Zero Common-Mode Voltage Model Predictive Torque Control Based on Virtual Voltage Vectors for the Dual Three-Phase PMSM Drive

Abstract

For the multiphase motor drive system, current harmonic components in stator windings and the common-mode voltage are the main factors affecting the control performance. In this paper, a novel model predictive torque control (MPTC) considering both harmonic and common-mode voltage suppression is proposed to improve the control performance. First, with the vector space decoupling (VSD) theory, 12 virtual voltage vectors are constructed based on the principle that the amplitudes of the vectors in the x-y harmonic subplace are zero to achieve the harmonic suppression. Then, these 12 virtual vectors are further simplified to six vectors to realize the common-mode voltage suppression, and they are taken as the candidate vectors to be rolling optimized to output the optimal voltage vector. This novel MPTC strategy can reduce the computational burden and avoid the weight factor design for the traditional multi-objective optimization. The effectiveness of this novel MPTC strategy was verified by simulations and experiments in comparison with normal MPTC methods, and it can save over 44% of the execution time of the traditional MPTC method and provide better suppression of the harmonic current components and the common-mode voltage.

1. Introduction

The multiphase permanent magnet synchronous motor (PMSM) has been widely studied for years due to its high fault tolerability, higher power, and smaller torque ripples, and it can better satisfy high-reliability applications, including aerospace, electric vehicles, ship propulsion, and so on [1,2,3,4]. Among these multiphase motors, the dual three-phase motor has been widely studied and used because of its close relationship with the traditional three-phase motor and its advantages as a multiphase motor [5,6]. The harmonic problem is one of the key problems for the multiphase motor control for the harmonic current components in stator windings; it causes additional losses, torque ripples, and vibrations, resulting in lower efficiency and worse performance [7,8]. The common-mode voltage (CMV) is also a pertinent issue for the motor drive, reducing both the system stability and the lifetime of the motor [9,10]. It is of great significance to study harmonic and common-mode voltage suppression strategies to improve the control performance of the multiphase motor drive.

For multiphase motor drives, there are two main control methods. One is the field-oriented control algorithm based on multiple d-q frames derived from the traditional three-phase motor control [11]. Another well-known control algorithm is the vector space decomposition (VSD) method, in which the fundamental, harmonic, and zero-sequence components are decomposed onto three orthogonal subplaces, and based on this, various field-oriented direct torque control (DTC) and modified control algorithms have been widely researched [12,13]. In these two kinds of methods, harmonic and common-mode voltage suppression are usually achieved by optimizing the candidate vectors according to the relationship between vectors mapping on different subplaces, or by introducing harmonic closed-loop control. For example, a novel space vector modulation (SVPWM) strategy was proposed in [13] to suppress the harmonic components and ensure the torque control. Additional harmonic current controllers were studied in [14,15] to suppress the low-order harmonics, and a feedforward compensation method was proposed in [16] to suppress the harmonic current and reduce the output torque fluctuation. In [17], a switching table-based DTC strategy with the consideration of harmonic components’ compensation was proposed for a dual three-phase motor drive. In the research of common-mode voltage suppression, some novel SVPWM technologies with a lower common-mode voltage for multiphase motor drives were proposed and studied in detail in [18,19].

In recent years, model predictive control (MPC) has been rapidly developed as an effective control strategy for various drive systems for its fast response and because it is suitable for multivariable and non-linear systems [20,21]. For example, in [8], a simplified MPC scheme was designed for a dual three-phase PMSM to reduce not only the harmonic currents, but also the computational burden. In [22], a multi-vector-based model predictive torque control (MPTC) was proposed to reduce both the current harmonics and the torque ripple, along with a constant switching frequency. In order to obtain better current and torque performance, a three-vector-based MPTC scheme was designed in [23] for PMSM drives. With the concept of virtual vectors, Refs. [24,25] proposed novel MPC strategies to obtain satisfactory performance of the current and torque ripple suppression. It is worth noting that although the MPC strategy is suitable for the multi-objective optimization system, the design of weight coefficients in multi-objective constraint optimization constitute a research challenge.

In view of the above, we propose a novel MPTC strategy for a dual three-phase PMSM drive. Firstly, with the VSD analysis, 12 virtual voltage vectors were constructed to achieve harmonic suppression; then, these 12 virtual vectors were simplified to six vectors to realize the common-mode voltage suppression. Lastly, these six virtual vectors are taken as the candidate vectors to be rolling optimized to output the optimal voltage vector. This novel MPTC strategy can reduce the computational burden and avoid the weight factor design of the traditional multi-objective optimization. The effectiveness of this novel MPTC strategy was verified by simulations and experiments in comparison with normal MPTC methods.

