Switching Sequence Model Predictive Direct Torque Control of IPMSMs for EVs in Switch Open-Circuit Fault-Tolerant Mode

: A switching sequence model predictive direct torque control (MPDTC) of IPMSMs for EVs in switch open-circuit fault-tolerant mode is studied in this paper. Instead of selecting one space vector from the possible four space vectors, the proposed MPDTC method selects an optimized switching sequence from two well-designed switching sequences, including three space vectors, according to a new designed cost function of which the control objectives have been transferred to the dq -axes components of the stator ﬂux-linkage under the maximum-torque-per-ampere control. The calculation method of the durations of the adopted space vectors in the optimized switching sequence is studied to realize the stator ﬂux-linkage reference tracking. In addition, the capacitor voltage balance method, by injecting a dc o ﬀ set to the current of fault phase, is given. Compared with the conventional MPDTC method, the complicated weighting factors designing process is avoided and the electromagnetic torque ripples can be greatly suppressed. The experimental results prove the e ﬀ ectiveness and advantages of the proposed scheme.


Introduction
The drive system of interior permanent magnet synchronous motors (IPMSMs) [1,2] based on two-level voltage source inverters (2L-VSIs) [3,4] has become one of the mainstream speed control schemes for electric vehicles (EVs) due to advantages, such as high efficiency, excellent speed regulation performance, and high power density. In order to improve the security of EVs, the functional safety of the drive system has gradually become one of the research hotspots. According to the failure mode analysis of the drive system [5], it is noted that the switch open-circuit faults of the 2L-VSI can have catastrophic consequences. In some hazardous environments, such as on highway or crowded roads, it is desirable that the 2L-VSI can continuously operate in the case of switch open-circuit failures. Thus, the fault-tolerant control of the 2L-VSI, which allows the EVs to work in the limp-home mode [6,7], is one of the key issues to ensure functional safety.
In the switch open-circuit fault-tolerant mode, the remedial methods consist of hardware and software reconfigurations. For the hardware reconfigurations, an auxiliary fourth leg is added to the 2L-VSI topology [8]. However, the cost, volume, and weight are increased, which are limited in the EV applications. Three-phase and four-switch inverters (3P4SIs) [9] can realize the switch open-circuit fault-tolerant control without increasing the cost, volume, and weight, and thus, it has

Conventional MPDTC of IPMSM for EV in Switch Open-Circuit Fault-Tolerant Mode
The speed regulation system of EVs on the basis of IPMSMs is shown in Figure 1a, where Vdc is the dc-link voltage; C1 and C2 are the upper and lower capacitors; and Sj and Sj1 (j = a, b, c) are the up and low switches, which are composed of insulated gate bipolar transistors (IGBTs) and antiparallel diodes. The energy of the lithium battery pack is transmitted to the IPMSM through the 2L-VSI. To realize the fault-tolerant control of switch-open-circuit fault, three bi-directional thyristors Ka, Kb, and Kc are added between the middle point of the dc-link capacitors ('o' as defined in Figure 1a) and the output terminal of the VSI. In normal operation mode, Ka, Kb, and Kc are all in off state, and when one of the phases has an open-circuit fault, the bidirectional thyristor of the corresponding phase is closed, and the output terminal of the fault phase is connected to the middle point of the dc-link capacitors to realize the circuit reconstruction. As an example, the equivalent fault-tolerant circuit in the case where either Sa or Sa1 is open is shown in Figure 1b, which is the so-called 3P4SI. In Figure 1b, the terminal voltage of phase a is 0 because it is directly connected to the midpoint of the dc-link capacitors, while for phases b and c, the terminal voltage is equal to the voltage of C1 (Vc1) with Sj switching on, and it is equal to the negative voltage of C2 (-Vc2) with Sj switching off. The switching states are defined as '1' and '0' for the former and latter cases, respectively. The combination of the switching states of phases b and c, as shown in Figure 2b, can form four space vectors in the two-phase static coordinate system (α-β), i.e., V1-V4. In Figure 2b, it is assumed that Vc1 is equal to Vc2. Taking V2 as an example, the corresponding switching state of phase b is 1 and the one of phase c is 0. By comparing the space vector diagrams in normal mode and in switch-open faulttolerant mode as shown in Figure 2, the number of space vectors was decreased from 7 to 4. The voltage and flux equations of IPMSMs are given in (1) and (2), where ud/uq, id/iq, Ld/Lq, Ψd/Ψq, are the dq-axes voltages, currents, inductances, and stator flux-linkages, respectively, Rs is the stator resistance, ωe is the electric angular velocity, and Ψf is the permanent magnet flux linkage: In Figure 1b, the terminal voltage of phase a is 0 because it is directly connected to the midpoint of the dc-link capacitors, while for phases b and c, the terminal voltage is equal to the voltage of C 1 (V c1 ) with S j switching on, and it is equal to the negative voltage of C 2 (-V c2 ) with S j switching off. The switching states are defined as '1' and '0' for the former and latter cases, respectively. The combination of the switching states of phases b and c, as shown in Figure 2b, can form four space vectors in the two-phase static coordinate system (α-β), i.e., V 1 -V 4 . In Figure 2b, it is assumed that V c1 is equal to V c2 . Taking V 2 as an example, the corresponding switching state of phase b is 1 and the one of phase c is 0. By comparing the space vector diagrams in normal mode and in switch-open fault-tolerant mode as shown in Figure 2, the number of space vectors was decreased from 7 to 4.
Energies 2020, 12, x FOR PEER REVIEW  3 of 14 flux-linkage reference tracking. In addition, the capacitor voltage balance method by injecting a dc offset to the current of fault phase is studied. Compared with the conventional MPDTC method, the complicated weighting factors designing process is avoided and the electromagnetic torque ripples can be greatly suppressed. An experimental prototype is established, and the experimental results prove the effectiveness and advantages of the proposed scheme.

