Improved SVPWM Fault-Tolerant Control Strategy for Five-Phase Permanent-Magnet Motor

Abstract

Multiphase permanent-magnet motors have received a lot of attention in the past few years owing to the merits of high power density, high efficiency and high fault-tolerant capability. Particularly, high fault tolerance is very desirable for safety-critical applications. This paper proposes an improved space vector pulse-width modulation (SVPWM) fault-tolerant control for five-phase permanent-magnet motors. First, generalized five-phase SVPWM fault-tolerant control is deduced and analyzed based on single-phase open-circuit fault, thus obtaining various SVPWM fault-tolerant control strategies and yielding a greatly increased capacity to enhance fault-tolerant performance of motor. Then, an improved SVPWM fault-tolerant control strategy with increased DC bus voltage utilization and reduced current harmonics is proposed and compared with the traditional one. Last, effectiveness and superiority of the proposed control strategy is verified by both simulation and experimental results on a five-phase permanent-magnet motor.

1. Introduction

Nowadays, permanent-magnet (PM) motor drives have been widely used in a variety of fields, owing to its high efficiency and high power density [1,2,3]. Also, in order to maintain reliability, fault-tolerant technologies are increasingly adopted into PM motors. Multiphase PM motors have attracted more and more attention because of their superior fault-tolerant capability to traditional three-phase PM motors [4,5,6,7,8,9]. A variety of multiphase PM motors with high reliability have been proposed for safety-critical applications, such as electric vehicles, electric ship propulsion and aerospace industries. Several fault-tolerant control strategies also have been proposed for multiphase PM motors when motors operate under one phase or two phase fault condition [10,11,12,13,14,15]. It was found that these strategies obtain satisfactory fault tolerant performance. Although some superior properties for fault tolerant operation have been attained by these strategies, there are still some other problems such as high switching losses of inverter and stator current distortion.

Generally, current hysteresis band pulse-width modulation methods can achieve excellent results and have been introduced into fault-tolerant control [15]. Nevertheless, high switching losses of inverter caused by high switching frequency should not be neglected. In order to avoid high switching losses, space vector pulse-width modulation (SVPWM) technique has been implemented in multiphase motor drives [16,17,18]. In these techniques, the adjacent four voltage vectors method was generally adopted for five-phase PM motor drives because of its high voltage utilization and low third-harmonic component [16]. However, only normal condition was taken into consideration in terms of SVPWM techniques rather than fault-tolerant condition [16,17,18]. When motors run under faulty condition, the normal voltage vector diagram and many system parameters will not be available probably owing to greatly reducing of voltage vectors. In [19], a SVPWM fault-tolerant control strategy was applied successfully for a five-phase motor. It was shown that improved efficiency is achieved with the proposed control strategy. However, this control strategy was derived from the perspective of current hysteresis band pulse-width modulation fault-tolerant control. Therefore, the control strategy has a less degree of freedom in terms of SVPWM fault-tolerant control. Moreover, the issues of DC-bus voltage utilization and harmonic current suppression under fault-tolerant condition were not raised more attention.

The purpose of this paper is to put forward a generalized SVPWM fault-tolerant control strategy for a five-phase PM motor with the loss of one phase, thus yielding an increased capacity to enhance fault-tolerant performance of the motor. The inverter model and voltage matrices under normal condition will be introduced in Section 2. Then, in Section 3, generalized five-phase SVPWM fault-tolerant control strategy is deduced and analyzed. In Section 4, based on the analysis of DC bus voltage utilization and harmonic current suppression, an improved SVPWM fault-tolerant control strategy is proposed and compared with the traditional one. In Section 5, simulation and experimental results are provided to demonstrate the validity and superiority of the improved SVPWM fault-tolerant control strategy. Finally, some conclusions are summarized in Section 6.

