Improved Deadbeat Predictive Current Control Strategy ... - IEEE Xplore

Improved Deadbeat Predictive Current Control Strategy for Permanent Magnet Motor Drives Hongjia Wang, Ming Yang, Li Niu, Dianguo Xu Institute of Power Electronics & Electrical Drive Harbin Institute of Technology Harbin, China [email protected]

Abstract—The predictive current control algorithm of permanent magnet synchronous motor (PMSM) drives has precise current tracking, constant switching frequency and is suitable for digital implementation. However, the conventional deadbeat predictive current control algorithm is sensitive to the stator inductance parameter mismatch. This paper presents an improved deadbeat predictive current control algorithm while achieving significantly increased robustness to inductance parameter mismatch. The current offset constraint and output voltage prediction method were modified. The relationship between system stability and inductance mismatch was analyzed using root locus method. In addition, the proposed algorithm is very simple and can be effectively implemented. Simulation results demonstrate the effectiveness of the proposed predictive current control algorithm. The system robustness is proven. Keywords- predictive current control; permanent magnet synchronous motor (PMSM); deadbeat; inductance mismatch

I.

INTRODUCTION

The permanent magnet synchronous motor (PMSM) has been used widely in industrial applications. It has advantages of high torque to inertia ratio, high efficiency, and superior power density [1]. The field orientation control is usually applied for a PMSM, this technique make a PMSM be like as torque control performance to a DC motor [2]. So, it is necessary to design a high performance current controller to obtain an efficient motor drives. In the current control for PMSM drives, The major techniques to regulate the stator current include either a variable switching frequency, such as the hysteresis control, or fixed switching frequency schemes, such as the ramp comparison control, stationary and synchronous frame proportional integral (PI) control, and deadbeat predictive current control [3, 4]. The hysteresis current control scheme has advantages such as a fast transient response and simplicity in implementation, but shows a high and non-constant inverter switching frequency. The ramp comparison control method has the advantages of limiting the maximum switching frequency and producing well-defined harmonics [5]. However, it has the disadvantages of steady-state phase delay errors and sensitivity to system parameters. The synchronous frame PI control does not have phase lag. And the compensation of the back EMF

and cross-coupling terms gives fast transient response and zero steady-state error irrespective of operating conditions [6]. Currently, there is a strong trend toward fully digital control of PMSM drives based on deadbeat current control (D-PCC) techniques [7–9]. The D-PCC algorithm has the potential for achieving the faster transient response, zero steady-state error, more precise current tracking, and suitable for digital implementation. However, the D-PCC method is sensitive to the motor and inverter parameters, the robustness range will reduce when motor inductance mismatch occurs. This paper presents a robust improved deadbeat predictive current control algorithm for PMSM drives. The current offset constraint and output voltage prediction method are modified. The algorithm remains stable while inductance mismatch exists. The proposed algorithm guarantees the system robustness as well. The paper is organized as follows. In Section II, the dynamic model of PMSM is illustrated. Section III describes the D-PCC algorithms and the proposed improved algorithm in detail. Section IV is the analysis of algorithm robustness. MATLAB simulation is shown to validate the effectiveness of the proposed algorithm in Section V. Conclusions are drawn in section VI. II.

The stator voltage equations of a PMSM in the synchronous reference frame are described as follows:

uq

Rs iq Lq

ud

Rs id Ld

diq

eq

did ed dt

dt

where uq and ud are the q-axis and d-axis voltages, iq are id the q-axis and d-axis currents, eq

ed

Project Supported by National Natural Science Foundation of China (51007012) and the Power Electronics Science and Education Development Program of Delta Environmental & Educational Foundation.

c 978-1-4244-8756-1/11/$26.00 2011 IEEE

DYNAMIC MODEL OF PMSM

The model of the PMSM forms the basis for the motor drive. The nonlinearities due to saturation of the iron and location of the stator winding in slots are neglected. The stator windings are sinusoidally distributed and displaced by 120 degrees.

