## 1 Introduction
This is definitely reasonable, given that the theoretical expectation is that the power of a wind turbine is (Equation (1)):  
\[P=\frac{1}{2}\pi R^{2}\rho v^{3}C_{p}(\beta,\lambda). \tag{1}\]  
In Equation (1), \(P\) is the produced power and depends on the rotor radius \(R\), the air density \(\rho\), the wind speed \(v\), and the power factor \(C_{p}\), which depends on the blade pitch angle \(\beta\) and the tip-speed ratio \(\lambda\) (or, in other words, the rotational speed \(\omega\)). The role of the rotational speed and blade pitch in data-driven models for the power was explored in detail in [5]: in that study, the rotor speed and blade pitch were added once at a time to a Gaussian process regression for the power of a wind turbine. The main result was that the inclusion of both variables reduced the error metrics and the rotational speed was slightly more influential than the blade pitch. The use of these two variables was discussed also in [29; 31; 35]. A further relevant technical development for multivariate wind turbine power curve models is data clustering [36]. Actually, it is likely that employing more than one sub-model might be more convenient than employing one model for all the power curve span. This is reasonable, because between the cut-in and the rated wind speed, it is possible to distinguish three operation regions, which have different features:Near the cut-in (approximately between 3 m/s and 5 m/s of wind intensity), the rotational speed of a wind turbine is practically fixed and the blade pitch varies;
* In the full aerodynamic load region (approximately between 5 m/s and 9 m/s of wind speed), the wind turbine attains the maximum possible aerodynamic efficiency by varying the rotational speed and holding the blade pitch practically fixed;
* In the partial aerodynamic load region (approximately above 9 m/s of wind intensity), the rotational speed is held fixed at the rated speed and the load is varied by regulating the blade pitch. Based on these considerations, the objective of the present study is formulating a multivariate method for data-driven wind turbine power curve analysis, which can be easily implemented in industrial applications. Therefore, each building block of a good multivariate wind turbine power curve model was simplified as much as possible. This was performed as follows:  
* The selection of the input variables was drastically simplified. Based on considerations similar to [5], the selected covariates of the model are wind speed, blade pitch, and rotor speed;
* The structure of the model was selected according to a compromise. It is evident that a linear model would be too simple: this was observed also in the recent study [31]. Nevertheless, it is worth exploring if it is possible to account for non-linearity in a simplified way, namely through a polynomial in the above indicated input variables. Therefore, a polynomial LASSO regression was selected. The advantage of this kind of model structure is that a selection of the covariates is substantially performed when the coefficients of the polynomial are set. A covariate, which was excluded from the model, has a vanishing coefficient in the polynomial;
* The data clustering was performed using the \(k\)-means algorithm on a reference data set: it was selected because it is a very well-established method, which can easily be implemented in industrial applications. The number of clusters was set automatically by computing the average silhouette score for each cluster number arrangement. The peculiarity and the innovativeness of the present study can therefore be individuated in a complex application (data-driven multivariate wind turbine power curve), characterized by several critical points, which must be mastered in depth, in order to provide solutions to each building block of the problem, which can be as simple as possible.