2. Model and Voltage Matrices

The star type connection of half bridge structure is adopted in inverter topology of a five-phase PM motor drive system. Voltages of inverter at phase coordinate are described as (1) under normal condition. Besides, the five-phase voltages composed to zero as a generalized principle is expressed as (2).

$$\{\begin{array}{l}{U}_a=U_{a{N}^{\prime }}+U_{N^{\prime }N}=\frac{U_{dc}}{2}(2S_a-1)+U_{N^{\prime }N}\\ U_b=U_{b{N}^{\prime }}+U_{N^{\prime }N}=\frac{U_{dc}}{2}(2S_b-1)+U_{N^{\prime }N}\\ U_c=U_{c{N}^{\prime }}+U_{N^{\prime }N}=\frac{U_{dc}}{2}(2S_c-1)+U_{N^{\prime }N}\\ U_d=U_{d{N}^{\prime }}+U_{N^{\prime }N}=\frac{U_{dc}}{2}(2S_d-1)+U_{N^{\prime }N}\\ U_e=U_{e{N}^{\prime }}+U_{N^{\prime }N}=\frac{U_{dc}}{2}(2S_e-1)+U_{N^{\prime }N}\end{array}$$

(1)

$$U_a+U_b+U_c+U_d+U_e=0$$

(2)

where U_a–U_e are the five-phase voltage, S_a–S_e are the five-phase switches, U_dc is the DC-bus voltage, U_xN′ is the voltage between inverter output terminal and bus neutral point and U_N′N is the voltage between bus neutral point and motor neutral point. S_x represents switch status, where “1” indicates the upper switch is on, while “0” indicates the lower switch is on.

Since five-phase voltages are strongly coupled at phase coordinate, it is necessary for voltages to be mapped at the α-β stationary coordinate with extended Clarke transformation matrices. Given as (3), voltages are decoupled expressed as a transformation matrix on α₁-β₁ and α₃-β₃ spaces.

$$\left[\begin{array}{l}{U}_{\alpha 1}\\ U_{\beta 1}\\ U_{\alpha 3}\\ U_{\beta 3}\\ U_0\end{array}\right]=\frac{2}{5}\left[\begin{array}{ccccc}1& c_1& c_2& c_2& c_1\\ 0& s_1& s_2& -s_2& -s_1\\ 1& c_2& c_1& c_1& c_2\\ 0& -s_2& s_1& -s_1& s_2\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right]\cdot \left[\begin{array}{l}{U}_a\\ U_b\\ U_c\\ U_d\\ U_e\end{array}\right]$$

(3)

where c₁ = cosα, c₂ = cos2α, s₁ = sinα, s₂ = sin2α and α = 2π/5 is the electrical angle of two adjacent phases. The position of phase A coincides with α-axis at the α-β stationary coordinate.

Substituting (1) and (2) into (3), the decoupled voltage matrix can be expressed as follows:

$$\left[\begin{array}{l}{U}_{\alpha 1}\\ U_{\alpha 3}\end{array}\right]=\frac{2}{5}\left[\begin{array}{l}\frac{5}{4}{U}_a+U_{dc}[c_1{S}_b+c_2{S}_c+c_2{S}_d+c_1{S}_e+\frac{1}{4}(S_b\\ +S_c+S_d+S_e)]+(c_1+c_2+\frac{1}{2})\cdot (2U_{N^{\prime }N}-U_{dc})\\ \frac{5}{4}{U}_a+U_{dc}[c_2{S}_b+c_1{S}_c+c_1{S}_d+c_2{S}_e+\frac{1}{4}(S_b\\ +S_c+S_d+S_e)]+(c_1+c_2+\frac{1}{2})\cdot (2U_{N^{\prime }N}-U_{dc})\end{array}\right]$$

(4)

$$\left[\begin{array}{l}{U}_{\beta 1}\\ U_{\beta 3}\end{array}\right]=\frac{2}{5}\left[\begin{array}{l}{U}_{dc}(s_1{S}_b+s_2{S}_c-s_2{S}_d-s_1{S}_e)\\ U_{dc}(-s_2{S}_b+s_1{S}_c-s_1{S}_d+s_2{S}_e)\end{array}\right]$$

(5)

As revealed in (4) and (5), it can be found that the voltage matrix ofα-axis is rather complex compared with that of β-axis. However, since c₁ = cosα, c₂ = cos2α, s₁ = sinα, s₂ = sin2α, and α = 2π/5 are constant values under normal condition, (4) is difficult to simplify.

