## 3 Operational Data Analysis: Methods and Results  
The data sets at disposal are the following:  
* \(D_{bef}\) goes from 1 September 2017 to 1 March 2018. It is a period during which the standard rotor rpm-power curve was operating. * \(D_{aft}\) goes from 1 March 2018 to 1 July 2018. It is a period during which T7 was operating with the improved rotor rpm-power curve. In Figure 2, the power-rotor rpm curve for, respectively, the data sets \(D_{bef}\) and \(D_{aft}\) is reported. The difference between the two is appreciable: After the upgrade, a lower rotational speed was needed to extract the same amount of power as before the upgrade. Figure 1: The layout of the wind farm. Unfortunately, it was impossible to study the impact of the upgrade using the power curve analysis. As discussed, for example, in Reference [30], the T7 wind turbine underwent an aerodynamic retrofitting: Vortex generators and passive flow control devices were installed on the blades. After this intervention, prior to the control upgrade, the wind speed measurements at turbine T7 provided by the control system were biased: This is discussed in Reference [30] through the study of the power factor \(C_{p}\). Therefore, without a reliable wind speed measurement at turbine T7, it was impossible to study the power curve and the effect of the upgrade must be estimated using other methods. The objective is therefore to estimate the power output of T7, denoted as \(y\) in the following, as a function of on-site conditions. Given the fact the wind speed measurements at T7 are unreliable and the operating parameters of T7 cannot be used in this task (because they change for given wind conditions, after the control upgrade), the critical point is deciding what variables opportunely define the on-site conditions and the relation between them and the power of T7. The decision is possibly to employ all the other 16 wind turbines in the farm as reference; the following measurements are available and can in principle be selected as input variables for modeling the power output of T7:  
* The nacelle wind speed,
* the power output,
* the individual blade pitch angles,
* the rotor revolutions per minute,
* the generator revolutions per minute,
* the high speed rotor temperature. The rationale is that if one trains a model with pre-upgrade data and validates it against a pre- and a post-upgrade data set, a difference should arise in the behavior of the residuals if the upgrade is really effective. One of the most important points in this kind of procedure is having an argument for selecting the most appropriate input variables for the model among all the possible measurements at our disposal in the wind farm. This problem is solved based on the stepwise regression algorithm [31]: It is an iterative algorithm for automatically selecting the most appropriate inputs for a multilinear regression task. Starting from an initial model, at each iteration of the algorithm, an input term is added to or removed from the model. The decision about keeping it, removing or restoring it is based on the \(p\)-value of an \(F\)-_statistics_, i.e., by testing the performance of the model with and without that potential  
Figure 2: The rotor rpm–power curves before and after the upgrade. term. If one input variable has a \(p\)-value which is greater than an exit tolerance (that is, if it is unlikely that the hypothesis of a zero coefficient in the linear model can be rejected), that variable is discarded. Therefore, it is important to study the results of the stepwise regression algorithm: Basically, the idea is that a loose bound (high values) on the \(p\)-value leads to larger models that reliably produce lower errors but are less solid, while a strict bound (small values) on the \(p\)-value leads to smaller models that can produce higher errors but are more solid. In this case, selecting a compromise is quite easy based on \(K\)-fold cross-validation [35] for the sensitivity study of the \(p\)-value selection.