Fault-Tolerant Direct Torque Control of Five-Phase Permanent Magnet Synchronous Motor under Single Open-Phase Fault Based on Virtual Vectors

Abstract

In the existing literature, direct torque control (DTC) by synthesizing virtual vectors can effectively suppress low-order harmonic currents under the single open-phase fault (OPF) of the five-phase permanent magnet synchronous motor (PMSM), but the sectors and the look-up tables need to be redesigned, which makes the control process more complicated. In order to solve this problem, an indirect correction method of virtual vectors is proposed, and the amplitudes of the virtual vectors are maximized. The fault-tolerant DTC strategy under the OPF ensures that there is no need to re-divide the sectors under the fault. And the selection rules of the look-up tables are consistent with the healthy operation. The difference is that the amplitudes of ten virtual vectors in the faulty operation are reduced, which simplifies the control process and is easy to implement. Finally, the correctness and effectiveness of the proposed control strategy were verified by experiments.

1. Introduction

In recent years, multiphase PMSMs have received extensive attention from industry and academia. Compared with traditional three-phase motors, multiphase motors have the advantages of low-voltage high power, low torque pulsation, and high reliability. They have good application prospects in the fields of aerospace, ship propulsion, and wind power generation [1,2,3,4,5]. The good fault-tolerant ability is a significant advantage for multiphase motors. Most open-circuit and short-circuit faults in multiphase machine drive systems can be converted into the OPF. Therefore, the research of fault-tolerant control strategies for multiphase machines is mainly focused on the OPF. There is no need to change any hardware and only the need to select an appropriate fault-tolerant control strategy to derate and continue to run when a single OPF has occurred. Hence, shutdown or system reconfiguration can be avoided [6,7]. At present, the common control strategies for multiphase motors include field-oriented control (FOC) [8,9,10,11], model predictive control (MPC) [12,13,14,15], and direct torque control (DTC) [16,17,18,19,20,21,22,23,24]. Among them, DTC has the advantages of simple structure, low sensitivity to motor parameters, and fast torque response. Thus, it is widely researched for multiphase motor drive systems.

In order to suppress low-order harmonic currents and reduce output torque ripple when DTC is used in the healthy operation of multiphase machines, the concept of virtual vectors are proposed in many studies [18,19], i.e., synthesizing basic voltage vectors into a new virtual voltage vector during the sampling period, and the average amplitudes of virtual vectors are zero in the harmonic subspace. The virtual vectors need to be reconstructed when an OPF occurs in the multiphase motor drive system. A fault-tolerant DTC strategy for five-phase induction motor under the single-OPF condition was proposed in [20]. The ten virtual vectors during the healthy operation were transformed into eight virtual vectors during the faulty operation, and the sectors were divided into eight. The look-up table was reconstructed under the faulty operation. In order to reduce the torque ripple and the third harmonic current caused by the single OPF of the five-phase PMSM, a fault-tolerant control strategy considering the third harmonic suppression was proposed in [21]. According to the effect of the virtual vectors in different sectors on the torque and flux linkage, the look-up tables under healthy and faulty operation were established, respectively. A DTC method for a five-phase induction motor during low-speed operation was proposed in [22], and its low-speed performance was improved by deriving a new look-up table. In [23], eight virtual voltage vectors were synthesized from fundamental wave subspace under the single OPF of the five-phase induction motor. The DTC based on virtual vectors was compared with rotor field oriented control, and the DTC could reduce the amount of calculation and showed strong robustness. To eliminate the stator harmonic current under the single OPF of the five-phase induction motor, a DTC strategy with an improved look-up table was proposed in [24]. The influence of virtual voltage vectors on torque and flux response under different speed and load conditions was analyzed. The virtual vectors have also been applied to the space vector pulse width modulation (SVPWM) [25] and predictive current control [26,27] of the multiphase motor. However, after the OPF of multiphase motors, there is a common problem that the spatial phases of virtual vectors are changed [20,23,24]. Therefore, it is necessary to redesign the stator flux partitions and look-up tables, which makes the control process more complicated.

