A Harmonic Suppression SVPWM Strategy for the Asymmetric Six-Phase Motor Fed by Two-Level Six-Phase VSI Operating in the Overmodulation Region

Abstract

For the asymmetric six-phase motor fed by a two-level six-phase voltage source inverter (VSI), when the modulation index is less than 0.5774, the motor operates in the linear region, and the harmonics in the Z₁ − Z₂ subspace can be suppressed to zero. When the modulation index is beyond 0.5774 and less than 0.6221, the motor operates in overmodulation regions, and the harmonics in the Z₁ − Z₂ subspace cannot be suppressed to zero. To minimize the harmonics in the Z₁ − Z₂ subspace for these regions, a harmonic suppression strategy, namely, harmonic suppression overmodulation strategy (HSOS), is proposed in this paper. The lower-order harmonics of the Z₁ − Z₂ subspace, namely, the 5th, 7th, 17th and 19th, are considered. Compared with the traditional four-vector overmodulation strategy (TFOS), the content of the 5th harmonic is reduced by about 20%, and the total harmonic distortion (THD_Z1Z2) of these four kinds of harmonics is reduced by about 21% in the proposed HSOS. Finally, the simulation and experiment are carried out to verify the effectiveness of the proposed strategy.

1. Introduction

Compared with traditional three-phase machines, multi-phase machines have many outstanding advantages, such as low-voltage high-power output, less torque ripple, favourable fault tolerance characteristics and higher reliability at the system level. They are widely used in various applications, such as electric locomotive traction, shipboard power propulsion and wind power generation [1,2]. The asymmetric six-phase motor in which two sets of three-phase windings are shifted by 30 electrical degrees is widely studied [3].

In recent decades, various technologies of the motor have been quickly developed. Some research is on the controller to improve the mechanical performance of the motor [4,5]. A controller of DC-link voltage of the three-phase PWM rectifier has been proposed in [4]. In this controller, an Artificial Bee Colony (ABC) optimization method is used to optimize the type-2 fuzzy neural network (T2FNN) controller. As a result, the proposed controller has a better dynamic response than the traditional T2FNN controller. Based on the proportional + derivative type-2 fuzzy neural network (PD-T2FNN), a controller is proposed in [5] for DC-link voltage control of the PWM rectifier. Compared with traditional PD, proportional + integral, and T2FNN controllers, the proposed controller has superior performance in the transient and steady-state responses under various working conditions. The pulse width modulation (PWM) strategy is another research hotspot for motor drive systems, which directly influences the inverter system efficiency and the quality of the output waveform [6,7]. The overmodulation PWM strategy is widely applied in electric vehicles, mining equipment and other motor drive fields. For example, an overmodulation control strategy has been used in the medium-speed range for motor drive control of the TOYOTA hybrid system II (THS II), and the output of the shaft torque has been increased by a maximum of approximately 30% in this range [8]. For a single-phase inverter operating in the overmodulation region, an improved SPWM method is proposed in [9] to eliminate the third harmonic. Thus, the total harmonic distortion is reduced and current quality is improved. For three-phase motors, an overmodulation space vector pulse width modulation (SVPWM) strategy based on Fourier series expansion of the reference voltage is proposed in [10], and maximum utilization of the dc-link voltage can be achieved. However, the calculation amount is large. Another SVPWM strategy based on the superposition principle for a three-phase motor operating in the overmodulation range is developed in [11]. This strategy does not need to calculate the control angle or the hold angle, and it is easy to implement digitally. However, the voltage harmonic components in the overmodulation region are large. Modulation functions of carrier-based PWM which are equivalent to this SVPWM strategy are investigated in [12]. However, these modulation functions are based on the superposition principle and are not universal. Ref. [13] has proposed a method that combines Space-Vector Modulated Direct Torque Control (DTC-SVM) and conventional DTC. In the linear region, the DTC-SVM is used, which provides low THD and low torque ripple, while in the overmodulation region, the conventional DTC is applied. The transition between these two methods is smooth, and the DC bus voltage is fully employed. In [14], an SVPWM strategy exploiting the Lagrangian method to minimize the harmonic voltage in the overmodulation range is proposed for a five-phase voltage source inverter (VSI). However, this method adopts suboptimal solutions under specific conditions instead of the optimal solution. The comprehensive relationship between the carrier-based PWM and SVPWM techniques for a five-phase motor operating in the linear modulation range is explained in [15]. For six-phase motors, there are two major models [16]: the double synchronous reference frame (double d-q) model [17,18] and the vector space decomposition (VSD) model [19,20]. In the double d-q model, the SVPWM strategy is similar to the traditional three-phase VSI. In [17], the proposed current control scheme based on the double d-q model uses four identical PI regulators for all current loops. In this way, the two sets of three-phase stator currents are independently controlled and kept balanced. However, the decoupling voltage required by this method is complex. An indirect field-oriented control (FOC) scheme for a six-phase induction machine is proposed in [18]. The unbalanced current between the two sets of three-phase windings is eliminated, and this scheme is applicable to a six-phase machine with any arbitrary angle of displacement between the two sets. However, many sensors are used in this method, which increases the cost of industrial application. For the VSD model, two-vector SVPWM is the simplest modulation strategy, which will generate many 6n ± 1 (n = 1, 3, 5, …) harmonic voltage components [19]. The commonly used SVPWM strategies are four-vector modulation techniques. An SVPWM strategy synthesizing the reference voltage vector using four adjacent large voltage vectors (4L) is proposed in [20], which can keep the harmonics in the Z₁ − Z₂ subspace at a minimum level within the linear modulation range. However, the harmonic distortion factor of this strategy has not been studied. Based on [20], several new discontinuous SVPWM strategies for six-phase VSIs have been proposed, and the harmonic current of each strategy has been completely studied [21]. The results demonstrate that the performance of these strategies is affected by the parameter k_σxy, which is defined as the ratio of the total (α − β) referred-to-stator leakage inductance to the (Z₁ − Z₂) leakage inductance. However, only the 4L strategy has been analysed, not other strategies; for example, the two large vectors plus two medium vectors (2L + 2M) strategy. The double zero-sequence injection PWM method is studied for the asymmetrical six-phase motors in [22]. This method can reduce the calculation amount, but the flux harmonic distortion factor (HDF) in the Z₁ − Z₂ subspace is higher than that in the SVPWM strategies. Using the SVPWM method, a zero CMV (ZCMV) strategy is proposed in [23] to eliminate the common mode voltage (CMV) for asymmetric six-phase motors. In this strategy, the turn-on/off moment of each phase is shifted, and the peak value of total CMV is suppressed to 0 in theory. To reduce the CMV for asymmetric six-phase motors, three SVPWM schemes, namely, CMV2, CMV3, and DCMV3, are proposed in [24]. Although their CMV suppression effects are the same, their harmonic performances are different. The DCMV3 has better harmonic performance in the high modulation region for leakage coupling coefficients ξ = 1 to 2. If ξ = 6.2, the CMV2 performs better in the higher modulation region, while the DCMV3 generates superior quality waveforms in the lower modulation region. A modified SVPWM technique that divides the α − β subspace into 24 sectors is proposed in [25]. It provides lower current THD, but it only focuses on the linear range and does not involve the overmodulation range.

