The lat part of thi work i devoted to the operating ytem without mechanical enor, uing a KALMAN filter a a peed oberver. Simulation reult are preented to verify the effectivene of the propoed approach. Keyword: DTC, Senor, KALMAN filter, induction machine, aturated model, peed oberver.. Introduction To tudy the control of any ytem, one of the mot important part i the ytem modelling. The induction machine i not a imple ytem, becaue of numerou complicated phenomena which affect it operation, uch a aturation, eddy current, kin effect etc... However, firtly thee phenomena will not be taken into account; thi allow obtaining imple equation which reflect accurately the machine operation []. 2. Modeling of the induction motor The application of Concordia tranformation to the rotor and tator winding reult on the following equation of the induction machine in the d-q reference frame [, 3]: 25 JARST. All right reerved dφ V = R I + + jω Φ () dt dφr Vr = Rr Ir + + jωl Φ r (2) dt Φ = Lσ I + Φ m, m Lm Im Φ r = Lσr Ir + Φ m, Im I Ir Φ = (3) = + (4) T = P( Φ I Φ I ) (5) em d q q d * Correponding author. E-mail: djelloulitaha@yahoo.fr (Djellouli T.). Addre: LREA, Univerity Dr Yahia Fare, Ain D heb, 26, Medea, ALGERIA 82
Djellouli T. et al., Journal of Advanced Reearch in Science and Technology, 25, 2(), 82-89. R, R tator and rotor reitance; r Lσ, Lσ r tator and rotor leakage inductance; mutual inductance; L m Φ, Φ tator and rotor flux vector; r V, V tator and rotor voltage vector; r I, I tator and rotor current vector; r Φ, magnetizing flux and current vector; m Im ω, ω ynchronou and lip angular peed; l T em electromagnetic torque. method and can be preented by the following matrix form a [3, 4]: V L I R I [ ] = [ ][ ] + [ ][ ] (6) [ V ] = V, V,, T [ ],,, [ L] Lσ + M d M dq M d M dq M dq Lσ + M q M dq M q = M d M dq Lσr + M d M dq M dq M q M dq Lσr + M q [ R] R ωl ωlm ωl R ωlm = ω l Lm Rr ω l Lr ωl Lm ωl Lr Rr M d =L mdy co 2 α+l m in 2 α : Mutual inductance of the axi d Mq=L m co 2 α+l mdy in 2 α: Mutual inductance of the axi q d q T = d q dr qr I I I I I M dq =(L mdy -L m )co α inα : Term explaining the cro effect between the axi in quadrate 25 JARST. All rightreerved M dy and L m are the dynamic and the tatic mutual inductance', repectively. α i the angle between the d axi and the magnetizing current I m. A can be een in the Fig., in teady tate, the difference i clear between the aturated model and the linear model. The intantaneou torque i maximal at tarting, after that it i tabilized to compenate the loe at no-load. From Fig., we can oberve that there i a difference between the torque of aturated model compared to linear model, which explain the lower tranient of the peed with thi model compared to the linear model. 83
Djellouli T. et al., Journal of Advanced Reearch in Science and Technology, 25, 2(), 82-89. 6 4 45 4 2 35 3 Speed(Rad/) 8 6 4 2 Torque (N.m) 25 2 5 5-2.2.4.6.8.2.4.6.8 2-5.2.4.6.8.2.4.6.8 2 3 6 2 aturated model 4 aturated model 2 Stator current (A) - Stator current (A) -2-4 -2-6 -3.2.4.6.8.2.4.6.8 2 Time().25.22.225.23.235.24.245.25 Time() Fig.. Start up following by a load application of an induction motor: linear model (blue curve), aturated model (green curve). 3. Direct torque control (DTC) Direct Torque Control (DTC) of an induction machine i baed on adequate voltage ource inverter. In a tator reference frame, the intantaneou value of tator flux and electromagnetic torque are etimated from the tator magnitude. Uing hyterei comparator, the flux and the torque are controlled directly and independently with an appropriate election of voltage vector impoed by the inverter. The inverter provide eight voltage vector. Thee vector are choen from a witching table baed on error of flux and torque and the tator flux vector poition (Tab.). Application of a tator voltage vector which make poible to decreae or to increae the tator flux and the electromagnetic torque in the ame time Tab. Switching table uing hyterei comparator of torque and flux N φ Cem 2 3 4 5 6 V 2 V 3 V 4 V 5 V 6 V V 7 V V 7 V V 7 V - V 6 V V 2 V 3 V 4 V 5 25 JARST. All rightreerved V 3 V 4 V 5 V 6 V V 2 V V 7 V V 7 V V 7 - V 5 V 6 V V 2 V 3 V 4 The bloc diagram of the DTC i hown in Fig.2. 84
