A robust control for a variable wind speed conversion system energy based on a DFIG using Backstepping

Abstract

This paper presents a modeling and robust control of the DFIG (doubly fed induction generator) used in the wind energy conversion system (WECS). We started by using the MPPT method to extract the maximum power in the WECS, modeling the double-fed inductor generator, and then applying the Backstepping controller to control the reactive power and electromagnetic torque in order to test the performance and the robustness of the system. All is simulated and presented in MATLAB/SIMULINK software.

Keywords

Wind turbine MPPT control modeling DFIG Backstepping

Introduction

Recently, three-bladed horizontal axis wind turbines based on a doubly fed induction generator (DFIG) are the most commonly used in the wind power system industry (Singh et al., 2019a, 2019c), the structure of these systems is based on the installation of two electronic power converters, a converter of generator-side power and a grid-side power converter, these are used to allow the DFIG to operate at variable speeds (Singh et al., 2018).

The modeling of any system is an important and necessary phase for the study and control of its operation (Chetouani et al., 2021; Jenkal et al., 2020), in this work we will approach the modeling of the elements of the wind system, necessary for the transition from a kinetic movement of the wind to electrical energy. All the constituent elements of the wind power chain have been developed and modeled to simulate them in the Simulink software, to visualize the temporal evolution of its various representative variables, to study the behavior of the wind power system, and to be able to decide control laws and control techniques that will be adapted to the system studied.

There is the PI method for its reliability and its simplicity of implementation in different control boards, so it keeps its performance (Poitier, 2003) for the cases of linear systems with constant parameters, but for nonlinear systems which require a certain precision and stability (Singh and Sundaram, 2021), it remains insufficient and suffers from instability during a large fluctuation of the system parameters (Singh et al., 2019d). For these reasons, in our thesis work, we have used a second command as a solution to the problems.

Relating to the first, the nonlinear Backstepping command is one of the commands that have proven their stability, performance, and robustness against disturbances.

We will discuss our first research activity devoted to the study of turbine control by the MPPT method to recover the maximum power produced and optimize energy efficiency.

During the second part, we will first talk about the theoretical tools necessary to present control algorithms such as the notions of stability in the sense of Lyapunov and the Backstepping algorithm. Then, we will detail the mathematical development which led to the synthesis of the Backstepping’s command laws, we will finish by representing the results of simulation for the possibilities of fluctuations of the generator’s internal parameters.

Wind turbine

Currently, the variable-speed wind system based on a double-fed induction generator (DFIG) is the most widely used (Singh et al., 2021). Its main advantage is to have its three-phase static converters sized for part of the nominal power of the DFIG, which generates a significant economic benefit compared to other solutions used (e.g. synchronous generator with permanent magnets) (Anaya-Lara et al., 2009; Baroudi et al., 2007; Singh and Sundaram, 2022). The wind turbine drives via a speed multiplier, where it’s connected directly to the electrical network by the stator and by the rotor through two three-phase static converters controlled in pulse width modulation (PWM) (Singh and Sundaram, 2021); one on the rotor side of DFIG (Singh et al., 2019b) called rotor side converter (RSC) and the other on the power grid side called grid side converter. In this part, we model the wind conversion chain, and we present the DFIG model in the Park model (d, q).

Wind turbine modeling

The wind power corresponding to an undisturbed wind is expressed in the equation below (Errami et al., 2020):

P_{v} = \frac{ρ s v^{3}}{2}

(1)

The aerodynamic torque appearing at the turbine is a function of this power:

\begin{matrix} P_{a e r o} = C_{p} (λ, β) * P_{v} = \frac{1}{2} * π * R^{2} * v^{3} * C_{p} (λ, β) \end{matrix}

(2)

The evolution of the power coefficient $C_{p}$ depends on the aerodynamic characteristics of the wind turbine as well as the operating conditions (Poitier, 2003). For a fixed pitch angle $β$ , the power coefficient $C_{p}$ can be expressed as a function of the speed ratio $λ$ . For a variable pitch angle, the power coefficient can be expressed as a function of $λ$ and $β$ , which is given as (Yessef et al., 2022c):

