**2.1 Wind turbine model**

The wind turbine aerodynamic modeling can be determined based on the power speed characteristics [12]. The mathematical expression of the mechanical power from wind turbine to the aerodynamic rotor is set forward (Eq. (1)).

$$P\_m = \frac{1}{2}\rho \text{ R } \mathbb{C}\_p \text{ } (\lambda, \rho) \text{V}^3 \tag{1}$$

The power coefficient can be expressed in terms of the tip speed ratio and the pitch angle as follows (Eq. (2)):

$$\mathbf{C}\_{p}(\lambda, \mathfrak{f}) = \mathbf{C}\_{1} \left( \frac{\mathbf{C}\_{2}}{\lambda\_{i}} - \mathbf{C}\_{3} \mathfrak{f} - \mathbf{C}\_{4} \right) \mathbf{e}^{\frac{-\mathbf{C}\_{5}}{\lambda\_{i}}} + \mathbf{C}\_{6} \lambda\_{i} \tag{2}$$

Where the power coefficients are *C***<sup>1</sup>** = 0.5176, *C***<sup>2</sup>** = 116, *C***<sup>3</sup>** = 0.4, *C***<sup>4</sup>** = 5, *C***<sup>5</sup>** = 21, *C***<sup>6</sup>** = 0.0068 [12].

*λ<sup>i</sup>* can be expressed by the following equation (Eq. (3)):

$$\frac{1}{\lambda\_i} = \frac{1}{\lambda + \mathbf{0.08\beta}} - \frac{\mathbf{0.035}}{\beta^3 + \mathbf{1}} \tag{3}$$

The tip speed ratio is given by (Eq. (4)):

$$
\lambda = \frac{\mathbf{R} \mathbf{\mathcal{Q}}\_t}{\mathbf{V}} \tag{4}
$$

The aerodynamic torque, the generator torque and mechanical speed appearing on the shaft of the generator [11, 12] are represented respectively by (Eq. (5)–(7)).

$$\mathbf{C}\_{air} = \frac{P\_m}{\mathcal{Q}\_t} \tag{5}$$

$$\mathbf{C}\_m = \frac{\mathbf{C}\_{air}}{\mathbf{G}} \tag{6}$$

$$
\mathfrak{Q}\_t = \frac{\mathfrak{Q}\_{mec}}{\mathbf{G}} \tag{7}
$$

In fact, the power of the used generator is low, the use of the pitch control can increase the cost of the whole system. Therefore, in this work, the pitch control does

**Figure 2.** *Power coefficient for a pitch angle β* ¼ **0***.*

*Nonlinear Control Strategies of an Autonomous Double Fed Induction Generator Based Wind… DOI: http://dx.doi.org/10.5772/intechopen.94757*

not prove to be a relevant solution to achieve our purpose and consequently, the pitch angle is maintained fixed (*β* ¼ 0), which is a valid hypothesis for low and medium wind speeds [9]. **Figure 2** illustrates the power coefficient *Cp* variation versus the tip speed ratio *λ* for a specific chosen value of the pitch angle *β* ¼ 0.

#### **2.2 Standalone DFIG model**

In this section, we attempt to analyze properly the DFIG in autonomous mode. The modeling of the three phase DFIG is carried out in a (d, q) reference frame. Using the generator convention, the Park model of the DFIG describing the functioning of this machine [22] in both stator and rotor side is given below respectively (Eqs. (8) and (9)):

$$\begin{cases} \frac{d}{dt}\boldsymbol{\psi}\_{sd} = -R\_{l}\boldsymbol{I}\_{sd} - \boldsymbol{V}\_{sd} + \boldsymbol{\alpha}\_{1}\boldsymbol{\psi}\_{sq} \\\\ \frac{d}{dt}\boldsymbol{\psi}\_{sq} = -R\_{s}\boldsymbol{I}\_{sq} - \boldsymbol{V}\_{sq} - \boldsymbol{\alpha}\_{1}\boldsymbol{\psi}\_{sd} \end{cases} \tag{8}$$

$$\begin{cases} \frac{d}{dt}\boldsymbol{\psi}\_{rd} = -R\_{r}\boldsymbol{I}\_{rd} + \boldsymbol{V}\_{rd} + \boldsymbol{\alpha}\_{2}\boldsymbol{\psi}\_{rq} \\\\ \frac{d}{dt}\boldsymbol{\psi}\_{rq} = -R\_{r}\boldsymbol{I}\_{rq} + \boldsymbol{V}\_{rq} - \boldsymbol{\alpha}\_{2}\boldsymbol{\psi}\_{rd} \end{cases} \tag{9}$$

As the d and q axis are magnetically decoupled, stator and rotor machine flux are expressed as (Eqs. (10) and (11)):

$$\begin{cases} \boldsymbol{\Psi}\_{sd} = \boldsymbol{L}\_{s}\boldsymbol{I}\_{sd} + \boldsymbol{M}\boldsymbol{I}\_{rd} \\\\ \boldsymbol{\Psi}\_{sq} = \boldsymbol{L}\_{s}\boldsymbol{I}\_{sq} + \boldsymbol{M}\boldsymbol{I}\_{rq} \end{cases} \tag{10}$$

$$\begin{cases} \boldsymbol{\Psi}\_{rd} = \boldsymbol{L}\_{r}\boldsymbol{I}\_{rd} + \boldsymbol{M}\boldsymbol{I}\_{sd} \\\\ \boldsymbol{\Psi}\_{rq} = \boldsymbol{L}\_{r}\boldsymbol{I}\_{rq} + \boldsymbol{M}\boldsymbol{I}\_{sq} \end{cases} \tag{11}$$

