Model predictive torque control of permanent magnet synchronous motor system driven by matrix converter

TENG Qing-fang, LU Chang

(1. School of Automation and Electrical Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China；=2. Key Laboratory of Opto-Technology and Intelligent Control of Ministry of Education,Lanzhou Jiaotong University, Lanzhou 730070, China;=3. Rail Transit Electrical Automation Engineering Laboratory of Gansu Province,=Lanzhou Jiaotong University, Lanzhou 730070, China)

Abstract： Employing matrix converter (MC) as driving mode, the strategy of model predictive torque control (MPTC) is proposed for three-phase permanent magnet synchronous motor (PMSM) system. MC is applied instead of conventional AC-DC-AC converter to increase the power factor (PF) of the system input side. MPTC is used to select optimal voltage space vector to enable the system to have satisfactory torque and flux control effect. The resultant MPTC strategy not only makes the MC-fed PMSM system operate reliably and have perfect control performance, but also makes the PF of the system input side be 1. Compared with direct torque control (DTC), the proposed MPTC strategy guarantees that MC-fed PMSM has better command-following characteristics in the presence of variation of load torque and tracking reference speed. Simulation results verify the feasibility and effectiveness of the proposed strategy.

Key words： permanent magnet synchronous motor (PMSM); matrix converter (MC); model predictive torque control (MPTC)

0　Introduction

The AC-AC converter is a common power electronic device. Although such a traditional AC-AC converter can realize the frequency conversion with only one commutation and its energy can be bidirectional, it uses many thyristors and its power factor (PF) is low. Therefore, in order to solve the above problems, a matrix converter (MC) occurs. The MC is a new type of direct AC-AC converter topology. Because of the characteristics of small size, simple structure, adjustable input power factor, input current and output voltage with the shape of sine wave, and low harmonic pollution, MC is widely used in such areas as motor drives, mobile power supplies and power generation systems, and has attracted the attention of many scholars all over the world[1-5].

Permanent magnet synchronous motor (PMSM) drive nowadays is widely used in the industry applications due to high efficiency and high power/torque density[6-9]. As for high performance PMSM drive systems, the popular control strategies include two types: field oriented control (FOC) based on the stator current control[10]and direct torque control (DTC) based on switching table control[11]. In addition, in recent years model predictive control (MPC) as a rising control strategy has been widely used. MPC has the ability of constraint processing and nonlinear processing[12]. Compared with FOC, MPC does not need the current loop and parameter tuning. On the other hand, compared with DTC, MPC is more accurate and effective in the switch state selection[13].

In recent years, MPC strategy has been introduced into the PMSM system driven by MC[14-18]. In Ref.[14], the fixed switching frequency has been ensured without calculating the duty cycle, so that the torque ripples can be minimized. In Ref.[15], by calculating the desired voltage vector angle and determining its sector, the computational complexity can be reduced and the system control performance can be improved. In Ref.[16], by means of a look-up table, the candidate voltage vectors can be reduced. In Ref.[17], a multi-objective optimization based on a ranking approach has been proposed instead of a cost function. In Ref.[18], the comparison between predictive current control-based MC and predictive torque control-based MC has been studied and their characteristics have been pointed out. What the above-mentioned research has in common is that they employ MPC method directly to select the switch state of MC to realize the control of PMSM.

For the PMSM system fed by MC, different from the existing direct MPC methods, this paper uses the indirect MPC strategy. Firstly, MC is equated into two parts: the virtual DC-AC converter and AC-DC converter. Secondly, MPC-based model predictive torque control (MPTC) is used to select the voltage space vector of the virtual DC-AC converter to reduce the torque ripple and flux pulsation. Thirdly, phase hysteresis comparator is applied to select the current space vector of the virtual AC- DC converter to make the PF of the system input side be 1. Finally, based on the combination of the voltage space vector and current space vector, the switch state of MC is determined, thus the PMSM system is controlled.

