## 4 Results
### Polynomial LASSO Regression  
In Table 3, the coefficients of the LASSO regression are reported: as indicated in Section 3.3, they were computed from the reference data set \(D_{1}\). It is interesting to notice that for Cluster 1, there is a weak dependence of the power \(P\) on the rotor speed \(\omega\), and this is consistent because the wind turbine basically operates at a fixed rotor speed in that region. As regards Cluster 3, the coefficient of the \(v_{c}^{3}\) covariate was set to zero, which is reasonable because, near the rated power, the power \(P\) grows more slowly than \(v_{c}^{3}\). The results for the \(K\)-fold cross-validation for the three clusters are reported in Table 4, in the form of the average RMSE and associated standard deviation. From Table 4, the proposed model behaved better in the near-rated region (Cluster 3) with respect to the moderate wind intensity region of Cluster 2: this is a non-trivial result because in previous studies [31], it has been highlighted that the near-rated regime is particularly critical to simulate. In Figure 6, the measured and simulated \(D_{2}\) data are reported, grouped into clusters. Figure 7 reports a zoom on an excerpt of the data from Figure 6, in order to appreciate the capability of the model in reproducing the behavior of the measured data. In Table 5, the results are reported as regards the model errors when simulating the output on the target data set \(D_{2}\): the main result is that on average, the single measurement of the power \(P\) can be simulated with a 7.2% percentage error. An interesting aspect is that the absolute errors do not increase from Cluster 2 to Cluster 3 and decrease in percentage (up to 2.8% on average), consistent also with Table 4; this is noticeable because, typically, the most challenging operation region for modeling the power \(P\) is the near-rated power (as discussed for example in [31]). The overall result for the \(MAE\) is noticeable as well, because it means that the power \(P\) can on average be predicted with an absolute error of 12 kW, which is 1.4% of the rated power of the test case wind turbine. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Cluster** & **Average RMSE (kW)** & **Std. Dev of RMSE (kW)** \\ \hline
1 & 15.5 & 9.2 \\ \hline
2 & 31.4 & 4.1 \\ \hline
3 & 28.7 & 6.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of the \(K\)-fold cross-validation for the three clusters: D1 data set. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Covariate** & **Cluster 1** & **Cluster 2** & **Cluster 3** \\ \hline \(x_{1}\) & \(-\)4.5 & 150.3 & \(-\)0.12 \\ \hline \(x_{2}\) & \(-\)7.9 & \(-\)18.6 & 1.8 \\ \hline \(x_{3}\) & 0 & \(-\)123.9 & \(-\)956.2 \\ \hline \(x_{4}\) & 5.5 & 0 & 3.1 \\ \hline \(x_{5}\) & \(-\)0.9 & \(-\)3.1 & \(-\)Figure 8 allows a visualization of the model properties depending on the operation regime: it consists o the plot (for data set \(D_{2}\)) of the residuals \(R\) (Equation (8)), averaged in intervals of measured power \(P\) whose amplitude is 10% of the rated. The average absolute residual was in general of the order of 5 kW and increased at the crossroad between Cluster 2 and Cluster 3; this is reasonable because, in this region, the wind turbine control switches from variable rotor speed and fixed pitch to rated rotor speed and pitch control. It is reasonable that the weak point of the proposed method is cluster merging; multi-dimensional measurements at the crossroad of two clusters should likely be assigned partially to both of them, for example through a fuzzy membership function. Nevertheless, this was considered out of the scope of the present study because simple methods, easily implementable in the industry, were selected. Figure 6: Measured and simulated power \(P\) for the data set \(D_{2}\); data are divided according to the three clusters. Figure 7: Measured and simulated power \(P\) for the data set \(D_{2}\): zoom on a data subset.