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# High performance of variable-pitch wind system based on a direct matrix converter-fed DFIG using third order sliding mode control

Amal Dendouga, Abdelhakim Dendouga, Najib Essounbouli

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Amal Dendouga, Abdelhakim Dendouga, Najib Essounbouli. High performance of variable-pitch wind system based on a direct matrix converter-fed DFIG using third order sliding mode control. Wind Engineering, 2024, 48 (3), pp.325-348. 10.1177/0309524X231199435. hal-04297359

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https://hal.science/hal-04297359v1

Submitted on 26 Sep 2024

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# High performance of variable-pitch wind system based on a direct matrix converter-fed DFIG using third order sliding mode control

Amal Dendouga<fnref n="1" />,*, Abdelhakim Dendouga<fnref n="2" /> and Najib Essounbouli<fnref n="3" />

## Abstract

In this paper, a full nonlinear control of a variable-pitch wind system (VPWS) based on the doubly fed induction generator (DFIG) fed by a direct matrix converter (DMC) has been presented. In this context, The MPPT has been implemented using the third order sliding mode control (TOSMC) in order to ensure maximum power provided by the wind turbine on the one side, on the other side the pitch control has been implemented in order to limit the power extracted to its nominal value. Moreover, a TOSMC has been incorporated into the direct flied-oriented control (DFOC) to ensure high-performance control of the active and reactive power of DFIG. To examine the performance of the TOSMC, a comparative study was performed between this last type and the first and second order sliding mode controllers. The obtained results affirmed the high performance provided by the TOSMC compared to lower order sliding mode controllers.

## Keywords

Direct flied-oriented control, direct matrix converter, doubly fed induction generator, Maximum Power Point Tracking, third order sliding mode control, variable-pitch wind system, pitch control

## Introduction

Currently, wind energy is one of the most widely used renewable sources of electrical energy generation, due to its less damaging environmental influences and its inexhaustible source. During the last decades, wind power systems have undergone a swift development thanks to the considerable development in the field of power electronics and control strategies (Kaloi et al., 2016; Soomro et al., 2021).

The researchers have given great importance to the variable speed wind system compared to the fixed speed system due to its ability to operate under different wind speeds. Thus, the Maximum Power Point Tracking (MPPT) strategy can be applied to extract the maximum power from the wind turbine. In addition, the pitch angle control of the blades can be designed in order to protect the wind turbine from higher wind speeds (Dahbi et al., 2016).

In this context, the DFIG is quite popular in the variable speed wind energy conversion system because of its high performance, the possibility to control stator active and reactive powers, and the most important advantage of DFIG is that the operating in a wide speed range (±30%) this means reduced power converter size and reduced maintenance cost (Patel et al., 2021; Soomro et al., 2021).

Actually, the extensive development in the field of microelectronics and information technologies has allowed the appearance of new topologies of power converters more suitable for the supply of DFIG such as matrix

## Corresponding author:

Amal Dendouga, MSE Laboratory, Department of Electrical Engineering, University of Biskra, BP 145, Biskra 07000, Algeria.

Email: am.dendouga@univ-biskra.dz

<footnote n="1">¹Laboratory of Energy Systems Modeling, Department of Electrical Engineering, University of Biskra, Biskra, Algeria</footnote>

<footnote n="2">²I3CUB Research Laboratory, Department of Electrical Engineering, University of Biskra, Biskra, Algeria</footnote>

<footnote n="3">³CReSTIC IUT of Troyes, University of Reims Champagne-Ardenne, Troyes, France</footnote>
converters instead of traditional indirect AC/DC/AC converters (Nguyen et al., 2020). Therefore, the DMC allows bi-directional power conversion between the DFIG and the grid without a DC link (Aydogmus et al., 2022; Dendouga and Dendouga, 2022).

To improve the input current waveform, therefore reducing the harmonic distortion rate THD of the input current in DMC, a passive filter must be connected between the DMC and the grid. Several passive filter topologies have been proposed in the research works; however, the passive filter with a damping resistor connected in parallel with the inductor has been considered a suitable solution (Dendouga, 2020; Dendouga and Dendouga, 2022).

According to the literature, Field Oriented Control (FOC) using PI controllers is the most popular method used to control the active and reactive power of DFIG-based WECS (Alhato and Bouallègue, 2019). It is very used due to its simplicity and reliability, but the main problem of this controller is the presence of a chattering phenomenon and a high THD of the current due to its linearity, especially during the rapid variation in wind speed.

For this reason, several studies have been performed to provide highly effective performances of control loops for the DFIG-based WECS; such as (Medjber et al., 2016) proposed neural networks and fuzzy logic controllers to control the active and reactive powers of DFIG, and the maximum power point tracking strategy is designed to extract the maximum power from the wind turbine. In El Mourabit et al. (2019), a nonlinear backstepping control is applied to a variable speed wind energy conversion system based on PMSG. A simulation using Matlab/Simulink and the control implementation in the DS1104R&D Controller Board for experimental validation are presented. Rezaei (2018) presented a nonlinear adaptive backstepping control method for MPPT of DFIG-based wind energy conversation systems. The laws of backstepping control do not require knowledge of the machine parametric characteristics. Taraft et al. (2015) proposed the sliding mode control (SMC) method to control active and reactive powers of DFIG supplied by matrix converter for wind turbine system. The Perturbation and Observation maximum power point tracking is designed to extract the maximum power from the wind turbine. Chojaa et al. (2021) proposed an integral sliding mode control (ISMC) to control the active and reactive powers of a doubly fed induction generator (DFIG) based wind turbine. And as well an artificial Neural Network Control (ANNC) was presented for maximum power point tracking in order to extract the maximum power. Kelkoul and Boumediene (2021) presented a nonlinear, super twisting sliding mode controller to control the active and reactive powers of DFIG-based WECS. The performance of the proposed technique was compared to that of the sliding mode controller in terms of parameter variations. In Dursun and Kulaksiz (2020), second-order Siding mode control (SO-SMC) is designed to capture maximum power from the WECSs, it is validated by two different wind speed profiles. A comparative study between the proposed approach SOSMC and conventional SMC is presented.

In this paper, a full control of a variable-pitch wind system based on a direct matrix converter-fed DFIG is designed using TOSMC to improve performances of the system under the stochastic variations in the wind speed.