Conventional MPDTC of IPMSM for EV in Switch Open-Circuit Fault-Tolerant Mode
The speed regulation system of EVs on the basis of IPMSMs is shown in Figure 1a, where Vdc is the dc-link voltage; C1 and C2 are the upper and lower capacitors; and Sj and Sj1 (j = a, b, c) are the up and low switches, which are composed of insulated gate bipolar transistors (IGBTs) and antiparallel diodes. The energy of the lithium battery pack is transmitted to the IPMSM through the 2L-VSI. To realize the fault-tolerant control of switch-open-circuit fault, three bi-directional thyristors Ka, Kb, and Kc are added between the middle point of the dc-link capacitors ('o' as defined in Figure 1a) and the output terminal of the VSI. In normal operation mode, Ka, Kb, and Kc are all in off state, and when one of the phases has an open-circuit fault, the bidirectional thyristor of the corresponding phase is closed, and the output terminal of the fault phase is connected to the middle point of the dc-link capacitors to realize the circuit reconstruction. As an example, the equivalent fault-tolerant circuit in the case where either Sa or Sa1 is open is shown in Figure 1b, which is the so-called 3P4SI. In Figure 1b, the terminal voltage of phase a is 0 because it is directly connected to the midpoint of the dc-link capacitors, while for phases b and c, the terminal voltage is equal to the voltage of C1 (Vc1) with Sj switching on, and it is equal to the negative voltage of C2 (-Vc2) with Sj switching off. The switching states are defined as '1' and '0' for the former and latter cases, respectively. The combination of the switching states of phases b and c, as shown in Figure 2b, can form four space vectors in the two-phase static coordinate system (α-β), i.e., V1-V4. In Figure 2b, it is assumed that Vc1 is equal to Vc2. Taking V2 as an example, the corresponding switching state of phase b is 1 and the one of phase c is 0. By comparing the space vector diagrams in normal mode and in switch-open faulttolerant mode as shown in Figure 2, the number of space vectors was decreased from 7 to 4.  (1) and (2), where ud/uq, id/iq, Ld/Lq, Ψd/Ψq, are the dq-axes voltages, currents, inductances, and stator flux-linkages, respectively, Rs is the stator resistance, ωe is the electric angular velocity, and Ψf is the permanent magnet flux linkage:  The voltage and flux equations of IPMSMs are given in (1) and (2), where u d /u q , i d /i q , L d /L q , Ψ d /Ψ q , are the dq-axes voltages, currents, inductances, and stator flux-linkages, respectively, R s is the stator resistance, ω e is the electric angular velocity, and Ψ f is the permanent magnet flux linkage: The current predictive equations can be obtained by discretizing (1) and they are given in (3) and (4), where T s is the sampling period and k is the number of T s : Substituting (3) and (4) into (2), the stator flux-linkage predictive equations are given in (5) and (6): Then, the amplitude of the stator flux-linkage is calculated as follows: In addition, the electromagnetic torque can be predicted according to (8), where P n is the pole pairs: Because the midpoint of the dc-link capacitors is connected with phase a, the difference between the voltage of C 1 and C 2 (V c_e ) can be calculated by (9): where V c1 , V c2 are the voltage of C 1 and C 2 , and i a is the current of phase 'a'. Then, the predictive equation of V c_e is given in (10): For the conventional MPDTC of IPMSMs for EVs in switch-open fault-tolerant mode, the controlled variables, including the stator flux, the electromagnetic torque, and the difference between the voltage of C 1 and C 2 for the four space vectors as shown in Figure 2b, are predicted according to (7), (8), and (10), respectively. Then, the cost function as given in (11) is calculated for each prediction, where T * e , Ψ * s are the reference of electromagnetic torque and stator flux; and λ 1 , λ 2 , and λ 3 are the weighting factors. At last, the space vector that minimizing the cost function is selected as the optimized solution: Energies 2020, 13, 5593