Djellouli T. et al., Journal of Advanced Reearch in Science and Technology, 25, 2(), 82-89. Fig.4 how the imulation reult of the direct torque control applied to a aturated induction machine for a nominal reference peed and nominal load application. The real peed i obtained from the mechanical enor. The obtained imulation reult how that the DTC i a robut control. The flux and the torque are decoupled and follow their reference. The real peed track it reference in good agreement. In addition, even with the preence of magnetic aturation, the DTC i operate correctly and there i no need to be modified Fig.2. General tructure of the DTC with mechanical enor 6 6 4 4 2 Refecence 2 Speed(Rad/) 8 6 4 Stator Current (A) -2-4 2-6 -8-2.2.4.6.8.2.4.6.8 2 Time().3.4.5.6.7.8.9.2.2 3 25 2 25 JARST. All rightreerved -5.2.4.6.8.2.4.6.8 2 Time() Fig.3. DTC of the induction motor: nominal reference peed and nominal load application Torque(N.m) 5 5 85
Djellouli T. et al., Journal of Advanced Reearch in Science and Technology, 25, 2(), 82-89. 4. DTC Without peed enor by EKF The machine peed i obtained through a mechanical peed enor. However, thi enor require a place for it intallation and lead to difficultie in it mounting; it i enitive to noie and vibration. Several trategie have been propoed in the literature to eliminate thi mechanical enor. Among thee trategie, there i the etimation by the extended Kalman filter (EKF). The Kalman filter i an oberver for nonlinear cloed-loop whoe gain matrix i variable. At each calculation tep, the Kalman filter predict the new value of tate variable of the induction machine (current, flux and peed). Thi prediction i made by minimizing the noie effect and modelling error of the parameter or the tate variable. The noie are uppoed to be white, Gauian and not correlated with the etimated tate [5]. The extended Kalman filter a any other oberver i baed on the ytem model. The output equation i: L I (R ) r L r α + Tr ωσ Tr ω Iα σl I β L I β d ωσ r L (R + ) ω r u α Tr Tr α L α φ = φ + σ dt L σ LR u φ σ β φβ β σ LR Ω Ω Iα I αs I βs αs βs Iβ = φ φ Ω T (7) 25 JARST. All rightreerved The equivalent dicrete filter i neceary for the implementation of the EKF in real time. It i aumed that the control input U(kT) i contant between the actual ampling intant [kt] and the previou ampling intant [(k +)T]. Thu, the dicrete model of the machine in extended form become: I α (k+ ) ( + at) a2p ΩT a3t a I 4pΩ T α (k) I β(k + ) a I 2p T ( at) a4p T a3t (k) Ω + Ω β φ α (k+ ) = a5t φα (k) φ β(k + ) a5t φβ (k) (k ) Ω + Ω(k) σl (8) u α (k) + T u β(k) σl I α (k+ ) = I α (k) I β (k) φ α (k) φ β (k) Ω(k) I βs (k ) + a t = R 5 (3) T Tr R + L a = a2 = a3 = a4 = σ LTr σ LTr σl It i aumed that the matrix of the tate vector P and the matrice Q & R of the meaurement noie are diagonal. 86
Djellouli T. et al., Journal of Advanced Reearch in Science and Technology, 25, 2(), 82-89. There are two tep to implement the EKF algorithm, the firt i the prediction, the econd i the correction, and thee two tep are introduced by an initialization of tate vector X and the covariance matrix P, Q and R. The firt etimation of the tate vector at time (k+) i: X(k ˆ + ) = f X(k ˆ / k), U(k), k (9) Thu, thi meaure tate allow the prediction of the output: Y(k ˆ + /k) = CX(k ˆ + /k) () The prediction covariance matrix of the filter i given by the following formula: Then: P(k+/k)=A(k)P(k/k)A T (k)+q () ( + at) ap 2 Ω(k/k)T at 3 ap 4 Ω (k/k)t (ai 2βS (k) + a 4φ β (k))t ap 2 (k/k)t ( at) ap 4 (k/k)t at 3 (ai 2 (k) a 4 (k))t Ω + Ω α + φα A(k) = at 5 at 5 Finally, the new value of the etimated tate vector at time (k +) i given by: X(k ˆ + / k + ) = X(k ˆ + / k) + K(k + ) Y(k + ) Y(k ˆ + / k) (2) The calculation of the error covariance i a follow: P(k+/k+) = {I - K (k+) C} P (k+/k) (3) Therefore, in the DTC without mechanical enor, the etimated peed i ued only for the control. The Kalman filter alo etimate the electromagnetic torque and the component, the magnitude and the ector of tator flux. Thi allow the complete elimination of the two etimator of torque and flux preented previouly. Thu, Only the KALMAN filter which give all the etimated quantitie that the DTC need. Fig.4 illutrate the cheme of thi control. 25 JARST. All rightreerved 87