λ = \frac{Ω_{t} * R}{ν}

(3)

where, $P_{v}$ , is the captured power of the wind turbine, ρ, is the air density, $ν$ , is the wind speed, Ω, is the angular speed (m/s) and R is the length of the blade (m)

Albert Betz calculated the maximum of $C_{p}$ as following (Boualouch et al., 2015):

C_{p max} (λ, β) = \frac{16}{27} = 0.5926

(4)

The expression of this power coefficient has been approached for this type of turbine, by the following equation (Mahvash et al., 2019):

C_{p} (λ, β) = (0.5 - 0.0167 * (β - 2)) * \sin (\frac{π * (λ + 0.1)}{18.5 - 0.3 * (β - 2)}) - 0.00184 * (β - 2) * (λ - 3)

(5)

The aerodynamic torque is given as:

C_{t} = \frac{P_{a e r o}}{Ω_{t}} = C_{p} (λ, β) * \frac{π * R^{2} * v^{3}}{2 * Ω_{t}}

(6)

The gearbox is utilized as a connection, to adjust the speed of the turbine to that of the generator. The energy losses (Bouderbala et al., 2019), versatility and grating in the gearbox are disregarded so:

Ω_{m e c} = Ω_{t} * G

(7)

C_{t} = C_{g} * G

(8)

The fundamental equation of dynamics is giving as (Yessef et al., 2022b):

T_{m e c} = J * \frac{d Ω_{m e c}}{dt} = T_{g} - T_{e m} - f * Ω_{m e c}

(9)

MPPT control

The variable-speed wind turbine increases the energy efficiency and improves the quality of the energy produced compared to that operating at a fixed speed (Errami et al., 2020; Lamnadi et al., 2016). To maximize the power captured, the maximum power extraction technique (MPPT) is applied.

Figure 1 shows the principle of MPPT control of wind turbine, the ratio of the speed of wind λ must be maintained at its optimum value $λ = λ_{o p t}$ on a certain wind speed range. Thus, the power coefficient would be maintained at its maximum value. The control objective is to optimize the capture wind energy by tacking the optimal torque:

T_{e m} = \frac{1}{2} * π * R^{5} * \frac{C_{p max}}{λ_{opti}^{3}}

(10)

Figure 1.

Diagram of the wind turbine and MPPT control.

DFIG modeling

The modeling of the DFIG is similar to the induction generator. The DFIG dynamic equation of a three-phase can be written in an asynchronous rotation direct-quadrature (d-q) reference frame as (Jenkal et al., 2020; Mensou et al., 2018; Yessef et al., 2022a):

The electrical equations:

{\begin{matrix} V_{s d} = R_{s} I_{s d} + \frac{d ϕ_{s d}}{dt} - ω_{s} ϕ_{s q} \\ V_{s q} = R_{s} I_{s q} + \frac{d ϕ_{s q}}{dt} + ω_{s} ϕ_{s d} \\ V_{r d} = R_{r} I_{r d} + \frac{d ϕ_{r d}}{dt} - ω_{r} ϕ_{r q} \\ V_{r q} = R_{r} I_{r q} + \frac{d ϕ_{r q}}{dt} - ω_{r} ϕ_{r d} \end{matrix}

(11)

{\begin{matrix} I_{s d} = \frac{1}{σ L_{s}} ϕ_{s d} - \frac{M}{σ L_{s} L_{r}} ϕ_{r d} \\ I_{s q} = \frac{1}{σ L_{s}} ϕ_{s d} - \frac{M}{σ L_{s} L_{r}} ϕ_{r d} \\ I_{r d} = - \frac{M}{σ L_{s} L_{r}} ϕ_{s d} + \frac{1}{σ L_{r}} ϕ_{r d} \\ I_{r q} = - \frac{M}{σ L_{s} L_{r}} ϕ_{s q} + \frac{1}{σ L_{r}} ϕ_{r q} \end{matrix}

