1 Introduction

Developments in power electronic equipment and microprocessors during the last three decades have led to improvements in the methods of controlling electrical machines. Control of several motors simultaneously are used in many industrial uses, such as paper or steel rolling mills and electrical systems of railroad [1]. The use of single inverter to control several motors is preferred in the cases that precised speed synchronization of motors is not important. Altough, complexity of system increases because the use of single inverter instead of mulitple inverters but it reduces the costs, weight, required space, and the number of switches. This system increases reliability compared to those with mechanical components such as gears and line shafts [2]. The parallel and series connected motors can be separately controlled on the basis of the number of stator phases. It means that the multiphase motors can be connected in series, except three-phase motors [3].

The most limitations in single inverter of dual motor drives are the following: (1) when the single voltage vector is applied to both motors, stator flux is instantantly put in the same direction, and (2) when unbalanced load condition takes place, the speed of the motors does not reach the desired values [4, 5]. The Control of dual motors using multiple-leg inverters has recently garnered much interest. For instance, the vector control method [6] uses a four-leg inverter while the approach proposed in [7, 8], uses a five-leg inverter. Such inverters can feed the two motors with different voltage vectors. However, if the number of motors is increased, the number of switches also must be increased [9]. Also another convertors well known the matrix convertors can be used for feeding a large number of voltage vectors. Some of these vectors, called active vectors, have constant direction but variable amplitude; however, some others, called rotation vectors [10], have constant amplitude but variable direction (Table 1). The use of both groups of vectors in single inverter dual motors creates a continuous vector space. The two main properties of the matrix convertors are its capability for generating sinusoidal input–output voltage and current and to adjust input power factor [11, 12]. The matrix convertors are preferred for extreme temperatures and critical volume-weight [13]. The control and the modulation are two major subjects in the single inverter dual motors. The present control strategies are basically of two main types: (1)The master–slave pattern, which is used in the same machines. Here, to ensure that the stator flux keeps within its nominal range, the motor with the greater amplitude of the stator flux is set as the master, and the behavior of the slave motor is not checked. However, a restriction of this arrangement is that the behavior of the slave motor may not be acceptable in some cases [2, 4]. (2)The mean control pattern, where the average and differential values of the speed and current of the motors are calculated. The main disadvantage of this pattern is that the stator flux of either motor may go beyond the reference value [4]. Moreover, a third pattern would be to mix the mentioned patterns [2].

Table 1 The vector space of a \(3\times 3\) matrix convertor
Full size table

Each of the mentioned patterns can be implemented through the use of either vector control or direct torque control (DTC) methods. For example, a sensorless vector control method was propounded by [14] based on the mean control pattern and the sum of the torque values of two induction motors controlled by a three-leg inverter. By [15] made a comparison between the P controller and the PI controller in terms of their effect on the torque control loop. In addition, in [1, 16] propounded the weighted vector control method for improving the speed function and startup conditions of a dual-motor drive and considered differences in torque values. A straightforward method presented by [17] improved controlling of single inverter in drives of dual induction motors.

The use of the DTC method for controlling single inverter in drives of dual motors has been less studied compared to the vector control method. By [18] proposed a method according to the master–slave pattern and based on the classical DTC method to control a system consisting of two permanent magnet synchronous motors fed by a three-leg inverter. The DTC approach allows easier torque control and a fast dynamic response, that does not need the transformation of rotary coordinates, and is independent of motor parameters [19]. However, the DTC method has more torque and current ripple than does the vector control approach [20]. The previous researchs indicate that using the master–slave pattern implemented via the DTC approach achieves the better results than does the vector control method. The opposite is true of the mean control pattern [2]. In the conventional DTC method, the vector space is divided to six sectors of 60° which results in eight voltage vectors. Considering the resultant discrete voltage space, it is impossible to accurately produce the voltage vector required to generate flux and torque [21]. To solve this problem, different approaches have been propounded which increased complexity of the DTC approach [10]. One of them is space vector modulation (SVM) [22], that provides continuous vector space for voltage modulation [23]. The SVM method works by removing the hystersis blocks and the lookup table, but it increases the complexity of computations. Also there is another configuration of DTC based methods which enhance the lookup table in order to select the optimal switching state. These methods result acceptable outputs and at the same time, it provides simplicity of control [21, 24]. The matrix convertor can produce 27 voltage vectors that creates a continuous voltage space. In the conventional DTC approach, only 18 vectors have a constant direction, and this implies the loss of the advantages of the matrix convertor [10]. To solve this problem, in [25] developed a modified DTC method, in which the circular vector space is divided up into twelve sectors of 30°. This modified technique investigates the effect of all the voltage vectors of the matrix convertor on the changes in flux, torque, and input power factor.