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Zr Ld id Zr\ f and

Zr Lq iq are the q-axis and d-axis back EMF voltages. Rs

is the stator phase resistance, Lq and Ld are the q-axis and d-

where Lp is the estimated stator inductance. Equation (3) and

axis stator inductances. For surface mounted permanent magnet machine Lq Ld . The rotor speed Zr and magnetic flux \ f

(4) can be rewritten in matrix form

A common topology of a current controlled three-phase PWM inverter-fed PMSM digital control system is depicted in Fig. 1. The control operation is synchronized with the interrupt signal generated by a pulse-width modulation (PWM) generator. The DC link voltage, stator currents and the rotor position are sensed at each carrier period. With the interrupt signal, the current control algorithm calculation starts to obtain the command voltages applied to the PWM generator. The main calculation is executed by a digital signal processor (DSP). The six PWM switching signals are applied to gate drivers to drive the three-phase inverter. PWM inverter

U DC

Rs

ea ec

Ls Gate drivers PWM generator i

Current reference

iq

Current controller

Tr

PREDICTIVE CURRENT CONTROL ALGORITHM WITH IMPROVED ROBUSTNESS

A. Conventinal D-PCC Algorithm At the beginning of the kth carrier period ª¬ kTs , k 1 Ts º¼ , calculate the optimal voltage reference U k , in terms of the actual stator current I k , output voltage reference U

* k

and

current error ǻI k . This voltage reference is then applied to the stator using space vector PWM method. And the actual current will follow the reference current at k 1 Ts , ǻI k 1 0 . This is the main idea of the D-PCC, where Ts is the discrete-time control sampling period. For digital implementation of the control algorithm, the stator voltage equations can be described as

u qk

u dk

Rs iqk Rs idk

Ts Lp Ts

i

qk 1

iqk eq

idk 1 idk ed

T

¬ª iqk

ª Lp « « Ts « « 0 «¬

0º ,M Rs »¼

idk ¼º , º 0 » », e Lp » » Ts »¼

ª eq « ¬0

0º » ed ¼

If the stator current can track the reference current for the system, (5) can be restated as

U k*

RI k* M I k* 1 I k* e

U k* M ǻI k 1 ǻI k

Uk

Encoder

Lp

ª Rs «0 ¬

u dk ¼º , I k

Then applying the deadbeat condition, i.e., ǻI k 1 (7). This results in the algorithm

Zr

Figure 1. PMSM digital control system with current controlled voltage source PWM inverter

III.

R

T

¬ª u qk

Rs

e jT r

id

*

Ls

Rs

ia ib ic

Uk

where U k* is the ideal inverter output voltage that achieves the current reference. Subtracting (5) from (6) and neglect the resistance, yields

Ls eb

where

PMSM Current sensor

RI k M I k 1 I k e

Uk

can be considered as constant during the sampling period.

U k* MǻI k

So, if the inverter output voltage given by (8) is applied over the period ª¬ kTs , k 1 Ts º¼ , the current error should return to zero at k 1 Ts . At the beginning of the kth carrier period ª¬ kTs , k 1 Ts º¼ , the estimate of U k* can be obtained from the Lagrange interpolation formula. Two typical implemented forms are U k*

U

* k

2U

U k* 1 * k 1

U

* k 2

According to (7), we obtain U k* 1

U k* 2

U k 1 M ǻI k ǻI k 1

U k 2 M ǻI k 1 ǻI k 2

Substituting (9)í(12) into (8), the output voltage can be given by

Uk

0 as in

Uk

Uk

U k 1 M 2 ǻI k ǻI k 1

2U k 1 U k 2 M 3 ǻI k 3 ǻI k 1 ǻI k 2

Note that in (13) and (14) the current error ǻI k must be available at the end of the period ª¬ k 1 Ts , kTs º¼ . For the current sensing and inverter output voltage applying time

2011 6th IEEE Conference on Industrial Electronics and Applications

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sequence relations in digital control scheme, the current error must be forward predicted by one sampling period. The estimate of ǻI k need to use U k* 1 U k* 2 and U k* 1 2U k* 2 U k* 3 , respectively. Then, substituting the estimate of ǻI k into (13) and (14), we obtain two forms of the D-PCC algorithms

