Dual three-phase permanent magnet synchronous motors model predictive torque control based on zero common-mode voltage

This article proposes a multi-vector model predictive torque control strategy based on zero common-mode voltage to suppress common-mode voltage and harmonic currents in a dual three-phase permanent magnet synchronous motor. Firstly, 20 zero common-mode voltage vectors are selected from the traditional 64 basic voltage vectors. By synthesizing these 20 vectors into 6 virtual voltage vectors (VVV), control over the harmonic plane is achieved, reducing harmonic currents and eliminating common-mode voltage. Then, the control error of the flux is reduced by increasing the number of vectors to achieve the purpose of reducing the flux ripple. Finally, through duty cycle modulation, multi-vector model predictive torque control extends the modulation range to the entire sector plane. Simulation results confirm that this strategy can effectively suppress harmonic currents and common mode voltage.


Introduction
In recent years, multi-phase motors have received increasing attention [1] .Dual three-phase permanent magnet synchronous motors (DTP-PMSM) have lower phase currents, higher torque densities, more stable output torque, more control degrees of freedom, and more flexible and versatile control [2] , and thus are widely used in electric vehicles, ship power transport, aerospace, and aviation [3] .
DTP-PMSM are challenged by harmonic currents [4] and common mode voltages [5] in practical applications, which not only affect the efficiency and performance of the motors but may also adversely affect the stability and service life of the motors.Therefore, it is crucial to explore effective strategies for suppressing common mode voltage and harmonic current to improve the control performance of DTP-PMSM.
By synthesizing the VVV from the effective voltage vector, [6] proposes new MPC strategies for better performance of current and torque.
In [7], a simplified model predictive control (MPC) scheme is designed for DTP-PMSM, which restrains harmonic currents and reduces computation time.[8] focuses on improving the current and torque performance of PMSM.A three-vector drive scheme is presented.
Analyzing the voltage vectors, [9] selects the set of voltage with zero-common-mode characteristics as the model prediction vector set, and further simplifies the selected set of zero-common-mode voltage vectors.
Similarly, [10] constructs 12 VVV for harmonic suppression and further simplifies these 12 virtual vectors into 6 vectors for common mode voltage suppression.However, no further duty cycle calculation or multi-vector optimization is performed on the VVV.
In this paper, a multi-vector MPTC strategy based on zero-common-mode voltage (ZCMV) is proposed to achieve common-mode voltage (CMV) and harmonic current suppression of DTP-PMSM MPTC.Twenty ZCMVs are filtered in conventional 64 fundamental voltage.To control harmonic planes, the reduction of harmonic currents is achieved by synthesizing the 20 zero-common-mode voltage vectors into 6 VVV.Then, the control error of flux is reduced by increasing the number of vectors to achieve the purpose of reducing the flux pulsation.Finally, multi-vector MPTC is achieved by duty cycle modulation, which extends the modulation range to the whole sector plane.The effectiveness of the algorithm on common mode voltage suppression and harmonic current suppression is verified.

Model of DTP-PMSM
In the system of synchronous rotating coordinates, the voltage equation of DTP-PMSM is: The equation of flux is: The electromagnetic torque is: Tn ii ξξ < where ud, id, and Ld represent voltages, currents, and inductance of the d-axis; uq, iq, and Lq represent voltages, currents, and inductance of the q-axis; ωe represents rotor electrical angular velocity.

VVV Control Set
The conventional VVV control set consists of 12 VVV, each of which is synthesized from the outermost 12 effective voltage in the αβ plane and the second outermost 12 effective voltage vectors in the same direction, respectively, as shown in Figure 1.
where ψs(k+1) is the predicted value of stator flux at the moment k+1, and Te(k+1) is predicted value of torque at the moment k+1.

Value function
The conventional three-phase PMSM MPTC takes the tracking error between the predicted and reference values of the stator flux and torque as the control objective.Therefore, the value function is usually expressed as: where λ is the coefficient of the stator flux magnitude error term; Te* and Te(k+1) are the reference values and predicted values of the torque; ψs* and ψs(k+1) are the reference values and predicted values of the flux.

Common mode voltage
The CMV of DTP-PMSM is: From Equation (1), by ensuring that three of the six-phase voltages remain high and three remain low during each switching cycle, Ucmv can be made to remain 0. Based on such a principle, 20 ZCMV satisfying the requirement can be found in the αβ place and xy place, which are V64, V62, V61, V54, V52, V51, V46, V45, V43, V34, V32, V31, V26, V25, V23, V16, V15, V13, V07, and V70, where V07 and V70 are zero vectors.These 20 zero common-mode voltage vectors are marked in red on the αβ plane, as shown in Figure 2.

