A Low-Complexity Double Vector Model Predictive Current Control for Permanent Magnet Synchronous Motors

Abstract

Compared to the conventional finite control set model predictive control (FCS-MPC), the double vector model predictive current control (DVMPCC) for permanent magnet synchronous motors (PMSMs) has a better steady-state performance without significantly increasing the switching frequency. However, determining optimal vectors with their dwell times requires a high computational burden. A low-complexity DVMPCC in the steady state was proposed in this study to address this problem. Firstly, the operating state of the motor was judged according to the speed error. During steady-state operation, the first optimal active vector was selected from three candidate vectors adjacent or identical to the active vector applied in the previous control period, reducing the number of comparisons by half. Next, the second optimal vector was selected from the other two active vectors, and the zero vector, the second optimal vector with the duty cycle, was determined according to the deadbeat condition of the q-axis current and cost function minimization. Finally, simulation and experimental results proved that the proposed low-complexity DVMPCC for surface-mounted permanent magnet synchronous motors is practical and feasible.

1. Introduction

The motor drive system, composed of an inverter and a motor, plays an essential role in many fields, such as rail transit, electric vehicles, and aerospace [1]. According to the energy characteristics of its DC-side, an inverter is categorized as a voltage source inverter (VSI) or a current source inverter (CSI) [2,3]; the topology of a VSI is simpler than that of a CSI [4]. Compared to other motors, permanent magnet synchronous motors (PMSMs) have the advantages of high power density, low cost, high efficiency, and high reliability [5]; so the PMSM drive system fed by a VSI has been widely studied and applied. Its topology is shown in Figure 1.

The main control strategies used for high-performance control of permanent magnet synchronous motor drives include field-oriented control (FOC) and direct torque control (DTC) [6]. In recent years, model predictive control (MPC) has been widely studied with regard to the control of power converters and electric drives due to the development of digital signal processors (DSPs) [7]. Compared to FOC and DTC, MPC has many advantages, including a simple structure, fast dynamic response, flexible control, refraining from tuning inner loop controller parameters, and ease of dealing with system nonlinear constraints [8]. MPC can be categorized as continuous control set model predictive control (CCS-MPC) and finite control set model predictive control (FCS-MPC) [9]. The candidate voltage vectors of FCS-MPC are limited, so the torque ripple and the current distortion are slightly significant [10].

To improve steady-state performance, CCS-MPC that applies two or three vectors is more suitable when the switching loss of double-vector MPC is less than that of three-vector MPC [11]. Many double-vector MPC methods have been proposed in succession. In [12], the second voltage was restricted to the zero vector, which covers a limited area, and the torque ripple was still significant; therefore, the second voltage vector selection range should be broadened. In [13], the first and second vectors were expanded to any vector combinations, and redundant vector combinations were removed. However, the simplified algorithm must still be calculated 28 times in a control period. The selected double vectors in [14] were not restricted to adjacent, but the design of the pulse generator was complicated. In [15], the second alternative voltage vector was expanded from the zero vector to three vectors adjacent to the first voltage vector, and the dwell time was calculated through the cost function. The torque ripple was reduced compared to that in [12]. In [15], the amplitude and angle of the reference voltage vector were derived according to the deadbeat condition and cost function, and the method of vector selection and dwell time calculation were presented to minimize the error from the reference voltage vector. In [16], a switching table was established to select the first voltage vector. Next, the second voltage vector was selected from all vectors for better control performance. The computational complexity and comparison numbers of the above methods are too high, so it is necessary to simplify double vector MPC.

According to different control objects, MPC can be categorized as model predictive torque control (MPTC) or model predictive current control (MPCC) [9]. The cost function of MPTC includes torque error and flux error. As the dimensions are different, a weighting factor is introduced into the cost function of MPTC [17]. However, tuning the weighting factor is complex and lacks theoretical guidance [18], so MPCC was adopted in this study, considering that its cost function consists of d-q axis currents, and no weighting factor was required.

This paper presents a low-complexity double vector MPCC (DVMPCC) for a PMSM drive system fed by VSI. As the reference voltage vector has a slight variation between adjacent control periods, the range of candidate vectors can be reduced to the active vectors adjacent or identical to the first optimal active vector in the previous control period, and the number of comparisons can be reduced. On this basis, the principle of double voltage vector selection and the calculation method of the dwell time are presented, as are their effectiveness and feasibility, which were certified using simulation and experimental results.