When a motor operates in the overmodulation region, low-order voltage harmonic components in the Z₁ − Z₂ subspace are unavoidable. However, harmonic impedance in the Z₁ − Z₂ subspace is only composed of the stator resistance and leakage inductance; thus, even low-level harmonic voltages can induce large harmonic currents. Therefore, it is necessary to suppress the harmonic components in the Z₁ − Z₂ subspace. In [26], a general n-phase SVPWM technique is proposed for overmodulation regions, which forces the zero-vector duty cycle δ₀ to zero for the entire sector, and it effectively suppresses voltage harmonic components in the Z₁ − Z₂ subspace. However, this technique is only applicable to odd n. An SVPWM strategy for six-phase voltage source-inverter-fed six-phase split-phase induction motors is presented in [27]. The approach is based on three-phase space vector modulators that can be operated in both the linear and overmodulation regions. However, it does not consider harmonic suppression in the Z₁ − Z₂ subspace. Through extending the three-phase VSI overmodulation method to the six-phase VSIs, Ref. [28] has proposed an SVPWM strategy that divides the overmodulation regions into two sections. In section I, the amplitude of reference voltage is proportional to the modulation index. In section II, due to the phase correction algorithm, the distortion of phase voltage is reduced. Although the traditional four-vector overmodulation strategy (TFOS) [29] can suppress the harmonic voltage components in the overmodulation range, the harmonic component content is still quite large and will be compared with the proposed strategy.

The control objective of this paper is to synthesize the reference voltage in the α − β subspace and simultaneously minimize the harmonic components in the Z₁ − Z₂ subspace when a six-phase asymmetric motor fed by two-level six-phase VSI operates in the overmodulation range. A novel SVPWM strategy, namely, harmonic suppression overmodulation strategy (HSOS), is proposed in this paper, which can effectively achieve this objective. The main innovation is that the optimization models consisting of an objective function and constraint conditions are built, and the external point method is adopted to minimize the harmonics in the Z₁ − Z₂ subspace. The main contribution is that the proposed HSOS can reduce the content of the 5th harmonic by about 20% and the THD_Z1Z2 of the four harmonics (namely, the 5th, 7th, 17th and 19th) by about 21% when it is compared with the TFOS. The rest of this paper is organized as follows: The basic theory of the six-phase two-level VSI model is described in Section 2. In Section 3, the existing method TFOS is briefly introduced at first, and then the proposed HSOS is deduced in detail. The simulation and experiment are carried out in Section 4 and Section 5, respectively. In these two sections, harmonic content of line voltage and current in these two strategies are compared, and the validity of the proposed HSOS is verified.

2. The Six-Phase Two-Level VSI Model

The typical topology of an asymmetric six-phase motor with two isolated neutral points fed by six-phase two-level VSI is shown in Figure 1.

It is known from vector space decomposition (VSD) theory that the 64 switching states of a six-phase VSI can be mapped into three mutually perpendicular subspaces, i.e., the α − β, Z₁ − Z₂ and O₁ − O₂ subspaces. The α − β subspace mainly contains the fundamental and 12K ± 1 (K = 1, 2, 3, …) harmonics, and the Z₁−Z₂ subspace mainly contains the 6K ± 1 (K = 1, 3, 5, …) harmonics. In the O₁ − O₂ subspace, the 6K ± 3 (K = 1, 3, 5, …) harmonics are contained. However, when the neutral points N and N’ are isolated from each other, all the voltage vectors of the O₁ − O₂ subspace are mapped to the origin.

The voltage vectors corresponding to each switching state in the α − β subspace and Z₁ − Z₂ subspace are determined by (1) and (2), respectively.

$$V_{\alpha \beta }=\frac{1}{3}{U}_{dc}\left(S_A+S_B{a}^4+S_C{a}^8+S_{X}a+S_Y{a}^5+S_Z{a}^9\right)$$

(1)

$$V_{Z_1{Z}_2}=\frac{1}{3}{U}_{dc}\left(S_A+S_B{a}^8+S_C{a}^4+S_X{a}^5+S_{Y}a+S_Z{a}^9\right)$$

(2)

where $a=e^{j{30}^{°}}$, and S_K (K = A, B, C, X, Y, Z) is the switching state of each branch of the inverter. S_K = 1 means that the top transistor of leg K is ON and the bottom is OFF. The opposite is true when S_K = 0. The corresponding switching states of the six branches constitute a set of binary numbers, which, from high to low, are A, B, C, X, Y and Z. Each set of binary numbers corresponds to an octal number that labels a voltage vector. Therefore, voltage vector diagrams of the α − β subspace and the Z₁ − Z₂ subspace can be obtained, as shown in Figure 2. Each subspace contains 60 effective voltage vectors and 4 zero-voltage vectors. These 60 effective voltage vectors can be classified into four groups according to their amplitudes, i.e., large vector V_L, medium vector V_M, small vector V_S and extra small vector V_E. The amplitude of each group of vectors is shown in

Table 1. Amplitude of space vector for six-phase VSI.

Voltage Vector	Amplitude
VL	0.644 Udc
VM	0.471 Udc
VS	0.333 Udc
VE	0.173 Udc

. As mentioned previously, the inverter output voltage vector in its natural reference frame can be mapped into three orthogonal planes, namely, the α − β, Z₁ − Z₂ and O₁ − O₂ subspaces, by (3).

$${\left[V_{\alpha }{V}_{\beta }{V}_{Z_1}{V}_{Z_2}{V}_{O_1}{V}_{O_2}\right]}^T=T_6^{-1}\cdot {\left[V_A{V}_B{V}_C{V}_X{V}_Y{V}_Z\right]}^T$$

(3)

where

$$T_6^{-1}=\frac{1}{3}\left[\begin{array}{cccccc}1& -\frac{1}{2}& -\frac{1}{2}& \frac{\sqrt{3}}{2}& -\frac{\sqrt{3}}{2}& 0\\ 0& \frac{\sqrt{3}}{2}& -\frac{\sqrt{3}}{2}& \frac{1}{2}& \frac{1}{2}& -1\\ 1& -\frac{1}{2}& -\frac{1}{2}& -\frac{\sqrt{3}}{2}& \frac{\sqrt{3}}{2}& 0\\ 0& -\frac{\sqrt{3}}{2}& \frac{\sqrt{3}}{2}& \frac{1}{2}& \frac{1}{2}& -1\\ 1& 1& 1& 0& 0& 0\\ 0& 0& 0& 1& 1& 1\end{array}\right]$$

(4)

The modulation index m is defined as the ratio of the phase voltage fundamental amplitude |V₁| and the dc-link voltage U_dc:

$$m=\frac{\left|V_1\right|}{U_{dc}}$$

(5)

3. The Proposed HSOS

3.1. The Traditional Four-Vector Overmodulation Strategy (TFOS)

For the four-vector SVPWM strategy, when the modulation index is less than 0.5774, harmonic voltage components in the Z₁ − Z₂ subspace can be suppressed to zero. However, when it is between 0.5774 and 0.6221, the harmonic components in the Z₁ − Z₂ subspace cannot be suppressed to zero [29]. The TFOS is one of the strategies used in this overmodulation range. According to this strategy, the reference voltage vector V_r* is synthesized from four adjacent large voltage vectors, and their duty cycles change with the angle change of V_r*. To reduce the harmonics, V_r* is regulated to fall on the boundary of a regular dodecagon by adjusting the value of parameter a, as shown in (6). Figure 3 shows the TFOS diagram of voltage vectors where two intermediate vectors V₁ and V₂ are defined. Vector V₁ is synthesized by V_a, V_b and V_c, and its position angle is the same as that of V_b. The duty cycle of V₁ is defined as η₁, and the duty cycle of V_b is a·η₁. Vector V₂ is synthesized from V_b, V_c and V_d, and its position angle is the same as that of V_c, as shown in Figure 3. The duty cycle of V₂ is defined as η₂, and the duty cycle of V_c is a·η_2. According to the volt-second balance rule, (7) can be obtained. Therefore, the duty cycle of each vector can be derived, as shown in (8).

