2. Related Research

2.2. Metaheuristic Search Algorithm Design

Metaheuristic search (MHS) algorithms are optimization algorithms designed to solve complex problems that are difficult to solve using traditional mathematical methods. These algorithms are inspired by natural or social phenomena and use heuristic search techniques to explore the solution space and find near-optimal or satisfactory solutions. The general steps of the search process in the MHS algorithm [Reference Kahraman, Aras and Gedikli24] are as follows (Algorithm 1):

Algorithm 1: General steps of the search process in MHS algorithms.

The Bat search algorithm (BSA) is a metaheuristic search algorithm inspired by the echolocation behavior of bats. In BSA, the generation of the population involves the following steps:

• Random initialization: Similar to other metaheuristic algorithms, BSA starts by randomly initializing the population with a set of candidate solutions.

• Frequency and velocity update: Each bat in the population updates its frequency and velocity based on the previous iteration’s information. This update allows bats to explore the solution space dynamically.

• Local search: Bats perform a local search around their current positions to improve their solutions. This local search helps to refine the solutions and enhance their quality.

• Solution update: Bats update their positions and solutions based on their updated frequencies, velocities, and the results of the local search. This step allows bats to move towards promising regions of the solution space.

By iteratively performing these steps, the population of bats in BSA explores the solution space, adjusts their positions, and refines their solutions in search of the optimal or near-optimal solutions for the given problem.

3. SFBA Algorithm

The first version of the BAS algorithm just requires one long beetle, and it is also able to produce appropriate answers to several straightforward optimization issues. However, optimization issues can be found in many other sectors, and they are becoming increasingly difficult to solve as a result of factors such as the increasing complexity of optimization variables. The BAS algorithm is frequently unable to produce a solution that satisfies the requirements when it is used to problems of this nature. In order to accomplish this goal, this paper creates a cluster of these beetles and applies the moth-flame optimization (MFO) algorithm on the population structure as a point of reference. These sorts of enhancements have the potential to render the algorithm more stable and, to some extent, boost its capacity for optimization.

The elite individual matrix E and its fitness value F_E are respectively shown in the formula. In addition, the following provisions are made in terms of population structure:

1. (1) In the elite individual matrix E, each beetle is sorted according to the increasing fitness value, and F_E is also sorted in this order. For minimization optimization problems, the smaller the fitness value, the better. Therefore, the first one is the best individual so far.

2. (2) Each individual beetle can only update its place based on the elite people in E who have the same order as it does. The positions of the remaining beetles are updated using the final individual in E as a reference point.

According to the regulations presented above, the beetle is permitted to hunt in the vicinity of the elite solution. Additionally, the beetle does not only search around a predetermined answer, which to some degree enhances the algorithm’s ability to explore the search space. The link that exists between insects and elite persons after they have been updated is depicted in Figure 1.

Figure 1 Updated relationship diagram of beetles and elite individuals.

If n beetles are updated each time relative to a set number of elite solutions and the elite individuals are always changing, then the capacity of the algorithm to produce potentially optimum solutions will be lowered when each iteration is updated relative to various elite solutions. This is because the ability of the algorithm to develop possible optimal solutions depends on the elite individuals. It is for this reason that, throughout the iterative process, the number of retained elite individuals undergoes a constant reduction:

(1) $\begin{array}{}{n}^{\prime }=round\left(n-t\cdot \frac{n-1}{T}\right)\text{,}\end{array}$

where t is the current number of iterations, T is the maximum number of iterations, round() is the function that symbolizes rounding, n is the number of long beetles, n′ is the number of retained elite individuals, t is the current number of iterations, and T is the maximum number of iterations.

This paper adopts cat chaotic map with relatively good ergodicity and uniformity. The classical cat chaotic map is a two-dimensional reversible chaotic map, and its dynamic equation can be described as follows:

(2) $\begin{array}{}\left(\begin{array}{c}s{x}_{m+1}\\ s{y}_{m+1}\end{array}\right)=\left(\begin{array}{c}11\\ 12\end{array}\right)\left(\begin{array}{c}s{x}_m\\ s{y}_m\end{array}\right)\mathrm{mod}1\text{.}\end{array}$

Among them, sx_m is the value of the sx sequence at the mth iteration. The structure of this chaotic map is simple, so the computation is fast. In addition, the chaotic sequence generated by it is also more uniformly distributed in the interval [0, 1]. If it is assumed that the size of the beetle population is n and the dimension of the variable is d, the resulting chaotic sequences are sx = {sx_j, j = 1, 2, …, d} and sy = {sy_j, j = 1, 2, …, d}, where sx_j = {sx_{i, j}, i = 1, 2, …, n} and sy_j = {sy_{i, j}, i = 1, 2, …, n}. It should be noted that the chaotic sequence needs to be given an initial value on the interval [0, 1]. Then, for each dimension in the population, the cat chaos map can be expressed as follows:

(3) $\begin{array}{}\left(\begin{array}{c}s{x}_{i+1,j}=\left(s{x}_{i,j}+s{y}_{i,j}\right)\mathrm{mod}1\text{,}\\ s{y}_{i+1,j}=\left(s{x}_{i,j}+2s{y}_{i,j}\right)mod1\text{.}\end{array}\right)\end{array}$

It should be noted that since the cat chaotic map has two sequences sx and sy, only sy is selected in this paper. After that, it is necessary to map the chaotic sequence into the search space to obtain the represented population X. We take x_i,j as an example and calculate it by the following formula:

(4) $\begin{array}{}{x}_{i,j}=l{b}_j+s{y}_{i,j}\cdot \left(u{b}_j-l{b}_j\right)\text{,}\end{array}$

where x_i,j is the jth dimension of the ith beetle and ub_j and lb_j are the upper and lower bounds of the jth dimension of the variable, respectively.

