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Two Flux Weakening Schemes for Surface-Mounted Permanent-Magnet Synchronous Drives - Design and Transient Response Considerations Dragan S. Marit*, Silva Hiti"', Constantin C. S t a n d * ,James M. Nagashima"*,and David B. Rutledge' * California Institute of Technology, Dept. of Electrical Engineering, M/S 136-93, Pasadena, CA 91 125 Dragan@caltech.edu ** General Motors, ATV - Torrance, 3050 W. Lomita Blvd., PO Box 2923, Torrance, CA 90509-2923 HitiS@pcssmtp.hac.com - Two recently published flux weakening schemes for surface-mounted permanent-magnet synchronous (SMPMS) drives have offered estimable advantages over previous solutions. The first scheme uses values of the voltage reference to generate an adequate flux weakening component of the stator current, whereas the second method detects an increase of the tracking error in torque-producing current component and uses the error signal for the flux weakening control. The viability of the algorithms has been demonstrated and the schemes have been compared and analyzed in steady state operation. The transient behavior is very important for electric vehicles and other highperformance applications. Here, we analyze the transient response characteristics of both schemes when a sudden change in the torque reference is received. The paper also describes an approach to practical regulator design and parameter selection for the flux weakening control section. suggested in [I] to sudden torque changes is sluggish and can even result in instability due to the loss of the current control at high-torque operating points. The error detection algorithm has much faster transient response to sudden torque changes, but results in a constantly present steady state error in the torque regulation. In [l], the schemes have been compared and analyzed in steady state operation. Here, the flux weakening schemes described in [ l ] are compared and evaluated during torque transients in the flux weakening region. The transient behavior to sudden torque changes is very important for high-performance applications such as electric propulsion. The procedure for selection of the parameters in the flux weakening section has not been addressed thus far. This paper provides guidelines for the regulator design and selection of the parameters in the flux weakening for improved steady state and transient characteristics of the drive system. Both schemes use space vector modulation (SVM). If the over-modulation and sixstep PWM inverter modes are utilized, [3], the performance degradation was observed due to the high harmonic content in the stator currents. The problem was thoroughly addressed and a remedy suggested in [4]. As a result of this work, we show that the voltage reference scheme ought to be used only if a rapid transient torque response is not required during the flux weakening operation. Otherwise, the faster algorithm of the second scheme should be employed. An approach to the practical design of the flux weakening control sections is described and the avenues for possible application of other results of the control theory are opened. The simulation results are provided to illustrate the transient characteristics of the schemes. Abstract 1. INTRODUCTION Due to their distinct characteristics, but also because of improvements in and reduced cost of permanent magnet (PM) technologies, PM machines have been used in many electrical drive applications. Some applications, like electrical propulsion, require a wide operating range above the motor base speed, i.e., a wide range of flux weakening operation. The straightforward approach to flux weakening operation is to calculate the magnetizing current reference from the surface-mounted permanent magnet synchronous (SMPMS) machine equations, assuming that all machine parameters are known. Limits for the magnetizing and torque current in the flux weakening region are calculated with a presumption that a SMPMS drive operates in the voltage or in the voltage and current limits. However, the method is very sensitive to uncertainties related to the system parameters. Recently published flux weakening schemes, [I], [2], [7], offered important advantages. The schemes in [ l ] and [2] do not use any machine parameters for calculations in the flux weakening region and DC bus voltage measurement is not necessary for the flux control. The scheme described in [l] uses closed-loop control of the phase voltage magnitude to generate a magnetizing current reference for the flux weakening operation. The approach described in [2] detects the steady-state error in the torque current and, then, uses the