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OPTIMAL EFFICIENCY BASED PREDICTIVE CONTROL OF INDUCTION MACHINE INCLUDING MAGNETIC SATURATION AND SKIN EFFECT K. Barra*, T.Bouktir* and K.Benmahammed** * Department of Electrical Engineering, University of Larbi Ben M’hidi, Oum El Bouaghi, 04000 Algeria. ** Department of Electronics, University of Setif, 19000 Algeria. Abstract- In this paper a minimum-loss control algorithm MLCA is applied to improve high efficiency of an induction motor. The efficiency optimisation is done by adjusting the rotor flux level to an optimal value as a function of the operating conditions (such as torque and speed). Magnetic saturation and skin effect are taken into account and their influence is discussed. The application of the generalized predictive control GPC algorithm is based on the input-output linearised model. Simulation studies show the performance of the proposed algorithm. Key-Words : minimum loss control, magnetic saturation, skin effect, Generalized predictive control, Direct field oriented control method, induction motor. In addition to its advantages, such as speed capability, robustness, cheapness and ease of maintenance, when used with a field oriented control scheme, the induction machine is the most widely used electrical machines. The usually used direct field oriented control methods keeps rotor flux at constant rated value and ignores the machine losses. Consequently, the efficiency of the motor is poor especially when the load torque is low or zero. The main losses, about 90 % of the total losses are copper, core losses and stray-load losses [1-4]. A good machine design dictates that operation around rated operating condition corresponds to high efficiency operating point. Whenever it is possible to obtain the required torque with less than the rated flux, it is possible to reduce the flux decreasing the magnetising current . A reduction in the magnetic flux results in a reduction in the core losses and a decrease in the magnetising current results in a decrease in the stator and rotor copper losses, due to the smaller magnetising current needed [2],[3]. One way to reduce the losses is to adjust the flux level as a function of the operating conditions such as torque and speed. 2. Induction motor model It is well known that the linear model of a squirrel-cage induction motor neglect the equivalent core losses resistance (generally connected in parallel with the statorrotor mutual inductance), with that model, all parameters are considered constant and the core losses are not taken into account. In order to consider such losses, the proposed model in [2] is used, where the equivalent core losses resistance Rfs is connected in series with the stator-rotor mutual inductance. The core loss resistance is a function of frequency and flux level, but the change of Rfs value depends on the variation of frequency larger than that of rotor flux [1]. Therfore Rfs is approximately a function of frequency only. The stator equivalent core loss resistance is determined from the classical experimental no-load test data as shown in figure1. by : R fs = a. fs + b. fs 2 (1) a and b are the coefficients of hysteresis and eddy current losses respectively. fs is the stator flux frequency. At low speed, Rfs can be neglected comparing to stator resistance but at high speed, the stator flux frequency is almost the same as the rotor speed frequency (the slip frequency is nearly zero and the rotor core losses can be neglected). 