Control Strategy for a Five-Leg Inverter Supplying Dual Three-Phase PMSM

This study presents a novel control strategy for a five-leg inverter-fed dual three-phase permanent-magnet synchronous motor (DT-PMSM)system. Three different five-leg inverter topologies (operating modes) can be reconfigured by choosing a pair of phases for sharing the common leg. The five-leg inverter operation mode with 150° phase difference can achieve full torque output, but the utilization of DC bus voltage is low. The operation mode with 30° phase difference has higher DC bus voltage utilization but lower output torque. The modified double zero-sequence injection (DZI) PWM strategy based on carrier-based PWM (CBPWM) for five-leg inverter operating modes is proposed. This strategy can achieve the maximum DC bus voltage utilization. The five-leg inverter operating modes can be used as a fault-tolerant solution for the occurrence of a fault in one leg of the six-leg inverter. Compared with the commonly used open phases fault-tolerant control (FTC) strategies, this strategy has the advantages of without increasing the stator copper, reducing the torque ripples, and implementing easily. Experimental results verify the effectiveness and feasibility of the proposed strategy.


I. INTRODUCTION
Multiphase machines enjoy the advantages of low voltage high power, low torque pulsation and good fault tolerant ability [1]- [3]. They have good application prospects in high-power driving applications such as electric vehicles, marine electric propulsion, and wind power generation. The phase-shifted 30 • dual three-phase motor (the asymmetric six-phase motor) has a greater application advantage due to the cancellation of the sixth torque harmonic among the various types of multiphase machines [4], [5].
The high fault-tolerant capability is an important application feature of multiphase machines. Various open-circuit and short-circuit faults in the machine drive system can be converted into the open-phase fault through the fault isolation. Therefore, the research of fault-tolerant control strategies for multiphase machines are mainly focused on the open-phase fault [6]- [20]. When an open-phase fault occurs, The associate editor coordinating the review of this manuscript and approving it for publication was Alfeu J. Sguarezi Filho .
to suppress the torque ripple, the postfault phase current references need to be modified to maintain the same rotating magnetomotive force as in prefault situation [6]. Decreasing the stator winding losses and increasing the torque operation range are two significative optimization objectives of the current references, so the minimum loss (ML) strategy (different amplitudes of currents) and the maximum torque (MT) strategy (equal amplitudes of currents) are two main strategies for the postfault current-reference generation. The full-range minimum losses (FRML) strategy which minimizes the losses in the whole torque operation range is proposed in [11]- [13]. To realize the field-oriented control operation in the open-phase fault situation, the vector space decomposition (VSD) can be implemented by using the normal decoupling transformation matrices [14]- [17] or the reduced-order transformation matrices [18]- [20]. The inner current control loops can been realized using dual PI [15] and model predictive controllers [9].
In recent years, the five-leg inverter supplied dual-machine drives have been extensive researched for reducing switching devices [21]- [31]. It has been experimentally proved that the two three-phase machines can be controlled independently by the five-leg inverter. The five-leg inverter supplied dual three-phase machine system can be regarded as the specific case of the five-leg inverter supplied dual-machine drive system. Compared with the six-leg inverter, the lower DC bus voltage utilization and the higher common leg current amplitude are two main drawbacks of the five-leg converter. When the six-phase inverter has a faulty inverter-leg, the five-leg inverter operation can be used as an alternative fault-tolerant control solution of dual three-phase machine. In this paper, three different five-leg inverter topologies (operating modes) have been studied for the dual three-phase machine, and the performance indexes are compared with the six-leg inverter healthily operation and the open-phase fault-tolerant operation. The modified DZI PWM strategy for five-leg inverter operating modes is investigated.
The paper is organized as follows. Section II describes the model of DT-PMSM. Section III describes the topology of the five-leg inverter fed dual three-phase machine system. Section IV discusses the DC bus voltage utilization. Next, the modified DZI PWM strategy for the five-leg inverter is presented in Section VI. Experimental results are presented in Section VII. Finally, concluding remarks are summarized in Section VIII.

