# A Hybridized Flower Pollination Algorithm and Its Application on Microgrid Operations Planning
## 3 Hybrid Flower Pollination Algorithm (HSFPA)
### Testing Function Results  
This subsection verifies the performance and potential of the proposed method of the HSFPA. Twenty-three selected essential functions from CEC2017 [40; 49] are used to test the proposed algorithm's performance. The chosen functions include the types of features, e.g., single-modal, multi-modal, and fixed-dimensional multi-modal. The experimental testing results of the HSFPA are compared with not only the regional methods, e.g., FPA [34], and SCA [35] algorithms but also with the other popular metaheuristic algorithms, e.g., PSO [28], MFPA [36], GA [26], DE [32], BA [33] algorithms, and the hybrid methods, e.g., BA-ABC [37], PSO-ABC [38], and GWO-FPA [39]. The proposed and comparative algorithms are set with the same conditions, e.g., population size \(N\) is set to _80_, maximum iteration times \(T\) is set to _1500_, and number runs is set to _25_. The setting experiment is the upper and lower bounds \([L_{b},U_{b}]\) of the initial value, and dimension \(D\) of the test functions are set according to the benchmark functions [49]. Table 1 lists the primary setting parameters of the algorithms. Table 2 depicts the conducted results of the proposed HSFPA compared with the FPA, PSO, MFPA, and SCA algorithms for the selected testing functions. \begin{table}
\begin{tabular}{l l l} \hline \multicolumn{2}{c}{**Algorithms**} & \multicolumn{2}{c}{**Parameters Setting**} \\ \hline HSFPA & \(p=0.72,\Gamma=0.01;\,\,\,\lambda=1.5,\,q=0.6,\,\,\omega=0.5,\,l_{w},h_{w}=[0. 21,0.91],\) \\ \hline FPA [34] & \(p=0.72,\Gamma=0.01;\,\,\,\lambda=1.5,\) \\ \hline SCA [35] & \(\alpha=1.5,\,\,[L_{b},U_{b}]=[-100,100]\) \\ \hline BA [33] & \(A_{0}=0.9,\,\,r_{0}\!\!=0.15,\,\alpha=0.25,\,\,\gamma=0.16,\,f_{min},\,f_{ max}=[0,0.5]\) \\ \hline MFPA [36] & \(p=0.72,\Gamma=0.01;\,\,\,\lambda=1.5,\,q=0.6,\,\,\omega=0.6\) \\ \hline PSO [28] & \(V_{max}=10,V_{min}=-10,\omega=0.9\,\,to\,0.4,c_{1}=c_{2}=1.49455,\) \\ \hline GA [26], & \(R_{min}=0.1,\,\,\,\,R_{cr}=0.9\) \\ \hline DE [32] & \(F=0.7,\,\,R=0.15\) \\ \hline BA-ABC [37] & \(A_{0}=0.7,\,\,r_{0}\!\!=0.15,\,\alpha=0.25,\,\,\gamma=0.16\) \\ \hline PSO-ABC [38] & \(a=-1,\,\,b=1,V_{max},V_{min}=[-10,10],\omega=0.9\,\,to\,0.4,c_{1}=c_{2}=1.49455\) \\ \hline GWO-FPA [39] & \(a=2\) to \(0,b=1,1=[-1.1][r1,\,\,r2]=[2,0.001],\) \\  & \(p=0.72,\Gamma=0.01;\,\,\,\lambda=1.5\) \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters’ settings for the algorithms. In Table 2, the symbol '+' indicates the \"better\" of the proposed method produces a good result compared to other algorithms. Making the same comparison, the symbol '\(-\)' means that the comparison results are weaker, and the symbol '\(-\)' indicates that the comparison results are similar. The other columns in the table are averaged values of the effects of algorithms running independently 25 times for test functions to produce the global best. Observing Table 2, it can be seen that the number of winners' symbols '+'s belonging to the HSFPA are more than the loser or approximation. The optimal results of the proposed HSFPA are generally better than the FPA, PSO, MFPA, and SCA algorithms in seeking function accuracy for 23 selected essential test functions. Figure 3 shows the comparison of the convergence curve of the proposed HSFPA with the other algorithms, e.g., the FPA, MFPA, and PSO algorithms for the selected test benchmark functions. Both the curve and its function's view space of (\(F_{1}(x)\), \(F_{4}(x)\), \(F_{6}(x)\), \(F_{8}(x)\), \(F_{9}(x)\), and \(F_{10}(x)\)) are depicted for viewing comparisons. Figure 3 also shows the function view space on each left side of the subplots. Several expression models in the chosen functions from the CEC2017 are used in testing models: F1-F3: unimodal, F4-F10: multimodal, F11-F17: hybrid, and F18-F23: compound test functions. It is seen that function F1 is the convex and unimodal function.