# Optimal Fractional PID Controller for Buck Converter Using Cohort Intelligent Algorithm
## 4 Results and Discussion
### Performance Comparison of CI  
The efficiency of the CI algorithm for optimization of the FOPID parameters was compared with the genetic algorithm (GA) [26], particle swarm optimization (PSO) [25], artificial bee colony (ABC) [24], and the simulated annealing (SA) [53; 54] methods for the same cost function ISE. The PSO and the ABC methods are swarm intelligence methods inspired by the collective behavior of animals or insect colonies, whereas the Genetic algorithm is an evolutionary search algorithm based on natural selection process and genetics. The SA optimization method can be used for multi-objective optimization and works like the annealing process used in metallurgy for heating and cooling of materials. All are meta-heuristic search algorithms like the CI and hence are used for comparison. The response of the FOPID controlled buck converter system was optimized using different algorithms, and is shown in Figures 14 and 15.  
Figure 14: System response for optimization of ISE and ITSE.  
Figure 13: CI optimized system response to load variations.  
A comparison of the time domain specifications and the cost function of the CI method with the other optimization methods is shown in Tables 4-7. The rise time and settling times are comparable in all the methods, but the overshoot and the computational time is least for the CI method for the cost function \(J_{1}=ISE\). The average number of functional evaluations is much lesser in the case of the CI method, which indicates a lesser control effort. The value of the minimized objective function is also least for CI with \(J_{1}\).  
For the cost function \(J_{2}\), the overshoot is small for CI and PSO methods, with the latter having the least average overshoot. The cost function value is small for the PSO and ABC methods, but computational time is very high, increased by 57.4% in PSO and by 96.4% in the ABC method. For the cost function \(J_{2}\), the cost function of the PSO is the best and the CI value is worst. However, the computational time is still the least in the case of the CI, being much lesser than all the other methods.  
For the cost function \(J_{4}\), the cost function values are comparable for all the methods. The simulated annealing method fails to reach the steady state value even after 200 iterations, and hence cannot be suitable for this index. The overshoot is increased in all the methods, with the least overshoot and cost function given by the ABC. Even though the computation time has increased for the CI, it continues to take lesser time for convergence compared to all the other methods.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & \multicolumn{5}{c}{\(J_{1}=ISE\)} \\ \cline{2-6}  & **CI** & **GA** & **PSO** & **ABC** & **SA** \\ \hline Avg. Overshoot \% & 5.6 & 14.6 & 16.2 & 18 & 21 \\ \hline Rise Time (ms) & 0.037 & 0.038 & 0.036 & 0.035 & 0.035 \\ \hline Settling time (ms) & 0.265 & 0.19 & 0.167 & 0.24 & 0.32 \\ \hline Cost function & 0.006 & 0.0057 & 0.0063 & 0.0067 & 0.0064 \\ \hline Avg. computation Time (s) & 542 & 1600 & 720 & 1100 & 1800 \\ \hline Avg. Function evaluations & 84 & 242 & 88 & 100 & 400 \\ \hline Avg. No. of iterations & 21 & 60 & 22 & 25 & 100 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of different optimization methods for the cost function ISE.  
Figure 15: System response for optimization of ITAE and IAE.