error to generate the magnetizing current reference. This principle was modified and further developed in [ 11 for use with current control in the synchronous reference frame. The voltage reference control method proved to be robust, without steady state error present in the second scheme, but computationally more complex. Also, the response of the first algorithm 0-7803-5662-4199/$10.00 01999 IEEE 11. SMPMS MODELING AND FLUX CONTROL As a review and to clarify the notation used henceforward, the SMPMS machine equations in the synchronous reference frame, [5], are given: di = R i + L - - dw L i v d 673 d dt e 4 (1) ISIE'99 - Bled, Slovenia where R, L are the stator resistance and inductance, we is the electrical speed of magnetic field, P i s the number of poles, ;1,is the flux linkage of permanent magnets, id is the flux generating component of the stator current, and iq is the torque generating component of the stator current. Maximal allowable motor phase voltage and phase current are determined by the inverter and machine ratings and by the available DC link voltage vdc. The following inequalities must be satisfied at any time: 2 ' d 2 id + 2 ' q "ma, + i,2 _< 2 2 I,,, .(4-a) Fig. 1: SMPMS voltage and current limit circles. (4-b) In (4-a,b), V,, is the maximal available phase voltage amplitude at the fundamental frequency and I,, is the maximal phase current. Typically, SMPMS control strategy at low speeds is chosen to maximize the torque per ampere ratio, i.e., id is set to zero if back EMF is sufficiently smaller than V,, (U,&<< V,,). Operation above the base speed, Vm,//zm, is enabled by allowing negative id current to flow and decrease the total flux in the machine air gap. It should be noted that, above the base speed, the drive always operates in the voltage limit. However, maximal torque in the flux weakening region is obtained when SMPMS drive operates in the voltage and current limits. In Fig. 1, this is a trajectory from point A to B when the speed increases from ul to u 2 . If the effect of the stator resistance is neglected, RzO in equations (1) and (2), and if the equality sign is used in (4-a,b), the mrfximal torque achievable and the corresponding reference i d are obtained as in [ 13 and [7]: . v (9) where T" is the value of the torque reference. 111. DESCRIPTION OF FLUX WEAKENING CONTROL SCHEMES Block-diagrams for the two flux weakening schemes are shown in Figs. 2 and 3. In both schemes, two antiwindup proportional-integral (PI) controllers, implemented in the synchronous reference frame, are used for current control, [l]. A space vector modulator (SVM), [3], is employed to generate IGBT gate signals. Transition into the full six-step operating mode during the flux weakening is enabled. The first flux weakening method was described in [11. It uses closed loop co~trolof the phase voltage magnitude Vphuseto generate i d , Fig.2. Flux weakening mode is entered when VPh,, approaches V,,, where v,,=o.577vdc for sine wave mode and V,,,,=0.637Vdc for full six-step operation. Transformed phase voltages in the synchronous reference frame, vd and vq, can be written as follows: iy i, = J I : ~- ~ where dd and dq are the outputs of current Further, we have: Solutions for id. and iq*for torque smaller than T,,, also be obtained from (1) - (4): ii = T * / K t o r q u e , can (8) 674 d 2 2 2 = d d +d,. _ . ., ..... Flux weakening ........... i Processor Fig. 2. Flux weakening control scheme with closed loop voltage control. Fig. 3. Flux weakening control scheme with current error detection. It follows from (12) and (13), that the boundary of the sine wave region is reached when d=0.867, whereas the six-step mode is reached at d=O.956. The d2 value is used as a feedback variable in the flux weakering loop, and compared to the onset modulation index &-,[I]. The error d'-d,,,' is regulated by an anti-windup PI controller, whose output is limited (to avoid irreversible demagnetization) between 0 and minimal allowable id*= -&,,,U . 