$$a=12\frac{\left|V_r{}^{*}\right|}{U_{dc}}\cos \theta -2\sqrt{3}-3$$

(6)

$$\{\begin{array}{l}{\eta }_2=\frac{\left(\sqrt{2}+\sqrt{6}\right)}{2\cos \theta }\sin \left(30°-\psi \right)\\ {\eta }_1=\frac{\left(\sqrt{2}+\sqrt{6}\right)}{2\cos \theta }\sin \psi \end{array}$$

(7)

$$\{\begin{array}{l}{\eta }_a=0.5\left(1-a\right){\eta }_1\\ {\eta }_b=0.5\left(1-a\right){\eta }_2+a{\eta }_1\\ {\eta }_c=a{\eta }_2+0.5\left(1-a\right){\eta }_1\\ {\eta }_d=0.5\left(1-a\right){\eta }_2\end{array}$$

(8)

It is worth noting that this strategy suppresses the harmonics only to a certain extent, and the harmonic components are still relatively large.

3.2. The Proposed HSOS Strategy

In the proposed HSOS strategy, the same four adjacent large voltage vectors are exploited to synthesize V_r*, and the α − β subspace is also divided into 12 sectors, as shown in Figure 2a. Assuming that V_r* is located in the first sector, its neighbouring four large voltage vectors are V_a, V_b, V_c and V_d, and this sector is divided into four regions (Z₁, Z₂, Z₃ and Z₄), as shown in Figure 4.

Each region will be studied in detail as follows, where the duty cycle of every vector and the borders of each region are deduced.

3.2.1. Modulation in Linear Region Z₁

According to the volt-second balance rule, (9) can be obtained.

$$\left[\begin{array}{cccc}{V}_{a\alpha }& V_{b\alpha }& V_{c\alpha }& V_{d\alpha }\\ V_{a\beta }& V_{b\beta }& V_{c\beta }& V_{d\beta }\\ V_{a{z}_1}& V_{b{z}_1}& V_{c{z}_1}& V_{d{z}_1}\\ V_{a{z}_2}& V_{b{z}_1}& V_{c{z}_2}& V_{d{z}_2}\end{array}\right]\cdot \left[\begin{array}{c}{\eta }_a\\ {\eta }_b\\ {\eta }_c\\ {\eta }_d\end{array}\right]=\left[\begin{array}{c}{V}_{\alpha }^{\ast }\\ V_{\beta }^{\ast }\\ V_{z_1}\\ V_{z_2}\end{array}\right]$$

(9)

When the modulation index is less than 0.5774; the harmonic voltage components in the Z₁ − Z₂ subspace can be suppressed to zero, and V_r* is located in the Z₁ region of Figure 4. In this region, harmonic voltage components can be expressed as (10).

$$\{\begin{array}{l}{V}_{z_1}=0\\ V_{z_2}=0\end{array}$$

(10)

Substituting (10) into (9), the duty cycle of each effective vector can be expressed as (11).

$${\left[\begin{array}{cccc}{\eta }_a& {\eta }_b& {\eta }_c& {\eta }_d\end{array}\right]}^T=C_1\cdot {\left[\begin{array}{cccc}{V}_{\alpha }^{\ast }& V_{\beta }^{\ast }& 0& 0\end{array}\right]}^T$$

(11)

where

$$\begin{array}{l}{C}_1={\left[\begin{array}{cccc}{V}_{a\alpha }& V_{b\alpha }& V_{c\alpha }& V_{d\alpha }\\ V_{a\beta }& V_{b\beta }& V_{c\beta }& V_{d\beta }\\ V_{a{z}_1}& V_{b{z}_1}& V_{c{z}_1}& V_{d{z}_1}\\ V_{a{z}_2}& V_{b{z}_1}& V_{c{z}_2}& V_{d{z}_2}\end{array}\right]}^{-1}\\ =\left[\begin{array}{cccc}0.2321& 0.866& -3.2321& -0.866\\ 0.6340& 0.6340& 2.3660& 2.3660\\ 0.6340& -0.6340& 2.3660& -2.3660\\ 0.2321& -0.866& -3.2321& 0.866\end{array}\right]\end{array}$$

(12)

Therefore, the duty cycle of the zero-voltage vector is obtained, as shown in (13).

$${\eta }_0=1-\left({\eta }_a+{\eta }_b+{\eta }_c+{\eta }_d\right)$$

(13)

On the border of the linear region, the duty cycle of zero vector is zero, and both values of the harmonic voltages V_z1 and V_z2 are also zero. In addition, the synthesized voltage mapping into the α and β axes must equal to $V_{\alpha }^{\ast }$ and $V_{\beta }^{\ast }$, respectively. So, there are five constraint equations and four duty cycles which are the functions of $V_{\alpha }^{\ast }$ or $V_{\beta }^{\ast }$. The coefficient matrix consists of five rows and four columns, which is not convenient to calculate. Therefore, those five constraint equations are divided into two parts, shown as (14) and (15).

$$\left[\begin{array}{cccc}1& 1& 1& 1\\ V_{a\beta }& V_{b\beta }& V_{c\beta }& V_{d\beta }\\ V_{a{z}_1}& V_{b{z}_1}& V_{c{z}_1}& V_{d{z}_1}\\ V_{a{z}_2}& V_{b21}& V_{c{z}_2}& V_{d{z}_2}\end{array}\right]\cdot \left[\begin{array}{c}{\eta }_a\\ {\eta }_b\\ {\eta }_c\\ {\eta }_d\end{array}\right]=\left[\begin{array}{c}1\\ V_{\beta }^{\ast }\\ 0\\ 0\end{array}\right]$$

(14)

According to (14), the duty cycles which are functions of $V_{\beta }^{\ast }$ are obtained. Then, the constraint condition of $V_{\alpha }^{\ast }$ is realized by (15).

$$V_{\alpha }^{\ast }=\left[\begin{array}{cccc}{V}_{a\alpha }& V_{b\alpha }& V_{c\alpha }& V_{d\alpha }\end{array}\right]\cdot {\left[\begin{array}{cccc}{\eta }_a& {\eta }_b& {\eta }_c& {\eta }_d\end{array}\right]}^T$$

(15)

$V_{\alpha }^{\ast }$ is a function of $V_{\beta }^{\ast }$, which consists of the linear region border function. Substituting the specific value of each vector into (15), the border function V_αL12 can be expressed as

$$L_{12}:V_{\alpha L_{12}}=0.5774$$

(16)

Line L₁₂ intersects with the lines of vectors V_c and V_b at points P₁ and P₂, respectively.

$$P_1:\left\{V_{\alpha P_1},V_{\beta P_1}\right\}=\left\{0.5774,-0.1547\right\}$$

(17)

$$P_2:\left\{V_{\alpha P_2},V_{\beta P_2}\right\}=\left\{0.5774,0.1547\right\}$$

(18)

3.2.2. Overmodulation Region

When the value of the modulation index is greater than 0.5774, the harmonic voltage components in the Z₁ − Z₂ subspace cannot be suppressed to zero. Under this condition, the harmonics should be suppressed as much as possible. Therefore, an optimization model that consists of an objective function and constraint conditions is constructed, as shown in (19) and (20). The objective function aims at minimizing the square of the magnitude of the actually synthesized voltage vector in the Z₁ − Z₂ subspace.

