A Hybridized Flower Pollination Algorithm and Its Application on Microgrid Operations Planning

Abstract

The meta-heuristic algorithms have been applied to handle various real-world optimization problems because their approach closely resembles natural human thinking and processing relatively quickly. Flowers pollination algorithm (FPA) is one of the advanced meta-heuristic algorithms; still, it has suffered from slow convergence and insufficient accuracy when dealing with complicated problems. This study suggests hybridizing the FPA with the sine–cosine algorithm (call HSFPA) to avoid FPA drawbacks for microgrid operations planning and global optimization problems. The objective function of microgrid operations planning is constructed based on the power generation distribution system’s relevant economic costs and environmental profits. Adapting hop size, diversifying local search, and diverging agents as strategies from learning SCA are used to modify the original FPA equation for improving the HSFPA solutions in terms of diversity pollinations, increasing convergence, and preventing local optimal traps. The experimental results of the HSFPA compared with the other algorithms in the literature for the benchmark test function and microgrid operations planning problem to evaluate the proposed scheme. Compared results show that the HSFPA offers outstanding performance compared to other competitors for the testing function. The HSFPA also delivers efficient optimal performance in microgrid optimization for solving the operations planning problem.

1. Introduction

Microgrid operation planning is considered one of the ways to promote the national policy to increase the variety of power generating resources for remote or island areas [1]. Microgrid operations planning is vital in increasing energy to raise the share of clean energy power, safe power among scattered natural power sources, and ensure the power system’s stability [2]. Due to the increased energy demand of consuming power in areas for industry and citizens, the national strategies for localities on proactively developing diversified clean energy promote growing diversity of power generating recourses [3,4].

The microgrid satisfies the dynamic load and power quality because of its power generation, transmission, and distribution capabilities that are considered a combination of an internal power system and an internal load [5,6]. Microgrids have the potential to improve power supply quality in remote places such as mountains, islands, and other areas, as well as efficiently prevent large-scale power outages caused by accidents and disasters [7]. Many local governments will regard the microgrid as the research focus of their power growth in the coming years due to its qualities of practical use and optimal coordinated control of multiple dispersed power sources [1,3]. Solar and wind power generation is needed to employ an enhanced gravitational search method with the multi-objective problem [8]. A new energy power generation in islands with grid-connected microgrids is optimized using an improved real-number coding adaptive evolutionary algorithm [9]. As energy demand rises, grid dependability and power quality make the power distribution operations planning challenge more complex [10,11].

According to characterized grid planning operations, two types of community microgrids are supportive and economic collaborative operation categories [12,13]. The critical supportive microgrid category focuses on essential supporting technologies, e.g., controlling, clustering transactions, and directing. It means that support plans are not affected much by the factor of the investor’s profits or stakeholders. Several features of this group are a single entity [14], central management [15], and top-down control [16,17]. On the other hand, the collaborative operation schemes focus on investing profits or interested organizations to increase mutual energy assistance amongst microgrids to enable flexible transactions, e.g., the multi-agent management, the cooperative strategy, and effective stakeholder economics operations. Several features of this group are dynamic [18,19] and flexible trans [20] stakeholder relationships [21,22].

Because of the large-scale nonlinear problem, typical methods for operations planning difficulties, such as linear programming, and least-squares equations, would suffer from, for example, high complicated computational costs [23]. One of the most promising methods for dealing with large-scale nonlinear issues is the meta-heuristic algorithm. The metaheuristic algorithms often use random or stochastic variables for exploring and exploiting search processes that can deal with nonlinear problems. In contrast, the exact methods often use gradient or derivative techniques that can apply to solve linear issues. The meta-heuristic algorithm has advanced quickly with applications in engineering, security, and finance that have become one of the most popular strategies to solve complex optimization problems [24,25].

The metaheuristic algorithms are classified with nature-inspired as the intelligent natural characteristics: natural selection, social, wild swarm behavior, and autocatalytic physical phenomenon. The popular algorithms are representative examples, e.g., Genetics-algorithms (GA) [26,27], Particles swarm algorithm optimizations (PSO) [28], Artificial bees colony-algorithm (ABC) [29], Ants colony optimization (ACO) [30], Cats swarms optimization algorithm (CSO) [31], Differential evolution (DE) [32], Bats algorithm (BA) [33], Flowers pollination algorithm (FPA) [34], Sine–Cosines algorithm (SCA) [35], etc.

Flowers pollination algorithm (FPA) is one of the advanced meta-heuristic algorithms with effective intelligent optimization, with a few simple parameters that have attracted the attention of many science and engineering researchers in recent years [34,36]. Nevertheless, it has suffered from slow convergence, falling in the optimal local trap, and insufficient accuracy when dealing with complicated problems like the operations planning problem.

Hybridizing two algorithms is one of the suggested ways of dealing with these issues, for example of hybrid methods, e.g., Bats algorithm with Artificial bee colony (BA-ABC) [37], hybridized Particles swarm optimization with Artificial bee colony optimization (PSO-ABC) [38], and hybridized Grey wolf optimization with Flower pollination algorithm (GWO-FPA) [39].

This study proposes a Hybridizing FPA with SCA (namely HSFPA) to enhance the flowers pollination algorithm for dealing with the complicated problem of microgrid operations planning. 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 HSFPA is implemented to enhance diversity agents, increase convergence and prevent local optimal traps. The suggested method’s performance efficiency is verified using the selected benchmark functions in CEC2017 [40] and the microgrid operations planning test [41]. The suggested approach contributions are highlighted as follows:

• Suggesting strategies to enhance the flowers pollination algorithm (HSFPA) based on hybridizing FPA with SCA for the power microgrid operations planning;
• Evaluating the suggested method’s performance by testing the selected benchmark functions in CEC2017 and comparing the proposed method’s results with the other algorithms in the literature;
• Establishing the multi-object approach for the suggested method (MHSFPA) based on the Pareto optimality for planning the grid operations with generation resources.

The rest of the paper is arranged as follows: Section 2 presents the microgrid operations planning statement and reviews the FPA and SCA. Section 3 proposes an HSFPA by hybridizing FPA with SCA; and uses the critical test function to verify the viability of the proposed algorithm. Section 4 describes the HSFPA for tackling the operations planning issue. The ending is discussed as the conclusions in Section 5.

2. Related Work

The section will present the microgrid operation statement and review the original algorithm of the FPA and SCA algorithms. Presentation details are stated in the following subsections.

2.1. Microgrid Operation Planning Statement

The grid operation planning model is one of the most effective alternative ways can generate the service-consuming power with better quality power supply, as demand for improved power supply quality in isolated or distant places grows [2,42]. Figure 1 displays a typical structure of a microgrid with power sources distribution. The expression balance of the system power load in period t is figured out as an equality constraint:

$$P_{total}\left(t\right)=P_{ES}\left(t\right)+P_{WT}\left(t\right)+P_{FC}\left(t\right)+P_{MT}\left(t\right)+P_{PV}\left(t\right)+u·P_{EX}\left(t\right)$$

(1)

where $P_i\left(t\right)$ are the components power outcomes of the energy supply system; $i=1.2\dots n$: corresponds to the following characters: ES—Energy storage batteries; WT—Wind turbines; FC—Fuel cells; MT—Micro-gas turbines; PV—Photovoltaic power, EX—Extra grid; $P_{total}\left(t\right)$ is the power load of the system for the period of t as times [43].

The generated operation cost of ES is assumed as charging and discharging power with a constant variable at the end of the period (t) connected to the next period at the end of the period (t − 1) that allows for brief dispatching interval characteristics. The link between energy storage capacity and charge–discharge power in the electric energy storage model is expressed as follows:

$$C_{ES}\left(t\right)=\left(1-\tau \right)P_{ES}\left(t-1\right)+\left[P_{ES,ch}\left(t\right){\eta }_{ES,ch}-\frac{P_{ES,dis}\left(t\right)}{{\eta }_{ES,dis}}\right]\times \Delta t,$$

(2)

where $C_{ES}\left(t\right)$ is a relevant operation cost of the ES in period t; $P_{ES}\left(t\right)$, $P_{ES,ch}\left(t\right)$, and $P_{ES,dis}\left(t\right)$ are the power of storage capacity, charge, and discharge of the ES in period t, respectively; ${\eta }_{ES,ch}$, ${\eta }_{ES,dis}$ are coefficients of the ES efficiency adjustment, respectively, t time; $\tau$ is a variable rate of the energy self-discharge.

