## 2 Mathematical Models of the WT and of the Doubly Fed Induction Generator  
### WT Mathematical Model and Parameters  
The wind system control is focused on the main variables that define its functioning at the WT's MPP: the angular speed \(n[\) rps\(]\), which must track the optimal value \(n_{OPTIM}\), and the stator current \(I_{S}\), which must be constrained at the nominal value \(I_{N}=724.64\ [\mathrm{A}]\) for the analyzed WT model. In the deep continental Oravita area, the wind speeds typically reach 25 [m/s] during summer windstorms. Here, the Fuhrlander FL-MD-70 WTs of 1.5 [MW] capacity is three-bladed with 70 [m] diameter and with an equivalent inertial moment of \(J=136\ [\mathrm{kgm}^{2}]\). These WTs automatically start whenever the wind speed increases beyond 3 [m/s]. The WT is meant to function in dynamic regime at variable wind speeds [2; 3; 4; 5; 6; 7]. Given the dynamical process [2; 3; 4; 5; 6; 7], the kinetic moment equation is  
\[J\frac{d\omega}{dt}=M_{TV}-M_{GE}, \tag{1}\]  
where \(\omega\) is the mechanical angular speed (MAS) at the electric generator's shaft, \(J\) is the equivalent moment of inertia, \(M_{TV}\) is the torque of the WT at the generator's shaft and \(M_{GE}\) is the electromagnetic torque of the generator's shaft. In terms of powers, the equivalence of the previous equation becomes  
\[J\frac{d\omega}{dt}\omega=P_{TV}-P_{GE}, \tag{2}\]  
with \(P_{TV}\) being the WT power and \(P_{GE}\) being the generator electromagnetic power. Expressing \(\omega=2\pi n=6.28n\) and considering the inertial moment \(J=136\ [\mathrm{kgm}^{2}]\), Equation (2) is particularized at \(\overline{J}=5369.1\ \mathrm{[kgm}^{2}]\) as  
\[\overline{J}\frac{dn}{dt}n=P_{TV}-P_{GE}. \tag{3}\]  
The term \(P_{TV}\) in the previous equation generally depends on the wind speed and the MAS, while \(P_{GE}\) depends on the MAS, on the voltage and on the frequency of the generator's stator and rotor, respectively. When the WT is started, it is required to bring it to the maximum power point as soon as possible. After coupling the generator to the power grid by WT MAS control, the MPP should be tracked at all times. Up to a maximal wind speed value of \(V_{MAX}=10.788\) [m/s], \(P_{TV}\) can be absorbed by the generator functioning at nominal flux and current values on the stator. Beyond the mentioned wind speed value, the power captured by the generator will saturate from above. The WT mathematical model (MM) is completely defined by the characteristic \(P_{TV}(n,V,\beta)\)[1; 2; 3; 4; 5]. Based on data collected in the Oravita location, the \(P_{TV}(n,V,\beta)\) characteristic was fitted to a widely used literature model, \(P_{TV}(n,V,\beta,\theta)=aV^{3}e^{-c\frac{V}{\pi}}(\frac{1}{2\pi}\frac{V}{n}e^ {-d\beta}-b)\) with parameters \(\theta=[a,b,c,d]\):  
\[P_{TV}(n,V,\beta,\theta)=5.692\times 10^{5}V^{3}e^{-\frac{6.6041V}{\pi}}\left(0.1591 5\frac{V}{n}e^{-3.0303\times 10^{-2}\times\beta}-1.7488\times 10^{-2}\right). \tag{4}\]  
Generally, the parameters of the model \(P_{TV}\) can be identified in two steps. First, for \(\beta=0\), the main parameters \(a,b,c\) can be identified through nonlinear regression on (4) about the values \(n\), \(V\) measured at the time points when \(\frac{dn}{dt}=0\) and using measured \(P_{GE}\), for which it holds that \(P_{TV}=P_{GE}\). This approach avoids knowing \(dn/dt\), which is difficult to obtain in practice, as the sampling period is not sufficiently high. However, if \(dn/dt\) is available, then \(P_{TV}\) can be computed at any point in time, accepting a known measured \(P_{GE}\). Then, there are several solutions for finding parameter \(d\). For example, when the WT works under no load, the stationary condition \(P_{TV}(n,V,\beta)=0\) leads to \(\beta=d\ln\left(\frac{9.1005V}{n_{MAX}}\right)\), meaning that, by measuring the blade's pitch angle, the maximum MAS value \(n_{MAX}\), at some wind speed value \(V\), the coefficient \(d\) can be determined.