## 2 Wind Turbine Generator Systems  
The turbines are composed of the rotor of blades that convert the kinetic energy of the wind into mechanical energy, the gearbox that multiplies the speed of the rotor and transmits it to the axis of rotation of the generator, the electric generator, control system and the instrumentation [18]. Figure 1 shows a general diagram of wind turbine. Figure 1: Parts of a wind turbine. To obtain the mathematical model of the wind power extracted by the rotor, an analysis of the wind passing between the blades in the swept area is made. The power extracted by the rotor is described with the following equation [19; 20]:  
\[P_{rotor}=\frac{1}{2}\rho\ A\ V^{3}\ Cp \tag{1}\]  
where the wind density is \(\rho\), the sweeping area is \(A\), wind before the turbine is V and Cp is a power coefficient. To obtain the value of \(Cp\), an experimental method can be used, dependent of \(\lambda\), pitch angle of the blade \(\beta\) and \(\alpha\) constant [20; 21]. The value of \(\alpha\) is obtained of Table 1. The curve fit is a good approximation for values of \(3<\lambda<15\), which are suitable for all blade configurations and models. \[Cp=\left(\sum_{i=0}^{4}\sum_{j=0}^{4}\alpha_{i,j}\beta^{j}\lambda^{i}\right) \tag{2}\]  
\[\lambda=\frac{w_{r}R}{V}=\frac{2\pi nR}{60V} \tag{3}\]  
where \(\mathrm{w_{r}}\) is the rotation frequency (rad/s), n is the speed of rotation (rpm) and R is the radius of the rotor (m). The mechanical system can be modeled as a two-mass system without losing precision, which is shown in Figure 2. \begin{table}
\begin{tabular}{c c c c c} \hline
**i** & **j** & \(\alpha_{\mathrm{ij}}\) & **i** & **j** & \(\alpha_{\mathrm{ij}}\) \\ \hline
4 & 4 & \(4.9686\times 10^{-10}\) & 4 & 3 & \(-7.1535\times 10^{-8}\) \\
4 & 2 & \(1.6167\times 10^{-6}\) & 4 & 1 & \(-9.4839\times 10^{-6}\) \\
4 & 0 & \(1.4787\times 10^{-5}\) & 3 & 4 & \(-8.9194\times 10^{-8}\) \\
3 & 3 & \(5.9924\times 10^{-6}\) & 3 & 2 & \(-1.0479\times 10^{-4}\) \\
3 & 1 & \(5.7051\times 10^{-4}\) & 3 & 0 & \(-8.6018\times 10^{-4}\) \\
2 & 4 & \(2.7937\times 10^{-6}\) & 2 & 3 & \(-1.4855\times 10^{-4}\) \\
2 & 2 & \(2.1495\times 10^{-3}\) & 2 & 1 & \(-1.0996\times 10^{-2}\) \\
2 & 0 & \(1.5727\times 10^{-2}\) & - & - & - \\
1 & 4 & \(-2.3895\times 10^{-5}\) & 1 & 3 & \(1.0683\times 10^{-3}\) \\
1 & 2 & \(-1.3365\times 10^{-4}\) & 1 & 1 & \(6.0405\times 10^{-2}\) \\
1 & 0 & \(-6.7606\times 10^{-2}\) & 0 & 4 & \(1.1524\times 10^{-5}\) \\
0 & 3 & \(-1.3365\times 10^{-4}\) & 0 & 2 & \(-1.2406\times 10^{-2}\) \\
0 & 1 & \(2.1808\times 10^{-1}\) & 0 & 0 & \(-4.1909\times 10^{-1}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Cp coefficients \(\alpha_{\mathrm{ij}}\)[21]. Figure 2: Two masses model for wind turbine. In the mathematical model, the torque of the rotor and the electromechanical torque of the generator act in opposition and are the inputs to the model, the rotation speeds are the output. The mathematical model is represented by the following equations [22; 23]:  
\[\frac{dw_{gen}}{dt}=\frac{1}{2H_{gen}}\left[-\frac{P_{elec}}{w_{gen}+w_{0}}-D_{ tg}\Big{(}w_{gen}-w_{rot}\Big{)}-k_{tg}\Delta\theta_{m}\right] \tag{4}\]  
\[\frac{dw_{rot}}{dt}=\frac{1}{2H_{rot}}\left[-\frac{P_{mech}}{w_{rot}+w_{0}}+D_{ tg}\Big{(}w_{gen}-w_{rot}\Big{)}+k_{tg}\Delta\theta_{m}\right] \tag{5}\]  
\[\frac{d(\Delta\theta_{m})}{dt}=w_{base}\Big{(}w_{g}-w_{t}\Big{)} \tag{6}\]  
where the inertial constant depends on the geometrical distribution of the mass. Inertia represents the time in which the generator reaches its rated power with the kinetic energy of its mass in rotation [24]. Inertial moment is calculated according to:  
\[H_{rotor}=\frac{J_{rotor}\;\;w_{rotor}^{2}}{2\;P_{n}}\;\;\;\;\;\;\;\phi\;\;\;\; \;\;\;H_{gen}=\frac{J_{gen}\;\;w_{gen}^{2}}{2\;P_{n}} \tag{7}\]  
In the case of the wind rotor, the inertia can be approximated according to:  
\[J_{rotor}=\frac{1}{8}m_{r}R^{2} \tag{8}\]  
where m\({}_{r}\) represents the mass of the rotor (includes the blades) and R is the radius of the rotor.