## 5 Conclusions  
In this study, a linear model is obtained by using the Bladed model of the Supergen Wind Energy Consortium (Supergen) 5 MW exemplar turbine model in different modes. These linear models are complex; hence, the realization and tuning of optimal controllers can be more challenging. Thus, the models must be reduced in order to design optimal controllers. Therefore, the Match-DC-Gain model reduction algorithm is used to reduce the model. The resulting models are similar to the original models in the meaningful frequency range. Based on these linear models MPC, \(H_{\infty}\), and LQG controllers are synthesized in each mode and compared with each other and with the standard PID controller, which is also included for comparison. Simulation results demonstrate that in, mode 1, all optimal controllers show satisfactory performances not only in time and frequency domain but also in the torque/speed plane. In mode 2, the open-loop frequency responses demonstrate that the optimal controllers, especially the MPC and \(H_{\infty}\) controllers, outperform the standard PID controllers. In more detail, the PID controller requires a drive-train damper to remove oscillation due to the drive-train mode, and designing such a damper could be challenging. However, when the optimal controllers are used, the oscillation due to the drive-train mode is removed without having to incorporate a drive-train damper. In mode 3, only the MPC and \(H_{\infty}\) controllers can be designed without having to incorporate a drive-train damper. The LQG controller needs to incorporate a drive-train damper similar to the PID controller, making the LQG controller less competitive than the other optimal controllers. Furthermore, the \begin{table}
\begin{tabular}{l c c c} \hline \hline  & **MPC** & \(H_{\infty}\) & **LQG** \\ \hline Mean power & 99.11 & 98.50 & 98.14 \\ efficiency in Mode 1 & & & \\ \hline RMSE of & & & \\ \(\omega(t)-\omega_{0}^{*}\) in Mode 2 & 2.55 & 3.523 & 2.023 \\ \hline RMSE of & & & \\ \(P(t)-P_{0}^{*}\) in Mode 3 & 1.9310 \(\times\) 10\({}^{5}\) & 1.8972 \(\times\) 10\({}^{5}\) & 3.0723 \(\times\) 10\({}^{5}\) \\ \hline \(\omega_{0}^{*}\): constant generator speed, 120 rad/s; \(P_{0}^{*}\): rated power, 5 MW. & & \\ \end{tabular}
\end{table} 
Table 2: Controllers performance in each mode. Figure 16: Open–loop frequency responses of PID, MPC, LQG, and \(H_{\infty}\) controllers based on Bladed linearized model in mode 3. MPC and \(H_{\infty}\) controllers are more sensitive than those in mode 2; hence, extra care should be taken when designing MPC and \(H_{\infty}\) controllers in mode 3. In summary, the MPC and \(H_{\infty}\) controller perform most satisfactorily because they can achieve performance on a par with the PID controller, without incorporating a drivetrain damper that is compulsory in the PID design. The LQG controller performs as well as the other optimal controllers only in mode 2, but it is more sensitive and harder to tune than the other controllers overall. As discussed in Section 1, wind turbine control involves both the determination of the operating strategy of the controller and its synthesis. This paper only discusses synthesis, which is concerned with designing SISO linear controllers at different operating points or modes. Further research will involve the operating strategy, which will allow the linear controllers designed in this paper to be combined by introducing an appropriate switching method to obtain the full envelope controller that covers the full operational envelope of wind speed. Finally, in this paper, the optimal controllers are designed as SISO controllers, but designing the optimal controllers as an MIMO controllers, which could be more efficient, will also be considered. Conceptualization, S.-h.H. and Y.-s.R. ; methodology, S.-h.H. ; software, Y.-s.R. ; validation, S.-h.H. and Y.-s.R. ; formal analysis, S.-h.H. ; investigation, Y.-s.R. and S.-h.H. ; resources, S.-h.H. and Y.-s.R. ; data curation, Y.-s.R. ; writing--original draft preparation, Y.-s.R. and S.-h.H.