# Robust Control Examples Applied to a Wind Turbine Simulated Model
## 4 Simulations and Comparisons
### Controller Performance Tests  
Regarding the fuzzy modeling and identification method, the GK clustering algorithm recalled in Section 3 with a number \(M=3\) of clusters and delays \(n=2\). These variables were applied for clustering the first data set consisting of \(\left\{P_{gk},\,\omega_{gk},\,\beta_{rk}\right\}\). A number of samples \(k=1\), \(2\),..., \(N\) were considered with \(N=440\times 10^{3}\). The same number of clusters and shifts were exploited for clustering the second data set \(\left\{P_{gk},\,\omega_{gk},\,\tau_{gk}\right\}\). After this procedure, the structures of the TS prototypes were derived for each output \(y_{k}\) equal to \(P_{gk}\) and \(\omega_{gk}\). In this way, the two continuous-time outputs \(y(t)=\left[\omega_{g}(t),\,\tau_{g}(t)\right]\) of the wind turbine continuous-time model of Equation (9) are approximated by two TS fuzzy prototypes of Equation (14). The performances of the fuzzy models that are derived using the procedure described above can be evaluated using the so-called Variance Accounted For (VAF) parameter [6]. In particular, the TS fuzzy model reconstructing the first output has a VAF index bigger than 90%, whilst for the second one, it was higher than 99%. This means that the fuzzy prototypes are able to describe the behavior of the controlled process with very good precision. These estimated TS fuzzy models have been used for the derivation of the fuzzy controllers and applied to the considered wind turbine benchmark. Two fuzzy controllers with 2 inputs and 1 output have been used for the control of the wind turbine system. As shown in Figure 4, these controllers are both fed by the sampled signals \(P_{gk},\,\omega_{gk}\) (i.e., the outputs of the wind turbine system) for the generation of the sampled signals \(\beta_{k}\) and \(\tau_{r}(t)\) (i.e., the control inputs for the wind turbine system). By using the inverse model principle, they were estimated exploiting the methodology recalled in Section 3.1. Again, the GK fuzzy clustering method has lead to two fuzzy regulators applied to the data sets \(\left\{\beta_{k},\,P_{gk},\,\omega_{gk}\right\}\) and \(\left\{\tau_{gk},\,P_{gk},\,\omega_{gk}\right\}\), respectively, with \(M=3\) clusters and \(n=3\) lagged signals. The controller performances were verified and validated via extensive simulations by considering different data sequences generated via the wind turbine simulator. Table 4 reports the values of the delay time \(T_{d}\), settling time \(T_{a}\), maximum overshoot \(S\%\) and the per-cent Normalized Sum of Squared tracking Error (\(NSSE\%\)) index defined in Equation (45):  
\[NSSE\%=100\sqrt{\frac{\sum_{k=1}^{N}\left(r_{k}-y_{k}\right)^{2}}{\sum_{k=1} ^{N}r_{k}^{2}}} \tag{45}\]  
Note that in partial load operation (Region 1), the performance is represented by the comparison between the power produced by the generator, \(y_{k}=P_{gk}\), with respect to the theoretical maximum power output, \(r_{k}=P_{r}\). On the other hand, in full load operation (Region 2), the tracking error is given by the difference between the generator speed, \(y_{k}=\omega_{\hat{g}k}\) and its nominal value, \(r_{k}=\omega_{nom}\). The achieved results are shown in Figure 7 for the case of the identified fuzzy controllers. Figure 7 depicts the signal representing the controlled generator speed \(\omega_{g}\) and the generated power \(P_{g}\) in gray bold gray line with respect to their desired values \(\omega_{nom}\) and \(P_{r}\) in dashed black line, respectively. It can be noted that in both partial and full load conditions, the fuzzy controllers are able to track the reference signals, as recalled in Section 2. Note that the performance of the fuzzy regulators are better than those achieved via the baseline governors, which were tuned with frequency approaches described in [23; 24].