$$\underset{\begin{array}{l}{\eta }_a{\eta }_b\\ {\eta }_c{\eta }_d\end{array}}{{\displaystyle \min }}f\left({\eta }_a,{\eta }_b,{\eta }_c,{\eta }_d\right)=\left(V_{z_1}^2+V_{z_2}^2\right)$$

(19)

$$s.t.\{\begin{array}{l}{V}_{a\alpha }\cdot {\eta }_a+V_{b\alpha }\cdot {\eta }_b+V_{c\alpha }\cdot {\eta }_c+V_{d\alpha }\cdot {\eta }_d=V_{\alpha }^{\ast }\\ V_{a\beta }\cdot {\eta }_a+V_{b\beta }\cdot {\eta }_b+V_{c\beta }\cdot {\eta }_c+V_{d\beta }\cdot {\eta }_d=V_{\beta }^{\ast }\\ {\eta }_a+{\eta }_b+{\eta }_c+{\eta }_d\le 1\\ -{\eta }_a\le 0\\ -{\eta }_b\le 0\\ -{\eta }_c\le 0\\ -{\eta }_d\le 0\end{array}$$

(20)

where η_a is the duty cycle of vector V_a, V_aβ is the projection of voltage vector V_a on the β axis, and the other symbols are similar. V_α* and V_β* are the projections of V_r* on the α and β axes, respectively. V_z1 and V_z2 can be expressed as

$$\{\begin{array}{l}{V}_{z_1}=V_{a{z}_1}\cdot {\eta }_a+V_{b{z}_1}\cdot {\eta }_b+V_{c{z}_1}\cdot {\eta }_c+V_{d{z}_1}\cdot {\eta }_d\\ V_{z_2}=V_{a{z}_2}\cdot {\eta }_a+V_{b{z}_2}\cdot {\eta }_b+V_{c{z}_2}\cdot {\eta }_c+V_{d{z}_2}\cdot {\eta }_d\end{array}$$

(21)

The formulated problem has equality and inequality constraints. To solve such a restricted optimization problem, the inequality constraints are relaxed, and only the equality constraints are solved first. The external point method [30] is adopted to convert the equality constraints into an unconstrained problem. Therefore, the optimal value of the objective function can be obtained. The corresponding inequality constraints will be activated when the solution violates the constraints. Consequently, the sector will be divided into different regions given by borders under the constraints. As a result, in different regions of each sector, the vectors used are different, so the penalty function is also different. Each region will be specifically described as follows.

Modulation in Region Z₂

In the Z₂ region, the zero-voltage vectors do not modulate V_r*, and thus the first inequality in (20) turns into an equality, i.e., ${\eta }_a+{\eta }_b+{\eta }_c+{\eta }_d=1$. The corresponding penalty function [30] F_Z2 is obtained, which is shown by (22), where the parameter μ is a penalty factor, with a value of 10⁷. The partial derivative for each duty cycle of penalty function F_Z2 is calculated and defined as zero. Thus, (23) is obtained.

$$\begin{array}{l}{F}_{z_2}\left(\eta ,\mu \right)=f\left(\eta \right)+{\displaystyle \sum _{j=1}^3\mu \cdot h_j^2}\left(\eta \right)\\ ={\left(V_{a{z}_1}{\eta }_a+V_{b{z}_1}{\eta }_b+V_{c{z}_1}{\eta }_c+V_{d{z}_1}{\eta }_d\right)}^2+{\left(V_{a{z}_2}{\eta }_a+V_{b{z}_2}{\eta }_b+V_{c{z}_2}{\eta }_c+V_{d{z}_2}{\eta }_d\right)}^2\\ +\mu {\left(V_{a\alpha }{\eta }_a+V_{b\alpha }{\eta }_b+V_{c\alpha }{\eta }_c+V_{d\alpha }{\eta }_d-V_{\alpha }^{*}\right)}^2+\mu {\left(V_{a\beta }{\eta }_a+V_{b\beta }{\eta }_b+V_{c\beta }{\eta }_c+V_{d\beta }{\eta }_d-V_{\beta }^{*}\right)}^2\\ +\mu {\left({\eta }_a+{\eta }_b+{\eta }_c+{\eta }_d-1\right)}^2\end{array}$$

(22)

$$\begin{array}{l}[\begin{array}{cccc}2\left(V_{a{z}_1}^2+V_{a{z}_2}^2+\mu V_{a\alpha }^2+\mu V_{a\beta }^2\right)+2\mu & 2\left(V_{a{z}_1}^{}{V}_{b{z}_1}^{}+V_{a{z}_2}^{}{V}_{b{z}_2}^{}+\mu V_{a\alpha }^{}{V}_{b\alpha }^{}+\mu V_{a\beta }^{}{V}_{b\beta }^{}\right)+2\mu & 2\left(V_{a{z}_1}^{}{V}_{c{z}_1}^{}+V_{a{z}_2}^{}{V}_{c{z}_2}^{}+\mu V_{a\alpha }^{}{V}_{c\alpha }^{}+\mu V_{a\beta }^{}{V}_{c\beta }^{}\right)+2\mu & \\ 2\left(V_{a{z}_1}^{}{V}_{b{z}_1}^{}+V_{a{z}_2}^{}{V}_{b{z}_2}^{}+\mu V_{a\alpha }^{}{V}_{b\alpha }^{}+\mu V_{a\beta }^{}{V}_{b\beta }^{}\right)+2\mu & 2\left(V_{b{z}_1}^2+V_{b{z}_2}^2+\mu V_{b\alpha }^2+\mu V_{b\beta }^2\right)+2\mu & 2\left(V_{b{z}_1}^{}{V}_{c{z}_1}^{}+V_{b{z}_2}^{}{V}_{c{z}_2}^{}+\mu V_{b\alpha }^{}{V}_{c\alpha }^{}+\mu V_{b\beta }^{}{V}_{c\beta }^{}\right)+2\mu & \\ 2\left(V_{a{z}_1}^{}{V}_{c{z}_1}^{}+V_{a{z}_2}^{}{V}_{c{z}_2}^{}+\mu V_{a\alpha }^{}{V}_{c\alpha }^{}+\mu V_{a\beta }^{}{V}_{c\beta }^{}\right)+2\mu & 2\left(V_{c{z}_1}^{}{V}_{b{z}_1}^{}+V_{c{z}_2}^{}{V}_{b{z}_2}^{}+\mu V_{c\alpha }^{}{V}_{b\alpha }^{}+\mu V_{c\beta }^{}{V}_{b\beta }^{}\right)+2\mu & 2\left(V_{c{z}_1}^2+V_{c{z}_2}^2+\mu V_{c\alpha }^2+\mu V_{c\beta }^2\right)+2\mu & \\ 2\left(V_{a{z}_1}^{}{V}_{d{z}_1}^{}+V_{a{z}_2}^{}{V}_{d{z}_2}^{}+\mu V_{a\alpha }^{}{V}_{d\alpha }^{}+\mu V_{a\beta }^{}{V}_{d\beta }^{}\right)+2\mu & 2\left(V_{b{z}_1}^{}{V}_{d{z}_1}^{}+V_{b{z}_2}^{}{V}_{d{z}_2}^{}+\mu V_{b\alpha }^{}{V}_{d\alpha }^{}+\mu V_{b\beta }^{}{V}_{d\beta }^{}\right)+2\mu & 2\left(V_{c{z}_1}^{}{V}_{d{z}_1}^{}+V_{c{z}_2}^{}{V}_{d{z}_2}^{}+\mu V_{c\alpha }^{}{V}_{d\alpha }^{}+\mu V_{c\beta }^{}{V}_{d\beta }^{}\right)+2\mu & \end{array}\\ \begin{array}{c}2\left(V_{a{z}_1}^{}{V}_{d{z}_1}^{}+V_{a{z}_2}^{}{V}_{d{z}_2}^{}+\mu V_{a\alpha }^{}{V}_{d\alpha }^{}+\mu V_{a\beta }^{}{V}_{d\beta }^{}\right)+2\mu \\ 2\left(V_{b{z}_1}^{}{V}_{d{z}_1}^{}+V_{b{z}_2}^{}{V}_{d{z}_2}^{}+\mu V_{b\alpha }^{}{V}_{d\alpha }^{}+\mu V_{b\beta }^{}{V}_{d\beta }^{}\right)+2\mu \\ 2\left(V_{c{z}_1}^{}{V}_{d{z}_1}^{}+V_{c{z}_2}^{}{V}_{d{z}_2}^{}+\mu V_{c\alpha }^{}{V}_{d\alpha }^{}+\mu V_{c\beta }^{}{V}_{d\beta }^{}\right)+2\mu \\ 2\left(V_{d{z}_1}^2+V_{d{z}_2}^2+\mu V_{d\alpha }^2+\mu V_{d\beta }^2\right)+2\mu \end{array}]\cdot \left[\begin{array}{c}{\eta }_a\\ {\eta }_b\\ {\eta }_c\\ {\eta }_d\end{array}\right]=\left[\begin{array}{c}2\mu \left(V_{\alpha }^{*}{V}_{a\alpha }+V_{\beta }^{*}{V}_{a\beta }\right)+2\mu \\ 2\mu \left(V_{\alpha }^{*}{V}_{b\alpha }+V_{\beta }^{*}{V}_{b\beta }\right)+2\mu \\ 2\mu \left(V_{\alpha }^{*}{V}_{c\alpha }+V_{\beta }^{*}{V}_{c\beta }\right)+2\mu \\ 2\mu \left(V_{\alpha }^{*}{V}_{d\alpha }+V_{\beta }^{*}{V}_{d\beta }\right)+2\mu \end{array}\right]\end{array}$$

