## 2 Materials and Methods
### Wind Turbine (WT) Modeling
#### 2.1.4. Maximum Power Point Tracking  
VSWTs can change their rotational speed in response to wind variation for wind power extraction and maintain the tip speed ratio (TSR) [70]. To maintain this TSR during wind fluctuations, the rotational speed must be adjusted to extract maximum wind power [79]. This is performed by an MPPT system. An MPPT is an algorithm used to extract maximum power irrespective of the wind speed. In other words, the MPPT is a way to determine the optimal generator speed that ensures maximum power yield during wind variations [70; 80]. Thus, according to [81], the dynamics of the MPPT curve are given by Equations (6)-(9), and their graphical representation is shown in Figure 9. Here, \(K_{opt}\) is called the optimization constant, which depends on turbine characteristics. \(\omega_{min}\) and \(\omega_{max}\) refer to the minimum and maximum rotor speeds against the cut-in wind speed \(V_{w(cut-in)}\) and cut-off wind speed \(V_{w(cut-off)}\), respectively. Under \(V_{w(cut-in)}\), the mechanical power is so low that it is not worth working, so it is preferred to shut down the turbine [82]. Similarly, beyond \(V_{w(cut-off)}\), also called the furling wind speed, the turbine should be stopped to avoid structural damage unless an advanced control of blade pitch angle acts on it.  
\[P_{g}^{*}=\frac{K_{opt}\omega_{0}}{(\omega_{0}-\omega_{min})}\big{(}\omega_{g} -\omega_{min}\big{)}\;when\;\omega_{min}\leq\omega_{g}\leq\;\omega_{0} \tag{6}\]  
\[P_{g}^{*}=K_{opt}\omega_{g}^{3}\;when\;\omega_{0}\leq\omega_{g}\leq\;\omega_{1} \tag{7}\]  
Figure 8: Simplified MATLAB Simulink representation of DFIG and converters.  
Figure 7: Mechanical rotor system of VSWT: (**a**) one-mass model and (**b**) Simulink representation of one-mass model mechanical rotor system of VSWT.  
Section PQ in Figure 9 shows that the output power of the turbine follows a linear trend under low-wind conditions, represented by Equation (6). Similarly, section QR shows the MPPT region or optimization region in which the wind turbine provides power according to Equation (7). The wind turbine will operate under this criterion until the V\({}_{\rm w}\) reaches the pre-defined optimal value of V\({}_{\rm w,base}\). Furthermore, section RS refers to constant speed and will maintain the turbine mechanical power at P\({}_{\rm max}\), as shown in Equation (8). In addition, this region is delimited by the maximum speed \(\omega_{\rm max}\) specified for the wind turbine within the thermal limits of the electrical generator. Equation (9) shows the constant power region with maximum power P\({}_{\rm max}\), when V\({}_{\rm w}\) exceeds the rated value, so that the generator and turbine are not overloaded and the dynamic load does not result in mechanical failure [70]. Blade pitch angle control, as discussed below (Section 2.1.5), will be activated in this region.  
The MPPT ensures maximum power is generated according to wind variations. However, to achieve this, a speed controller is needed that ensures equilibrium is maintained between the actual generated available power and the reference power generated by the MPPT. Therefore, a PI-based speed controller is considered--it is widely used in wind generation applications [83]. Figure 10 shows the MATLAB Simulink representation of the speed controller exhibiting constants k\({}_{\rm psc}\) and k\({}_{\rm isc}\), whose values are given in Table 2. The output of this speed controller is a reference torque signal, which is an input for the DFIG, whose values are limited within a specific range according to the thermal limits of the generator.  
Figure 9: Generated power versus rotational speed curve.