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However, there are some drawbacks for the conventional MPDTC method. Firstly, the number of space vectors has been decreased from 7 to 4 in the switch-open fault-tolerant mode, and the torque and flux ripples become a serious problem owing to the number of the selectable space vectors to optimize the three objectives in (11) has been decreased by 42.85%. Secondly, the design process of λ 1 , λ 2 , and λ 3 is complicated owing to the dimensions of T e , Ψ s , and V c_e not being identical. Thirdly, the switching frequency of the conventional MPDTC is usually far smaller than the sampling frequency (1/T s ) and it is not fixed, which makes the filter design complicated.

Stator Flux-Linkage Calculation under MTPA Control
The electromagnetic torque equation is given in (12) with a simplified form, where T * en , i dn , and i qn can be calculated by (13). The base value of the current and electromagnetic torque is defined in (14): The efficiency of the speed control system is one of the key indicators of EVs, which is of great significance to the improvement of driving mileage and energy saving. Efficiency optimization of IPMSMs is achieved through MTPA control, of which the relationship between T en and i dn can be obtained in (15) by making ∂T * en / ∂i dn equal to 0: To realize the MTPA control, the inverse function of (15) is needed. However, it is hard to obtain the analytic formula owing to (15) is a high-order nonlinear equation. A fitting function as shown in (16) is given to calculate i dn with a specific T * en . Both the MTPA and the fitting curves are plotted in Figure 3, and the two curves are nearly in coincidence, which indicates the effectiveness of the fitting function: Energies 2020, 13, 5593  After calculating idn, iqn can be obtained by (17) according to (12): Furthermore, the stator flux-linkage references in the synchronous rotating coordinate axes for a specific * can be calculated by (18) and (19) to realize the MTPA control: After calculating i dn , i qn can be obtained by (17) according to (12): Furthermore, the stator flux-linkage references in the synchronous rotating coordinate axes for a specific T * e can be calculated by (18) and (19) to realize the MTPA control: That is to say, the control objectives can be transferred to Ψ * d and Ψ * q from T * e and Ψ * s under the MTPA control.

Switching Sequence Selection
In Figure 2b, the space vector diagram can be divided into 2 sectors, i.e., S 1 and S 2 . Sector S 1 is above the α-axis, and it consists of V 1 , V 2 , and V 3 . Sector S 2 is below the α-axis, and it consists of V 1 , V 3 , and V 4 . As shown in Figure 4b,c, the positions of V 1 and V 3 are also lined in the α-axis in the case where V c1 is not equal to V c2 , and thus, the sector definition method can be applied in the three cases shown in Figure 4. (20) That is to say, the control objectives can be transferred to * and * from * and * under the MTPA control.

