## 3 Wind System Control at Variable Wind Speeds from 3 [m/s] up to \(V_{MAX}\) [m/s]  
With the determined mathematical model of the WT, there are two operation ranges (ORs) under consideration that depend on the wind speed value. The first OR--named OR1--is for \(3\leq V(t)\leq V_{MAX}=10.788\ [\mathrm{m/s}]\), where the wind power from the WT can still be captured at the generator that works at nominal stator flux and stator current values. Here, the control objective is to regulate the MAS \(n\) to the value \(n_{OPTIM}\), where the latter depends linearly on the wind speed value \(V(t)\). This control is achievable through the generator frequency \(f\). The second OR--called OR2--is for \(V_{MAX}\leq V(t)\leq 21\ [\mathrm{m/s}]\),where the operation forces the generator power to saturate and the objective is to regulate \(P_{TV}\) through the blade pitch angle \(\beta\), which will be analyzed later.  
For wind speeds up to the value \(V_{MAX}\), the MPP must ensure **optimal operation** for the wind system based on the following equations that define the mechanical Equation (3), considered for the case \(\beta=0\) specific to OR1:  
\[\begin{split} P_{TVMAX}(V)=2442.3V^{3},\\ P_{GE}=-\frac{571.32\times\frac{\left(2.3805\cdot 10^{-2}+R_{ad} \right)}{1-\frac{20}{7}}}{\left(\frac{2.3805\cdot 10^{-2}+\frac{\left(2.3805\cdot 10^{ -2}+R_{ad}\right)}{1-\frac{20}{7}}}{\left(\frac{2.3805\cdot 10^{-2}+\frac{\left(2.3805\cdot 10^{ -2}+R_{ad}\right)}{1-\frac{20}{7}}}{\left(\frac{2.3805\cdot 10^{-2}+\frac{\left(2.3805\cdot 10^{ -2}+R_{ad}\right)}{1-\frac{20}{7}}}\right)^{2}}+(2.0618\cdot 10^{-3})^{2}},\\ I_{N}=724.64\\ =\frac{13.8}{\sqrt{\left(\frac{2.3805\cdot 10^{-2}+\frac{\left(2.3805 \cdot 10^{-2}+R_{ad}\right)}{1-\frac{20}{7}}}{\left(\frac{2.3805\cdot 10^{-2}+ \frac{\left(2.3805\cdot 10^{-2}+R_{ad}\right)}{1-\frac{20}{7}}}\right)^{2}}+(2.0618 \cdot 10^{-3})^{2}}},\\ P_{TV}(n,V)=5.692\times 10^{5}V^{3}e^{-\frac{6641V}{n}}\left(0.15915 \frac{V}{n}-1.7488\times 10^{-2}\right)\end{split} \tag{12}\]  
where \(P_{TVMAX}(V)\) is obtained from the MPP on the \(P_{TV}(n,V)\) characteristic, at which point the optimal MAS setpoint is \(n=n_{OPTIM}=3.8269\times V\).  
Using the converter block displaced between the generator and the electrical grid, the optimal values of the stator frequency and of the additional rotor resistance \(f_{OPTIM}\) and \(R_{ad-OPTIM}\) are realized, respectively. From the prescribed ratio \"\(U/f\)\", the stator voltage \(U_{S}\) follows immediately, with a value that makes the flux constant. The generator's electrical power \(P_{GE}\) injected to the grid is the summed effect of the stator and rotor powers, that is, \(P_{GE}=Pstator+Protor\) (see Figure 2).  
From a control analysis perspective, the stator current needs to flow at the nominal value \(I_{N}=724.64\)\(\left[\mathrm{A}\right]\) and the MAS \(n\) must track the optimal value \(n_{OPTIM}\). To smartly decouple the control problem, we extract the solution for \(R_{ad}\) that ensures \(I_{S}=I_{N}\) from the previous equation, which results in \(R_{ad}\) as a function of \(n\) and \(V\):  
\[\begin{split} R_{ad}(n,V)=R_{ad-OPTIM}(n,V)\\ =\left(\frac{2}{7}n-1\right)\left(1.8932\times 10^{-2}f-\frac{2.3805 \times 10^{-2}}{7}\right)\\ +2.3805\times 10^{-2},\end{split} \tag{13}\]  
Figure 2: The wind system control architecture at WT’s maximum power point. The doubly fed induction generator has the stator and rotor voltages controlled by converters.  
Then, the remaining control objective concerns the control of the MAS value \(n\) seen as the controlled output, through the stator frequency \(f\) seen as the control input. With the resulting values \(f,n\), the \(R_{ad-OPTIM}\) above ensures nominal stator current value.