(12)

The magnetic equations:

\begin{matrix} ϕ_{s d} = L_{s} I_{s d} + L_{m} I_{r d} \\ ϕ_{s q} = L_{s} I_{s q} + L_{m} I_{r q} \\ ϕ_{r d} = L_{r} I_{r d} + L_{m} I_{s d} \\ ϕ_{r d} = L_{r} I_{r q} + L_{m} I_{s q} \end{matrix}

(13)

The stator/rotor active and reactive power are giving as:

\begin{matrix} P_{s} = V_{s d} I_{s d} + V_{s q} I_{s q} \\ Q_{s} = V_{s q} I_{s d} - V_{s d} I_{s q} \\ P_{r} = V_{r d} I_{r d} + V_{r q} I_{r q} \\ Q_{r} = V_{r q} I_{r d} - V_{r d} I_{r q} \end{matrix}

(14)

The electromagnetic torque is expressed as:

T_{e m} = p * (I_{s q} * ϕ_{s d} + I_{s d} * ϕ_{s q})

(15)

Where:

$V_{s d}, V_{s q}$ : stator voltages in the d-q axis reference frame.

$V_{r d}, V_{r q}$ : rotor voltages in the d-q axis reference frame.

$I_{s d}, I_{s q}$ : stator currents in the d-q axis reference frame.

$I_{r d}, I_{r q}$ : rotor currents in the d-q axis reference frame.

$L_{s}, L_{r}, L_{m}$ : self and mutual inductances of stator and rotor windings, respectively.

$R_{r}, R_{s}$ : the stator and rotor resistance.

$φ_{s d}, φ_{s q}$ : stator flux linkages in d-q axis reference.

$φ_{r d}, φ_{r q}$ : rotor flux linkages in d-q axis reference.

p: number of machine pole pairs.

Power control strategy

To simplify the study of the power control strategy, we use a vector control of DFIG based on Stator Field Oriented (SFO), by setting the stator field matched up with d-axis as shown in Figure 2.

Choice of reference:

Figure 2.

abc to dq.

By using the stator flux-oriented principle, the stator flux is oriented on the d-axis, then the flux q-axis component $ϕ_{s d} = ϕ_{s}$ and $ϕ_{s q} = 0$ . Thus, the stator resistance was neglected a realistic hypothesis for high-power machines used for wind production, the equations of the machine’s stator voltages are reduced to:

\begin{matrix} V_{s d} = \frac{d ϕ_{s d}}{dt} \\ V_{s q} = ω_{s} * ϕ_{s} \end{matrix}

(16)

With the hypothesis of constant stator flux, one obtains:

\begin{matrix} V_{s d} = 0 \\ V_{s q} = ϕ_{s} * ω_{s} \end{matrix}

(17)

For the control of the generator, expressions are established showing the relationship between the currents and the rotor tensions that will be applied to it:

\begin{matrix} V_{r d} = R_{r} I_{r d} + (L_{r} - \frac{L_{m}^{2}}{L_{s}}) * \frac{d I_{r d}}{dt} - (L_{r} - \frac{L_{m}^{2}}{L_{s}}) * ω_{r} I_{r q} \\ Vq = R_{r} I_{r q} + (L_{r} - \frac{L_{m}^{2}}{L_{s}}) * \frac{d I_{r q}}{dt} - (L_{r} - \frac{L_{m}^{2}}{L_{s}}) * ω_{r} I_{r d} + ω_{r} * \frac{L_{m}}{L_{s}} * ϕ_{s} \end{matrix}

(18)

The statoric power is controlled by the rotor voltages (Figure 3), it is an independent control of active and reactive powers on the d-q reference frame, we can write as:

\begin{matrix} P_{s} = V_{s d} I_{s d} + V_{s q} I_{s q} = - V_{s} * (\frac{L_{m}}{L_{s}}) * I_{r q} \\ Q_{s} = V_{s q} I_{s d} - V_{s d} I_{s q} = V_{s} * \frac{ϕ_{s}}{L_{s}} - V_{s} * (\frac{L_{m}}{L_{s}}) * I_{r d} \end{matrix}

(19)

Figure 3.

