## 2 Dynamic Model of Wind Turbine
### Dynamic Model of Wind Turbine  
The drivetrain can be modeled as a two-inertia system in which rotor blades and generator are combined in one axis through a gearbox. Since the moment of inertia of the rotor blade is very high, the influence of the moment of inertia on the gearbox can be neglected [28]. Therefore, the influence of the gearbox is reflected on the generator side, and the wind turbine is expressed as a two-inertia model. Assuming that the shaft is thin and long, a two-inertia system can be modeled as a system with torsion, connected by a spring and a damper. It is assumed that the rotor has inertia \(J_{r}\) and the generator has inertia \(J_{g}\). They are connected through a torsional spring with spring constant \(K_{sh}\) and a torsional damper with damping constant \(D_{sh}\), as illustrated in Figure 1. The aerodynamic torque applied to the drivetrain is denoted by \(T_{a}\) and the generator torque is denoted by \(T_{g}\). \(T_{sh}\) stands for the torsional torque developed in the shaft. \(\omega_{r}\) and \(\omega_{g}\) are the angular velocities corresponding to rotor and generator, respectively. For other types of drivetrain models, see, e.g., [29; 30; 31; 32; 33].  
The dynamics of the drivetrain are derived as [34]  
\[\begin{split} T_{a}&=J_{r}\dot{\omega}_{r}+T_{sh} \\ T_{sh}&=J_{g}\dot{\omega}_{g}+T_{g}\\ T_{sh}&=Q_{s}+Q_{d}=K_{sh}(\theta_{r}-\theta_{g}) +D_{sh}(\omega_{r}-\omega_{g}).\end{split} \tag{1}\]  
We define the state vector \(x=[\omega_{r},\omega_{g},Q_{s}]^{\top}\) and express the dynamics (1) in the state space as  
\[\begin{split}\dot{x}&=A_{s}x+B_{s}u\\ y&=C_{s}x,\end{split} \tag{2}\]  
where \(u=[T_{a},T_{g}]^{\top}\) is the input; \(y\) is the system output; and \(A_{s}\), \(B_{s}\), \(C_{s}\) are system matrices given by  
\[A_{s}=\begin{bmatrix}-D_{sh}/J_{r}&D_{sh}/J_{r}&-1/J_{r}\\ D_{sh}/J_{g}&-D_{sh}/J_{g}&1/J_{g}\\ K_{sh}&-K_{sh}&0\end{bmatrix},B_{s}=\begin{bmatrix}1/J_{r}&0\\ 0&-1/J_{g}\\ 0&0\end{bmatrix},C_{s}=\begin{bmatrix}0&1&0\end{bmatrix}.\]  
Figure 1: Two-inertia model of the drivetrain.  
The turbine power \(P_{t}\) of the wind turbine and aerodynamic torque \(T_{a}\) applied to the wind turbine are given by  
\[\begin{split} P_{t}&=\frac{1}{2}\rho\pi R^{2}C_{p}( \lambda,\beta)V^{3}\\ T_{a}&=\frac{1}{2}\rho\pi R^{3}C_{q}(\lambda,\beta)V^ {2},\end{split} \tag{3}\]  
where \(\rho\) is the air density, \(R\) is the radius of the blade, \(V\) is the wind speed, \(C_{p}(\lambda,\beta)\) is the power coefficient, and \(C_{q}(\lambda,\beta)\) is the torque coefficient. \(C_{p}(\lambda,\beta)\) and \(C_{q}(\lambda,\beta)\) depend on the tip-speed-ratio (TSR) \(\lambda\) defined by \(\lambda=\omega_{r}R/V\) and the pitch angle \(\beta\)[31; 32; 35]; \(C_{p}\) and \(C_{q}\) are related as \(C_{q}=C_{p}/\lambda\). We take a widely used model of \(C_{p}\) given by  
\[C_{p}(\lambda,\beta)=c_{1}\bigg{(}c_{2}\frac{1}{\Lambda}-c_{3}\beta-c_{4}\beta ^{c_{5}}-c_{6}\bigg{)}e^{-c_{7}\frac{1}{\Lambda}}, \tag{4}\]  
where the parameter \(\frac{1}{\Lambda}\) is defined as  
\[\frac{1}{\Lambda}=\frac{1}{\lambda+0.08\beta}-\frac{0.035}{1+\beta^{3}},\]  
and \(c_{1},\ldots,c_{7}\) are constants that depend on system parameters [34; 36]. Since \(C_{p}(\lambda,\beta)\) is related to the wind-rotor aerodynamic characteristics [37], the numerical values of \(c_{1},\ldots,c_{7}\) depend on the wind turbine under consideration. In this paper, the parameters of the T-100 of Argolabe. S.L. Engineering Company are used, where \(C_{p,max}\) is 0.4728 and \(\lambda\) is 6 [38]. The \(C_{p}(\lambda,\beta)\) curve is extracted from Matlab/Simulink simulations with zero pitch angle (\(\beta=0^{\circ}\)). Through simulations, the parameters of \(c_{1},\ldots,c_{7}\) are chosen as \(c_{1}=0.29\), \(c_{2}=115\), \(c_{3}=0.5\), \(c_{4}=0\), \(c_{5}=0\), \(c_{6}=6\), and \(c_{7}=13.1\).