The operations relevant economic costs of the WT are related to the power outcome determined [15] by the natural wind speed, primarily the output of wind turbines:

$$C_{WT}\left(t\right)=\frac{1}{{\eta }_{WT}\left(t\right)}\times P_{WT}\left(t\right),$$

(3)

where $C_{WT}\left(t\right)$ and $P_{WT}\left(t\right)$ are relevant operation costs and output power capacity of the WT in period t; ${\eta }_{WT}\left(t\right)$ is a converted coeffect parameter that is related to unit prices of some kinds of operations, e.g., investment, depreciation, maintenance. ${\eta }_{WT}$ is set to as expression of $0.057+0.34\left(\frac{P_{WT}\left(t\right)}{35}\right)-0.3095{\left(\frac{P_{WT}\left(t\right)}{33}\right)}^2+0.033{\left(\frac{P_{WT}\left(t\right)}{33}\right)}^3$ in the experiment section [2,44]. The output power comprises numerous factors of the wind turbine’s power output connected to wind speed:

$$P_{WT}\left(t\right)=\{\begin{array}{c}\begin{array}{c}0, 0\le v\left(t\right)<v_{ci} \\ A+B\times v\left(t\right)+C\times v{\left(t\right)}^2, v_{ci}\le v\left(t\right)\le v_r \end{array}\\ \begin{array}{c}{P}_{v\left(t\right)}, v_r\le v\left(t\right) \le v_{co } \\ 0, v_{co}<v\left(t\right) \end{array}\end{array},$$

(4)

where $v\left(t\right)$ is $v_{ci}$, $v_r$, and $v_{co}$ are the wind-cut, wind-rated, and wind-cutting velocities; the WT parameters of $A,B,C$ are coeffective of the power characteristic curve; $P_{v\left(t\right)}$ is the rated power of the fan.

The energy FC operating cost for consumption generated power during the operation [16] is expressed as follows:

$$C_{FC}\left(t\right)=\frac{G_{price}}{L_{gas}}\times \frac{P_{FC}\left(t\right)\Delta t}{{\eta }_{FC}\left(t\right)},$$

(5)

where $P_{FC}\left(t\right)$ is the output power of the fuel cell and ${\eta }_{FC}\left(t\right)$ is the efficiency in the period of t [17]; $L_{gas}$ is natural gas’s low calorific value. A relationship between $P_{FC}\left(t\right)$ and ${\eta }_{FC}\left(t\right)$ of the FC power generation efficiency and outcome power is expressed as the following function of ${\eta }_{FC}\left(t\right)=0.6735-0.0023\times P_{FC}\left(t\right)$ [2,44].

The operation relevant economic costs of the PV are related to the solar cell sampling point’s ambient temperature value depending on the formula of the power outcome given:

$$C_{pv}\left(T\right)=P_{pv}\left(t\right)\times {\eta }_{pv}=P_{rpv}\frac{I_F\left(t\right)}{800}\left[1+{\lambda }_t\left(T\left(t\right)-T_{st}\right)\right],$$

(6)

where $I_F\left(t\right)$ and ${\lambda }_t$ are practical illumination brilliance and temperature coefficient at measuring or sampling point t of power-temperature in the optimized operation; $T\left(t\right)$ and $T_{sc}$ are the PV ambient temperature value with sampling point at time t, and the under common test of temperature value, respectively; ${\eta }_{pv}$ is the output efficiency of PV power supply. The relationship of the $T\left(t\right)$ and $T_{sc}$ is given as $T\left(t\right)=25.6+\frac{I_F\left(t\right)}{800}\left(T_{sc}-20\right)$ in experiment [2,44]. The operation cost of a thermal generator of MT is expressed as follows:

$$C_{MT}\left(t\right)=\frac{G_{price}}{L_{gas}}\times \frac{P_{MT}\left(t\right)}{{\eta }_{MT}\left(t\right)},$$

(7)

where $P_{MT}\left(t\right)$ is the MT’s output power (unit kW) in period t; $C_{MT}\left(t\right)$ and ${\eta }_{MT}\left(t\right)$ are the cost function of miniature MT and the coeffective parameter of adaptive function efficiency in period t; $L_{gas}$ and $G_{price}$ are natural gas’s low calorific value (normal is set to $13.5 \mathrm{kWh}/\mathrm{kg}$) and natural gas price (unit is $$/{\mathrm{m}}^3$, normal is set to 169.0 $$/{\mathrm{m}}^3$) [45].

In the experiment part, the relationship function between $C_{MT}\left(t\right)$ and ${\eta }_{MT}\left(t\right)$ is expressed in MT output power and power generation efficiency as follows: ${\eta }_{MT}\left(t\right)=0.1068+0.4174\left(\frac{P_{MT}\left(t\right)}{65}\right)-0.3095{\left(\frac{P_{MT}\left(t\right)}{65}\right)}^2+0.0753{\left(\frac{P_{MT}\left(t\right)}{62}\right)}^3$ [2,46]. The safety grid and power balance must be considered when conducting microgrid planning, including the output power constraint of distributed power supply, the climbing rate limits of unit generators, and power interaction constraints.

The output power constraint of the distributed power supply is expressed as follows:

$$P_{i,min}\le P_i\left(t\right)\le P_{i,max}$$

(8)

where $P_{i,min}$,$P_{i,max}$ are the minimum and maximum active power output of the i-th power supply.

The FC technology efficiently converts a form of chemical energy into electric power through acting chemical reactions, which is an efficiency of power generation as substantially microgrid applications. The constraint of the unit climbing rate of FC is expressed as follows:

$$\{\begin{array}{c}\left|P_{FC}\left(t\right)-P_{FC}\left(t-1\right)\right|\le P_{FC}^{max}\\ \left|P_{MT}\left(t\right)-P_{MT}\left(t-1\right)\right|\le P_{MT}^{max}\end{array},$$

(9)

where $P_{FC}\left(t\right)$, $P_{FC}\left(t-1\right)$ are the active power output of the fuel cell in period t and period t − 1, respectively, and $P_{FC}^{max}$ is the upper power (max) limit of the fuel cell under climbing constraints.

The power interaction constraints when the microgrid and large grid are connected are:

$$P_{EX,min}\left(t\right)\le P_{EX}\left(t\right)\le P_{EX,max}\left(t\right),$$

(10)

where $P_{EX}\left(t\right)$ is the power of points of common coupleing of exchanged grids between the microgrid and the large grid; $P_{EX,min}\left(t\right)$ and $P_{EX,max}\left(t\right)$ are the minimum and maximum powers of an exchanged grid in the period of t.

2.2. Flower Pollination Algorithm (FPA)

Flowers pollination algorithms (FPA) [34,36] are inspired by the pollination of plants and flowers in nature that has gained many academics’ attention because of their advantages of ease of implementation, fewer parameters, and quick changes. The FPA algorithm is based on two different pollination strategies: local pollination and global pollination. Four principles followed are figured out to mimic the two pollination procedures to develop a better mathematical model:

✓

For the algorithm’s global pollination process, Biotic or cross-pollination is an exploring component in which pollinators can carry pollen and move far away distance that obeys Levy flights;

✓

For the local pollination, the plant self-pollination is considered the exploiting component of the algorithm;

✓

Insects that suck the nectar of flowers are considered pollinators to develop flowers constancy equivalent to a reproduction probability, which is the proportional similarity of two flowers relatives;

✓

For the diversity between the local pollination and global pollination, a parameter is used to control switching between them; a range of p, where $p\in \left(0,1\right)$, it can be adjusted for slightly biased towards local pollination.