The purpose of the study reported here is threefold: (1) to introduce a novel drive system with a matrix convertor for two parallel-connected induction motors through the enhanced switching tables. Also the proposed control method is explianed. (2) to lower the torque ripple more than does the conventional DTC; and (3) to maintain the stability of motor speed under unbalanced load circumstances. The remainder of the paper is as follows. In Sect. 2, the proposed control method is delineated. Section 3 presents and discusses the simulation results. Finally, conclusions are given in Sect. 4.

2 Materials and methods

2.1 Modeling of dual induction motors in the proposed method

If the voltage drop of the stator resistance is disregarded, the flux equation of induction motors will be as follows:

$$\overrightarrow {{\Delta \psi_{s} }} = \overrightarrow {{V_{s} }} \cdot \Delta t$$
(1)

The radial and tangential components of the stator voltage vector alter the amplitude and angle of the stator flux vector, respectively. These components are expressed in a synchronous reference frame as follows [10]:

$$V_{d}^{s} = V_{\alpha } \cos \theta + V_{\beta } \sin \theta$$
(2)
$$V_{q}^{s} = - V_{\alpha } \sin \theta + V_{\beta } \cos \theta$$
(3)

In the single inverter dual motor drives, a same voltage vector is applied to both motors simultaneously. In this case, variations in the amplitude of the stator flux for each motor are described as follows:

$$\Delta \left| {\vec{\psi }_{s1} } \right| = V_{d}^{s} \cdot \Delta t$$
(4)
$$\Delta \left| {\vec{\psi }_{s2} } \right| = V_{d}^{s} \cdot \Delta t$$
(5)

By applying Eqs. (1) to (5) to each motor and using the mean control pattern, the flux average and torque average are expressed as follows:

$$\frac{{\left| {\vec{\psi }_{s1} } \right| + \left| {\vec{\psi }_{s2} } \right| }}{2} = \left| {\overline{{\psi_{s} }} } \right|$$
(6)
$$\frac{{T_{e1} + T_{e2} }}{2} = \overline{T}_{e}$$
(7)

By applying Eqs. (4) and (5) to each motor and considering Eq. (2), the effect of the radial component of the applied voltage vector on the variations in the average flux of stator in the interval [t1, t2] can be described as follows:

$$\Delta \left| {\overline{\psi }_{s} } \right| = \left| {\overline{\psi }_{st2} } \right| - \left| {\overline{\psi }_{st1} } \right| \cong \overline{V}_{d} \cdot \Delta t \Rightarrow \frac{{d\left| {\vec{\overline{\psi }}_{s} } \right|}}{dt} = \overline{V}_{d} = V_{\alpha } \cos \theta + V_{\beta } \sin \theta$$
(8)

The torque of the induction motor can be expressed using Eq. (9) [10]:

$$T_{e} = \frac{3}{2} \cdot \frac{P}{2} \cdot \frac{{L_{m} }}{{\sigma \cdot L_{s} \cdot L_{r} }} \cdot \left| {\vec{\psi }_{r} } \right| \cdot \left| {\vec{\psi }_{s} } \right| \cdot \sin \gamma$$
(9)

where \(\sigma\) and \(\gamma\) are leakage factor and load angle, respectively.

By substituting Eqs. 6 and 7 in Eq. 9 for the single inverter dual motor drives, average torque will be expressed as below:

$$\overline{T}_{e} = \frac{3}{2} \cdot \frac{P}{2} \cdot \frac{{L_{m} }}{{\sigma \cdot L_{s} \cdot L_{r} }} \cdot \left| {\overrightarrow {{\overline{\psi }}}_{r} } \right| \cdot \left| {\overrightarrow {{\overline{\psi }}}_{s} } \right| \cdot \sin \gamma$$
(10)

