Single- and Two-Phase Open-Circuit Fault Tolerant Control for Dual Three-Phase PM Motor Without Phase Shifting

A fault tolerant control strategy for dual three-phase permanent magnet synchronous motor (DTP-PMSM) with 0° phase shifting between two windings is proposed in this paper. When a DTP-PMSM suffers from single- or two-phase open-circuit fault, to maintain the torque performance, current in faulty phase must be compensated by other healthy phases, which causes asymmetric self- and mutual inductances. Therefore, the 2nd harmonic component can be observed in the dq-axis currents and torque ripple also increases. To analyse this harmonic component in two types of faults, post-fault model of no-phase-shifting DTP-PMSM based on vector space decomposition (VSD) method is built in this paper. Then, proportion-integral-resonant (PIR) controller which can be used to suppress the specific periodic disturbance is applied to compensate the 2nd harmonic components. As the resonant controller is rarely dependent on motor parameters, phase and amplitude of current harmonic, the PIR controller can be applied in no-phase-shifting DTP-PMSM to suppress the harmonic caused by single- and two-phase open-circuit faults. The experimental results validate the effectiveness of the proposed fault tolerant control under single- and two-phase open-circuit faulty conditions.


I. INTRODUCTION
Multiphase machines are widely used in industrial applications such as aerospace and traffic because of excellent fault tolerant ability which ensures multiphase machines to work reliably. One of the most studied and applied multiphase machines is the dual three-phase permanent-magnet synchronous motor (DTP-PMSM). A DTP-PMSM contains two sets of armature windings with a specific angle of phase shifting. DTP-PMSMs with 30 • phase shifting between two sets of windings [1], [2] were widely researched. However, DTP-PMSMs with 0 • phase shifting [3]- [5] obtaining high winding factor and weak coupling between two set of windings [4], [5] were rarely studied in previous research.
Vector space decomposition (VSD) method decoupling torque-producing and flux-producing components achieves high dynamic torque performance, so it was widely used The associate editor coordinating the review of this manuscript and approving it for publication was Hao Luo .
for DTP-PMSMs with 30 • phase shifting [6]- [8]. However, it cannot be applied to DTP motors with 0 • phase shifting because fundamental and harmonic sub-planes transformed by VSD method are dependent in these motors. Another control method widely used for DTP-PMSMs is the two-individual current control method [9], [10], in which two sets of windings are controlled separately. This control method works properly under healthy condition, especially for DTP-PMSM with 0 • phase shifting, since the mutual inductances between two sets of windings are small. However, for multiphase motors, the possibility of failures including open-and short-circuit faults increases with more power devices [11]. When these faults occur, this method is only suitable for DTP-PMSMs with 30 • phase shifting [12], while it is unsuitable for DTP-PMSMs with 0 • phase shifting because two sets of windings are in phase. Therefore, the VSD method should be modified to be suited for DTP-PMSMs with 0 • phase shifting under both healthy and faulty conditions. VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ An additional leg was applied in [13] to maintain the torque performance under faulty condition, but the topological structure of the driver is complicated and neutral points of the motor must be led out. Therefore, another feasible method is connecting two neutral points [14], and then the current in faulty phase can be compensated by currents in another set of windings. In this way, no additional hardware is introduced in driver system. The latter method is exploited in this paper.
However, this method causes the unbalanced currents between two sets of windings, which leads to asymmetric self-and mutual inductances. The asymmetric inductances cause harmonic disturbance in the dq-axis currents. The harmonic components of current and torque in multiphase motors were analysed in [15]- [20]. Harmonic components in the dq-axis currents under open-circuit fault mainly contain the 2 nd harmonic caused by unbalance inductances [15], [16]. Other harmonics such as the 4 th harmonic [17] and the 6 th harmonic [19], [20] were analysed separately. However, the harmonics of DTP-PMSM with 0 • phase shifting under faulty condition are lack of researching in literature. To suppress current harmonics and torque ripple, the virtual synchronous reference frame method [16], the feedforward compensation method [18] or the genetic algorithm [21] were introduced in fault tolerant control method.
In multiphase motors, the single-and two-phase open-circuit fault tolerant control methods were widely researched [11]. These different types of open-circuit fault can occur on any phase. Therefore, the virtual synchronous reference frame method and the feedforward compensation method which are related to the phase and amplitude of current harmonics need be modified under different open-circuit faulty conditions. Meanwhile, the inaccurate motor inductance measurement can influence the performance of current harmonic suppression in the feedforward compensation method or the genetic algorithm. However, the proportionintegral-resonant (PIR) controller [22], [23] can suppress the periodic disturbance and obtains strong robustness which means that the performance of the PIR controller is barely affected by the phase and amplitude of the current harmonics. The PIR controller can be employed into current loop of vector control [24], [25] and suppress the current harmonic under different open-circuit faulty conditions. However, it lacks research in no-phase-shifting DTP-PMSM fault tolerant control. Therefore, in this paper, the PIR controller is applied in DTP-PMSM with 0 • phase shifting fault tolerant control to suppress the current harmonics and torque ripple caused by single-and two-phase open-circuit faults.
In this paper, single-and two-phase open-circuit fault tolerant control strategy for no-phase-shifting DTP-PMSM is proposed. Firstly, in Section II, a new decoupling method based on VSD method is established and then, pre-and postfault models are built. The 2 nd harmonic components are revealed in the post-fault model. Meanwhile, the currents in healthy phase which should be modified to maintain the total magnetic motive force (MMF) under faulty conditions are deduced in Section III. The modification of currents causes asymmetric inductance and leads to the 2 nd harmonic dq-axis currents which are suppressed by the PIR controller in Section IV. The comparison between the feedforward compensation method and the PIR controller is provided to show the effectiveness and advantage of the PIR controller. Finally, experimental results are provided to demonstrate that the proposed fault tolerant control is effective under single-and two-phase open-circuit faulty conditions.