In order to solve the above problems in fault-tolerant DTC, an indirect correction method for virtual vectors under the OPF of the five-phase motor is herein proposed. And a fault-tolerant DTC strategy based on virtual vectors without redesigning sectors and look-up tables is constructed. The paper is organized as follows. In Section 2, the voltage vectors distributions of the five-phase PMSM in healthy and faulty operation are introduced. The DTC strategy based on virtual voltage vectors in the healthy operation is introduced in Section 3. Next, an indirect correction method of virtual vector with the A phase OPF is proposed in Section 4. The experimental results are presented in Section 5. Finally, the conclusions are presented in Section 6.

2. Voltage Vectors Distribution of Five-Phase PMSM

2.1. Voltage Vectors Distribution in the Healthy Operation

According to the vector space decomposition modeling theory, the static transformation matrix of five-phase PMSM can be expressed as follows:

$${\mathit{T}}_{5\mathrm{s}}=\frac{2}{5}\left[\begin{array}{ccccc}1& \cos \gamma & \cos 2\gamma & \cos 3\gamma & \cos 4\gamma \\ 0& \sin \gamma & \sin 2\gamma & \sin 3\gamma & \sin 4\gamma \\ 1& \cos 3\gamma & \cos 6\gamma & \cos 9\gamma & \cos 12\gamma \\ 0& \sin 3\gamma & \sin 6\gamma & \sin 9\gamma & \sin 12\gamma \\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right]$$

(1)

where γ = 2π/5. The first and second rows correspond to the α₁-β₁ subspace and participate in the electromechanical energy conversion. The third and fourth rows correspond to the α₃-β₃ subspace and do not participate in the electromechanical energy conversion. The last row is zero sequence component, which is always zero.

The voltages in the natural frame [A-B-C-D-E] are transformed into the static frame [α₁-β₁-α₃-β₃-o] by the static transformation matrix (1), and the voltages can be further expressed as follows:

$$\left[\begin{array}{l}{U}_{\alpha 1}\\ U_{\beta 1}\\ U_{\alpha 3}\\ U_{\beta 3}\\ U_{\mathrm{o}}\end{array}\right]={\mathit{T}}_{5\mathrm{s}}\left[\begin{array}{l}{U}_A\\ U_B\\ U_C\\ U_D\\ U_E\end{array}\right]$$

(2)

where U_A, U_B, U_C, U_D, and U_E are phase voltages of the five-phase PMSM. U_α1, U_β1, U_α3, U_β3, and U_o are the voltage components in the static frame.

The voltage vectors are mapped to α₁-β₁ and α₃-β₃ subspaces, as shown in Figure 1. Each subspace contains 30 basic vectors and two zero vectors (V₀ and V₃₁). Basic vectors can be divided into large vectors V_L, medium vectors V_M, and small vectors V_S according to the amplitudes of basic vectors, and the amplitudes are 0.6472U_dc, 0.4U_dc, and 0.2472U_dc, respectively.

2.2. Voltage Vectors Distribution under the Single Open-Phase Fault

Taking the A phase OPF as example, and the main circuit topology of the five-phase PMSM drive system is shown in Figure 2.

When the A phase OPF occurs, the static transformation matrix in the healthy operation is no longer applicable, and the reconstructed reduced-order static transform matrix can be expressed as follows:

$${\mathit{T}}_{4\mathrm{s}}=\frac{2}{5}\left[\begin{array}{cccc}\cos \gamma +e_0& \cos 2\gamma +e_0& \cos 3\gamma +e_0& \cos 4\gamma +e_0\\ \sin \gamma & \sin 2\gamma & \sin 3\gamma & \sin 4\gamma \\ \sin 3\gamma & \sin 6\gamma & \sin 9\gamma & \sin 12\gamma \\ 1& 1& 1& 1\end{array}\right]$$

(3)

where e₀ = 0.25, and e₀ is the correction coefficient.