The current work examines three main topics:

The first topic was to design the MPPT strategy using the TOSMC to optimize the exploitation of wind energy. As well as, the pitch angle control technique based on the PI controller was applied when the wind speed exceeds its nominal value in order to limit the power extracted to its nominal value and also avoid malfunctioning of the system. The second topic was to show that a DMC structure can be used for a VPWS without DC-link problems. A DMC provides bidirectional power flow from the grid to DFIG or vice versa. The DMC needs an input filter to enhance the input current waveform and decrease the harmonics injected by the system into the grid; therefore, a damped RLC passive input filter topology was designed. A nonlinear control of active and reactive power of a DFIG using of the TOSMC in order to improve the performance and efficiency of the system was designed as the third topic of this work.

The overall structure of the paper is organized as follows: The mathematical modeling of wind turbine and of DFIG is provided briefly in section “Modeling of the variable-pitch wind system.” Section “MPPT strategy with speed control” introduces the maximum power point tracking strategy with speed control, while the control of blade angle (pitch angle control) is described in section “Control of the blade angle.” Section “Control of DFIG” includes the DFOC control scheme with designed controllers (PI, FOSMC, SOSMC, and TOSMC). Section “Venturini switching algorithm for DMC” describes Venturini switching algorithm for direct matrix converter. Damped passive input filter is designed in section “Damped passive RLC input filter.” The simulation results with a comparative study are presented and discussed in section “Results and discussion.” Finally, conclusions of this study are given in section “Conclusion.”
<figure><img src="image_3.png" /><figcaption>Figure 1. Descriptive diagram of the variable speed wind turbine system based on a DFIG.</figcaption></figure>

## Modeling of the variable-pitch wind system

The structure studied in this work is based on a variable-pitch wind system using a DFIG with Gearbox; the stator of the generator is connected directly to the grid, while the rotor is connected to the grid via a DMC. Figure 1 shows the structure of the whole system.

### Model of wind turbine

When winds with speed $V$ and air density $\rho$ pass through a swept area by the blades $S$ of a wind turbine, the wind power is (Khan, 2022; Soomro et al., 2021):

$$P_{wind} = \frac{1}{2} \rho s v^3 \quad (1)$$

The aerodynamic power generated by the wind turbine is related to the wind power through a power conversion coefficient $C_p$ as follows (Khan, 2022; Wang et al., 2021):

$$P_{aer} = \frac{1}{2} \rho \pi R^2 v^3 C_p \quad (2)$$

The tip speed ratio $\lambda$ is expressed by the following expression (Adouni et al., 2016; Kaloi et al., 2016):

$$\lambda = \frac{\Omega_t R}{v} \quad (3)$$

Where: $\rho$ is the density of the air ($\rho \approx 1.22 \text{ kg/m}^3$ at atmospheric pressure at 20°C); $S$ is the swept area by the wind turbine ($s = \pi R^2$); $R$ is the blade length; $V$ is the wind speed; and $\Omega_t$ is the angular speed of the wind turbine.

The power coefficient $C_p(\lambda, \beta)$ is a function of the pitch angle ($\beta$) and tip speed ratio ($\lambda$), which can be defined by the following equation (Amrane and Chaiba, 2016; Yaichi et al., 2019a):

$$C_p(\lambda, \beta) = (0.35 - 0.0167(\beta - 2)) \sin\left[\frac{\pi(\lambda + 0.1)}{14.43 - 0.3(\beta - 2)}\right] - 0.00184(\lambda - 3)(\beta - 2) \quad (4)$$

The aerodynamic torque imparted on the wind turbine is given by the equation (5)
$$T_{aer} = \frac{P_{aer}}{\Omega_t} = C_p(\lambda, \beta) \frac{\rho s v^3}{2} \frac{1}{\Omega_t} \quad (5)$$

The dynamic equation of the system can be written as follows (Chojaa et al., 2021; Kelkoul and Boumediene, 2021):

$$J \frac{d\Omega_{mec}}{dt} = T_g - T_{em} - f\Omega_{mec} \quad (6)$$

With:

$J = \frac{J_t}{G^2} + J_g$ is a total inertia of the wind turbine and generator, $f = \frac{f_t}{G^2} + f_g$ is a viscous total friction coefficient, $\Omega_{mec} = G \Omega_t$ is the rotation speed of the generator, $T_g = \frac{T_{mec}}{G}$ is the generator torque, $G$ is the transmission ratio of the gearbox, and $T_{em}$ is the electromagnetic torque of generator.

## Model of DFIG

Due to the strongly coupled dynamics in DFIG, the modeling and control of DFIG in the three-phase system (abc) are quite difficult (Yaichi et al., 2019b; Sami et al., 2020).

Therefore, the dynamical model of the DFIG in the d, q Park reference frame is given by the following electrical equations (Amrane et al., 2022; Djilali et al., 2020):

$$\begin{aligned}
v_{sd} &= R_s I_{sd} + \frac{d}{dt} \phi_{sd} - \omega_s \phi_{sq} \\
v_{sq} &= R_s I_{sq} + \frac{d}{dt} \phi_{sq} - \omega_s \phi_{sd} \\
v_{rd} &= R_r I_{rd} + \frac{d}{dt} \phi_{rd} - (\omega_s - \omega_r) \phi_{rq} \\
v_{rq} &= R_r I_{rq} + \frac{d}{dt} \phi_{rq} + (\omega_s - \omega_r) \phi_{rd}
\end{aligned} \quad (7)$$

Where the expressions of flux are given by (Morshed and Fekih, 2017; Saihi et al., 2019):

$$\begin{aligned}
\phi_{sd} &= L_s I_{sd} + M_{sr} I_{rd} \\
\phi_{sq} &= L_s I_{sq} + M_{sr} I_{rq} \\
\phi_{rd} &= L_r I_{rd} + M_{sr} I_{sd} \\
\phi_{rq} &= L_r I_{rq} + M_{sr} I_{sq}
\end{aligned} \quad (8)$$

The expressions of active and reactive power can be given by equations (9) and (10) (Benbouhenni and Bizon, 2021b):

$$P_s = \frac{3}{2} (v_{sd} I_{sd} + v_{sq} I_{sq}) \quad (9)$$

$$Q_s = \frac{3}{2} (v_{sq} I_{sd} - v_{sd} I_{sq}) \quad (10)$$

The electromagnetic torque is related to a stator flux and to a rotor currents as follows (Soomro et al., 2021):

$$T_{em} = \frac{3}{2} n_p \frac{M_{sr}}{L_s} (\phi_{sd} I_{rq} - \phi_{sq} I_{rd}) \quad (11)$$

Where: $\omega_s$, $\omega_r$ are stator and rotor pulsation, $R_s$ and $R_r$ are the stator and rotor resistances, $L_s$ and $L_r$ are stator and rotor inductances, $M_{sr}$ is mutual inductance and $n_p$ is the number of pole pairs.

