Model predictive flux control of permanent magnet synchronous motor driven by three-level inverter based on fine-division strategy

MIAO Zhongcui, LI Haiyuan, HE Yangyang, WANG Yunkun

(School of Automation & Electrical Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China)

Abstract： Aiming at the difficulty of setting the weight coefficient in the value function of model predictive torque control (MPTC) for permanent magnet synchronous motor (PMSM) driven by three-level inverter, a fine-division model predictive flux control (MPFC) method is proposed. First, establish a mathematical model between the motor torque and the stator flux linkage according to the mathematical equations of PMSM. Thus, the control of the motor torque and stator flux linkage in the MPTC is transformed into the control of a single stator flux linkage vector, omitting the cumbersome weight setting process in the traditional MPTC. The midpoint potential control strategy is proposed, which uses the characteristics of redundant small vectors to balance the midpoint potential. After that, a fine-division strategy is proposed, which effectively reduces the number of candidate vectors and the computational burden of the system. Finally, the proposed MPFC is compared with MPTC by simulation. The results show that the proposed fine-division MPFC effectively reduces the system calculation, and has the advantages of simple principle and better dynamic and steady-state control performance. The feasibility of the control strategy is verified.

Key words： permanent magnet synchronous motor (PMSM)； three-level inverter； model predictive flux control (MPFC)； weight coefficient； midpoint potential

0 Introduction

Permanent magnet synchronous motors (PMSM) have been widely used in many fields such as high-end computerized numerical control (CNC) machine tools, new energy vehicles, aerospace, high-speed rail traction systems and other fields because of their high efficiency, high power factor, high reliability and easy maintenance[1]. In the field of modern motor control, traditional control strategies mainly include direct torque control (DTC) and field oriented control (FOC)[2-3]. Although it have better steady-state performance by FOC, its dynamic performance still needs further improvement[4]. In comparison, DTC can directly control the amplitude of torque and flux linkage. It has the advantages of fast response speed and simple structure[5], however, it also has some disadvantages such as large torque fluctuation, high current harmonic content and large calculation amount[6-7].

Model predictive control (MPC) can be divided into continuous control set MPC (CCS-MPC) and finite control set MPC (FCS-MPC) according to the type of voltage vector control set[8]. The control variable in CCS-MPC is applied to the controlled object through pulse width modulation, which greatly increases the switching frequency and switching loss of the switch tube. And the amount of calculation of CCS-MPC is large, so it has not been widely used[9-10]. In the prediction process of FCS-MPC, it only predicts the voltage vector that the inverter can directly output. The algorithm is relatively simple, easy to implement, and the dynamic response speed is fast. It has become a research hotspot of MPC today[11]. However, due to the finiteness of the space voltage vector and the fixed amplitude and direction, the current pulsation is relatively large. A lag compensation control strategy is proposed for the traditional single vector model predictive control[12]. Nowadays, FCS-MPC has been developed from traditional single vector MPC to multi vector MPC control.

In the field of MPC motor control, the value function is usually used to select the optimal voltage vector, and the value function usually includes flux ripple and torque ripple[13-15]. In order to achieve the expected control requirements, it is necessary to add a weight coefficient to balance their weight in the value function[16].

Compared with the two-level inverter, the neutral-point-clamped (NPC) three-level inverter has a higher output level than the traditional two-level inverter. Therefore, the sine of the output voltage and current waveform is better, and the voltage stress of each power tube on each phase bridge arm is smaller, which is suitable for high-voltage and large-capacity applications[17-18]. However, if the NPC three-level inverter does not solve the problem of midpoint voltage fluctuations, it may make the entire system out of control. The characteristics of redundant small vectors are used, and the midpoint potential is controlled by judging the direction of the midpoint current and the magnitude of the midpoint potential[19].

In addition, because there are 27 voltage vectors in the NPC three-level inverter, 27 judgments are required when selecting the optimal voltage vector, which will greatly increase the amount of calculations, cause a certain processing delay in the system, and ultimately lead to inaccurate predictions[20]. Therefore, it is necessary to research how to reduce the amount of calculation and the computational burden of the system.

In order to eliminate the tedious weight coefficient setting process in the three-level model predictive torque control (MPTC) system, PMSM is taken as the research object, and a PMSM three-level vector subdivision model predictive flux control strategy is proposed in this paper. First, the MPTC prediction model is established by using Euler’s discrete equation, and the relationships among the stator flux vector, the torque and the amplitude are derived, and the relationships are transformed into the constraint of flux linkage vector. At the same time, the midpoint potential control strategy[21]is used to solve the midpoint voltage fluctuation problem of the NPC three-level inverter. On this basis, in order to reduce the computational burden of the system, a three-level vector fine partition strategy is proposed. The basic space vector can be divided into 6 large sectors, and each large sector can be subdivided into 4 small sectors. Finally, the number of candidate vectors is reduced to 3.