This paper is organized as follows. The system structure and mathematical models are presented in Section 2. The traditional model predictive torque control strategy is presented in Section 3. The proposed MPTC based on virtual voltage vectors and the simulation experimental results are presented in Section 4 and Section 5, respectively. Finally, conclusions are given in Section 6.

2. System Structure and Mathematical Models

2.1. Mathematical Models of the Dual Three-Phase PMSM

Figure 1 shows the distortion of stator windings of the studied dual three-phase PMSM, in which ABC is the first set of three-phase stator windings and UVW is another set of windings, and the neutral points of each are isolated. The electrical angle between the two sets of windings is 30°.

Without the consideration of zero-sequence components, the voltage, flux, and torque equations in fundamental d-q coordinates and harmonic x-y coordinates can be written as in Equations (1)–(4), based on the vector space decomposition (VSD) theory.

$$\left[\begin{array}{c}{u}_{\mathrm{d}}\\ u_{\mathrm{q}}\end{array}\right]=\left[\begin{array}{cc}{R}_{\mathrm{s}}& 0\\ 0& R_{\mathrm{s}}\end{array}\right]\cdot \left[\begin{array}{c}{i}_{\mathrm{d}}\\ i_{\mathrm{q}}\end{array}\right]+\left[\begin{array}{cc}{L}_{\mathrm{d}}& 0\\ 0& L_{\mathrm{q}}\end{array}\right]\cdot \frac{\mathrm{d}}{\mathrm{d}t}\left[\begin{array}{c}{i}_{\mathrm{d}}\\ i_{\mathrm{q}}\end{array}\right]+\left[\begin{array}{c}-{\omega }_{\mathrm{e}}{L}_{\mathrm{q}}{i}_{\mathrm{q}}\\ {\omega }_{\mathrm{e}}{L}_{\mathrm{d}}{i}_{\mathrm{d}}+{\omega }_{\mathrm{e}}{\psi }_{\mathrm{f}}\end{array}\right]$$

(1)

$$\left[\begin{array}{c}{u}_{\mathrm{x}}\\ u_{\mathrm{y}}\end{array}\right]=\left[\begin{array}{cc}{R}_s& 0\\ 0& R_s\end{array}\right]\cdot \left[\begin{array}{c}{i}_{\mathrm{x}}\\ i_{\mathrm{y}}\end{array}\right]+\left[\begin{array}{cc}{L}_{\mathrm{z}}& 0\\ 0& L_{\mathrm{z}}\end{array}\right]\cdot \frac{\mathrm{d}}{\mathrm{d}t}\left[\begin{array}{c}{i}_{\mathrm{x}}\\ i_{\mathrm{y}}\end{array}\right]$$

(2)

$$\left[\begin{array}{l}{\psi }_{\mathrm{d}}\\ {\psi }_{\mathrm{q}}\\ {\psi }_{\mathrm{x}}\\ {\psi }_{\mathrm{y}}\end{array}\right]=\left[\begin{array}{llll}{L}_{\mathrm{d}}& 0& 0& 0\\ 0& L_{\mathrm{q}}& 0& 0\\ 0& 0& L_{\mathrm{z}}& 0\\ 0& 0& 0& L_{\mathrm{z}}\end{array}\right]\left[\begin{array}{l}{i}_{\mathrm{d}}\\ i_{\mathrm{q}}\\ i_{\mathrm{x}}\\ i_{\mathrm{y}}\end{array}\right]+\left[\begin{array}{l}1\\ 0\\ 0\\ 0\end{array}\right]{\psi }_{\mathrm{f}}$$

(3)

$$T_{\mathrm{e}}=3p_{\mathrm{n}}{i}_{\mathrm{q}}\left[i_{\mathrm{d}}(L_{\mathrm{d}}-L_{\mathrm{q}})+{\psi }_{\mathrm{f}}\right]$$

(4)

where u_d, u_q, u_x, and u_y are the voltage components in d-q and x-y coordinates, respectively, and i_d, i_q, i_x, and i_y are the corresponding current components. L_d and L_q are the inductances in d-q coordinates (here, L_d = L_q = L for the surface-mounted structure) and L_z is the leakage inductance. R_s is stator resistance, ω_e is rotor electric angular velocity, ψ_f is permanent magnet flux, T_e is output electromagnetic torque, and p_n is the number of poles.