Diagram block of DFIG.

Backstepping control technique

The technique of BACKSTEPPING consists in making a nonlinear complex dynamic system equivalent to simple subsystems of order 1, in cascade and stable in the sense of LYAPUNOV, which gives them a good quality of robustness (Bossoufi et al., 2021). The BACKSTEPPING command is a multi-step method, in each step, a virtual command is generated and intermediate command laws are developed to ensure the convergence of the system toward a stable equilibrium state, this can be achieved from the functions of LYAPUNOV which ensure step by step the stabilization of each synthesis step.

To facilitate the application of this technique, it is necessary to make the model of the system in strict parametric form as shown in Figure 4, that is to say, that the derivative of each variable of the state vector must be a function of the preceding variables and which depends thereafter on the next variable.

The system should take the following form:

\begin{matrix} x_{1}^{•} = f_{1} (x_{1}) + g_{1} (x_{1}) x_{2} \\ x_{2}^{•} = f_{2} (x_{1}, x_{2}) + g_{2} (x_{1}, x_{2}) x_{3} \\ x_{3}^{•} = f_{3} (x_{1}, x_{2} x_{3}) + g_{3} (x_{1}, x_{2}, x_{3}) x_{4} \end{matrix}

(20)

x_{n}^{•} = f_{n} (x_{1}, x_{2} x_{3} \dots x_{n}) + g_{n} (x_{1}, x_{2}, x_{3} \dots x_{n}) u

Figure 4.

Diagram block of Backstepping Control of the DFIG.

With:

x(t) represents the n state variables.

u represents the control variable.

To illustrate the principle of Backstepping, we consider a time-varying, dynamic and non-linear system of order 2, which is given by:

\begin{matrix} x_{1}^{•} = f_{1} (x_{1}) + g_{1} (x_{1}) x_{2} \\ x_{2}^{•} = f_{2} (x_{1}, x_{2}) + g_{2} (x_{1}, x_{2}) x_{3} \\ y = x_{1} \end{matrix}

(21)

The purpose of this method is to control the quantity $x_{1}$ , and to forward the output $y = x_{1}$ of the system to a reference noted $y_{ref} (t)$ , the design of the command will be developed in two stages according to the order number of the system (Loucif, 2021).

First step:

We choose the first equation, where the variable $x_{2}$ is estimated as a virtual control variable:

\begin{matrix} x_{1}^{•} = f_{1} (x_{1}) + g_{1} (x_{1}) x_{2} \\ y = y_{ref} = (x_{1})_{d} \end{matrix}

(22)

We then define the error variable $e_{1}$ :

e_{1} = (x_{1})_{d} - x_{1}

(23)

We calculate its derivative:

e_{1}^{•} = (x^{•})_{d} - x_{1}^{•}

(24)

Consider the LYAPUNOV control function obtained by the equation below:

V_{1} (e_{1}) = \frac{1}{2} e_{1}^{2}

(25)

We then calculate its total derivative:

\begin{matrix} V_{1}^{•} = e_{1} e_{1}^{•} = e_{1} ({(x_{1}^{•})}_{d} - x_{1}^{•}) = e_{1} ({(x_{1}^{•})}_{d} - f_{1} (x_{1}) - g_{1} (x_{1}) x_{2}) \\ W i t h V_{1}^{•} (e_{1}) = - k_{1} e_{1}^{2} \end{matrix}

(26)

To guarantee the stability of the control, the derivative of the error must be negative:

(x_{1}^{•})_{d} - f_{1} (x_{1}) - g_{1} (x_{1}) x_{2} < 0

(27)