An improved set of VVV
From Figure 2, in the αβ plane, there are a total of 18 effective zero common-mode voltage vectors, of which every third one is in the same direction.
Under the premise of considering the utilization rate of bus voltage, 18 zero-common-mode voltages are synthesized into 6 VVV by using the opposite characteristics of the largest vector and the second largest vector in the same direction on the αβ plane, so that the vectors in opposite directions on the xy plane cancel each other.The distribution of the resultant VVV on the αβ plane is shown in Figure 3.

Multi-VVV modulation
On top of constructing the virtual vectors, this paper combines the zero vector within the same sector with two adjacent VVVs to form a brand new desired voltage vector.Through the synthesis of two voltage vectors, six ideal desired voltage vectors are finally obtained.The modulation range of the desired voltage vectors is extended from six discrete points to be able to cover any direction and amplitude.After evaluating the candidate vectors VVi and VVi+1 by minimizing the value function twice, it is necessary to calculate the adjacent VVV duty cycle.This modulation method is obtained through the principle of differential beat-free control of the stator torque and flux.The predicted values of torque and stator flux at the moment k+1 are: The action time of vectors VVi, VVi+1, and zero vector can be derived from Equation ( 7):

Simulation verification
Based on the introduction of the improved MPTC strategy in Section 3, simulations are performed in Matlab/Simulink.To better reflect the effect of the improved MPTC, the improved MPTC is compared with the traditional MPTC strategy.The value of weight factor λ in Equation ( 5) is taken as 80000.The DTP-PMSM parameters are shown in Table 1.
The simulation results of the two groups are respectively shown in Figure 4 and Figure 5 under the steady state condition with a rotational speed of 50 r/min and a torque of 150 N•m.In addition, Figures 5(c) and 5(d) show the current waveform and the harmonic spectrum analysis of the conventional MPTC, and the total harmonic distortion(THD) of the current is about 11.55%.Figures 6(c) and 6(d) show the harmonic spectrum analysis and current waveform of the improved MPTC, and the current THD is 6.16%.The current pulsation of the improved MPTC is reduced, the waveform is smoother, and the total harmonic distortion is reduced by 5.39%.(Figures 5(d In summary, the improved MPTC can significantly reduce the torque ripple and CMV compared with the conventional MPTC.Meanwhile, the method has a smaller THD value of A-phase current, a better control effect, and a more stable system.The reason is that the new synthesis vector generated by the zero common mode voltage principle may have a better control effect.In other words, using the proposed method eliminates the effects caused by CMV and controls the unreached region of existing methods.

Conclusions
A multi-vector MPTC strategy based on ZCMV is proposed for DTP-PMSM facing problems such as harmonic currents and common-mode voltages in traditional model predictive torque control.After simulation verification, the strategy can effectively suppress CMV and current harmonics and extend the modulation range of motor control.

Figure 1 .
Figure 1.Basic control set of conventional MPTC.Figure 2. Zero common-mode voltage in αβ plane.

Figure 2 .
Figure 1.Basic control set of conventional MPTC.Figure 2. Zero common-mode voltage in αβ plane.
In the above equation, Sψs_1 and Sψs_2 denote the change rate of stator Flux linkage magnitude after the vectors VVi and VVi+1 are applied to the inverter, respectively; STe_1 and STe_2 denote the rate of change of torque after the vectors VVi and VVi+1 are applied to the inverter, respectively.The change rate of flux linkage magnitude Sψs and the rate of change of torque magnitude STe are shown below: ) and 5(a), the standard deviation of the torque pulsation of the conventional MPTC is 1.75 when the torque is 150 N•m, while the standard deviation of the torque pulsation of the improved MPTC is 1.08, which reduces the magnitude of the torque pulsation of the improved MPTC by 38.3% compared to the conventional MPTC.As shown in Figures5(b) and 6(b), the CMV of the conventional MPTC is -41.7 V-41.7 V, while the CMV of the improved MPTC is 0, which can indicate that the improved MPTC eliminates the CMV.
According to the model of DTP-PMSM, the method of forward Euler discretization is used to obtain the discrete equation of the motor variable at the next time as follows:

Table 1 .
Detail parameters of the DTP-PMSM.