14 Stator c urrent v alues 12 Equivalent c ore los ses res is tanc e [Oh m] 1. Introduction fit ted curve 0.25 A 0.5 A 1A 10 8 6 4 2 0 0 5 10 15 20 25 30 frequency [Hz] 35 40 45 50 Fig. 1. Stator core losses equivalent resistance versus stator current frequency. The induction motor model used under field oriented control FOC can be expressed in the synchronously rotating d-q reference frame as follows: σ r .R fs di .i ds + σ .L s . ds − σ .L s .w s .i qs + V ds = R s + dt 1+ σ r R fs dφ dr (1 − σ )(. 1 + σ s ) .φ dr + dt (1 + σ r ).L m di σ r .R fs .i qs + σ .L s . qs + σ .L s .w s .i ds + V qs = R s + dt 1+ σ r (2) (1 − σ )(. 1 + σ s ).ws .φ dr dφ Tr . dr + φ dr = L m .i ds dt w s = w + w sl L m .i qs w sl = Tr .φ dr Tem = p.(1 − σ )( . 1 + σ s )φ dr .i qs Assuming that none of the motor parameters has any dependence on the rotor flux, under the constant load torque, we can obtain the rotor flux providing the minimum loss by setting the derivative of the equation (8) to zero ∂Ploss (9) =0 ∂φ dr The relationship that minimises the criterion Ploss as function of torque and speed can be obtained by: ∗ 1/ 2 (φ dr * ) S = k opt . ( Tem ) (10) with : 1/ 2 kopt where σs = ls lr 1 , σr = , σ = 1[(1 + σ s ).(1 + σ r )] Lm Lm 1/ 2 σ r .R fs Rr R + + s Lm ( 1 + σ r )2 (1 + σ r ) = Rs + R fs .1+σs ) p.(1 − σ )( (11) 400 - proposed 350 . conventional ( P js = R s . i ds 2 + i qs 2 ) (3) Power losses [W] 300 2.1 Induction machine losses model Stator copper losses : Rotor copper losses: ) Stator core losses: ( Rr (1 + σ r ) ) 2 2 φ . dr − i ds + i qs 2 Lm R fs φ Pfs = .σ r . ids 2 + iqs 2 + dr .ids 1+σr Lm 0 L2m ) 1 1.5 Fig. 2. Loss map (5) 1.5 1 Rotor speed 0.5 75 rpm 750 rpm 1500 rpm 0 0 0.5 1 1.5 Torque [p.u] Fig. 3. Appropriate rotor flux versus torque 0.9 1500 rpm 0.8 0.7 0.6 750 rpm Efficiency Pin = Vds .ids + Vqs .iqs Pout = Tem .Ω + R fs 0.5 Torque where s 75 rpm 750 rpm 0 Ploss = Pin − Pout (R 150 50 (4) 2.2 Losses minimisation In conventional direct field-oriented control method ids* is the reference flux current and iqs* is the reference torque current and the rotor flux level is maintained constant at its rated value for every value of the load torque. Consequently, the core losses are ignored and the efficiency of the motor is poor especially when the load torque is low. In steady state, assuming that the machine control ensures no static error, then we can write: dφ dr =0 dt (6) Tem i qs = p.(1 − σ ).(1 + σ s ).φ dr Total losses is the input power minus the output power Ploss = 1500 rpm 200 100 Appropriate rotor flux [p.u] ( P jr = R r . i dr 2 + i qr 2 = 250 (7) σ r .R fs Rr + Rs + 2 ( 1 + σ r ) 2 −2 ( ) σ + 1 r .φdr2 + .Tem .φdr 2 . 1 + σ s )] [ p.(1 − σ )( (8) - proposed . conventional 0.5 0.4 75 rpm 0.3 0.2 0.1 0 0 0.5 1 Torque [p.u] Fig. 4. Efficiency map 1.5 6 5 Rotor rés is tanc e [Ohm ] The total losses for the proposed method is lower than that for the conventional method, which is constant rated rotor flux control, in the region of low torque as it is shown in figure 2. The appropriate rotor flux for maximum operating efficiency given by (10) is plotted in figure 3. Figure 4. shows an efficiency map. It is clear that the proposed method is superior to the conventional method over a wide range of torques. 4 3 2 Fitted curve measured points 1 3. Magnetic saturation and skin effect Rotor flux variation given with (2) , is valid only under the assumption of linear magnetic circuit. Such an assumption in real and exact cases is not admissible because inductances are nonlinear functions of currents. It is therfore necessary to modify the equation of the rotor flux in (2), so that it accounts for the main flux saturation. The synchrounous speed test gives the stator total inductance ( Ls=Lm+ls ) by measuring the reactive power. Figure 5. shows the stator total inductance as function of stator current . 