Switching Sequence Selection
In Figure 2b, the space vector diagram can be divided into 2 sectors, i.e., S1 and S2. Sector S1 is above the α-axis, and it consists of V1, V2, and V3. Sector S2 is below the α-axis, and it consists of V1, V3, and V4. As shown in Figure 4b,c, the positions of V1 and V3 are also lined in the α-axis in the case where Vc1 is not equal to Vc2, and thus, the sector definition method can be applied in the three cases shown in Figure 4.
The switching sequences in S1 and S2 can be designed as Figure 5 according to the nearest-threevectors (NTVs) principle. In the sector S1, V1, V2, and V3 are selected as the NTVs, and the corresponding switching sequence is shown in Figure 5a. However, in the sector S2, V1, V3, and V4 are selected as the NTVs, and the corresponding switching sequence is shown in Figure 5b. The two cases in Figure 5a,b are defined as the switching sequence I and II, respectively.
The switching sequences in S 1 and S 2 can be designed as Figure 5 according to the nearest-three-vectors (NTVs) principle. In the sector S 1 , V 1 , V 2 , and V 3 are selected as the NTVs, and the corresponding switching sequence is shown in Figure 5a. However, in the sector S 2 , V 1 , V 3 , and V 4 are selected as the NTVs, and the corresponding switching sequence is shown in Figure 5b. The two cases in Figure 5a,b are defined as the switching sequence I and II, respectively. The switching sequences in S1 and S2 can be designed as Figure 5 according to the nearest-threevectors (NTVs) principle. In the sector S1, V1, V2, and V3 are selected as the NTVs, and the corresponding switching sequence is shown in Figure 5a. However, in the sector S2, V1, V3, and V4 are selected as the NTVs, and the corresponding switching sequence is shown in Figure 5b. The two cases in Figure 5a,b are defined as the switching sequence I and II, respectively. In Figure 5, t1-t4 are the durations of V1-V4, and tb and tc are the durations of state '1' of phase 'b' and 'c', respectively. For the switching sequence I, tb is bigger than tc and the size-relation of tb and tc is opposite for the switching sequence II.
It can be seen from Figure 5 that both the switching sequences adopt V1 and V3, the difference is that the switching sequence I adopts V2 while the switching sequence II adopts V4. The optimized switching sequence can be selected according to (20) where a new evaluation criteria (g1) is defined in (21). If S is I, the switching sequence I is selected; otherwise, the switching sequence II is selected: = ( * − ( + 1)) + * − ( + 1) . In Figure 5, t 1 -t 4 are the durations of V 1 -V 4 , and t b and t c are the durations of state '1' of phase 'b' and 'c', respectively. For the switching sequence I, t b is bigger than t c and the size-relation of t b and t c is opposite for the switching sequence II.
It can be seen from Figure 5 that both the switching sequences adopt V 1 and V 3 , the difference is that the switching sequence I adopts V 2 while the switching sequence II adopts V 4 . The optimized switching sequence can be selected according to (20) where a new evaluation criteria (g 1 ) is defined in (21). If S is I, the switching sequence I is selected; otherwise, the switching sequence II is selected: According to the part A of this section, the control objectives of tracking of T * e and Ψ * s under the MTPA control can be transferred to the tracking of Ψ * d and Ψ * q . Thus, the optimized switching sequence selecting method in (20) can ensure the optimization of the electromagnetic torque and stator flux.

Duration Calculation
After determining the switching sequence type, the durations of the adopted space vectors should be calculated to realize the tracking of Ψ * d and Ψ * q . Taking the switching sequence I as an example, Figure 6 shows one of the possible trajectories of Ψ d and Ψ q . According to the part A of this section, the control objectives of tracking of * and * under the MTPA control can be transferred to the tracking of * and * . Thus, the optimized switching sequence selecting method in (20) can ensure the optimization of the electromagnetic torque and stator flux.

Duration Calculation
After determining the switching sequence type, the durations of the adopted space vectors should be calculated to realize the tracking of * and * . Taking the switching sequence I as an example, Figure 6 shows one of the possible trajectories of and . With the switching sequence I, the difference between the reference and predicted stator flux in the synchronous rotating coordinate can be calculated by (22): where k1d, k2d, and k3d are the slopes of with V1, V2, and V3 and k1q, k2q, and k3q are the slopes of with V1, V2, and V3. They can be calculated by (23) and (1): With the switching sequence I, the difference between the reference and predicted stator flux in the synchronous rotating coordinate can be calculated by (22): where k 1d , k 2d , and k 3d are the slopes of Ψ d with V 1 , V 2 , and V 3 and k 1q , k 2q , and k 3q are the slopes of Ψ q with V 1 , V 2 , and V 3 . They can be calculated by (23) and (1): where j = 1, 2, 3, 4 and V id , V iq are the d,q-axes component of space vector V j . According to Figure 5a, (24) can be obtained: By substituting (24) in (22), (22) can be written as (25): The target of the duration calculation is to determine the durations t a and t b in order to minimize the ripples of Ψ d and Ψ q under MTPA control. The optimal t b and t c can be obtained by minimizing the function g 2 defined in (26): (27) Furthermore, the optimal t b and t c can be calculated by (27) and the results are given in (28): where the variables in (28) satisfy (29): For the switching sequence II, the optimal t b and t c can be calculated following the same principle and the results are given in (30) and (31):