The $C_p$ value is proportional to the tip speed ratio $\lambda$, Figure 2 shows the variation of $C_p$ under different pitch angles $\beta$.
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Power coefficient $C_p$ ($\lambda, \beta$) curves for several values of pitch angles $\beta$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 3.** Power characteristics of VPWS under different values of wind speeds.</figcaption></figure>

From Figure 2 it can be seen that, the power coefficient reaches the maximum value $C_{p\_max} = 0.35$ at the optimal tip speed ratio $\lambda_{opt} = 7.1$ which corresponds to $\beta = 2^\circ$.

Figure 3 illustrates the $P_{mec}$-$\Omega_{mec}$ curve of the wind turbine under different wind speeds. From this figure, it is clear that for each wind speed, there is a maximum mechanical power captured by the wind turbine. The intersection points of the pink line with every curve present the maximum power that has to be extracted from the wind turbine to achieve the optimum operating point continuously.

The tracking of a maximum power point of a wind turbine may encounter some problems so, our purpose is the application of the MPPT strategy with different controllers for tracking the maximum power of a wind turbine under different wind speeds.

According to the wind speed, two techniques of control are distinguished: the first is the maximum power point tracking (MPPT) technique to maximize the extracted power, and the second is the pitch angle control technique in order to limit the mechanical power to its nominal value when the wind speed exceeds its nominal value.
<figure><img src="image_6.png" /><figcaption>Figure 4. VPWS operating zones.</figcaption></figure>

## MPPT strategy with speed control

The MPPT strategy is designed in the WEC system as an optimization solution that aims to enhance the efficiency of the VPWS by exploiting the maximum energy from the wind whatever the wind speed; it involves the ongoing search for the maximum power from the wind when the available power is lower than the nominal generator power (Dahbi et al., 2016). This is obtained if the power coefficient $C_p$ is at its maximum value by adjusting the mechanical speed $\Omega$ at the reference which corresponds to an optimum value Figure 4 (Zone II) (Medjber et al., 2016).

To achieve this step, the expression of reference speed is set at the maximum value given by (Hamzaoui et al., 2016):

$$ \Omega_{t\_ref} = \frac{v\lambda_{opt}}{R} \qquad (12) $$

And the maximum value of power is written as follows (Dahbi et al., 2016):

$$ P_{aer} = \frac{1}{2} \rho s C_{p\_max}(\lambda_{opt}, \beta) \left( \frac{R\Omega_{t\_ref}}{\lambda_{opt}} \right)^3 \qquad (13) $$

The expression of reference electromagnetic torque is as follows:

$$ T_{em\_ref} = C_{\Omega}(\Omega_{mec\_ref} - \Omega_{mec}) \qquad (14) $$

With:

$C_{\Omega}$ is the controller of speed, $T_{em\_ref}$ is the reference electromagnetic torque, $\Omega_{mec\_ref} = G\Omega_{t\_ref}$.

The MPPT strategy with speed control is represented in the form of a simplified block diagram given in Figure 5.

In this paper, three types of controllers are studied to control the mechanical speed in order to follow its reference such as first order sliding mode control FOSMC, second order sliding mode control SOSMC based on a super twisting algorithm STA, and third order sliding mode control TOSMC.

## First order sliding mode controller FOSMC

Sliding mode control is a nonlinear control method developed to control the mechanical speed of the wind turbine due to its simplicity and robustness; where its structure consists of two parts, one concerning the equivalent control ($u_{eq}$) and the other the switching component ($u_n$) (Dursun and Kulaksiz, 2020).

The principle of this method depends on keeping the system states on the sliding surface to achieve the stability of a system.

Consider the following sliding surface of mechanical speed:
<figure><img src="image_7.png" /><figcaption>Figure 5. Scheme of MPPT strategy with speed control.</figcaption></figure>

$$S_{\Omega_{mec}} = \Omega_{mec\_ref} - \Omega_{mec} \qquad (15)$$

The derivative of equation (15) is deduced as follows:

$$\dot{S}_{\Omega_{mec}} = \dot{\Omega}_{mec\_ref} - \frac{1}{J}(T_g - T_{em} - f\Omega_{mec}) \qquad (16)$$

According to equation (16), the expression of reference electromagnetic torque is:

$$T_{em\_ref} = T_{em\_eq} + T_{em\_n} \qquad (17)$$

Where $T_{em\_eq}$ and $T_{em\_n}$ are the equivalent and switching components of the control, respectively.

During the sliding mode and in permanent mode, we have $S_{\dot{\Omega}_{mec}} = 0$ (Dendouga, 2020).

From equation (16), the expression of equivalent component is obtained as follows:

$$T_{em\_eq} = T_g - J\dot{\Omega}_{mec\_ref} - f\Omega_{mec} \qquad (18)$$

The switching component is adopted as:

$$T_{em\_n} = K_{\Omega_{mec}}\mathrm{sign}(S_{\Omega_{mec}}) \qquad (19)$$

Thus, the total control $T_{em\_ref}$ is expressed by:

$$T_{em\_ref} = T_g - J\dot{\Omega}_{mec\_ref} - f\Omega_{mec} + K_{\Omega_{mec}}\mathrm{sign}(S_{\Omega_{mec}}) \qquad (20)$$

In the convergence mode, consider the following the Lyapunov function (Taraft et al., 2015; Yaichi et al., 2019):

$$\dot{V} = S_{mec}\dot{S}_{mec} < 0 \qquad (21)$$

Equation (22) is considered

$$\dot{S}_{\Omega_{mec}} = \frac{K}{J} K_{\Omega_{mec}} \mathrm{sign}(S_{\Omega_{mec}}) \qquad (22)$$

The $K_{\dot{\Omega}_{mec}}$ must be negative.
## Second order sliding mode controller SOSMC

To ensure good tracking and accurate response the second order sliding mode control SOSMC is designed. The super twisting algorithm STA insures all the properties of first order sliding mode control and reduces chattering in the system.