The voltages in the natural frame [B-C-D-E] are transformed into the static frame [α₁-β₁-β₃-o] by the reduced-order static transformation matrix, and the voltage can be expressed as given below:

$$\left[\begin{array}{l}{U}_{\alpha 1}\\ U_{\beta 1}\\ U_{\beta 3}\\ U_{\mathrm{o}}\end{array}\right]={\mathit{T}}_{4s}\left[\begin{array}{l}{U}_B\\ U_C\\ U_D\\ U_E\end{array}\right]$$

(4)

Under the A phase OPF, the voltage vectors’ distribution in α₁-β₁ subspace and β₃ pulse subspace is shown in Figure 3. Each subspace contains fourteen basic vectors and two zero vectors (V₀ and V₁₅). The amplitude and phase angle of each voltage vector in the α₁-β₁ subspace and β₃ pulse subspace are shown in

Table 1. The amplitude and phase angle of each voltage vector in the α₁-β₁ subspace and β₃ pulse subspace with the A phase OPF.

Voltage Vectors	α1-β1 Subspace	β3 Pulse Subspace
V0 (0000)	0∠0°	0
V1 (0001)	0.4412Udc∠−59.55°	0.2351Udc
V2 (0010)	0.3245Udc∠−133.56°	−0.3804Udc
V3 (0011)	0.6155Udc∠−90°	−0.1453Udc
V4 (0100)	0.3245Udc∠133.56°	0.3804Udc
V5 (0101)	0.1453Udc∠−90°	0.6155Udc
V6 (0110)	0.4472Udc∠180°	0
V7 (0111)	0.4412Udc∠120.45°	0.2351Udc
V8 (1000)	0.4412Udc∠59.55°	−0.2351Udc
V9 (1001)	0.4472Udc∠0°	0
V10 (1010)	0.1453Udc∠90°	−0.6155Udc
V11 (1011)	0.3245Udc∠−46.44°	−0.3804Udc
V12 (1100)	0.6155Udc∠90°	0.1453Udc
V13 (1101)	0.3245Udc∠46.44°	0.3804Udc
V14 (1110)	0.4412Udc∠120.45°	−0.2351Udc
V15 (1111)	0∠0°	0

.

3. The DTC Strategy Based on Virtual Voltage Vectors in Healthy Operation

It can be seen from Figure 1 that the amplitudes of the basic voltage vectors in the α₃-β₃ subspace are not zero in the healthy operation of the five-phase PMSM. Because the harmonic impedance of the α₃-β₃ subspace is only constituted with the stator resistance and the leakage inductance, a very low harmonic voltage can generate large harmonic currents, which can increase the system loss and damage the motor. Therefore, the voltage vectors need to be corrected. The large and medium vectors in the same direction in the α₁-β₁ subspace correspond to the small and medium vectors in the opposite direction in the α₃-β₃ subspace. Therefore, as long as the action time of the large and medium vectors in the α₁-β₁ subspace are in inverse proportion to the amplitudes of the small and medium vectors in a pulse width modulation (PWM) control period T_s, it can be ensured that the synthesized voltage vector in the α₃-β₃ subspace is zero.

The new vectors synthesized are defined as the virtual vectors VV_n (n = 1~10). The control period is T_s. It is assumed that the action time of the large vector V_L is λT_s, and the action time of the medium vector V_M is (1 − λ)T_s. The amplitude of the synthesized virtual vector in the α₁-β₁ subspace is |V_α1-β1|, and the amplitude in the α₃-β₃ subspace is |V_α3-β3|. In a control period T_s, it is obtained by the volt-second balance principle:

$$\left\{\begin{array}{l}\left|{\mathit{V}}_{\alpha 1-\beta 1}\right|T_{\mathrm{s}}=\lambda \left|{\mathit{V}}_{\mathrm{L}}\right|T_{\mathrm{s}}+(1-\lambda )\left|{\mathit{V}}_{\mathrm{M}}\right|T_{\mathrm{s}}=0.6472U_{\mathrm{dc}}\lambda T_{\mathrm{s}}+0.4U_{\mathrm{dc}}(1-\lambda )T_{\mathrm{s}}\\ \left|{\mathit{V}}_{\alpha 3-\beta 3}\right|T_{\mathrm{s}}=\lambda \left|{\mathit{V}}_{\mathrm{S}}\right|T_{\mathrm{s}}-\left(1-\lambda \right)\left|{\mathit{V}}_{\mathrm{M}}\right|T_{\mathrm{s}}=0.2472U_{\mathrm{dc}}\lambda T_{\mathrm{s}}-0.4U_{\mathrm{dc}}(1-\lambda )T_{\mathrm{s}}\end{array}\right.$$

(5)

Setting |V_α3-β3| = 0, λ = 0.618 and |V_α1-β1| = ($\sqrt{5}$/5)U_dc = 0.5527U_dc can then be obtained. Ten virtual vectors are obtained from the above principles, and the α₁-β₁ subspace is divided into ten sectors, as shown in Figure 4.

According to the effect of the radial and tangential components of the virtual vectors in different sectors on the torque and flux linkage, the voltage vector look-up table under the healthy operation of the five-phase PMSM is obtained, as shown in

Table 2. Voltage vector look-up table in the healthy operation.

Fψ	FT	Sectors
		1	2	3	4	5	6	7	8	9	10
1	1	VV3	VV4	VV5	VV6	VV7	VV8	VV9	VV10	VV1	VV2
	−1	VV9	VV10	VV1	VV2	VV3	VV4	VV5	VV6	VV7	VV8
	0	V0	V31	V0	V31	V0	V31	V0	V31	V0	V31
−1	1	VV4	VV5	VV6	VV7	VV8	VV9	VV10	VV1	VV2	VV3
	−1	VV8	VV9	VV10	VV1	VV2	VV3	VV4	VV5	VV6	VV7
	0	V31	V0	V31	V0	V31	V0	V31	V0	V31	V0

.

4. Indirect Correction Method of Virtual Vectors with Single Open-Phase Fault

4.1. Equal-Amplitude Virtual Vectors

The number of voltage vectors is changed from 32 to 16 when a single OPF occurs in the five-phase PMSM, and thus, the virtual vector after the OPF needs to be re-corrected. As shown in Figure 3, the voltage vector amplitudes no longer meet the specific proportional relationship after the A phase OPF, and the spatial distribution is no longer uniform. Thus, it is difficult to adopt the method of direct synthesis that is used in the healthy operation for virtual vector correction. An indirect virtual vector correction method is proposed in this paper.

To ensure that the synthesized voltage vectors in the β₃ pulse subspace are zero under the A phase OPF, the reference voltage V* = [U*_α1 U*_β1 U*_β3]^T can be expressed as V* = [μcosθ μsinθ 0]^T, where μ is the amplitude of the reference voltage in the α₁-β₁ subspace, and θ is the angle between the reference voltage vector and the α₁-axis. Since the zero sequence voltage is no longer injected after the single OPF, the phase voltage is equal to the modulation voltage (voltage between inverter output and DC bus voltage midpoint, such as U_AO in Figure 2), so the amplitudes of phase voltages need to be less than 0.5U_dc. The phase voltage reference values can be obtained by the inverse transformation of the reconstructed reduced-order static transform (4). The waveform of the remaining reference phase voltages when the linear modulation ratio m = 0.3406 is shown in Figure 5. It can be seen that the maximum amplitude of the corresponding reference phase voltages is 0.5U_dc. Therefore, the 0.3406U_dc amplitude voltage vectors can be synthesized in α₁-β₁ subspace at any phase angle, ensuring that the synthesized voltage vectors in the β₃ pulse subspace are zero.