While functioning as a generator, the electromagnetic torque produced by the DFIG can be represented in terms of stator and rotor currents and flux as follows (Eq. (12)):

$$\mathbf{C}\_{em} = \frac{3}{2} p \frac{\mathbf{M}}{L\_s} \left( I\_{rd} \boldsymbol{\upmu}\_{sq} - I\_{rq} \boldsymbol{\upmu}\_{sd} \right) \tag{12}$$

Neglecting the machine viscous friction phenomenon, the electromechanical equation is given by (Eq. (13)):

$$\frac{dw\_r}{dt} = \frac{p}{J} \left(\mathbf{C}\_m - \mathbf{C}\_{em}\right) \tag{13}$$

#### **2.3 Load model**

In stand-alone technology of DFIG based WECS, the stator of the machine is not connected to the grid but supplies an isolated load. Different kinds and values of loads can be connected to the stator terminals. The connected load is detected ð Þ *RLP*, *LLP* in the following work. This couple ð Þ *RLP*, *LLP* depends mainly on load demand power percentage noted *LPd* [12]. Based on *LPd*, the connecting load can be computed using the following equations (Eq. (14) and (15)).

*Entropy and Exergy in Renewable Energy*

$$R\_{LP} = \frac{3V\_{1n}^2}{LP\_d.P\_{1n}\sqrt{1 + (\text{tg}\,\rho^2)}}\tag{14}$$

$$L\_{LP} = \frac{R\_{LP} \text{tg} \,\rho}{\alpha\_{1n}} \tag{15}$$

Where *P*1*n*, *V*1*n*, and *w*1*<sup>n</sup>* represent respectively nominal power, voltage and pulsation of the machine.

Electrical equations on the stator side can be rewritten as follows (Eq. (16)) [12]:

$$\begin{cases} V\_{sd} = R\_{LP} i\_{sd} + L\_{LP} \frac{d}{dt} i\_{sd} - \alpha\_1 L\_{LP} i\_{sq} \\\\ V\_{sq} = R\_{LP} i\_{sq} + L\_{LP} \frac{d}{dt} i\_{sq} + \alpha\_1 L\_{LP} i\_{sd} \end{cases} \tag{16}$$

#### **2.4 Converter model**

For the considered power schema shown in **Figure 1**, the voltages across a, b, c rotor weddings of the DFIG are constructed as follows (Eq. (17)) [17].

$$
\begin{bmatrix} V\_{an} \\ V\_{bn} \\ V\_{cn} \end{bmatrix} = \frac{V\_{dc}}{3} \begin{bmatrix} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{bmatrix} \begin{bmatrix} f\_1 \\ f\_2 \\ f\_3 \end{bmatrix} \tag{17}
$$

Where *f* <sup>1</sup>, *f* <sup>2</sup> *and f* <sup>3</sup> represent the control signals and *Vdc* is the DC-link voltage referring to [17].

### **3. Controllers synthesis**

The target of the proposed DFIG based WECS control is to keep the stator voltage amplitude and frequency constant and equal to their nominal values namely 220 *V* and 50*Hz* versus the load variations and wind speed fluctuations. Accordingly, the integration of a controller inside the studied system seems crucial. A few technologies about voltage and frequency control in an autonomous system based on DFIG are studied in literature [15]. However, in terms of complexity, these techniques exhibit many disadvantages in practice. In this work, we attempt to explore three types of controllers. In a first step, the implementation of the classical PI control technique is presented. Then, thanks to its advantages, the Back-Stepping controller is modeled and integrated into the system. Finally, a new technique of control is studied known as Sliding Mode controller.

In addition, the synthesis of different control strategies is based on choosing a synchronously dq reference frame with the stator voltage that is oriented with the d axis [23]. Consequently, we can formulate (Eq. (18)):

$$\begin{cases} \mathbf{V}\_{sd} = \mathbf{0} \\ \mathbf{V}\_{sq} = \mathbf{V}\_s \end{cases} \tag{18}$$

#### **3.1 PI controller parameters calculation**

In order to ensure the convergence conditions of the proposed system and to obtain good responses, PI controller parameters should be chosen properly. This *Nonlinear Control Strategies of an Autonomous Double Fed Induction Generator Based Wind… DOI: http://dx.doi.org/10.5772/intechopen.94757*

**Figure 3.** *The control diagram.*

section describes a simple method for PI parameters computing. In general, the control diagram is presented as shown in **Figure 3**.