675 To achieve a rapid transient response during the step changes in the torque command, the scheme given in Fig. 3 is used, [I]. The error signal for the q axis current is compared with the predetermined threshold value. When the error increases above the threshold value, an appropriate id* is generated. id* is proportional to the filtered absolute value of the q axis current error, and it is limited between 0 and &, . The sum of squares, \ir\, I> *') i d - f i , -, is compared to the maximal squared current value, z : ~ from (4-b). If the current limit constraint is satisfied, then i,' and id' are unchanged and used as references for t p current regulators. Otherwise, (6) is used to calculate i, . The low-pass filter is used to prevent unnecessary current injection in the d axis due to highfrequency noise. IV. ANALYSIS AND DESIGN SUGGESTIONS It is known, [ 5 ] , that feed-fonvard compensation of the cross-coupling terms in (1) and (2) is necessary for good current regulation. The compensation is presented in Fig. 4 for the case of the voltage control scheme, [l]. Analogously, it is implemented in the second algorithm too. The compensation also offers an opportunity to design controller parameters using results from the linear systems control theory and to accomplish excellent overall system performance. The design task for the voltage control flux weakening scheme requires that the proportional and integral gains of two PI current controllers, as well as the gains for the flux weakening section, are determined. When feed-forward terms are used, the voltage equations of the SMPMS machine reduce to (14) and (15). di vd = R i d + L d dt In the Laplace domain, these equations can be transformed into the following form: llmlter where, Ks=2Vdc/3KA.Now, any method from control systems theory can be applied to design the current control. However, a relatively simple but effective method is used here. The parameters of PI controllers are selected using compensation by left-half-plane (stable) pole-zero cancellation. The block-diagram of the current control is shown in Fig. 5. The compensation implies that, ideally, the following relationship is satisfied: After the compensation, the closed-loop current transfer function becomes, according to Fig. 5 and using equations (17) and(18): (14) di v, = Ri, iL4 dt ;1 If the gain of the current sensors is KA and if we use equations (10) and (1 l), we can write this as: where wc=KpKs/Lis the closed-loop bandwidth. Now, we first choose the desired bandwidth. This sets forth the value for Kp. Then, Kiis calculated from (18). In practice, the compensation is achieved only approximately, but it still results in very good performance, when the aforementioned feed-forward terms are employed. We notice that the controller gains are simple functions of v d c . As a result, the current controI can be further improved by including, if necessary, a simple algorithm for the adaptation of the PI coefficients when the DC bus voltage varies. Design of the PI controller in the flux weakening section is more convoluted because of an additional nonlinearity that comes from the modulation index function, (13). We are interested in designing the parameters in such PI Fig. 4. The feed-forwardcompensationof the cross-couplingterms in equations (1) and (2), the voltage scheme. Fig. 5. Equivalentblock-diagramfor decoupled current control using proportional-integrallaw. 676 a way that the transfer function from the iq* to the output of the flux weakening section, id*, provides an adequate response at elevated speeds. From Fig. 5 , we can derive that the transfer hnction between the current command input and the duty cycle in the corresponding axis is: Fig. 6 Simplified block-diagram for the design of the flux weakening PI controller. We can now reduce the block-diagram shown in Fig. 4 to the structure given in Fig. 6. Bearing in mind that in the flux weakening operation the square of the reference duty cycle stays close to d,', the d2 function can be linearized around the d,,, by retaining only the fist order terms in the Taylor expansion for d'=f(dd,dJ. The linearization is performed around the midpoint of the possible range for dd and d, , when: The feed-forward decoupling method is used in the second scheme too. By analogy, the current regulation satisfies the equations (14) - (17). The design follows equations (18) and (19). Then, the iq sensitivity transfer function is derived from (19) and Fig. 5 : 1+- S Therefore, the trayfer function for filtered used to generate id in Fig. 3, is now: iF,which is S Now, if we use Ko=(ddol,we obtain: S 1+- GFwo (s)= K o K , K , . -a S 1+ @M Kfi/KP! S S 1+- (25) O C If we apply the compensation again, so that K,dKpfw=wc, and if we use KFW'KdCMKifi, the equation (25) becomes: S 1+G F w o = KFW O M S The next step is to choose KFw,i.e., K+v so that the flux weakening response is optimized. It is important to see that an exessively high increase o f K F w is restricted by presence of the limiter, Figs. 2 and 4. Once K,& is determined, the proportional gain is K&= &/uC. Having in mind the final value theorem, [6], we see from (28) that the error would exist only during transients and would go to zero as the steady state is approached. However, the threshold condition, Fig. 3, results in a steady state error in q axis current. The