0 0 5 10 15 20 25 30 Rotor current frequency [Hz] 35 40 45 50 Fig. 6. Rotor resistance versus rotor current frequency Figure 7. shows the relationship given by (8) for various load torques. For the points noted by asterisk, the minimum loss point correspond to operating point that satisfy the relation (10) in which magnetic saturation is ignored. The points noted by circle correspond to the case considering magnetic saturation. 1100 1000 0.6 Fitted curve Measured points 0.5 . ignoring saturation Power losses [W] 800 0.4 0.3 Stator induc tanc e [H ] - considering saturation 900 0.2 700 600 torque 1.5 p.u 500 400 1 p.u 300 0.5 p.u 200 0.1 100 0 0 0 0.5 1 1.5 2 2.5 Stator c urrent [ A] 3 3.5 4 0 0.2 0.4 0.6 0.8 1 Rotor flux [p.u] 1.2 1.4 1.6 1.8 Fig. 7. Total losses versus rotor flux Fig. 5. Stator inductance as function of stator current An acceptable approximation is that, two inductances are defined: an unsaturated inductance Ls0 which applies for stator current lower than Is0, and a saturated one for higher Is current. L I s < I s0 L s = s 0 (12) L f I I I s > I s 0 ( ) − − s s0 s0 The rotor resistance and the leakage inductances are determined by locked-rotor tests with stator frequencies from 10 to 50 Hz, and the determined constants are extrapolated down to a few hertz to take into account the skin effect in the rotor. On figure 6, one can see that the rotor resistance changes considerably due to skin effect [2]. 4. Generalized predictive controller Generalized predictive control (GPC) introduced by CLARKE ( 1987-1988 ) can be summarized through the three following points. If we want to make the plant output coincide in the future with a setpoint or a known trajectory, it is necessary: First to predict the plant output over a • defined horizon. then to calculate the future control • values which will minimize the errors between the predicted outputs and the setpoints values. Finally to apply only the first optimal • control value, and repeat all this at the next sampling period, with a receding horizon. 4.1 Model and cost function The Controlled AutoRegressive Intergrated Moving Average model (CARIMA) is commonly used in GPC, as it is applicable to many single-input single output plants with dead-time equal to one :[6] A(q −1 ) y (t ) = q −1 B(q −1 )u (t ) + C ( q −1 ) ∆( q −1 ) ξ (t ) (13) ~ U = [∆u (t ) , ... , ∆u (t + N u − 1)]T where u (t ) , y (t ) are the input and output of the plant at the sample instance t, ξ (t ) is an uncorrelated random sequence and the use of −1 ∧ ∧ ∧ Y = y (t + N1 ) ,... , y (t + N 2 ) −1 the operator ∆(q ) = 1 − q ensures an integral control law. A(q −1 ) , B(q −1 ), C (q −1 ) are polynomials in the backward g N1 −1 .... 0 g N1 .... 0 g N1 +1 g N1 M .... .... 0 G= .... .... g1 g N u .... .... M M g N 2 g N 2 −1 .... g N 2 − N u +1 A(q −1 ) = 1 + a1 q −1 + ... + a na q − na B(q −1 ) = b0 + b1 q −1 + ... + bnb q − nb The criterion which is minimized is a weighted sum of squares of predicted future errors and increments of control values: 2 (14) so that: [ −1 .G T .(W − F) . [ the first line of the matrix G T .G + λ.I N u (21) ] −1 .G T has to 4.3 Choice of different parameters the backward shift operator q-1 and obtained by solving the following Diophantine equations : ∆ (q −1 ). A(q −1 ).E j (q −1 ) + q − j .F j (q −1 ) = 1 G j (q −1 ) + q − j .H j (q −1 ) = B (q −1 ).E j (q −1 ) [ [ [ (16) ] ] ] degree E j (q −1 ) = degree G j (q −1 ) = j − 1 degree F j (q −1 ) = degree A(q −1 ) degree H j (q −1 ) = degree B ( q −1 ) − 1 The first term of equation (15) is called ‘free response’, as it represents the plant predicted output ω(t+j), when there is no future control action. The second term is called ‘forced response’ , as it represents the output prediction due to the future control actions u (t + j − 1) , j ≥ 1 . If we define the following vector formed with the polynomial solutions of equations (16) : F = [ f (t + N 1 ) , ... , f (t + N 2 )]T and, if we denote : ] ∂J ~ =0, ∂U (15) for N1 ≤ j ≤ N 2 , where F j , G j , H j are polynomials in ] ] ] Uˆ opt = GT .G + λ .I N u (20) computed off line and only the first control value of the sequence is applied on the system. forced response [ [ [ (19) and the optimal control law comes from G j (q ).