Where *H s*ð Þ represents the transfer function of the system which is given by (Eq. (19)):

$$H(s) = \frac{K\_{b0}}{1 + T\_{b0}s} \tag{19}$$

*Kp* and *Ki* are the PI controller parameters for proportional and integral actions respectively. These parameters are computed based on DFIG parameters so as to ensure quick and convergent response of the DFIG based WECS subsystems. *Ki* is determined using the pole compensation method and *Kp* is deduced so as to ensure a fast response of the DFIG subsystems.

#### **3.2 PI controller design**

The main principle of the PI control topology is to control and regulate different physical parameters of the system using closed loops control. In this context, while applying this controller, in order to obtain the final control signals (*Urd* and *Urq*), two control loops are required. The computing of reference rotor currents (*Ird* and *Irq*) is carried out in a first step based on the calculation of the difference between reference and measured value of stator voltage (*Usd* and *Usq*). Calculating the output rotor voltages is performed in a second step by minimizing the error between reference and measured rotor currents already calculated in the first step. Therefore, stator and rotor voltages can be reformulated as follows (Eq. (20) and (21)):

$$\begin{cases} \begin{aligned} \mathbf{V}\_{sd} &= \left[ R\_s I\_{sd} + L\_s \frac{d\mathbf{I}\_{sd}}{dt} - \alpha\_s L\_s I\_{sq} \right] + M \frac{d\mathbf{I}\_{rd}}{dt} - \alpha\_1 M \mathbf{I}\_{rq} \\\\ \mathbf{V}\_{sq} &= \left[ R\_s I\_{sq} + L\_s \frac{d\mathbf{I}\_{sq}}{dt} + \alpha\_1 L\_s I\_{sd} \right] + M \frac{d\mathbf{I}\_{rq}}{dt} + \alpha\_1 M \mathbf{I}\_{rd} \end{aligned} \end{cases} (20) $$
 
$$\begin{cases} \mathbf{V}\_{rd} = R\_r I\_{rd} + L\_r \frac{d\mathbf{I}\_{rd}}{dt} + M \frac{d\mathbf{I}\_{sd}}{dt} - \alpha\_2 L\_r I\_{rq} - \alpha\_2 M \mathbf{I}\_{sq} \\\\ \mathbf{V}\_{rq} = R\_r I\_{rq} + L\_r \frac{d\mathbf{I}\_{rq}}{dt} + M \frac{d\mathbf{I}\_{sq}}{dt} + \alpha\_2 L\_r I\_{rd} + \alpha\_2 M \mathbf{I}\_{sd} \end{aligned} \tag{21}$$

The orientation of stator voltages with the d axis leads to (Eq. (22)):

$$\begin{cases} \mathbf{0} = \mathbf{R}\_s \mathbf{I}\_{sd} + \mathbf{L}\_s \frac{d\mathbf{I}\_{sd}}{dt} + \mathbf{M} \frac{d\mathbf{I}\_{rd}}{dt} - \alpha\_1 \mathbf{L}\_s \mathbf{I}\_{sq} - \alpha\_1 \mathbf{M} \mathbf{I}\_{rq} \\\ \mathbf{V}\_s = \mathbf{R}\_s \mathbf{I}\_{sq} + \mathbf{L}\_s \frac{d\mathbf{I}\_{sq}}{dt} + \mathbf{M} \frac{d\mathbf{I}\_{rq}}{dt} + \alpha\_1 \mathbf{L}\_s \mathbf{I}\_{sd} + \alpha\_1 \mathbf{M} \mathbf{I}\_{rd} \end{cases} \tag{22}$$

Departing from these equations and omitting coupling terms, reference rotor currents can be expressed in terms of stator voltages.

Moreover, referring to (Eq. (22)), the first derivative of the stator current can be written as (Eq. (23)):

$$\begin{cases} \frac{dI\_{sd}}{dt} = \frac{1}{L\_s} \left[ -R\_s I\_{sd} - M \frac{dI\_{rd}}{dt} + \alpha\_1 L\_s I\_{sq} + \alpha\_1 M I\_{rq} \right] \\\\ \frac{dI\_{sq}}{dt} = \frac{1}{L\_s} \left[ V\_s - R\_s I\_{sq} - M \frac{dI\_{rq}}{dt} - \alpha\_1 L\_s I\_{sd} - \alpha\_1 M I\_{rd} \right] \end{cases} \tag{23}$$

Referring to (Eq. (21)) and (Eq. (23)), reference rotor voltages are given as (Eq. (24)):

$$\begin{cases} \mathbf{V}\_{rd} = \mathbf{R}\_{r}\mathbf{I}\_{rd} - \alpha\_{2}\mathbf{L}\_{r}\mathbf{I}\_{rq} - \alpha\_{2}\mathbf{M}\mathbf{I}\_{sq} + \mathbf{L}\_{r}\frac{d\mathbf{I}\_{rd}}{dt} + \frac{\mathbf{M}}{\mathbf{L}\_{s}}\mathbf{L}\_{d} \\\\ \mathbf{V}\_{rq} = \mathbf{R}\_{r}\mathbf{I}\_{rq} + \alpha\_{2}\mathbf{L}\_{r}\mathbf{I}\_{rd} + \alpha\_{2}\mathbf{M}\mathbf{I}\_{sd} + \mathbf{L}\_{r}\frac{d\mathbf{I}\_{rq}}{dt} + \frac{\mathbf{M}}{\mathbf{L}\_{s}}\mathbf{L}\_{q} \end{cases} \tag{24}$$