1　Mathematical model of PMSM system driven by converter

1.1　Dynamic model of PMSM

The mathematical model of surface-mounted PMSM is considered indqsynchronous reference frame (dqframe). Assuming that the magnetic circuit is not saturated, the spatial magnetic field is sinusoidal distribution, and the hysteresis loss and eddy current loss are neglected, the stator current equations of PMSM indqframe can be expressed as

(1)

whereud,uq,id,iq,LdandLqaredandqaxis stator voltage, stator current, stator inductance indqframe, respectively;Rsis stator resistance;ψfis permanent magnet flux;pis the number of pole pairs; andωris rotor mechanical angular velocity.

The mechanical equation of PMSM is given by

(2)

whereJis moment of inertia;T1is load torque;Tfis coulomb friction torque;Bmis viscous friction coefficient; andTeis electromagnetic torque, which is expressed in thedqframe as

Te=1.5p[ψfiq+(Ld-Lq)idiq].

(3)

The flux linkage equations are written as

(4)

(5)

whereψdandψqaredandqaxis stator flux indqframe, respectively; andψsis stator flux.

1.2　Topological structure of MC and its mathematical model

In this paper, a three-phase MC with 3×3 switches, (totally 9 bidirectional switches) is employed. Each switch can be turned on or off. The MC input side is a three-phase voltage source, and the output side is connected to three-phase PMSM. The principle diagram of MC-fed PMSM is shown in Fig.1.

Fig.1　Topology of matrix converter

The relationship between input side voltageUi, currentIiand output side voltageUo, currentIoas shown in Fig.1 can be expressed as

Uo=[uAuBuC]T=M[uaubuc]T=MUi,

(6)

Ii=[iaibic]T=MT[iAiBiC]T=MTIo,

(7)

whereSmn(m∈{A,B,C},n∈{a,b,c}) is the switching state of MC;Smn=1 andSmn=0 are the “on” and “off” states of MC, respectively;Mis the transfer function matrix of MC from its input side to the output side. The operation of MC should follow the criterion that any two phases of the input side cannot short-circuited and any phase of the output side cannot be broken. Therefore, the following relations can be obtained as

Sma+Smb+Smc=1,m∈{A,B,C}.

(8)

According to the constraint condition given by Eq.(8), the 27 states of 9 bidirectional switches of MC can be yielded, among which except 6 kinds of switching states, the remaining 21 kinds of switching states can be effectively used to regulate the output voltage. Compared with the traditional voltage source inverter, MC can output more voltage vectors. As a result, it can achieve better control effect. In order to facilitate the in-depth study of the

system, MC can be equivalent to two parts: the virtual AC-DC converter and the virtual DC-AC converter, which are shown in Fig.2.

Fig.2　Equivalent AC-DC-AC structure of matrix converter

2　Design of MPTC system for PMSM driven by MC

The objective of MPTC PMSM driven by MC is that on the one hand the PF of input side is 1, on the other hand, PMSM system can work reliably and its speed and torque can be controlled not only to have satisfactory performance but also to be sufficiently robust against load variation. The diagram of the proposed control system is shown in Fig.3.

Fig.3　Block diagram of MPTC system for PMSM driven by MC

The MPTC is used to select the voltage space vectorUoof the virtual DC-AC converter. The phase hysteresis comparator is employed to select the current space vectorIiof the virtual AC-DC converter. The selection list is to determine the switching state of the MC according to the combination ofUoandIi.

2.1　Voltage space vector selection based on MPTC

MPTC combines the advantages of MPC and DTC. In the dynamic process of controlling the torque and stator flux, DTC uses online look-up table to select the voltage vector, while MPTC evaluates the cost function to select the optimal voltage vector. Therefore MPTC can effectively reduce torque and stator flux ripples and obviously improve the control performance of PMSM system.

2.1.1Model predictive torque control principle

For MPTC, the cost function of model prediction is defined as

s.t.Ui∈{U0,U1,…,U7},i=0,1,…,7,

(9)

(10)

The output voltage vector of virtual DC-AC converter is selected according to the principle of MPTC. In each sampling period, the corresponding cost function values of the 8 voltage vectors are calculated respectively. The 8 different cost function values are compared, and the switching vector corresponding to the minimum cost function value is chosen as the switching state of the virtual DC-AC converter in one sampling period.