The second derivative of the sliding surface of mechanical speed $S_{\Omega_{mec}}$ gives (Djilali et al., 2020):

$$ \ddot{S}_{\Omega_{mec}} = G_1 + G_2 \dot{T}_{em\_ref} \quad (23) $$

Where: $|G_1| \le C$, $C>0$, $0 < K_m \le G_2 \le K_M$.

According to the STA algorithm, the expression of the reference control variable $T_{em\_ST}$ is defined by the following equation (Borlaug, 2017; Dendouga, 2020):

$$ T_{em\_ST} = K_1 |S_{\Omega_{mec}}|^{0.5} \mathrm{sign}(S_{\Omega_{mec}}) + K_2 \int \mathrm{sign}(S_{\Omega_{mec}})dt \quad (24) $$

Where $K_1$ and $K_2$ are determined according to the inequalities presented as follows (Dendouga, 2020; Djilali et al., 2020):

$$ K_1^2 \ge \frac{4C K_M (K_2 + C)}{k_m^2 K_m (K_2 - C)}, K_2 > \frac{C}{K_m} \quad (25) $$

$K_1$ and $K_2$ are negative constants when the Lyapunov condition is verified ($\dot{V} = S_{mec}\dot{S}_{mec} < 0$). The reference torque is expressed by:

$$ T_{em\_ref} = T_{em\_eq} + T_{em\_ST} \quad (26) $$

Where: $T_{em\_eq}$ is the same obtained by equation (18).

## Third order sliding mode controller TOSMC

To overcome the main drawbacks of FOSMC and SOSMC (the chattering phenomena and the ripples); the TOSMC was proposed (Borlaug, 2017). The TOSMC method is one of the nonlinear methods that have been suggested for controlling electrical systems (Benbouhenni and Bizon, 2021; Borlaug, 2017). As it can be used in the case of linear and non-linear systems due to the simplicity of its algorithm.

The control law of the proposed control TOSMC is given as follows:

$$ \begin{aligned} U_{TOSMC} &= U_{ST} + U_{eq} + U_3 \\ U_{ST} &= U_1 + U_2 \end{aligned} \quad (27) $$

With:

$$ \begin{aligned} U_1 &= K_1 |S|^{0.5} \mathrm{sign}(S) \\ U_2 &= K_2 \mathrm{sign}(S) \\ U_3 &= K_3 \int \mathrm{sign}(S)dt \end{aligned} $$

In this paper, the TOSMC is used to force the mechanical speed to track its reference, the sliding surface of the mechanical speed $S_{\Omega_{mec}}$ is defined as the error between the desired and real value, as expressed in the following equation:

$$ S_{\Omega_{mec}} = \Omega_{mec\_ref} - \Omega_{mec} \quad (28) $$

The output signal for the mechanical speed controller is given as follows:

$$ T_{em\_TOSMC} = K_1 |S_{\Omega_{mec}}|^{0.5} \mathrm{sign}(S_{\Omega_{mec}}) + K_2 \mathrm{sign}(S_{\Omega_{mec}}) + K_3 \int \mathrm{sign}(S_{\Omega_{mec}})dt + T_{em\_eq} \quad (29) $$
<figure><img src="image_8.png" /><figcaption>Figure 6. Scheme of pitch angle control strategy.</figcaption></figure>

Where:

The expression of $T_{em\_eq}$ is the same given by equation (18).

The constants $K_1$, $K_2$, and $K_3$ are used to enhance the performance of the TOSMC controller.

To guarantee the stability condition, the Lyapunov equation must be satisfied (Dendouga, 2020):

$$ \dot{v} = S_{\Omega_{mec}} \dot{S}_{\Omega_{mec}} < 0 \qquad (30) $$

## Control of the blade angle

The gusts of wind with a speed higher than its rated value can damage the wind energy conversion system. Therefore, it is necessary to use the pitch angle control of blades to protect the wind turbine from high power by limiting the power at its nominal value (Dahbi et al., 2016; Hamzaoui et al., 2016). The block diagram of pitch control is shown in Figure 6.

The principle of Pitch angle control is based on comparing the generated power from the wind turbine with the reference power, after that the error between them sent to the PI controller which generates in the output reference value of the angle $\beta$ ref.

The following equation expresses the relationship between the pitch angle $\beta$ and its reference value $\beta_{ref}$:

$$ \beta = \frac{1}{\tau_{pitch} S} \beta_{ref} \qquad (31) $$

With:

$\tau_{pitch}$: is a time constant.

## Control of DFIG

### Direct field-oriented control DFOC

DFOC is one of the most widely strategies to control the DFIG, especially in industrial applications. The objective of this strategy is to control the DFIG in a way that makes its behavior similar to that of the DC motor with separate excitation (Benbouhenni and Bizon, 2021; Taraft et al., 2015).

The principle of a DFOC strategy depends to orient the stator flux along the d axis, to simplify the control of stator power as follows (Benamor et al., 2019; Benbouhenni and Bizon, 2021):

$$ \phi_{sd} = \phi_s \text{ and } \phi_{sd} = 0 \qquad (32) $$

Hence, direct and quadrature stator voltages can be expressed as:

$$ v_{sd} = 0 \text{ and } v_{sq} = v_s = \omega_s \phi_s \qquad (33) $$

After simplifications, the expressions of stator current of the DFIG are given by:
$$ \left\{ \begin{aligned} I_{sd} &= \frac{v_s}{\omega_s L_s} \frac{M_{sr}}{L_s} I_{rd} \\ I_{rq} &= -\frac{M_{sr}}{L_s} I_{rq} \end{aligned} \right. \qquad (34) $$

The expression of stator active and reactive powers are given by (Dekali et al., 2021):

$$ \left\{ \begin{aligned} P_s &= -\frac{3}{2}v_s \frac{M_{sr}}{L_s} I_{rq} \\ Q_s &= \frac{3}{2} \left( \frac{v_s^2}{\omega_s L_s} - v_s \frac{M_{sr}}{L_s} I_{rd} \right) \end{aligned} \right. \qquad (35) $$

The equation (36) represent the electromagnetic torque.

$$ T_{em} = -\frac{3}{2} n_p \frac{M_{sr}}{L_s} \phi_{sd} I_{rq} \qquad (36) $$

The expressions of direct rotor voltage and quadrature rotor voltage of the DFIG are given by the following equations (Dekali et al., 2021):

$$ \left\{ \begin{aligned} V_{rd} &= R_r I_{rd} + L_r \sigma_{dI_{rd}}^d - g \omega_s L_r \sigma I_{rq} \\ V_{rq} &= R_r I_{rq} + L_r \sigma_{dI_{rq}}^d + g \omega_s L_r \sigma I_{rd} + g \frac{v_s M_{sr}}{L_s} \end{aligned} \right. \qquad (37) $$

Where:

$\sigma = 1 - \frac{M_{sr}}{L_r L_s}$ is the dispersion coefficient of the DFIG, $g = \frac{\omega_s - \omega_r}{\omega_s}$ is the slip.