Capacitor Voltage Balance
It can be seen from (10) that the capacitor voltage balance can be realized by injecting a dc offset to i a . For the space vector V 1 and V 3 , the output voltage of phase 'a' (V an ) is 2/3V c2 and −2/3V c1 , respectively. Thus, increasing the duration of V 1 (t 1 ) is equivalent to injecting a positive dc component into i a , and increasing the duration of V 3 (t 3 ) is equivalent to injecting a negative dc component into i a . According to Figure 5, an offset (∆t), as given in (32), can be added to t b and t c to adjust the durations of V 1 and V 3 : If ∆t is positive, t 1 is decreased and t 3 is increased; it is equivalent to injecting a negative dc component into i a and it is helpful to decrease V c_e . On the contrary, it is helpful to increase V c_e if ∆t is negative. Accordingly, the control diagram of the capacitor voltage balance by adjusting ∆t can be designed as shown in Figure 7. The dc offset of V c_e is obtained by a low pass filter (LPF), and a proportional integral (PI) controller is designed to make it 0.

Capacitor Voltage Balance
It can be seen from (10) that the capacitor voltage balance can be realized by injecting a dc offset to ia. For the space vector V1 and V3, the output voltage of phase 'a' (Van) is 2/3Vc2 and −2/3Vc1, respectively. Thus, increasing the duration of V1 (t1) is equivalent to injecting a positive dc component into ia, and increasing the duration of V3 (t3) is equivalent to injecting a negative dc component into ia. According to Figure 5, an offset (∆ ), as given in (32), can be added to tb and tc to adjust the durations of V1 and V3: If t Δ is positive, t1 is decreased and t3 is increased; it is equivalent to injecting a negative dc component into ia and it is helpful to decrease Vc_e. On the contrary, it is helpful to increase Vc_e if ∆ is negative. Accordingly, the control diagram of the capacitor voltage balance by adjusting ∆ can be designed as shown in Figure 7. The dc offset of Vc_e is obtained by a low pass filter (LPF), and a proportional integral (PI) controller is designed to make it 0. The whole control diagram of the switching sequence MPDTC is shown in Figure 8. With the process of stator flux calculation under MTPA control as shown in the part A, the control objectives can be transferred to * and * from * and * , which can avoid the complicated process of adjusting , in the conventional MPDTC. In addition, there is no need to design a cost function for the proposed capacitor voltage balance method, i.e., the design process of in the conventional MPDTC can also be avoided. Thus, the problem of designing the weighting factors in the conventional MPDTC can be solved. Instead of selecting one space vector from V1-V4 as shown in Figure 4, the switching sequence with three space vectors is selected for the proposed method. The switching frequency of the proposed switching sequence MPDTC is fixed and it is equal to the sampling frequency, which is helpful to decrease the torque and flux ripples. The whole control diagram of the switching sequence MPDTC is shown in Figure 8. With the process of stator flux calculation under MTPA control as shown in the part A, the control objectives can be transferred to Ψ * d and Ψ * q from T * e and Ψ * s , which can avoid the complicated process of adjusting λ 1 , λ 2 in the conventional MPDTC. In addition, there is no need to design a cost function for the proposed capacitor voltage balance method, i.e., the design process of λ 3 in the conventional MPDTC can also be avoided. Thus, the problem of designing the weighting factors in the conventional MPDTC can be solved. Instead of selecting one space vector from V 1 -V 4 as shown in Figure 4, the switching sequence with three space vectors is selected for the proposed method. The switching frequency of the proposed switching sequence MPDTC is fixed and it is equal to the sampling frequency, which is helpful to decrease the torque and flux ripples.

