## 2 Mathematical Models of the WT and of the Doubly Fed Induction Generator
### Mathematical Model of the Slip Ring Induction Generator  
The Fuhrhander FL-MD-70 models have the following nominal data: equivalent moment of inertia \(J=136\left[\mathrm{kgm}^{2}\right]\), nominal power \(P_{N}=P_{GE}=1.5\left[\mathrm{MW}\right]\), nominal voltage \(U_{N}=690\)\(\left[\mathrm{V}\right]\), nominal current \(I_{N}=724.64\)\(\left[\mathrm{A}\right]\), pole number equal to four (\(2p_{1}=4\)), nominal MAS \(n_{N}=1518.3\)\(\left[\mathrm{rpm}\right]\), nominal slip \(s_{N}=-1.2231\times 10^{-2}\)[24]. The doubly fed slip ring induction wound rotor generator parameters are as follows [43]. The short circuit impedance \(Z_{sc}=0.1058\)\(\left[\Omega\right]\), and the short circuit resistance is \(R_{sc}=R_{s}+R_{R}=4.7610\times 10^{-2}\)\(\left[\Omega\right]\), where \(R_{S}=R_{R}=2.3805\times 10^{-2}\)\(\left[\Omega\right]\) are the stator and rotor resistances, respectively. The short circuit nominal reactance is \(X_{sc}^{m}=0.10309\)\(\left[\Omega\right]\) at 50 \(\left[\mathrm{Hz}\right]\), which at different frequencies \(f\) becomes \(X_{sc}=X_{sc}^{m}\cdot\frac{f}{50}[\Omega]\). At the stator voltage \(\underline{U}_{S}\) (\(\underline{o}\) means phasor notation for \"\(o\)\") and frequency \(f\), the generator stator and rotor currents \(\underline{I}_{s}\), \(\underline{I}_{R}\) are (at negligible magnetization current)  
\[ \begin{split}\underline{I}_{S}=\frac{\underline{U}_{S}}{R_{S}+ \frac{R_{R}}{s}+jX_{Sc}}=\frac{\underline{U}_{S}}{2.3805\times 10^{-2}+\frac{2.3805 \times 10^{-2}}{s}+j\frac{0.10309f}{50}},\\ \underline{I}_{R}=-\frac{\underline{U}_{S}}{2.3805\times 10^{-2}+ \frac{2.3805\times 10^{-2}}{s}+j\frac{0.10309f}{50}}.\end{split}  \tag{5}\]  
where \(s\) is the slip at non-nominal operation. Note that the slip depends on the frequency \(f\) and MAS value \(n\) according to  
\[s=\frac{n_{1}-n}{n1}=\frac{\frac{f}{p_{1}}-n}{\frac{f}{p_{1}}}=\frac{\frac{f}{ 2}-n}{f/2}=\frac{f-2n}{f}=1-\frac{2n}{f}, \tag{6}\]  
The generator's nominal power at the shaft is equivalently expressed as  
\[P_{GE}=3\frac{p_{1}\underline{U}_{S}^{2}(R_{R}/s)}{\left(R_{S}+\frac{R_{R}}{ s}\right)^{2}+(2.0618\times 10^{-3})^{2}f^{2}}, \tag{7}\]  
where \(2p_{1}\) is the number of poles. With a single stator converter in the generator, wind energy can only be captured within a reduced wind speed domain. Thus, we propose an additional rotor converter within the architecture, which maximizes the wind energy capture over the full interval of wind speeds under which the WT operates. With a wound rotor, the rotor voltage \(\underline{U}_{R}\) can be used to control the desired MAS as follows. Through a rotor converter, it is \(\underline{U}_{R}\) that can change the rotor parameters \(R_{R}\) through an additional additive resistance defined as  
\[R_{ad}=\underline{U}_{R}/\underline{I}_{R}. \tag{8}\]  
Therefore, by proper adaptation of the rotor additional resistance \(R_{ad}\), we can change \(R_{R}+R_{ad}\), in turn modifying the equivalent generator's characteristic to make it depend on the wind speed value \(V\). As a consequence, since \(R_{ad}\) will be more significant than \(R_{R}\), the rotor electric power \(P_{rotor}\) injected into the grid will be approximated with \[P_{rotor}=3R_{ad}I_{R}^{2}. \tag{9}\]  
Ultimately, it is the rotor converter that changes \(R_{ad}\) through \(\underline{U}_{R}\), to result in \(P_{rotor}\).