2.2. Dual Three-Phase Two-Level Inverter

A dual three-phase two-level inverter (Figure 2) is taken as the drive inverter, and there are 2⁶ = 64 output voltage vectors, whose distributions on fundamental αβ and harmonic x-y subplaces are shown in Figure 3 (in which the state of each phase is represented with an octal number), and the corresponding synthesis vector can be expressed as

$${\mathit{v}}_{\mathsf{\alpha }\mathsf{\beta }}=\frac{1}{3}{V}_{\mathrm{dc}}(S_{\mathrm{A}}+S_{\mathrm{B}}{e}^{\mathrm{j}\frac{2\pi }{3}}+S_{\mathrm{C}}{e}^{\mathrm{j}\frac{4\pi }{3}}+S_{\mathrm{U}}{e}^{\mathrm{j}\frac{\pi }{6}}+S_{\mathrm{V}}{e}^{\mathrm{j}\frac{5\pi }{6}}+S_{\mathrm{W}}{e}^{\mathrm{j}\frac{3\pi }{2}})$$

(5)

$${\mathit{v}}_{\mathrm{xy}}=\frac{1}{3}{V}_{\mathrm{dc}}(S_{\mathrm{A}}+S_{\mathrm{B}}{e}^{\mathrm{j}\frac{4\pi }{3}}+S_{\mathrm{C}}{e}^{\mathrm{j}\frac{2\pi }{3}}+S_{\mathrm{U}}{e}^{\mathrm{j}\frac{5\pi }{6}}+S_{\mathrm{V}}{e}^{\mathrm{j}\frac{\pi }{6}}+S_{\mathrm{W}}{e}^{\mathrm{j}\frac{3\pi }{2}})$$

(6)

It can be seen from Figure 3 that there are 60 non-zero and 4 zero voltage vectors (V₀₀, V₀₇, V₇₀ and V₇₇) in each subplace. The non-zero voltage vectors can be divided into four categories: 12 large vectors (outermost layer), 12 middle vectors (second outer layer), 12 small vectors (innermost layer), and 24 vectors between the second layer and the innermost layer. The specific amplitudes of these four types of vectors are listed in

Table 1. Four types of voltage vectors.

Vector Types	Amplitudes
Large vectors Vmax	$\left|V_{\max }\right|=\frac{\sqrt{2}(\sqrt{3}+1)}{6}{V}_{\mathrm{dc}}\approx 0.644V_{\mathrm{dc}}$
Middle vectors Vmidl	$\left|V_{\mathrm{midl}}\right|=\frac{\sqrt{2}}{3}{V}_{\mathrm{dc}}\approx 0.471V_{\mathrm{dc}}$
Sub-small vectors Vmids	$\left|V_{\mathrm{mids}}\right|=\frac{1}{3}{V}_{\mathrm{dc}}\approx 0.333V_{\mathrm{dc}}$
Small vectors Vmin	$\left|V_{\min }\right|=\frac{\sqrt{2}(\sqrt{3}-1)}{6}{V}_{\mathrm{dc}}\approx 0.173V_{\mathrm{dc}}$

.

In addition, the same voltage has a contrary mapping location on different subplaces.

3. Traditional Model Predictive Torque Control Strategy

Considering that only the fundamental components are involved in the electromechanical energy conversion, the forward Euler formula is used to discretize the motor models in the fundamental subplace, and Equation (1) can be discretized as

$$\{\begin{array}{l}{i}_{\mathrm{d}}^{k+1}=i_{\mathrm{d}}^k+\frac{T_{\mathrm{s}}}{L}\left[u_{\mathrm{d}}^k-R_{\mathrm{s}}{i}_{\mathrm{d}}^k+{\omega }_{\mathrm{e}}L{i}_{\mathrm{q}}^k\right]\\ i_{\mathrm{q}}^{k+1}=i_{\mathrm{q}}^k+\frac{T_{\mathrm{s}}}{L}\left[u_{\mathrm{q}}^k-R_{\mathrm{s}}{i}_{\mathrm{q}}^k-{\omega }_{\mathrm{e}}L{i}_{\mathrm{d}}^k-{\omega }_{\mathrm{e}}{\psi }_{\mathrm{f}}\right]\end{array}$$

(7)

where T_s is the sampling period, k is the kth sampling time, and ω_e can be considered to be constant when the T_s is small.

