Model predictive control method with common-mode voltage suppression for dual three-phase permanent magnet synchronous motor

Abstract

Recently, the interest in model predictive control (MPC) and dual three-phase drives has been growing rapidly. Due to the high redundancy of voltage vector in the system composed of dual three-phase permanent magnetic synchronous motor (PMSM) and six-phase inverter, the computational complexity and current harmonics of MPC are high. In addition, the zero vector has been used by traditional MPC, which will cause higher common-mode voltage. In this paper, a novel MPC method with twice predictions and synthetic vectors is proposed which can not only suppress common-mode voltage, but also reduce computational complexity and current harmonics. The mathematical model of a dual three-phase PMSM are verified by the experimental results under the common-mode voltage suppression.

Keywords

Model predictive control dual three-phase PMSM common-mode voltage synthetic vectors

1. Introduction

The dual three-phase permanent magnet synchronous motor, a typical multiphase PMSM, has attracted a great attention which has low torque ripple, high power under low DC power and strong fault-tolerant capability [1–3]. According to inherit the well-known three-phase technology, the dual three-phase PMSM has advantages over five-phase PMSM and seven-phase PMSM. Traditional field oriented control (FOC) is widely applied, in which response of the controller is influenced by PI parameters. PI parameters cannot give consideration to the requirement of rapid response of the system and small steady-state error [4,5]. In addition, parameters of the motor and uncertainties in the design process can also affect the stability of the system [6].

While in recent times, the MPC has become the main competitor to FOC due to its inherent advantages such as intuitive implementation, fast dynamic response and easy inclusion of nonlinearities and constraints [7,8]. It has been well known that the traditional MPC suffers from the heavy computation burden, especially for multiphase machines where the voltage vectors increase exponentially with the number of phases. So, in [9], only the largest voltage vectors are applied in the control process to reduce computational complexity. However, the method abandons the advantage of abundant voltage vectors and the current harmonic is high. A restrained search predictive control method is presented in [10], which reduces the computing effort with restricted available voltage vectors. However, intensive calculation are involved in this method. In addition, high-frequency components and large amplitudes of the common-mode voltage (CMV) at the motor neutral point generate high-frequency common-mode current (CMC) to the ground path and induce voltage on the shaft, which eventually reduces the machine life [11]. There are few researches on common-mode voltage suppression for the MPC method of dual three-phase motors.

This paper proposes a novel MPC method with twice predictions and synthetic vectors, which is an improvement on the basis of traditional MPC. The application of the synthetic vectors can effectively reduce the redundancy of the voltage vector, so the purposes of reducing computational complexity and suppressing current harmonics can be achieved. Twice predictions can be applied to reduce the common-mode voltage, because the zero vector is not required in the control process.

2. Model of dual three-phase PMSM

2.1. Space voltage vectors

The dual three-phase PMSM is one of the multiphase PMSM which consists of two sets of three-phase windings. The two sets of three-phase windings which are called ABC windings and DEF windings are spatially shifted by 30 electrical degrees. The power is supplied from a six-phase voltage source inverter (VSI). According to the vector space decomposition (VSD) method [12], the machine model is decoupled into three orthogonal subplanes. Normally, the neutral points of the two stator windings of dual three-phase PMSM are isolated to cancel the zero sequence current. The z1–z2 subplane contains current harmonics that cause loss, so the current in z1–z2 subplane should be suppressed.

There are 64 switching states in the entire inverter. For the 𝛼–𝛽 and z1–z2 subplanes, there are four switching states corresponding to the zero voltage vector, i.e. V₀₀, V₀₇, V₇₀ and V₇₇. The remaining 60 active switching states are represented by 48 different voltage vectors in 𝛼–𝛽 and z1–z2 subplanes. Figure 1 shows the voltage space vectors in the 𝛼–𝛽 and z1–z2 subplanes for a six-phase VSI.

Fig. 1.

Voltage space vectors in the 𝛼–𝛽 and z1–z2 subplanes for a six-phase VSI.