II. MACHINE MATHEMATICAL MODEL A. MODEL UNDER ORIGINAL REFERENCE FRAME
The structure of windings and drive system topology of DTP-PMSM with 0 • phase shifting are shown in Fig. 1(b) and Fig. 2, respectively, where six phases are named as phase A, B, C, X, Y, Z . Phases A and X , phases B and Y , phases C and Z are correspondingly in phase, so the model in the original reference frame can be expressed as: where u s = [u A u B u C u X u Y u Z ] T represents the terminal voltage vector of six phases, i s and ψ s , similar to u s , represent the current and flux linkage vectors, R s , L s and ψ fm represent the resistance, inductance and PM flux linkage matrix of six phases, respectively, and [5]  where where L 1 represents the phase self-inductance, M 1 , M 2 represent the mutual inductances in one set of windings. Two sets of individual windings can be parallel connected considering that the phase shifting between two sets of windings is 0 • . In this way, the DTP-PMSM is equivalent to a single three-phase PMSM and the drive system topology is shown in Fig. 3. The three phases of equivalent PMSM are defined as phase A', B' and C'. The equivalent inductance matrix is L 0 . Flux linkage of the parallel windings can be calculated by the average of two individual windings, which can be written as: The inductance relationship between the inductance of the parallel windings and individual windings can be derived as: The equivalent PMSM will be further used to analyse the 2 nd harmonic in faulty model.