Experimental Prototype
To validate the effectiveness of the proposed switching sequence MPDTC of IPMSM for EV in switch-open fault-tolerant mode, an experimental setup as shown in Figure 9 was established. A 320 V dc-link voltage is obtained by the PWM rectifier controlled by the controller 1 to simulate the lithium battery packs. IPMSM1 and IPMSM2 are coaxially connected. IPMSM1 is connected with the 3P4SI, and the proposed control strategy is implemented by the controller 3. IPMSM2 is the load motor, which is fed by the inverter 2 with controller 2. The main parameters of the 3P4SI and IPMSM1 are given in Table 1.

Experimental Prototype
To validate the effectiveness of the proposed switching sequence MPDTC of IPMSM for EV in switch-open fault-tolerant mode, an experimental setup as shown in Figure 9 was established. A 320 V dc-link voltage is obtained by the PWM rectifier controlled by the controller 1 to simulate the lithium battery packs. IPMSM1 and IPMSM2 are coaxially connected. IPMSM1 is connected with the 3P4SI, and the proposed control strategy is implemented by the controller 3. IPMSM2 is the load motor, which is fed by the inverter 2 with controller 2. The main parameters of the 3P4SI and IPMSM1 are given in Table 1.

Results and Discussion
In the experiment, the speed of IPMSM1 is controlled as 750 r/min and the load torque (Tl) is set at 50 and 100 Nm at different periods. The curves of stator flux-linkage and electromagnetic torque for the conventional method as given in [22] and the proposed MPDTC are shown in Figures 10 and  11, respectively. In the middle of Figures 10 and 11, the load torque changes from 50 to 100 Nm. It can be seen from Figure 10 that the peak-to-peak values of the stator flux-linkage ripple for the conventional MPDTC are 0.041 and 0.046 Wb with Tl set at 50 and 100 Nm, respectively. With the