## First order sliding mode control (FOSMC)

In order to track the stator powers of DFIG to their references, the sliding surface of stator active and reactive powers are expressed by (Bounar et al., 2019; Yaichi et al., 2019):

$$ \left\{ \begin{aligned} S(P_s) &= P_{s\_ref} - P_s \\ S(Q_s) &= Q_{s\_ref} - Q_s \end{aligned} \right. \qquad (38) $$

The derivative of (38) are given by (Bounar et al., 2019; Benbouzid et al., 2014):

$$ \left\{ \begin{aligned} \dot{S}_{P_s} &= \dot{P}_{s\_ref} + \frac{3}{2} V_s \frac{M_{sr}}{L_r L_r \sigma} (V_{rq} - R_r I_{rq}) \\ \dot{S}_{Q_s} &= \dot{Q}_{s\_ref} + \frac{3}{2} V_s \frac{M_{sr}}{L_r L_r \sigma} (V_{rd} - R_r I_{rd}) \end{aligned} \right. \qquad (39) $$

The expression of reference control variable is (Benbouzid et al., 2014; Dendouga, 2020):

$$ V_{r_{d,q}} = V_{r_{d,q_{eq}}} + V_{r_{d,q_n}} \qquad (40) $$

In steady state, $\dot{S}_{P_s} = 0$ and $\dot{S}_{Q_s} = 0$ (Dendouga, 2020).

The equivalent voltage component of control will be as follow:

$$ \left\{ \begin{aligned} V_{rq_{eq}} &= R_r I_{rq} - \frac{2}{3} P_{s\_ref} \frac{L_s L_r \sigma}{V_s M_{sr}} \\ V_{rd_{eq}} &= R_r I_{rd} - \frac{2}{3} Q_{s\_ref} \frac{L_s L_r \sigma}{V_s M_{sr}} \end{aligned} \right. \qquad (41) $$

The switching component is given by:

$$ \left\{ \begin{aligned} V_{rq_n} &= K_{P_s} \mathrm{sign}(S_{P_s}) \\ V_{rd_n} &= K_{Q_s} \mathrm{sign}(S_{Q_s}) \end{aligned} \right. \qquad (42) $$

Finally, the total control is written as follows:

$$ \left\{ \begin{aligned} V_{rq_{ref}} &= R_r I_{rq} - \frac{2}{3} \dot{P}_{s\_ref} \frac{L_s L_r \sigma}{V_s M_{sr}} + K_{P_s} \mathrm{sign}(S_{P_s}) \\ V_{rd_{ref}} &= R_r I_{rd} - \frac{2}{3} \dot{Q}_{s\_ref} \frac{L_s L_r \sigma}{V_s M_{sr}} + K_{Q_s} \mathrm{sign}(S_{Q_s}) \end{aligned} \right. \qquad (43) $$
The Lyapunov equation can be given as follows (Bounar et al., 2019):

$$ \dot{v} = S_{Ps, Qs} \dot{S}_{Ps, Qs} < 0 \qquad (44) $$

The conditions (41) are satisfied when:

$$ \text{If } \operatorname{sign}(S_{Ps, Qs}) > 0 \dot{S}_{Ps, Qs} < 0 \qquad (45) $$

$$ \text{If } \operatorname{sign}(S_{Ps, Qs}) < 0 \dot{S}_{Ps, Qs} > 0 \qquad (46) $$

Thus, $K_{Ps}$ and $K_{Qs}$ must be positive.

## Second order sliding mode control (SOSMC)

Equation (47) give the second derivative of the sliding surface of stator powers $S_{Ps}$ and $S_{Qs}$ (Benbouzid et al., 2014; Djilali et al., 2020):

$$ \left\{ \begin{aligned} \ddot{S}_{Ps} &= G_1 + G_2 \dot{V}_{rq\_ref} \\ \ddot{S}_{Qs} &= G_1 + G_2 \dot{V}_{rd\_ref} \end{aligned} \right. \qquad (47) $$

Where: $|G_1| \le C$, $C>0$, $0 < K_m \le G_2 \le K_M$.

When the STA algorithm is applied, the reference control variable of the stator powers is written as in the following equations (Dendouga, 2020; Kelkoul and Boumediene, 2021):

$$ \left\{ \begin{aligned} V_{rqST} &= K_1 |S_{Ps}|^{0.5} \operatorname{sign}(S_{Ps}) + K_2 \int \operatorname{sign}(S_{Ps}) dt \\ V_{rdST} &= K_1 |S_{Qs}|^{0.5} \operatorname{sign}(S_{Qs}) + K_2 \int \operatorname{sign}(S_{Qs}) dt \end{aligned} \right. \qquad (48) $$

Where $K_1$ and $K_2$ are:

$$ K_1^2 \ge \frac{4C K_M (K_2 + C)}{k_m^2 K_m (K_2 - C)}, \quad K_2 > \frac{C}{K_m} \qquad (49) $$

$K_1$ and $K_2$ are positive constants when the Lyapunov condition is verified.

Equation (50) give the expression of the reference voltages generated by the second order sliding regulator as follows:

$$ \left\{ \begin{aligned} V_{rqref} &= V_{rqeq} + V_{rqST} \\ V_{rdref} &= V_{rdeq} + V_{rdST} \end{aligned} \right. \qquad (50) $$

Where: $V_{rd_{eq}}$ and $V_{rq_{eq}}$ are expressed by the same obtained in equation (41).

## Third order sliding mode control (TOSMC)

Between different techniques of control used to improve the performances of the system (chattering reduction, tracking and robustness), the third order sliding mode control TOSMC is proposed to control the stator powers.