B. PROPOSED VECTOR DECOUPLING METHOD
The transformation matrix of the VSD method can be written as (7), shown at the bottom of the page, where ϕ represents the phase shifting between two windings.
The matrix transforms the six-phase components into the fundamental and harmonic components. However, ϕ is equal to zero in DTP-PMSMs with 0 • phase shifting and this transformation matrix is unsuitable since the vector α is dependent on the vector z 1 , which means that the fundamental and harmonic components are not decoupled.
Therefore, a modified vector decoupling method suitable for DTP-PMSMs with 0 • phase shifting is proposed. The fundamental components related to torque performance are deduced from VSD method. The rest components can be defined as unbalanced components which represent the unbalance between two sets of windings. The transformation matrix can be expressed as: The currents of the α-β sub-plane produce rotating MMF and participate in electromechanical energy conversion. The z 0 component is zero-order component. The unbalanced components z 1 represent the unbalance between phase A and X , which are in phase. The unbalanced components z 2 and z 3 can be defined in the same way. These components are used to solve current sharing problem, but they do not participate in energy conversion. A rotation transformation matrix is used to convert the α-β components to the synchronous reference frame d-q components whereas the unbalanced components remain unchanged. The transformation matrix can be written as: The fundamental and unbalanced components are decoupled under healthy condition. The 2 nd harmonic components do not exist in the dq-axis currents theoretically. The voltage and electromagnetic equations are expressed as: VOLUME 8, 2020 where I 2×2 represents the 2 nd order identity matrix.
where p is the number of pole pairs. As shown in (11), the components within unbalanced components are not decoupled, which means that one unbalanced component can affect others. However, they cannot affect the fundamental components, and hence the torque performance is immune to the unbalanced components.

C. MODEL UNDER FAULTY CONDITION
The fundamental components are coupled with unbalanced components when the DTP-PMSM suffers from open-circuit fault. The αβ-axis self-inductances under healthy condition are constant and equal due to the non-salient rotor structure, whereas they are asymmetric under open-circuit faulty condition. As a consequence, when they are transformed into the dq-axis self-inductances, the 2 nd harmonic components appear. By way of example, this paper assumes that phase A is open-circuit under single-phase faulty condition and phases A and B are open-circuit under two-phase faulty condition, while the conclusion is generally applicable.
The flux linkage of the equivalent single three-phase motor based on (5) can be written as: where ψ A represents the flux linkage of equivalent three phase motor, i A , i B , i C represent the three phase currents of equivalent motor, respectively, and i A = i A +i X , i B = i B +i Y , i C = i C + i Z . ψ B and ψ C can be deduced similarly.
When single-phase open-circuit fault occurs, i A = 0 and ψ A = ψ X . The flux linkage of the equivalent three phase PMSM can be expressed as: The inductances in each set of windings are asymmetric shown in (3). However, the inductances of equivalent PMSM are symmetric because the coupling between two sets of windings compensates the asymmetric inductances, and the symmetrical inductances in equivalent PMSM obtains only when currents in two sets of windings are symmetrical. Thus, the equivalent inductance can be affected by current distribution in two sets of windings. When current amplitudes of phase B, C, Y, Z are equal as these current amplitudes under healthy condition, the self-inductance of phases B' and C' and mutual inductance in equivalent motor are unchanged, while self-inductance of phase A' changes to L 1 which is the phase self-inductance of DTP-PMSM. The matrix of flux linkage can be expressed as: The single-phase open-circuit fault causes asymmetric self-inductance in equivalent motor shown in (17). The voltage equation of the dq-axis is expressed in (27), where L represents the α-axis self-inductance under healthy condition and equals to 2(L 0 − M 0 ) 3, L represents the asymmetric self-inductance and equals to 2(L 1 − L 0 ) 3, R s represents the stator resistance, ω, θ and fd represent the electric angular velocity, electric angle of the rotor position and the flux linkage of PM, respectively.
The equivalent PMSM flux linkage equation under twophase-open-circuit fault condition can be deduced as: The equivalent PMSM under two-phase open-circuit faulty condition shows the asymmetric self-and mutual inductances. The voltage equation can be expressed as (28), where L α , L β and M αβ represent the self-inductances of the α-axis, β-axis and the αβ-axis mutual inductance under faulty condition, respectively.
Post-fault models of DTP-PMSM have been built in this section. Both single-and two-phase open-circuit faults cause asymmetric inductances which can further bring the 2 nd harmonic components. Additionally, torque performance which is directly related to the q-axis current deteriorates because of the 2 nd harmonic disturbance.

