# Robust Control Examples Applied to a Wind Turbine Simulated Model
## 2 Wind Turbine Simulated Model
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variable** & \(R\) & \(\chi\) & \(\omega_{\eta}\) & \(B_{dt}\) & \(B_{r}\) \\ \hline
**Nominal value** & 57.5 m & 0.6 & 11.11 rad s\({}^{-1}\) & 775.49 N m s rad \({}^{-1}\) & 7.11 N m s rad \({}^{-1}\) \\ \hline
**Variable** & \(B_{g}\) & \(K_{dt}\) & \(\eta_{dt}\) & \(j_{g}\) & \(j_{r}\) \\ \hline
**Nominal value** & 45.6 N m s rad \({}^{-1}\) & \(2.7\cdot 10^{9}\) N m rad \({}^{-1}\) & 0.97 & 390 kg m\({}^{2}\) & \(55\cdot 10^{6}\) kg m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Wind turbine benchmark parameters. It is worth noting also that the nonlinearity represented by Equations (1) and (3) is sketched in Figure 3, for different values of \(\lambda(t)\) (i.e., \(v(t)\)) and \(\beta(t)\). Figure 3 also shows the power curve that highlights the the so-called partial load (Region 1) and full load (Region 2) working conditions of a wind turbine [22]. In fact, the baseline controller implemented in the wind turbine simulator works in these two operating conditions. A schematic diagram of the baseline wind turbine controller system is detailed in Figure 4.  
In particular, Figure 4 highlights that in partial load working condition, the optimal tracking is achieved without any pitching of the blades, which are fixed to zero degrees. In this case, \(\lambda\) is constant at its optimal value \(\lambda_{opt}\), that is defined by the maximal value of the power coefficient map \(C_{p}\) when \(\beta=0\) degrees, as shown in Figure 3a. Therefore, this working condition is completely defined by setting \(\tau_{g}=\tau_{r}\) (i.e., the generator torque is equal to the required reference value) with pitch angle \(\beta=0\) degrees. Figure 4: The details of the baseline wind turbine controller. Figure 3: Examples of power coefficient function (**a**) and power curve (**b**). Figure 2: The wind speed sequence \(v(t)\) used in the benchmark model. The reference torque signal \(\tau_{r}\) shown in Figure 1 is computed as:  
\[\tau_{r}=K_{opt}\,\omega_{g}^{2} \tag{10}\]  
where:  
\[K_{opt}=\frac{1}{2}\,\rho\,A\,R^{3}\,\frac{C_{p_{\max}}}{\lambda_{opt}^{3}} \tag{11}\]  
with \(C_{p_{\max}}\) the maximal value of \(C_{p}\), related the to \(\lambda_{opt}\), i.e., the optimal tip-speed ratio, as sketched in Figure 3a. When the power reference \(P_{r}=4.8\) MW is achieved by the wind turbine system (it corresponds to the so-called rated power) [22], and the wind speed \(v(t)\) increases, the controller is switched to the control Region 2 (full load condition). In this working condition (Region 2), the control objective consists of tracking the power reference \(P_{r}\), obtained by regulating \(\beta\), such that the \(C_{p}\) is decreased, as shown in Figure 3a. In a traditional industrial control scheme, usually a PI controller is used to keep \(\omega_{g}\) at the prescribed value by changing \(\beta\); the second input of the controller is \(\tau_{g}\). The baseline controller considered in this work was implemented with a sample frequency at 100 Hz, i.e., \(T_{s}=0.01\) s. In full load conditions, i.e., in Region 2, the actuated input \(\beta\) is controlled via the relations of Equation (12) [25]:  
\[\left\{ \begin{array}{rcl}\beta_{k}&=&\beta_{k-1}+k_{p}\,e_{k}+\left(k_{i}\,T _{s}-k_{p}\right)\,e_{k-1}\\ \epsilon_{k}&=&\omega_{g_{k}}-\omega_{nom}\end{array} \right. \tag{12}\]  
with the sample index \(k=1,\,2,\,\ldots,\,N\). The parameters for this PI speed controller are \(k_{i}=0.5\) (\(1/\)s) and \(k_{p}=3\) (\(\mathrm{deg}/\mathrm{rad}/\mathrm{s}\)), with sampling time \(T_{s}=0.01\) s, as reported in [25]. For the case of the wind turbine system considered in this paper, with rated power \(P_{r}=4.8\) MW, the constant \(\omega_{nom}\) is equal to 162 (\(\mathrm{rad}/\mathrm{s}\)) [22]. The control of the further input \(\tau_{g}\) shown in Figure 1, a second PI regulator is used, in the form of Equation (13):  
\[\left\{