Djellouli T. et al., Journal of Advanced Reearch in Science and Technology, 25, 2(), 82-89. DC Voltage Inverter IM Concordia Tranform Switching Table KALMAN FILTER Speed Controller Fig.4. General tructure of the direct torque control without mechanical enor by the ue of the extended KALMAN filter The imulation reult of the DTC without mechanical enor control i hown in the Figure 5 and 6. The inertion of the Kalman filter in the DTC, give good performance, the etimated quantitie follow perfectly the reference quantitie with a light error of etimation in tranient. The Sytem behaviour i good even in the preence of magnetic aturation; however, the mutual inductance value adaptation i introduced inide the EKF algorithm. 6 6 4 2 4 Speed (Rad/) 8 6 4 Reference peed Etimated peed Real peed Stator Current (A) 2-2 2-4 -2.2.4.6.8.2.4.6.8 2-6.4.5.6.7.8.9 3.5 25 Real Torque Etimated Torque 2 Fig.5. without peed enor DTC by EKF applied to the linear model 25 JARST. All rightreerved.5 Torque (N.m) 5 Fi Béta (Wb) -.5 5 - -5.2.4.6.8.2.4.6.8 2 -.5 -.5 - -.5.5.5 Fi alpha (Wb) 88
Djellouli T. et al., Journal of Advanced Reearch in Science and Technology, 25, 2(), 82-89. 6 4 6 Speed (Rar/) 2 8 6 4 2 Reference peed Etimated peed Real peed Stator Current (A) 4 2-2 -4-6 -2.2.4.6.8.2.4.6.8 2.2.22.23.24.25.26 Time() 3.5 25 2 Real Torque Etimated Torque.5 Torque (N.m) 5 Fi Béta (Wb) -.5 5 - -5.2.4.6.8.2.4.6.8 2 -.5 -.5 - -.5.5.5 Fi alpha (Wb) Fig.6. without peed enor DTC by EKF applied to the aturated model 5. Concluion Thi paper preent an approach for the induction machine modeling including magnetic aturation. The aturated model i more accurate than the implified model in every facet of the prediction of machine performance. The main baic concept of direct torque control DTC are preented. Thi control can be performed by uing a uitable choice of inverter voltage vector. Simulation reult chow the robutne and the advantage of thi control, uch a the no need of the magnetic aturation compenation. The application of the Extended KALMAN Filter (EKF) for the direct torque control (DTC) without peed enor give an excellent performance; the machine electrical quantitie are perfectly etimated. The reult obtained how the need for the adaptation of the mutual value function of the aturation level inide Kalman filter algorithm. 6. Reference [] Va. P 99 Vector control of AC machine, Oxford/UK, Clarendon Pre, [2]. Levi. E, Wang. M 22 - A peed etimator for high performance enorle control of induction motor in the field weakening region, IEEE tran. on Power Electronic, vol.7, no. 3, pp. 365-378. 25 JARST. All rightreerved [3]. Moulahoum S., Baghli L, Rezzoug. A,Touhami O,28- "Senorle Vector Control of a Saturated Induction Machine accounting for iron lo", European Journal of Electrical Engineering, EJEE, Lavoiier, Hermè Science, Vol :, N :4/5, pp 5-543, [4]. Boldea. I, Naar. S.A,987- Unified treatment of core loe and aturation in orthogonal axi model of electric machine, Proc. IEE, vol. 34, pt. B, pp. 353-363, [5].KIM.Y.R,.SUL.S.K,PARK. M.H,994 - Speed enorle vector control of induction motor uing extended Kalman filter, IEEE Tran. on Indutry Application, vol.ia-3, no 3, pp. 225-233. [6].Zhong. L, Rahman. M.F, Hu. W.Y,and Lim. K.W,997."Analyi of direct torque control in permanent magnet Synchronou motor drive, "IEEE Tran. Power Electron.,vol.2,pp.528-536. [7]. Zhengyun. R, Huade. L, and Shujin. C,26 -Application of Optimized EKF in Direct Torque Control Sytem of induction motor. -7695-266-/6 $2., IEEE 89