Equation (51) give the sliding surface of the stator active and reactive powers, which are selected as the error between the desired and real dynamics as follows (Benbouhenni and Bizon, 2021):

$$ \left\{ \begin{aligned} S(P_s) &= P_{s\_ref} - P_s \\ S(Q_s) &= Q_{s\_ref} - Q_s \end{aligned} \right. \qquad (51) $$

From equation (28), it can be deduced that the control laws applied to control the stator powers are expressed by the reference rotor voltage components as follows (Benbouhenni and Bizon, 2021; Borlaug, 2017):

$$ \left\{ \begin{aligned} V_{rqref} &= K_1 |S_{Ps}|^{0.5} \operatorname{sign}(S_{Ps}) + K_2 \int \operatorname{sign}(S_{Ps}) dt + K_{Ps} \operatorname{sign}(S_{Ps}) + V_{rqeq} \\ V_{rdref} &= K_1 |S_{Qs}|^{0.5} \operatorname{sign}(S_{Qs}) + K_2 \int \operatorname{sign}(S_{Qs}) dt + K_{Qs} \operatorname{sign}(S_{Qs}) + V_{rdeq} \end{aligned} \right. \qquad (52) $$
The expressions of $V_{rd_{eq}}$ and $V_{rq_{eq}}$ are the same given by equation (41).

To guarantee the stability condition, the Lyapunov equation must be verified:

$$ \dot{v} = S_{Ps, Qs} \dot{S}_{Ps, Qs} < 0 \qquad (53) $$

Thus the constants $K_1$, $K_2$, and $K_3$ are positive.

## Venturini switching algorithm for DMC

In this section, the Venturini switching algorithm for the DMC will be presented. In the power electronic laws the voltage sources must never be in a short and the current sources must never be in an open circuit (Dendouga, 2020), therefore possesses 27 possible combinations of switching for DMC (Dendouga and Dendouga, 2019).

The conduction time $t_{ij}$ of the switch defined by (Dendouga, 2010):

$$ t_{Aj} + t_{Bj} + t_{Cj} = T_{seq} \qquad (54) $$

With: $T_{seq}$ the switching sequence of the MC, and $0 < t_{ij} < T_{seq}$.

The duty cycle can be defined as (Hamane et al., 2015):

$$ m_{ij}(t) = \frac{t_{ij}}{T_{seq}} \quad \text{where: } 0 < m_{ij} < 1 $$

According to equation (54) one can give (Casadei et al., 2002):

$$ m_{Aj} + m_{Bj} + m_{Cj} = 1 \qquad (55) $$

Considering the ratio between the output voltage and the input voltage of the DMC (Chaoui et al., 2016; Hamane et al., 2015).

$$ q = \frac{v_o}{v_i} = \frac{i_i}{i_o} \qquad (56) $$

Furthermore, suppose that the desired output voltage is expressed by (Hamane et al., 2015):

$$ V_o(t) = qV_{im} \begin{bmatrix} \cos(w_o t) \\ \cos(w_o t + \frac{2\pi}{3}) \\ \cos(w_o t + \frac{4\pi}{3}) \end{bmatrix} \qquad (57) $$

From Venturini and Alesina (1980) the expression of the modulation matrix $m$ is given by (Casadei et al., 2002; Rodríguez et al., 2005):

$$ m_{ij} = \frac{1}{3} \left[ 1 + \frac{2V_i V_j}{V_{im}^2} \right] \qquad (58) $$

## Damped passive RLC input filter

The input passive filter is an essential component must be placed between the DMC and the grid to generate sinusoidal input currents, to therefore reduce the harmonic distortion rate.

In this paper, the passive filter with a damping resistor connected in parallel with the inductor proposed to increase the damping factor of the LC filter illustrated by Figure 7 (Nguyen et al., 2016).

The voltage and current transfer function of the input filter are given by the following expressions by (Dendouga, 2020; She et al., 2010):

$$ \left\{ \begin{aligned} V_A(S) &= \frac{(L_f S + R_d + R_f) V_{gA}(S) - R_d (L_f S + R_f) I_A(S)}{R_d L_f C_f S^2 + (R_d R_f C_f + L_f) S + (R_d + R_f)} \\ I_{gA}(S) &= \frac{(L_f C_f S^2 + (R_d + R_f) C_f) V_{gA}(S) + (L_f S + R_d + R_f) I_A(S)}{R_d L_f C_f S^2 + (R_d R_f C_f + L_f) S + (R_d + R_f)} \end{aligned} \right. \qquad (59) $$
<figure><img src="image_9.png" /><figcaption>Figure 7. Single-phase circuit of damped input filter RLC for DMC.</figcaption></figure>

The characteristic frequency $\xi$ and the damping factor $\omega_n$ of transfer functions are defined as follows (Dendouga, 2020; She et al., 2010):

$$ \left\{ \begin{aligned} \omega_n &= \sqrt{\frac{R_d + R_f}{R_d L_f C_f}} \\ \xi &= \frac{R_d R_f C_f + L_f}{2 \sqrt{R_d L_f C_f (R_d + R_f)}} \end{aligned} \right. \qquad (60) $$

Considering that damping resistor $R_d \gg (R_d \to \infty)$, and as the resistance of the inductor $R_f$ is usually few ohms; the expression (60) could be simplified as (Dendouga, 2020; She et al., 2010):

$$ \left\{ \begin{aligned} \omega_n &= \frac{1}{\sqrt{L_f C_f}} \\ \xi &= \frac{1}{2R_d} \sqrt{\frac{L_f}{C_f}} \end{aligned} \right. \qquad (61) $$

## Results and discussion

The simulation of variable-pitch wind system (VPWS) based on DFIG is carried out using Matlab/Simulink to validate the performance of the proposed control strategy. Table 1 displays parameters of the DFIG-based VPWS.

In this study, the efficiency of DFIG-based VPWS is investigated under the following two cases:

A. Operating of the variable-pitch wind system in zone 2;

B. Operating of the variable-pitch wind system in all zones.

### Simulation results of control of the variable-pitch wind system in zone 2

In this case, the simulation of the MPPT strategy was implemented to enhance the efficiency of the VPWS by exploiting the maximum energy from the wind. In order to validate the efficiency of the considered wind system with different proposed control strategies, a comparative study was carried out.