And $k_{1}$ >0

Which will allow us to calculate the virtual control variable $x_{2}$ :

(x_{2})_{d} = \frac{k_{1} e_{1} - f_{1} (x_{1}) + {(x_{1}^{•})}_{d}}{g_{1} (x_{1})}

(28)

Second step:

We now choose the overall system:

\begin{matrix} x_{1}^{•} = f_{1} (x_{1}) + g_{1} (x_{1}) x_{2} \\ x_{2}^{•} = f_{2} (x_{1}, x_{2}) + g_{2} (x_{1}, x_{2}) u \end{matrix}

(29)

Now the real command u, will appear in this step, we must follow the same steps to calculate the real control variable u, but this time with a new Lyapunov function, which is of the form (Mensou, 2017):

V (e_{1} e_{2}) = \frac{1}{2} e_{1}^{2} + \frac{1}{2} e_{2}^{2}

(30)

The total derivative of Lyapunov’s function:

V^{•} (e_{1} e_{2}) = e_{1} e_{1}^{•} + e_{2} e_{2}^{•} = - k_{1} e_{1}^{2} - k_{2} e_{2}^{2}

(31)

With $V^{•} (e_{1} e_{2}) < 0$ and $k_{2}$ >0

We can deduce the true control variable u:

u = \frac{k_{2} e_{2} - f_{2} (x_{1}, x_{2}) + {(x_{2}^{•})}_{d}}{g_{2} (x_{1}, x_{2})}

(32)

Stator reactive power and torque control

Torque and reactive power errors:

\begin{matrix} e_{1} = T_{e m}^{*} - T_{e m} \\ e_{2} = Q_{s}^{*} - Q_{s} \end{matrix}

(33)

The derivative of torque and reactive power errors:

\begin{matrix} e_{1}^{•} = T_{e m}^{• *} - T_{e m}^{•} \\ e_{2}^{•} = Q_{s}^{• *} - Q_{s}^{•} \end{matrix}

(34)

\begin{matrix} e_{1}^{•} = T_{e m}^{• *} + \frac{p L_{m} V_{s}}{L_{s} ω_{s}} * (- B I_{r q} - ω_{r} I_{r d} - C ϕ_{s} ω_{r} + A V_{r q}) \\ e_{2}^{•} = Q_{s}^{• *} + \frac{L_{m} V_{s}}{L_{s}} * (- B I_{r q} + ω_{r} I_{r q} + A V_{r d}) \end{matrix}

(35)

We use the “Lyapunov” function, to ensure the stability of the control (Mensou, 2017):

V_{1} = \frac{1}{2} e_{1}^{2} + \frac{1}{2} e_{2}^{2}

(36)

The derivative of Lyapunov’s candidate function:

V_{1}^{•} = e_{1} (T_{e m}^{• *} + \frac{p L_{m} V_{s}}{L_{s} ω_{s}} * (- B I_{r q} - ω_{r} I_{r d} - C ϕ_{s} ω_{r} + A V_{r q})) + e_{2} (Q_{s}^{• *} + \frac{L_{m} V_{s}}{L_{s}} * (- B I_{r q} + ω_{r} I_{r q} + A V_{r d}))

(37)

To guarantee the stability and good control of the system, the derivative of the Lyapunov function must be negative and the gains $k_{1}$ , $k_{2}$ positive (Drhorhi, 2020):

V_{1}^{•} (e_{1}) = - k_{1} e_{1}^{2} - k_{2} e_{2}^{2} < 0

(38)

Virtual rotor current control variables:

\begin{matrix} I_{r q}^{*} = (\frac{L_{s} ω_{s}}{Bp V_{s} L_{m}} (T_{e m}^{*} + k_{1} e_{1}) + \frac{1}{R_{r}} (V_{r q} - \frac{ω_{r}}{A} I_{r d} - \frac{C}{A} ϕ_{s} ω_{r})) \\ I_{r d}^{*} = (\frac{L_{s}}{B V_{s} L_{m}} (Q_{S}^{*} + k_{2} e_{2}) + \frac{1}{R_{r}} (V_{r d} + \frac{ω_{r}}{A} I_{r q})) \end{matrix}