2.2. Common-mode voltage of VSI

Common-mode voltage is generated by the pulse width modulation (PWM) of VSI. Common-mode voltage can be represented by the voltage between neutral points of the two stator windings and DC link neutral point which are given in Eq. (1): $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}\displaystyle u_{N_{g}}={\displaystyle \frac{u_{A_{g}}+u_{B_{g}}+u_{C_{g}}}{3}}\\[12.0pt] \displaystyle u_{N_{g}^{\prime }}={\displaystyle \frac{u_{D_{g}}+u_{E_{g}}+u_{F_{g}}}{3}}.\end{array}\right. & & \displaystyle\end{eqnarray}$ (1)

The common-mode voltage of each set winding has 4 states which are −U_dc∕2, −U_dc∕6, U_dc∕6 and U_dc∕2. It can be noted that common-mode voltage is high such as ±U_dc∕2 when voltage vector V₀₀, V₀₇, V₇₀ and V₇₇ are utilized.

3. Model predictive control of dual three-phase PMSM

3.1. Predictive model of dual three-phase PMSM

For small values of a control sampling time T_s, Euler’s approximation as Eq. (2) can be used for discretization. $\begin{eqnarray}\displaystyle {\displaystyle \frac{di}{dt}}\approx {\displaystyle \frac{i(k+1)-i(k)}{T_{s}}}. & & \displaystyle\end{eqnarray}$ (2)

The prediction values of stator currents at t (k +1) can be obtained by Euler’s approximation as follow [13]: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \left\{\begin{array}{@{}l@{}}\displaystyle \hat{i} _{d}(k+1)=i_{d}(k)+\frac{T_{s}}{L_{d}}[u_{d}(k)-R_{s}i_{d}(k)+E_{d}(k)]\\[12.0pt] \displaystyle \hat{i} _{q}(k+1)=i_{q}(k)+\frac{T_{s}}{L_{q}}[u_{q}(k)-R_{s}i_{q}(k)+E_{q}(k)]\end{array}\right.\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \left\{\begin{array}{@{}l@{}}\displaystyle \hat{i} _{z1}(k+1)=i_{z1}(k)+{\displaystyle \frac{T_{s}}{L_{ls}}}[u_{z1}(k)-R_{s}i_{z1}(k)]\\[12.0pt] \displaystyle \hat{i} _{z2}(k+1)=i_{z2}(k)+{\displaystyle \frac{T_{s}}{L_{ls}}}[u_{z2}(k)-R_{s}i_{z2}(k)]\end{array}\right.\end{eqnarray}$ (4) where $\hat{i} _{d}(k+1)$ , $\hat{i} _{q}(k+1)$ , $\hat{i} _{z1}(k+1)$ and $\hat{i} _{z2}(k+1)$ are prediction values of stator currents at next sampling time in 𝛼–𝛽 and z1–z2 subplanes respectively. i_d(k), i_q(k), i_z1(k) and i_z2(k) are feedback values of stator currents at current sample time in 𝛼–𝛽 and z1–z2 subplanes respectively. E_d(k), E_q(k) are d, q-axis back electromotive forces (BEMF) at current sample time.

3.2. Traditional MPC method of dual three-phase PMSM

The general structure of MPC method for dual three-phase PMSM drives maintains the outer speed loop with a PI controller and replaces the inner current loops using the predictive approach. From the point of view of the control process, in order to minimize the quality function g which is shown in Eq. (5): $\begin{eqnarray}\displaystyle g\text{} & =\text{} & \displaystyle K_{1}|\hat{i} _{q}(k+1)-i_{q}^{\ast }(k+1)|+K_{2}|\hat{i} _{d}(k+1)-i_{d}^{\ast }(k+1)|\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼K_{3}|\hat{i} _{z1}(k+1)-i_{z1}^{\ast }(k+1)|+K_{4}|\hat{i} _{z2}(k+1)-i_{z2}^{\ast }(k+1)|\end{eqnarray}$ (5) where K_i (i =1,2,3,4) are coefficients which are the weighting factors for each component. The value of these coefficients must be selected according to the control objective and drive features. Although different coefficients K_i can change the importance of the both subplanes, there are still three problems because it is impossible to satisfy the requirements of both subplanes simultaneously in a sampling period. First, there are 49 voltage vectors corresponding to the switching state, which will lead to high computational complexity. Secondly, current harmonics in the z1–z2 subplane can not be suppressed. Finally, the zero vector is repeatedly used, which will cause higher common-mode voltage.

