## 4 Results
### Comparison against Benchmark Models  
In order to support the usefulness of the approach proposed in this work, in this subsection, the comparison against benchmark models is pursued. The same kind of procedure is adopted, as depicted in Figure 1: the unique difference is given by the fact that the LASSO regression of Section 3.3 is substituted by other types of regression. For brevity, the reported information about this benchmark regressions is essential, since the selected models are well known in the literature. The benchmark models are the following:  
* Univariate LASSO: For each cluster, a LASSO regression similar to the one in Section 3.3 was set up. The set of possible covariates from which the regression starts is given by \(\left(v_{c},v_{c}^{2},v_{c}^{3}\right)\). Given that, as supported also in [32], the wind speed is the most important input variable for a power curve model; this benchmark was selected in order to analyze how much improvement is obtained when the same kind of regression is maintained and the most important operation variables are included (as in Section 3.3);
* Multivariate support vector regression: This benchmark is obtained by setting up a multivariate SVR for each cluster. The input variables are the corrected wind speed \(v_{c}\), the blade pitch \(\beta\), and the rotor speed \(\omega\). In this case, the non-linearity is accounted for implicitly by the model, and there is no relevance in feeding as the input higher powers of the input variables;
* Multivariate Gaussian process regression: This benchmark is similar as the previous, but the regression is based on the Gaussian process. The two multivariate benchmarks were selected because they have been widely employed for multivariate wind turbine power curve models [5, 18, 31], and it is therefore interesting to inquire to what extent the error metrics worsen when the simplifications of this work are adopted. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Metric** & **Overall** & **Cluster 1** & **Cluster 2** & **Cluster 3** \\ \hline \(\bar{\Delta}\) (\%) & 0.0002 & 0.003 & \(-\)0.006 & 0.002 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results for the average difference of the \(MAPE\) when dividing \(D_{2}\) into two random subsets. Figure 8: Average residual \(R\) (Equation (8)) as a function of the measured power \(P\): \(D_{2}\) data set. Data clustering: The well-established \(k\)-means algorithm was employed to divide the multi-dimensional data appropriately, and a separate sub-model was set up for each cluster. The method was tested on real-world data from a Vestas V52 wind turbine owned by the Lucky Wind company and sited in Southern Italy. The reference data set was employed for training the \(k\)-means algorithm and selecting the optimal data clustering (based on the silhouette coefficient): \(k=3\) was selected, and this qualitatively coincides with the main different control regions of a modern wind turbine. Subsequently, a polynomial LASSO regression for the power of the wind turbine was performed for each obtained cluster: the reference data were used for selecting the input variables and setting the regression coefficients through the \(K\)-fold cross-validation. The performance of the model was quantified by analyzing the discrepancy between the measurement and simulation in the target data set. It resulted that the mean absolute error of the model in the validation data set was 12 kW (1.4% of the rated power), corresponding to a mean absolute percentage error of 7.2%. The absolute errors did not increase in the near-rated region with respect to the moderate wind speed region (Cluster 3 against Cluster 2), and therefore, the percentage errors decreased, reaching 2.8% on average. This result is interesting because it is typical that, vice versa, the near-rated region is the most critical as regards power monitoring.