(39)

Control of rotor currents

Rotor current errors:

\begin{matrix} e_{3} = I_{r q}^{*} - I_{r q} \\ e_{4} = I_{r d}^{*} - I_{r d} \end{matrix}

(40)

The derivative of the errors of the currents:

\begin{matrix} e_{3}^{•} = I_{r q}^{• *} - I_{r q}^{•} \\ e_{4}^{•} = I_{r d}^{• *} - I_{r d}^{•} \end{matrix}

(41)

We replace the derivative of the currents by their expressions:

\begin{matrix} e_{3}^{•} = I_{r q}^{• *} - (α V_{r q} - δ I_{r q} - ω_{r} I_{r d} - γ ϕ_{s} ω_{r}) \\ e_{4}^{•} = I_{r d}^{• *} - (α V_{r d} - δ I_{r d} + ω_{r} I_{r q})) \end{matrix}

(42)

We use the “Lyapunov” function, to ensure the stability of the control:

V_{2} = \frac{1}{2} e_{1}^{2} + \frac{1}{2} e_{2}^{2} + \frac{1}{2} e_{3}^{2} + \frac{1}{2} e_{4}^{2}

(43)

The derivative of Lyapunov’s candidate function:

\begin{matrix} V_{2}^{•} = e_{1} e_{1}^{•} + e_{2} e_{2}^{•} + e_{3} e_{3}^{•} + e_{4} e_{4}^{•} \\ V_{2}^{•} = e_{1} e_{1}^{•} + e_{2} e_{2}^{•} + e_{3} (I_{r q}^{• *} - (A V_{r q} - B I_{r q} - ω_{r} I_{r d} - C ϕ_{s} ω_{r})) + e_{4} (I_{r d}^{• *} - (A V_{r d} - B I_{r d} + ω_{r} I_{r q})) \end{matrix}

(44)

With $k_{1}$ , $k_{2}$ , $k_{3}$ and $k_{4}$ as positive constants

\begin{matrix} V_{2}^{•} = - k_{1} e_{1}^{2} - k_{2} e_{2}^{2} - k_{3} e_{3}^{2} - k_{4} e_{4}^{2} \leq 0 \\ V_{r q}^{*} = \frac{1}{α} (\frac{L_{s} ω_{s}}{δ p V_{s} L_{m}} (T_{e m}^{• *} + k_{1} e_{1}) + \frac{1}{R_{r}} (V_{r q} - \frac{ω_{r}}{A} I_{r d} - \frac{γ}{A} ϕ_{s} ω_{r}) + k_{3} e_{3} + A) \\ V_{r d}^{*} = = \frac{1}{α} (\frac{L_{s}}{δ V_{s} L_{m}} (Q_{s}^{• *} + k_{2} e_{2}) + \frac{1}{R_{r}} (V_{r d} + \frac{ω_{r}}{A} I_{r q}) + k_{4} e_{4} + B) \end{matrix}

(45)

Simulations and results

Backstepping control DFIG results

We apply a random wind profile, which changes between 8.5 and 13 m/s. It shows clearly in Figure 5. The parameters of the system are given in Table 2.

Figure 5.

Wind speed (m/s).

For the Figures 6 to 9, the control has as input the torque error, the reactive stator power error and the rotor current error $I_{r}$ in d-q. The output makes it possible to evaluate the reference vector to drive the torque, the reactive stator power, and the rotor currents $I_{r}$ according to d-q toward their reference values during a period of time. In this simulation results, we observe that the errors converge toward zero, which means that our method control is robust.

Figure 6.

The electromagnetic torque error.

Figure 7.

Reactive stator power error.

Figure 8.