∆u (t + j − 1) 144424443 with : ∧ ~ ~ ~ ~ J = (G.U + F - W)T .(G.U + F - W) + λ .U T .U ∧ free response The output prediction has the following form : The criterion J can be rewritten in a matrix form The previous CARIMA model is now used to elaborate the predicted outputs under the form:[5],[6] −1 (18) Y = G.Uˆ + F with the assumption : ∆u (t + j ) ≡ 0 for j ≥ N u where : N1, N2 are the costing horisons Nu is the control horison λ is the control weighting factor w(t + j ) is the future setpoint. 4.2 Minimization of the criterion y (t + j ) = F j (q −1 ). y (t ) + H j (q −1 ).∆u (t − 1) + 14444442444444 3 (17) W = [w(t + N1 ) , ... , w(t + N 2 )]T shift operator q-1 and in most cases C (q −1 ) = 1 . ∧ Nu N2 J = ∑ . w(t + j ) − y (t + j ) + λ ∑ ∆u 2 (t + j − 1) j = N1 j = N1 T There is in fact no particular rule that enables an optimal choice of N1,N2,Nu and λ . Morever it is possible to note the four following points : [5],[8],[9] * It is better to choose N1 so that at least one element of the first row of G is nonzero; that implies that N1 should be greather than the maximum expected dead-time of the process.7 * N2 should be chosen in order to satisfy N2.T equal to the time response of the process (T sampling period). * Very often Nu is chosen so that Nu<<N2 and we previously stressed the fact Nu =1 is very interesting. * λ is often hard to determine a priori. If the matrix Gt.G is itself invertible, even λ=0 gives a solution. But in most cases, it seems better to choose λ from the empiric relation: ( ) λ = tr G t .G (22) 4.4 Identification of the system Before the design of the control law, an identification of the system is necessary to define the CARIMA model. The identification algorithm can be executed off-line at the beginning of the test, or when an important change of the system is detected. The basic idea is to generate a pseudo random binary sequences (P.R.B.S) to calculate the correlation functions. The model that we are looking for has the following form : q −1 B ( q −1 ) A(q −1 ) 200 (23) where na and nb are chosen a priori and can be changed if the results are not satisfying, then a recursive least squares method provides the unknown parameters [− a1 ,.....,−a na , b0 , b1 ,....., bnb ] . 180 160 140 motor speed [rd/s] G (q −1 ) = 120 100 80 60 5 Simulation results 40 To prove the interest of the MLCA, a GPC control law has been applied both to the speed loop and rotor flux loop and the tuning that has been chosen is: (sampling period =5ms) Speed loop: 20 0 0 5 10 15 time [s] Fig. 8. Motor speed and speed reference [N11, N12 , Nu1, λ1 ] = [1 , 48 , 5 , 1165.4] flux loop: 15 [N 21, N 22 , Nu 2 , λ2 ] = [1 , 37 , 1 , 1.364] In this paper, an IM model in direct field oriented frame including core losses, magnetic saturation and skin effect have been presented. Based on this model, a loss minimisation algorithm was proposed and some simulated results have been presented, showing the validity of the proposed algorithm. The method is based on the choice of an optimal value of the rotor flux level depending on the operating conditions (torque, speed) and taking into account the inductances and the rotor resistance variations. Steady