(23)

Substituting the specific values of these vectors, as shown in Table 1, into (23) and calculating it, the duty cycle of each vector can be expressed as (24).

$$\left[\begin{array}{c}{\eta }_a\\ {\eta }_b\\ {\eta }_c\\ {\eta }_d\end{array}\right]=\left[\begin{array}{ccc}-3& 0.866& 1.866\\ 3& 0.634& -1.366\\ 3& -0.634& -1.366\\ -3& -0.866& 1.866\end{array}\right]\cdot \left[\begin{array}{c}{V}_{\alpha }^{\ast }\\ V_{\beta }^{\ast }\\ 1\end{array}\right]$$

(24)

The values of duty cycles must be positive, which means that they are effective on the border defined by the activated constraints. If the solutions have any negative value, then they are regarded as invalid, and V_r* is outside of the Z₂ region. Therefore, an additional equality constraint must be activated; thus, regions Z₃ and Z₄ are defined.

In order to make sure all duty cycles are positive, η_a, η_b, η_c and η_d in (24) are defined as zero, respectively. Then, four functions are obtained and corresponding lines can be drawn in Figure 4. However, only the lines obtained by ${\eta }_a=0$ and ${\eta }_d=0$ cross the Z₂ region. Therefore, the borders are obtained.

The Z₂ and Z₃ boundary function $V_{\beta L_{23}}$ can be obtained by setting ${\eta }_a=0$ in (24), as shown in (25).

$$V_{\beta L_{23}}=3.4642\cdot V_{\alpha }-2.1547$$

(25)

Line L₂₃ intersects with the ${\alpha }_k$ axis and the line of vector V_c at points P₃ and P₁, respectively.

$$P_3:\left\{V_{\alpha P_3},V_{\beta P_3}\right\}=\left\{0.6220,0\right\}$$

(26)

The Z₂ and Z₄ boundary function $V_{\beta L_{24}}$ can be obtained by setting ${\eta }_d=0$ in (24), as shown in (27).

$$V_{\beta L_{24}}=-3.4642\cdot V_{\alpha }+2.1547$$

(27)

Line L₂₄ intersects with the ${\alpha }_k$ axis and the line of vector V_b at points P₃ and P₂, respectively.

Modulation in Region Z₃

In the Z₃ region, the zero vectors and vector V_a do not modulate V_r*, so the first and second inequalities of (20) turn into equalities, i.e., ${\eta }_a+{\eta }_b+{\eta }_c+{\eta }_d=1$ and $-{\eta }_a=0$. The corresponding penalty function F_Z3 is obtained and shown as (28). Similar to the Z₂ region, the partial derivative for each duty cycle of penalty function F_Z3 is calculated and defined as zero. Then, (29) is obtained.

$$\begin{array}{l}{F}_{z_3}\left(\eta ,\mu \right)=f\left(\eta \right)+{\displaystyle \sum _{j=1}^4\mu \cdot h_j^2}\left(\eta \right)\\ ={\left(V_{a{z}_1}{\eta }_a+V_{b{z}_1}{\eta }_b+V_{c{z}_1}{\eta }_c+V_{d{z}_1}{\eta }_d\right)}^2+{\left(V_{a{z}_2}{\eta }_a+V_{b{z}_2}{\eta }_b+V_{c{z}_2}{\eta }_c+V_{d{z}_2}{\eta }_d\right)}^2\\ +\mu {\left(V_{a\alpha }{\eta }_a+V_{b\alpha }{\eta }_b+V_{c\alpha }{\eta }_c+V_{d\alpha }{\eta }_d-V_{\alpha }^{*}\right)}^2+\mu {\left(V_{a\beta }{\eta }_a+V_{b\beta }{\eta }_b+V_{c\beta }{\eta }_c+V_{d\beta }{\eta }_d-V_{\beta }^{*}\right)}^2\\ +\mu {\left({\eta }_b+{\eta }_c+{\eta }_d-1\right)}^2+\mu {\eta }_{{}_a}^2\end{array}$$