3. Generalized Five-Phase SVPWM Fault-Tolerant Control

In this study, the sinusoidal back electromagnetic force distribution method is chosen for the five-phase PM motor system, since the studied motor is designed with the sinusoidal back electromagnetic force. Generalized five-phase SVPWM fault-tolerant control can be realized by the following three-step procedures:

• Simplification of voltage vector matrices and diagrams under single-phase open-circuit fault;
• Voltage vector selection and calculation;
• Analysis of the parameter α₁. α₁ is the electrical angle from α-axis to phase B counterclockwise.

3.1. Voltage Vector Matrices

Assuming single open-circuit fault happens on phase A, the voltage reduced-order transformation matrix derived from (3) can be further expressed as (6):

$$\left[\begin{array}{l}{U}_{\alpha 1}\\ U_{\beta 1}\\ U_{\alpha 3}\\ U_{\beta 3}\\ 0\end{array}\right]=\frac{2}{5}\left[\begin{array}{cccc}\cos {\alpha }_1& \cos {\alpha }_2& \cos {\alpha }_3& \cos {\alpha }_4\\ \sin {\alpha }_1& \sin {\alpha }_2& \sin {\alpha }_3& \sin {\alpha }_4\\ \cos 3{\alpha }_1& \cos 3{\alpha }_2& \cos 3{\alpha }_3& \cos 3{\alpha }_4\\ \sin 3{\alpha }_1& \sin 3{\alpha }_2& \sin 3{\alpha }_3& \sin 3{\alpha }_4\\ 1& 1& 1& 1\end{array}\right]\cdot \left[\begin{array}{l}{U}_b\\ U_c\\ U_d\\ U_e\end{array}\right]$$

(6)

where α₁, α₂, α₃ and α₄ are the electrical angles from α-axis to phase B, phase C, phase D, and phase E counterclockwise.

Substituting (1) and (2) into (6), the decoupled voltage matrix of α-axis can be expressed as follows:

$$\left[\begin{array}{l}{U}_{\alpha 1}\\ U_{\alpha 3}\end{array}\right]=\frac{2}{5}\left[\begin{array}{l}\frac{1}{4}{U}_a+U_{dc}[\cos {\alpha }_1{S}_b+\cos {\alpha }_2{S}_c+\cos {\alpha }_3{S}_d\\ +\cos {\alpha }_4{S}_e+\frac{1}{4}(S_b+S_c+S_d+S_e)]+(\cos {\alpha }_1\\ +\cos {\alpha }_2+\cos {\alpha }_3+\cos {\alpha }_4+1)\cdot (U_{N\prime N}-\frac{1}{2}{U}_{dc})\\ \frac{1}{4}{U}_a+U_{dc}[\cos 3{\alpha }_1{S}_b+\cos 3{\alpha }_2{S}_c+\cos 3{\alpha }_3{S}_d\\ +\cos 3{\alpha }_4{S}_e+\frac{1}{4}(S_b+S_c+S_d+S_e)]+(\cos 3{\alpha }_1\\ +\cos 3{\alpha }_2+\cos 3{\alpha }_3+\cos 3{\alpha }_4+1)\cdot (U_{N^{\prime }N}-\frac{1}{2}{U}_{dc})\end{array}\right]$$

(7)

3.2. Matrices Simplification

In this case, in order to keep the magnetomotive force unchanged, shifts of voltage phasors have to be taken into consideration. Hence, values of parameters α₁, α₂, α₃ and α₄ are diverse, which differs from constant value of α = 2π/5. Since the parameters α₁, α₂, α₃ and α₄ lack a fixed relationship, it is still difficult to simplify (7). In order to establish the fixed relationship among α₁, α₂, α₃ and α₄, cosine function rule with electrical angles α_x (0 < α_x < 2π) is depicted as Figure 1. It is benefit to simplify (7) when the sum of α₁ and α₄ is equal to 2π and the sum of α₂ and α₃ is equal to 2π. As shown in Figure 1, cosα₁ is equal to cosα₄ and cosα₂ is equal to cosα₃ in this case. As a result, voltage phasors are symmetrical by α-axis.

