# A Hybridized Flower Pollination Algorithm and Its Application on Microgrid Operations Planning
## 3 Hybrid Flower Pollination Algorithm (HSFPA)
### Enhanced Flower Pollination Optimization  
The lack of diversity agents, optimal local trap, and low search efficiency existed in the FPA algorithm as its drawbacks when dealing with complicated optimization problems. Strategies selected from the SCA method, e.g., adapting hop size, diversifying local search, and diverging agents, are used as the specific description of the proposed method in combining advantages to avoid the FPA algorithm drawbacks. The first strategy, adapting hop size, is used as the so-called step size for adding to Levy flight as the exploring pollination update formula to speed up the convergence. Since the pollen work in FPA relies on the rule of Levy flight to update the global pollination value, the Levy flight formula that would affect convergence speed by \(\mu\times L\) can be known, where \(\mu\) is a parameter presenting as the following step size:  
\[\mu=h_{w}-(h_{w}-l_{w})\cdot\frac{ite}{IterMax} \tag{16}\]  
where \(h_{w}\) and \(l_{w}\) are the weight coefficient constants of the control step length coefficient (in the experiment, \(h_{w}\) is set to 0.9 and \(l_{w}\) is set to 0.2; \(ite\) is the current iteration; \(IterMax\) is the total number of iterations). The uniform mutation operator of the global pollinations process by the step size parameter concerned leads to a significant adjusting hop in the early stage which can make the algorithm converge to the optimal value quickly. Therefore, the renewal mechanism of global pollination has become as follows:  
\[S_{i}^{t+1}=S_{i}^{t}+\mu\times L\big{(}g_{best}-S_{i}^{t}\big{)}+rand\Big{(}S _{j}^{t}-S_{k}^{t}\Big{)}, \tag{17}\]  
where \(S_{i}^{t+1}\) and \(S_{i}^{t}\) are the position of pollen solution \(i\) at the current iteration \(t\); \(S_{j}^{t}\) and \(S_{k}^{t}\) are the pollen solution at \(j\) and \(k\) generated randomly; \(g_{best}\) is the optimal global pollen solution. The second strategy, diversifying local search, is used as the mutation mechanism with a small probability that can be adapted to make it easier to jump out of the optimal local solution. According to the advantages of the mutation mechanism, some adjusting and modifying processes are carried out in the algorithm's local search, with carrying out half of the optimal solution and mutating boundary. The formula of changing half of the optimal solution is expressed as follows:  
\[S_{i}^{t+1}=\left(\frac{S_{i}^{t}+g_{best}}{2}\right)rand+\varepsilon\big{(}S_{ a}^{t}-S_{b}^{t}\big{)}\,, \tag{18}\]  
where \((S_{i}^{t}+g_{best})/2\) preserves the beneficial information of the current optimal individual and the valuable information of the individual \(i\) that is guided to the better optimal position; \(S_{i}^{t}\) is the operator itself; \(rand\) is a random number; \(S_{a}^{t}\) and \(S_{b}^{t}\) are random vectors. A small probability of mutation is used in checking the boundary value to jump out of the optimal local solution to the better promising area according to the fitness value. The boundary check value that achieves the purpose of one mutation in the local search is expressed as follows. If the agents are over the upper boundary of below the lower border, it will be placed in the radius of the space search as updating the following equations:  
\[S_{i}^{t}= \begin{cases}U_{b}\cdot(1-rand),&\text{if iteration is }even\\ L_{b}\cdot(1+rand),&otherwise,\end{cases} \,, \tag{19}\]  
where \(L_{b}\) and \(U_{b}\) are the lower and upper boundaries of the space search in the problem; \(rand\) is a random number. The different randomness neighbor search can be applied to reduce the low search efficiency issue caused. The third strategy, variants agents, is used for varying local search in exploiting the search phase, increasing convergence. The commonly used variation method is the different strategy extracted from the local differential search that significantly enhances the optimization algorithm. The variation strategy is expressed as follows:  
\[S_{i}^{t+1}=\left\{