Torque changes resulting from the applied \(\vec{V}_{s}\) are as follows:

$$\Delta \overline{T}_{e} = k \cdot \left| {\vec{\overline{\psi }}_{r} } \right| \cdot \left( {\frac{{d\left| {\vec{\overline{\psi }}_{s} } \right|}}{dt} \cdot \sin \gamma + \frac{d\gamma }{{dt}} \cdot \cos \gamma \cdot \left| {\vec{\overline{\psi }}_{s} } \right|} \right) \cdot \Delta t$$
(11)

where K is a constant defined as Eq. (12) below:

$$k = \frac{3}{2} \cdot \frac{P}{2} \cdot \frac{{L_{m} }}{{\sigma \cdot L_{s} \cdot L_{r} }}$$
(12)

Hence:

$$\Delta \gamma = \Delta \theta - \Delta \zeta$$
(13)

where \({\Delta }\theta\) and \({\Delta }\zeta\) are the change in the angles of \(\vec{\overline{\psi }}_{s}\) and \(\vec{\overline{\psi }}_{r}\), respectively, that results from the application of the voltage vector.

The following equation indicates the change in the amplitude of \(\vec{\overline{\psi }}_{s}\):

$$\Delta \left| {\vec{\overline{\psi }}_{st} } \right| = \Delta \theta \cdot \left| {\vec{\overline{\psi }}_{s1} } \right| \cong V_{q} \cdot \Delta t$$
(14)

Therefore:

$$\frac{d\theta }{{dt}} = \frac{{V_{q} }}{{\left| {\vec{\overline{\psi }}_{s1} } \right|}}$$
(15)

Given that the rotor flux rotates at synchronous speed,

$$\Delta \zeta = \omega_{s} \cdot \Delta t \Rightarrow \frac{d\zeta }{{dt}} = \omega_{s}$$
(16)

Substituting (3), (15), and (16) in (13) results:

$$\frac{d\gamma }{{dt}} = \frac{d\theta }{{dt}} - \frac{d\zeta }{{dt}} = \frac{{ - V_{\alpha } \sin \theta + V_{\beta } \cos \theta }}{{ \left| {\vec{\overline{\psi }}_{s1} } \right|}} - \omega_{s}$$
(17)

Also, if (8) and (17) are substituted in (11), the change in the average torque is expressed as follows:

$$\Delta \overline{T}_{e} = k \cdot \left| {\vec{\overline{\psi }}_{r} } \right| \cdot \left( {V_{d} \cdot \sin \gamma + \left( {V_{q} - \omega_{s} \cdot \left| {\vec{\overline{\psi }}_{s} } \right| } \right) \cdot \cos \gamma } \right) \cdot \Delta t$$
(18)

The flux evaluation function (λ), torque evaluation function (τ), and the back-EMF evaluation function (e) of the matrix convertor are defined as follows [10]:

$$\lambda = \frac{{V_{q} }}{{2/\sqrt 3 V_{in} }}$$
(19)
$$\tau = \frac{{V_{d} }}{{2/\sqrt 3 V_{in} }}$$
(20)
$$e = \frac{{\omega_{s} \cdot \left| {\vec{\overline{\psi }}_{s1} } \right|}}{{2/\sqrt 3 V_{in} }}$$
(21)

where \(V_{in}\) is the amplitude of the input voltage vector. By substituting (19), (20), and (21) in (8) and (18), the rate of the effect of each voltage vector on the change of the average torque and the average flux are estimated as below:

$$\frac{d}{dt}\overline{T}_{e} \propto \tau + \lambda - e$$
(22)
$$\frac{d}{dt} \left| {\overrightarrow {{\overline{\psi }}}_{s1} } \right| \propto \lambda$$
(23)

The d and q voltage components of the matrix convertor are substituted in (19) and (20). The \(\tau\) functions and the \(\lambda\) functions are presented in Table 2. Results indicate that these \(\tau\) and \(\lambda\) functions are dependent on the angle of the input voltage and the angle of the average flux vector, respectively. The input reactive power and input current vector can be used to obtain the evaluation function of input power factor (Eq. (24)). The \({\upeta }\) functions of all voltage vectors of the matrix convertor are given in Table 2.

$$\eta = \frac{{I_{q} }}{{2/\sqrt 3 I_{in} }}$$
(24)

where \(I_{in}\) is the amplitude of the input current vector.

Table 2 Evaluation functions of torque, flux and input power factor
Full size table

As it can be seen in Table 2, the evaluation functions \((\eta )\) are the functions of the angle of the input current and the angle of the voltage vector, respectively.