Taking into account the digital delay in the model predictive control, a forward beat delay compensation for Equation (7) is performed and then converted into the state space equation, as

$${\mathit{i}}^{k+2}=A{\mathit{i}}^{k+1}+B{\mathit{u}}^{k+1}+H$$

(8)

where

$$\begin{gathered} A=\left[\begin{array}{cc}1-\frac{R_{\mathrm{s}}{T}_{\mathrm{s}}}{L}& {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\\ -{\omega }_{\mathrm{e}}{T}_{\mathrm{s}}& 1-\frac{R_{\mathrm{s}}{T}_{\mathrm{s}}}{L}\end{array}\right],B=\frac{T_{\mathrm{s}}}{L}\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],C=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right], \\ H=\left[0 \frac{-{\omega }_{\mathrm{e}}{\psi }_{\mathrm{f}}{T}_{\mathrm{s}}}{L}\right], \mathit{u}={[u_{\mathrm{d}}^{} u_{\mathrm{q}}^{}]}^T,\mathit{i}={[i_{\mathrm{d}}^{} i_{\mathrm{q}}^{}]}^T. \end{gathered}$$

Then, the stator flux ψ_s and the output torque T_e at (k + 2)th can be presented as

$$\{\begin{array}{l}{T}_{\mathrm{e}}^{k+2}=3p_{\mathrm{n}}{i}_{\mathrm{q}}^{k+2}{\psi }_{\mathrm{f}}\\ {\psi }_{\mathrm{s}}^{k+2}=\sqrt{{(L{i}_{\mathrm{d}}^{k+2}+{\psi }_{\mathrm{f}})}^2+{(L{i}_{\mathrm{q}}^{k+2})}^2}\end{array}$$

(9)

The cost function is designed as

$$J_k={(T_{\mathrm{e}}^{*}-T_{\mathrm{e}}^{k+2})}^2+{\lambda }_{\mathrm{F}}{({\psi }_{\mathrm{s}}^{*}-{\psi }_{\mathrm{s}}^{k+2})}^2$$

(10)

where λ_F is the weight factor, which can be expressed as λ_F = T_eN/|ψ_sN| according to the electromagnetic relationship, and T_eN and ψ_sN represent the rated torque and stator flux.

Since there are four categories and a total of 64 switching states for the six-phase two-level inverter, the selection of the candidate voltage vectors can be executed according to different control objectives.

For example, to improve the utilization of the DC-link voltage, the 12 large vectors (V_max in Table 1) in the outermost layer can be selected as the candidate vectors for rolling optimization. To suppress the harmonic current components, these vectors with small amplitudes in the harmonic x-y subplace can be selected as the candidate vectors, but this may contradict the fundamental current control.

4. Proposed Model Predictive Torque Control Based on Virtual Voltage Vectors

Assuming that the harmonic suppression constraint term was added to Equation (11), the suppression performance can be improved to a certain extent, but it would bring difficulties in the design of weighting factors. In this paper, virtual voltage vectors are constructed to consider both the fundamental components’ control and the harmonic components’ suppression, based on which the MPTC strategy is derived to ensure control effect and to avoid the weight factor design problem of the multi-objective constraint control.

4.1. Construction of the Virtual Voltage Vectors

It can be seen from Figure 3 that these 12 large vectors V_max in the αβ subplace are mapped to the 12 small vectors V_min in the x-y subplace; contrarily, these 12 small vectors V_min in the αβ subplace are mapped to the 12 large vectors V_max in the x-y subplace. The mapping relationship of the same vectors in different subplaces is shown in

Table 2. Mapping relationship of the same vectors in different subplaces.

Vector Types in αβ Subplace	Vector Types in x-y Subplace
Large vectors Vmax	Small vectors Vmin
Middle vectors Vmidl	Middle vectors Vmidl
Sub-small vectors Vmids	Sub-small vectors Vmids
Small vectors Vmin	Large vectors Vmax

.

To improve the utilization of DC-link voltage, reduce the harmonic interference, and consider the role of redundant vectors, these 12 large and 12 middle vectors in the αβ subplace were selected to construct the virtual voltage vectors.

Taking as examples the large vector V₄₄ and middle vector V₆₅, which can be constructed as the virtual vector V_m1-αβ, as shown in Figure 4a, this virtual vector should have an amplitude of zero in the x-y subplace to suppress the harmonic components, i.e., |V_m1-x-y| = 0, as shown in Figure 4b.

