# Robust Control Examples Applied to a Wind Turbine Simulated Model
## 4 Simulations and Comparisons
### Robustness Analysis  
This section summarizes further simulation results that concern the evaluation of the achieved characteristics of the developed control strategies when the effects of uncertainty and disturbance are taken into account. In particular, the wind turbine benchmark in the MATLAB(r) and Simulink(r) environments can vary the variables and the parameters of the simulated process in a statistical way. In this way, it is possible to analyze the effects of the model-reality mismatch and the measurement errors on the designed controllers. Moreover, a Monte Carlo analysis is also considered since it represents a practical approach for validating and verifying the features of the developed control schemes when applied to the considered wind turbine process. The same approach was for suggested for the first time by the same authors in [40] and applied to a different simulated system. The Monte Carlo tool is very useful in this case since the behavior of control strategies designed assuming the nominal plant depends on both the model-reality mismatch and the measurement errors. Under these considerations, the uncertainty values of the parameters and variables of the wind turbine simulator considered in this work are summarized in Table 5. Therefore, the Monte Carlo analysis was achieved by modeling these parameters and variables as Gaussian stochastic processes, with mean values equal to the nominal ones, and standard deviations corresponding to realistic error values, typical of wind turbine models [3]. Therefore, for the evaluation of the reliability and robustness characteristics of the designed control schemes, the average values of the \(NSSE\%\) index were computed and evaluated in simulation via 1000 Monte Carlo runs. Note that, as already remarked in Section 3, proper algorithms were exploited for guaranteeing the derivation of controller models that lead to stable closed loops. On the other hand, the stability of the closed loop system when adaptive algorithms are exploited was investigated in [37]. Note however that, if unstable models should be obtained due to large uncertainties of Table 5, gain and phase margin requirements have to be included in the controller design, as described, e.g., in [41], and the controller parameters can be computed using the \(w\)-plane design. In this case, it can be shown that the model of Equation (15) with \(n_{a}=n_{b}=n=2\) is transformed into its equivalent description of Equation (46):  
\[\frac{T_{s}\left(b_{1}\,T_{s}-b_{2}\,T_{s}\right)w^{2}+\left(-2\,b_{1}\,T_{s}+ 2\,b_{2}\,T_{s}+T_{s}\left(2\,b_{1}+2\,b_{2}\right)\right)\,w-4\,b_{1}-4\,b_{ 2}}{\left(a_{1}\,T_{s}^{2}-T_{s}^{2}-a_{2}\,T_{s}^{2}\right)w^{2}+\left(4\,a_{ 2}\,T_{s}-4\,T_{s}\right)w-4-4\,a_{2}-4\,a_{1}}\approx\frac{k}{w\,\tau_{o}-1} \,e^{-w\,L} \tag{46}\]  
Under the validity of the approximation of Equation (46), which neglects the fast dynamic stable modes, and consider only the unstable pole \(\tau_{o}\) with the effective delay \(L\) (see, e.g., the approximation in [36]), the adaptive controller parameters are computed via the relations of Equation (47): \begin{table}
\begin{tabular}{c c} \hline \hline
**Model Variable/Parameter** & **Standard Deviation** \\ \hline \(\beta(t)\) & 11\% \\ \(\omega_{g}(t)\) & 18\% \\ \(\tau_{g}(t)\) & 21\% \\ \(\tilde{P}_{g}(t)\) & 20\% \\ Pitch 2 nd order model & \\ natural frequency and damping ratio & 49\% \\ Drive train model efficiency & 5\% \\ Converter 1 st order model time constant & 50\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Wind turbine uncertain variables. \[\left\{ \begin{array}{lcl}K_{p}&=&\dfrac{\delta\,\tau_{o}}{A_{m}\,k}\\ T_{l}&=&\dfrac{1}{\frac{\pi}{2}\,\delta-\delta^{2}\,L-\frac{1}{\tau_{o}}}\\ \delta&=&\dfrac{A_{m}\,\phi_{m}+\frac{\pi}{2}\,A_{m}\,(A_{m}-1)}{(A_{m}^{2}-1) \,L}\end{array} \right.