3.3. MPC using synthetic voltage vectors

It can be found that voltage vectors can be classified in small, medium, medium-large and large voltage vectors in 𝛼–𝛽 and z1–z2 subplanes as shown in Fig. 1 When the synthetic vectors are selected, not only the vector sum of the z1–z2 subplane should be zero to reduce the current harmonics, but also the vector magnitude of the 𝛼–𝛽 subplane should be as large as possible to improve the DC bus voltage utilization. Therefore, the two vectors have opposite directions in the z1–z2 subplane, and have the same direction and large amplitude in the 𝛼–𝛽 subplane. As shown in Fig. 2, 13 synthetic vectors are finally created by one large and one medium-large voltage vector. Compared with the original 49 voltage vectors, the computational complexity can be effectively reduced.

The application time of the medium-large and large vector should be different. For example, synthetic vector V₁ is formed by the voltage vector V₄₄ and V₆₅. The application time of V₄₄ and V₆₅ is T₄₄ and T₆₅ respectively. Calculation equation of application time in the z1–z2 subplane based on “Volt-Second” relation is shown in Eq. ((6)): $\begin{eqnarray} \displaystyle V_{44(z1-z2)}\cdot T_{44}+V_{65(z1-z2)}\cdot T_{65}=0\cdot T_{m}. & & \displaystyle \end{eqnarray}$ (6) After calculation, the application time must be T₄₄ = 0.73 ⋅ Tm and T₆₅ = 0.27 ⋅ Tm. And the general form of the synthetic vector is the following: $\begin{eqnarray}\displaystyle V_{\text{syn}}=0.73\cdot V_{44}+0.27\cdot V_{65}=0.598U_{dc}. & & \displaystyle\end{eqnarray}$ (7) The application of these synthetic vectors in MPC strategy provides zero vector sum in z1–z2 subplane with no consideration of these components into quality function g. Hence, a new quality function g is shown in ((8)): $\begin{eqnarray}\displaystyle g=|\hat{i} _{q}(k+1)-i_{q}^{\ast }(k+1)|+|\hat{i} _{d}(k+1)-i_{d}^{\ast }(k+1)|. & & \displaystyle\end{eqnarray}$ (8)

Fig. 2.

Applied synthetic vectors in 𝛼–𝛽 subplane.

3.4. MPC using twice predictions and synthetic vectors

It can be seen (1) that the use of zero vectors causes higher common-mode voltage. Without the use of zero vectors, the number of candidate synthetic vectors is reduced to 12, which will increase the error between reference values and actual values of current. Under the premise of not using the zero vector, the application of the twice predictions and synthetic vectors can make up for the shortcomings caused by the reduction in the number of synthetic vectors, and can also reduce the common-mode voltage.

First, predict the future values of output current of 12 synthetic vectors and select the optimal synthetic vector, recorded as V_s1. Second, predict again based on the value of previous step. It is worth noting that the synthetic vector selected in the first prediction is not considered in the second prediction. The second prediction abandons the synthetic vector which application time is not in the range of 0 ∼ T_s. Hence, the combination of two same vectors has been considered. Finally, application time of two synthetic vectors is calculated. Define a time coefficient 𝛾(0 < 𝛾 <1) and application time of two synthetic vectors is 𝛾T_s and (1 −𝛾)T_s respectively. The final state of system is equivalent to the linear sum of the effects of two synthetic vectors affect independently.

The current vector in dq reference frame is shown in (9): $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}\overrightarrow{{\varepsilon}i_{s1}}=\left[\begin{array}{@{}c@{}}{\varepsilon}i_{ds1}\\ {\varepsilon}i_{qs1}\end{array}\right]=\left[\begin{array}{@{}c@{}}i_{ds1}^{\ast }(k+1)-\widehat{i}_{ds1}(k+1)\\ i_{qs1}^{\ast }(k+1)-\widehat{i}_{qs1}(k+1)\end{array}\right]\\[12.0pt] \overrightarrow{{\varepsilon}i_{s2}}=\left[\begin{array}{@{}c@{}}{\varepsilon}i_{ds2}\\ {\varepsilon}i_{qs2}\end{array}\right]=\left[\begin{array}{@{}c@{}}i_{ds2}^{\ast }(k+1)-\widehat{i}_{ds2}(k+1)\\ i_{qs2}^{\ast }(k+1)-\widehat{i}_{qs2}(k+1)\end{array}\right]\end{array}\right. & & \displaystyle\end{eqnarray}$ (9) where ϵ is the error representing the differences between reference values and predicted values of current. Hence 𝛾 can be calculated as shown in Eq. (10): $\begin{eqnarray}\displaystyle {\gamma}={\displaystyle \frac{{\varepsilon}i_{ds1}^{2}+{\varepsilon}i_{qs1}^{2}-{\varepsilon}i_{ds1}{\varepsilon}i_{ds2}-{\varepsilon}i_{qs1}{\varepsilon}i_{qs2}}{({\varepsilon}i_{ds1}^{2}-{\varepsilon}i_{qs1}^{2})^{2}+({\varepsilon}i_{ds2}^{2}-{\varepsilon}i_{qs2}^{2})^{2}}}. & & \displaystyle\end{eqnarray}$ (10)