2. Math Model and Cost Function

In two rotating frames (d-q axis), the voltage equations of a PMSM are expressed as

$$u_d=R_s{i}_d+L_d\frac{d{i}_d}{dt}-{\omega }_e{L}_q{i}_q$$

(1)

$$u_q=R_s{i}_q+L_q\frac{d{i}_q}{dt}+{\omega }_e(L_d{i}_d+{\psi }_f)$$

(2)

The first-order Euler formula is applied to (1) and (2), and the discrete current equations of i_d and i_q are expressed as

$$i_d(k+1)=(1-\frac{R_s{T}_s}{L_d})i_d(k)+\frac{T_s}{L_d}{\omega }_e{L}_q{i}_q(k)+\frac{T_s}{L_d}{u}_d(k)$$

(3)

$$i_q(k+1)=(1-\frac{R_s{T}_s}{L_q})i_q(k)-\frac{T_s}{L_q}{\omega }_e{L}_d{i}_d(k)-\frac{T_s}{L_q}{\omega }_e{\psi }_f+\frac{T_s}{L_q}{u}_q(k)$$

(4)

where i_d(k) and i_q(k) are the sampling values of i_d and i_q at kT_s time, u_d(k) and u_q(k) are the values of u_d and u_q at kT_s time, and i_d(k+1) and i_q(k+1) are the predictive values of i_d and i_q at (k+1)T_s time, respectively.

The cost function of MPCC is shown as (5). According to (3) and (4), the predicted current values i_d(k + 1) and i_q(k + 1) can be obtained employing each voltage vector. Next, i_d(k + 1) and i_q(k + 1) are substituted into (5), and the voltage vector that minimizes the cost function is obtained.

$$g=\left|i_{dref}-i_d(k+1)\right|+\left|i_{qref}-i_q(k+1)\right|$$

(5)

3. Conventional Double Vectors MPCC Based on the Deadbeat of q-Axis Current

Conventional q-axis current deadbeat double vectors MPCC (CQCD-MPCC) adopts the zero vector as the second vector, according to the deadbeat condition of the q-axis current, and the first optimal vector with the duty cycle is determined from the six basic voltage active vectors; the space distribution is shown in Figure 2 [19].

The diagram of QCD-MPCC is shown in Figure 3; u₂ and u₀ are the optimal vectors. The q-axis current deadbeat (i_q(k + 1) = i_qref) can be achieved by allocating the appropriate dwell time.

According to (3) and (4), the current rate of change of i_d and i_q caused by the zero vector can be derived and expressed as

$$s_{q0}=(1-\frac{R_s{T}_s}{L_q})i_q(k)-\frac{T_s}{L_q}{\omega }_e{L}_d{i}_d(k)-\frac{T_s}{L_q}{\omega }_e{\psi }_f$$

(6)

$$s_{d0}=(1-\frac{R_s{T}_s}{L_d})i_d(k)+\frac{T_s}{L_d}{\omega }_e{L}_q{i}_q(k)$$

(7)

The current rate of change of i_d and i_q caused by the i-th active vector can be derived and expressed as

$$s_{qi}=s_{q0}+\frac{T_s}{L_q}{u}_{qi}(k)$$

(8)

$$s_{di}=s_{d0}+\frac{T_s}{L_d}{u}_{di}(k)$$

(9)

where i = 1, 2, …, 6. The current prediction equation after introducing the duty cycle of the active vector can be rewritten as

$$i_q(k+1)=i_q(k)+s_{qi}{d}_i{T}_s+s_{q0}(1-d_i)T_s$$

(10)

$$i_d(k+1)=i_d(k)+s_{di}{d}_i{T}_s+s_{d0}(1-d_i)T_s$$

(11)

According to the deadbeat condition of q-axis current (i_q(k+1) = i_qref), d_i can be derived and expressed as

$$d_i=\frac{i_{qref}-i_q(k)-s_{q0}{T}_s}{(s_{qi}-s_{q0})T_s}$$

(12)

Finally, the range of d_i should be corrected: if d_i < 0, let d_i = 0; if d_i > 1, let d_i = 1. It should be noted that CQCD-MPCC can only minimize the error of i_q. The selected double vectors with the duty cycle cannot achieve the minimum error of the cost function (5).