It is assumed that in one cycle T_s, the action time of V₆₄ is λT_s and the action time of V₄₆ is (1 − λ)T_s; the synthesis formulas in different subplaces can be obtained according to the principle of volt-second theory.

$$\{\begin{array}{l}\left|V_{\mathrm{m}1-\mathsf{\alpha }\mathsf{\beta }}\right|=\lambda \left|V_{64-\mathsf{\alpha }\mathsf{\beta }}\right|+(1-\lambda )\left|V_{46-\mathsf{\alpha }\mathsf{\beta }}\right|=\frac{2}{3}{V}_{\mathrm{dc}}\left[\lambda \cos \frac{\pi }{12}+(1-\lambda )\cos \frac{\pi }{4}\right]\\ \left|V_{\mathrm{m}1-\mathrm{xy}}\right|=\lambda \left|V_{64-\mathrm{xy}}\right|+(1-\lambda )\left|V_{46-\mathrm{xy}}\right|=\frac{2}{3}{V}_{\mathrm{dc}}\left[\lambda \cos \frac{5\pi }{12}+(1-\lambda )\cos \frac{\pi }{4}\right]\end{array}$$

(11)

To satisfy |V_m1-x-y| = 0, the solution results of Equation (11) can be obtained as

$$\lambda =\sqrt{3}-1$$

(12)

$$\left|V_{\mathrm{m}1-\mathsf{\alpha }\mathsf{\beta }}\right|=\left(\sqrt{2}-\sqrt{6}/3\right)V_{\mathrm{dc}}$$

(13)

According to the above method, the 12 constructed virtual voltage vectors composed of 12 large vectors and 12 middle vectors can be derived, as shown in Figure 5.

4.2. Common-Mode Voltage Suppression

For the motor control, the common-mode voltage should also be restricted to reduce shaft voltage and leakage current components, which would result in electromagnetic interference and affect the normal operation.

The common-mode voltages of this dual three-phase PMSM can be expressed as

$$\begin{array}{ll}{u}_{\mathrm{CMV}}& =\frac{V_{\mathrm{A}}+V_{\mathrm{B}}+V_{\mathrm{C}}+V_{\mathrm{U}}+V_{\mathrm{V}}+V_{\mathrm{W}}}{6}\\ & =\frac{1}{6}{V}_{\mathrm{dc}}(S_{\mathrm{A}}+S_{\mathrm{B}}+S_{\mathrm{C}}+S_{\mathrm{U}}+S_{\mathrm{V}}+S_{\mathrm{W}})\end{array}$$

(14)

Taking 64 switch states as the input, the common-mode voltages with different voltage vectors are summarized in

Table 3. Common-mode voltages with different voltage vectors.

UCMV	Corresponding Voltage Vectors	Number
+Vdc	V77	1
+2Vdc/3	V37, V57, V67, V73, V75, V76	6
+Vdc/3	V17, V27, V33, V35, V36, V47, V53, V66, V55, V56, V63, V65, V71, V72, V74	15
0	V07, V13, V15, V16, V23, V25, V26, V31, V32, V34, V43, V45, V46, V51, V52, V54, V61, V62, V64, V70	20
−Vdc/3	V03, V05, V06, V11, V12, V14, V21, V22, V24, V30, V41, V42, V44, V50, V60	15
−2Vdc/3	V02, V04, V10, V20, V40, V01	6
−Vdc	V00	1

.

It can be seen from Table 3 that to suppress the common-mode voltage as much as possible, these 20 basic voltage vectors shown in Table 3 with zero common-mode voltage should be selected as the candidate vectors, and these 20 basic vectors include two zero vectors, six small vectors, six middle vectors, and six large vectors. Therefore, the six virtual voltage vectors composed of six large vectors and six middle vectors are lastly constructed to be the candidate vectors for the rolling optimization of the MPTC strategy, as shown in Figure 6.

4.3. Overall Control

According to the previous design process, the proposed MPTC strategy considering the harmonic components and common-mode voltage suppression for the dual three-phase PMSM in this paper is shown in Figure 7, and the implementation flowchart is shown in Figure 8.

5. Simulation and Experimental Results

5.1. Simulation Results

Simulations were carried out to verify the effectiveness of the proposed MPTC strategy in the environment of MATLAB/Simulink, and the comparisons were conducted with traditional MPTC, virtual vector-based MPTC without the suppression of common-mode voltage (in term of MPTC1), and the proposed MPTC method with the suppression of common-mode voltage (termed MPTC2). The detail parameters of the dual three-phase PMSM (used for experiments) are listed in

Table 4. Detail parameters of the dual three-phase PMSM.