4. Experimental results

Experimental platform of dual three-phase PMSM system is shown in Fig. 3. The power part of the six-phase inverter is refitted from the power part of the two three-phase inverters. The system DC voltage is 300 V, the sample time is 100 μs. The rotor speed reference is 1500 rpm, and the load torque is 80 N ⋅ m. The detailed dual three-phase PMSM motor parameters are listed in Table 1.

Table 1
Parameter of dual three-phase PMSM

Parameter Value Parameter Value

Rated power (kW) 25 Rated voltage (V) 220

Number of pole pair 4 Permanent magnet flux (Wb) 0.1213

d-axis inductance (mH) 0.445 q-axis inductance (mH) 1.003

Stator resistance (𝛺) 0.03 Inertia (kg ⋅ m²) 0.02

Parameter	Value	Parameter	Value
Rated power (kW)	25	Rated voltage (V)	220
Number of pole pair	4	Permanent magnet flux (Wb)	0.1213
d-axis inductance (mH)	0.445	q-axis inductance (mH)	1.003
Stator resistance (𝛺)	0.03	Inertia (kg ⋅ m²)	0.02

Fig. 3.

Experimental platform of dual three-phase PMSM system.

Fig. 4.

Phase A current waveform of two MPC methods.

Fig. 5.

FFT analysis results of two MPC methods.

Fig. 6.

Common-mode voltage waveform of two MPC methods.

Figure 4(a) is the phase A current waveform of traditional MPC method and Fig. 4(b) is the phase A current waveform of new MPC method. A comparison of the two figures shows that the current waveform of new MPC method is more sinusoidal. Figure 5(a) and (b) show the FFT analysis results of traditional MPC method and new MPC method respectively. It can be seen from these figures that new MPC method maintains the vector sum of the z1–z2 sub-plane to zero, so its third and fifth harmonics are significantly reduced compared to traditional MPC method. Although the degree of reduction of higher harmonics is not as obvious as that of lower harmonics, the content of higher harmonics is negligible compared with the content of lower harmonics. These experimental results can prove that the new MPC method can effectively suppress current harmonics by using synthetic vectors.

Figure 6(a) shows the common-mode voltage of traditional MPC method which the peak is above 150 V and below −150 V and appears in each cycle. As can be seen from the figure, since the traditional MPC method uses the zero vector multiple times, its common-mode voltage waveform has a large fluctuation range and many spikes, which will cause serious electromagnetic interference. Figure 6(b) shows the common-mode voltage of new MPC method which is generally between −50 V and +50 V and change times in each cycle is also reduced. The comparison of the two pictures shows that the new MPC method does not use the zero vector, so the amplitude of the common mode voltage is significantly reduced and the waveform is significantly improved. This experimental result can prove that the new MPC method can effectively reduce the common mode voltage by using two predictions.

5. Conclusion

This paper investigates the MPC method with twice predictions and synthetic vectors of dual three phase PMSM considering the common-mode voltage suppression. The mathematical model of 12 synthetic vector control system is established. The MPC method with twice predictions and synthetic vectors are analyzed, which is effectively reduce the complexity of mathematical models, reduce current harmonics, and suppress common-mode voltage. It shows that the common-mode voltage(−50 V∼50 V) and change rate is significantly reduced in experiment, verifying the validity of the proposed method.

In the future study, the response speed and accuracy of the new MPC method will be studied in depth when there is a disturbance in the control system.

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