Then, the voltage reduced-order transformation matrix can be expressed as (8).

$$\left[\begin{array}{l}{U}_{\alpha 1}\\ U_{\beta 1}\\ U_{\alpha 3}\\ U_{\beta 3}\\ 0\end{array}\right]=\frac{2}{5}\left[\begin{array}{cccc}{c}_1& c_2& c_2& c_1\\ s_1& s_2& -s_2& -s_1\\ c_2& c_1& c_1& c_2\\ -s_2& s_1& -s_1& s_2\\ 1& 1& 1& 1\end{array}\right]\cdot \left[\begin{array}{l}{U}_b\\ U_c\\ U_d\\ U_e\end{array}\right]$$

(8)

where c₁ = cosα₁, c₂ = cosα₂, s₁ = sinα₁, s₂ = sinα₂ and α₁, α₂ satisfy 0 < α₁ < π/2, π/2 < α₂ < π, respectively.

Substituting (1) and (2) into (8), the decoupled voltage matrix of α-axis can be expressed as follows:

$$\left[\begin{array}{l}{U}_{\alpha 1}\\ U_{\alpha 3}\end{array}\right]=\frac{2}{5}\left[\begin{array}{l}\frac{1}{4}{U}_a+U_{dc}[c_1{S}_b+c_2{S}_c+c_2{S}_d+c_1{S}_e+\frac{1}{4}(S_b\\ +S_c+S_d+S_e)]+(c_1+c_2+\frac{1}{2})\cdot (2U_{N^{\prime }N}-U_{dc})\\ \frac{1}{4}{U}_a+U_{dc}[c_2{S}_b+c_1{S}_c+c_1{S}_d+c_2{S}_e+\frac{1}{4}(S_b\\ +S_c+S_d+S_e)]+(c_1+c_2+\frac{1}{2})\cdot (2U_{N^{\prime }N}-U_{dc})\end{array}\right]$$

(9)

In order to further simplify the voltage vector matrix of α-axis, (9) can be divided into two components, as follows:

$$\left[\begin{array}{l}{U}_{\alpha 1\_basic}\\ U_{\alpha 3\_basic}\end{array}\right]=\frac{2}{5}\left[\begin{array}{l}{U}_{dc}(c_1{S}_b+c_2{S}_c+c_2{S}_d+c_1{S}_e)\\ U_{dc}(c_2{S}_b+c_1{S}_c+c_1{S}_d+c_2{S}_e)\end{array}\right]$$

(10)

$$\left[\begin{array}{l}{U}_{\alpha 1\_remaining}\\ U_{\alpha 3\_remaining}\end{array}\right]=\frac{2}{5}\left[\begin{array}{l}\frac{1}{4}[U_a+U_{dc}(S_b+S_c+S_d+S_e)]\\ +(c_1+c_2+\frac{1}{2})\cdot (2U_{N^{\prime }N}-U_{dc})\\ \frac{1}{4}[U_a+U_{dc}(S_b+S_c+S_d+S_e)]\\ +(c_1+c_2+\frac{1}{2})\cdot (2U_{N\prime N}-U_{dc})\end{array}\right]$$

(11)

where U_α1__basic and U_α1__remaining are termed as the basic and remaining components of the voltage vector matrix of α-axis in the fundamental plane, respectively. Similarly, the parameters U_α3__basic and U_α3__remaining are the basic and remaining components in the harmonic plane.