III. CURRENTS UNDER FAULTY CONDITION
The rotation MMF due to the armature windings currents under faulty condition should maintain unchanged as healthy condition to preserve the torque performance, Therefore, the current of faulty phase should be compensated by other healthy phase currents.
The rotation MMF of stator currents under healthy condition can be expressed as:  (19) where ω represents the electrical angular velocity. N and I m represent the winding turns of each phase and the phase current amplitude, respectively. The rotation MMF under single-phase fault changes as: To make sure that the MMF does not change under faulty condition, the healthy five phase currents should be modified as: The relationship explains that the current in phase A is compensated by currents in phase X , since two phases are in phase. Hence, the current in phase X must double to maintain the MMF unchanged.
Similarly, stator currents under two-phase open-circuit fault should be, (24)-(28), as shown at the bottom of the page.

A. THE FEEDFORWARD COMPENSATION
The 2 nd harmonic component is the dominant harmonic component in both single-and two-phase open-circuit faults and it leads to periodic disturbance in the dq-axis currents. The feedforward compensation method was used to compensate the 2 nd harmonic component in [17], but it is only suited for five-phase PM motor. Therefore, the feedforward compensation method suitable for DTP-PMSM with 0 • phase shifting should be deduced. As shown in (27)  The 2 nd harmonic voltages are related to the motor speed ω and asymmetric inductances. Therefore, the accuracy of the feedforward compensation method is influenced by the measured speed and inductance. However, the measured i X = 2i m cos(ωt) (24) i Y = 2i m cos(ωt − 2π/3) (25) inductance is commonly inaccurate, which makes the 2 nd harmonic currents difficult to compensate completely. Furthermore, the phase of the 2 nd harmonic components changes with the different faulty phase, thus the controller requires detecting the faulty phase and then adjust the control algorithm, which increases the complexity of the harmonic suppression method.

B. THE 2 nd PIR CONTROLLER
Vector control in healthy condition can decouple the dq-axis flux linkage completely and voltage equation does not obtain the 2 nd harmonic component. Therefore, the PI controller can be applied under healthy condition to track the DC components in the dq-axis currents. However, when the dq-axis currents contain specific harmonic components, the PI controller cannot track on and compensate these components in steady state. A PIR controller is applied by connecting a resonant controller paralleled with a PI controller to suppress the 2 nd disturbance in the dq-axis currents, and the structure of PIR controller is shown in Fig. 5. The transfer function of the ideal resonant controller can be expressed as: where K r is the resonant gain and ω 0 is the electrical angular velocity of rotor. The ideal resonant controller is difficult to apply in practice due to its narrow bandwidth, which leads to poor frequency robustness [26]. Thus, in this paper, the quasi-resonant controller is introduced and the transfer function is: where the ω c is the cutoff frequency of the resonant controller. The bode plots of the ideal and quasi-resonant controllers are shown in Fig. 6, where the bandwidth of the quasiresonant controller is wider than the ideal resonant controller. The resonant controller can produce a gain which is high enough at the resonant frequency to accurately track the sinusoidal signal. Therefore, the PI controller acts as the main controller and can track the DC components in the dq-axis currents. The resonant controller can track and compensate the 2 nd harmonic disturbance in the dq-axis currents.
The PIR controller is of high robustness which means that the compensation accuracy is little affected by the change of motor parameters such as inductance. Moreover, different types of fault such as single-and two-phase open-circuit   faults change the phase and amplitude of the 2 nd harmonic components. The simulation result is shown in Fig. 7(a). However, the PIR controller with the same parameters can adapt to both single-and two-phase open-circuit faulty conditions which is shown in Fig. 7(b). Therefore, the PIR controller can cope with different types of fault effectively without modifying fault tolerant control algorithm. The motor control system schematic is shown in Fig. 9, where θ r and ω r represent the rotor electrical angle and electrical speed, respectively.

