# Fractional Order Controller Design for Wind Turbines
## 4 Conclusions  
The first objective of this study was to develop a two-mass wind turbine model in order to derive a transfer function that accurately described the behavior of a regular wind turbine motor. After acquiring the wind turbine model, the next objective was to develop controllers that could extract the maximum amount of energy from wind. After analyzing the properties of many types of controllers, we concluded that an FOPID controller would be the best choice. As FOPID controllers involve complex calculus, the parameters (\(K_{P}\), \(K_{I}\), \(K_{D}\), \(\lambda\), \(\mu\)) of the FOPID controllers were tuned using optimization methods in order to reduce the required human effort. Three optimization algorithms (GA, MOGA and PSO) were chosen to investigate which produced the best performance for our model. The controllers had to ensure a steady-state error that was equal to zero, a low overshoot, a small settling time and robustness to noise and disturbances. In Section 3, our results and discussion about the optimized FOPID controllers showed that the MOGA-FOPID controller produced the best overall performance compared to all of the other controllers (PSO-FOPID, GA-FOPID, ABC-PID [4], PSO-PID [4], FOPID [4] and PI [4]). Additionally, this outcome highlighted the improved performances that were achieved in this study compared to those in [4]. In conclusion, in order to control the speed of a two-mass wind turbine system, the fractional order PID controller that was tuned using a multi-objective genetic algorithm (MOGA-FOPID) was the best choice thanks to the ease of implementing the controller and tuning method in the MATLAB environment and also due to its remarkable results. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Controller** & \begin{tabular}{c} **Settling** \\ **Time (s) +** \\ **50\% B\({}_{u}\)** \\ \end{tabular} & \begin{tabular}{c} **Overshoot** \\ **(\%)** \\ \end{tabular} & \begin{tabular}{c} **Rise Time (s) +** \\ **50\% K\({}_{u}\)** \\ \end{tabular} & \begin{tabular}{c} **Overshoot** \\ **(\%)** \\ \end{tabular} & \begin{tabular}{c} **Rise Time (s) +** \\ **(\%)** \\ \end{tabular} & \
\begin{tabular}{c} **Overshoot** \\ **(\%)** \\ \end{tabular} &
\begin{tabular}{c} **Rise Time (s)** \\ **(\%)** \\ \end{tabular} \\ \hline GA-FOPID & 0.02 & 0 & 4.675 \(\times\) 10\({}^{-3}\) & 0.022 & 0 & 4.398 \(\times\) 10\({}^{-3}\) & 0.025 & 0 & 3.275 \(\times\) 10\({}^{-3}\) \\ MOGA-FOPID & 0.025 & 0.7 & 2.127 \(\times\) 10\({}^{-3}\) & 0.024 & 0 & 4.398 \(\times\) 10\({}^{-3}\) & 0.025 & 0 & 2.625 \(\times\) 10\({}^{-3}\) \\ PSO-FOPID & 0.015 & 7.2 & 2.181 \(\times\) 10\({}^{-3}\) & 0.015 & 7.2 & 4.398 \(\times\) 10\({}^{-3}\) & 0.013 & 5.7 & 2.580 \(\times\) 10\({}^{-3}\) \\ PSO-PID (IAE [4]) & 0.02 & 0 & 4.744 \(\times\) 10\({}^{-3}\) & 0.01 & 0 & 5.021 \(\times\) 10\({}^{-3}\) & 0.03 & 0 & 3.450 \(\times\) 10\({}^{-3}\) \\ ABC-FOPID & 0.025 & 0 & 5.436 \(\times\) 10\({}^{-3}\) & 0.025 & 0 & 4.675 \(\times\) 10\({}^{-3}\) & 0.03 & 0 & 3.450 \(\times\) 10\({}^{-3}\) \\ FOPID & 0.015 & 20.8 & 2.181 \(\times\) 10\({}^{-3}\) & 0.02 & 20.6 & 2.389 \(\times\) 10\({}^{-3}\) & 0.015 & 13.4 & 2.850 \(\times\) 10\({}^{-3}\) \\ PI (IAE [4]) & 0.005 & 5.2 & 1.004 \(\times\) 10\({}^{-3}\) & 0.005 & 5.2 & 9.349 \(\times\) 10\({}^{-4}\) & 0.005 & 4.2 & 1.525 \(\times\) 10\({}^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The effects of system parameter uncertainties on controller performance.