1 Mathematical model of PMSM

The embedded PMSM is taken as the research object and the mathematical model is established in thedqcoordinate system.

The stator flux equation is

(1)

The stator voltage equation is

(2)

The electromagnetic torque equation is

(3)

whereψdandψqare thedq-axis components of the stator flux linkage；LdandLqare thedq-axis inductance components；udanduqare thedq-axis components of the stator voltage；idandiqare thedq-axis components of the stator current；Ris the stator resistance of each phase；ωis the electrical angular velocity of the rotor；ψfis the permanent magnet flux linkage；Teis electromagnetic torque；npis the number of pole pairs.

2 Model predictive torque control

2.1 Control system structure

The structure diagram of the PMSM model predictive torque control system is shown in Fig.1.

The three-level inverter adopts a midpoint clamp topology. First, the system measures the motor current and speed signals, and calculates the actual value of the stator flux and torque. At the same time, after the speed error passes through the PI regulator, the expected value of the stator flux and torque is obtained. According to the actual value of the system variable, the model predicts the torque controller to predict the running trend of the motor on different voltage vectors. After the prediction process is completed, the value function is used to evaluate the error between the predicted result and the expected value of the system variable. Finally, the voltage vector with the smallest comprehensive error is selected as the output.

Fig.1 MPTC block diagram of permanent magnet synchronous motor

2.2 Predictive model

Substituting Eq.(1) into Eq.(2), the stator instantaneous current expression is obtained by

(4)

Discretizing the stator instantaneous current expression, the current prediction model is obtained by

(5)

The value function in MPTC usually takes the torque and the amplitude of the stator flux linkage as the control target, but the magnitude of the two is different, and a weight coefficient needs to be introduced for adjustment. The general MPTC value function expression is

(6)

In order to achieve the optimal control effect in actual operation, it is necessary to debug the error weight coefficientλf, which is very cumbersome. Therefore, the selection of the weight coefficient in the MPTC value function is still a difficult point. How to effectively omit the design process of the weight coefficient has become an important issue in the field of motor control.

3 Model predictive flux control

3.1 Control system structure

The block diagram of the predictive flux linkage control of a PMSM driven by a three-level inverter is shown in Fig.2.

Fig.2 Three-level MPFC block diagram of permanent magnet synchronous motor

Firstly, the error between the given speed and the actual speed is obtained through the speed loop PI regulator to obtain the electromagnetic torque reference value, and it is sent to the conversion module together with the flux linkage reference value to obtain the flux linkage reference value at timek+1. Then the predicted value of the flux linkage and the reference value are sent to the value function for rolling optimization, and the action vector that minimizes the value function is obtained. Then the action time calculation module judges the voltage vector after the neutral point potential is balanced, and the corresponding action time is obtained. Finally, the corresponding switching sequence is output and sent to the inverter.

3.2 Predictive model

The expression of thedqaxis component of the stator flux linkage at timekis

(7)

Substituting Eq.(7) into Eq.(3), it can be obtained that

(8)

Take the derivative ofδand rewrite it into incremental form, and finally get

(9)

In order to improve the efficiency of motor control, controlid=0, and then

(10)

(11)

Finally, according to the predicted value of the flux linkage at timek+1 and the reference value of the flux linkage, the MPFC value function without the weight coefficient can be established as

(12)

By comparing with the value function in MPTC, it can be seen that the value function in MPFC transforms the control of the torque and the amplitude of the flux into the control of the stator flux vector. Thereby the weight coefficientλfis eliminated.

3.3 Midpoint potential control strategy

By analyzing the DC side circuit, the equivalent model of the circuit is shown in Fig.3.

Fig.3 NPC three-level inverter circuit equivalent model diagram

The current passing through the midpoint is callediNP. Regarding the direction flowing into the motor windings as the positive direction, the two capacitors areC1andC2, respectively. And assuming that the two capacitors are equal in value, both areC. The currents of passing through the two capacitors areiC1andiC2, respectively. Δvnpis the voltage difference between the two capacitors. The state average method is adopted, and the average voltage in a period is taken as the voltage difference at this moment, and the relationship between capacitor voltage and current is expressed as

(13)

It can be seen that the voltage difference between the two capacitors will produce a midpoint current. As shown in Fig.4, when the small vector OPO acts,iNPflows into the midpoint, makingvc0rise. As shown in Fig.5, when the small vector NON acts,iNPflows out of the midpoint, makingvc0drop. Therefore, when using small vectors, the characteristics of positive and negative redundant small vectors can be used to complete the adjustment of the midpoint potential.