Where

$$\begin{cases} L\_d = \left[ -R\_s I\_{sd} - M \frac{dI\_{rd}}{dt} + o\_1 L\_t I\_{sq} + o\_1 M I\_{rq} \right] \\\\ L\_q = \left[ V\_s - R\_s I\_{sq} - M \frac{dI\_{rq}}{dt} - o\_1 L\_t I\_{sd} - o\_1 M I\_{rd} \right] \end{cases}$$

(Eq. (24)) can be rewritten as follows:

$$\begin{cases} \mathbf{V}\_{rd} = \left[ \mathbf{R}\_r \mathbf{I}\_{rd} + \left( L\_r - \frac{\mathbf{M}^2}{L\_s} \right) \frac{d\mathbf{I}\_{rd}}{dt} \right] + T\_d \\\\ \mathbf{V}\_{rq} = \left[ \mathbf{R}\_r \mathbf{I}\_{rq} + \left( L\_r - \frac{\mathbf{M}^2}{L\_s} \right) \frac{d\mathbf{I}\_{rq}}{dt} \right] + T\_q \end{cases} \tag{25}$$

Where

$$\begin{cases} T\_d = -\left(\alpha\_2 L\_r - \frac{M^2}{L\_s} \alpha\_1\right) I\_{rq} - M(\alpha\_2 - \alpha\_1) I\_{sq} - \frac{R\_s M}{L\_s} I\_{sd} \\\\ T\_q = \left(\alpha\_2 L\_r - \frac{M^2}{L\_s} \alpha\_1\right) I\_{rd} + M(\alpha\_2 - \alpha\_1) I\_{sd} - \frac{R\_s M}{L\_s} I\_{sq} + \frac{M}{L\_s} V\_s \end{cases}$$

#### **3.3 Backstepping controller design**

In this paper, we aim at improving performances of the studied system. In this context, in order to answer this need and to respond to our objective, we are basically interested in developing control strategies resting on linearization of the autonomous WECS based DFIG.

With the presence of many kinds of uncertainties, the back-stepping controller is able to linearize effectively a nonlinear system. In fact, during stabilization, unlike other techniques of linearization, this control technique has the flexibility to keep useful nonlinearities [23].

*Nonlinear Control Strategies of an Autonomous Double Fed Induction Generator Based Wind… DOI: http://dx.doi.org/10.5772/intechopen.94757*

The stabilization of the virtual control state stands for the main purpose of the Backstepping controller. Therefore, this control strategy rests on the stabilization of a variable error by selecting carefully the suitable control inputs which are obtained from the analysis of Lyapunov function [24, 25].

In order to regulate effectively the output reference rotor voltages, the reference rotor currents are obtained based on a PI controller. Then, the rotor voltages are obtained by using a back-stepping controller.

Indeed, the first step of the Backstepping control is meant to identify the tracking errors by Eq. (26).

$$\begin{cases} \mathbf{e\_1} = \mathbf{I\_{rd}^\*} - \mathbf{I\_{rd}} \\\\ \mathbf{e\_2} = \mathbf{I\_{rq}^\*} - \mathbf{I\_{rq}} \end{cases} \tag{26}$$

Tracking errors first derivative can be written as Eq. (27):

$$\begin{cases} \dot{\mathbf{e}}\_1 = \dot{\mathbf{I}}\_{rd}^\* - \dot{\mathbf{I}}\_{rd} \\\\ \dot{\mathbf{e}}\_2 = \dot{\mathbf{I}}\_{rq}^\* - \dot{\mathbf{I}}\_{rq} \end{cases} \tag{27}$$

The derivative of the rotor currents can be obtained referring to Eq. (24) and can be written as follows (Eq. (28)):

$$\begin{cases} \frac{d\mathbf{I}\_{rd}}{dt} = \frac{\mathbf{V}\_{rd}}{L\_r} - \frac{\mathbf{R}\_r}{L\_r}\mathbf{I}\_{rd} - \frac{\mathbf{M}}{L\_r}\frac{d\mathbf{I}\_{sd}}{dt} + \alpha\_2\mathbf{I}\_{rq} + \alpha\_2\frac{\mathbf{M}}{L\_r}\mathbf{I}\_{sq} \\\\ \frac{d\mathbf{I}\_{rq}}{dt} = \frac{\mathbf{V}\_{rq}}{L\_r} - \frac{\mathbf{R}\_r}{L\_r}\mathbf{I}\_{rq} - \frac{\mathbf{M}}{L\_r}\frac{d\mathbf{I}\_{sq}}{dt} - \alpha\_2\mathbf{I}\_{rd} - \alpha\_2\frac{\mathbf{M}}{L\_r}\mathbf{I}\_{sd} \end{cases} (28)$$

To ensure the convergence and the stability of the system, the Lyapunov function is chosen to be a quadratic function (defined as a positive function) and is given by Eq. (29).