Rotor current $I_{r d}$ error.

Figure 9.

Rotor current $I_{r q}$ error.

The reactive stator power (Figure 10) has a zero value, to ensure a unity factor on the network side, we also notice that the electromagnetic torque (Figure 11) follows its reference setpoint, which is extracted from the MPPT control, in addition the electromagnetic torque varies in the same way as the wind speed to make our machine run at a speed optimal which allows obtaining the maximum power, and in Figure 12 we can see that the current $I_{r q}$ follows its reference.

Figure 10.

Reactive stator power and its reference.

Figure 11.

The electromagnetic torque and its reference.

Figure 12.

Current roto $I_{r q}$ and its reference.

Backstepping control performance and Robustness

To evaluate the robustness of the proposed control command, the parameters of the generator were varied, the resistance values $R_{r}$ and $R_{s}$ are increased by 400% of their nominal value.

Figure 13 shows the variation of the reactive stator power, current rotor $I_{r q}$ , and the torque. In these results, we can observe that the fluctuations of the generator parameters have kept the same performance in terms of precision and tracking of the setpoint. It can therefore be seen that the Backstepping control is robust against changes in the parameters of the DFIG.

Figure 13.

Robustness test: (a) reactive stator power, (b) the electromagnetic torque, and (c) current rotor Irq.

To demonstrate the advantages and the validity of the proposed control it is interesting to compare it with other techniques in the literature. It is noted here that these techniques were not performed under the same conditions because it is very difficult to find different results under the same conditions. From Table 1, it can be seen that the proposed control-based offers a low total harmonic distortion (THD = 7.51%) compared with the other techniques.

Table 1.

Comparison of different technical methods.

Publication paper	Control strategy	The rise time (s)	The response time (s)	Error (%)	THD (%)
Bossoufi et al. (2014)	Fuzzy SMC	-	0.32	-	-
Chakib (2018)	SMC based backstepping	-	0.5	-	2.98
Yang et al. (2019)	DTC	-	0.2	-	18.8
Yang et al. (2019)	FSC	-	0.16	-	8.26
Yang et al. (2019)	MPDC	-	0.15	-	8.17
Alami et al. (2022)	Sliding mode control	-	0.015	0.15	1.25
Rhaili et al. (2020)	Sliding mode control for PMSG	-	0.1	-	-
Proposed technique	Backstepping	0.01	0.045	0.42	7.51

Table 2.

Parameter of wind turbine and DFIG.

Denotation	The numerical value of the parameter
Rated power	P = 1500w
Stator resistance	$R_{s}$ = 1.18Ω
Rotor resistance	$R_{r}$ = 1.66Ω
Stator inductance	$L_{s}$ = 0.20H
Rotor inductance	$L_{r}$ = 0.18H
Mutual inductance	M = 0.17H
Frequency	f = 50HZ
Gain multiplier	G = 2
Inertia total moment	J = 0.04kg.m²
Radius of the turbine	R = 1m
Optimal tip speed ratio	$λ_{o p t}$ = 802
Maximal power coefficient	$c_{p max}$ = 0.5

In addition, it presents acceptable results interns of rapidity and precision in comparison to other methods in the literature which are characterized by error = 0.42% and the response time = 0.045 s.

Conclusion

This work aims to obtain better energy efficiency, we were devoted to control the DFIG applied in a wind energy conversion system, by modeling the whole system (the wind turbine and the Doubly Fed Induction Generator). The method of controlling the turbine using Maximum Power Point Tracker (MPPT) control has been described. And then the Backstepping control method follows the command setpoints calculated by the MPPT command, these studied techniques were modeled and simulated using the SIMULINK environment with a generator of the nominal power of 1.5 kW, the simulation results observed clearly illustrate the performance of the Backstepping technique in terms of stability, speed, and precision, as well as its robustness concerning fluctuations in the internal parameters of the DFIG.

Footnotes

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

Hasnaa Jenkal

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