## 3 System Description
### Equations  
The quantity of power that a wind turbine captures from the wind is defined as a factor, C\({}_{\rm P}\), multiplied by the power of the wind entering the swept area [17; 43]. Nevertheless, this factor depends on the TSR (usually stated as \(\lambda\)) and on the pitch (\(\beta\)) non-linearly, also involving nine wind turbine constants that generally are not known:  
\[{\rm C_{P}}(\lambda,\ \beta)=\frac{{\rm P_{harvested}}}{{\rm P_{wind}}} ={\rm C_{1}}\bigg{(}\frac{{\rm C_{2}}}{\lambda_{1}}-{\rm C_{3}} \beta-{\rm C_{4}}\beta^{2}-{\rm C_{5}}\bigg{)}{\rm e}^{-\frac{{\rm C_{6}}}{ \lambda_{1}}}+{\rm C_{7}}\lambda, \tag{1}\]  
\[\frac{1}{\lambda_{1}}=\bigg{(}\frac{1}{\lambda-{\rm C_{8}}\beta}-\frac{{\rm C _{9}}}{\beta^{3}-1}\bigg{)}. \tag{2}\]  
In the equation above, \(\beta\) is a controlled variable, while the TSR is defined as the ratio between linear velocity at the tip of the blades and that of the wind:  
\[\lambda= \frac{{\rm V_{tip}}}{{\rm V_{wind}}}=\frac{\Omega_{\rm rotor}{\rm R }}{{\rm V_{wind}}}. \tag{3}\]  
C\({}_{\rm P}\) and \(\lambda\) are key parameters in the control of wind turbines as system operators modify them, changing the energy captured by the wind turbine, to achieve a longer lifespan of the equipment and a better exploitation of the wind energy.  
The power of the wind is a function of the air density, the swept area of the rotor, and the wind speed:  
\[{\rm P_{wind}}=\frac{1}{2}\rho_{\rm air}{\rm A_{rotor}}{\rm V_{wind}^{3}}. \tag{4}\]  
The torque of the rotor produced by the wind (usually noted as T\({}_{\rm Aero}\)) can be found as a function of the harvested power and the rotation speed of this low-speed shaft (\(\omega_{\rm LSS}\)). In  
\begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Nominal power & 5 MW \\ Hub and rotor diameters & 3, 126 m \\ Hub height & 90 m \\ Cut-in, nominal, cut-out wind speed & 3, 11.4, and 25 m/s \\ Number of blades, rotor topology & 3, upwind \\ Rotor, generator nominal speed & 12.1, 1173.7 rpm \\ Efficiency & 94.4\% \\ Control & Collective pitch, variable speed \\ \hline \end{tabular}
\end{table}
Table 1: Key specifications of the NREL 5 MW [19].  
turn, this rotation speed can be obtained from the high-speed shaft, usually a measured variable, and the gear ratio N\({}_{\text{gear}}\):  
\[\text{T}_{\text{Aero}}^{\text{LSS}}=\frac{\text{P}_{\text{harvested}}}{\omega_{ \text{LSS}}}=\frac{\text{P}_{\text{wind}}\times\text{C}_{\text{P}}(\lambda,\ \beta)}{\text{N}_{\text{gear}}}. \tag{5}\]  
On the other hand, there is a control or generator torque. This variable can be controlled by the wind turbine operator. It often equals to a constant value in the nominal operation of the wind turbine, i.e., the nominal generator torque:  
\[\text{T}_{\text{Generator}}^{\text{HSS}}=\frac{\text{P}_{\text{harvested}}}{ \omega_{\text{HSS}}}=\frac{\text{P}_{\text{wind}}\times\text{C}_{\text{P}}( \lambda,\ \beta)}{\omega_{\text{LSS}}\times\text{N}_{\text{gear}}}. \tag{6}\]  
Therefore, considering the ideal rotation of the low-speed axis, the speed is constant in the nominal state. It should be taken into account that the control torque is related to the inertia of the high-speed shaft, while the wind torque is related to the low-speed shaft:  
\[\text{T}_{\text{Aero}}^{\text{LSS}}-\text{T}_{\text{Generator}}^{\text{LSS}} =\text{I}_{\text{Drivetrain}}\times\alpha^{LSS}, \tag{7}\]  
\[\text{T}_{\text{Generator}}^{\text{LSS}}=\frac{\text{T}_{\text{Generator}}^{ \text{HSS}}}{\text{N}_{\text{gear}}}. \tag{8}\]  
Again, the inertias must be carefully handled. In the case of the previous equations, I\({}_{\text{Drivetrain}}\) is related to the low-speed shaft, as the two torques are also calculated with respect to that speed axis. To close, \(\alpha\) states the angular acceleration of the chosen axis.  
As a result, the performance of the wind turbine in terms of harvested power, speed, etc., can be modified using the control variables, i.e., T\({}_{\text{Generator}}^{\text{HSS}}\) and \(\beta\) (pitch).