4. Low-Complexity Double Vector Model Predictive Current Control

4.1. Low-Complexity Determined Method for the First Active Vector

In CQCD-MPCC, six calculations and comparisons are required to select the active vector, and the computational burden is too high. This section proposes a low-complexity double vector MPCC (LCDB-MPCC) that can reduce times required for calculation and comparison when a PMSM operates in a steady state. The control diagram is shown in Figure 4. As the reference voltage vector has little angle variation during a control period, the range of candidate vectors can be reduced to the active vectors adjacent or identical to the first optimal active vector in the previous control period, and the number of comparisons can be reduced. The operating state of a PMSM is judged according to the speed error ∆ω_r. If ∆ω_r is less than one rad/s, it is considered as the steady state in this paper. The specific selection steps for the first active vector u_opt1 are as follows.

Step 1: Initialize the voltage vector in the previous period, u_old = u₀.

Step 2: If ∆ω_r > 1 rad/s or u_old = u₀, u_opt1 is selected from all basic active vectors and applied in the whole control period.

Step 3: If ∆ω_r < 1 rad/s and u_old ≠ u₀, u_opt1 is selected from three active vectors, which are adjacent or identical to u_old. The candidate vectors corresponding to different u_old are shown in Table 1.

Step 4: Substitute the candidate vectors into the cost function, respectively; u_opt1 is determined according to the minimization of the cost function.

Figure 4. Diagram of the whole control scheme.

Figure 4. Diagram of the whole control scheme.

Table 1. Candidate vectors corresponding to different u_old.

uold	Candidate Vectors
u1	u6(101), u1(100), u2(110)
u2	u1(100), u2(110), u3(010)
u3	u2(110), u3(010), u4(011)
u4	u3(010), u4(011), u5(001)
u5	u4(011), u5(001), u6(101)
u6	u5(001), u6(101), u1(100)

Table 1. Candidate vectors corresponding to different u_old.

uold	Candidate Vectors
u1	u6(101), u1(100), u2(110)
u2	u1(100), u2(110), u3(010)
u3	u2(110), u3(010), u4(011)
u4	u3(010), u4(011), u5(001)
u5	u4(011), u5(001), u6(101)
u6	u5(001), u6(101), u1(100)

4.2. The Principle of Determining the Second Vector

The second optimal vector u_opt2 is selected from the other two candidate vectors of u_opt1 and the zero vector. As the speed and torque are adjusted by i_q, u_opt2 is determined according to the deadbeat condition of i_q. The flow chart is shown in Figure 5, and the specific steps are as follows.

Step 1: According to (8), i_q(k + 1) under each candidate vector applied to u_opt2 can be obtained as follows

$$i_q(k+1)=i_q(k)+s_{qopt1}{t}_{opt1}+s_{qj}(T_s-t_{opt1})$$

(13)

where j = 0, 1, 2. It is noted that s_qj represents the current rate of Δi_q caused by the zero vector and the other two candidate vectors; i_q(k + 1) is replaced with i_qref, and t_opt1 can be obtained as follows

$$t_{opt1}=\frac{i_{qref}-i_q(k)-s_{qj}{T}_s}{s_{qopt1}-s_{qj}}$$

(14)

Step 2: The range of t_opt1 should be corrected: if t_opt1 > T_s, let t_opt1 = T_s. At this time, i_q(k + 1) should be recalculated using (13).

Step 3: According to (11), calculate i_d(k + 1). Next, substituting i_d(k + 1) and i_q(k + 1) into (5), the value of the cost function when the second voltage is u_j is obtained and denoted as g_j.

Step 4: Compare g₀, g₁, and g₂, u_j, and select the minimum g as the u_opt2.

Figure 5. Flow chart of determining the second vector.

Figure 5. Flow chart of determining the second vector.

