# Adaptive Pitch Controller of a Large-Scale Wind Turbine Using Multi-Objective Optimization
## 4 Proposed Design Evaluation
### Simulation with Step Change Wind Speed Profile  
In the first simulation, the wind speed profile is composed of different upward and downward step changes. It starts at 12 m/s and increases its value +2 m/s every 100 s until reaching 22 m/s; then, the rest of wind profile is a descending mirror of this part, as shown in Figure 10. This wind profile allows evaluating the control response of system through the six operation points used in the design stage. Figure 10 shows the responses of the proposed GSC obtained for the satisfactory solution (satisfactory GSC) in previous section and those achieved by the BSC. The wind speed profile v\({}_{\text{w}}\), generator speed \(\omega_{\text{g}}\), tower f-a displacement x\({}_{\text{t}}\), and pitch angle \(\beta\) responses are depicted. The simulation results show that the proposed GSC performs better than the baseline controller for all wind speeds. In Figure 11, a part of these responses is zoomed to obtain a better appreciation of the improvement of the proposed design. The GSC achieves a faster wind speed disturbance rejection improving the tracking of \(\omega_{\text{g}}\), and the fluctuations in the controlled variable x\({}_{\text{t}}\) are smaller. Figure 9: Baseline controller structure within the nominal. Figure 11: Zoomed simulation responses for a wind speed step change from 12 to 14 m/s. Figure 10: Simulation responses for first simulation. In order to analyze a fair comparison, the obtained GSC tuned with the satisfactory solution parameters is also compared with this same GSC tuned with the parameters of the two optimal solutions based on each objective functions \(\mathsf{f}_{1}=\text{IAE}_{\text{avg}}\) and \(\mathsf{f}_{2}=\text{CVR}_{\text{std}}\). Different performance indices are calculated from the simulation data for the comparative quantitative analysis. These indices are the following: the two conflicting objectives used for optimization in the proposed designs (IAE\({}_{\text{avg}}\) and CVR\({}_{\text{xt}}\)), the cumulative variation rate of the pitch angle CVR\({}_{\beta}\), and the cumulative variation rates CVR\({}_{\text{Mfa}}\) and CVR\({}_{\text{Mss}}\) of the lifetime damage equivalent moments at the tower base in the fore-aft and side-to-side direction (Mfa and Mss), respectively [30, 69]. The expressions of the three last indices are calculated from the simulation data according to (26). These last indices have not been used in the optimization process; however, they have been calculated to perform a more complete quantitative analysis. Table 4 shows all these indices for the four controllers: satisfactory GSC, IAE\({}_{\text{avg}}\) optimal GSC, CVR\({}_{\text{xt}}\) optimal GSC, and BC. \[ \begin{array}{c}\text{CVR}_{\beta}=\int_{0}^{\mathsf{k}}|\beta(\mathsf{k} _{\text{k}})-\beta(\mathsf{k}_{\text{k}-1})|\;\text{dt}\\ \text{CVR}_{\text{Mfa}}=\int_{0}^{\mathsf{k}}|\mathsf{Mfa}(\mathsf{k}_{\text{ k}})-\mathsf{Mfa}(\mathsf{k}_{\text{k}-1})|\;\text{dt}\\ \text{CVR}_{\text{Mss}}=\int_{0}^{\mathsf{k}}|\mathsf{Mss}(\mathsf{k}_{\text{ k}})-\mathsf{Mss}(\mathsf{k}_{\text{k}-1})|\;\text{dt}\\ \end{array}  \tag{26}\]  
The IAE\({}_{\text{avg}}\) index related to the error has been improved at the expense of a greater control effort CVR\({}_{\beta}\) in the manipulated variable \(\beta\). It is important to clarify that the variations of step changes in wind speed are unrealistic. However, this is a very strong requirement for the control system, which must overcome the abrupt wind change. Observing the qualitative and quantitative results with step wind, the following points can be highlighted: the proposed GSC tuned with satisfactory solution parameters obtains a better CVR\({}_{\text{xt}}\) index value, reducing its index by 42.3% compared to that obtained by the baseline controller. With respect to the IAE\({}_{\text{avg}}\) value, the designed controller reduces this index by 93.3% in comparison with that obtained using the baseline controller. Figure 12 shows the spider diagrams obtained for the previously performed simulations.