In order to facilitate the implementation, it is assumed that the direction of virtual vectors under the A phase OPF is consistent with that under the healthy operation. Taking the virtual voltage vector $V{V}_2^{*}$ as an example, μ = 0.3406U_dc, and θ = π/5. The reference voltage V* can be expressed as follows:

$$V^{*}={\left[0.3406U_{\mathrm{d}\mathrm{c}}cos\left(\pi /5\right) 0.3406U_{\mathrm{d}\mathrm{c}}sin\left(\pi /5\right) 0\right]}^T$$

(6)

The phase voltage reference values can be obtained by the inverse transformation of (4). U_B, U_C, U_D, and U_E are 0.4985U_dc, −0.1904U_dc, −0.4257U_dc, and 0.1177U_dc, respectively. Because the zero sequence voltage is no longer injected after the single OPF, the phase voltage is equal to the modulation voltage. The relationship between the modulation voltages of remaining phases and the phase duty cycles under the single OPF can be written as given below:

$${U_{\mathrm{x}}}^{\ast }=\left(D_{\mathrm{x}}-0.5\right)U_{\mathrm{dc}}$$

(7)

where U_x^* stands for the modulation voltages of remaining phases, and D_x is the duty cycle of each phase. X stands for the phase B, C, D, or E.

The duty cycle of each remaining phase with the phase A OPF can be obtained by the expression (7). D_B, D_C, D_D, and D_E are 0.9985, 0.3096, 0.0743, and 0.6177, respectively. Then, the PWM waveform corresponding to the virtual vector $V{V}_2^{*}$ is shown in Figure 6.

It can be seen from Figure 6 that $V{V}_2^{*}$ is synthesized by three basic vectors (V₈, V₉, and V₁₃) and two zero vectors (V₀ and V₁₅), and the proportions of voltage vectors are 0.3808, 0.3081, 0.2353, 0.0015, and 0.0743, respectively. The voltage vectors and their proportions corresponding to the ten corrected virtual vectors after the phase A OPF can be obtained by using the same method, as shown in

Table 3. The voltage vectors and their proportions corresponding to the same amplitude virtual vectors after the A phase OPF.

Virtual Vector	Voltage Vector and Its Proportion				Amplitude of the Virtual Vector
$V{V}_1^{*}$	V0, V15		V9		0.3406Udc
	0.2384		0.7616
$V{V}_2^{*}$	V0, V15	V8	V9	V13	0.3406Udc
	0.0758	0.3808	0.3081	0.2353
$V{V}_3^{*}$	V0, V15	V8	V12	V13	0.3406Udc
	0.2662	0.3530	0.2631	0.1177
$V{V}_4^{*}$	V0, V15	V4	V12	V14	0.3406Udc
	0.2662	0.1177	0.2631	0.3530
$V{V}_5^{*}$	V0, V15	V4	V6	V14	0.3406Udc
	0.0758	0.2353	0.3081	0.3808
$V{V}_6^{*}$	V0, V15	V6			0.3406Udc
	0.2384	0.7616
$V{V}_7^{*}$	V0, V15	V2	V6	V7	0.3406Udc
	0.0758	0.2353	0.3081	0.3808
$V{V}_8^{*}$	V0, V15	V2	V3	V7	0.3406Udc
	0.2662	0.1177	0.2631	0.3530
$V{V}_9^{*}$	V0, V15	V1	V3	V11	0.3406Udc
	0.2662	0.3530	0.2631	0.1177
$V{V}_{10}^{*}$	V0, V15	V1	V9	V11	0.3406Udc
	0.0758	0.3808	0.3081	0.2353

. It can be seen that the virtual vectors $V{V}_1^{*}$ and $V{V}_6^{*}$ only use one basic voltage vector because the voltage vectors V₉ and V₆ do not generate voltage components in the β₃ pulse subspace.