Figure 2 depicts an illustration of types of pollination: self and cross-pollinations. Based on these four rules, the mathematical model established the FPA algorithm, which is described in the equations as follows. The global pollination is expressed with the updating equation as the following formula:

$$x_i^{t+1}=x_i^t+L\times \left(g_{best}-x_i^t\right)$$

(11)

where $g_{best }$denotes the optimal global solution obtained under the current generation according to the fitness function value, and $x_i^t$ indicates the position of pollen individual $i^{th}$ in the $t^{th}$ iteration, and $L$ is the parameter of Levy flight in rule (1), which is a parameter of the gamma function that can control the distance of pollen propagation:

$$L~\frac{\lambda \Gamma \left(\lambda \right)sin(\frac{\pi \lambda }{2})}{\pi }·\frac{1}{s^{1+\lambda }}$$

(12)

The local pollination expressed with the updating equation based on the plant itself is a self-pollination exploiting component of the algorithm. Γ(λ) is expressed as the standard gamma function. The updating formula of local pollination is expressed as:

$$x_i^{t+1}=x_i^t+\epsilon (x_j^t-x_k^t)$$

(13)

where$\epsilon$ is a random number between (0, 1) and $x_j^t$,$x_k^t$ can be regarded as different individuals from the same population; and j and k are generated randomly $\in \left[1, N_p\right]$.

2.3. Sine and Cosine Algorithm (SCA)

The SCA [35] is a newly developed metaheuristic optimization algorithm different from FPA with updating mechanism is inspired by the natural population. In the SCA, the search attributes of the algorithm global and local components are expressed with updating equations through the sine and cosine function as the following description formula:

$$X_i^{t+1}=\{\begin{array}{c}{X}_i^t+r_1\times \mathit{sin}\left(r_2\right)\times \left|r_3{P}_i^t-X_i^t\right| r_4<0.5\\ X_i^t+r_1\times \mathit{cos}\left(r_2\right)\times \left|r_3{P}_i^t-X_i^t\right| r_4\ge 0.5\end{array}$$

(14)

where $r_1$, $r_2$,$r_3,$ and $r_4$ are four random parameters in the position formula; $r_{1 }$ is considered as controlling the moving direction of mining itself or exploring outwards, $r_2$ is decided on the direction of movement of mobile location, $r_3$ is considered as a parameter of the goal to give a random weighting for random enhancing or weakening the influence of the objective to define the distance. Finally, by setting a particular parameter $r_4$ ([0, 1]), the function realizes the equivalent switch between sine and cosine:

$$r_1=a-t\frac{a}{T}$$

(15)

where $t$ is the current iteration, $T$ is the maximum generation for loops, and $a$ is a constant.

The significant steps of the algorithm process are comprised as follows:

Step 1.

Initialization of population agents random;

Step 2.

Evaluation of each of the search agents based on the fitness of objective function;

Step 3.

Updating the best solution obtained based on sorting fitness-function values so far for local search;

Step 4.

Generating new solution set; check a new solution and updating the position-of-search agents with global attribute search;

Step 5.

Checking termination; output the final best solution obtained.

3. Hybrid Flower Pollination Algorithm (HSFPA)

This section presents a new hybridizing FPA with the SCA algorithm for global optimization to enhance the flower pollination algorithm (HSFPA). The performance and potential of the proposed method of the HSFPA are verified in the following subsections.

3.1. 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 μ is a parameter presenting as the following step size:

$$\mu =h_w-\left(h_w-l_w\right)·\frac{ite}{IterMax}$$

(16)

where $h_w$and $l_w$ are the weight coefficient constants of the control step length coefficient (in the experiment, $h_w \mathrm{is} \mathrm{set} \mathrm{to} 0.9$ and $l_w \mathrm{is} \mathrm{set} \mathrm{to} 0.2$; $ite$ 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:

$$S_i^{t+1}=S_i^t+\mu \times L\left(g_{best}-S_i^t\right)+rand\left(S_j^t-S_k^t\right),$$

(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:

$$S_i^{t+1}=\left(\frac{S_i^t+g_{best}}{2}\right)rand+\epsilon \left(S_a^t-S_b^t\right),$$

(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:

$$S_i^t=\{\begin{array}{c}{U}_b·\left(1-rand\right), \mathrm{if} \mathrm{iteration} \mathrm{is} even\\ L_b·\left(1+rand\right), otherwise, \end{array},$$

(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:

$$S_i^{t+1}=\{\begin{array}{c}{S}_i^t+\epsilon \times sin\left(\alpha \right)\times \left|r_1{g}_{best}-S_i^t\right| if \omega \le 0.5\\ S_i^t+\epsilon \times cos\left(\alpha \right)\times \left|r_1{g}_{best}-S_i^t\right| otherwise\end{array},$$

(20)

where $\alpha$ is a random number that is set to $2·\pi ·rand\left(0,1\right)$; $\epsilon$ is the scaling factor; $r_1$ is rand number arange [0, 1]; $\omega$ is a selecting coefficient quality pollination agent that is given as follows:

$$\omega =1-\frac{fit\left(i\right)-bestfit}{worsefit-bestfit},$$

(21)

where $fit \left(i\right)$ is the fitness value of the objective function obtained corresponds to the $i$-th pollination; $bestfit$ and $worsefit$ are the best fitness and the worst fitness value obtained values currently. Algorithm 1 shows the pseudo-code of the proposed method (namely HSFPA).

Algorithm 1: Pseudo-code of the HSFPA scheme
Input: Output:	Np, fit, and parameters: $U_b, L_b$, p, q, $l_w,h_w$, and MaxIter Global best-solution
1:	Determine the objective function ($fit$) and the boundary conditions of the task$\left(U_b, L_b\right)$.
2:	According to the boundary conditions and dimensions of the function, initialize a population of $n$ flowers with a random solution.
3:	Set the conversion and variation parameters $p$ and $q$ are set to 0.4 and 0.7, respectively, and the step size parameter $l_w,h_w$.
4:	Find the $g_{best}$ based on the initialization population by the fit.
5:	$\mathit{For} t=1: MaxIter$ $\mathit{For} i=1:n$ $\mathit{If} rand<p$ Set the step size parameter via Equation (16)       Draw a step vector L which obeys a Levy distribution Global pollination via Equation (17) $\mathit{Else} \mathit{if} rand<q$       Generate $\epsilon$ random number between (0, 1)       Do local pollination via Equations (18) and (19)  Else New solution pollination is generated with Equation (20)  $\mathit{End} \mathit{if}$
6:	Evaluate $\mathrm{new} \mathrm{solutions}$.  If new solutions are better than the previous ones,           update them in the population.
7:	Find the current best solution $g_{best}$  $\mathit{End} \mathit{for}$ $\mathit{End} \mathit{for}$
8:	Output the best-solution found

The algorithm convergence is necessary for finding the optimal result that allows for “jumping from” local optimal points to better ones based on the fitness values. The terms of convergence are also considered a good measure to get a “good” solution and are probably found in comparison algorithms for the optimal solution. The HSFPA is constructed based on the FPA and some strategy updating from the SCA algorithm. Pollination diversity has increased, and randomly produced segments are used to generate novel solutions. The convergence of the HSFPA would confirm with the inheritance algorithm of both proven convergence algorithms from FPA [47] and SCA [48] algorithms.

The conditions for convergence of the algorithm are considered as follows:

Criteria 1: If $fit\left(Algorithm\left(S, \xi \right)\right)\le fit\left(S\right)$ and $\xi \in \mathsf{\Omega }$, then $fit\left(Algorithm\left(S, \xi \right)\right)\le fit\left(\xi \right)$,

where $fit$ is the objective function, $S$ is a solution, at $t$-th iteration a new solution: $S^{t+1}=Algorithm\left(S^t,\xi \right)$, ξ is the visited solutions over the iterative process, and $\mathsf{\Omega }$ is the feasible solutions of the problem search space.

Criteria 2: For $\forall B\in \mathsf{\Omega }$ subject to $\left(B\right)>0,$ ${\displaystyle \prod }_{t=0}^{\infty }\left(1-u_t\left(B\right)\right)=0$,

where $u_t\left(B\right)$ is a measure of the probability on $B$ at $t$-th iteration of the algorithm process.

A converge global algorithm has to satisfy the two mentioned conditions with the search sequence. If $fit$ is measurable and the feasible solution space Ω is a measurable subset on, then $\underset{t\to \infty }{\lim }P\left(S_t\in R_{ϵ,M}\right)=1$, where $P\left(S_t\in R_{ϵ,M}\right)$ is the probability measure of the $t$-th solution on $R_{ϵ,M}$ at the iteration.

