Recent Developments of Electrical Drives - Part 27 ppt

II-10. Impact of Magnetic Saturation of Induction Motor 253 Taking into account equations (18), the position controller and the rotor flux linkage controller are given by (19). v α = k α0 e 1 + k α1 ˙ e 1 + k α2 ¨ e 1 + y ··· ∗ 1 v β = k β1 e 2 + k β2 ˙ e 2 + ¨ y ∗ 2 (19) After inserting (19) in the linearized system (15), the tracking error dynamics of the closed loop system is given by (20) e ··· 1 + k α2 ¨ e 1 + k α1 ˙ e 1 + k α0 e 1 = 0 ¨ e 2 + k β2 ˙ e 2 + k β1 e 2 = 0 (20) with k ( · ) being positive constants. The desired dynamics of the tracking errors e 1 and e 2 is assured by selecting the corresponding eigenvalues λ ( · ) of the characteristic equations (21). λ 3 + k α2 λ 2 + k α1 λ + k α0 = 0 λ 2 + k β2 λ + k β1 = 0 (21) Observer design The state variables of the selected IM model are necessary to realize control, based on the described input-output linearization. The corresponding observer, similar to the one presented in [15], is given by (22). It is based on the electromagnetic subsystem of the two-phase i s , Ψ r state-space IM model (3) in the αβ reference frame. The coefficients k ( · ) are determined in the literature [14]. d dt ⎡ ⎢ ⎢ ⎢ ⎣ ˆ i sα ˆ i sβ ˆ ψ rα ˆ ψ rβ ⎤ ⎥ ⎥ ⎥ ⎦ = (C +ω r W) ⎡ ⎢ ⎢ ⎢ ⎣ ˆ i sα ˆ i sβ ˆ ψ rα ˆ ψ rβ ⎤ ⎥ ⎥ ⎥ ⎦ + D  u sα u sβ  + ⎡ ⎢ ⎢ ⎢ ⎣ k 1 −ω r k 2 −ω r k 2 k 1 k 3 −ω r k 4 −ω r k 4 k 3 ⎤ ⎥ ⎥ ⎥ ⎦  ˆ i sα ˆ i sβ  −  i sα i sβ   (22) The symbol ( ˆ· ) denotes the observed values. Experimental results The experiments have been performed to test the proposed input-output linearizing tracking control. The elements of the experimental system are the three-phase Semikron IGBT inverter, the three-phase 3 kW IM Sever with wound rotor, whose parameters are given in AppendixC,andtheDCmotorMavilorMo2000withan InfranorDCpowerconverter,asthe dynamic load. The control algorithm was executed on the dSPACE DS1103 microcontroller board. A block diagram of the proposed IM drive’s tracking control that includes magnetic saturation is presented in Fig. 2. Experiments were done using the reference value  2∗ r = 1.6 ( Vs ) 2 . The smooth refer- ence trajectories for the position θ r and the speed ω r were generated from the kinematic model and are shown in Figs. 3(a) and 4(a). The step changes of the load torque t l vs. time are shown in Fig. 3(d). The results of the input-output linearizing tracking control with the included saturation were compared with the results obtained with the same type of control VSI Encoder E (x) −1 D(x) Observer Inductances calculation (saturation) Load torque estimator T(x) Control IM (y 1 , y 1 ,y 1 , y 1 )* (y 2 , y 2 ,y 2 )* - - v sβ u sβ u 2 2 3 3 sα v sα ω r i a i b i c − Figure 2. Block diagram of the IM’s input-output linearizing tracking control. Figure 3. Reference and measured rotor position trajectory  ∗ r and  r : (a) saturation is not included, (b) saturation is included, (c) difference  r =  ∗ r −  r without and with saturation, and (d) load torque t l . II-10. Impact of Magnetic Saturation of Induction Motor 255 Figure 4. Reference and measured rotor speed trajectory ω ∗ r and ω r : (a) saturation is not included, (b) saturation is included, and (c) difference ω r = ω ∗ r − ω r without and with saturation. without included saturation [11]. The settings of the controllers were equal in both cases: k α0 = 750,000, k α1 = 25,000, k α2 = 275, k β1 = 90,000, and k β2 = 600. An analysis of the results showed that the position error