Fig.4 OPO circuit topology diagram

Fig.5 NON circuit topology diagram

Among them, the corresponding relationship between the positive and negative small vectors and the generated midpoint current in the three-level inverter is shown in Table 1.

In the midpoint potential control strategy, the switching state is further adjusted by judging the direction of the midpoint current and the current switching state at the moment, and comparing it with the current midpoint voltage.

Table 1 Correspondence table of positive and negative small vectors and midpoint currents

2) The midpoint currentiNPflows out of point O.iNPis greater than 0.vc0decreases, and is represented by |iNP|=P. The midpoint currentiNPflows into point O,iNPis less than 0,vc0increases, and is represented by |iNP|=N.

3) Using -h≤vc0≤hto indicate the reasonable range of the midpoint voltage, and by defining |vc0|=P asvc0>h, it means thatvc0is too large. Define |vc0|=O as -h≤vc0≤h, which means thatvc0is within a reasonable range, and there is no need to balance the midpoint potential. Define |vc0|=N asvc0<-h, which meansvc0is too small.

The specific judgment process is shown in Fig.6.

Fig.6 Flow chart of midpoint potential control strategy

3.4 Vector fine partition strategy

For the traditional three-level inverter, in order to improve the performance of the PMSM, all the space voltage vectors must be sent to the value function for rolling optimization, and the action vector with the smallest value function is obtained. But the three-level inverter has a total of 27 basic voltage vectors (6 large vectors, 6 medium vectors, 12 small vectors and 3 zero vectors). Therefore all the 27 basic voltage vectors needs to be judged by the traditional method, which greatly increases the calculation amount of the system and causes the calculation burden to the system. Therefore, in order to reduce the computational burden of the system, a three-level vector fine partition screening strategy is proposed to reduce the number of selectable voltage vectors.

Fig.7 NPC three-level basic space vector distribution map

Firstly, judge the large sector where the reference voltage vector is located. According to the flux equation expression under theαβcoordinate, the expression of the angle betweenψαandψβsynthesized flux and theα-axis is obtained as

(14)

(15)

(16)

In order to further reduce the number of candidate vectors and reduce the computational burden of the system, the large sector is further divided into four small sectors by dotted lines. As shown in Fig.8, the four small sectors are numbered, respectively. The judgment of the small sector where the reference voltage vector is located is more complicated, and the angle and mode length of the reference voltage need to be considered. Due to the symmetry of the three-level basic space vector distribution, the identification process of the small sector where the reference voltage vector is located in the large sector Ⅰ is taken as an example.

Fig.8 Fine vector partition distribution map

According to the dividing line equation, the specific judgment process is shown as 1)-4).

Fig.8 shows an example where the reference voltage vector is located in the large sector Ⅰ. When the small sector where the reference voltage vector is located isA, according to the vector proximity principle, at this time, only the zero vector and the small vector ONN and OON are available for selection. When the small sector of the reference voltage vector isB, the available vectors are small vector ONN, OON and medium vector PON. When the small sector where the reference voltage vector is located isC, the available vectors are small vector ONN, medium vector PON and large vector PNN. When the small sector of the reference voltage vector isD, the available vectors are small vector OON, medium vector PON and large vector PNN. Candidate vector table of the six large sectors of the basic voltage space vector can be obtained by analogy. Therefore, through the fine partition screening strategy, the number of candidate vectors can be reduced from 27 to 3 in the end.

Finally, according to the judgment of the sector where the reference voltage vector is located and the selection of the basic vector, the action time of each basic vector in the selected basic vector combination is allocated to synthesize the required reference voltage vector.

Due to the symmetry of the basic vector distribution in the three-level space, as shown in Fig.8. As an example, three basic vectorsv0,v1andv2are used for synthesis in sectorA. The following volt-second balance equations should be satisfied within one sampling period.

(17)

wherev0,v1andv2are the selected basic vectors, respectively；T0,T1andT2are the action time of the corresponding basic vectors, respectively.

Table 2 Alternative vector table of each sector

The action time of the three basic vectors is gotten by solving Eq.(17).

(18)

In the same way, the basic vector action time in each sector shown in Fig.8 can be obtained.

Selectv1,v2andv4as the basic vectors in sectorB, and the action time is

(19)

Selectv1,v3andv4as the basic vectors in sectorC, and the action time is

(20)

Selectv2,v4andv5as the basic vectors in sectorD, and the action time is

(21)

Other sectors are similar to sector I.

Finally, according to the synthesized optimal voltage vectors, a switching sequence that is advantageous for suppressing midpoint potential fluctuations is obtained, and sent to the inverter to drive the permanent magnet synchronous motor.

