## 4 Proposed Load Scheduling Model
### Artificial Hummingbird Optimization Algorithm  
AHA is represented as one of the bio-inspired optimization algorithms, which mimics the intelligent behavior of a hummingbird. AHA was found by [PERSON] et al. (2022) in 2022 as a global optimization algorithm. The AHA includes three essential components, which are food sources, hummingbirds, and visit tables. The hummingbird chooses a suitable food source by evaluating nectar quality, the last time a source has been visited, and the nectar-refilling rate. In AHA, the food source is represented by an individual (solution) vector, while the nectar-refilling rate is considered by the fitness function of the optimization problem. It is worth mentioning that the highest nectar-refilling rate points to the leading fitness function value. In AHA, the position of a hummingbird and food source are the same because each hummingbird is assigned to a specific food source. The information about the food source of the position, nectar-refilling rate, and visit time are stored in the mind of a hummingbird. Moreover, the information is shared with other hummingbirds in the population. The time period since a hummingbird reached a food source represents the visit level (visit time). The visit level of every food source regarding various hummingbirds is registered into a visit table. The hummingbirds prefer visiting a food source with a high visit level, because it has a high nectar-refilling rate as compared to others. Thus, the visit table is used as a guide for hummingbirds to specify the appropriate food source. In each iteration, the visit levels in the visit table are updated for each food source. The AHA comprises four main stages, namely, initialization, guided foraging, territorial foraging, and migration foraging. The stages will be discussed in detail in the following section. Figure 1 shows the general flow chart of the AHA algorithm. _Initialization_  
Similar to other metaheuristic algorithms, the AHA uses the random distribution for initializing \(N_{P}\) solutions of a population (\(POP\)) within the search space \(D_{P}\) of decision variables and is as follows ([PERSON], 2017):  
\[X_{\bar{\mu}}^{0}=X_{\bar{\mu}}^{Low}+rand\left(X_{\bar{\mu}}^{Up}-X_{\bar{ \mu}}^{Low}\right)\text{, where}j=1,2,\ldots,D_{P},i=1,2,\ldots,N_{P} \tag{5}\]  
where \(X_{\bar{\mu}}^{0}\) is the initial value of \(j^{th}\) decision variable of \(i^{th}\) solution that refers to the \(i^{th}\) food source position. \(X_{\bar{\mu}}^{Low}\)and\(X_{\bar{\mu}}^{Up}\) are the lower and upper bounds of search space of \(j^{th}\) decision variable of \(i^{th}\) solution, respectively. \(rand\) is a random number, which is chosen within \([0,1]\). In this stage, the visit table that includes the visit level of each food source is initialized by  
\[VT_{ki}^{G}=\left\{ \begin{array}{ll}0&ifi\
eq k\\ null&ifi=k\end{array} \right.\text{, where}i=1,2,\ldots,N_{P},k=1,2,\ldots,N_{P} \tag{6}\]where \(VT_{ki}^{G}=0\) refers to the \(i^{th}\) food source that has just been visited by \(k^{th}\) hummingbird for \(G^{th}\) iteration. _Flight modes_  
The hummingbird can use one of three different flight modes for flying towards the food sources. The flight modes are axial, diagonal, and omnidirectional. In the axial flight mode, the hummingbird flies on one coordinate axis. In the meantime, with the diagonal flight mode, the hummingbird flies on two coordinate axes. On the other hand, in the omnidirectional flight mode, the hummingbird flies in three coordinate axes. The choosing or switching between various flight modes is achieved by a directional switching vector. The axial, diagonal, and omnidirectional flight patterns for \(D_{P}-D\) space can be presented, respectively, as follows [21]:  
Figure 1: AHA algorithm flowchart. \[D_{j}=\left\{ \begin{array}{ll}1&ifj=rand_{j}([1,D_{P}])\\ 0&else\end{array} \right.,\text{where}j=1,2,\ldots,D_{P} \tag{7}\]  
\[D_{j}=\left\{ \begin{array}{ll}1&ifj=P(r),r\in[1,K],P=randperm(K),K\in[2, \lceil r3.(d-2)\rceil+1]\\ 0&else\end{array}