(28)

$$\begin{array}{l}\left[\begin{array}{cccc}2\left(V_{a{z}_1}^2+V_{a{z}_2}^2+\mu V_{a\alpha }^2+\mu V_{a\beta }^2\right)+2\mu & 2\left(V_{a{z}_1}^{}{V}_{b{z}_1}^{}+V_{a{z}_2}^{}{V}_{b{z}_2}^{}+\mu V_{a\alpha }^{}{V}_{b\alpha }^{}+\mu V_{a\beta }^{}{V}_{b\beta }^{}\right)& 2\left(V_{a{z}_1}^{}{V}_{c{z}_1}^{}+V_{a{z}_2}^{}{V}_{c{z}_2}^{}+\mu V_{a\alpha }^{}{V}_{c\alpha }^{}+\mu V_{a\beta }^{}{V}_{c\beta }^{}\right)& \\ 2\left(V_{a{z}_1}^{}{V}_{b{z}_1}^{}+V_{a{z}_2}^{}{V}_{b{z}_2}^{}+\mu V_{a\alpha }^{}{V}_{b\alpha }^{}+\mu V_{a\beta }^{}{V}_{b\beta }^{}\right)& 2\left(V_{b{z}_1}^2+V_{b{z}_2}^2+\mu V_{b\alpha }^2+\mu V_{b\beta }^2\right)+2\mu & 2\left(V_{b{z}_1}^{}{V}_{c{z}_1}^{}+V_{b{z}_2}^{}{V}_{c{z}_2}^{}+\mu V_{b\alpha }^{}{V}_{c\alpha }^{}+\mu V_{b\beta }^{}{V}_{c\beta }^{}\right)+2\mu & \\ 2\left(V_{a{z}_1}^{}{V}_{c{z}_1}^{}+V_{a{z}_2}^{}{V}_{c{z}_2}^{}+\mu V_{a\alpha }^{}{V}_{c\alpha }^{}+\mu V_{a\beta }^{}{V}_{c\beta }^{}\right)& 2\left(V_{c{z}_1}^{}{V}_{b{z}_1}^{}+V_{c{z}_2}^{}{V}_{b{z}_2}^{}+\mu V_{c\alpha }^{}{V}_{b\alpha }^{}+\mu V_{c\beta }^{}{V}_{b\beta }^{}\right)+2\mu & 2\left(V_{c{z}_1}^2+V_{c{z}_2}^2+\mu V_{c\alpha }^2+\mu V_{c\beta }^2\right)+2\mu & \\ 2\left(V_{a{z}_1}^{}{V}_{d{z}_1}^{}+V_{a{z}_2}^{}{V}_{d{z}_2}^{}+\mu V_{a\alpha }^{}{V}_{d\alpha }^{}+\mu V_{a\beta }^{}{V}_{d\beta }^{}\right)& 2\left(V_{b{z}_1}^{}{V}_{d{z}_1}^{}+V_{b{z}_2}^{}{V}_{d{z}_2}^{}+\mu V_{b\alpha }^{}{V}_{d\alpha }^{}+\mu V_{b\beta }^{}{V}_{d\beta }^{}\right)+2\mu & 2\left(V_{c{z}_1}^{}{V}_{d{z}_1}^{}+V_{c{z}_2}^{}{V}_{d{z}_2}^{}+\mu V_{c\alpha }^{}{V}_{d\alpha }^{}+\mu V_{c\beta }^{}{V}_{d\beta }^{}\right)+2\mu & \end{array}\right]\\ \left[\begin{array}{c}2\left(V_{a{z}_1}^{}{V}_{d{z}_1}^{}+V_{a{z}_2}^{}{V}_{d{z}_2}^{}+\mu V_{a\alpha }^{}{V}_{d\alpha }^{}+\mu V_{a\beta }^{}{V}_{d\beta }^{}\right)\\ 2\left(V_{b{z}_1}^{}{V}_{d{z}_1}^{}+V_{b{z}_2}^{}{V}_{d{z}_2}^{}+\mu V_{b\alpha }^{}{V}_{d\alpha }^{}+\mu V_{b\beta }^{}{V}_{d\beta }^{}\right)+2\mu \\ 2\left(V_{c{z}_1}^{}{V}_{d{z}_1}^{}+V_{c{z}_2}^{}{V}_{d{z}_2}^{}+\mu V_{c\alpha }^{}{V}_{d\alpha }^{}+\mu V_{c\beta }^{}{V}_{d\beta }^{}\right)+2\mu \\ 2\left(V_{d{z}_1}^2+V_{d{z}_2}^2+\mu V_{d\alpha }^2+\mu V_{d\beta }^2\right)+2\mu \end{array}\right]\left[\begin{array}{c}{\eta }_a\\ {\eta }_b\\ {\eta }_c\\ {\eta }_d\end{array}\right]=\left[\begin{array}{c}2\mu \left(V_{\alpha }^{*}{V}_{a\alpha }+V_{\beta }^{*}{V}_{a\beta }\right)\\ 2\mu \left(V_{\alpha }^{*}{V}_{b\alpha }+V_{\beta }^{*}{V}_{b\beta }\right)+2\mu \\ 2\mu \left(V_{\alpha }^{*}{V}_{c\alpha }+V_{\beta }^{*}{V}_{c\beta }\right)+2\mu \\ 2\mu \left(V_{\alpha }^{*}{V}_{d\alpha }+V_{\beta }^{*}{V}_{d\beta }\right)+2\mu \end{array}\right]\end{array}$$

(29)

By substituting the values of these vectors into (29) and calculating it, the duty cycle of each vector can be expressed as (30).

$$\left[\begin{array}{c}{\eta }_a\\ {\eta }_b\\ {\eta }_c\\ {\eta }_d\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\\ -5.1962& 3& 3.7321\\ 11.1962& -3& -6.4641\\ -6& 0& 3.7321\end{array}\right]\cdot \left[\begin{array}{c}{V}_{\alpha }^{*}\\ V_{\beta }^{*}\\ 1\end{array}\right]$$

(30)

Modulation in Region Z₄

In the Z₄ region, the zero vectors and vector V_d do not modulate V_r*, so the first and fifth inequalities of (20) turn into equalities, i.e., ${\eta }_a+{\eta }_b+{\eta }_c+{\eta }_d=1$ and $-{\eta }_d=0$. The corresponding penalty function F_Z4 is obtained and written as (31). Similar to the Z₂ and Z₃ regions, (32) can be obtained.

$$\begin{array}{l}{F}_{z_4}\left(\eta ,\mu \right)=f\left(\eta \right)+{\displaystyle \sum _{j=1}^4\mu \cdot h_j^2}\left(\eta \right)\\ ={\left(V_{a{z}_1}{\eta }_a+V_{b{z}_1}{\eta }_b+V_{c{z}_1}{\eta }_c+V_{d{z}_1}{\eta }_d\right)}^2+{\left(V_{a{z}_2}{\eta }_a+V_{b{z}_2}{\eta }_b+V_{c{z}_2}{\eta }_c+V_{d{z}_2}{\eta }_d\right)}^2\\ +\mu {\left(V_{a\alpha }{\eta }_a+V_{b\alpha }{\eta }_b+V_{c\alpha }{\eta }_c+V_{d\alpha }{\eta }_d-V_{\alpha }^{*}\right)}^2+\mu {\left(V_{a\beta }{\eta }_a+V_{b\beta }{\eta }_b+V_{c\beta }{\eta }_c+V_{d\beta }{\eta }_d-V_{\beta }^{*}\right)}^2\\ +\mu {\left({\eta }_a+{\eta }_b+{\eta }_c-1\right)}^2+\mu {\eta }_d^2\end{array}$$