state losses minimisation method is an important way to minimise losses when the motor torque is low or zero or when the motor operates at low speeds. The efficiency of the steady state optimised MLCA (varying flux) trajectories for the whole cycle is better than the optimal constant flux usually used in FOC. torque [N. m] load torque 5 0 0 5 10 15 time [s] Fig.9. Motor torque and load torque 3.5 3.4 3.3 Rotor res is tanc e [Ohm ] 6 Conclusions 10 3.2 3.1 3 2.9 2.8 2.7 2.6 0 5 10 15 time [s] Fig. 10. Rotor resistance 0.65 0.6 0.55 0.5 Stator to tal induc tanc e [H] The proposed MLCA is applied to the direct control method of induction motor drive where rotor flux reference is used from (10). The chosen example of motor speed reference and load torque trajectories are given by figure 8. and figure 9. On these figures, we can see that motor speed and torque follow their references without steady state errors nor overshoot. Figure 10. and figure 11. show both the rotor resistance and stator total inductance variations. We can see the variations of rotor resistance due to skin effect and stator inductance due to saturation effect as given by figure 5. and figure 6. On figure 12., It is clear that the optimal rotor flux follows the torque and speed variations as given by equation (10). At t=0, the initial rotor flux is not zero in order to deliever high torque value when the machine starts. motor torque 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0 5 10 time [s) Fig. 11. Stator total inductance 15 1.4 [4]. K.Matsuze, T. Yoshizumi and S.Katsuta, High reponse flux control of direct field oriented induction motor with high efficiency taking core loss into account, IAS Annual meetings, pp.410-417,1997. rotor flux norm rotor flux reference 1.2 Rotor flux [Wb] 1 0.8 [5]. D.Dumur and P.Boucher, Predictive control Application in the machine-tool field,Advances in ModelBased Predictive control, Oxford Science Publications, Oxford University Press, 1994, pp.471-482. 0.6 0.4 0.2 0 0 5 10 15 time [s] Fig. 12. Rotor flux norm [7]. Mademlis C. and N. Margaris, Loss minimization in vector–controlled interior permanent synchronous motor drives, IEEE Tran. on Ind. Electronics, Vol. 49, N°.6, December 2002. speed signal c ontrol 3 speed loop 2 (incréments de commande) 1 0 -1 0 5 10 15 time [s] 0.6 rotor flux loop flux s ignal control [6]. Bitmead. R.R., M. Gevers and V. Wertz, Adaptive Optimal Control: The thinking Man’s GPC, Prentice Hall International, 1990. 0.4 0.2 [8]. D.Dumur, P.Boucher and J. Roder, Design of an open architecture structure for implementation of Predictive controllers for motor drives, Proc. Of the 1998 IEEE, Int. Conf. On Cont. Applications, pp.1307-1311, 1998. 0 -0.2 0 500 1000 1500 time [s] 2000 2500 3000 Fig. 13. Signal control Motor rated data: 1.1KW, 1500rpm, 220/380V, 3.4A, 7 N.m, Rs=8Ω, Rr=3.1Ω, Ls=Lr=0.47H, Lm=0.443H, J=0.06kg. m 2 , f=0.0042kg. m 2 , φr 0 =1.14Wb. References [1] S.Taniguchi,T.Yoshizumi and K.Matsuze, A method of speed sensorless control of direct-field-oriented induction motor operating at high efficiency with core loss consideration, Proc.of the IEEE/IECON’ 2001. [2]. E.Mendes, A.Baba and A.Razek, Losses minimization of a field oriented controlled induction machine, Proc. of the ‘Electrical machines and drives‘ conf.,pp.310-314. [3]. R.D.Lorenz and S.M.Yang, Efficiency optimized flux trajectories for closed-cycle operation of field orientation induction machine drives, IEEE Trans.on Ind. App.,1992, vol.28, n°3, pp.574-580. [9] D.W. Clarke, C. Mohtadi and P.S. Tuffs, Generalized predictive control – Part I. The basic algorithm, Part II. Extensions and interpretations, Automatica, pp.23-2:137160,1987.