II. THE MODEL OF DT-PMSM
Six-leg inverter fed dual three-phase motor system is shown in Fig.1. Assume that the windings of the DT-PMSM are sinusoidal, the magnetic saturation effect of the core and the mutual leakage between the windings are ignored.
According to the VSD theory, all variables of the DT-PMSM can be mapped to the α-β, z 1 -z 2 , and o 1 -o 2 three mutually orthogonal subspaces. Among them, only the α-β subspace is related to electromechanical energy conversion, and the o 1 -o 2 subspace is a zero-sequence subspace. The static decoupling transformation matrix is 0 It is only necessary to transform α-β components into the general synchronous reference frame d-q components. The rotating transformation matrix is The voltage, flux, and torque equations are where L D , L Q are d-and q-axes inductances, L aal is stator selfleakage inductance. p n is the number of pole pairs. ψ f is the permanent magnet flux linkage.

III. THE TOPOLOGY OF THE FIVE-LEG INVERTER FED DUAL THREE-PHASE MOTOR SYSTEM
When one leg of the six-phase inverter fails, the faulty phase should share the common leg with others. Taking the F-phase as an example, the fault-tolerant topology of the fiveleg inverter is illustrated in Fig.2. There are three different operating modes: CF-sharing mode (30 • phase difference), BF-sharing mode (150 • phase difference), and AF-sharing mode (90 • phase difference). When the F-phase leg fails, the T4 is switched off for isolating. The switch T1 is switched on, then phase C and F will share the common leg C (CF-sharing mode). Similarly, the switch T2 or T3 is switched on, corresponding to the BF-sharing or AF-sharing mode, respectively. Since two phases as a pair to share the common leg, the common leg current increases, so the common leg current needs to be discussed. When the dual three-phase motor operates healthy, the phase currents are: where I m is the phase current amplitude, and θ i is the phase angle of the A phase current.
(1) CF-sharing mode The common leg current i COM is The amplitude of the common leg current is 1.932 times of the normal leg current, so the CF-sharing mode can provide 0.518 times of the rated torque of the six-leg inverter operation.
(2) BF-sharing mode The common leg current i COM is The common leg current amplitude is 0.517 times of the normal leg current, so the full rated torque can be provided.
(3) AF-sharing mode The common leg current i COM is The common leg current amplitude is 1.414 times of the normal leg current, so only 0.707 times of the rated torque can be provided.
The CF-sharing mode of five-leg inverter fed dual threephase machine system is shown in Fig. 3.

IV. DC BUS VOLTAGE UTILIZATION ANALYSIS
For a five-leg inverter fed dual three-phase machine system, since two phases as a pair to share the common leg, the control freedom is reduced, and the bus voltage utilization rate is also inevitably lowered, so the DC bus voltage utilization in different operating modes needs to be discussed.
In order to facilitate the analysis of the bus voltage utilization, the reference voltage of the z 1 -z 2 subspace is set to zero, i.e., the ideal sine reference phase voltage. The modulation The reference voltages of each phase of the dual three-phase machine can be expressed as: For the dual three-phase machine system with the isolated neutrals, the line voltages between two sets of three-phase windings have no practical significance. We only need to consider the internal line voltages of one three-phase winding. Consistent with the commonly used three-phase machine, taking the line voltage u AB as an example, u AB can be expressed as: According to the symmetry of the three-phase system, the remaining line voltage can also be expressed in the form of (12). The magnitude of the line voltage must be less than the DC bus voltage: where m ≤ 1/ √ 3, so the dual three-phase machine system fed by the six-leg inverter has the same linear modulation range 0.577U dc with the three-phase machine system.
(2) BF-sharing mode Using the same method as CF-sharing mode, the maximum amplitude of the line voltages is u AE 3 cos(π/12)mU dc sin(θ − π/4) It needs to meet where m ≤ 0.299, so the maximum linear modulation range in the BF-sharing mode is 0.299U dc , which is 51.8% of the six-leg inverter operation.
(3) AF-sharing mode Similarly, the maximum amplitude of the line voltage is u CD 3 cos(π/12)mU dc sin(θ + π/4) It needs to meet where m ≤ 0.299, so the maximum linear modulation range is 0.299U dc , which is 51.8% of the six-leg inverter operation.
The above analyses all assume that the reference voltage of the z 1 -z 2 subspace is zero, which is corresponding to the two-dimensional current control. However, in actual applications, the reference voltage of the z 1 -z 2 subspace is not zero due to the four-dimensional current control, so the the linear modulation range is slightly reduced.
From the above analyses, the performance indexes under the same load condition in different control methods are listed in Table 1. The minimum loss strategy with the isolated neutrals [15] is used for the open-phase fault-tolerant operation in Table 1.
Compared with the commonly used open phases fault-tolerant control strategies, the stator copper loss of the five-leg inverter operation is the same as the healthy six-leg inverter operation. Since two phases as a pair to share the common leg in the five-leg inverter operation, the maximum amplitude of leg currents increases, and the DC bus voltage utilization rate is also inevitably lowered. The higher the DC bus voltage utilization, the higher the motor speed can be achieved.
As can be seen from Table 1, in the CF-sharing mode, the motor can achieve the highest speed which is 0.707 times of the rated speed, but only 0.518 times of the rated torque can be provided. In the BF-sharing mode, the motor can provide full rated torque, but only 0.518 times of the rated speed can be achieved. There is no advantage to using the AF-sharing mode. Therefore, only CF-sharing or BF-sharing mode can be selected in the actual application. The following analysis does not consider the AF-sharing mode.