Results and Discussion
In the experiment, the speed of IPMSM1 is controlled as 750 r/min and the load torque (T l ) is set at 50 and 100 Nm at different periods. The curves of stator flux-linkage and electromagnetic torque for the conventional method as given in [22] and the proposed MPDTC are shown in Figures 10 and 11, respectively. In the middle of Figures 10 and 11, the load torque changes from 50 to 100 Nm. It can be seen from Figure 10 that the peak-to-peak values of the stator flux-linkage ripple for the conventional MPDTC are 0.041 and 0.046 Wb with T l set at 50 and 100 Nm, respectively. With the proposed MPDTC, the stator flux-linkage ripple has been greatly reduced. As shown in Figure 11, the peak-to-peak values of the stator flux-linkage ripple for the proposed MPDTC are 0.004 and 0.004 Wb with T l setting at 50 and 100 Nm, respectively. They were reduced by 90.2% and 91.3% in the two cases. proposed MPDTC, the stator flux-linkage ripple has been greatly reduced. As shown in Figure 11, the peak-to-peak values of the stator flux-linkage ripple for the proposed MPDTC are 0.004 and 0.004 Wb with Tl setting at 50 and 100 Nm, respectively. They were reduced by 90.2% and 91.3% in the two cases.    In addition, the peak-to-peak values of the electromagnetic torque ripple for the conventional MPDTC, as shown in Figure 10, are 54.7 and 61.3 Nm with Tl set at 50 and 100 Nm, respectively. With the proposed MPDTC, the electromagnetic torque ripple was also greatly reduced. As shown in Figure 11, the peak-to-peak values of the electromagnetic torque ripple for the proposed MPDTC are 5.1 and 5.1 Nm with Tl set at 50 and 100 Nm, respectively. They were reduced by 90.7% and 91.7% in the two cases. In addition, both the stator flux-linkage and electromagnetic torque can be fast tracked in the load sudden change case, and thus the proposed MPDTC has an excellent dynamic performance.
The curves of the phase current with Tl set at 100 Nm for the conventional and the proposed MPDTC are shown in Figures 12 and 13, respectively. It is obvious that the current ripples were greatly reduced. The spectra of the phase current are shown in Figures 14 and 15. For the conventional MPDTC, the total harmonic distortion (THD) of the phase current is 10.35%, and it was reduced to 4.14% with the proposed MPDTC method. As shown in Figure 14, the harmonic components of the In addition, the peak-to-peak values of the electromagnetic torque ripple for the conventional MPDTC, as shown in Figure 10, are 54.7 and 61.3 Nm with T l set at 50 and 100 Nm, respectively. With the proposed MPDTC, the electromagnetic torque ripple was also greatly reduced. As shown in Figure 11, the peak-to-peak values of the electromagnetic torque ripple for the proposed MPDTC are 5.1 and 5.1 Nm with T l set at 50 and 100 Nm, respectively. They were reduced by 90.7% and 91.7% in the two cases. In addition, both the stator flux-linkage and electromagnetic torque can be fast tracked in the load sudden change case, and thus the proposed MPDTC has an excellent dynamic performance.
The curves of the phase current with T l set at 100 Nm for the conventional and the proposed MPDTC are shown in Figures 12 and 13, respectively. It is obvious that the current ripples were greatly reduced. The spectra of the phase current are shown in Figures 14 and 15. For the conventional MPDTC, the total harmonic distortion (THD) of the phase current is 10.35%, and it was reduced to 4.14% with the proposed MPDTC method. As shown in Figure 14, the harmonic components of the phase current concentrate on the low frequency range, mainly owing to the switching frequency of the conventional MPDTC being far smaller than the sampling frequency and it is not fixed. On the contrary, the low frequency harmonic components as shown in Figure 15 were decreased, and the harmonic order concentrate on 200, which is equal to the switching frequency of the proposed MPDTC.
Energies 2020, 12, x FOR PEER REVIEW 11 of 14 phase current concentrate on the low frequency range, mainly owing to the switching frequency of the conventional MPDTC being far smaller than the sampling frequency and it is not fixed. On the contrary, the low frequency harmonic components as shown in Figure 15 were decreased, and the harmonic order concentrate on 200, which is equal to the switching frequency of the proposed MPDTC.          The curves of the sector and line-to-line voltage (Vab, Vac, Vbc) with Tl set at 100 Nm for the proposed MPDTC are shown in Figure 16. The line-to-line voltage curve between the two fault-free phases, i.e., Vbc as shown in Figure 16, is similar as the one of the 2L-VSI, while the line-to-line voltage curves between the fault-free and the fault phase were changed as Vac and Vab in Figure 16. The curves of the sector and line-to-line voltage (V ab , V ac , V bc ) with T l set at 100 Nm for the proposed MPDTC are shown in Figure 16. The line-to-line voltage curve between the two fault-free phases, i.e., V bc as shown in Figure 16, is similar as the one of the 2L-VSI, while the line-to-line voltage curves between the fault-free and the fault phase were changed as V ac and V ab in Figure 16. The curves of the sector and line-to-line voltage (Vab, Vac, Vbc) with Tl set at 100 Nm for the proposed MPDTC are shown in Figure 16. The line-to-line voltage curve between the two fault-free phases, i.e., Vbc as shown in Figure 16, is similar as the one of the 2L-VSI, while the line-to-line voltage curves between the fault-free and the fault phase were changed as Vac and Vab in Figure 16.    The curves of the sector and line-to-line voltage (Vab, Vac, Vbc) with Tl set at 100 Nm for the proposed MPDTC are shown in Figure 16. The line-to-line voltage curve between the two fault-free phases, i.e., Vbc as shown in Figure 16, is similar as the one of the 2L-VSI, while the line-to-line voltage curves between the fault-free and the fault phase were changed as Vac and Vab in Figure 16.   Accordingly, the above analysis indicates that the proposed MPDTC method is effective. Compared with the conventional method, the ripples of the stator flux-linkage, electromagnetic torque, and phase current were greatly reduced with the same sampling period.

Conclusions
A switching sequence MPDTC of IPMSM for EV in switch open-circuit fault-tolerant mode was studied. The control objectives were transferred to Ψ * d and Ψ * q from T * e and Ψ * s under the MTPA control. Instead of selecting one space vector from the possible four space vectors, the proposed MPDTC method selects an optimized switching sequence including three space vectors and the calculation method of the durations of the adopted space vectors is given to realize the tracking of Ψ * d and Ψ * q . The capacitor voltage balance method, by injecting a dc offset to the current of the fault phase, is also given. The experimental results indicate the effectiveness of the proposed method and the electromagnetic torque ripples were decreased by more than 90% compared with the conventional method, which is helpful to maintain the driving comfort in the open-circuit fault-tolerant mode. In the future research, the smooth transition strategy from a healthy to open-circuit fault state with a model predictive controller will be investigated.