The HSFPA’s iteration process always keeps/updates the current global best solution for the whole population that meets the first critical. It means that the algorithm meets the first convergence requirement mentioned according to condition 1.

As Markov chains with a definition of the state and state-space: The pollen and global best (best) solution in the search history form the states of pollen: $y=\left(S, g_{best}\right)$, where $S, g_{best} \in \Omega$ and f$\left(g_{best}\right) \le f\left(S\right)$ for minimization problems. The state and state-space are symbols as follows:

$$Y=\left\{y=\left(S, g_{best}\right)|S, g_{best} \in \mathsf{\Omega }, fit\left(g_{best}\right)\le fit\left(S\right)\right\}.$$

(22)

Y is state and state-space. The states of the pollen known as population/group form a state-space containing the historical global best solution denoted by

$$Z=\left\{z=\left(y_1, y_2,\dots , y_{N_p}\right), y_i \in Y, 1 \le i \le N_p\right\},$$

(23)

where Z is the historical global best solution. The best among all $g_{bes{t}_i}$ is known as the global best solution, so that f($g_{best}^{*}$) = min(f($g_{bes{t}_i}$)), $1 \le i \le n$.

The population/group state sequence will converge to the optimal set over a sufficient number of iterations; as a result, the probability of failing to find the globally optimal solution is 0; thus, the second convergence condition is satisfied. As a result, the HSFPA has ensured a convergence in the direction of its global optimality.

3.2. 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. Parameters’ settings for the algorithms.

Algorithms	Parameters Setting
HSFPA	$p=0.72,\Gamma =0.01; \lambda =1.5, q=0.6, \omega =0.5, l_w,h_w=\left[0.21,0.91\right]$,
FPA [34]	$p=0.72,\Gamma =0.01; \lambda =1.5,$
SCA [35]	$\alpha =1.5$$, [L_b,U_{b]}=\left[-100,100\right]$
BA [33]	$A_0=0.9, r_0$$=0.15, \alpha =0.25, \gamma =0.16,$$f_{min}, f_{max}=\left[0,0.5\right]$
MFPA [36]	$p=0.72,\Gamma =0.01; \lambda =1.5, q=0.6, \omega =0.6$
PSO [28]	$V_{max}=10,V_{min}=-10,\omega =0.9 to 0.4,c_1=c_2=1.49455,$
GA [26],	$R_{mu}=0.1, R_{cr}=0.9$
DE [32]	$F=0.7$$, R=0.1$5
BA-ABC [37]	$A_0=0.7, r_0$$=0.15, \alpha =0.25, \gamma =0.16$
PSO-ABC [38]	$a=-1, b=1$$,V_{max},V_{min}=\left[-10,10\right],\omega =0.9 to 0.4,c_1=c_2=1.49455$
GWO-FPA [39]	$a=2 \mathrm{to} 0,b=1,l=\left[-1,1\right]\left[r1, r2\right]=\left[2,0.001\right],$ $p=0.72,\Gamma =0.01; \lambda =1.5$

lists the primary setting parameters of the algorithms.

Table 2. Comparison of the HSFPA for 23 selected testing functions with the FPA, PSO, MFPA, and SCA algorithms, respectively.

Functions	HSFPA	FPA	$\mathit{r}$	PSO	$\mathit{r}$	MFPA	$\mathit{r}$	SCA	$\mathit{r}$
$F_1\left(x\right)$	9.24 × 10−53	1.98 × 10−01	+	1.69 × 10−08	+	1.10 × 10−01	+	7.15 × 10−06	+
$F_2\left(x\right)$	3.70 × 10−22	4.69 × 10−01	+	2.08 × 10−04	+	4.06 × 10−01	+	8.22 × 10−02	+
$F_3\left(x\right)$	4.79 × 10−11	1.48 × 10−01	+	2.79 × 10−01	+	5.53 × 10−02	+	5.40 × 10−05	+
$F_4\left(x\right)$	3.21 × 10−19	3.27 × 10−02	+	3.56 × 10−01	+	3.78 × 10−01	+	3.61 × 10−02	+
$F_5\left(x\right)$	3.31 × 10−02	3.69 × 10−03	−	3.32 × 10−02	~	3.29 × 10−03	−	1.84 × 10−01	+
$F_6\left(x\right)$	4.04 × 10−04	1.89 × 10−01	+	8.30 × 10−02	+	6.55 × 10−05	−	9.85 × 10−02	+
$F_7\left(x\right)$	−3.37 × 10+00	−3.21 × 10+00	+	−2.99 × 10+00	+	−3.28 × 10+0	+	−3.51 × 10+00	~
$F_8\left(x\right)$	−10.40 × 10+0	−10.2 × 10+00	~	−8.63 × 10+00	+	−9.40 × 10+0	+	−5.08 × 10+00	+
$F_9\left(x\right)$	0.01 × 10+00	1.01 × 10−01	−	3.01 × 10+01	+	4.81 × 10+00	+	1.03 × 10+00	+
$F_{10}\left(x\right)$	8.88 × 10−16	4.67 × 10−01	+	6.21 × 10−02	+	2.64 × 10−01	+	6.41 × 10−02	+
$F_{11}\left(x\right)$	0.00 × 10+00	4.41 × 10−02	+	1.33 × 10−09	+	9.63 × 10−03	−	3.29 × 10−07	+
$F_{12}\left(x\right)$	1.27 × 10−03	3.81 × 10−02	+	9.10 × 10−03	~	3.70 × 10−02	+	3.29 × 10−03	~
$F_{13}\left(x\right)$	3.19 × 10−04	3.49 × 10−04	+	5.69 × 10−03	+	3.57 × 10−04	~	9.65 × 10−04	+
$F_{14}\left(x\right)$	−3.89 × 10−12	−4.49 × 10−12	+	−3.90 × 10−11	+	3.92 × 10−11	+	3.97 × 10−11	+
$F_{15}\left(x\right)$	−6.16 × 10−01	−7.92 × 10−01	+	−7.02 × 10−01	+	−7.02 × 10−01	+	−7.02 × 10−01	+
$F_{16}\left(x\right)$	1.29 × 10−01	1.19 × 10−01	~	2.80 × 10−01	+	2.80 × 10−01	+	2.80 × 10−01	+
$F_{17}\left(x\right)$	2.12 × 10−01	3.08 × 10−01	+	7.01 × 10−01	+	7.11 × 10−02	−	7.11 × 10−01	+
$F_{18}\left(x\right)$	2.31 × 10−02	4.58 × 10−02	+	2.31 × 10−01	~	2.81 × 10−01	+	2.29 × 10−02	−
$F_{19}\left(x\right)$	9.81 × 10−01	1.64 × 10−01	+	1.36 × 10−01	−	1.36 × 10−01	+	1.36 × 10−01	+
$F_{20}\left(x\right)$	1.89 × 10−01	2.18 × 10−01	+	2.15 × 10−02	−	2.15 × 10−01	+	2.15 × 10−01	+
$F_{21}\left(x\right)$	−2.11 × 10−01	−2.13 × 10−01	+	−2.12 × 10−01	~	−2.10 × 10−01	~	−2.20 × 10−01	+
$F_{22}\left(x\right)$	−7.57 × 10−01	−6.27 × 10−01	+	−7.48 × 10−01	~	−7.68 × 10−01	+	−7.68 × 10−01	~
$F_{23}\left(x\right)$	3.44 × 10−01	3.50 × 10−01	+	3.95 × 10−01	+	3.96 × 100−01	~	3.96 × 10−01	+
Summary: paired HSFPA comparison		19+ 2− 2~		16+ 2− 5~		15+ 5− 3~		18+ 1− 3~

depicts the conducted results of the proposed HSFPA compared with the FPA, PSO, MFPA, and SCA algorithms for the selected testing functions.

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 (F₁(x), F₄(x), F₆(x), F₈(x), F₉(x), and F₁₀(x)) are depicted for viewing comparisons. Figure 3 also shows the function view space on each left side of the subplots. Several expression modals in the chosen functions from the CEC2017 are used in testing modals: 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. F4, F6, and F9 are functions with peaks as many local optimizations. The processes converge swiftly in an almost exponential way. The convergence rate improves exponentially with a higher slope when the search passes through a valley region between iterations. The multimodal, hybrid, and compound type modals are complex functions with more convex ridges.