θ r in Fig. 3 and the rotor speed error ω r in Fig. 4 are considerably smaller when magnetic saturation is included in the control algorithm, observer, and load torque estimator than in the case when magnetic saturation is neglected. It is obvious from the results in Fig. 5 that tracking control with the included magnetic saturation performed the position task with a slightly higher stator current i s =  i 2 sα +i 2 sβ , than the one without saturation. In contrast, tracking control without any included mag- netic saturation required smaller stator current to perform the same task at no-load, but it responded with a much higher increase in stator current i s , when the motor was loaded with step changes of the load torque (Fig. 5a). The reason for the described behavior of the controlled IM in Fig. 5 can be explained if the controlled system is analyzed together with the observer and, if only for explanation, the stator currents i sα , i sβ are transformed to the common dq reference frame, i.e. to the stator currents i sd , i sq (Fig. 6). The observer of electromagnetic state variables, with included magnetic saturation yields a smaller rotor flux linkage module  r for equal stator current value than the linear observer introduced in [15]. Accordingly, the input-output linearizing tracking control with included magnetic saturation increases the magnetizing stator current i sd in the direction of the rotor flux linkage vector, to achieve the reference value of the rotor flux linkage module. Therefore, the IM with the proposed input-output linearizing control is going to be magnetized in the best possible way to ensure the proper stiffness and optimal dynamic response. When the step changes of the load torque are applied on 256 Dolinar et al. Figure 5. Stator currents i sα , i sβ , and i s =  i 2 sα −i 2 sβ : (a) saturation is not included and (b) saturation is included. the shaft, the input-output linearizing control with included magnetic saturation performs much better than the control with neglected saturation, requiring smaller stator current i sq to produce the necessary torque with the rotor flux linkage vector. The transformed currents are shown in Fig. 6. The measured stator current i sd in the case of the input-output linearizing tracking control with and without included magnetic saturation agrees with the corresponding value of i sd determined from the nonlinear andlinearized magnetizing curve of the IM, used in the observer with and without included magnetic saturation (Fig. C1, Appendix C). Figure 6. Stator currents i sd and i sq . II-10. Impact of Magnetic Saturation of Induction Motor 257 Conclusion Consideration of magnetic saturation in the IM model substantially improves its accuracy, leading to a more efficient and consistent synthesis of the control algorithm, observer, and estimator of load torque. The proposed input-output linearizing control of IM with included magnetic saturation improves the dynamic performance of the drive. It gives smaller rotor position and speed errors, as well as a higher stiffness and a better load torque rejection, which results in a smaller stator current, when the load torque is introduced. An important reason for the improved behavior of the controlled IM is more adequately calculated value of the rotor flux linkage when magnetic saturation is considered in the observer design. Appendix A Elements of matrices C, Z, W, D L im = L 2 l − L l L 2 rl  1 L rl + L m + 1 L rl + L  + L ¸ 4 rl ( L rl + L