The relationship between \(\eta\) and the input reactive power of single inverter dual motor can be expressed as follows [10];

$$\frac{d}{dt} Q \propto \eta$$
(25)

In order to extract the unit input power factor, input reactive power should be kept close to zero. Evaluation functions are proportional to the rate of the changes in torque, flux, and input reactive power.

2.2 Enhanced switching table of matrix convertor

According to Table 2, it is seen that the evaluation functions are dependent to \(\theta_{s}\) and \(\alpha_{i}\). The area between 0 and 2π is divided into 12 sectors of 30 degrees and the mean change rate of every evaluation functions, based on the changes of \(\theta_{s}\) and \(\alpha_{i}\) is calculated in each sector that called impact factor. As it mentioned, the variation of impact factors, directly will effect on the change of average flux and torque parameters respectively. So, impact factors for every switching states and each evaluation function are calculated by the following relations and are indicated in Tables 3, 4, 5, 6.

$${\mathcal{R}}_{\tau } = round \left[ {\frac{k}{{\left( {\pi /6} \right)^{2} }}\mathop \int \limits_{{\frac{\pi }{6}\left( {n_{\alpha } - 1} \right)}}^{{\frac{\pi }{6}n_{\alpha } }} \mathop \int \limits_{{\frac{\pi }{6}\left( {n_{\theta } - 1} \right)}}^{{\frac{\pi }{6}n_{\theta } }} \tau d\theta d\alpha } \right]$$
(26)
$${\mathcal{R}}_{\lambda } = round \left[ {\frac{k}{{\left( {\pi /6} \right)^{2} }}\mathop \int \limits_{{\frac{\pi }{6}\left( {n_{\alpha } - 1} \right)}}^{{\frac{\pi }{6}n_{\alpha } }} \mathop \int \limits_{{\frac{\pi }{6}\left( {n_{\theta } - 1} \right)}}^{{\frac{\pi }{6}n_{\theta } }} \lambda d\theta d\alpha } \right]$$
(27)
$${\mathcal{R}}_{\eta } = round \left[ {\frac{k}{{\left( {\pi /6} \right)^{2} }}\mathop \int \limits_{{\frac{\pi }{6}\left( {n_{\alpha } - 1} \right)}}^{{\frac{\pi }{6}n_{\alpha } }} \mathop \int \limits_{{\frac{\pi }{6}\left( {n_{\theta } - 1} \right)}}^{{\frac{\pi }{6}n_{\theta } }} \eta d\alpha d\beta } \right]$$
(28)
$${\mathcal{R}}_{e} = round \left( {10e} \right)$$
(29)
Table 3 Impact factor of torque and flux for constant vectors of matrix convertor
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Table 4 Impact factor of torque and flux for rotation vectors of matrix convertor
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Table 5 Impact factor of input power factor for constant vectors of matrix convertor
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Table 6 Impact factor of input power factor for rotation vectors of matrix convertor
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The Eq. (29) is definition of the impact factor of back emf [10].

The output of functions is rounded to the nearest integer that will be between -9 and 9. In other words, the effect of every evaluation functions is categorized in to 18 levels.

With regards to (22), (26) and (27), the average value of torque variations stated in each cell of table is as follows:

$$\frac{d}{dt} T_{e} \approx avg\left( {\frac{d}{dt} T_{e} } \right) \propto {\mathcal{R}}_{\tau } + {\mathcal{R}}_{\lambda } - {\mathcal{R}}_{e}$$
(30)

At the same way for the flux variations and input power factor variations are calculated:

$$\frac{d}{dt} \left| {\overrightarrow {{\overline{\psi }}}_{s1} } \right| \approx avg\left( {\frac{d}{dt} \left| {\overrightarrow {{\overline{\psi }}}_{s1} } \right|} \right) \propto {\mathcal{R}}_{\lambda }$$
(31)
$$\frac{d}{dt} Q \approx avg\left( {\frac{d}{dt} Q} \right) \propto {\mathcal{R}}_{\eta }$$
(32)

Finally for all 27 states, the impact factors are calculated. The impact factors of ± 10, ± 11 and ± 12 vectors in some areas will be unaffected. In these circumstances, the effect of these factors on variations of torque, flux and input power factor in single inverter dual motor drives is considered to be zero.