4 Simulation

The feasibility of the fine-division MPFC control strategy of PMSM motor system is verified by Matlab/Simulink, and compared with the traditional MPTC control strategy. When using Matlab to simulate, it is found that when the weight coefficientλf=100 in traditional MPTC, the control performance reaches the optimal control effect. If no special instructions are given, the following MPTC selects the weight coefficientλf=100 for comparison with MPFC.

Table 3 Main parameters of PMSM

4.1 Steady-state control performance comparison

First, the motor runs under the conditions of 1 000 r/min and a load of 2 N·m, and the MPTC under the two weighting coefficients is compared with the fine-division MPFC proposed in this paper. The final experimental results are shown in Fig.9. When the weight coefficient isλf=50, the torque ripple and flux ripple of the traditional MPTC are relatively large, and the harmonic content of the three-phase current THD=6.56%. When the weight coefficient isλf=100, the torque pulsation and the stator flux pulsation are relatively balanced, close to the optimal state, and the harmonic content of the three-phase current THD is reduced from 6.56% to 1.23% (Fig.9(b)).

(a) MPTC, λf=50

It can be seen from Fig.9(c) that the fine-division MPFC can have better steady-state performance than that without adjusting the weight coefficient under the same operating conditions. It has lower torque ripple and flux ripple and better current sinuosity, and the harmonic content of three-phase current THD is reduced to 0.38%, which is lower than MPTC under optimal state. In summary, it can be seen that the steady-state performance of the fine-division MPFC strategy is better.

4.2 Comparison of midpoint potential control performance

The midpoint voltage balance control simulation is shown as Fig.10, the working condition isω=1 000 r/min, and the load torque is 1 N·m.

After 0.3 s, the midpoint voltage control strategy is adopted. Due to the fluctuation of the neutral point potential of the divider capacitor in the three-level inverter, it can be seen in Fig.10(a) that when the neutral point potential control strategy is not adopted for MPFC,vC1andvC2seriously deviate from the normal value of 200 V before 0.3 s, and rapidly tend to 200 V after 0.3 s, the neutral point voltage decreases rapidly, and finallyvc0tends to balance at 0.338 s, and is controlled near 0 V. The traditional MPTC control strategy achieves the balance of neutral point voltage after 0.393 s. Due to the smaller deviation of MPFC’s neutral point potential, MPFC’s regulation time is faster and the current waveform is better. In conclusion, the fine-division MPFC has better performance in neutral point potential control.

(a) MPFC

4.3 Dynamic control performance comparison

Fig.11 shows that the speed changes from 1 000 r/min to 1 500 r/min att=0.2 s. It can be seen that in the case of a sudden change in the speed, both the traditional MPTC and the fine-division MPFC can track the speed command well, and the response time is basically the same. Moreover, the electromagnetic torque can respond quickly, and quickly return to a stable level. However, the rotational speed under the fine-division MPFC strategy basically has no overshoot, and the torque ripple caused by the sudden change of rotational speed is also smaller. And since the midpoint potential control strategy is added to the fine-division MPFC, it can be seen that the suppression of the midpoint potential is more effective and stable. In summary, it can be seen that the dynamic performance of the fine-division MPFC is better under the condition of sudden changes in speed.

(a) MPTC

Fig.12 shows the change from 1 N·m to 2 N·m att=0.2 s under the condition of sudden torque change. It can be seen from the simulation waveform that in the case of a sudden change in torque, both the traditional MPTC and the fine-division MPFC can better track the speed command, the response time is basically the same, and the torque can quickly return to a stable level. However, under the fine-division MPFC, the overshoot of the speed is smaller, the torque ripple is smaller, and the suppression of the midpoint potential is more effective and stable. In summary, it can be seen that under the condition of sudden torque change, the dynamic performance of the fine-division MPFC is better.

(a) MPTC

5 Conclusions

The fine-division MPFC is proposed for a permanent magnet synchronous motor driven by a three-level inverter. The control of the stator flux linkage amplitude and torque is transformed into the control of a single stator flux linkage vector, omitting the weight setting process in the traditional MPTC value function. The midpoint voltage control strategy is proposed according to the characteristics of positive and negative redundant small vectors to solve the problem of midpoint potential imbalance in three-level inverter. In order to further reduce the computational burden of the system, the fine-division strategy for voltage vectors is proposed, which reduces the number of candidate vectors from 27 to 3. Finally, a detailed comparison between the fine-division MPFC and the traditional MPTC control strategy is carried out through simulation. The results show that the proposed fine-division MPFC has the advantages of simple principle and better dynamic and steady-state control performance.