The curve of the convergence of the proposed HSFPA observed in Figure 3 shows that the fast one against the other algorithms belongs to the HSFPA. It means that the convergence rate of the proposed HSFPA is quick to find out the global optimization.

Table 3. The obtained results of the HSFPA are compared with the other algorithms, e.g., GA, BA, and DE algorithms, for the selected testing functions.

Test Func.	HSFPA		GA		BA		DE
	Best.	Std.	Best.	Std.	Best.	Std.	Best.	Std.
$F_1\left(x\right)$	9.74 × 10−53	6.85 × 10−05	4.67 × 10−02	4.65 × 10−02	4.56 × 10−05	3.98 × 10−03	7.15 × 10−06	7.92 × 10−05
$F_2\left(x\right)$	3.17 × 10−22	4.49 × 10−03	3.69 × 10−04	3.46 × 10−03	3.08 × 10−04	3.69 × 10−03	2.22 × 10−22	2.88 × 10−02
$F_3\left(x\right)$	4.79 × 10−10	9.58 × 10−03	3.38 × 10−02	5.35 × 10−01	3.79 × 10−10	3.48 × 10−04	5.40 × 10−05	5.92 × 10−06
$F_4\left(x\right)$	3.61 × 10−19	4.42 × 10−02	3.27 × 10−02	4.04 × 10−03	4.26 × 10−02	4.37 × 10−01	3.61 × 10−02	4.31 × 10−01
$F_5\left(x\right)$	4.86 × 10−02	7.72 × 10 00	4.69 × 10+01	3.69 × 10+01	4.82 × 10−02	3.69 × 10−02	4.84 × 10+01	4.48 × 10 00
$F_6\left(x\right)$	4.04 × 10−04	58.08 × 10−03	4.89 × 10−01	5.66 × 10−01	5.34 × 10−02	5.56 × 10−01	9.85 × 10−02	7.56 × 10−01
$F_7\left(x\right)$	−3.30 × 10+01	−6.60 × 10−01	−3.28 × 10+0	−3.30 × 10+0	−2.99 × 10+0	−2.28 × 10+01	−2.30 × 10+01	−2.31 × 10+01
$F_8\left(x\right)$	−1.02 × 10+01	−1.01 × 10+01	−2.12 × 10+01	−2.05 × 10+01	−2.26 × 10+01	−2.20 × 10+01	−2.32 × 10+01	−2.10 × 10+01
$F_9\left(x\right)$	1.01 × 10−01	1.01 × 10+01	3.11 × 10+01	3.11 × 10+01	3.89 × 10+01	2.01 × 10+02	2.03 × 10+00	4.04 × 10+01
$F_{10}\left(x\right)$	4.08 × 10−06	4.78 × 10−01	5.67 × 10−01	5.44 × 10+00	6.21 × 10−02	6.67 × 10−01	6.41 × 10−02	6.87 × 10−01
$F_{11}\left(x\right)$	0.02 × 10+01	0.02 × 10+01	4.41 × 10−01	2.03 × 10−01	1.33 × 10−09	4.41 × 10−01	3.29 × 10−03	3.76 × 10−02
$F_{12}\left(x\right)$	1.27 × 10−03	2.54 × 10−01	3.81 × 10−02	1.47 × 10−01	5.10 × 10−03	3.81 × 10−01	3.29 × 10−03	1.52 × 10−01
$F_{13}\left(x\right)$	3.01 × 10−04	6.20 × 10−02	3.21 × 10−04	2.33 × 10−02	1.63 × 10−03	3.21 × 10−01	9.45 × 10−04	1.29 × 10−01
$F_{14}\left(x\right)$	5.19 × 10−04	4.55 × 10−02	4.29 × 10−04	5.42 × 10−01	5.19 × 10−04	4.49 × 10−02	5.19 × 10−04	4.49 × 10−01
$F_{15}\left(x\right)$	−3.89 × 10−12	−4.49 × 10−02	−3.89 × 10−12	−4.49 × 10−01	−3.89 × 10−12	−4.49 × 10−01	−3.89 × 10−12	−4.49 × 10−01
$F_{16}\left(x\right)$	−6.16 × 10−11	−7.92 × 10−04	−6.16 × 10+01	−7.92 × 10−10	−6.16 × 10+00	−7.92 × 10−10	−6.16 × 10+01	−7.92 × 10+01
$F_{17}\left(x\right)$	1.29 × 10−04	1.29 × 10−3	1.29 × 10−3	1.19 × 10−01	1.29 × 10−4	1.39 × 10−03	1.49 × 10−04	1.49 × 10−03
$F_{18}\left(x\right)$	2.12 × 10−01	3.08 × 10+01	2.02 × 10−01	3.08 × 10+02	2.13 × 10−01	3.08 × 10+02	2.38 × 10−01	3.08 × 10+01
$F_{19}\left(x\right)$	1.31 × 10−02	4.28 × 10−01	2.51 × 10−02	4.58 × 10−01	2.81 × 10−02	4.58 × 10−01	2.51 × 10−01	4.88 × 10 00
$F_{20}\left(x\right)$	9.21 × 10−11	8.64 × 10−02	9.81 × 10−10	8.64 × 10−02	8.51 × 10−10	6.64 × 10−01	9.51 × 10−10	8.64 × 10−01
$F_{21}\left(x\right)$	1.89 × 10−05	2.18 × 10−01	1.89 × 10−04	2.18 × 10−04	1.89 × 10−01	2.18 × 10−01	1.89 × 10−02	2.18 × 10−01
$F_{22}\left(x\right)$	−2.11 × 10−12	−2.13 × 10−01	−2.11 × 10−11	−2.13 × 10−01	−2.11 × 10−11	−2.13 × 10−11	−2.11 × 10−02	−2.13 × 10−02
$F_{23}\left(x\right)$	−7.57 × 10−03	−9.27 × 10−01	−7.57 × 10−03	−9.27 × 10−02	−7.57 × 10−01	−9.27 × 10−02	−7.57 × 10−04	−9.27 × 10−02
+/−/~			19/1/3	18/1/4	19/2/2	18/1/5	16/2/5	14/5/4

and

Table 4. The obtained results of the HSFPA are compared with hybrid methods, e.g., BA-ABC, PSO-ABC, and GWO-FPA algorithms, for 23 selected testing functions.

