## 4 Simulation Results
Furthermore, these peaks are eliminated by employing a drive-train damper [17]; i.e., the PID controller requires a drive-train damper to maintain the peak at about 14.13 rad/s below 0 dB (Figure 15). Furthermore, the red and blue plots represent the open-loop frequency responses without and with drive-train damper, respectively. Figure 15 shows the open-loop frequency response of the PID controller when a drive-train damper (in blue) is used, compared with the open-loop frequency response of the PID controller when a drive-train damper (in red) is not used. Figure 15 shows that the damper removes the peak at 14.13 rad/s, which is due to the drive-train mode. Thus, oscillation due to the drive-train mode is removed. However, the drive-train damper is not required when the optimal controllers are in place (Figure 14), which is a significant advantage over standard PID controllers. Figure 11: Time responses of power in modes 1, 2, and 3: MPC, LQG, and \(H_{\infty}\) based on Bladed linearization model; (**a**) mode 1, (**b**) mode 2, (**c**) mode 3. In summary, the time responses shown in Figure 9b, results presented in Table 1, and open-loop frequency responses shown in Figure 14 show that the overall performance of these optimal controllers are satisfactory. In contrast to the PID controller, the MPC and \(H_{\infty}\) controllers exhibit better performance than the LQG controller because the peak still remains when the LQG controller is in place although it is maintained below 0 dB. Figure 13b shows the corresponding control input, i.e., torque demand in mode 2. Generator speed control is achieved by adjusting torque in mode 2 as aforementioned. Figure 9c shows that the time responses for the MPC, LQG, and \(H_{\infty}\) controllers at a mean wind speed of 16 m/s are adequate in terms of fluctuations as they persist under 12%. Table 1 summarizes the results for generator speed fluctuations of all optimal controllers in mode 3. It depicts that fluctuations are adequate as they persist below 12% which is within the controller design specifications mentioned in Section 3.1. Figure 11c illustrates the corresponding time response of the power output of the MPC, LQG, and \(H_{\infty}\) controllers at a mean wind speed of 16 m/s (above-rated wind speeds, i.e., mode 3). Furthermore, modes 1 and 2 controller performances are also interpreted in the frequency domain. Figure 16 shows that, the gain crossover frequencies of the controllers (MPC, LQG, and \(H_{\infty}\)) at a mean wind speed of 16 m/s are in the range of 0.6 to 2 rad/s. Note that the PID controller is included here for comparison only. The results imply that these controllers are neither extremely aggressive nor slow. However, the peak corresponding to the drive-train mode for the PID and LQG controllers remain mostly above 0 dB at about 13.75 rad/s (higher frequency range), indicating that the controller's sensitivity to noise and uncertainty is high. These peaks are removed by incorporating a drive-train damper [17]; i.e., the PID controllers require a drive-train damper to maintain the peak at around 13.75 rad/s below 0 dB (Figure 15). However, MPC and \(H_{\infty}\) controllers achieve the \begin{table}
\begin{tabular}{c c c c} \hline
**Operating Mode** & **MPC** & \(H_{\infty}\) & **LQG** \\ \hline Mode 2 & 5.36 & 7.36 & 9.45 \\ \hline Mode 3 & 4.49 & 3.91 & 9.39 \\ \hline \end{tabular}
\end{table} 
Table 1: Mean of percentage of speed fluctuations. Figure 12: Open–loop frequency responses of PID, MPC, LQG, and \(H_{\infty}\) based on Bladed linearized model in mode 1.  
same results without incorporating such dampers, i.e., in contrast to modes 1 and 2, the peak remains present but is below 0 dB. Nonetheless, these controllers in mode 3 are more sensitive than those in mode 2.