m )( L rl + L ) c 11 =− 1 L im  R s  L l − L 2 rl L qq  + R r L m L r  L l − L 2 rl  1 L rl + L m + 1 L rl + L  − L sl L rl L dd + L 3 rl ( L rl + L m )( L rl + L )  c 12 = 1 L im  R s L 2 rl L dq − R r L m L r L sl L rl L d q  c 13 = 1 L im  R r L r  L l − L 2 rl  1 L rl + L m + 1 L rl + L  − L sl L rl L dd + L 3 rl ( L rl + L m )( L rl + L )  c 14 =− 1 L im  R r L r L sl L rl L dq  , c 21 = c 12 , c 23 = c 14 c 22 =− 1 L im  R s  L l − L 2 rl L dd  + R r L m L r  L l − L 2 rl  1 L rl + L m + 1 L rl + L  − L sl L rl L qq + L 3 rl ( L rl + L m )( L rl + L )  c 44 =− R r L r = c 33 c 24 = 1 L im  R r L r  L l − L 2 rl  1 L rl + L m + 1 L rl + L  − L sl L rl L qq + L 3 rl ( L rl + L m )( L rl + L )  c 31 = R r L m L r = c 42 , c 33 =− R r L r = c 44 , c 42 = R r L m L r = c 31 258 Dolinar et al. z 12 = 1 L im  L l − L 2 rl L r  L l − L 2 rl L qq  z 13 =− 1 L im  L m L r L 2 rl L dq + L sl L rl L dq  z 21 =− 1 L im  L l − L 2 rl L r  L l − L 2 rl L dd  , z 22 = z 11 z 14 =− 1 L im  − L m L r  L l − L 2 rl L qq  +  L l − L 2 rl  1 L rl + L m + 1 L rl + L  − L sl L rl L dd + L 3 rl ( L rl + L m )( L rl + L )  z 23 =− 1 L im  L m L r  L l − L 2 rl L dd  +  −L l + L 2 rl  1 L rl + L m + 1 L rl + L  + L sl L rl L qq − L 3 rl ( L rl + L m )( L rl + L )  z 24 = z 13 , z 34 = 1, z 43 =−1, w 13 = 1 L im  L sl L rl L dq  , w 13 = 1 L im  L sl L rl L dq  w 14 = 1 L im  L l − L 2 rl  1 L rl + L m + 1 L rl + L  − L sl L rl L dd + L 3 rl ( L rl + L m )( L rl + L )  w 23 =− 1 L im  L l − L 2 rl  1 L rl + L m + 1 L rl + L  − L sl L rl L qq + L 3 rl ( L rl + L m )( L rl + L )  w 24 =−w 13 , w 34 =−1, w 43 = 1 d 11 = 1 L im  L l − L 2 rl L qq  , d 12 = 1 L im  L 2 rl L dq  d 21 = d 12 , d 22 = 1 L im  L l − L 2 rl L dd  Appendix B Lie derivatives L 3 f φ 1 = ∂ ∂x  L 2 f φ 1  dx dt =∇  L 2 f φ 1  [ f + Gu ] = p L m L r 1 J  c 22 + c 33 − f J  i sβ ψ rα −  c 11 + c 44 − f J  i sα ψ rβ +c 12  i sα ψ rα −i sβ ψ rβ  − ω r  i sα ψ rα +i sβ ψ rβ  + ( c 24 + c 13 − 2w 13 ω r ) ψ rα ψ rβ + ( c 23 + w 23 ω r ) ψ 2 rα − ( c 14 + w 14 ω r ) ψ 2 rβ  + f J 2 t l + f 2 J 2 ω r L 2 f φ 2 = ∂ ∂x  L f φ 2  dx dt =∇  L f φ 2  [ f + Gu ] = 2R r L m L r  c 11 + c 33 − 2 c 31 L m  i sα ψ rα +  c 22 + c 44 − 2 c 31 L m  i sβ ψ rβ II-10. Impact of Magnetic Saturation of Induction Motor 259 +c 12  i sβ ψ rα +i sα ψ rβ  + ω r  i sβ ψ rα −i sα ψ rβ  + ( c 14 + c 23 + w 14 ω r + w 23 ω r ) ψ rα ψ rβ +  c 13 − 2 c 33 L m  ψ 2 rα +  c 24 − 2 c 33 L m  ψ 2 rβ + w 31 ω r  ψ 2 rα − ψ 2 rβ  + c 31  i 2 sα +i 2 sβ   L gα L 2 f φ 1 = ∂ ∂x  L f φ 1  g α = p L m L r 1 J  d 21 ψ rα − d 11 ψ rβ  L gβ L 2 f φ 1 = ∂ ∂x  L f φ 1  g β = p L m L r 1 J  d 22 ψ rα − d 12 ψ rβ  L gα L f φ 2 = ∂ ∂x  L f φ 2  g α = 2R r L m L r  d 11 ψ rα + d 21 ψ rβ  L gβ L f φ 2 = ∂ ∂x  L f φ 2  g β = 2R r L m L r  d 12 ψ rα + d 22 ψ rβ  Appendix C Table 1. Parameters of the 3 kW induction motor with wound rotor Sever ZPD112MK4: R s 1.976  L m 0.223 H L s 0.2335 H f 0.0007 Nms/rad R r 2.91  L r 0.2335 H J 0.031 kgm 2 T n 15 Nm Figure C 1. Rotor flux linkage and corresponding stator current in the case of linear and nonlinear magnetizing curve. 