2.3 System configuration

The proposed method consists of four main steps (Fig. 1). First, the values of stator flux and torque of motors are estimated [10] and the average values of these parameters are calculated via Eq. (13). Then input reactive power is calculated via Eq. (24). Subsequently, the reference impact values are calculated via Eqs. (33), (34) and (35) according to \(\theta_{s}\), \(\alpha_{i}\) and \(\beta_{o}\) that are applied to switching table. Finally, the best switching state that leading to minimize the objective function according to Eq. (36) is selected. The values of reference impact factors to extract average torque, average flux and input reactive power are calculated as follow:

$${\mathcal{R}}^{*}_{\tau } = \frac{{\left( {\overline{T}_{e}^{*} - \overline{T}_{e} } \right)}}{{k_{t} }} + {\mathcal{R}}_{e} - {\mathcal{R}}_{\lambda }$$
(33)
$${\mathcal{R}}^{*}_{\lambda } = \frac{{\left( {\left| {\overrightarrow {{\overline{\psi }}}_{s} } \right|^{*} - \left| {\overrightarrow {{\overline{\psi }}}_{s} } \right|} \right)}}{{k_{\psi } }}$$
(34)
$${\mathcal{R}}^{*}_{\eta } = \frac{Q}{{K_{Q} }}$$
(35)

where \(k_{Q} , k_{\psi } \,{\text{and}}\,k_{\tau }\) are coefficients of input reactive power, flux and positive torque respectively. The values of these parameters will have a significant impact on the variation rate of torque, flux and input power factor. The appropriate adjustment of these coefficients is also dependent on the steady state operating conditions and the variation rate of torque and flux of motors than nominal values [10].

Fig. 1
figure 1

Proposed DTC-SVM for dual ınduction motors fed by MC

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2.4 Objective function

The objective function must be able to choice the best among 27 switching states that lead to the minimization of the Eq. (36). The impact factors of torque, flux and input reactive power in single inverter dual motor drives are related to each other by weighted coefficients \(a_{\tau } , a_{\psi } , a_{Q}\) respectively. The values of these coefficients determine the amount of the effectiveness of impact factors in Eq. (36).

$$F = a_{\tau } \left| {{\mathcal{R}}^{*}_{\tau } - {\mathcal{R}}_{\tau } } \right| + a_{\psi } \left| {{\mathcal{R}}^{*}_{\lambda } - {\mathcal{R}}_{\lambda } } \right| + a_{Q} \left| {{\mathcal{R}}^{*}_{\eta } - {\mathcal{R}}_{\eta } } \right|$$
(36)

3 Results

Speed and torque are the most important controllable parameters in all motor drive methods. In this study, it was assumed that both motors are the same in terms of electrical parameters. According to the strategy used in this research, the drive system consists of two motors, was modeled like a single motor drive system. According to the reviewed articles, the problem of these systems occurs in situations where there is an imbalance in the load of the motors. Under this condition, and according to the relationships in the DTC method, each motor will need a different voltage vector, which in turn leads to differences in motor speeds, and in theory, motors speed may not follow the reference value. Therefore, the focus of the research is on showing the efficiency of the proposed method in unbalanced load conditions. Of course, the results were examined under normal conditions too. Relationships 1 to 5 show the effect of the selected voltage vector on the changes in the magnetic flux of the motors stator. If there are more vectors in the 360-degree space, the vector space will be more continuous and creating the conditions that the changes in magnetic flux are also small. Therefore, the practical results of using 27 matrix converter voltage vectors in DTC-SVM method leads to a decrease in magnetic flux ripples. According to the relationship between stator flux and motor speed, the proposed method leads to less fluctuations in motors speed and torque, which has been investigated in the simulation results.

The proposed method was compared with the conventional DTC via some numerical simulations in order to study the steady-state and dynamic performances. The machines under investigation were 200 HP, 4 poles, 460 VL, and 60 Hz squirrel cage induction motors where stator resistance is 0.0148 Ω, rotor resistance is 0.0092 Ω, leakage inductance of stator is 0.003 H, leakage inductance of rotor is 0.003 H and magnetizing inductance of stator is 0.01 H.