Then, different values of α₁ and α₂ are substituted into (10) and (11). When the values of α₁ and α₂ are satisfied with (12), voltage phasors are symmetrical by β-axis and (13) is obtained in the form of equality.

$${\alpha }_2=\pi -{\alpha }_1$$

(12)

$$c_1+c_2=\cos {\alpha }_1+\cos (\pi -{\alpha }_1)=0$$

(13)

Also, it can be found that (11) is simplified to zero and (9) is equal to (10). Thus, the voltage matrix on α₁-β₁ and α₃-β₃ spaces can be simplified as:

$$\left[\begin{array}{l}{U}_{\alpha 1}\\ U_{\beta 1}\\ U_{\alpha 3}\\ U_{\beta 3}\end{array}\right]=\frac{2}{5}\left[\begin{array}{l}{U}_{dc}(c_1{S}_b+c_2{S}_c+c_2{S}_d+c_1{S}_e)\\ U_{dc}(s_1{S}_b+s_2{S}_c-s_2{S}_d-s_1{S}_e)\\ U_{dc}(c_2{S}_b+c_1{S}_c+c_1{S}_d+c_2{S}_e)\\ U_{dc}(-s_2{S}_b+s_1{S}_c-s_1{S}_d+s_2{S}_e)\end{array}\right]$$

(14)

where c₁ = cosα₁, c₂ = cos(π − α₁), s₁ = sinα₁ and s₂ = sin(π − α₁).

As calculated by (14), post-fault voltage vector diagrams of the generalized five-phase SVPWM fault-tolerant control are obtained, as shown in Figure 2.

3.3. Vector Selection and Operation Time Calculation

In order to further investigate the generalized five-phase SVPWM fault-tolerant control, the appropriate voltage vectors will be selected and operation time will be calculated based on the simplified post-fault voltage vector matrices and diagrams. When an open-circuit fault occurs in phase A, S_a is invalid, since the five-phase inverter has been a four-phase one. Besides, there are only 2⁴ = 16 different switching states after fault, while there are 2⁵ = 32 switching states before fault. During fault-tolerant condition, the values of 16 voltage vectors can be derived. It can be calculated by (15) that the amplitudes of voltage vectors U₅ (0101) and U₁₀ (1010) are zero. However, the currents of U₅ and U₁₀ are not zero from phase B to phase E. Thus, both vectors U₅ and U₁₀ are not really zero vectors. Also, they are not suitable to compose the reference vector.

$$U_x=\frac{2}{5}{U}_{dc}\cdot [S_b\cdot e^{j\alpha }+S_c\cdot e^{j(\pi -\alpha )}+S_d\cdot e^{-j(\pi -\alpha )}+S_e\cdot e^{-j\alpha }]$$

(15)

where U_x is the values of voltage vectors. Twelve non-vanishing and two vanishing vectors can be utilized, except for U₅ and U₁₀. Each sector selects three non-vanishing vectors and two vanishing vectors to compose the reference vector. The first and last vector is U₅ or U₁₀, respectively. For example, the reference vector is composed by three non-vanishing vectors U₈ (1000), U₉ (1001) and U₁₃ (1101), and two vanishing vectors U₀ (0000) and U₁₅ (1111) in sector 1. Moreover, assuming actuation durations of vectors U₈, U₉ and U₁₃ are t₁, t₂, and t₃, the relationships among them and reference vector can be given as (16) according to the sine thermo.

$$\frac{U_{ref}}{\sin (\pi -{\alpha }_1)}=\frac{\frac{t_1}{T_s}{U}_8+\frac{t_3}{T_s}{U}_{13}}{\sin \theta }=\frac{\frac{t_2}{T_s}{U}_9}{\sin ({\alpha }_1-\theta )}$$

(16)

where U_ref is the reference vector, T_s is the period of the pulse-width modulation, and θ is the angle between U₉ and U_ref.