In this context, the pitch angle of the blades for the wind turbine is constant $\beta = 2^\circ$ in this case. Figure 8 shows the wind speed profile, which is a random profile changes between 8 and 13 m/s.

When the wind speed is less than the nominal wind speed, the MPPT control strategy is necessary to extract the maximum power from the wind turbine. According to Figure 9, the maximum value of power coefficient $cp$ ($\lambda, \beta$) is approximately 0.35, which corresponds to the optimum value of the tip speed ratio $\lambda_{opt} = 7.1$ for a pitch angle $\beta = 2^\circ$ in spite of the sudden variations of the wind speed.
<table><caption>**Table I.** VPWS-DFIG parameters.</caption><thead><tr><td>Parameters of VPWS-DFIG</td><td>Value</td></tr></thead><tbody><tr><td>Rated power Pn</td><td>7.5 kW</td></tr><tr><td>Number of pair of poles P</td><td>2</td></tr><tr><td>Stator rated voltage Vs</td><td>220/380 V</td></tr><tr><td>Rotor resistance Rr</td><td>0.62 Ω</td></tr><tr><td>Stator resistance Rs</td><td>0.45 Ω</td></tr><tr><td>Stator inductance Ls</td><td>0.084 H</td></tr><tr><td>rotor inductance Lr</td><td>0.081 H</td></tr><tr><td>Mutual inductance Msr</td><td>0.078 H</td></tr><tr><td>Friction coefficient fr</td><td>0.0054 N.m.s<sup>-1</sup></td></tr><tr><td>P_turbine</td><td>7.5 kW</td></tr><tr><td>Rotor radius R</td><td>2.25 m</td></tr><tr><td>Gearbox gain G</td><td>5</td></tr><tr><td>Stator rated frequency fs</td><td>50 Hz</td></tr></tbody></table>

<figure><img src="image_10.png" /><figcaption>**Figure 8.** Random wind speed profile.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 9.** Represents the power coefficient in Zone 2.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>**Figure 10.** Represents the Rotor speed in Zone 2.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Figure 11.** Represents the mechanical power in Zone 2.</figcaption></figure>

The change of the power coefficient confirms that the three controllers tracked the maximum value $C_{pmax} = 0.35$. However, the TOSMC demonstrated faster response time and precision compared to the SOSMC and FOSMC.

The rotor speed of the turbine is shown in Figure 10 which is controlled in order to optimize the power in zone 2. It can be said that the speed is variable and takes the same shape as the wind speed change. As well the mechanical speed perfectly tracks its reference value for the three controllers, while the TOSMC has a fast response time compared to the other controllers.

Figure 11 shows the mechanical power generated by the turbine using three different types of controllers for the MPPT strategy, as it can be noted that all controllers attained the maximum exploitation of the power in spite of variations in the wind speed. However, the TOSMC has fast response time compared to the SOSMC and FOSMC.

Consequently, the obtained results clearly indicate the effectiveness of the dynamic performance of the TOSMC over the others controller under the stochastic variations of the wind speed.

## Simulation results of full control of the variable-pitch wind system in all zones

In this case, the whole system of the variable-pitch wind energy system based on a doubly-fed induction generator fed by a direct matrix converter was carried out under MATLAB/Simulink environment.

The control system relies on the following approaches:
<figure><img src="image_14.png" /><figcaption>**Figure 12.** Random profile of high wind speed.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Figure 13.** Represents the power coefficient in all Zones.</figcaption></figure>

In the first approach, the MPPT strategy was proposed in order to extract the maximum power from the wind turbine when the wind speed is lower than the nominal speed on the one side. On the other side, the Pitch control was designed to adjust the blade angle so that the maximum rated power for the wind turbine is not exceeded. In this way, the protection of the system against high wind speeds is ensured. For this reason, a TOSMC controller was used to control the speed of the turbine due to its exceptional advantages compared to other controllers.

The second approach consists of the direct flied-oriented control (DFOC) of active and reactive powers for the DFIG using third order sliding mode controller TOSMC, while controlling the direct matrix converter by Venturini modulation algorithm.

The following comparative study giving below were done to test the robustness and performance of the proposed TOSMC controller under the stochastic variation of the wind speed.

In order to control the functioning of the wind turbine in all zones of operating and validate the dynamic performances of the system studied, a random wind speed was applied, in which the wind speed profile simulated varies between 8 and 16 m/s as shown in Figure 12.

It can be seen clearly from Figure 13, that the power coefficient cp attains its maximum value $cp\_max = 0.35$ with applied the MPPT strategy (Zone 2) when the wind speed is lower than the rated value ($V < Vn = 13$ m/s). While it decreases when the wind speed exceeds the rated speed ($V > Vn = 13$ m/s) by applying the pitch control technique (Zone 4).

Figure 14 illustrates the mechanical speed of the wind turbine which is controlled in order to maximize the power generated by the turbine using the MPPT strategy in Zone 2 and to be rated by applying the pitch control in Zone 4.
<figure><img src="image_16.png" /><figcaption>**Figure 14.** Represents the Rotor speed in all Zones.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Figure 15.** Represents the mechanical power in all Zones.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**Figure 16.** Pitch angle.</figcaption></figure>

Figure 15 shows the power generated by the wind turbine. One can notice that the pitch angle control of the blades (β) intervenes (in Zone 4) when the wind speed is higher than the rated wind speed to limit the power to the nominal value (7.5 kW).

From Figure 16, one can see that the pitch angle value increases when the wind speed exceeds the rated value (13 m/s), in order to protect the wind turbine by limiting the generated power to its nominal value (7.5 kW). While, when the wind speed is lower than the rated wind speed the pitch angle value is equal 2°.
<figure><img src="image_19.png" /><figcaption>**Figure 17.** Stator active power of DFIG.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>**Figure 18.** Stator reactive power of the DFIG.</figcaption></figure>

The active and reactive powers generated by the DFIG are presented in Figures 17 and 18 respectively, the Ps and Qs perfectly follow their references for the three controllers. While it can be seen that the TOSMC type controller gives good tracking of the stator powers to their references with low overshoot compared to FOSMC and SOSMC controllers with considerable overshoot for the different control modes (MPPT and Pitch Control). In order to optimize the quality of the electrical energy generated and to ensure operation with a unity power factor on the stator side, the reactive stator power Qs is kept at null value (Qs = 0 (VAR)). Additionally, the active power takes the same shape as the wind speed change, while the Qs is not affected by the wind speed change.