(31)

$$\begin{array}{l}[\begin{array}{cccc}2\left(V_{a{z}_1}^2+V_{a{z}_2}^2+\mu V_{a\alpha }^2+\mu V_{a\beta }^2\right)+2\mu & 2\left(V_{a{z}_1}^{}{V}_{b{z}_1}^{}+V_{a{z}_2}^{}{V}_{b{z}_2}^{}+\mu V_{a\alpha }^{}{V}_{b\alpha }^{}+\mu V_{a\beta }^{}{V}_{b\beta }^{}\right)+2\mu & 2\left(V_{a{z}_1}^{}{V}_{c{z}_1}^{}+V_{a{z}_2}^{}{V}_{c{z}_2}^{}+\mu V_{a\alpha }^{}{V}_{c\alpha }^{}+\mu V_{a\beta }^{}{V}_{c\beta }^{}\right)+2\mu & \\ 2\left(V_{a{z}_1}^{}{V}_{b{z}_1}^{}+V_{a{z}_2}^{}{V}_{b{z}_2}^{}+\mu V_{a\alpha }^{}{V}_{b\alpha }^{}+\mu V_{a\beta }^{}{V}_{b\beta }^{}\right)+2\mu & 2\left(V_{b{z}_1}^2+V_{b{z}_2}^2+\mu V_{b\alpha }^2+\mu V_{b\beta }^2\right)+2\mu & 2\left(V_{b{z}_1}^{}{V}_{c{z}_1}^{}+V_{b{z}_2}^{}{V}_{c{z}_2}^{}+\mu V_{b\alpha }^{}{V}_{c\alpha }^{}+\mu V_{b\beta }^{}{V}_{c\beta }^{}\right)+2\mu & \\ 2\left(V_{a{z}_1}^{}{V}_{c{z}_1}^{}+V_{a{z}_2}^{}{V}_{c{z}_2}^{}+\mu V_{a\alpha }^{}{V}_{c\alpha }^{}+\mu V_{a\beta }^{}{V}_{c\beta }^{}\right)+2\mu & 2\left(V_{c{z}_1}^{}{V}_{b{z}_1}^{}+V_{c{z}_2}^{}{V}_{b{z}_2}^{}+\mu V_{c\alpha }^{}{V}_{b\alpha }^{}+\mu V_{c\beta }^{}{V}_{b\beta }^{}\right)+2\mu & 2\left(V_{c{z}_1}^2+V_{c{z}_2}^2+\mu V_{c\alpha }^2+\mu V_{c\beta }^2\right)+2\mu & \\ 2\left(V_{a{z}_1}^{}{V}_{d{z}_1}^{}+V_{a{z}_2}^{}{V}_{d{z}_2}^{}+\mu V_{a\alpha }^{}{V}_{d\alpha }^{}+\mu V_{a\beta }^{}{V}_{d\beta }^{}\right)& 2\left(V_{b{z}_1}^{}{V}_{d{z}_1}^{}+V_{b{z}_2}^{}{V}_{d{z}_2}^{}+\mu V_{b\alpha }^{}{V}_{d\alpha }^{}+\mu V_{b\beta }^{}{V}_{d\beta }^{}\right)& 2\left(V_{c{z}_1}^{}{V}_{d{z}_1}^{}+V_{c{z}_2}^{}{V}_{d{z}_2}^{}+\mu V_{c\alpha }^{}{V}_{d\alpha }^{}+\mu V_{c\beta }^{}{V}_{d\beta }^{}\right)& \end{array}\\ \begin{array}{c}2\left(V_{a{z}_1}^{}{V}_{d{z}_1}^{}+V_{a{z}_2}^{}{V}_{d{z}_2}^{}+\mu V_{a\alpha }^{}{V}_{d\alpha }^{}+\mu V_{a\beta }^{}{V}_{d\beta }^{}\right)\\ 2\left(V_{b{z}_1}^{}{V}_{d{z}_1}^{}+V_{b{z}_2}^{}{V}_{d{z}_2}^{}+\mu V_{b\alpha }^{}{V}_{d\alpha }^{}+\mu V_{b\beta }^{}{V}_{d\beta }^{}\right)\\ 2\left(V_{c{z}_1}^{}{V}_{d{z}_1}^{}+V_{c{z}_2}^{}{V}_{d{z}_2}^{}+\mu V_{c\alpha }^{}{V}_{d\alpha }^{}+\mu V_{c\beta }^{}{V}_{d\beta }^{}\right)\\ 2\left(V_{d{z}_1}^2+V_{d{z}_2}^2+\mu V_{d\alpha }^2+\mu V_{d\beta }^2\right)+2\mu \end{array}]\left[\begin{array}{c}{\eta }_a\\ {\eta }_b\\ {\eta }_c\\ {\eta }_d\end{array}\right]=\left[\begin{array}{c}2\mu \left(V_{\alpha }^{*}{V}_{a\alpha }+V_{\beta }^{*}{V}_{a\beta }\right)+2\mu \\ 2\mu \left(V_{\alpha }^{*}{V}_{b\alpha }+V_{\beta }^{*}{V}_{b\beta }\right)+2\mu \\ 2\mu \left(V_{\alpha }^{*}{V}_{c\alpha }+V_{\beta }^{*}{V}_{c\beta }\right)+2\mu \\ 2\mu \left(V_{\alpha }^{*}{V}_{d\alpha }+V_{\beta }^{*}{V}_{d\beta }\right)\end{array}\right]\end{array}$$

(32)

Then, the duty cycles in this region are obtained and expressed as (33).

$$\left[\begin{array}{c}{\eta }_a\\ {\eta }_b\\ {\eta }_c\\ {\eta }_d\end{array}\right]=\left[\begin{array}{ccc}-6& 0& 3.7321\\ 11.1962& 3& -6.4641\\ -5.1962& -3& 3.7321\\ 0& 0& 0\end{array}\right]\cdot \left[\begin{array}{c}{V}_{\alpha }^{*}\\ V_{\beta }^{*}\\ 1\end{array}\right]$$

(33)

To better present this strategy, a flowchart is shown in Figure 5.

4. Simulation Results

To verify the proposed SVPWM strategy, a six-phase two-level VSI model is built on the MATLAB/Simulink platform. Simulation results of the proposed strategy are compared with that of TFOS. The simulation parameters are set as follows: output voltage frequency f = 50 Hz, carrier frequency f_c = 10 kHz, and carrier period T_c = 1/f_c. The dc-link voltage is set to 400 V. Harmonic voltages u_z1 and u_z2 are normalized by the dc-link voltage of 400 V. Simulation results are shown in Figure 6.

The V_r* crosses different regions due to different modulation index values. When the modulation index m is set to 0.585, it crosses the Z₁ and Z₂ regions. When the m is set to 0.605 or 0.615, it crosses the Z₂, Z₃ and Z₄ regions. V_r* crosses the Z₃ and Z₄ regions when the m is set to 0.622. Harmonic voltages in the Z₁ − Z₂ subspace are zero in the Z₁ region, whereas they are not zero in the Z₂, Z₃ and Z₄ regions. Figure 6 shows that the voltage harmonic components of the proposed strategy are smaller than those of the TFOS.

To further study the harmonic components content of voltage u_z1, the u_z1 waveforms of the two strategies for one fundamental period and corresponding spectrum analysis are given in Figure 7, Figure 8, Figure 9 and Figure 10. The percentage of fundamental magnitude is used as a unit in the spectrum analyses, which is consistent with the following experiment.

In the simulation, the switching frequency is 10 kHz and the harmonic analysis is in the range of up to 1 kHz. The main harmonics in the Z₁ − Z₂ subspace within 1 kHz are the 5th, 7th, 17th and 19th. Based on Figure 7, Figure 8, Figure 9 and Figure 10, the content of these harmonics is listed in

Table 2. Simulation results of the two strategies.

	The TFOS Strategy (% of Fundamental)				The HSOS Strategy (% of Fundamental)
	5th	7th	17th	19th	5th	7th	17th	19th	5th Reduction (%)
m = 0.585	0.69	0.21	0.6	0.18	0.51	0.47	0.22	0.17	26.09
m = 0.605	6.2	2	1	0.7	4	0.8	0.7	0.5	35.48
m = 0.615	9.5	3.7	0.8	0.4	8	1.7	0.7	0.6	15.79
m = 0.622	11.7	4.65	0.55	0.2	11	4	0.2	0.1	5.98

.

