In this section, the WTGS and its functional operation are discussed. Figure 1 shows the one-line diagram of a wind turbine generation system connected to an infinite grid.
The WTGS is model such that maximum power (torque) is extracted from the turbines for a given wind speed using a maximum power point tracking (MPPT) algorithm.
The torque is then applied to a 2 MVA doubly fed induction generator (DFIG), with rotor-side and grid-side controllers. Therefore, representing a type 3 WTGS.

Figure 1: Wind Turbine Generation System

System Configuration and DFIG Modeling

The system configuration of a grid-connected doubly fed induction generator is shown in Figure 2.
The system comprises a wind turbine, a DFIG, a DC-link capacitor, and the back-to-back (Type of devices: IGBTs) three-phase pulse width modulated (PWM) voltage source converter with their respective controllers.
The stator of the DFIG is supplied by the three-phase voltages directly from the grid at constant amplitude and frequency, creating the stator magnetic field.
The rotor of the DFIG is supplied by three-phase voltages that takes a different amplitude and frequency at steady state to reach different operating condition of the machine (speed, torque).

Figure 2: Investigated DFIG Wind Turbine System Configuration

With,

: Wind speed in m/s

: Turbine angular speed in rad/s

: Torque produced by the turbine in N.m.

: Mechanical power produced by the turbine in W. 

: Rotor angular speed in rad/s

: Torque received by the rotor in N.m.

: Mechanical power received by the rotor in W. 

: Reduction ratio of the gear box. 

: Efficiency ratio of the gear box.  

: Active and reactive power of the stator and rotor sides.

: Active and reactive power of the grid side.

 Wind Turbine Modeling

The kinetic energy of the wind is converted to rotational energy in the form of a mechanical torque by a wind turbine (WT).
The power extracted by the turbine from available power in the wind is given in equation (1), as follows:

Where,

: Tip speed ratio.

: Blade radius (m).

: Wind speed (m/s).

: Air density.

: Pitch angle.

: Power coefficient, and it is a function of the tip speed ratio  and the the pitch angle .

The equation of the power coefficient is given in (3), and it reflects the efficiency if the conversion form wind energy to mechanical energy.

Where, through  are the characteristic coefficients of the wind turbine, and is the rotational speed of the wind turbine (rad/s) as in [3].
The characteristic curve ( vs ) for different values of the pitch angle () is shown in Figure 3; and it can be observed that the optimum power coefficient is obtained when  is equal to zero degree (curve in blue). 
The maximum power extraction from the WT can be achieved when the turbine operates at the optimum .
Figure 4 illustrates the maximum power tracking curve for the DFIG. The optimum power to be extracted from the WTGS can be expressed as follows:

Figure 3: vs Characteristics of the Wind Turbine for Various Pitch Angles

Figure 4: WTGS Characteristics for DFIG with Maximum Power Point Tracking

DFIG Modeling

In synchronous  reference frame rotating at a speed, the modeling of the DFIG is given by the following equations:

• Stator voltage components

• Rotor voltage components