The electromagnetic torque of the DFIG illustrates in Figure 19, for (FOSMC, SOSMC, and TOSMC controllers) and its reference obtained from the MPPT strategy based on TOSMC. It can be seen that the TOSMC makes it possible to obtain good tracking of the electromagnetic torque at its reference for the different modes of control (MPPT and Pitch Control).

From Figure 20, it can be seen that the waveform of rotor voltage per phase (at the output of the DMC) is formed by a succession of input voltage pulses (at the input of the DMC) by using a Venturini switching algorithm.

Figures 21 and 22 illustrate the stator current of DFIG and grid current, It can see that the three phases are sinusoidal with a frequency equal to $f = 50$ (Hz). However, the TOSMC controller improves the waveform compared to FOSMC and SOSMC controllers.

Figure 23 shows the THD of stator current obtained for the three controllers. It can see that the fundamental frequency is equal to 50 Hz and the THD is acceptable in the case of the TOSMC controller (1.06%; compliant with IEEE standard) compared to the other controllers: FOSMC (1.38%; compliant with IEEE standard) and
<figure><img src="image_21.png" /><figcaption>Figure 19. Electromagnetic torque of the DFIG.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>Figure 20. Rotor voltage $V_{ra}$: (a) FOSMC, (b) SOSMC, and (c) TOSMC.</figcaption></figure>

SOSMC (1.28%; compliant with IEEE standard). Which makes it possible to inject clean energy with a low THD into the grid.

The waveform of the input current of the DMC illustrates in Figure 24, which is deduced by using the Venturini switching algorithm.

Table 2 presents a comparative summary of the proposed TOSMC strategy and other strategies used in this work.

According to the results illustrated in Table 2, it is confirmed that the TOSMC strategy is the most efficient in terms of good tracking and minimizing ripples of different curves (powers, electromagnetic torque, and stator currents) compared to the other strategies of control such as FOSMC and SOSMC. In addition, the TOSMC strategy
<figure><img src="image_23.png" /><figcaption>Figure 21. Stator currents Is: (a) FOSMC, (b) SOSMC, and (c) TOSMC.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>Figure 22. Grid currents: (a) FOSMC, (b) SOSMC, and (c) TOSMC.</figcaption></figure>
<figure><img src="image_25.png" /><figcaption>Figure 23. Harmonic spectrum of stator current: (a) FOSMC, (b) SOSMC, and (c) TOSMC.</figcaption></figure>

ensures a good quality of power conversion between the DFIG and the grid with a reduced total harmonic distortion rate THD under the sudden variations in the wind speed.

## Conclusion

This article deals with the development, design and implementation of the TOSMC strategy for the control of active and reactive powers generated by the variable pitch wind system based on DFIG.

In this context, full control of the wind turbine has been developed, as a first approach; the MPPT strategy based on a third-order sliding mode controller (TOSMC) has been designed in order to ensure maximum exploitation of the wind energy, and as a second approach; control of the pitch angle has been implemented in order to limit the power extracted to its maximum value for the safe operating of the system when the wind speed exceeded to the rated speed of the wind turbine.

In order to ensure direct energy conversion without needing a DC-link between the grid and the DFIG as in traditional power converters, a direct matrix converter controlled by the strategy of Venturini was proposed. Moreover, the RLC input passive filter with a damping resistor connected in parallel with the inductor was designed to obtain an almost sinusoidal waveform for the current with a low rate of total harmonic distortion (THD).

The TOSMC strategy has been tested and evaluated by comparing it with the other control techniques carried out using a random profile of the wind speed which changes between 8 and 16 m/s under Matlab/Simulink.

The simulation results demonstrate that the proposed TOSMC controller gives good performances compared to the FOSMC and SOSMC in terms of the dynamic response, tracking reference, precision, and THD of the injected currents into the grid under random wind speed.

The outcomes describe the robustness of the proposed MPPT technique using TOSMC combined with pitch angle control at different operating wind speed ranges, which enhances the efficiency of the power generation in the wind turbine system. As well as, the simulation study has confirmed that the Venturini modulation algorithm has high effectiveness in terms of the acceptable waveform, high transformation ratio (0.86), and simplicity of
<figure><img src="image_26.png" /><figcaption>**Figure 24.** Input DMC current IA: (a) FOSMC, (b) SOMC, and (c) TOSMC.</figcaption></figure>

<table><caption>**Table 2.** Comparative study between the proposed TOSMC strategy and other strategies of control.</caption><thead><tr><th>Performance criteria</th><th>FOSMC</th><th>SOSMC</th><th>TOSMC</th></tr></thead><tbody><tr><td>Reactive and active power tracking</td><td>Good</td><td>Good</td><td>Excellent</td></tr><tr><td>Minimization of reactive and active power ripples</td><td>Good</td><td>Good</td><td>Very good</td></tr><tr><td>Minimization of stator current ripples</td><td>Good</td><td>Good</td><td>Very good</td></tr><tr><td>Torque tracking</td><td>Good</td><td>Good</td><td>Excellent</td></tr><tr><td>Dynamic response</td><td>Medium</td><td>Medium</td><td>Fast</td></tr><tr><td>MPPT tracking</td><td>Good</td><td>Very good</td><td>Excellent</td></tr><tr><td>THD (%)</td><td>1.38% (&lt;5%)</td><td>1.28% (&lt;5%)</td><td>1.06% (&lt;5%)</td></tr><tr><td>Quality of stator current</td><td>Good</td><td>Very good</td><td>Excellent</td></tr><tr><td>Performance</td><td>Medium</td><td>Medium</td><td>High</td></tr></tbody></table>

implementation. The use of a damped RLC passive filter combined with the direct matrix converter minimizes the rate of harmonics THD hence, it is possible to ensure a clean production of electrical energy from the wind turbine system.

Future research needs to focus on the control of the power converter, improving wind turbine system efficiency using developed control strategies and more efficient and robust. Furthermore, an experimental implementation of the proposed drive system can be accomplished.
**Declaration of conflicting interests**

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

**Funding**

The author(s) received no financial support for the research, authorship, and/or publication of this article.

**ORCID iD**

Amal Dendouga https://orcid.org/0000-0003-2202-2035

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