$$\mathbf{V\_1 = \frac{1}{2}e\_1^2 + \frac{1}{2}e\_2^2} \tag{29}$$

The expression of Lyapunov derivative function is defined as negative function and it is expressed as follows (Eq. (30)).

$$
\dot{\mathbf{V}}\_1 = -\mathbf{K}\_1 \mathbf{e}\_1^2 - \mathbf{K}\_2 \mathbf{e}\_2^2 \tag{30}
$$

Eq. (29) can be rewritten as:

$$
\dot{V}\_1 = \mathbf{e}\_1 \dot{\mathbf{e}}\_1 + \mathbf{e}\_2 \dot{\mathbf{e}}\_2 \tag{31}
$$

In order to guarantee a stable tracking, the Back-stepping gain coefficients *K***<sup>1</sup>** and *K***<sup>2</sup>** need to be positive [25].

Referring to Eqs. (30) and (31), it can be concluded that (Eq. (32)):

$$\begin{cases} \mathbf{e\_1}\dot{\mathbf{e\_1}} = -\mathbf{K\_1}\mathbf{e\_1^2} \\ \mathbf{e\_2}\dot{\mathbf{e\_2}} = -\mathbf{K\_2}\mathbf{e\_2^2} \end{cases} \tag{32}$$

Consequently, we can obtain (Eq. 33):

$$\begin{cases} -\mathbf{K}\_1 \mathbf{e}\_1 = \dot{\mathbf{I}}\_{rd}^\* - \dot{\mathbf{I}}\_{rd} \\\\ -\mathbf{K}\_2 \mathbf{e}\_2 = \dot{\mathbf{I}}\_{rq}^\* - \dot{\mathbf{I}}\_{rq} \end{cases} \tag{33}$$

Based on Eqs. (28) and (33), we can obtain (Eq. (34)):

$$\begin{cases} -K\_1 \mathbf{e}\_1 = \dot{I}\_{rd}^\* - \frac{V\_{rd}}{L\_r} + \frac{R\_r}{L\_r} I\_{rd} + \frac{M}{L\_r} \frac{dI\_{sd}}{dt} - \alpha\_2 I\_{rq} - \alpha\_2 \frac{M}{L\_r} I\_{sq} \\\\ -K\_2 \mathbf{e}\_2 = \dot{I}\_{rq}^\* - \frac{V\_{rq}}{L\_r} + \frac{R\_r}{L\_r} I\_{rq} + \frac{M}{L\_r} \frac{dI\_{sq}}{dt} + \alpha\_2 I\_{rd} + \alpha\_2 \frac{M}{L\_r} I\_{sd} \end{cases} (34)$$

Finally, the rotor control voltages are given by (Eq. (35)):

$$\begin{cases} \mathbf{V}\_{rd} = \mathbf{L}\_r \left[ \mathbf{K}\_1 \mathbf{e}\_1 + \dot{\mathbf{I}}\_{rd}^\* + \frac{\mathbf{R}\_r}{L\_r} \mathbf{I}\_{rd} + \frac{\mathbf{M}}{L\_r} \frac{d\mathbf{I}\_{sd}}{dt} - \boldsymbol{\alpha}\_2 \mathbf{I}\_{rq} - \boldsymbol{\alpha}\_2 \frac{\mathbf{M}}{L\_r} \mathbf{I}\_{sq} \right] \\\\ \mathbf{V}\_{rq} = \mathbf{L}\_r \left[ \mathbf{K}\_2 \mathbf{e}\_2 + \dot{\mathbf{I}}\_{rq}^\* + \frac{\mathbf{R}\_r}{L\_r} \mathbf{I}\_{rq} + \frac{\mathbf{M}}{L\_r} \frac{d\mathbf{I}\_{sq}}{dt} + \boldsymbol{\alpha}\_2 \mathbf{I}\_{rd} + \boldsymbol{\alpha}\_2 \frac{\mathbf{M}}{L\_r} \mathbf{I}\_{sd} \right] \end{cases} (35)$$

#### **3.4 Sliding mode controller design**

Thanks to its advantages such as the simplicity of the implementation, the stability and the insensitivity to external disturbances, the sliding mode controller is a widely used strategy [26]. Similar to the back-stepping controller, the aim of this strategy is to stabilize a chosen virtual control state. It rests on the stabilization of a variable error, defined as a sliding surface, by selecting the suitable control inputs. In fact, the output control parameters are obtained by the determination of two components: Ueq and U<sup>N</sup> as given in Eq. (36).