5. Simulation Results

The PMSM drive system fed by VSI was established in a Matlab/Simulink environment. The simulation and comparison were completed using three DVMPCC methods, which are as follows:

(1)

Conventional q-axis current deadbeat double vectors model predictive current control (CQCD-MPCC): As introduced in Section 3, the first optimal vector with the duty cycle is determined according to the deadbeat condition of the q-axis current, and the second optimal vector is restricted to the zero vector.

(2)

Improved q-axis current deadbeat double vectors model predictive current control (IQCD-MPCC): Compared to CQCD-MPCC, the second optimal vector is expanded to the six basic active vectors from the only zero vector.

(3)

Proposed low-complexity double vectors model predictive current control (LCDVMPCC): When the PMSM operates in the steady-state, the first optimal vector is determined from three candidate vectors, and the second optimal vector with the duty cycle is determined according to the deadbeat condition of the q-axis current and minimization of the cost function.

The parameters of the PMSM are shown in

Table 2. Parameters of the PMSM.

Parameters	Value
Rated voltage	300 V
Rated power	4 kW
Rated speed	3000 rpm
Rated torque	10 N·m
Rated current	18 A
Pole pairs	4
Stator resistance	0.15 Ω
Stator inductance	1.625 mH

.

Simulation Results for the Steady-State

When the PMSM operated at 1600 rpm and 10 N·m, the simulation waveform of the stator current and the FFT results of CQCD-MPCC, IQCD-MPCC, and LCDV-MPCC were as shown in Figure 6. The proposed LCDV-MPCC had the least harmonic distortion. The THD was 3.94%, less than those of CQCD-MPCC and IQCD-MPCC.

Figure 7 shows the ripple of i_q, i_d, and T_e under different speeds when the PMSM operated at 5 N·m. As the deadbeat of i_q is considered to be the principle of the vector, it was determined using all three methods. Thus, the difference between i_{q_ripple} and T_{e_ripple} was slight. However, the second optimal vector of the proposed LCDV-MPCC was selected according to the minimization of the cost function, which contains the predictive error of i_d, so the i_{d_ripple} of LCDV-MPCC was less than those of CQCD-MPCC and IQCD-MPCC.

Figure 8 shows the ripples of i_q, i_d, and T_e under different load torques when the PMSM operated at 3000 rpm. As the load torque increased, the steady-state performance of the proposed LCDV-MPCC was significantly better than those of CQCD-MPCC and IQCD-MPCC, except for i_{d_ripple}, which could also be reflected from i_{q_ripple} and T_{e_ripple}.

The THD of the stator current under CQCD-MPCC, IQCD-MPCC, and LCDV-MPCC when parameter variations in inductance and resistance occurred are shown in Figure 9. As the PI controller was adopted to regulate the speed, there were no steady-state errors in the speed when parameter variations occurred. However, parameter variations may lead to incorrect vector selection or increase the duty cycle’s calculation error, so the stator current’s harmonic distortion is more severe as parameters vary. Figure 9 shows that the sensitivity of the control system to inductance is greater than it is to resistance because the discrete current equation only contains one item about resistance. Although the proposed LCDV-MPCC did not significantly inhibit parameter variations, the THD was less than those of CQCD-MPCC and IQCD-MPCC.

6. Experimental Results

An experimental platform of a PMSM drive system fed by VSI was established as shown in Figure 10. Experimental platform variations are shown in Figure 9, and parameters are shown in

Table 3. Parameters of the experimental platform.

Device	Type and Parameters	Manufacture
DC bus	DS1020	eTOMMENS/Dongguan, China
PMSM	180ST-M, 4.5 kW	Customed
Encoder	OIH, 2500 C/T	Tamagawa/Japan
Torque sensor	YH502, 0–50 N·m	ALIPO/Guangzhou, China
Magnetic powder brake	TS-PB-A, 0–50 N·m	YOUYAN/Wuxi, China
RTU-BOX	RTU-BOX204	Rtunit/Nanjing, China

.

