# Fractional Order Controller Design for Wind Turbines
## 3 Results and Discussion  
The simulation structure that was presented in Figure 3 was implemented in a blank MATLAB/Simulink workspace. The resultant controllers (GA-FOPID, MOGA-FOPID and PSO-FOPID), along with the controllers from [4] (PSO-PID, ABC-PID, FOPID and PI), were tested using different types of references, which are discussed in this section.  
Firstly, the controllers were tested using a classic step response with a reference of 1 rad/sec (normalized value). As can be seen in Figure 4, all of the controllers worked exceptionally well, with quite low overshoots, small settling times and zero steady-state errors. Figure 5 shows the corresponding control signals. Secondly, a disturbance was simulated using a step signal at 0.08 s (Figure 6). In all of the cases, very high overshoots could be noticed, although this could be overlooked due to the quick return to the reference signal.  
The next stage was to test the behavior of the controllers using a square wave signal with an amplitude of 0.5 and a frequency of 0.5 (rad/s). Figure 7 shows the same performance results that were obtained for the step input, both for positive and the negative slopes.  
As another test, a sine wave was used, which had an amplitude of 0.3, a bias of 0.45, a frequency of 2\(\uppi\)/60 (rad/s) and a phase of 0 rad. As shown in Figure 8, all of the controllers displayed an ideal overlap on the reference signal.  
Figure 8: The speed response of the wind turbine model for a sinusoidal reference.  
Figure 6: The behavior of the wind turbine model when a step disturbance occurred.  
Figure 7: The speed response of the wind turbine model for a square wave reference.  
For a final test, the performance of the controllers was tested using a wind gust reference, which was obtained from random values for a sine wave. As shown in Figure 9, the proposed PSO-FOPID had the fastest response.  
By taking into account all of these results (which can also be seen in Tables 6 and 7), a conclusion could be drawn. Based on Table 6, the fastest controller in terms of settling time was the PSO-FOPID and the lowest overshoots were produced by GA-FOPID, PSO-PID and ABC-PID. The lowest amplitude value was obtained by PSO-FOPID.  
As shown in Table 7, the fastest controllers to return to the reference signal after the simulated step disturbance were the MOGA-FOPID and PSO-FOPID, with very little difference between their results.  
From the obtained results, it could be observed that the controllers that were developed in this study produced better performances in terms of settling time, overshoot and the  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Controller Type** & **S****  
robustness of the control signal compared to the controllers that were developed in [4]. Which of the developed controllers was the best would depend on the needs of the client. PSO-FOPID would be an excellent choice for a fast system but GA-FOPID would be a better choice for a low overshoot. On the other hand, MOGA-FOPID would be the best option for a mid-range performance for both settling time and overshoot.  
Additionally, Tables 8 and 9 present the results for the behavior of the controllers when system parameter uncertainties of \(\pm\) 50% of their nominal values occurred. The changes in performance were minimal. In this robustness test, it could be concluded that GA-FOPID, PSO-PID and ABC-PID were the controllers with the most stable performance.