V. THE MODIFIED DZI PWM STRATEGY FOR THE FIVE-LEG INVERTER
For a five-leg inverter fed dual three-phase motor system, since two phases as a pair to share the common leg, the number of voltage vectors is reduced to 32. Taking the CF-sharing mode as an example, since the phase C and phase F share the common leg, the switching function S c = S F . The voltage vector diagrams of the α-β and z 1 -z 2 subspaces are shown in Fig.4.
It can be seen from Fig.4 that it is very difficult to analyze the PWM strategy based on SVPWM, so the DZI PWM strategy based on CBPWM is adopted in this paper. The four-dimensional current vector control is adopted to suppress the harmonic currents in this paper. The four-dimensional current vector control of the dual threephase motor system is shown in Fig.5.
The double zero-sequence injection PWM strategy is shown in Fig. 6. For the healthy six-leg inverter operation, the mean zero-sequence components injection is frequently  adopted. And the zero-sequence components can be written as: u o1 = −0.5 (u max 1 + u min 1 ) = 0.5u mid1 u o2 = −0.5 (u max 2 + u min 2 ) = 0.5u mid2 (23) where u max , u mid , and u min are maximum, middle, and minimum values of the three phase voltages, respectively [33]. The principles of the DZI PWM strategy used for the five-leg inverter operation and the healthy six-leg inverter operation are consistent, but the injected zero sequence voltages for two sets of three-phase system are no longer independent of each other. Taking the CF-sharing mode as an example, the modulation voltage (Inverter output terminal to DC bus midpoint) of the com leg must be equal, i.e., u * COM = u * C = u * F . So the zero sequence voltages can be defined as And the modulation voltages can be written as: The amplitudes of modulation voltages must be less than 0.5U dc , so the limited condition of the zero-sequence voltages in two three-phase systems can be expressed as The voltage is normalized by the DC bus voltage U dc . Substituting (24) into (26): The limited condition of the zero-sequence voltage u o is So the minimum and maximum values of u o can be written as The necessary and sufficient condition for the existence of u o is u o min ≤ u o max . consistent with the healthy six-leg inverter operation, frequently used mean continuous pulse width modulation (CPWM) is: For BF-sharing mode, it is only needed to change the u C in (24)-(29) to u B .
A five-leg voltage source inverter simulation model based on Matlab/Simulink is constructed to validate the DZI PWM strategy in this paper. The reference phase voltage frequency f is 10Hz. Fig.7 shows the voltage waveforms in the CF-sharing mode when the reference phase voltage amplitude is 0.408U dc . Fig.8 shows the voltage waveforms in the BF-sharing mode when the reference phase voltage amplitude is 0.299U dc .
According to Fig.7(a) and Fig.8(a), the maximum amplitudes of the modulation voltages exactly reach 0.5U dc , which indicates that DZI PWM strategy in this paper can achieve the maximum DC bus voltage utilization. Apart from the modified DZI PWM strategy, there is no difference in the control strategy of the five-leg inverter and the six-leg inverter.

