# Chaotic Evolutionary Programming for an Engineering Optimization Problem
## 6 Results and Discussion  
The standard test functions were considered to analyze the ability of the CEPPS solution approach. The results obtained by the proposed CEPPS variants were compared with that obtained from CEP. To compare the performance, seven standard test functions were solved using CEP and CEPPS variants. Table 2 shows the worst (W), average (A), and best (B) results for each test function after 30 trials. The comparison of the convergence plots for the investigated standard test functions is shown in Figure 2 using the log scale. Figure 2 represents the convergence behavior of [PERSON]'s function, which exhibits that CEPPS-1 and CEP-1 converge quickly in the starting phase, but result in premature convergence. However, CEPPS-2 and CEP-2 result in much better solutions, although initially, these were slow to converge. Finally, CEPPS-2 obtains the best solution compared to the others. \begin{table}
\begin{tabular}{c c c c c} \hline
**Case** & **Valve Point Loading** & **Ramp Rate** & **Prohibited Operating Zone** & **Transmission Loss** \\ \hline
1 & \(\times\) & \(\times\) & \(\times\) & \(\times\) \\
2 & \(\checkmark\) & \(\times\) & \(\times\) & \(\times\) \\
3 & \(\times\) & \(\times\) & \(\checkmark\) & \(\times\) \\
4 & \(\times\) & \(\times\) & \(\times\) & \(\checkmark\) \\
5 & \(\checkmark\) & \(\times\) & \(\checkmark\) & \(\checkmark\) \\
6 & \(\times\) & \(\times\) & \(\checkmark\) & \(\checkmark\) \\ \hline \end{tabular}
\end{table} 
Table 1: Multi-fuel system undertaken for the study, \(P_{D}=2700\) MW [30]. Figure 2: Convergence behaviors of the different test functions. The convergence behavior of [PERSON]'s function is depicted in Figure 2b. It shows the same trend as in the case of [PERSON]'s function. However, for the Rastrigin function, it is found that CEPPS-2 performs significantly better than CEP-2 and provides a better solution. The convergence behavior in Figure 2c shows that in the case of [PERSON]'s function, both CEPPS-1 and CEP-1 result in premature convergence, while the performance of CEP-2 is significantly better than CEPPS-2. Furthermore, the behavior of [PERSON]'s function in Figure 2d shows that CEPPS-1 results in the best solution of the problem, whereas both CEP-1 and CEP-2 are the worst performers. The result of the sphere function in Figure 2e shows that the CEPPS-2 algorithm has the best convergence rate. Furthermore, the convergence behavior comparison in Figure 2f,g for the step and Step 2 function, respectively, indicates that CEP-2 and CEPPS-2 have better performance than the others. Even CEP-2 performs better than CEPPS-2 for the Step 2 function. Thus, the convergence behavior plots show that the CEP-2 and CEPPS-2 variants perform better than CEP-1 and CEPPS-1. Their performance is judged either on the basis of the convergence rate or solution quality. It is also observed that CEPPS-1 performs better on some of the standard test problems, whereas CEP-2 and CEPPS-2 show supremacy for other standard test functions. The system depicted in Table 1 was solved for ELD by applying CEPPS-2. This system was considered for six different cases. In all the cases, the population size \(N_{P}\) were fixed at 50, and the number of iterations \(IT^{max}\) was fixed at 2000. The cost comparison was performed with other published work and is presented in Table 3. The results of biogeography based optimization (BBO), composite PSO (CPSO), GA with mutation update (CGA-MU), differential evolution (DE), the improved gravitational search algorithm (IGA), the improved genetic algorithm with multiplier update (IGA-MU), the enhanced augmented Hopfield neural network (ELHN), the hybrid of DE and BBO (DEBBO), krill herd optimization (KHA), the quadratic programming augmented Hopfield neural network (QP-ALHN), PSO, and synergic predator peg optimization (SPPO) were considered for the comparison of the solution quality.