Test Func.	HSFPA		BA-ABC		GWO-FPA		PSO-ABC
	Best.	Std.	Best.	Std.	Best.	Std.	Best.	Std.
$F_1\left(x\right)$	9.24 × 10−53	1.85 × 10−03	2.67 × 10−02	3.65 × 10−01	3.56 × 10−05	1.98 × 10−01	7.15 × 10−06	7.92 × 10−01
$F_2\left(x\right)$	3.70 × 10−22	7.40 × 10−04	4.69 × 10−01	4.46 × 10+01	3.08 × 10−04	4.69 × 10+04	8.22 × 10−02	1.88 × 10+01
$F_3\left(x\right)$	4.79 × 10−02	9.58 × 10−01	3.38 × 10−02	5.35 × 10−01	3.79 × 10−06	3.48 × 10−04	5.40 × 10−05	5.92 × 10−01
$F_4\left(x\right)$	3.21 × 10−19	6.42 × 10−22	3.27 × 10−02	4.04 × 10+01	4.26 × 10−02	3.37 × 10+01	3.61 × 10−02	1.31 × 10+01
$F_5\left(x\right)$	3.86 × 10−02	7.72 × 10−01	3.69 × 10−02	3.69 × 10+01	3.82 × 10−02	3.69 × 10−01	2.84 × 10+01	2.48 × 10+02
$F_6\left(x\right)$	4.04 × 10−04	8.08 × 10−01	4.89 × 10−01	4.66 × 10+01	8.34 × 10−02	2.56 × 10+04	9.85 × 10−02	7.56 × 10+02
$F_7\left(x\right)$	−3.30 × 10+01	−6.60 × 10+01	−3.28 × 10+01	−3.30 × 10+02	−2.99 × 10+01	−3.28 × 10+01	−3.30 × 10+01	−1.31 × 10+01
$F_8\left(x\right)$	−1.02 × 10+01	−1.01 × 10+01	−1.12 × 10+01	−1.05 × 10+02	−1.26 × 10+01	−1.20 × 10+02	−1.32 × 10+01	−1.10 × 10+02
$F_9\left(x\right)$	0.01 × 10−01	0.01 × 10+01	2.11 × 10+01	1.11 × 10+02	3.89 × 10+01	1.01 × 10+02	1.03 × 10+00	4.04 × 10+02
$F_{10}\left(x\right)$	8.88 × 10−16	1.78 × 10−01	4.67 × 10−01	4.44 × 10+01	6.21 × 10−02	4.67 × 10+04	6.41 × 10−02	1.87 × 10+02
$F_{11}\left(x\right)$	0.02 × 10−01	0.02 × 10+01	4.41 × 10−02	2.03 × 10+02	1.33 × 10−09	4.41 × 10−01	3.29 × 10−07	1.76 × 10−02
$F_{12}\left(x\right)$	1.27 × 10−03	2.54 × 10−01	3.81 × 10−02	1.47 × 10−01	9.10 × 10−03	3.81 × 10−01	3.29 × 10−03	1.52 × 10−01
$F_{13}\left(x\right)$	3.12 × 10−04	6.20 × 10−01	3.01 × 10−04	2.31 × 10−01	1.69 × 10−03	3.20 × 10+01	9.65 × 10−04	1.28 × 10+01
$F_{14}\left(x\right)$	3.09 × 10−05	4.55 × 10−02	4.29 × 10−04	5.42 × 10−01	5.19 × 10−04	4.49 × 10−01	5.19 × 10−04	4.49 × 10+01
$F_{15}\left(x\right)$	−3.89 × 10−12	−4.49 × 10−02	−3.89 × 10−12	−4.49 × 10−01	−3.89 × 10−12	−4.49 × 10+01	−3.99 × 10−12	−4.49 × 10−01
$F_{16}\left(x\right)$	−6.16 × 10−11	−7.92 × 10−01	−6.16 × 10−10	−7.92 × 10−01	−6.16 × 10−10	−7.92 × 10−01	−6.16 × 10−10	−7.92 × 10+01
$F_{17}\left(x\right)$	1.29 × 10−04	1.29 × 10−01	1.29 × 10−03	1.19 × 10−01	1.29 × 10−04	1.39 × 10−01	1.49 × 10−04	1.49 × 10−03
$F_{18}\left(x\right)$	2.12 × 10−13	3.08 × 10+01	2.02 × 10−14	3.08 × 10−02	2.13 × 10−12	3.08 × 10+03	2.38 × 10−11	3.08 × 10+03
$F_{19}\left(x\right)$	1.31 × 10−02	4.28 × 10−02	2.51 × 10−02	4.58 × 10−02	2.81 × 10−02	4.58 × 10−02	2.51 × 10−02	4.88 × 10+02
$F_{20}\left(x\right)$	9.21 × 10−11	8.64 × 10−10	9.81 × 10−10	8.64 × 10−10	8.51 × 10−10	6.64 × 10−10	9.51 × 10−10	8.64 × 10+03
$F_{21}\left(x\right)$	1.89 × 10−05	1.18 × 10+01	1.89 × 10−01	2.18 × 10+04	1.89 × 10−04	2.18 × 10+04	1.89 × 10−04	2.18 × 10+05
$F_{22}\left(x\right)$	−2.11 × 10−12	−2.13 × 10+01	−2.11 × 10−11	−2.13 × 10+1	−2.11 × 10−11	−2.13 × 10+01	−2.11 × 10−12	−2.13 × 10+01
$F_{23}\left(x\right)$	−7.57 × 10−03	−9.27 × 10−01	−7.57 × 10+03	−9.27 × 10+04	−7.57 × 10−01	−9.27 × 10+02	−7.57 × 10−01	−9.27 × 10+02
+/−/~			19/1/3	18/1/4	19/2/2	18/1/5	16/2/5	16/3/4

depict comparison of the HFPA obtained results with the in pair compared other algorithms in the literature. The obtained results with the best and standard deviation (std.) values in Tables are compared in pairs for the HFPA with other algorithms. At the end of Table 3 and Table 4, we have briefly summarized results. The symbol ‘+’ indicates the “better” of the proposed method, producing 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. Table 3 displays the obtained results of the HSFPA in comparison with the other algorithms, e.g., GA, BA, and DE algorithms, for the selected testing functions. The estimation of the HSFPA offers better results than the other comparative methods with more highlighted bolds.

Furthermore, the experimental results of the proposed HSDPA are also compared with the other hybrid methods, e.g., BA-ABC [37], PSO-ABC [38], and GWO-FPA [39]. Table 4 depicts the comparison of the proposed HSFPA with hybrid methods, e.g., BA-ABC, PSO-ABC, and GWO-FPA algorithms for 23 selected testing functions.

It can be seen that, in the number of highlights in Table 4, most of them belonged to the proposed HSFPA that provides impressive competition optimization results for the selected test functions compared to the other approaches. Figure 4 displays the comparison of the convergence curve of the proposed HSFPA with hybrid methods, e.g., BA-ABC, PSO-ABC, and GWO-FPA algorithms selected test functions. The terms of convergence are necessary for finding the optimal results. That agent’s search allows for “jumping from” local optimal points to better ones based on the fitness values. The convergence is also considered a good measure to get a “good” solution and is probably found in comparison algorithms for the optimal solution. As observed, the proposed HSFPA produced the fastest convergence in comparison with the other algorithms. It can say that the proposed method is an alternative algorithm for competitors.

4. Application of the HSFPA for Microgrid Operations Planning

This section presents the multiobjective function for the microgrid operational planning issues, optimizing process steps, and analyzing and discussing the results. The detailed presentation is described as follows.

4.1. Multiobjective Model Construction

The proposed HSFPA would process in the objective function of the optimization based on finding out the feasible optimization area in the problem search space. In optimization processing, the objective function is critical in moving fitness ahead to the best global result produced from mathematical modeling. The variable electricity prices for active microgrid power are optimized by the HSFPA algorithm as the period changes synchronously. We construct the optimization problem with a mathematical multi-object model based on the relevant economic costs and environmental profits of a power activities microgrid system with a scheduling cycle. The goal function of microgrid optimization operations is a mathematical model developed using the microgrid’s minimum economic costs and the environmental pollution control’s minimum costs as follows:

$$min F={{\displaystyle \sum }}_{t=1}^T(F_1\left(t\right),F_2\left(t\right)),$$

(24)

where F is the multiobjective function for relevant total costs of microgrid-connected operations; $F_1$ and $F_2$ are the economic and environmental costs of power generation, and $T$ is the scheduling period $t,$ the multiobjective function subject to the constraints as mentioned in Section 2.1. Time interval and period use can be a year, month, day, or hour determined by the power supply outputs. The daily load curve’s unit and the generating plants’ output curve’s period, such as 24 h, are used, and the same with 12 months for a year and four seasons.