In this section, a detailed analysis of this controller is presented in order to obtain the control output rotor voltages which regulate the output voltage and frequency of the system and maintain them constant no matter which external disturbances occur.

$$\begin{cases} \mathbf{V}\_{rd} = \mathbf{V}\_{rd}^{eq} + \mathbf{V}\_{rd}^{N} \\\\ \mathbf{V}\_{rq} = \mathbf{V}\_{rq}^{eq} + \mathbf{V}\_{rq}^{N} \end{cases} \tag{36}$$

In our research, the sliding surface is chosen as follows (Eq. (37)):

$$\begin{cases} \mathbf{S}\_1 = I\_{rd}^\* - I\_{rd} \\\\ \mathbf{S}\_2 = I\_{rq}^\* - I\_{rq} \end{cases} \tag{37}$$

The derivative of sliding surfaces is given by (Eq. (38)):

$$\begin{cases} \dot{\mathbf{S}}\_1 = \dot{\mathbf{I}}\_{rd}^\* - \dot{\mathbf{I}}\_{rd} \\\\ \dot{\mathbf{S}}\_2 = \dot{\mathbf{I}}\_{rq}^\* - \dot{\mathbf{I}}\_{rq} \end{cases} \tag{38}$$

*Nonlinear Control Strategies of an Autonomous Double Fed Induction Generator Based Wind… DOI: http://dx.doi.org/10.5772/intechopen.94757*

Based on Eq. (28), Eq. (38) can be rewritten as follows (Eq. (39)):

$$\begin{cases} \dot{\mathbf{S}}\_1 = \dot{I}\_{rd}^\* - \frac{V\_{rd}}{L\_r} + \frac{R\_r}{L\_r} I\_{rd} + \frac{M}{L\_r} \frac{dI\_{sd}}{dt} - \alpha\_2 I\_{rq} - \alpha\_2 \frac{M}{L\_r} I\_{sq} \\\\ \dot{\mathbf{S}}\_2 = \dot{I}\_{rq}^\* - \frac{V\_{rq}}{L\_r} + \frac{R\_r}{L\_r} I\_{rq} + \frac{M}{L\_r} \frac{dI\_{sq}}{dt} + \alpha\_2 I\_{rd} + \alpha\_2 \frac{M}{L\_r} I\_{sd} \end{cases} (39)$$

In a first step, and during the sliding mode, in order to compute the first part of the control signal, we can set forward these hypotheses (Eq. (40)):

$$\begin{cases} \mathbf{S} = \mathbf{0} \\\\ \dot{\mathbf{S}} = \mathbf{0} \\\\ \mathbf{V}\_{rd}^{N} = \mathbf{0} \\\\ \mathbf{V}\_{rq}^{N} = \mathbf{0} \end{cases} \tag{40}$$

Subsequently, equivalent voltage expressions are formulated as follows (Eq. (41)):

$$\begin{cases} \mathbf{V}\_{rd}^{eq} = L\_r \left[ \dot{I}\_{rd}^\* + \frac{R\_r}{L\_r} I\_{rd} + \frac{M}{L\_r} \frac{dI\_{sd}}{dt} - \alpha\_2 I\_{rq} - \alpha\_2 \frac{M}{L\_r} I\_{sq} \right] \\\\ \mathbf{V}\_{rq}^{eq} = L\_r \left[ \dot{I}\_{rq}^\* + \frac{R\_r}{L\_r} I\_{rq} + \frac{M}{L\_r} \frac{dI\_{sq}}{dt} + \alpha\_2 I\_{rd} + \alpha\_2 \frac{M}{L\_r} I\_{sd} \right] \end{cases} (41)$$

In a second step, during the convergence mode, to ensure the condition *S*\_ *S*<0, we can suppose that (Eq. (42)):

$$\begin{cases} \mathbf{V}\_{rd}^{N} = \mathbf{K}\_{1} \operatorname{sign}(\mathbf{S}\_{1}) \\ \mathbf{V}\_{rq}^{N} = \mathbf{K}\_{2} \operatorname{sign}(\mathbf{S}\_{2}) \end{cases} \tag{42}$$

To guarantee a stable tracking, *K*<sup>1</sup> and *K*<sup>2</sup> are chosen positive constants [27].

### **4. Controllers management strategy**

With the rapid progress of control topologies, various nonlinear control strategies such as Backstepping and sliding mode controllers whetted the interest of many researchers who attempted to develop and further enhance them. These algorithms succeeded to improve different performances of the studied system, but they remain unable to ensure optimal and safe operation of the rotor side converter. In general, the rotor side converter integrated in a DFIG based WECS is estimated at 30% of the machine nominal power which presents the main advantage of the DFIG use [12]. However, with the presence of load demand power variations and wind speed fluctuations, the rotor demanded power may exceed 30% of the DFIG nominal power. Hence, to ensure a safe functioning of the RSC and to guarantee an optimal operation mode of the DFIG, a management strategy is proposed. The wind energy can be then used effectively in order to satisfy the demand of the connected load on the one hand and to ensure security of the DFIG rotor side converter with an optimal operation on the other hand.

Based on the captured wind velocity and the load demanded power, the proposed management algorithm computes and specifies secure operation boundaries (0*:*7 *ω*1*<sup>n</sup>* ≤*ω<sup>r</sup>* ≤1*:*3 *ω*1*<sup>n</sup>*).