error is necessary to provide i; during flux weakening and it is minimized by increasing K , On the other hand, the K , constant cannot be too high because of possible oscillations related to the presence of the Bmiter, Fig. 3. The calculation of i; begins in the flux weakening section of Fig. 3 quickly after the torque command is received, (28). Conversely, the voltage control scheme is expected to provide a more sluggish response, whose characteristics are dependent upon the additional processing in the closed loop regulation of .;i Consequently, although the first scheme provides better steady state, [ 11, much better transient behavior is expected from the second algorithm. The analysis presented here was used to choose the starting values for the control parameters in the simulation models. V. SIMULATION RESULTS The simulations were run at the constant speed of 0.7 p.u, i.e., deep in the flux weakening region. The torque current reference changed from 0 to 0.2 p.u, with a slew 677 rate of 50 p d s e c . The transition into six-step was disabled, i.e., d,,, set to 0.867, in order to observe i, and id responses in a clear manner. Namely, in the overmodulation and six-step modes, these current components contain the sixth harmonic, as discussed in [4]. The time domain response (P.u.) for the voltage control scheme is shown in Fig. 7. A certain delay in id response can be observed and an overshoot in i, is noticeable. An attempt to decrease the i, overshoot (by adjusting the control parameters results) in a larger id delay, i.e., a trade-off between these two characteristics must be made in a real drive system. We notice that the settling time is relatively long compared to the time duration of the reference transient and it is close to approximately five times the rise time. This response is also illustrated in Fig. 8, where iq versus id trajectory is presented in the d-q plane. There, we can clearly see that the iq overshoot is almost 20%. This imposes some qualitative limitations in electrical drives where this scheme is utilized and when a rapid transient response of the torque control loop is demanded. A similar simulation test was repeated for the second technique as well. The transient problems associated with the id delay and the overshoot in iq are not present, Figs.9 and 10. The corresponding settling time is much shorter than in Fig. 7. * 0.a o.ai o.ao5 0.3‘5 0.11 Time (sec) 0.315 o.3a Fig. 7. iq and id transient responses, the voltage control scheme. 0.25- 0.2 . 0.15. ’, 0.1 . 0.05- -0.3 -0.28 -0.26 jd -0.24 -0.22 -0.2 Fig. 10. Transient i,, versus id trajectory, the error detection scheme. The simulation results confirmed that the second scheme offers advantageous transient performance. However, the steady state error of the second scheme can be observed in Figs. 9 and 10. VII. CONCLUSION Two flux weakening schemes for SMPMS machine drives are discussed in terms of their transient response characteristics. Guidelines for the selection of the control parameters are suggested. Using the theoretical and simulation results presented here and experimental results given in [l], we conclude that if a fast transient response is not required, the voltage reference control scheme should be employed. It comes with high efficiency because id automatically adapts to the exact value needed during the flux weakening operation. There is no steady state error in the q axis. On the other hand, the second scheme offers a faster transient response and the steady state error is not critical for applications where the current control loop is an inner loop, e.g., in velocity control. The optimum efficiency of the voltage scheme is sacrificed in the error scheme and has to be addressed separately by considering some additional optimization routines as a part of the control software. REFERENCES ] ] : O n. D. S. Maric, S. Hiti, C.C. Stancu, and J. M. Nagashima, “Two Improved Flux Weakening Schemes for Surface Mounted Permanent Magnet Machine Drives Employing Space Vector Modulation”, in Proc IEEE IECON’9, vol. 1, pp. 508-512,1998. -0.3 -0.28 .0.26 id -0.22 -0.24 D. Sudhoff, K. A. Corzine and H. J. Hegner, “A Flux-Weakening Strategy for Current-Regulated Surface-Mounted Permanent-Magnet Machine Dnves”, IEEE Transactions on Energy Conservation, vol. 10,no. 3,pp. 431-437, Sep. 1995. -0.2 Fig. 8. Transient iq versus id trajectory, the voltage control scheme. J. Holtz, W. Lotzkat and A. Khambadkone, “On Continuous Control I I I I of PWM Inverters in the Over-Modulation Range Including the SixStep Mode,“ in Proc. IEEE IECON.92, pp. 307-312,1992. I I D. S. Maric, S. 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