The economic costs of the microgrid power generation are, e.g., operation and maintenance costs of micropower sources for gas, wind, solar turbines, and generators connected to the main grid power. The first objective function is modeled by calculating summarized economic costs of the ES, WT, FC, MT, PV, and EX operations. Each power source component’s expense, as mentioned in Section 2.1, is subject to its constraints. For example, the cost of WT is shown in Equation (3) is subject to Equation (4) constraints, and the other expenses of FC, MT, PV, and EX are subject to Equations (8)–(10) constraints. The first objective function $F_1$ is formulated as follows:

$$F_1\left(t\right)={{\displaystyle \sum }}_{t=1}^T\left[{\alpha }_{ES}{C}_{ES}\left(t\right)+{\alpha }_{WT}{C}_{WT}\left(t\right)+{\alpha }_{FC}{C}_{FC}\left(t\right)+{\alpha }_{MT}{C}_{MT}\left(t\right)+{\alpha }_{PV}{C}_{PV}\left(t\right)+\mu {\alpha }_{EX}{C}_{EX}\left(t\right)\right],$$

(25)

where $C_i\left(t\right)$ are relevant operations costs of ($i$-th is set of the ES, WT, FC, MT, PV, EX); ${\alpha }_i$ are the unit prices of some kinds of operations, e.g., maintaining, investing, depreciating, and interactive power with the grid in the period t, respectively; μ is a parameter if it is set 1, it means a state of grid-connected operation; otherwise, it is an off-grid operation. The variables, sets, and the bounds of the optimization constrained are set specified in the experimental section

Another function of the object is the environmental costs that are a conversion processing development effected around envirement. The related costs operation parameters are considered with each treatment and the emission coefficients of various pollutants at using period point prices. The cost of treating pollutants discharged $F_2$ with expression is calculated as follows:

$$F_2={{\displaystyle \sum }}_{t=1}^T{{\displaystyle \sum }}_{k=1}^K{b}_k\left({{\displaystyle \sum }}_{i=1}^N{a}_{i,k}·P_i\right),$$

(26)

where $b_k$ and $a_{i,k}$ are the treatment cost and coefficient of class K pollutants discharged, $/kg; $P_i$ is the i-th distributed power source, g/KWh; K is the serial number of pollutants [7]. The variable $P_i$ is the distributed power source that is an influential factor for optimal results. In the experiment section, we use two distributed power sources of MT and FC components, converting them to the variable costs via Equations (5) and (7) with specified values listed in

Table 5. The emission-related operations cost of pollution remediation and emission coefficients.

The Emissions		$\mathit{C}{\mathit{O}}_2$	$\mathit{S}{\mathit{O}}_2$	$\mathit{N}{\mathit{O}}_{\mathit{x}}$	$\mathit{C}\mathit{O}$
$\mathrm{The} \mathrm{coefficient}/(\mathrm{g}/{\mathrm{kW}}^{-1}$)	MT	179	0.00089	0.621	0.16
	FC	635	0	0.023	0.054
$\mathrm{Related} \mathrm{operations} \cos \mathrm{ts}/(\$/{\mathrm{kg}}^{-1}$)		0.0052	0.693	1.19	0.201

. Moreover, the microgrid objective function’s problem complexity depends on the energy source components of economic costs that, in some cases, is quadratic complexity over iteration.

4.2. Multi-Object HSFPA (MHSFPA) Configuring Steps

There are several techniques dealing with the multiobjective optimal operation model, e.g., the Pareto front [50] and weighting ways [51,52]. The Pareto optimality is used to figure out the feasible solutions of the microgrid optimization operation planning. Then, the best points are selected to provide the suitable weighting values for applying the weighting method objective function. The optimal dominance of the Pareto optimality is denoted as an $\overrightarrow{x}\prec \overrightarrow{y}$ that is determined in the feasible solutions [53]. An optimal dominance definition as $\overrightarrow{x}$ is the dominance solution ($\overrightarrow{y}$) iff: $F_1\left(\overrightarrow{x}\right)<=F_2\left(\overrightarrow{y}\right)$ for entire feasible sites and $F_1\left(\overrightarrow{x}\right)<F_2\left(\overrightarrow{y}\right)$ for at least a site in the feasible area.

The economic operation and environmental pollution costs are used to construct the multiobjective optimal operation model for the minimizing objective cost function. It means that $F_1$ and $F_2$ are two variables that are mutually exclusive. The environmental cost will rise as the economic cost decreases. The environmental benefits will be reduced if the economic cost is increased. However, the minimum economic cost and the maximum ecological benefit are mutually exclusive, so we use the minus technique for converting maximum environmental benefit into minimizing harm ones. The solution set in the object space is often built a ranging set 0 to 1, so a safe checking boundary is stated as the following expression:

$$\{\begin{array}{c}{S}_{F_1}=\frac{F_1-F_1^{min}}{F_1^{max}-F_1^{min}}\\ S_{F_2}=\frac{F_2-F_2^{min}}{F_2^{max}-F_2^{min}}\end{array},$$

(27)

where $S_{F_1},$ and $S_{F_2}$ are two solution sets of checking measure parameters ∈ [0, 1] as considering checking boundary factors. The meaning parameters are used with most satisfied and dissatisfied values ranging set 0 to 1 or [0, 1]. The weighting method for a multiobjective function can be expressed as follows:

$$Min F_3=\omega \times F_1+\left(1-\omega \right)\times F_2,$$

(28)

where $\omega$ is a variable of weight considering the proportion of the relevant economic cost and the practical environmental benefits, respectively. The comprehensive benefit of the optimal solution set generated by the optimization algorithm is the chosen best sites. The best points are selected from the Pareto optimal front to provide suitable weighting values for the weight responding, respectively, facts. Figure 5 depicts the HSFPA’s flowchart for planning microgrid operations.

The optimizing applying steps of the HSFPA for the microgrid operation planning problem is described as follows:

Step 1:

Input the microgrid model parameters: $P_d\left(t\right), P\left(t\right)$, and (T): daily load demand, power output curves, unit generating set respectively; $C_i\left(t\right)$: time-of-use energy costs; and $C{O}_2$,$C{O}_2$, $N{O}_x$, $CO$ for MT, FC: as various pollution cost treatment factors shown in Table 5 and Table 6.

Step 2:

Initialize random population Np; get the objective function F(t) value with Np; get new solution set by using the suggested strategies; the items with the best fitness value are selected from the complete collection of forwarding next iteration.

Step 3:

Rank the fitness to identify the best solution currently; the selected solutions are set to produce new solutions with pollination; set the step size parameter $\mu$.

Step 4:

A Levy distribution is drawn a step vector L; update the equation of global pollination with better fitness to obtain the most up-to-date areas with the feasible ones.

Step 5:

Compare the improving pollination locations: the global solution should be updated if the new site outperforms the previous ones.

Step 6:

Verify the boundary of the current solution to determine the pollination placements; evaluate new solutions with fitness value; preserve the optimal global historical values.

Step 7:

Check the termination condition, e.g., if not reach max-iteration, repeat steps 2 to 6; otherwise, feasible output area of the pollination locations and its best global outcome value.

Table 6. The relevant parameters of each unit of the operation model of the power sources.

Power Sources	kW-Power Capacity		Constraint of Climb Rate	$$\begin{gathered} \mathbf{Maintenance} \\ \mathbf{Cos}\mathbf{ts} (\$/\mathrm{kW}) \end{gathered}$$	$\mathbf{Install} \mathbf{Cos}\mathbf{ts} (\$/\mathrm{kW})$	Capacity Factors (%)	Service Life/Year
	Upper	Lower
ES	61.87	−61.500	0.0001	0.005	0.107	40	15
FC	69.45	5.150	1.9600	0.107	2.258	37	20
MT	73.92	18.450	12.3000	0.108	1.606	68	20
PV	35.52	0.000	0.0012	0.012	5.180	36	18
WT	39.69	0.000	0.0012	0.036	2.915	27	18
EX	74.29	−73.800	0.0012	0.001	0.0001	2	22

Table 6. The relevant parameters of each unit of the operation model of the power sources.

Power Sources	kW-Power Capacity		Constraint of Climb Rate	$$\begin{gathered} \mathbf{Maintenance} \\ \mathbf{Cos}\mathbf{ts} (\$/\mathrm{kW}) \end{gathered}$$	$\mathbf{Install} \mathbf{Cos}\mathbf{ts} (\$/\mathrm{kW})$	Capacity Factors (%)	Service Life/Year
	Upper	Lower
ES	61.87	−61.500	0.0001	0.005	0.107	40	15
FC	69.45	5.150	1.9600	0.107	2.258	37	20
MT	73.92	18.450	12.3000	0.108	1.606	68	20
PV	35.52	0.000	0.0012	0.012	5.180	36	18
WT	39.69	0.000	0.0012	0.036	2.915	27	18
EX	74.29	−73.800	0.0012	0.001	0.0001	2	22

4.3. Results Analysis and Discussion

Because the microgrid model has been incorporated with more new energy sources, the parameters of the model space are growing exponentially, causing optimization scheduling as a nonlinear issue to become more complex and difficult to calculate. The traditional methods, e.g., mixed integer programming or gradient methods with quadric solutions, are highest for economic dispatching with a single component of the microgrid system [18,19]. The traditional techniques are suffered from complex computation and massive executed computation times. This subsection verifies the suggested scheme HSFPA performance by analyzing according to the multi-objectives, Pareto optimal, and weighting selected points optimal for the issues of the microgrid operations planning. The multi-object HSFPA (MHSFPA) outcomes are compared with the popular multi-object methods, e.g., multi-object FPA (MOFPA) [54], multi-object SCA (MOSCA) [55], and multi-object evolution algorithm (MOEA/D) [56]. The setting parameters for the algorithms and environment for the microgrid system are conducted for the operations planning based on several scenarios of scheduling periods of the days, months, and years, e.g., the period day is divided into 24 h periods; 30 days/month, and 12 months/year. The selected point in Pareto optimality is used to conduct the weighting multi-object by applying the MHSFPA. The outcome results of the weighting MHSFPA are also compared with the other approaches in the literature, e.g., the MBGA [57], PSO [28], and FPA schemes.

