## 4 Numerical Analysis: Methods and Results  
As discussed in Section 1, this work deals with a twofold approach to the estimate of the power production improvement provided by wind turbine control upgrades. A validation on the basis of numerical simulations is provided: The aeroelastic code FAST v8 [37], developed at National Renewable Energy Laboratory, is selected. A detailed model of the wind turbine has been set up, in particular in terms of the aerodynamics of blades (that have been tuned thanks to the open source code QBlade [38]) and elastic properties of blades and tower, according to the information available in Reference [39] for the wind turbine model of interest for this study. The turning point for the purposes of this study is the representation of the control system (especially in terms of coupling with the electric generator), needed for the code to run the simulation. The approach called \"Simple Variable-Speed Torque Control\" has been chosen for modeling the relationship between torque and rpm: It is particularly useful, given the peculiarity of the data at our disposal (no reliable wind speed measurement is available at the site of the wind turbine of interest, as discussed in Section 3), because it does not require the relationship between wind and power. Actually, the relationship between generator rpm and produced power \(P\) (assuming an unitary efficiency, in order to use the relationship \(C_{m}=\frac{P}{\omega}=\frac{60P}{2\pi rpm}\) to compute the mechanical torque \(C_{m}\)) is taken from the operational data before and after the upgrade (data sets \(D_{bef}\) and \(D_{aft}\)) and is reproduced in the numerical simulations. The FAST code represents this relationship with approximations in three power regions. A quadratic polynomial from zero power to maximum-power control is used (called \"Region 2\"), while at rated power, a constant value of torque is set (called \"Region 3\"). These two regions are connected by a very short interval of linear transition (called \"Region 2\(\frac{1}{2}\)\"). The approximation functions and the values of transition between the regions are fitted to the operational measurements before and after the upgrade: The result of this procedure is reported in Figure 4, where the average experimental and simulated generator rpm-generator torque curves before and after the upgrade are shown. Notice that the post-upgrade average curve is more regular than the pre-upgrade and for this reason, the fit results to be more precise. This fact can also be seen from the operational data reported in Figure 2: Computing the average and the standard deviation of the rotor rpm per power intervals, the result was that the percentage standard deviations was lower after the upgrade. This means that the control upgrade has improved the stability of the curve through a more appropriate selection of the rotor rpm as a function of the tip speed ratio. Once the pre- and post- upgrade model have been tuned, the following step is the simulation of the wind speed and, consequently, of the electrical power. The decision has been to simulate several time series and the average wind intensities are selected to span from cut-in to rated with 0.5 m/s intervals. This has been done in order to resemble the typical wind speed spacing of the average power curve computed according to the IEC guidelines [26]. The time series were generated using the TurbSim package and an A turbulence class was selected because, on the basis of the data at disposal for this study (despite the fact that their sampling time is 10 min), it was evaluated that the site is considerably turbulent. A wind shear exponent of 0.2 was selected because it is a good approximation for onshore wind farms. Each run of the model simulates 400 s with a time step of the millisecond: In Figure 5, examples of wind intensity, rotor rpm, and power time series are reported. Figure 4: Numerical and experimental mechanical torque–rpm curves.