Predictive Current Control With Current-Error Correction ... - IEEE Xplore

7 downloads 0 Views 1MB Size Report
Abstract—The performance of the conventional predictive cur- rent control (PCC) when applied to a permanent-magnet brushless ac (PM BLAC) drive is ...
IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 42, NO. 4, JULY/AUGUST 2006

1071

Predictive Current Control With Current-Error Correction for PM Brushless AC Drives P. Wipasuramonton, Z. Q. Zhu, Senior Member, IEEE, and David Howe

Abstract—The performance of the conventional predictive current control (PCC) when applied to a permanent-magnet brushless ac (PM BLAC) drive is analyzed. A technique is proposed to reduce the current errors that arise due to the inaccuracies in the system parameters and the nonideal behavior of the inverter during a steady-state operation. In addition, a current-regulated delta modulator is employed to achieve a fast dynamic response during a transient operation. The performance of the proposed PCC, which is simple to implement on a low-cost fixed-point DSP, is demonstrated experimentally. Index Terms—Brushless ac (BLAC) drive, current control, permanent magnet, predictive control, space-vector modulation.

I. I NTRODUCTION

D

UE TO THEIR high efficiency and torque density, vector-controlled permanent-magnet brushless ac (PM BLAC) drives are being employed increasingly for the high-performance servo applications. Various current-control techniques have been developed to achieve an accurate reference-current tracking during the steady state and a fast dynamic response during the transient state. These are classified as the hysteresis, linear, and predictive controls [1]. However, while the conventional hysteresis control provides a good transient performance and is insensitive to the load variations, it can result in an uncontrollable switching-frequency variation and a current error that may exceed the hysteresis band as well as the limit-cycle phenomenon [1]. An alternative form of the hysteresis control is the current-regulated delta modulator (CRDM) [2], [3], in which the phase-current errors are sampled at regular intervals during a finite time period. Similar to the conventional hysteresis control, it exhibits a fast transient response, the maximum switching frequency being limited by the sampling frequency. However, during the steady state, it exhibits low-frequency harmonics and a fundamental current error. Linear controllers such as the ramp-comparison and synchronous PI controls are generally used with the voltagesource pulsewidth-modulation (PWM) converters [4]. Hence, they employ a constant switching frequency, so as to ensure low noise emissions. However, the ramp-comparison control has Paper IPCSD-06-018, presented at the 2005 IEEE International Electric Machines and Drives Conference, San Antonio, TX, May 15–18, and approved for publication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Industrial Drives Committee of the IEEE Industry Applications Society. Manuscript submitted for review October 15, 2005 and released for publication March 2, 2006. The authors are with the Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield S1 3JD, U.K. (e-mail: elp01pw@ sheffield.ac.uk; [email protected]; [email protected]). Digital Object Identifier 10.1109/TIA.2006.876085

an inherent regulation problem, which results in the amplitude and phase errors. Although the synchronous PI control can overcome such problems compared to the hysteresis control, its transient response is relatively slow. Predictive current control (PCC) also results in a constant switching frequency and a low current ripple, generally by employing the space-vector modulation [5]. The current error is predicted at the beginning of each sampling period on the basis of the actual error and the motor parameters [4], and the PWM voltage for the next modulation period is determined so as to minimize the anticipated current error. However, in order to achieve a satisfactory performance, the controller requires an accurate knowledge of the motor parameters, otherwise a significant steady-state current error and current oscillations may result. To overcome this, integral compensation was used in [6], while the disturbance compensation was employed in [7]. However, in [6], the integral gain needed to be carefully tuned in order to achieve an acceptable transient response without the overshoot. In addition to the parameter inaccuracies, nonideal behavior of the inverter, due to the switching dead time, device’s ON -state voltage drop, and dc-bus-voltage variations, which are unavoidable in the practical drive systems, result in the distorted output currents during the steady-state operation [8]–[11]. In this paper, the performance of a conventional PCC, when the parameter inaccuracies and a nonideal inverter behavior exist, is analyzed, and a modified PCC is presented. A currenterror-correction technique is proposed to reduce the current error during the steady-state operation, and a CRDM is employed during the transient state in order to obtain a fast dynamic performance. Implementation of the modified PCC is simple and is realized via a low-cost fixed-point DSP. The experimental results are presented to demonstrate the merits of the proposed PCC.