4.2. Maximum Amplitude Virtual Vectors

It can be seen from Table 3 that all the voltage vectors corresponding to the ten virtual vectors contain a certain proportion of zero vectors (V₀ and V₁₅). The existence of the zero vectors indicates that the system cannot achieve the maximum DC bus voltage utilization. Thus, in order to further improve the DC bus voltage utilization, the action time of the zero vectors are allocated to the basic vectors according to the corresponding proportions, which will increase the action time of basic vectors. Taking the virtual vector $\mathit{V}{\mathit{V}}_2^{*}$ as an example, the same amplitude virtual vectors $\mathit{V}{\mathit{V}}_2^{*}$ are synthesized by three basic vectors(V₈, V₉, and V₁₃) and two zero vectors (V₀ and V₁₅), and the proportions of voltage vectors are 0.3808, 0.3081, 0.2353, 0.0015, and 0.0743. Defining D₀ is the duty cycle of zero vectors (V₀ and V₁₅), and D₀ = 0.0758 in $\mathit{V}{\mathit{V}}_2^{*}$. The proportions of V₈, V₉, and V₁₃ allocated by D₀ are 0.0312, 0.0253, and 0.0193, respectively. By this way, the maximum amplitude virtual vectors $\mathit{V}{\mathit{V}}_2^{**}$ can be obtained, which are synthesized only by the three basic vectors V₈, V₉, and V₁₃, and the proportions of three voltage vectors are 0.4120, 0.3334, and 0.2546, respectively. By adopting this method, the amplitudes of virtual vectors can be maximized, and the following expression can be obtained:

$$\frac{0.3406U_{\mathrm{dc}}}{\left|V_{\max }\right|}=\frac{1-D_0}{1}$$

(8)

where |V_max| stands for the maximum amplitude of the virtual vector. D₀ is the duty cycle of zero vectors (V₀ and V₁₅).

Taking the virtual vector $\mathit{V}{\mathit{V}}_2^{*}$ as an example, D₀ = 0.0758, and then, |V_max| = 0.3685U_dc.

The maximum amplitudes of the other nine virtual vectors are obtained by using the same method. Finally, the voltage vectors and their proportions corresponding to the maximum amplitude virtual vectors after the A phase OPF are shown in

Table 4. The voltage vectors and their proportions corresponding to the maximum amplitude virtual vectors after the A phase OPF.

Virtual Vector	Voltage Vector and Its Proportion				The Maximum Amplitude of the Virtual Vector
$V{V}_1^{**}$	V0, V15	V9			0.4472Udc
	0	1
$V{V}_2^{**}$	V0, V15	V8	V9	V13	0.3685Udc
	0	0.4120	0.3334	0.2546
$V{V}_3^{**}$	V0, V15	V8	V12	V13	0.4642Udc
	0	0.4811	0.3585	0.1604
$V{V}_4^{**}$	V0, V15	V4	V12	V14	0.4642Udc
	0	0.1604	0.3585	0.4811
$V{V}_5^{**}$	V0, V15	V4	V6	V14	0.3685Udc
	0	0.2546	0.3334	0.4120
$V{V}_6^{**}$	V0, V15	V6			0.4472Udc
	0	1
$V{V}_7^{**}$	V0, V15	V2	V6	V7	0.3685Udc
	0	0.2546	0.3334	0.4120
$V{V}_8^{**}$	V0, V15	V2	V3	V7	0.4642Udc
	0	0.1604	0.3585	0.4811
$V{V}_9^{**}$	V0, V15	V1	V3	V11	0.4642Udc
	0	0.4811	0.3585	0.1604
$V{V}_{10}^{**}$	V0, V15	V1	V3	V11	0.3685Udc
	0	0.4120	0.3334	0.2546

.

Taking the maximum amplitude virtual vector $V{V}_2^{**}$ as an example, the synthesis processes of $V{V}_2^{**}$ in the α₁-β₁ subspace and β₃ pulse subspace are shown in Figure 7. Moreover, when the A phase OPF occurs in the five-phase PMSM drive system, the distribution diagram of ten maximum amplitude virtual vectors in the α₁-β₁ subspace can be obtained, which is shown in Figure 8.