Uk Uk

U k 1 2U k 2 M 3 ǻI k 1 2 ǻI k 2 U k 1 5U k 2 3U k 3 M 6 ǻI k 1 8 ǻI k 2 3 ǻI k 3

1 ǻI k 2

U k*

1 MǻI k 2

Meanwhile, to get a smoother estimate of U k* , the relationship below is used U k*

1 * U k 1 U k* 2 2

The estimate of U k*1 and U k* 2 can be obtained from (7) U k* 1

U k* 2

U k 1 M ǻI k ǻI k 1

U k 2 M ǻI k 1 ǻI k 2

O z

O z

O z

z4 6

z4

Uk

Similarly, the current error should be predicted and then substituted into (22) gives Uk

1 1 U k 1 U k 2 U k 3 2 2 1 1 §3 · M ¨ ǻI k 1 ǻI k 2 ǻI k 3 ¸ 2 2 ©2 ¹

' Ls 2 'L 'L z 8 s z 3 s Ls Ls Ls

1 3 3 ' Ls 2 1 ' Ls 1 ' Ls z z z 2 2 Ls 2 Ls 2 Ls

0.2 Ls d ' Ls d 0.25 Ls

0.06 Ls d ' Ls d 0.08 Ls

' Ls d 0.53 Ls

1

1

0.5

0.5

0

-0.5

This improved algorithm calculates inverter output voltage required for the interval ª¬ kTs , k 1 Ts º¼ to correct the previous current errors. The weighting factors are significantly decreased comparing with (15) and (16).

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' Ls 'L 2 s Ls Ls

z3 3

The root locus plots for the underestimated and overestimated stator inductance for the characteristic equations (25)–(27) are shown in Figs. 2–4, respectively. The Figs. 2–3 show that only a 20% or 6% error in the inductance can be tolerated before system instability occurs. The conventional DPCC algorithms are very sensitive to parameter mismatch. On the other hand, the robustness and stability ranges are up to 53% mismatch in the inductance when the proposed algorithm is used, as shown in Fig. 4. The stability ranges of inductance error for each case are

Imaginary Z axis

Ls 'Ls . Substitute (15), (16) and

(23) into (24), use Z-transform to find the transfer function and examine the characteristic polynomial of the system. The characteristic equations of (15), (16), and (23) are given by

Substituting (19), (20) and (21) into (18) gives 1 1 1 § · U k 1 U k 2 M ¨ ǻI k ǻI k 2 ¸ 2 2 2 © ¹

parameter Ls , where Lp

Imaginary Z axis

Uk

Substituting (17) into (7) gives

M 1 U k* U k ǻI k

ǻI k 1

Then focus on the algorithms robustness to the mismatch between stator inductance estimated parameter Lp and actual

made equal to half of the previous current error, i.e. ǻI k 1

ALGORITHM ROBUSTNESS COMPARISON

B. ImprovedD-PCC Algorithm The conventional D-PCC algorithm is based on the deadbeat condition. In this paper, the improved D-PCC algorithm is proposed with relaxing the deadbeat constraint. The current error at the end of carrier period ª¬ kTs , k 1 Ts º¼ is

IV.

For the predictive current control algorithms mentioned above, the actual current error at the end of the interval ¬ª kTs , k 1 Ts ¼º will be described as

-1 -1

0

-0.5

-0.5

0

0.5

Real Z axis (a) 0.2 Ls d ' Ls

1

-1 -1

-0.5

(b)

0

0.5

Real Z axis ' Ls d 0.25 Ls

Figure 2. Root locus plot in Z-domain, (15) and (25)


1

0.5

0.5

0

0.002

0.004 t [s] 0.006

0

0.5

1

-0.5

0

0.5

Real Z axis

Real Z axis

(a) 0.06 Ls d ' Ls

(b) ' Ls d 0.08 Ls

1

1

0.5

0.5

0

-0.5

1

0

0.5

1

Real Z axis (a) Ls d ' Ls

10 Fundamental (200Hz) = 6.623 , THD= 8.15% 5 0 0

20

40 60 Harmonic order

80

100

(b)

0

-1 -1

-0.5

0

0.5

1

Real Z axis

10 5 0 -5 -10 0

ia

0.002

0.004 t [s] 0.006

(b) ' Ls d 0.53 Ls


V.