II. C ONVENTIONAL PCC A simplified schematic of a voltage-source inverter-fed threephase BLAC drive is shown in Fig. 1. For a machine with a surface-mounted-magnet rotor, the phase-voltage equations may be expressed as dia − ψf ωe sin(θe ) dt dib − ψf ωe sin(θe − 2π/3) vb = Rib + L dt dic − ψf ωe sin(θe + 2π/3) vc = Ric + L dt

va = Ria + L

0093-9994/$20.00 © 2006 IEEE

(1) (2) (3)

1072

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 42, NO. 4, JULY/AUGUST 2006

TABLE I INFLUENCE OF THE STATOR-WINDING INDUCTANCE ON THE C URRENT R ESPONSE

Fig. 1. Voltage-source inverter-fed BLAC drive.

where va , vb , and vc are the phase voltages, R and L are the stator-winding resistance and inductance per phase, respectively, ψf is the flux linkage per phase due to the permanentmagnet rotor, and ωe and θe are the electrical rotor speed and position, respectively. Since the equations for all the three phases are similar and the variables in one phase can be derived from the other two phase equations, only the derivation of the PCC for phase-a will be presented. If the sampling period Ts is sufficiently small, (1) can be discretized into va (n) = Ria (n)+

L (ia (n+1)−ia (n))−ψf ωe (n) sin(θe (n)) . Ts (4)

Fig. 2.

Model of a conventional PCC.

as a disturbance, the transfer function relating the actual current to the desired current can be written as (L0 /L)z ia (z) = . i∗a (z) z + ((L0 /L) − 1)

(7)

In a conventional PCC, a motor model with nominal parameters is used to predict the voltage that is required to make the actual phase current ia (n) approach the desired reference value i∗a (n + 1). The required phase voltage can be calculated as

The output response is summarized in Table I. However, a voltage-fed inverter is never ideal. In practice, the switching dead time, the device’s ON-state voltage drop and the dc-bus voltage variations can adversely affect the control performance, particularly during the steady-state operation. The L0 va∗ (n) = R0 ia (n)+ (i∗a (n+1)−ia (n))−ψf 0 ωe (n) sin(θe (n)) most obvious effect is the distortion of the output current caused Ts (5) mainly by the low-order harmonics [10]. Therefore, the influence of the nonideal inverter operation can also be considered as where the subscript “0 ” denotes the nominal values for the another disturbance. As a consequence, the conventional PCC motor parameters, and the superscript “∗ ” denotes the desired can be modeled, as depicted in Fig. 2, and (6) can be modified to values of the electrical parameters. However, when the calcu1 lated voltage va∗ (n) is applied and assuming that the inverter is ia (n + 1) = [−∆Lia (n) + L0 i∗a (n + 1)] ideal, the actual phase current ia (n + 1) may not be the same as L Ts the desired value i∗a (n + 1), due to the parameter inaccuracies. − ∆ψf ωe (n) sin (θe (n)) + η(n) (8) Therefore, replacing va (n) in (4) with va∗ (n) in (5) gives L where η is the disturbance arising from the nonideal inverter operation.

ia (n + 1) =

1 [(Ts ∆R − ∆L)ia (n) + L0 i∗a (n + 1) L − Ts ∆ψf ωe (n) sin (θe (n))]

1 ∼ = [−∆Lia (n) + L0 i∗a (n + 1)] L L Ts since Ts  − ∆ψf ωe (n) sin(θe (n)) , L ∆R

III. P ROPOSED C URRENT -E RROR -C ORRECTED PCC

(6)

where ∆R = R0 − R, ∆L = L0 − L, and ∆ψf = ψf 0 − ψf . It is apparent from (6) that the current error caused by the parameter mismatches arises mainly from the inaccuracies in the values of the stator-winding flux linkage and inductance. The flux-linkage mismatch results in a current error that is influenced by the back-EMF, which has very slow dynamics in the current-control loop. By treating the back-EMF influence

With the conventional PCC, as stated earlier, the parameter inaccuracies and the nonidealized inverter operation cause the output current to deviate from the reference value during the steady-state operation. Moreover, a high-frequency current ripple inherently exists due to the PWM. Therefore, the phase-a current error, which results with a conventional PCC, can be expressed as i∗a − ia = iae = iaeh + iael

(9)

where iaeh is the high-frequency component of the current error, which results from the PWM switching frequency, and iael is the low-frequency component, which arises from the

WIPASURAMONTON et al.: PCC WITH CURRENT-ERROR CORRECTION FOR PM BLAC DRIVES

Fig. 3.

