# Fractional Order Controller Design for Wind Turbines
## 2 Materials and Methods
### Fractional Order PID (FOPID) Controller  
The electric system is the fastest component of a wind turbine system. Therefore, the design of the air turbine and generator can be separated into two individual control loops. The inner loop, which is the faster of the two loops, controls the electric generator, while the outer loop controls the air turbine, which generates the optimal power for the rotor speed. In this study, the outer control loop was our focus to ensure that the proposed performance could be achieved. \begin{table}
\begin{tabular}{l c} \hline \hline  & **Parameter** & **Constant Value** \\ \hline Air density & \(\rho=1.29\) kg/m\({}^{3}\) \\ Rotor radius & \(R=21.65\) m \\ Gearbox ratio & \(n_{g}=43.165\) \\ Turbine side inertia & \(J_{t}=3.2\times 10^{5}\) kg\(\cdot\)m\({}^{2}\) \\ Generator inertia & \(J_{g}=34.4\) kg\(\cdot\)m\({}^{2}\) \\ Wind turbine external damping coefficient & \(K_{t}=0\) \\ Generator external damping coefficient & \(K_{g}=0\) \\ Shaft stiffness coefficient & \(K_{ls}=2.691\times 10^{5}\) Nm/rad \\ Shaft damping coefficient & \(B_{ls}=9500\) Nm/rad/s \\ Electromagnetic torque coefficients & \(K_{1}=0.1082\) ; \(K_{2}=0.2147\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the two-mass wind turbine mathematical model. Figure 3: The control scheme for the two-mass wind turbine mathematical model. Figure 2: A Simulink block diagram of the two-mass wind turbine mathematical model. For this reason, fractional order PID (FOPID) controllers were used for the outer control loop. FOPID controllers are a generalization of classical PID controllers: their output is represented by a linear assembly of the inputs, a fractional integral of the inputs and a fractional derivative of the inputs. In many scientific articles, they have most often been used in the form PI\({}^{\lambda}\)D\({}^{\mu}\), where \(\lambda\) represents the fractional order of the integral and \(\mu\) indicates the fractional order of the derivative. When both values are equal to 1, a classic PID controller is obtained. Thus, their values are represented by the interval (0; 1), which offers an advantage for adjusting the dynamic properties of a system [18]. Thanks to the additional two degrees of freedom (\(\lambda\) and \(\mu\)), supplementary parameters can be monitored, such as robustness, system sensitivity and complementary sensitivity. Equation (12) shows the FOPID control equation that was used:  
\[u_{FOPID}(s)\ =K_{P}\cdot e(s)+K_{I}\cdot\frac{e(s)}{s^{\lambda}}\ \ +K_{D} \cdot s^{\mu}\cdot e(s) \tag{12}\]  
Nonetheless, tuning the parameters of FOPID controllers can be difficult and often requires continuous updating over time. This highlights the importance of tuning FOPID parameters to improve the rejection of external disturbances, the system robustness and its tracking precision [18]. The tuning of FOPID parameters can be performed in multiple ways. In this study, we chose three different optimization algorithms (GA, MOGA and PSO) to compare. Before explaining each of these three optimization algorithms, the cost function that was used to achieve the desired performance must be discussed. As mentioned earlier, the objective was to design FOPID controllers for the purpose of maximizing wind power extraction. Additionally, the controllers had to deliver good results for stability, steady-state error, settling time, overshoot and, of course, robustness. Specifications regarding the phase margin, sensitivity function and robustness constraints were considered with the aim of achieving the desired performance requirements. Moreover, it should be noted that the values that were chosen for the crossover frequency (\(\omega_{gc}\)) and the phase margin (\(\phi_{m}\)) were 1000 rad/s and 85\({}^{\circ}\), respectively. Firstly, the system and FOPID controllers had to operate in the frequency domain.