V. EXPERIMENTAL RESULT
In this paper, a 1.1 kW surface-mounted no-phase-shifting DTP-PMSM is used to verify the fault tolerant control strategy. Fig. 8 shows the prototype of the experimental system. It consists of the prototype machine, two threephase inverters, one dc power supply and the controller. IGBT modules FP75R12KT4 made by INFINEON are utilized as the power devices and its switching frequency is 10 kHz. The open-circuit fault is achieved by disconnecting the motor phase directly. In the experiment part of this paper, phase A is disconnected under single-phase faulty condition. Phases A and B are disconnected under two-phase faulty condition. The neutral points of two sets of windings are connected directly. The rotor position and speed are measured by Kubler A02H incremental encoder. The fault tolerant control strategy is conducted on dSPACE 1007 controller. The control cycle and sampling period of current, rotor position and speed are 100 µs. The reference of the speed is 80 rpm and the load torque is 50 Nm in the experiments. Design parameters of this motor are listed in TABLE 1.

A. HEALTHY CONDITION
In order to validate the proposed vector control method, the phase currents and the dq-axis currents are provided in Fig. 10. It can be observed in Fig. 10 that the currents of two windings are symmetrical and the amplitude of phase currents is about 1.1 A. Although the phase inductances are asymmetric in single windings, the equivalent three phase PMSM obtains symmetrical equivalent inductance under healthy condition and no obvious harmonic can be observed in the dq-axis currents.    dq-axis currents. The feedforward compensation and resonant controller are applied to suppress the 2 nd current harmonic which are shown in Figs. 12-13. It can be observed that the feedforward compensation can reduce the 2 nd harmonic in the dq-axis currents. However, it still exists and obvious due to the inaccurate inductance measurement. On the contrary, the resonant controller is much more effective even when the parameters are not accurate.     Fig. 15, it can be observed that two methods significantly suppress the 2 nd harmonic disturbances. However, the resonant controller is preferable for suppression of the 2 nd harmonic. The DC component is about 50.1 Nm, and the amplitude of the 2 nd harmonic torque is decreased from about 1.72 Nm to 0.43 Nm in the feedforward compensation and to 0.17 Nm in the resonant controller.   need modification. Fig. 17 shows the dq-axis currents applying feedforward compensation and the resonant controllers.
Figs. 18 and 19 compare the torque waveforms and spectra for three cases which are the same as Figs. 14-15. The resonant controller is again preferable for suppression of the 2 nd harmonic. The 2 nd harmonic is about 1.48 Nm under   two-phase fault, and it decreases to 0.57 Nm and to 0.14 Nm with applying the feedforward compensation and the resonant controller, respectively.
By comparing these two fault tolerant control methods, we can conclude that the 2 nd resonant controller has more advantages which can suppress the 2 nd harmonic disturbance better. Meanwhile, self-inductance is difficult to measure precisely and changes with temperature and current value considering core saturation in practice. Hence, the feedforward compensation related to inductance cannot always compensate the harmonic effectively. The experiments validate the robustness of the resonant controller which is less dependent on motor parameters. The 2 nd harmonic in the dq-axis currents and torque can be suppressed under single-and two-phase open-circuit faulty conditions with the proposed fault-tolerant control method.
The 2 nd harmonic components under single-and two-phase open-circuit faults are shown in Fig. 20. The phase and amplitude of the 2 nd harmonic components are different in Fig. 20(a). However, the PIR controller with the same parameters can adapt to both single-and two-phase opencircuit faulty conditions which is shown in Fig. 20(b). The q-axis current harmonics in both situations are suppressed effectively which validated the effectiveness of the PIR controller.

VI. CONCLUSION
The paper has proposed a fault tolerant control method for no-phase-shifting DTP-PMSM. The PIR controller is applied in fault tolerant control to suppress the 2 nd harmonic disturbance in the dq-axis currents caused by asymmetric inductance.