260 Dolinar et al. Acknowledgment This work was supported in part by the Slovene Ministry of Education, Science and Sport, Project No. P2-0115. References [1] P. Vas, Electrical Machines and Drives: A Space-Vector Theory Approach, Oxford: Oxford University Press, 1992. [2] J.C. Moreira, T.A. Lipo, Modelling of saturated ac machines including air gap flux harmonic components, IEEE Trans. Ind. Appl., Vol. 28, No. 2, pp. 343–349, 1997. [3] E. Levi, A unified approach to main flux saturation modelling in d-q axis models of induction machines, IEEE Trans. Energy Convers., Vol. 10, No. 3, pp. 455–461, 1995. [4] E. Levi, Impact of cross-saturation on accuracy of saturated induction machine models, IEEE Trans. Energy Convers., Vol. 12, No. 3, pp. 211–216, 1997. [5] R.D.Lorenz, D.W. Novotny,Saturation effects infield-orientedinduction machines, IEEETrans. Ind. Appl., Vol. 26, No. 5, pp. 283–289, 1990. [6] E. Levi, S. Vukosav´ıc, V. Vuˇckov´ıc, “Saturation Compensation Schemes for Vector Controlled Induction Motor Drives”, PESC’90 Record, San Antonio, TX, USA, pp. 591–598, 1990. [7] P. Vas, Sensorless Vector and Direct Torque Control, Oxford: Oxford University Press, 1998. [8] E.Levi, M. Sokola, S.N. Vukosav´ıc, A method for magnetizing curve identification in rotor flux oriented induction machines, IEEE Trans. Energy Convers., Vol. 15, No. 2, pp. 157–162, 2000. [9] Z. Krzeminski, A. Jaderko, “A Speed Observer System of Induction Motor with Magnetizing Curve Identification”, EPE-PEMC 2000, Kosice, Slovakia, 2000. [10] Z. Krzeminski, “Nonlinear Control of Induction Motor”, Proc. 10th IFAC World Congress, Munchen, Germany, 1987, pp. 349–354. [11] R. Marino, S. Peresada, P. Valigi, Adaptive nonlinear control of induction motors via extended matching, Lect. Note Contr. Inform. Sci., Vol. 160, pp. 1435–1454, 1991. [12] T. von Raumer, J.M. Dion, L. Dugard, “Adaptive Nonlinear Speed and Torque Control of IM”, Proceedings of European Control Conference, Gronigen, June 1993, pp. 592–596. [13] R.T. Novotnak, J. Chiasson, M. Bodson, High-performance motion control of an induction motor with magnetic saturation, IEEE Trans. Contr. Syst. Technol., Vol. 7, No. 3, pp. 315–327, 1999. [14] P. Ljuˇsev, “Analysis of Induction Motor Control Taking into Account Magnetic Saturation”, Master thesis, University of Maribor, Slovenia, 2002. [15] G.C.Verghese, S.R. Sanders, Observersforflux estimation in inductionmachines,IEEE Trans. Ind. Electron., Vol. 35, No. 1, pp. 85–94, 1988. II-11. DIRECT POWER AND TORQUE CONTROL SCHEME FOR SPACE VECTOR MODULATED AC/DC/AC CONVERTER-FED INDUCTION MOTOR M. Jasinski, M. P. Kazmierkowski and M. Zelechowski Warsaw University of Technology, Institute of Control & Industrial Electronics, ul. Koszykowa 75, 00-662 Warszawa, mja@isep.pw.edu.pl, mpk@isep.pw.edu.pl, zelechom@isep.pw.edu.pl WWW: http://www.ee.pw.edu.pl/icg Abstract. A novel control scheme for PWM rectifier-inverter system is proposed. Fast control strate- gies such as line voltage Sensorless Virtual Flux (VF) based Direct Power Control with Space Vector Modulator (DPC-SVM) for rectifier and Direct Torque Control with Space Vector Modulator (DTC- SVM) for inverter side are used. These strategies lead to good dynamic and static behaviour of the proposed control system—Direct Power and Torque Control- Space Vector Modulated (DPTSVM). Simulations and experiment results obtained show good performance of the proposed system. Addi- tional power feedforward loop from motor to rectifier control side improved dynamic behaviours of the power flow control. As a result, better input-output energy matching allows decreasing the size of the dc-link capacitor Introduction The adjustable speed drives (ASD) with diode rectifier nowadays is the most popular on the marked. Large electrolytic capacitor is used as an energy-storing device to decouple rectifier and the inverter circuits. The