6.1. Experimental Results for the Steady-State

Experimental results for the steady-state were obtained under the operating conditions of a speed of 1600 r/min with 10 N·m. Steady-state experimental waveforms under CQCD-MPCC, IQCD-MPCC, and LCDV-MPCC are shown in Figure 11, respectively. As the candidate vectors were limited and i_d was not considered in the vector selection process, Figure 11a shows significant low-order harmonic distortion in the stator current under CQCD-MPCC, and the THD is 5.26%. Figure 11b shows that after expanding the range of candidate vectors for u_opt2, the 5th and 7th harmonic contents in the stator current under IQCD-MPCC were reduced, and the THD was 4.78%. Severe distortion occurred on i_a under the control of TPTC. As the proposed LCDV-MPCC considers the control performance of i_d, Figure 11c shows that its ripple of i_d was less than those of CQCD-MPCC and IQCD-MPCC, and the THD was 4.17%.

Both experimental and numerical results demonstrate that the proposed LCDV-MPCC’s steady-state performance was better than those of CQCD-MPCC and IQCD-MPCC. However, under the same control method, the harmonic distortion in experimental results was more severe than it was in numerical results because there were nonlinear disturbances in the experimental results, such as dead time, a voltage drop in the switching tubes, and the parameters mismatch.

The harmonic analysis of the electromagnetic torque of PMSM under CQCD-MPCC, IQCD-MPCC, and LCDV-MPCC is shown in Figure 12. As shown in Figure 10, the 5th and 7th harmonics are the main harmonic components in the stator current, which generates the 6th harmonic component in i_q and T_e. The THD and 6th harmonic content of T_e under LCDV-MPCC were 3.25% and 1.22%, respectively. Both were less than those of CQCD-MPCC and IQCD-MPCC.

6.2. Experimental Results for the Dynamic-State

The dynamic performance of the proposed LCDV-MPCC is presented in this section. Figure 13 shows the torque, speed, and stator current waveforms when the speed reference was suddenly changed from 2000 r/min to 2500 r/min. Although a slight overshoot occurred in the dynamic process, the time that the speed tracked to the reference was less than 0.1 s.

Figure 14 shows experimental waveforms when the load torque decreased from 10 N·m to 8 N·m. The torque could smoothly transition with little fluctuation in the speed.

6.3. Execution Time of the Three MPCC Methods

Figure 15 shows the execution time for the three MPCC methods. The second vector of CQCD-MPCC was restricted to the zero vector. The number of calculations and comparisons used to select the first active vector was six, and the execution time was 15.7 µs. The second vector applied for CQCD-MPCC could be any vector. The number of calculations and comparisons used to select the second vector was three, so the execution time increased to 21.8 µs. However, for the proposed RCDC-MPCC, as the range of candidate vectors was reduced in the steady-state, there was a total of six times when the first and second optimal vectors were selected, which was equal to that of CQCD-MPCC, and the execution time was 15.3 µs.

6.4. Performance Comparisons between the Proposed Control and the Controls Mentioned in This Paper

Performance comparisons between CQCD-MPCC, IQCD-MPCC, and LCDV-MPCC are clearly shown in

Table 4. Performance comparisons between CQCD-MPCC, IQCD-MPCC, and LCDV-MPCC.

Comparative Items	CQCD-MPCC	IQCD-MPCC	LCDV-MPCC
Calculation number for the 1st vector	6	6	3
Calculation number for the 2nd vector	0	6	3
Calculation complexity	Low	High	Low
Steady-state performance	Low	Moderate	High
Dynamic performance	Moderate	Moderate	Moderate
Range of candidate vectors	Small	Wide	Wide

, according to the above simulation and experimental results.

7. Conclusions

This paper proposes a low-complexity double vector MPCC for a PMSM drive system fed by VSI. The corresponding principle of vector selection and the calculation method of the dwell time are also introduced in detail. A series of simulations and experiments were carried out to certify the feasibility and effectiveness of the proposed method. The conclusions are summarized as follows:

(1)

During steady-state operation, the number of candidate vectors is reduced from six to three, so the computational complexity can be reduced.

(2)

Compared to other double vectors MPCCs, the proposed method can expand the range of vector selection, and optimal vectors with the duty cycle are determined according to the deadbeat of q-axis current and minimization of the cost function so that a better steady-state performance can be obtained.

(3)

In the practical use of the PMSM drive system, parameter variations in inductance and resistance may occur, generating errors in vector selection and the duty cycle calculation. Therefore, parameter identification algorithms will be introduced in future research.