# Fractional Order Controller Design for Wind Turbines
## 4 Conclusions
\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) +** \\ **50\% K\({}_{u}\)** \\ \end{tabular} & \
\begin{tabular}{c} **Overshoot** \\ **(\%)** \\ \end{tabular} &
\begin{tabular}{c} **Rise Time (s) +** \\ **50\% \(I_{u}\)** \\ \end{tabular} \\ \hline GA-FOPID & 0.02 & 0 & 3.889 \(\times\) 10\({}^{-3}\) & 0.02 & 0 & 4.726 \(\times\) 10\({}^{-3}\) & 0.01 & 0 & 1.323 \(\times\) 10\({}^{-3}\) \\ MOGA-FOPID & 0.025 & 0.8 & 2.984 \(\times\) 10\({}^{-3}\) & 0.025 & 0.9 & 2.512 \(\times\) 10\({}^{-3}\) & 0.015 & 0.8 & 9.525 \(\times\) 10\({}^{-4}\) \\ PSO-FOPID & 0.01 & 7.2 & 2.437 \(\times\) 10\({}^{-3}\) & 0.01 & 7.2 & 2.512 \(\times\) 10\({}^{-3}\) & 0.005 & 9.1 & 9.525 \(\times\) 10\({}^{-4}\) \\ PSO-PID (IAE [4]) & 0.02 & 0 & 4.736 \(\times\) 10\({}^{-3}\) & 0.02 & 0 & 5.616 \(\times\) 10\({}^{-3}\) & 0.01 & 0 & 1.414 \(\times\) 10\({}^{-3}\) \\ ARG-PID (IAE [4]) & 0.025 & 0 & 4.736 \(\times\) 10\({}^{-3}\) & 0.023 & 0 & 5.616 \(\times\) 10\({}^{-3}\) & 0.02 & 0 & 1.511 \(\times\) 10\({}^{-3}\) \\ FOPID & 0.02 & 20.7 & 2.254 \(\times\) 10\({}^{-3}\) & 0.017 & 20.7 & 2.230 \(\times\) 10\({}^{-3}\) & 0.015 & 34.4 & 1.371 \(\times\) 10\({}^{-3}\) \\ PI (IAE [4]) & 0.005 & 5.2 & 1.283 \(\times\) 10\({}^{-3}\) & 0.005 & 5.2 & 1.264 \(\times\) 10\({}^{-3}\) & 0.003 & 5.5 & 6.734 \(\times\) 10\({}^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The effects of system parameter uncertainties on controller performance. Our future work will include implementation analyses using different platforms for the tuning process for these controllers. Conceptualization, I.P. and C.O.S. ; methodology, I.P. ; software, I.P. ; validation, I.P. and C.O.S. ; formal analysis, I.P. ; investigation, C.O.S. ; writing--original draft preparation, I.P. ; writing--review and editing, E.-H.D. ; supervision, E.-H.D. All authors have read and agreed to the published version of the manuscript. This work was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS-UEFISCDI, project number PN-III-P4-PCE-2021-0750, within PNCDI III. The authors declare no conflict of interest. The following abbreviations are used in this paper:  
IAE & Integral of absolute error  
ITAE & Integral of time-weighted absolute error  
ISE & Integral of squared error  
ITSE & Integral of time-weighted squared error  
FOPID & Fractional order proportional-integral-derivative controller  
GA & Genetic algorithm  
MOGA & Multi-objective genetic algorithm  
PSO & Particle swarm optimization  
ABC & Artificial bee colony  
\(A\) & Linear inequality constraint  
b & Linear inequality constraint  
Aeq & Linear equality constraint  
beq & Linear equality constraint  
lb & Lower bound  
ub & Upper bound  
C & Linear inequality constraint vector  
Ceq & Linear equality constraint vector  
**List of Symbols**  
The following symbols are used in this paper:  
\(v\) & Wind speed  
\(\rho\) & Density of air  
\(R\) & Rotor radius  
\(P_{t}\) & Aerodynamic power of rotor  
\(\lambda\) & Tip speed ratio  
\(\beta\) & Pitch angle of blade  
\(C_{p}(\lambda,\beta)\) & Power coefficient  
\(\omega_{t}\) & Rotor speed  
\(\omega_{ls}\) & Low-speed shaft  
\(\omega_{g}\) & Generator speed  
\(\omega_{upt}\) & Optimal value of \(\omega_{t}\)  
\(\theta_{t}\) & Rotor side angular deviation  
\(\theta_{ls}\) & Genarbox side angular deviation  
\(\theta_{g}\) & Generator side angular deviation  
\(T_{t}\) & Aerodynamic torque of wind turbine  
\(T_{ls}\) & Low-speed shaft torque  
\(T_{hs}\) & High-speed shaft torque  
\(T_{em}\) & Electromagnetic torque  
\(J_{t}\) & Turbine side inertia  
\(J_{g}\) & Generator side inertia