3.1 Unbalanced load test

The conventional [19] and proposed methods were subjected to the unbalanced load conditions. The speeds of the motors were set at 800 rpm, and both motors were started under no-load condition. After 1.2 s, torque values of 100 and 200 Nm were applied to the motor 1 and motor 2, respectively. The simulation results are presented in Figs. 2 and 3. Figures 4 and 5 show more details about the speed curves. As can be seen, in both methods, both motors followed the torque command. Furthermore, fluctuations can be observed in the torque curve for 0.2 s after step change in the load occurred. In the proposed method, these fluctuations occurred in the motor subjected to the lower load, while in the conventional approach the fluctuations occurred in the motor under higher load. It is also clear in Figs. 4 and 5 that in the proposed method, there are less oscilations in motor speed compared with the conventinoal approach. According to Figs. 2 and 3, torque ripple reduced significantly in the proposed approach.

Fig. 2
figure 2

Unbalanced load condition in conventional DTC: a Speed of motors; b Torque of motor; c current of motor 1; d current of motor 2

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Fig. 3
figure 3

Unbalanced load condition in the proposed DTC: a speed of motors; b Torque of motors; c Current of motor 1; d Current of motor 2

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Fig. 4
figure 4

Speed of motors under unbalanced load condition in conventional DTC

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Fig. 5
figure 5

Speed of motors under unbalanced load condition in the proposed DTC

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3.2 Change in speed step under unbalanced load condition

An investigation was made into the effect of changes in the reference speed on the transient and steady-state performance in the proposed DTC method. Both motors were started under no-load condition, but the load of motor 1 was changed to 400 Nm after 3 s. When the reference speed was changed from 1500 to 500 rpm, − 500 rpm and − 1500 rpm both motors followed the speed command closely. In this section, torque changes and motor speeds in braking mode are also examined. When the speed is negative, the motor's transient torque is visible. In this case, the motor with low load will have more fluctuations in torque and current. According to the equations extracted in the previous section, these fluctuations are predictable for low-load motors. Figure 6 illustrates the characteristics of both motors in this condition. Fluctuations can be seen in the torque curve of motor2 for 0.4 s, but fluctuations were damped in the steady-state condition.

Fig. 6
figure 6

Speed step command under unbalanced load condition in the proposed DTC: a motors speed; b motors torque; c stator current of motor 1; d stator current of motor 2

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3.3 Dynamic performance during a change in torque command

This subsection discusses the ability of the proposed method to generate electrical power when motors brake. According to Fig. 7, torque was changed from + 400 to − 400 Nm, with speed being set at 600 rpm. The speed and torque followed the reference values. As can be seen in Fig. 8, sinusoidal voltage is in phase with the corresponding current at the input side of matrix convertor, and this confirms the validity of the propounded DTC to operate under unit input power factor. When the torque command changes, the direction of motor current changes.According to Fig. 8, the phase angle difference between input current and corresponding voltage is about π radian, indicating that the machines now generate electrical power. In this case, motors speed is more than the reference speed because slip is more than 1.

Fig. 7
figure 7

Torque step command from + 400 Nm to -400 Nm at 600 rpm in the proposed DTC: a motor speed; b motor torque; c stator current of Motor 1; d stator current of Motor 2

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Fig. 8
figure 8

Input voltage and input current waveform at 600 rpm during torque step command in the proposed DTC

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4 Conclusions

In this study a new DTC control method based on mean control pattern for dual identical induction motors fed by a matrix convertor proposed. The main question of the research was whether the speed of two motors in unbalanced loads conditions can be controlled by a single convertor? Also, the effect of the proposed method on the current and motor torque ripple was another question. This method without adding complexity to DTC, effectively uses all matrix convertor switching states and combines the advantages of matrix convertor with the benefits of DTC for use in an single inverter dual motor drives system and hysteresis blocks are replaced with the PI controller.

The effect of each matrix convertor switching states on the change rates of the torque, flux and input power factor of single inverter dual motor drives systems was evaluated. Finally, by using an objective function and based on the switching tables the optimal voltage vector selected. The performance of proposed method has been tested by simulations and The results showed a significant improvement in torque speed and fluctuations compared to the basic method. Furthermore, the proposed method provides a safe condition for motors to operate during the regenerative braking condition.

According to the results of the present study, it is suggested that the Master–Slave strategy be used to simultaneous control of the two motors with a single convertor and the results of this strategy be compared with the proposed method. It is also suggested that the proposed strategy in this study be evaluated with the use of a five-leg inverter instead of a matrix convertor. Practical experiments to compare with simulations results will be effective in improving the proposed method.