1073

Modified PCC.

machine parameter inaccuracies and from the nonideal inverter operation. In the proposed PCC illustrated in Fig. 3, in which i∗∗ a represents the phase-a reference current in the conventional PCC, iael is extracted from the current error and fed back positively to the desired current i∗a . This leads to ∗ ∗ i∗∗ a − ia = (ia + iael ) − ia = iaeh + iael ⇒ ia − ia = iaeh . (10)

As will be evident, the low-frequency component of the current error is then eliminated. The time constant of the lowpass filter τ is simple to specify since the bandwidths of iaeh and iael are significantly different. In the frequency domain, the “current-error corrector” can be written as ∗ i∗∗ a (s) = ia (s) +

iae (s) . sτ

(11)

Therefore, the new reference phase current i∗∗ a for the conventional PCC is the desired phase current to which the integral of the current error, with an integral gain of 1/τ , is added. Since the value of τ is quite small (for instance, τ = 320 µs, in this paper), it provides a relatively high integral gain. In the discretetime domain, (11) can be approximated as ∗ ∼ ∗ i∗∗ a (n + 1) = ia (n + 1) + qa (n + 1) = ia (n + 1) + qa (n) (12)

where qa (n) =

n Ts  ∗ [i (j) − ia (j)] . τ j=0 a

(13)

It should be noted that the approximation of qa (n + 1) by qa (n) in (12) is necessary since the current error at the (n + 1)th sampling instant is not yet available. Since the phase-a reference current with a conventional PCC ∗∗ ∗ is now i∗∗ a , upon substituting ia by ia , (8) becomes i∗∗ a (n + 1) =

L ∆L ia (n) ia (n + 1) + L0 L 1 [Ts ∆ψf ωe (n) sin (θe (n)) − Lη(n)] . (14) + L0

By inserting (13) into (12) and treating the last term of (14) as a disturbance, the transfer function relating the output current to the desired phase current can be found using    L0 Ts ia (z) L ·z z− 1− τ     . (15)   = 2 i∗a (z) z + z LL0 1 + Tτs − 2 + 1 − LL0

Fig. 4. Loci of the poles with a variation of L0 /L.

It should be noted that, when τ → ∞, (15) approaches (7). Using Jury’s stability test [13], the condition for making the system stable is 0
0 if x ≤ 0

(20)

and Sa , Sb , and Sc are the switching-state signals. Otherwise, the current-error-corrected PCC mode is selected.

WIPASURAMONTON et al.: PCC WITH CURRENT-ERROR CORRECTION FOR PM BLAC DRIVES

1075

TABLE II PM BLAC MOTOR PARAMETERS

Fig. 7. Simulated currents (L0 /L = 1, ψf 0 /ψf = 1, and 300 r/min). (a) Conventional PCC, speed = 300 r/min. (b) Proposed PCC, speed = 300 r/min.

IV. S IMULATED AND E XPERIMENTAL R ESULTS The utility of the proposed PCC has been verified by both the simulations and the measurements, and its performance has been compared to that which results with the conventional PCC on a vector-controlled PM BLAC drive (Fig. 6). The drive comprises a surface-mounted-magnet BLAC motor, whose parameters are given in Table II, and a MOSFET voltage-source inverter with an 80-V dc bus. The sampling time Ts for the current loop was set at 80 µs, while the inverter dead time was set at ∼ 4 µs. Since the influence of the stator resistance is small, as stated in Section II and reported in [6] and [12], the R0 in the motor model was set to zero. For the value of Ts /τ = 0.25, which was selected for both the simulations and the experiments, a closed-loop pole for the proposed PCC is located on the unit circle at L0 /L = 1.78, compared to L0 /L = 2 for the conventional PCC.

Fig. 8. Simulated currents (L0 /L = 0.5 and ψf 0 /ψf = 1). (a) Conventional PCC, speed = 300 r/min. (b) Proposed PCC, speed = 300 r/min. (c) Conventional PCC, speed = 3000 r/min. (d) Proposed PCC, speed = 3000 r/min.

1076

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 42, NO. 4, JULY/AUGUST 2006

Fig. 10. Measured currents (L0 /L = 1, ψf 0 /ψf = 1, and 300 r/min). (a) Conventional PCC. (b) Proposed PCC.

Fig. 9. Simulated currents (L0 /L = 0.7 and ψf 0 /ψf = 0.5). (a) Conventional PCC, speed = 300 r/min. (b) Proposed PCC, speed = 300 r/min. (c) Conventional PCC, speed = 3000 r/min. (d) Proposed PCC, speed = 3000 r/min.