As can be seen from Table 2, the content of the 5th harmonic is the largest among these four kinds of harmonics, and it is averagely reduced by 20.84% by the HSOS strategy.

According to [31], the voltage total harmonic distortion (THD) in the Z₁ − Z₂ subspace within 1 kHz can be calculated as:

$$TH{D}_{Z_1{Z}_2}=\sqrt{U_5^2+U_7^2+U_{17}^2+U_{19}^2}/U_1$$

(34)

where U_j is the magnitude of the j-th-order harmonic. Therefore, the THD_Z1Z2 values of both strategies are obtained, as shown in

Table 3. Voltage THD_Z1Z2 of the two strategies in simulation.

	The TFOS Strategy (%)	The Proposed HSOS Strategy (%)	Reduction by HSOS (%)
m = 0.585	0.9575	0.7472	21.96
m = 0.605	6.628	4.1689	37.1
m = 0.615	10.2343	8.2304	19.58
m = 0.622	12.6038	11.7068	7.12

.

Table 3 shows that the THD_Z1Z2 of the proposed strategy is lower than that of the TFOS, especially at m = 0.605. The value of THD_Z1Z2 is averagely decreased by 21.44% with these four modulation levels.

Figure 11 shows the phase voltage total harmonic distortion (THD) values of the two-vector method, the TFOS and the proposed HSOS at different values of the modulation index values. It can be seen that the THD value of the two-vector method is the highest and is maintained at approximately 16.6% [28]. When the modulation index value is greater than 0.577, the THD values of the proposed HSOS are lower than that of TFOS.

5. Experiment

An experimental platform with the asymmetric six-phase induction motor fed by a two-level six-phase VSI is built to validate the performance of the proposed strategy, as shown in Figure 12. The platform mainly contains: a control board, a power board, a six-phase induction motor, a load motor, and a DC source. The control board is based on the Microzed evaluation board, equipped with the Xilinx Zynq System-on-Chip. It is composed of a floating-point digital signal processor (DSP) and a field programmable gate array (FPGA). The DSP is used to calculate, and the FPGA is responsible for the generation of PWM signals. The power board is implemented by the insulated gate bipolar transistor IHW30N120R2 (1200 V, 30 A, Infineon). A Switching Power Supply Weir 413D (7.5 V, 4 A) provides 5 V DC voltage for current sensors. The parameters of the six-phase induction motor are shown in

Table 4. Parameters of the experimental six-phase motor.

rated frequency	50 Hz
moment of inertia	0.112 kg·m2
stator resistance	1.2 Ω
rotor resistance	1.4 Ω
stator leakage inductance	0.0178 H
rotor leakage inductance	0.0178 H
mutual inductance	0.2987 H
synchronous speed	1500 r/min

. The dc-link voltage U_dc = 400 V, the dead time is 1.5 μs, and the carrier frequency f_c is 10 kHz. The motor starts with no-load, and a 5 N·m load torque is carried in stable motor operation. In the experiment, the modulation index m is set to 0.585, 0.605, 0.615 and 0.622, respectively. However, for the brevity of the paper, only experimental pictures of m = 0.605 and 0.615 are presented. Experimental results are shown in Figure 13, Figure 14, Figure 15 and Figure 16.

The harmonic content of line voltage and current of both strategies are listed in

Table 5. Experiment voltage values of two strategies.

	The TFOS Strategy (% of Fundamental)					The HSOS Strategy (% of Fundamental)
	THD	5th	7th	17th	19th	THD	5th	7th	17th	19th	5th Reduction (%)
m = 0.585	0.98	0.7	0.21	0.61	0.18	0.75	0.51	0.47	0.21	0.19	27.14
m = 0.605	6.68	6.15	2	1.1	0.75	4.19	4	0.8	0.75	0.55	34.96
m = 0.615	10.27	9.5	3.7	0.75	0.45	8.26	8	1.75	0.65	0.6	15.79
m = 0.622	12.65	11.7	4.7	0.6	0.2	11.71	11	4	0.25	0.15	5.98

and

Table 6. Experiment current values of two strategies.

	The TFOS Strategy (% of Fundamental)					The HSOS Strategy (% of Fundamental)
	THD	5th	7th	17th	19th	THD	5th	7th	17th	19th
m = 0.585	0.79	0.74	0.17	0.19	0.05	0.65	0.52	0.37	0.06	0.05
m = 0.605	7.55	7.3	1.7	0.3	0.2	4.69	4.5	0.8	0.2	0.1
m = 0.615	11.9	11.4	3.15	0.25	0.1	9.79	9.6	1.6	0.15	0.1
m = 0.622	15.03	14.4	4.2	0.2	0.1	13.79	13.3	3.4	0.1	0.05

.

As can be seen from Table 5, the 5th voltage harmonic is reduced by an average of 20.97% in the HSOS strategy. According to (34), the THD_Z1Z2 value of experiment voltage and current can be obtained, as shown in

Table 7. Experiment voltage and current values of THD_Z1Z2 of the two strategies.

	Current Values (THDZ1Z2)		Voltage Values (THDZ1Z2)
	TFOS (%)	HSOS (%)	TFOS (%)	HSOS (%)	Reduction by HSOS (%)
m = 0.585	0.7843	0.643	0.9667	0.7491	22.51
m = 0.605	7.504	4.576	6.6027	4.1839	36.63
m = 0.615	11.8303	9.7341	10.2325	8.2368	19.52
m = 0.622	15.0017	13.7282	12.6246	11.7083	7.26

.

Table 2 and Table 5 show that the content of harmonic voltage is very close between simulation and experiment. Therefore, the values of THD_Z1Z2 are almost the same, which can be seen from Table 3 and Table 7.

The THD value includes the harmonic distortion both in the α − β subspace and the Z₁ − Z₂ subspace [31]. Therefore, the THD_Z1Z2 values which are shown in Table 7 are smaller than the THD values shown in Table 5 and Table 6 at the same modulation index.

As can be seen from Table 7, the THD_Z1Z2 values of both voltage and current in the proposed HSOS are smaller than that of TFOS. At these four modulation levels, the voltage THD_Z1Z2 value of the proposed strategy has decreased by 21.475% on average, which is close to the simulation result. Therefore, the proposed HSOS strategy can effectively suppress the harmonics in the Z₁ − Z₂ subspace.

6. Conclusions

In this paper, an SVPWM strategy, namely, HSOS, is proposed for the asymmetric six-phase motor fed by two-level six-phase VSI operating in the overmodulation region. In this strategy, the reference vector is synthesized and the harmonics in the Z₁ − Z₂ subspace are suppressed simultaneously for the overmodulation region. To minimize the harmonics in the Z₁ − Z₂ subspace, the optimization models consisting of an objective function and constraint conditions are built, and an external point method is adopted to solve these models.

The low-order harmonics in the Z₁ − Z₂ subspace mainly include the 5th, 7th, 17th and 19th, and the content of the 5th harmonic is the largest. Compared with the TFOS in the simulation and experiment, the 5th harmonic content of the proposed HSOS is reduced by 20.84% and 20.97%, and the THD_Z1Z2 of these four harmonics is reduced by 21.44% and 21.475%, respectively. Therefore, the proposed HSOS can reduce the content of the 5th harmonic by about 20% and the THD_Z1Z2 of the four harmonics by about 21%.