capacitors have some drawbacks: low reliability, high size, weight and cost. Hence, reliability of the dc-link capacitor is the major factor limiting the lifetime of the ASD systems [1]. Development of control methods for Pulse Width Modulated (PWM) boost rectifier (active rectifier) was possible thanks to advances in power semiconductors devices and Digital Signal Processors (DSP). Therefore, the Insulated Gate Bipolar Transistors (IGBT) AC/DC/AC converter controlled by PWM is used in motor drive systems (Fig.1). Thanks to active rectifier the dc-link capacitor can be reduced [2]. Farther reduction of the capacitor can be achieved by power feedforward loop from motor side to the control of the PWM rectifier. A lot of works are given attention to reduce the dc-link capacitor. However, a small capacitance leadstoahighdc-voltage fluctuation. Toavoid thisdrawback variousdc-voltage control schemes have been proposed. Some of them take into account the inverter dynamics S. Wiak, M. Dems, K. Kom ˛ eza (eds.), Recent Developments of Electrical Drives, 261–274. C  2006 Springer. 262 Jasinski et al. Figure 1. Representation of three-phase PWM rectifier—inverter system; vector diagram and coor- dinate system for: a) PWM rectifier side b) inverter side. to improve the PWM rectifier current control by feedback linearization [3] and master-slave [1] manner. Another control methodology proposed a fast dc-link voltage controller which works with dc-voltage and motor variables as inputs [4]. Moreover, various methods of the output power estimation have been discussed in [5]. In the mentioned methods active and reactive powers of the PWM rectifier are indirectly controlled via current control loops. Besides, stator current controllers control the torque and flux of the motor too. In this paper a line voltage sensorless Virtual Flux (VF) based Direct Power Control with Space Vector Modulator (DPC-SVM) is applied to control of the PWM rectifier. The inverter with induction motor is controlled via Direct Torque Control with Space VectorModulator(DTCSVM).Contrar y tothescheme proposedin[6], oursolutionincludes not stator flux controller but space vector modulator. Hence, an AC/DC/AC converter of Fig. 1, is controlled by Sensorless Direct Power and Torque Control-Space Vector Modulated (DPT-SVM) scheme. In comparison to methods that control an active and reactive power, torque and flux in indirect manner the coordinates transformation and decoupling are not required. Moreover, the current control loops are avoided. In respect of dynamic, of dc-voltage control the power balance between line and motor is very important. Therefore, to improve instantaneous input/output power matching, the additional feedforward power control loop is introduced. Thanks to better control of the power flow the fluctuation of the dc-link voltages will be decrease. So the size of the dc-link capacitor can be reduced. . (eds.), Recent Developments of Electrical Drives, 261 274 . C  2006 Springer. 262 Jasinski et al. Figure 1. Representation of three-phase PWM rectifier—inverter system; vector diagram and coor- dinate. saturation modelling in d-q axis models of induction machines, IEEE Trans. Energy Convers., Vol. 10, No. 3, pp. 455–461, 1995. [4] E. Levi, Impact of cross-saturation on accuracy of saturated induction. control loop is introduced. Thanks to better control of the power flow the fluctuation of the dc-link voltages will be decrease. So the size of the dc-link capacitor can be reduced.