## 2 Materials and Methods
### Wind Turbine (WT) Modeling
#### 2.1.1 Aerodynamic Model  
To develop the aerodynamic model of a wind turbine, the mathematical representation of wind power (power extracted from the wind) is given in Equation (1) [65]. Here, \(K_{pow}=\frac{1/2\cdot p\cdot\Pi\cdot R^{2}}{\frac{1}{\text{base }[w]}}\) represents the power constant, \(v_{w}\) is the wind speed, and \(C_{\text{p}}\) is the power coefficient. Similarly, \(\rho\), \(R\), \(\lambda\), and \(\beta\) denote the air density (kg/m\({}^{3}\)), turbine radius (m), relative turbine speed (tip speed ratio), and blade pitch angle in degrees, respectively. \[P_{tur\ [p,u]}=K_{pow}v_{w}{}^{3}C_{p}(\lambda,\beta) \tag{1}\]  
where \(C_{\text{p}}\) is defined as the ratio of the extractable mechanical power to the power in the wind stream [66] or the power extraction efficiency of the wind turbine [67]. In [68,69], different models that represent \(C_{\text{p}}\) for three-bladed turbines are discussed. The most common is the model presented in Equation (3), where \(c_{1}\)-\(c_{6}\) are numerical constants, whose values are given in Table 10.  
\[C_{p}=c_{1}\left(\frac{c_{2}}{\lambda_{i}}-c_{3}\beta-c_{4}\right)^{-\frac{ \alpha 5}{3_{i}}}+c_{6}\lambda \tag{2}\]  
Here,  
\[\frac{1}{\lambda_{i}}=\frac{1}{\lambda+0.08\beta}-\frac{0.035}{\beta^{3}+1} \tag{3}\]  
Using the analytical method, it is found in the literature that \(C_{\text{p}}\) reaches its maximum at a particular air velocity ratio, i.e., v_2/v_1 = 1/3; therefore, \(C_{\text{p,max}}\) is 0.593 [66]. This ideal value of \(C_{\text{p,max}}\) is called the \"[PERSON] factor\", after [PERSON], who first derived this ideal value. Thus, it is concluded that under ideal conditions, only 59.3% of the wind power is extractable from the wind in terms of mechanical energy. In real terms, the maximum extractable mechanical power or \(C_{\text{p}}\) should be less than these ideal conditions. Furthermore, it is clear in Equation (2) that the power coefficient \(C_{\text{p}}\) is a nonlinear function and depends on two factors: specific (or relative) turbine speed, \(\lambda\), and blade pitch angle, \(\beta\). Relative  
Figure 3: Simplified schematic diagram of DFIG-based VSWT. turbine speed, \(\lambda\), is the ratio of the rotor blade tip speed to wind speed, \(V_{\rm w}\), as shown in Equation (7). \[\lambda=\frac{K_{\rm v}\ \omega_{t\ [p,u]}}{v_{w\ (ms^{-1})}} \tag{4}\]  
where \(K_{\rm v}\) denotes a speed constant, and it is the product of the base turbine speed, \(\omega_{\rm tbase\ [rad/s]}\), and the diameter of the turbine, \(D_{\rm[m]}\)[70]. The dependence of the speed constant, \(K_{\rm v}\), on turbine diameter indicates that \(\lambda\) and \(C_{\rm p}\) must be varied for each turbine due to the variation in turbine characteristics. Figure 4 shows the graphical representation of Equations (2)-(4) in terms of \(C_{\rm p}\), \(\beta\), and \(\lambda\) by using the values given in Table 10. It is clearly shown that the maximum achievable power coefficient value is 0.48 when the relative turbine speed and blade pitch angle of the turbine are 8.1 and 0 degrees, respectively. In other words, the maximum extractable mechanical power of the turbine from the available incident wind is 48%. It is important that \(C_{\rm p}\) should be maintained at or close to \(C_{\rm p,max}\) to obtain more power and capture more energy. In addition, Figure 5 shows the relationship between \(P_{\rm tur}\) and the variable range of wind speed and rotor speed. It clearly shows that at \(V_{\rm w}=12\) ms\({}^{-1}\), the optimum power is 0.73 pu, and the rotor speed is 1.2 pu. This rotor speed is the base rotor speed for this turbine. However, each curve in Figure 5 shows a specific optimal mechanical power and optimal rotor speed at defined wind speeds. It is important to consider that in all these curves, the blade pitch angle is zero. As described above, the VSWT DFIG model is represented per unit; therefore, \(\omega_{\rm t\ [p,u]}\) in Figure 6 and \(\omega_{\rm g\ [p,u]}\) in Figure 3 are equal.