Vector Types in α-β Subplace	Vector Types in x-y Subplace
Pole pairs np	5
Stator resistance Rs	0.08 Ω
d-axis inductance Ld	0.33 mH
q-axis inductance Lq	0.33 mH
PMSM magnetic flux ψf	0.01215 Wb
DC-link voltage	270 V
Rotary inertia Jm	72.96 × 10−6 kg·m2
Rated speed n	11,000 r/min
Rated stator current IN	12A
Rated torque TN	2.2 N∙m

.

First, the motor starts up to 5000 r/min without load; then, at t = 0.05 s, a 2.2 N∙m load is suddenly applied. The corresponding output torque T_e, the six-phase stator current i_a~i_w, the harmonic current components i_x and i_y, and the common-mode voltage u_CMV with different control methods are shown in Figure 9, Figure 10 and Figure 11.

It can be seen from the comparisons shown Figure 9 to Figure 11 that this proposed MPTC method (termed MPTC2) not only has a satisfactory harmonic output performance, but it also has zero common-mode voltage output. To verify the effectiveness of the proposed MPTC method, more simulations were carried out under different loads (rated load and half rated load) and speeds (3000 r/min and 7000 r/min). The hormonic current components in the x-y subplace and the common-mode voltages under different situations are similar to Figure 11c,d, and the output torque responses are shown in Figure 12 and Figure 13.

It can be seen from Figure 12 and Figure 13 that the proposed MPTC method studied in this paper has effective adaptability under different working conditions.

5.2. Experimental Results

Experiments were carried out on a five-pole dual three-phase PMSM. The experimental platform is shown in Figure 14, and the whole control algorithm was performed with a Xilinx Kintex-7-FPGA board. A magnetic particle brake was taken as the load.

The steady a-phase current i_a, harmonic current i_x, and the common-mode voltage u_CMV with different control strategies are shown in Figure 15, Figure 16 and Figure 17.

For the results of the traditional MPTC shown in Figure 15, the maximum amplitude of the harmonic current i_x can reach about ±10 A, and there is a severe distortion in the phase stator current. Moreover, the peak value of common-mode voltage can reach ±135 V, indicating that this traditional MPTC method cannot ensure the normal operation of the drive system.

With the virtual vector-based MPTC strategies, both the harmonic suppression and stator current control performance have been significantly improved, as shown in Figure 16 and Figure 17. However, there is still a large common-mode voltage (about ±45 V, as shown in Figure 16c), as the common mode voltage suppression is not considered in MPTC1. On the other hand, with the consideration of the common-mode voltage suppression in MPTC2, the u_CMV is almost zero, as shown in Figure 17c.

The performance comparisons of different algorithms are summarized in

Table 5. Comparisons of different algorithms.

Methods	THD of ia	Maximum uCMV (V)	Execution Time (μs)
Traditional MPTC	11.27%	±135	49.20
MPTC1	4.76%	±45	32.70
MPTC2	2.87%	0	27.50

.

Next, the output torque responses of different algorithms during dynamic loading are compared, as shown in Figure 18, where at t = 0 s, the PMSM starts up without load, and at t = 5 s, the rated load is added.

It can be seen from the dynamic output torque response in Figure 18 that the proposed MPTC strategy studied in this paper has better performance and less torque ripple than the other methods.

To further verify the superiority of this MPTC method, experiments in the full speed range were carried out at 1000 r/min, 3000 r/min, 5000 r/min, 7000 r/min, 9000 r/min, and 11,000 r/min under the loads of 1.1 N∙m and 2.2 N∙m. The quantitative comparisons of torque ripple and the THD comparisons are shown in Figure 19 and Figure 20.

The torque ripple and the THD performance of the proposed MPTC method are superior to other strategies in the full speed range and under different loads, confirming its feasibility and effectiveness.

6. Conclusions

We studied in detail a novel model predictive torque control strategy considering both harmonic and common-mode voltage suppression. Firstly, 12 virtual voltage vectors were constructed based on the VSD theory to satisfy the principle that the amplitudes of the vectors in the x-y harmonic subplace are zero to guarantee the harmonic suppression; then, six virtual vectors were selected as the candidates to ensure the common-mode voltage suppression. Compared with the traditional MPTC method, this novel MPTC can not only realize the harmonic and common-mode voltage suppression, but also avoid the weight factor design of the multi-objective optimization, with less computational burden. Lastly, the effectiveness and feasibility of the control method studied in this paper were verified by numerous comparative experiments with the traditional MPTC method.