Thus, in order to obtain nominal stator output voltage and frequency (*V* ¼ 220*V*, *F* ¼ 50*H*), both rotor voltage and frequency change with every detected change in load demand power (Eq. (43)) to maintain a constant electromagnetic torque (Eq. (44)). As it is shown in (Eq. (43)), the rotor voltage depends mainly on the load impedance whereas, the rotor pulsation depends basically on the rotational speed of the machine.

$$\begin{cases} \overline{V}\_r = \frac{j}{a\_1 M} \cdot \frac{\overline{Z} \cdot \overline{Z}\_{sT}}{\overline{Z}\_{ch}} \overline{V}\_s\\ a\_2 = a\_1 - a\_r \end{cases} \tag{43}$$

$$C\_{\epsilon} = \frac{3}{2} p \frac{\overline{V}\_{\epsilon}^{2}}{a\nu\_{1}} \Im \mathsf{mag} \left( j \frac{\overline{Z}\_{\epsilon T}}{\overline{Z}\_{ch}^{-2}} \right) \tag{44}$$

where *ZLP* ¼ *RLP* þ *jω*1*LLP*, *Zr* ¼ *Rr* þ *jω*2*Lr*, *ZsT* ¼ ð Þþ *Rs* þ *RLP jω*1ð Þ *Ls* þ *LLP* and *<sup>Z</sup>* <sup>¼</sup> *Zr* <sup>þ</sup> *<sup>ω</sup>*1*ω*2*M*<sup>2</sup> *ZsT* .

The proposed algorithm can be summarized by the flowchart displayed in **Figure 4**.

The handling of the algorithm detailed in **Figure 4** allows us towards the end to obtain the speed range for every connected load. Within this framework, the main idea of the developed strategy is to detect first the load demanded power. Based on the wind turbine parameters, the connected load value and the rotor speed limits (*wr* ¼ 0*:*7*wn* and *wr* ¼ 1*:*3*wn*), the wind speed limits are calculated (*Vmin* and *Vmax*). Otherwise, if the detected wind speed does not respect the given algorithm boundaries, the DFIG has to disconnect from the load unless the wind speed respects computed limits.

### **5. Simulation results and discussion**

In order to analyze the system modeling, to check performances of the studied controllers and to compare the system responses using each control strategy, the

#### *Nonlinear Control Strategies of an Autonomous Double Fed Induction Generator Based Wind… DOI: http://dx.doi.org/10.5772/intechopen.94757*

proposed stand-alone wind energy conversion system based on doubly fed induction generator is implemented and tested using Matlab/Simulink environment.

The studied system rests on a wind turbine, a doubly fed induction generator and a three-phase isolated load. The DFIG parameters are obtained experimentally in the LTI, Cuffies-Soissons, France laboratory. They are exhibited in **Table 1** [28].

To satisfy the convergence conditions of the proposed system, parameters of PI controller, Backstepping and Sliding mode controllers are selected properly.

In this paper, our intrinsic purpose is to maintain constant output stator voltage and frequency under sudden variations of wind speed and load demand. Therefore, a selected profile of wind speed deduced from the proposed management strategy and load demand is applied to the system. These profiles are chosen properly in such a way that rotor side power is limited under 30% of DFIG nominal power so as to ensure the safety of the rotor side converter.

In fact, to analyze the system performances, a comparison between the three proposed strategies of control is carried out. A set of different simulation tests have been performed and carried out for 20 seconds.

The profile of wind speed profile is presented in **Figure 5**. To check the performance of the proposed model, different sudden variations are applied to the system. The wind speed varies from 7*m=s* to 15*:*3 *m=s*. For 15 ≤*t*≤ 20*s*, the wind velocity increases to attend 15*:*3 *m=s*. At this moment, for safety reasons, the management and control strategy reacts in such a way it disconnects the DFIG from the load.

**Figure 6** represents the demand of the isolated load. The demanded power of the load presents many variations. For 0≤ *t*≤5*s*, the demand of the connected load is equal to the nominal power *Pn*. At *t* ¼ 5*s*, the demand of the load decreases and becomes 0*:*8*Pn*. However, at ¼ 10*s*, an increase in the demanded power from 0*:*7*Pn* to 1*:*2*Pn* is recorded. For 15≤*t* ≤20*s*, when the wind speed increases suddenly, the load is totally disconnected from the DFIG and the power transmitted to the isolated load becomes equal to zero.

To interpret properly the different results, to demonstrate the load and the wind speed variation effects on the system response and to show the response of each controller, two zooms are chosen to be presented in different figures. A zoom noted (a) is performed at *t* ¼ 5*s* to show the effect of the load demanded power variation. Moreover, at *t* ¼ 15*s* a zoom noted (b) is stated in different figures in order to



**Figure 5.** *Wind speed profile.*

**Figure 6.** *Load demand variations.*

demonstrate the effectiveness of the management strategy when the wind speed does not abide by computed limits.

**Figure 7** plots the output stator voltages (*Usd* and *Usq*). We notice that the stator voltage is maintained constant in case of load demand variations (at *t* ¼ 5*s* and *t* ¼ 10 *s*) and even when a wind speed fluctuation is detected (at *t* ¼ 15*s*).