3.4. Influence of Parameter α₁ on DC-Bus Voltage Utilization and Harmonic Current Suppression

By utilizing the generalized five-phase SVPWM fault-tolerant control, various SVPWM fault-tolerant control strategies with different values of parameter α₁ can be obtained. However, several previous studies investigate the fault-tolerant control by quoting the idea of current hysteresis band pulse-width modulation fault-tolerant control [19] called the traditional strategy, which the parameter α₁ is fixed to π/5 (36°). Obviously, this analogy has less generality.

On one hand, since the cost of DC power supply with a large capacity is expensive, low DC-bus voltage utilization is lack of economy in motor drive system. However, it is difficult for fault-tolerant control to maintain voltage utilization with fewer irregular vectors compared with normal control. In detail, vector diagrams of the five-phase SVPWM strategy have uniform sector size and cover three different sizes of non-vanishing vector: 0.6472U_dc, 0.4U_dc and 0.2472U_dc. As revealed in [20], voltage utilization can be represented by the inscribed circle radius of regular decagon for the SVPWM strategy, as shown in Figure 3a. When the adjacent four vectors SVPWM strategy is adopted, the maximum value of voltage utilization U_max is equal to 0.5257U_dc. However, vector diagram of the generalized five-phase SVPWM fault-tolerant control strategy is probably more irregular than that of the normal control strategy which has chaotic sectors. Similarly, as shown in Figure 3b, voltage utilization can be represented by the inscribed circle radius of diamond for the generalized fault-tolerant control strategy.

On the other hand, since the harmonics increase copper loss and decrease efficiency, specific analysis process of harmonics suppression is presented on post-fault voltage vector diagrams, as shown in Figure 4. Differing from vector diagrams on the SVPWM control strategy, not all vectors on α₁-β₁ space can be mapped to the α₃-β₃ space on fault-tolerant control strategies. For example, although U₃ and U₁₂ are utilized to compose reference vector on the α₁-β₁ space, they are absent from the α₃-β₃ space. As a result, there are only ten vectors remained except for U₀, U₃, U₅, U₁₀, U₁₂ and U₁₅.

Harmonic suppression is influenced by two parts. Firstly, applying times of vectors are analyzed. As shown in Figure 4b, third-harmonic component of voltage can be divided into α-axis and β-axis components, respectively. In the following discussion, Sector 1 is taken as an example. Since the amplitudes of voltage vectors U₈ and U₁₃ are equal, the β-axis component of third-harmonic voltage can be eliminated when t₁ is equal to t₃ as follows:

$$t_1=t_3$$

(17)

Although the β-axis component of third-harmonic voltage can be suppressed by appropriate vector applying time, the α-axis component of the third-harmonic voltage is difficult to suppress using (17) on α₃-β₃ space [21]. Secondly, locations of vectors are further analyzed. U₉ should be as short as possible and the angle between U₈ and β-axis should be as narrow as possible.

Based on the rules of voltage utilization calculation and harmonic current suppression above, different values of the parameter α₁ are introduced into voltage vector diagrams. The relationships of voltage utilization and third-harmonic component with different values of the parameter α₁ (0 < α₁ < π/2(90°)) can be drawn, as shown in Figure 5. It can be observed that the parameter α₁ has a great impact on voltage utilization and harmonics suppression. When the angle of α₁ is close to π/4 (45°), it is benefit to improve voltage utilization. In addition, when the angle of α₁ is as large as possible, it is benefit to suppress harmonic components. Thus, giving consideration to the two aspects, the α₁ = π/4 (45°) control strategy is select for further investigated.

4. Comparison and Evaluation

According to analysis of the generalized five-phase SVPWM fault-tolerant control strategy and inference of the parameter α₁, the improved SVPWM fault-tolerant control strategy (α₁ = π/4) is derived, which will be compared with the traditional SVPWM fault-tolerant control strategy (α₁ = π/5) in terms of voltage vector diagrams, voltage utilization and harmonics suppression.