VI. EXPERIMENTAL RESULTS
In order to verify the effectiveness of the five-leg control strategy in this paper, an experimental verification is carried VOLUME 8, 2020 out on a surface-modified DT-PMSM. The parameters of the DT-PMSM are listed as follows: P n = 3, R s = 1.4 , L D = 2.04mH, L Q = 2.04mH, and ψ fd = 0.28Wb. The prototype of the experimental system platform is shown in Fig. 9.
The CF-sharing mode is used in the experiment. The DC bus voltage is 200V. PWM switching frequency is 10kHz. The dead time is 1.5us, and the motor load is constantly at 7Nm. Phase currents i A , i B , i C , i D , and the phase current i C harmonic spectra under different control methods are shown in Figs. 10. The motor speed is 200r/min. Fig.10(a) is the current waveform under the twodimensional current control of the healthy six-leg inverter. The phase current only contain the 5th, 7th harmonics due to the effect of VSI dead-time and other nonlinear factors. Fig.10(b) is the current waveform under the two-dimensional current control of the five-leg inverter. In addition to containing 5th,7th harmonics, the phase currents also contain 3rd harmonic, which is due to the internal asymmetry of the two sets of three-phase system when phase C and F share a common leg. Fig.10(c) is the current waveform under the four-dimensional current control of the five-leg inverter. Phase currents contain almost no harmonic currents. The four-dimensional current control can effectively suppress harmonic currents due to the asymmetry of the five-leg inverter non-linearity [32]. Fig.11 shows the current waveforms of the α-β and z 1 -z 2 subspaces under different control methods. It can be seen that the difference among them is only the harmonic current of the z 1 -z 2 subspace. In addition to the 5th, 7th harmonics caused by the dead-time effect, the harmonic currents in the z 1 -z 2 subspaces are more serious under the two-dimensional current control of the five-leg inverter, because there is the  internal asymmetry of the two sets of three-phase system when phase C and F share a common leg. The harmonic currents are well suppressed through the currents closed loop control in the z 1 -z 2 subspace. It has no obvious difference in the α-β subspace, so the torque output effect has no obvious difference among three different control methods. Fig.12 shows modulation voltages and phase currents under different control methods. The motor speed is 300r/min. In the health six-leg inverter operation, modulation voltages of phase C and F are given. In the five-leg inverter operation, because modulation voltages of phase C and F are the same (phase COM), modulation voltages of the phase E and COM are given. The experimental results are consistent with the simulation analysis in the Fig.7(a).
In order to verify the dynamic performance of the five-leg inverter control in this paper, Fig.13 shows the motor starting waveforms of the healthy six-leg and five-leg operation under the same control parameters. The reference speed is 300r/min, It can be seen that there is no obvious difference in the speed response time between the five-leg inverter and the six-leg inverter operation, which indicates that the five-leg inverter control strategy in this paper has good dynamic response.

VII. CONCLUSION
In this paper, a novel control strategy for a five-leg inverter fed DT-PMSM system is investigated. Three different fiveleg inverter operating modes can be reconfigured by choosing a pair of phases for sharing the common leg. The BF-sharing mode can achieve full torque output, but the DC bus voltage utilization is reduced from 0.577U dc to 0.299U dc . The CF-sharing mode has higher DC bus voltage utilization 0.408U dc , but only 0.518 times of the rated torque can be provided. Apart from the modified DZI PWM strategy, there is no difference in the control strategy of the five-leg inverter and the six-leg inverter. The four-dimensional current vector control can effectively suppress harmonic currents due to the asymmetry of the five-leg inverter non-linearity.
The five-leg inverter operating modes can be used as a fault-tolerant solution for the occurrence of a fault in one leg of the six-leg inverter. Compared with the commonly used open phases fault-tolerant control strategies, this strategy has the advantages of without increasing the stator copper, reducing the torque ripples, and implementing easily.