The optimization of MPTC is shown in Fig.4, wherexrepresents the responses of torque and flux andTsis sampling period. The predicted valuexp4(k+1) is the closest to the reference valuex*, soU4is chosen as the control voltage vector at (k+1)th moment andU3is chosen as the control voltage vector at (k+2)th moment.

Fig.4　Optimization process of model predictive torque control

2.1.2Stator current prediction

Considering that thed-axis inductance of surface-mounted PMSM is approximately equal to theq-axis inductance, i.e.,Ld=Lq=L, according to Eq.(1), the prediction of the stator current at the next sampling instant can be expressed as

pLωr(k)iq(k)]+id(k)

pωr(k)(Lid(k)+ψf)]+iq(k),

(11)

whereid(k+1) andiq(k+1) are predicted values of stator current at (k+1)th moment.

2.1.3Torque and stator flux prediction models

According to Eq.(3), the prediction model of torque at the next sampling instant can be given by

Te(k+1)=1.5pψfiq(k+1).

(12)

Similarly, according to Eq.(5), the prediction model of flux at (k+1)th can be written as

(13)

2.2　Current space vector selection based on hysteresis comparator

The input voltage vector of MC, which rotates at a fixed frequency, is not influenced by the MC switching state; but the input current vector is affected by the MC switching state, and its phase is controllable, which provides the basis for the selection of input current vector.

The control procedure of virtual AC-DC converter is that firstly, input current vector phaseθiiand input voltage vector phaseθiuare detected and the sector of input voltage vectorMl(l=1,…,6) is determined. Secondly, the phase difference Δθbetweenθiuandθiiis measured and then fed into the hysteresis comparator which outputsδ. Finally, on the basis of combination ofδandMl, current space vector is selected. From foregoing description, it can be seen that an appropriate current space vectorIican be selected to reduce Δθ. If Δθis zero, input voltage vector and input current vector are in phase, the PF of system input side will be 1, which is the control criterion of the virtual AC-DC converter.

The current space vector of virtual AC-DC converter is expressed as

s.t.Ii∈{I1,I2,…,I9},i=1,2,…,9,

(14)

whereIdcis the output DC current of the virtual AC- DC converter;Si(i=a,b,c) is the switching state of the ith bridge arm.Si=1 orSi=-1 when the upper bridge arm is on or off.Si=0 when both upper and lower bridge arms are off. According to Eq.(14), there are 9 input current vectors, namelyI1,I2,I3,I4,I5,I6,I7,I8andI9, among whichI1-I6are six non-zero vectors, andI7-I9are three zero vectors. The layout of six non-zero current space vectors and corresponding six sectorsM1-M6are shown in Fig.5.

In order to make the input voltage vector and the input current vector be in-phase, the selection list of input current vectors can be obtained as shown in Table 1. The phase hysteresis comparator in Fig.3 is defined as

(15)

whereεθis the threshold value of phase hysteresis comparator.

Fig.5　Layout input space current vector of virtual AC-DC converter

Table 1　Selection list of input current vector of virtual AC-DC converter

2.3　Selection of MC switching states

The switching state of MC is determined by the virtual AC-DC converter and the virtual DC-AC converter. The switching state of virtual AC-DC depends on input current vector. At the same time, the switching state of virtual DC-AC converter depends on output voltage vector. On the basis of combination of input current vector with output voltage vector, selection list of switching states of MC can be obtained, as shown in Table 2.

Table 2　Selection list of switching states of MC

For example, symbol “acc” in Table 2 means that when output voltage vector of the virtual DC-AC converter isU1and input current vector of the virtual AC-DC converter isI1, and three switches such asSAa,SBcandSCcin MC are switched on (that is to say, the output terminalA,BandCare connected to the input terminalsa,bandc, respectively), and so forth.