4.1. Comparison of Voltage Vectors

Figure 6 presents the pre- and post-fault basic space vectors of the traditional and improved control strategies. As shown in Figure 6a, it can be found that the phasor angles of phases B and E shift π/5, while phases C and D remain the original position. In addition, as shown in Figure 6b, it can be found that all non-fault phasor angles are adjusted. In detail, the phasor angles of phases B and E shift 3π/20, while the phasor angles of phases C and D shift π/20.

As calculated by (14), vector diagrams of the traditional control strategy can be obtained which have chaotic sizes of sectors and cover three different sizes of non-vanishing vector: 0.6472U_dc, 0.4U_dc and 0.4702U_dc. Unlike the cases of the traditional control strategy, vector diagrams of the improved control strategy have uniform size of sectors and cover only two different sizes of non-vanishing vector: 0.5657U_dc and 0.4U_dc.

4.2. Comparison of DC-Bus Voltage Utilization

As described in Figure 7, the maximum values of DC-bus voltage utilization U_max can be represented by the inscribed circle radius of diamond based on five-phase motor with single-phase open-circuit fault. In detail, the maximum value of voltage utilization U_max is equal to 0.38U_dc when the traditional fault-tolerant control strategy is adopted, as shown in Figure 7a. Compared with the traditional fault-tolerant control strategy, the improved fault-tolerant control strategy possesses higher DC-bus voltage utilization, which is up to 0.4U_dc as shown in Figure 7b. Hence, the improved control strategy is superior to the traditional control strategy on DC-bus voltage utilization.

4.3. Comparison of DC-Bus Voltage Utilization

Since third-harmonic take account for the main proportion in total harmonic distortion in the five-phase motor with single-phase open-circuit fault, the third-harmonic component is analyzed as shown in Figure 8, where the third-harmonic component is represented by the length of composed reference vector mapped on the α₃-β₃ space.

In detail, the length of composed reference vector mapped on the α₃-β₃ space can be calculated by (18) when the traditional fault-tolerant control strategy is adopted as shown in Figure 8a. Also, as calculated by (19), the improved fault-tolerant control strategy offers lower third-harmonic component as compared with the traditional fault-tolerant control strategy, as shown in Figure 8b.

$$U_{\alpha 3\_tra\_ref}=0.6472U_{dc}+0.3236U_{dc}\cdot 2=1.2944U_{dc}$$

(18)

$$U_{\alpha 3\_imp\_ref}=0.5657U_{dc}+0.2829U_{dc}\cdot 2=1.1314U_{dc}$$

(19)

where U_{α3_tra_ref} and U_{α3_imp_ref} are termed as the length of composed reference vector mapped in the α₃-β₃ space in sector 1 with the traditional and improved control strategies, respectively. Clearly, the result of (19) is smaller than that of (18). It can be verified that the results above can be transplanted to all sectors. Hence, the improved control strategy is superior to the traditional control strategy on harmonic suppression.

5. Simulation and Experimental Verification

5.1. Simulation

As shown in Figure 9, the control system for a five-phase PM motor is established, where the fault-tolerant SVPWM generator is the core of the system for fault tolerant operation. The main simulation parameters are listed in

Table 1. Motor drive parameters.

Symbol	Simulation Parameters/Units	Values
Npm	Number of permanent magnet pole pairs	21
Is	Stator current rms/A	10
n	Rotating speed/(r/min)	600
Te	Average torque/(Nm)	38
Udc	DC-bus voltage/(V)	159

.