Fig. 7 shows the simulated results for both the conventional and the proposed PCC schemes at low speed (300 r/min), when L0 /L = 1 and ψf 0 /ψf = 1. It can be seen that, although the parameters that are used in the motor model are accurate, nonideal inverter operation causes a significant phase-current error during the steady-state operation, when the conventional PCC is employed, while the current error is reduced with the proposed PCC. Fig. 8 shows the simulated results at both the low and high speeds (300 and 3000 r/min), when L0 /L = 0.5 and ψf 0 /ψf = 1, which confirm that the influence of an error in the inductance and the influence of the dead time are significantly reduced with the proposed PCC scheme. Fig. 9 shows the simulated results at both the low and high speeds, for L0 /L = 0.7 and ψf 0 /ψf = 0.5. It is apparent that the performance, which results with the conventional PCC, is influenced heavily by the back-EMF, which increases with the speed, while in the proposed PCC the influence of the back-EMF is very small. Figs. 10–12 show the measured results that correspond to the simulated conditions shown in Figs. 7–9, respectively. As can be seen, the measurements agree well with the results from the simulations. Fig. 13 shows the measured current waveform, which results when L0 /L = 1.5, ψf 0 /ψf = 1.0, and the proposed PCC is employed. In this case, a close-loop pole is close to the edge of the unit circle. A similar phenomenon is observed when a conventional PCC is employed, although it occurs at a slightly higher L0 /L ratio. This confirms the analysis in

WIPASURAMONTON et al.: PCC WITH CURRENT-ERROR CORRECTION FOR PM BLAC DRIVES

Fig. 11. Measured currents (L0 /L = 0.5 and ψf 0 /ψf = 1). (a) Conventional PCC, speed = 300 r/min. (b) Proposed PCC, speed = 300 r/min. (c) Conventional PCC, speed = 3000 r/min. (d) Proposed PCC, speed = 3000 r/min.

1077

Fig. 12. Measured currents (L0 /L = 0.7 and ψf 0 /ψf = 0.5). (a) Conventional PCC, speed = 300 r/min. (b) Proposed PCC, speed = 300 r/min. (c) Conventional PCC, speed = 3000 r/min. (d) Proposed PCC, speed = 3000 r/min.

1078

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 42, NO. 4, JULY/AUGUST 2006

Fig. 13. Oscillations observed in a measured current waveform for the proposed PCC (L0 /L = 1.5, ψf 0 /ψf = 1.0, and speed = 300 r/min).

Fig. 15. Measured transient currents (L0 /L = 0.7 and ψf 0 /ψf = 0.5). (a) Conventional PCC. (b) Proposed PCC.

the previous section, in which it is preferable to assure the value of inductance in the motor model to be slightly smaller than the actual value. Figs. 14 and 15 compare the simulated and measured current responses, which result with both the conventional and the proposed PCC controllers when the current command is reversed (from i∗q = 2 to −2), for L0 /L = 0.7 and ψf 0 /ψf = 0.5. It can be seen from Fig. 14(a) that, although the conventional PCC exhibits a fast transient response, a significant steady-state current error exists, as mentioned earlier. Fig. 14(b) shows that, although the modified PCC reduces the steady-state phase-current error, it results in an unacceptable overshoot during the transient state. Fig. 14(c), however, shows that the proposed PCC results in a significantly smaller overshoot, since the peak current is clamped by the assigned boundary H in the CRDM operation, this being 0.45 A in both the simulation and experimental drives. V. C ONCLUSION

Fig. 14. Simulated transient currents during (L0 /L = 0.7 and ψf 0 /ψf = 0.5). (a) Conventional PCC. (b) Modified PCC. (c) Proposed PCC.

The conventional PCC results in a current error due to the motor parameter variations and the nonideal inverter operation. The proposed PCC overcomes this problem by incorporating a current-error-correction technique. Moreover, a good transient response is maintained by employing a CRDM scheme. The proposed control algorithm is simple to implement on a low-cost DSP.