The parameters for the algorithms refer to Table 2, and the other setting parameters, e.g., for MEA/D: the $P_c=0.3$, $P_m=0.01$; for PSO: inertia factor ${\omega }_{max}=0.95$, ω_min = 0.15, learning factor c₁ = c₄ = 1.687, the maximum number of iterations of the algorithms is 1000, and the population size is 60. The microgrid system’s setting environment is set, e.g., emission coefficient of various pollutants, treatment costs, and operations’ prices. Table 5 and Table 6 illustrate the emission-related pollution remediation and emission coefficients and the units of the operation power sources’ relevant operation price. Figure 6 displays the daily demanding grid voltage curves of the microgrid power loads.

In the grid-connected state, the operation cost of microgrid generating electricity could be more expensive than the large grid’s consumed electricity because of much large-scale consumption and convenience equipment. However, the microgrid effectively produces balance and service by supplying power to the major grid during peak load hours and peak load times to meet the load demands. The microgrid’s battery storage is charged at the trough hours, but it provides the power to meet the peak load hour demand. The battery storage can also be charged when the system creates excess electric energy to ensure the system’s continuous power supply. The generators of the miniature gas turbines, diesel engineering, and fuel cells in the microgrid operate at full power at peak demands. Whenever the grid is off with the primary grid, its battery storage is powered by battery discharge.

Table 7. The electricity consumption price on the electricity meter by measuring daily hours.

Periods of Times	Interval Hours	$\mathrm{Price}/(\$/\mathrm{kW}·\mathrm{h})$
Normal-period	07:30–10:30	0.51
	15:30–18:30
	21:30–23:00
Peak-period	10:30–15:30	0.84
	18:30–21:30
Trough-period	23:00–07:30	0.19

shows the electricity consumption price on the electricity meter by measuring daily hours.

Table 8. The output results of three multi-object approaches: the MHSFPA, MOFPA, MOCSA, and MOEA/D algorithm with a selected specific point.

Types of Energy Production Outcomes	MOFPA	MOSCA	MOEA/D	MHSFPA
Active Powers (kW)	3.96 × 10+02	3.95 × 10+02	3.96 × 10+02	3.96 × 10+02
Total powers (kW)	5.02 × 10+02	5.02 × 10+02	5.02 × 10+02	5.03 × 10+02
Grid network loss (kW)	9.83 × 10−01	9.85 × 10−01	1.45 × 10+00	9.03 × 10−01
Total costs ($/kW)	1.80 × 10−01	1.80 × 10−01	1.80 × 10−01	1.75 × 10−01
Maintance ($/kW)	1.78 × 10−03	1.76 × 10−03	1.77 × 10−03	1.68 × 10−03
Win\Loss\Draw	0/3/2	0/2/2	0/2/3	itself

shows a summary output results of three multi-object approaches: the MHSFPA, MOSCA, MOFPA, and MOPEA/D algorithms with a selected specific point [$S_{F_1}$, $S_{F_2}$] = ([0.151, 0.251], [0.5, 0.5]),([0.5, 0.5], [0.251, 0.151]) for the microgrid Pareto optimal frontier with day periods. Figure 7 displays the obtained result curves of the multiobjective optimal dispatching with Pareto optimal front in MHSFPA, MOSCA, MOFPA, and MOEA/D algorithms.

In Figure 7, the green curve is a set of Pareto optimal solutions as a Pareto frontier. The collection of optimal solutions on a Pareto border is not dominated by any other feasible solutions. Inside the arc are points that represent feasible choice points. In contrast, the infeasible points lie outside the convex curve. Selected points that are closer or lied on the frontier inside the curve would be a better solution because any other point does not strictly dominate them. The selection of a point on the Pareto front graph curve would affect the priority distribution of the objective function. A particular point is selected to prioritize the economic objective function over the environmental one. Indeed, the objective function of environmental interest conflicts with the objective function of economic interests. However, to simplify the model and balance the harmony between the economy and the environment, we only consider the relation between FC, MT, and their air factors.

Figure 8 and Figure 9 show the planned HSFPA’s curve results in scheduling intervals of daily and monthly cycles are compared to those produced by the MBGA [57], PSO [28], and FPA methodologies for the weighting multi-objective function in Equation (28) with the referred weight that ${\omega }_1$ = 0.6 and ${\omega }_2$ = 0.4, respectively. The weight is referred to the function in Equation (28) from the obtained Pareto optimality space as a selected point in Figure 7.

As observed from Figure 8 and Figure 9, it is seen that the proposed HSFPA’s optimization result curves have a faster-archived convergence speed than that of the other schemes, like the MBGA, PSO, and AOA methods in the selected specific point for the objective function in optimizing the microgrid operational planning. Figure 10, Figure 11 and Figure 12 show the microgrid component-distributed power sources’ output curves of grid-connected optimization according to the intervals cycle periods of the daily, monthly, and year power capacity loads.

The environmental seasons’ factors can impact the subsystem operations, e.g., rainy, windy, solar light, dry seasons, or capacity facilities, such as load shift on long-term capacity planning based on historical data and load demand curves. The other operation scenarios are divided into grid-connected and off-grid operations. Figure 13 and Figure 14 show the production graphs of the grid-connected and off-grid microgrid subsystem power supplies in the daily cycle load. Various configurations of the microgrid include grid-tied and off-grid variants. The grid-connected case uses the local utility grid to ensure households are never without electricity, as shown in Figure 13 with the grid-tied systems. If the domestic solar panel system generates more daily electricity than the family requires, this surplus energy can be exported back to the national grid. On the other hand, if they need more electricity than the solar panels have generated, the grid can supply this. In contrast, as shown in Figure 14, the off-grid case must generate electricity and typically feature a more extensive battery system and inverters to assure reliability. Due to the high fuel cost, many energy solution providers are turning to solar-based off-grid solutions, which use solar energy to generate electricity. In general, it can be said that the proposed HSFPA scheme products outperform the other schemes in comparison to performing solutions for the microgrid operation planning issue.

5. Conclusions

This paper investigated an improved variant of the Flower pollination algorithm (FPA) by hybridizing FPA with the sine–cosine algorithm (SCA) (namely HSFPA) to solve the microgrid operations planning and global optimization problems. The strategies of adapting hop size, diversifying local search, and diverging agents from the SCA are implemented in the HSFPA by adding expressions for enhancing diversity agents, increasing convergence, and preventing local optimal traps. The multi-objective function of microgrid planning is constructed based on the power grid system expressions of relevant economic costs and environmental profits. The experimental results of the HSFPA compared with the original FPA and SCA and the other algorithms in the literature, e.g., GA, PSO, DE, MPFA, and hybrid algorithms, e.g., PSO-ABC, GWO-FPA, and BA-ABC for the benchmark test function and microgrid operations planning problem to evaluate the proposed scheme. The results show that the HSFPA algorithm produces better optimization performance than the original algorithm and some algorithms in test functions. The HSFPA algorithm also provides practical and stable operations planning of the microgrid system. In future works, the introduced HSFPA will be applied to the job shop scheduling [58] and further active adaptation and efficient utilization of a high proportion of distributed renewable energy sources with economic and safe adjustment methods [59].