It can be seen that the directions of ten virtual vectors in Figure 8 (the faulty operation) are consistent with those in Figure 4 (the healthy operation), and the difference between the virtual vectors in the healthy and faulty operation is that the amplitudes of ten virtual vectors under the A phase OPF are reduced. Thus, there is no need to redivide the sectors and redesign the look-up table after the A phase OPF of five-phase PMSM drive system.

4.3. Implementation of the Proposed Fault-Tolerant Direct Torque Control Strategy

The block diagram of the fault-tolerant DTC for the five-phase PMSM based on virtual vectors is shown in Figure 9. The look-up table is consistent with the switch Table 2 during normal DTC control, but the virtual vectors are selected in Table 3 (the same amplitude) or Table 4 (the maximum amplitude). The same amplitude virtual vectors, which have equal but relatively small amplitudes, are suitable for the conditions of low speed and light load (THD and torque ripple are small). The maximum amplitude virtual vectors with unequal amplitudes can improve the utilization rate of the DC bus voltage and obtain large speed and torque output, which are suitable for the conditions of high speed and heavy load.

By using the back-EMF integration algorithm, the voltage model for the five-phase PMSM flux estimation can be expressed:

$$\left\{\begin{array}{l}{\psi }_{\alpha 1}={\displaystyle \int (U_{\alpha 1}-}{R}_{\mathrm{s}}{i}_{\alpha 1})\mathrm{d}t+{\psi }_{\alpha 1-0}\\ {\psi }_{\beta 1}={\displaystyle \int (U_{\beta 1}-}{R}_{\mathrm{s}}{i}_{\beta 1})\mathrm{d}t+{\psi }_{\beta 1-0}\\ \left|{\psi }_{\mathrm{s}}\right|=\sqrt{{{\psi }_{\alpha 1}}^2+{{\psi }_{\beta 1}}^2}\end{array}\right.$$

(9)

The torque estimation can be expressed as follows:

$$T_{\mathrm{e}}=\frac{5}{2}{n}_p({\psi }_{\alpha 1}{i}_{\beta 1}-{\psi }_{\beta 1}{i}_{\alpha 1})$$

(10)

The flux space position angle θ_s of the stator flux is given below;

$${\theta }_{\mathrm{s}}=\arctan (\frac{{\psi }_{\beta 1}}{{\psi }_{\alpha 1}})$$

(11)

where ψ_α1, ψ_β1 are the components of the flux linkage in the α₁-β₁ subspace. i_α1, i_β1 are the components of the stator current in the α₁-β₁ subspace. U_α1, U_β1 are the components of the stator voltage in the α₁-β₁ subspace. ψ_α1-0,ψ_β1-0 are the initial flux linkages of α₁ and β₁ axes, respectively. |ψ_s| is the estimated amplitude of the flux linkage. R_s is the stator resistance. T_e is the estimated torque value. n_p is the number of pole pairs.

5. Experimental Results

In order to verify the feasibility of the control strategy proposed in this paper, a five-phase PMSM experimental platform was developed, as shown in Figure 10. A DC motor was used as load, and XE164 of Infineon was used as the core controller. The five-phase voltage source inverter was composed of five Infineon IGBT half-bridge power modules FF300R12E4 in parallel. The parameters of the five-phase PMSM are listed as follows: R_s = 0.5 Ω, n_p= 4, L_d = L_q = 8.4 mH, and ψ_f = 0.32 Wb. The PWM control frequency was 10 kHz. The dead time was 2 μs. The motor speed was 200 rpm. The load torque was 7 N·m, and the DC bus voltage was 200V in the experiments. Due to the experimental conditions of low speed and light load, the subsequent experiments all adopt the method of the same amplitude virtual vectors (under the A phase OPF).

Figure 11 shows phase currents waveforms of the five-phase PMSM during the healthy operation. Under the healthy operation, the THD of the phase C current of the five-phase PMSM is 4.5%.