0.01

Figure 5. Steady-state phase current waveform and hamonic spectrum, (15)

-0.5

-0.5

0.008

15

Harmonic magnitude [% of Fundamental]

-0.5

-1 -1

Imaginary Z axis

Imaginary Z axis

ia

(a)


-1 -1

10 5 0 -5 -10 0

-0.5

SIMULATION RESULTS

The proposed predictive control strategy is evaluated through simulations using MATLAB/Simulink. The parameters of the PMSM are shown in Table I. DC bus voltage is 310 V, switching frequency is 10 kHz, and deadtime is 4 ȝs.

0.008

0.01

(a) Harmonic magnitude [% of Fundamental]

-1 -1

0

Phase current [A]

-0.5

Phase current [A]

1 Imaginary Z axis

Imaginary Z axis

1

15 10 Fundamental (200Hz) = 6.738 , THD= 12.03% 5 0 0

20


80

100

(b)

Rated power

750 W

Rated torque

2.39 N·m

Rated current

4.8 A

Rated speed

3000 r/min

Maximum speed

5000 r/min

Pole pairs

4

Stator resistance

0.45 ȍ

Stator inductance

3.9 mH

The stability and robustness of the predictive deadbeat current control are mainly affected by parameter variations. The performance of the conventional D-PCC algorithm and the proposed D-PCC algorithm are shown in Figs. 5–7. These results are obtained with 25% mismatch in inductance estimate under rated load torque condition. Figs. 5–6 show the timedomain current responses and the corresponding harmonic spectra of the conventional D-PCC algorithm. The results show that the conventional D-PCC algorithm is unstable and generates significant low-order harmonics, leading to a total harmonic distortion (THD) of 8.15% and 12.03%. In contrast, the proposed algorithm is still stable and generates the minimal low-order harmonics. The THD is 5.80% as depicted in Fig. 7.

Figure 6. Steady-state phase current waveform and hamonic spectrum, (16) Phase current [A]

PARAMETERS OF PMSM

10 5 0 -5 -10 0

ia

0.002

0.004 t [s] 0.006

0.008

0.01

(a) Harmonic magnitude [% of Fundamental]

TABLE I.

15 10 Fundamental (200Hz) = 6.612 , THD= 5.80% 5 0 0

20


80

100

(b) Figure 7. Steady-state phase current waveform and hamonic spectrum, (23)

Fig. 8 shows the result of a speed controller with the proposed current controller. The PI speed controller is employed. The speed command is changed from 3000 rpm to –3000 rpm under the 50% rated load torque condition. The speed shows almost no overshoot, while the torque transient is


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Speed [rpm]

adjusted to obtain nearly ideal speed tracking. The stator currents are highly sinusoidal.

parameter mismatch, the proposed algorithm is still stable and generates the minimal low-order harmonics.

3000

REFERENCES

Motor speed

0 -3000 0

Speed command

[1] 0.05

0.1 t [s]

0.15

0.2

3 phase current [A]

20

[2]

ia

10

[3]

ib

0 -10 -20 0.0

[4]

ic

0.05

0.1 t [s]

0.15

0.2

Figure 8. Dynamic response of the proposed predictive current controller in speed loop when speed command changes

VI.

CONCLUSION

In this paper, an improved deadbeat predictive current control strategy is introduced that has significantly increased robustness to stator inductance parameter mismatch. The conventional D-PCC algorithm is unstable when inductance mismatch occurs. The improvement is achieved by modifying the current offset constraint and output voltage prediction method. The theoretical analyses show that stability ranges are up to 53% mismatch in the stator inductance. The robustness range has been extended in comparison with conventional deadbeat predictive current control algorithm. Moreover, comparative evaluation simulations have been presented to demonstrate the effectiveness of the proposed control scheme. The results show that when the PMSM is rated operating with

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[5]

[6]

[7]

[8]

[9]

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