## 4 Results  
A preliminary result regards the positive effect of the use of PCA for the normal behavior model. Table 2 reports an estimation of the computational time required for the models with and without PCA. The improvement in computational time is appreciable. Figure 3 reports an example of a time series of simulated and measured current for the healthy Ref wind turbine. From Figure 3, it is possible to notice that the dimension reduction through the PCA does not significantly affect the behavior of the model estimates, whose error metrics for the test data set are summarized in Table 3. From Table 3, it arises that the error metrics on the test data set are acceptably low and therefore the model is \begin{table}
\begin{tabular}{c c c} \hline
**Target** & **Regressors** & \(\mathbf{r_{xy}}\) \\ \hline \(P\) & \(\{v\ \beta\ T_{b}\ \omega\ \Omega\ T_{1}\ T_{2}\ T_{3}\ V_{1}\ V_{2}\ V_{3}\}\) & \(\{0.97\ 0.61\ 0.89\ 0.90\ 0.90\ 0.85\ 0.85\ 0.85\ 0.72\ 0.67\ 0.75\}\) \\ \(I\) & \(\{v\ \beta T_{b}\ \omega\ \Omega\ T_{1}\ T_{2}\ T_{3}\ V_{1}\ V_{2}\ V_{3}\}\) & \(\{0.97\ 0.61\ 0.86\ 0.90\ 0.90\ 0.85\ 0.85\ 0.85\ 0.85\ 0.71\ 0.66\ 0.74\}\) \\ \(V\) & \(\{v\ T_{b}\ \omega\ \Omega\ T_{1}\ T_{2}\ T_{3}\ I_{1}\ I_{2}\ I_{3}\}\) & \(\{0.67\ 0.67\ 0.67\ 0.63\ 0.63\ 0.63\ 0.63\ 0.71\ 0.71\ 0.71\}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Selected features for the normal behavior models and Pearson coefficient \(r_{xy}\) with the target: \(D_{train}\) data set. \begin{table}
\begin{tabular}{c c c} \hline
**Target** & **Sim. Time PCA** & **Sim. Time No PCA** \\ \hline Power \(P\) & 84 s & 354 s \\ Current \(I\) & 91 s & 383 s \\ Voltage \(V\) & 170 s & 968 s \\ \hline \end{tabular}
\end{table} 
Table 2: Simulation time with and without the use of principal component analysis (PCA). potentially capable of detecting anomalies, as will be seen herein. In order to estimate margins for the reliability of the proposed model, a 10-fold cross validation was performed with 300 model runs: for each model run, the error metrics were computed and the results reported in Table 3 should be intended as the average metrics over the model runs, to which a standard deviation was associated. From Table 3 it arises that the average error metrics can be considered particularly robust because their standard deviation is very low. Furthermore, as regards the power \(P\), it is possible to compare with the literature: the order of the error metrics is similar to the results collected in [36] for the same test case. Therefore, it can be stated that the dimension reduction through the PCA does not remarkably affect the quality of the regression and helps in highlighting the dependence of the target on the orthogonalized input variable matrix. In Figure 4, qualitative results are reported that are meaningful for the identification of the incoming fault and which will be elaborated further on. The simulated power is reported as a function of the measured power for two data sets approaching the date of the fault. On the left, the plots for the reference wind turbine Ref are reported, while the plots for the target faulty wind turbine Tar are on the right. It arises that, approximately two weeks before the fault, it is possible to distinguish anomalous behavior at the Tar wind turbine: the simulated power is significantly higher than the measured value. This means that the input features are such that the extracted power should be higher, according to the normal behavior model. In Figure 5, a similar kind of plot is reported for the simulation of the current, approaching the fault (the same data set as Figure 4c,d); from this Figure, it arises that the fault onset can be individuated, also modeling directly the electric parameters of the generator.