Figure 12 shows phase currents waveforms of the five-phase PMSM after the A phase OPF. It can be seen from Figure 12 that the remaining four phase currents after the A phase OPF have good sine quality. Because the principle of the minimum copper losses is adopted in this paper, the current amplitudes of phase B and E are higher than those of phase C and D. Under the fault-tolerant DTC strategy, the THD of the phase C current is 5.7%. Figure 13 shows waveforms of phase currents, torque, and speed in the steady state under the A phase OPF. It can be seen from Figure 13 that the torque ripple is small, and the speed is relatively smooth.

Figure 14 shows the current waveforms of the α₁-β₁ subspace and β₃ pulse subspace under the A phase OPF. It can be seen from Figure 14 that the currents i_α1 and i_β1 in the α₁-β₁ subspace after the A phase OPF have good sine quality, and the harmonic current i_β3 in the β₃ pulse subspace is small. Thus, the steady-state performance under the A phase OPF is good.

Figure 15 shows the waveforms of phase currents, torque, and speed when the load torque is suddenly increased from 4 N·m to 7 N·m. Figure 16 shows the waveforms of phase currents, torque, and speed when the load torque is suddenly reduced from 7 N·m to 4 N·m. The corresponding local amplification waveforms are given in Figure 15 and Figure 16. It can be seen that the phase currents and torque can be quickly tracked. With the sudden increase of the torque, the amplitudes of the phase currents become larger, and the torque can be quickly stabilized at 7 N·m. With the sudden reduction of torque, the amplitudes of the phase currents become smaller, and the torque can be quickly stabilized at 4 N·m. When the load torque is suddenly increased or reduced, there is a slight sudden change in speed, and the speed can restore to the reference value quickly.

Figure 17 shows the waveforms of phase currents, torque, and speed when the speed is suddenly increased from 150 rpm to 300 rpm. The local amplification waveforms are given in Figure 17. It can be seen that the speed can be quickly tracked. With the sudden increase of speed, the speed can be quickly stabilized at 300 rpm. Therefore, the proposed control strategy has good dynamic response performance.

Figure 18 shows the experimental waveforms of the motor starting process. As can be seen in Figure 18, at the beginning of the starting process, the overshoot of the speed is small, and the speed and torque can quickly reach the reference value. Moreover, the currents of phase B and C can also reach stability quickly.

Figure 19 shows the experimental waveforms when the motor reverses from 200 rpm to −200 rpm. As can be seen in Figure 19, when the motor reverses, with the rapid adjustment, the torque and speed can reach stability quickly, proving that the proposed control strategy has good static and dynamic performances.

Figure 20 shows the switching waveforms of the five-phase PMSM from the healthy operation to the fault-tolerant DTC operation. It can be seen that the fault-tolerant DTC strategy proposed in this paper can achieve the smooth switching of the five-phase PMSM from the healthy operation to the fault-tolerant DTC operation.

6. Conclusions

This paper proposes an indirect correction method of virtual vectors for the five-phase PMSM drive system under single OPF. By the proposed indirect correction method, two kinds of virtual vectors can be obtained: the same amplitude and the maximum amplitude virtual vectors, where the maximum amplitude virtual vectors eliminate the zero vectors, and the maximum utilization rate of the DC bus voltage is realized. The difference between the virtual vectors in the healthy and faulty operation is that the amplitudes of ten virtual vectors under single OPF are reduced, and a fault-tolerant DTC strategy based on virtual vectors is proposed. This strategy can ensure that the sector distribution and the selection rules of look-up tables are consistent with those in the healthy operation, which simplifies the control process. In addition, the proposed fault-tolerant DTC strategy has good dynamic and static response performances. Without loss of generality, the virtual vectors’ indirect correction method proposed in this paper can also be extended to other multiphase motors and different types of OPFs. The virtual vectors can also be applied to the SVPWM and predictive current control of the multiphase motor under the OPF.