Figure 10, Figure 11, Figure 12 and Figure 13 compare the simulated results of the traditional (α₁ = π/5) and improved (α₁ = π/4) control strategies from normal SVPWM to fault-tolerant SVPWM. After open-circuit fault occurs at 0.1 s on phase A, the current of phase A becomes 0 A suddenly. Firstly, torque performances of two fault-tolerant control strategies are evaluated, as shown in Figure 10. It can be found that torque ripple of the improved control strategy is relatively lower than that of the traditional control strategy. Secondly, current waveforms of two fault-tolerant control strategies are compared, as shown in Figure 11. Obviously, it can be seen that the amplitudes of phase currents on both fault-tolerant control strategies increase. Thirdly, as shown in Figure 12, by Fast Fourier Transform analysis of simulated fault-tolerant current waveforms, it is shown that the amplitudes of fundamental waves with both fault-tolerant control strategies are similar. Also, it should be noted that the total harmonic distortion of the improved control strategy is lower than the traditional one. Finally, the DC-bus voltage utilizations with the two fault-tolerant control strategies are presented in Figure 13. The inner trace and the middle circle represent the output voltage of the motor under the fault-tolerant and normal condition, where the outer circle reflects on DC-bus voltage. Hence, it can be concluded that the improved control strategy can offer good performance on better harmonics suppression and higher voltage utilization than the traditional control strategy based on the simulated results.

5.2. Experimental Results

In order to further verify the theoretical results, a five-phase PM motor drive with a type of TMS320F2812-based digital signal processor is built, as shown in Figure 14. Experiments are carried out under non-rated condition limited by maximum capacity of inverter and torque sensor. During the experiment, the reference rotating speed and the initial load are set to 120 r/min and 5.6 Nm. Also, the DC-bus voltage is set as 27 V.

Firstly, measured torque and currents of the motor with normal control strategy, traditional (α₁ = π/5), and improved (α₁ = π/4) fault-tolerant control strategies for phase A being open-circuit fault are compared, as shown in Figure 15. It can be observed that the motor with fault remains almost the same average torque as that of normal control strategy by utilizing the generalized five-phase SVPWM fault-tolerant strategy, albeit with increasing a few torque ripples and phase current amplitudes. Hence, it can be verified that the generalized strategy is beneficial for the motor drive to offer high fault-tolerant capability. As presented in simulation results, phases C and E suffer from higher total harmonic distortion than other phases when the fault-tolerant control strategy are applied. Thus, the two phase currents are analyzed in experiments. Figure 15b,c gives the measured currents of phases C and E. It can be seen that with the improved (α₁ = π/4) control strategy, phase current harmonics are reduced as compared to that of the traditional (α₁ = π/5) control strategy.

Secondly, by Fast Fourier Transform analysis of measured fault-tolerant current waveforms of phases C and E, it can be clearly verified that total harmonic distortion of the phase current of the improved control strategy is lower than the traditional control strategy, as shown in Figure 16. Besides, the amplitudes of fundamental waves of the two fault-tolerant control strategies are similar.

Finally, the voltage utilizations of the traditional and improved fault-tolerant control strategies are measured, as shown in Figure 17. It can be found that the improved control strategy offers higher voltage utilization than the traditional control strategy.

6. Conclusions

In this paper, an improved SVPWM fault-tolerant control strategy for a five-phase PM motor has proposed to deal with the loss of one phase. Compared with the traditional matrices analysis method with the fixed α₁ = π/5, the matrices analysis method in this paper effectively extend the feasible matrices under the fault tolerant condition by adjusting the parameter α₁. Then, generalized five-phase SVPWM fault-tolerant control has deduced and analysed, which yields a greatly increased capacity to enhance fault-tolerant performances of the motor. Also, various SVPWM fault-tolerant control strategies with different values of the parameter α₁ can be deduced. It is found that the parameter α₁ has a great impact on harmonic current suppression and DC-bus voltage utilization. Then, giving consideration to both two aspects of the harmonic current suppression and DC-bus voltage utilization, an improved fault-tolerant control strategy with α₁ = π/4 has been proposed. In addition, the improved control strategy (α₁ = π/4) has been compared with the traditional control strategy (α₁ = π/5). It is shown that the improved control strategy offers lower current harmonics and higher voltage utilization. Both simulation and experimental results have verified the proposed control strategy. In the future, new fault tolerant control will be considered to further reduce current harmonics and improve efficiency of the motor system.