**Figure 8** describes the rotor voltages (*Urd* and *Urq*) which correspond to the control signals of the system. In each detected variation in load impedance and wind speed, controllers react by controlling rotor voltages which vary in order to maintain constant output voltage and frequency. For example, at *t* ¼ 5*s*, the load demanded power changes. So, the direct rotor voltage (*Urd*) decreases and the quadratic rotor voltage (*Urq*) increases instantaneously to regulate the stator outputs. In fact, when the load demand varies at *t* ¼ 10*s*, the control system reacts to obtain the same results

#### **Figure 7.**

*The stator output voltages responses using the PI controller, the Backstepping controller and the Sliding mode controller. Two zooms are done at t = 5 s noted (a) and at t = 15 s noted (b) to see the advantages and disadvantages of each controller.*

*Nonlinear Control Strategies of an Autonomous Double Fed Induction Generator Based Wind… DOI: http://dx.doi.org/10.5772/intechopen.94757*

#### **Figure 8.**

*The rotor output voltages responses using the PI controller, the Backstepping controller and the Sliding mode controller. Two zooms are done at t=5 s noted (a) and at t=15 s noted (b) to see the advantages and disadvantages of each controller.*

in stator side. Furthermore, at *t* ¼ 0*:*1*s*, in both **Figures 7** and **8**, we detect the existence of a transient regime uniquely in case a PI controller is used. At this starting time, some picks are presented. However, the response of the backstepping and the sliding mode controllers are smoother and more flexible. Thus, while comparing the response of the backstepping and the sliding mode controllers, it is obviously visible in **Figures 7** and **8** that the steady state regime is reached faster when the backstepping controller is used. In both **Figures 7** and **8**, the presence of two zooms (a and b) permits to check the performances of the studied system, to judge the response of each controller and to select the best one. Departing from those figures (**Figures 7(a)** and **(b) and 8(a)** and **(b)**), in case of implementing a PI controller, we notice the presence of overshoot in stator and rotor output voltages. Thus, the robustness of the backstepping and sliding mode controllers is demonstrated.

**Figure 9** highlights the evolution of the stator pulsation *w*1, the rotor pulsation *w*<sup>2</sup> and the electric angular speed of the DFIG *wr*. We infer that w2 and wr vary inversely to keep a constant stator pulsation and constant output frequency accordingly. Thus, the variation of the wind speed affects the rotor pulsation and the electric angular speed which is a normal process.

The stator output currents are displayed in **Figure 10**. It is deduced that when the power demanded by the load changes, the stator currents change in order to satisfy the demand of the load. Moreover, in zone (b), the wind speed increases and reaches 15*:*3 *m=s*. Therefore, for security reasons, the management strategy reacts to disconnect the DFIG from the load and subsequently, the power supplied to the load becomes equal to zero. At this moment, the stator currents increase and become equal to zero. However, the controller operates properly, and the stator voltage remains equal to the nominal value 220 *V* and remains ready for the next coupling of the DFIG and the load.

**Figure 9.** *Machine pulsations.*

#### **Figure 10.**

*The stator output currents responses using the PI controller, the Backstepping controller and the Sliding mode controller. Two zooms are done at t=5 s noted (a) and at t=15 s noted (b) to see the advantages and disadvantages of each controller.*

A scrutiny of **Figures 8**–**10** reveals that, by comparing these three strategies, the results obtained by using the Backstepping and the sliding mode controllers are faster and more flexible than these obtained by using the PI control topology.

In order to evaluate developed control strategy performances for each load and wind speed variations, the RMSE (Root-Mean-Square Error) is calculated as follows:

*Nonlinear Control Strategies of an Autonomous Double Fed Induction Generator Based Wind… DOI: http://dx.doi.org/10.5772/intechopen.94757*


#### **Table 2.**

*Vsd and Vsq curves errors (Time* ¼ ½ � 0*s* � 5*s ).*


#### **Table 3.**

*Vsd and Vsq curves errors (Time* ¼ ½ � 5*s* � 10*s ).*


#### **Table 4.**

*Vsd and Vsq curves errors (Time* ¼ ½ � 10*s* � 15*s ).*

$$RMSE = \sqrt{\frac{1}{N} \sum\_{i=1}^{N} (\mathbf{x}\_i - \overline{\mathbf{x}\_i})^2} \tag{45}$$

Where *N* is the number of obtained points, *xi* is the estimated value, *xi* is the observed value.

**Tables 2**–**4** sum up the RMSE of different algorithms for the same conditions (wind speed and load variations). As noticed, in each table, the RMSE of the Sliding mode controller is the smallest value. As a matter of fact, the different results obtained by this controller are close to the desired results which confirms the effectiveness of the sliding mode algorithm compared to others.

## **6. Conclusion**

The modeling and analysis of the isolated DFIG based on WECS have been presented. The main purpose of this chapter is on the one hand to regulate the output voltage and frequency, to maintain them constant and equal to their nominal values (220 V, 50 Hz) under wind speed fluctuations and load demand variations and on the other hand to guarantee a safe operation mode of the rotor side converter through limiting the rotor side power by around 30% of the machine power. Therefore, three different system control strategies are proposed and examined. Compared to the PI controller, a stable operation of the whole system is obtained with the application of back-stepping and sliding mode controllers. However, the Sliding mode controller presents more precision and its responses are much faster than the Backstepping controller. Thus, the system performances such as precision, stability, rapidity are improved. Analysis and simulation results prove the accuracy as well as the effectiveness of both the Back-stepping and sliding mode control strategies compared to classic control strategy.