WIPASURAMONTON et al.: PCC WITH CURRENT-ERROR CORRECTION FOR PM BLAC DRIVES

R EFERENCES [1] D. M. Brod and D. W. Novotny, “Current control of VSI-PWM inverters,” IEEE Trans. Ind. Appl., vol. IA-21, no. 4, pp. 562–570, May/Jun. 1985. [2] T. G. Habetler and D. M. Divan, “Performance characterization of a new discrete pulse modulated current regulator,” IEEE Trans. Ind. Appl., vol. IA-25, no. 6, pp. 1139–1148, Nov./Dec. 1989. [3] X. Xu and D. W. Novotny, “Bus utilization of discrete CRPWM inverters for field-oriented drives,” IEEE Trans. Ind. Appl., vol. 27, no. 6, pp. 1128–1135, Nov./Dec. 1991. [4] M. P. Kazmierkowski and L. Malesani, “Current control techniques for three-phase voltage-source PWM converters: A survey,” IEEE Trans. Ind. Electron., vol. 45, no. 5, pp. 691–703, Oct. 1998. [5] H. W. van der Broeck, H. Skudelny, and G. V. Stanke, “Analysis and realization of a pulsewidth modulator based on voltage space vectors,” IEEE Trans. Ind. Appl., vol. 24, no. 1, pp. 142–150, Jan./Feb. 1988. [6] H. Le-Huy, K. Slimani, and P. Viarouge, “Analysis and implementation of a real-time predictive current controller for permanent-magnet synchronous servo drives,” IEEE Trans. Ind. Electron., vol. 41, no. 1, pp. 110–117, Feb. 1994. [7] K. H. Kim and M. J. Youn, “A simple and robust digital current control technique of a PM synchronous motor using time delay control approach,” IEEE Trans. Power Electron., vol. 16, no. 1, pp. 72–82, Jan. 2001. [8] R. B. Sepe and J. H. Lang, “Inverter nonlinearities and discrete-time vector current control,” IEEE Trans. Ind. Appl., vol. 30, no. 1, pp. 62–70, Jan./Feb. 1994. [9] J. W. Choi and S. K. Sul, “A new compensation strategy reducing voltage/current distortion in PWM VSI systems operating with low output voltages,” IEEE Trans. Ind. Appl., vol. 31, no. 5, pp. 1001–1008, Sep./Oct. 1995. [10] D. Leggate and R. J. Kerkman, “Pulse-based dead-time compensator for PWM voltage inverters,” IEEE Trans. Ind. Electron., vol. 44, no. 2, pp. 191–197, Apr. 1997. [11] T. Sukegawa, K. Kamiyama, K. Mizuno, T. Matsui, and T. Okuyama, “Fully digital, vector-controlled PWM VSI-fed ac drives with an inverter dead-time compensation strategy,” IEEE Trans. Ind. Appl., vol. 27, no. 3, pp. 552–559, May/Jun. 1991. [12] S. M. Yang and C. H. Lee, “A deadbeat current controller for field oriented induction motor drives,” IEEE Trans. Power Electron., vol. 17, no. 5, pp. 772–778, Sep. 2002. [13] J. R. Leigh, Applied Digital Control: Theory Design and Implementation. Englewood Cliffs, NJ: Prentice-Hall, 1984.

P. Wipasuramonton received the B.Eng. degree in control engineering and the M.Eng. degree in electronic engineering from King Mongkut’s Institute of Technology, Ladkrabang, Thailand, in 1987 and 1994, respectively. He is currently working toward the Ph.D. degree in the Department of Electronic and Electrical Engineering, the University of Sheffield, Sheffield, U.K. From 1991 to 1993, he was an Electrical Engineer at Berli Jucker, Thailand, and from 1994 to 2001, he was a Researcher with the National Electronics and Computer Technology Center, Thailand. His research interests include the electrical drives and energy conversion.

1079

Z. Q. Zhu (M’90–SM’00) received the B.Eng. and M.Sc. degrees from Zhejiang University, Hangzhou, China, in 1982 and 1984, respectively, and the Ph.D. degree from the University of Sheffield, Sheffield, U.K., in 1991, all in electrical and electronic engineering. From 1984 to 1988, he was a Lecturer in the Department of Electrical Engineering, Zhejiang University. In 1988, he joined the University of Sheffield, where he has been a Professor of electrical engineering since 2000. His current major research interests include the application, control, and design of permanent-magnet machines and drives.

David Howe received the B.Tech and M.Sc. degrees from the University of Bradford, Bradford, U.K., in 1966 and 1967, respectively, and the Ph.D. degree from the University of Southampton, Southampton, U.K., in 1974, all in electrical power engineering. He has held academic posts at Brunel and Southampton Universities and spent a period in industry with NEI Parsons Ltd. working on electromagnetic problems related to turbo generators. He is currently a Professor of electrical engineering with the University of Sheffield, Sheffield, U.K., where he heads the Electrical Machines and Drives Research Group. His research activities span all facets of controlled electrical drive systems, with particular emphasis on permanent-magnet excited machines. Prof. Howe is a Chartered Engineer in the U.K., a Fellow of the Royal Academy of Engineering, and a Fellow of the Institution of Electrical Engineers (IEE), U.K.