3　Simulation and analysis

In order to verify the correctness and effectiveness of the designed system, the simulation is carried out by Matlab/simulink. The parameters of the PMSM used in the system are shown in Table 3.

Table 3　Parameters table of PMSM

In simulation, the three-phase AC input voltage is 220 V, the frequency is 50 Hz, and the sampling period is 10 μs. Threshold value in Eq.(15) is 0.05°. Two simulation schemes are given:

1) In order to check the PF of MC-fed MPTC PMSM system (abbreviated as MPTC system), the input voltage phase and input current phase are compared;

2) In order to verify better anti-disturbance ability of MPTC system than MC-fed DTC PMSM system (abbreviated as DTC system), their dynamic responses are compared and analyzed under the variation of load torque and reference speed.

3.1　Phase comparison between input voltage and input current of MC

The reference speed is set at 1 000 r/min and the load is set at 0 N·m. Since the amplitude of input voltage is much larger than that of input current, the actual input voltage is reduced by 300 times, so that they can be compared in the same coordinate system. Fig.6 shows the phaseavoltage of the input side which is reduced by 300 times, and the phaseacurrent of the input side.

It can be seen from Fig.6 that before 0.1 s the amplitude of the input phase-acurrent oscillates slightly; after 0.1 s the amplitude of input voltage and input current are stable, their phases are the same, and thus the PF of input side can reach 1. And the input waveform is roughly a sine wave.

Fig.6　Input voltage and input current of phase a

3.2　Comparison of anti-disturbance ability between DTC and MPTC systems

For fair comparison, the PI parameters of DTC and MPTC systems are the same and adjusted as

Kp=0.5,Ki=0.4.

3.2.1Comparison of system anti-load change capacity

In simulation, the reference speed is set at 1 000 r/min, and the load toque is increased from 0 to 2 N·m at 0.1 s. Figs.7 and 8 are the simulation results of the DTC and MPTC systems (including speed, torque, flux and stator current).

By comparing Fig.7(a) with Fig.8(a), it can be seen that when the load torques of these two systems increase, their speeds decrease at 0.1 s, but the speed decrease of DTC system is larger than that of MPTC system. From Fig.7(b) and Fig.8(b), it can be seen that the two systems have fast torque response, but MPTC system can effectively reduce torque ripple.

Comparison between Fig.7(c) and Fig.8(c) shows that MPTC system has satisfactory flux control effect. Comparing three-phase stator currents in Fig.7(d) with one in Fig.8(d) shows that MPTC system can effectively reduce the stator current ripple.

As a result, MPTC system is more powerful anti-load change capacity than DTC system.

3.2.2Comparison of system tracking change reference speed

The starting load torque is set at 1 and the speed is increased from 1 000 to 1 200 r/min at 0.1 s, and decreased to 900 r/min at 0.2 s.

The comparison of Fig.9(a) and Fig.10(a) shows that when the rotate speed suddenly changes, the two systems can reach the given reference speed in a short time, however, at 0.2 s, the speed response of DTC system is overshoot, while MPTC system has a smooth response without overshoot. Comparing the torque responses of these two systems in Fig.9(b) and Fig.10(b), the pulse generated by the MPTC system is smaller than that of DTC system, and the torque recovery is faster than that of DTC system. Therefore, compared with DTC system, MPTC system has a better ability of tracking reference speed.

Fig.7　Response of DTC system in case of load torque change

Fig.8　Response of MPTC system in case of load torque change

Fig.9　Dynamic responses of based DTC system in case of speed change

Fig.10　Dynamic responses of MPTC system in case of speed change

4　Conclusion

As for three-phase PMSM system fed by MC, MPTC strategy is designed. MC is applied to guarantee that the PF of system input side can be 1. MPTC strategy can make the system have good torque and flux control effect so as to improve the control performance of the system. For the PMSM system fed by MC, compared with DTC strategy, MPTC strategy can guarantee that the system has smaller torque ripple and better command-following characteristics in the presence of variation of load torque and reference speed. Simulation results verify the feasibility and effectiveness of the proposed strategy.