The fundamental BAS algorithm is quite sensitive to the settings of its parameters, and the use of alternative parameters can frequently provide substantially different optimization outcomes. Alternately, a certain combination of parameters may be appropriate for one problem, but it may not be possible to solve another problem in an efficient manner. The following formula is used to compute the distance along each dimension:

(5) $\begin{array}{}{D}_{i,j}^t=\left(\begin{array}{c}\left(e_{i,j}^t-x_{i,j}^t\right),i\le n^{\prime },\\ \left(e_{n^{\prime },j}^t-x_{i,j}^t\right),i>n^{\prime },\end{array}\right)\end{array}$

where x^t_i,j represents the jth dimension of the ith beetle at time t, e^t_i,j is the jth dimension of the ith current optimal individual at time t, D^t_i,j represents the distance between the ith beetle and the corresponding elite individuals in the jth dimension at time t, and n′ represents the number of elite individuals currently reserved.

After considering the random time delay, the step size is expressed as the following formula:

(6) $\begin{array}{}{step}_{i,j}^t=r_1\cdot D_{i,j}^t+M\cdot c\cdot r_2\cdot \left(b_{i,j}^{t-k}-x_{i,j}^t\right)\text{.}\end{array}$

M is the delay factor, which is used to decide whether to add the delay term, and c is the acceleration coefficient. Both of these factors are discussed further. The range of possible values for k, which denotes the delay time, is [0, t − 1]. The acceleration coefficient c found in the BAS-RDEO algorithm is conceptually comparable to the cognitive and social factors found in the PSO algorithm. An acceleration coefficient that varies over time is utilized in this piece of work. To be more explicit, it is diminishing in a linear fashion, which can be stated as follows:

(7) $\begin{array}{}c=c_i-\left(c_i-c_f\right)\cdot \frac{t}{T}\text{.}\end{array}$

Among them, c_i and c_f represent the initial value and termination value of the acceleration coefficient, respectively.

In this paper, the evolutionary factor is used to judge the evolutionary state of the population, which has been introduced. According to the size of the evolution factor and the characteristics of the search process, four evolution states can be defined: the convergence state, the development state, the exploration state, and the jumping state. Before calculating the evolution factor, the average distance I_i between each long beetle and other long beetles needs to be calculated by the following formula:

(8) $\begin{array}{}{I}_i=\frac{1}{n}\sum _{q=1}^n\sqrt{\sum _{j=1}^d{\left(x_{i,j}-x_{q,j}\right)}^2.}\end{array}$

Among these variables, n and d stand for the size of the population and the variable dimension, respectively. When all is said and done, the evolutionary factor known as EF can be defined as follows:

(9) $\begin{array}{c}EF=\frac{I_g-I_{\min }}{I_{\max }-I_{\min }},\xi \left(t\right)=\left(\begin{array}{c}\mathrm{1,0}\le EF<0.25,\\ \mathrm{2,0.25}\le EF<0.5,\\ \mathrm{3,0.5}\le EF<0.75,\\ \mathrm{4,0.75}\le EF\le 1.\end{array}\right)\end{array}$

Among them, ξ(t) = 1, 2, 3, 4 represents the convergence state, the development state, the exploration state, and the hop state, respectively. Transform the abovementioned formula into the following form:

(10) $\begin{array}{c}\left(\begin{array}{c}{x}_{il}^t=x_i^t+{step}_i^t\text{,}\\ x_{ir}^t=x_i^t-{step}_i^t\text{.}\end{array}\right)\left(\begin{array}{c}{f}_{x_i}^{t+1}=\min \left(f\left(x_{il}^t\right),f\left(x_{ir}^t\right)\right)\text{,}\\ x_i^{t+1}=\underset{x_i}{\mathrm{argmin}}{f}_{x_i}^{t+1}\text{.}\end{array}\right)\end{array}$

The concept of inverse learning has been explained. Elite reverse learning is a novel technology in the field of intelligent computing that is analogous to traditional reverse learning but is more effective. Following this, you will find a definition of the concept of elite reverse learning.

Definition (elite reverse learning): If $x_e^t=\left(x_{e,1}^t,x_{e,2}^t,\dots ,x_{e,d}^t\right)$ is an elite individual in the population at time t, then its elite reverse solution is ${oxx}_e^t=\left(o{x}_{e,1}^t,o{x}_{e,2}^t,\dots ,o{x}_{e,d}^t\right)$ . Each dimension of ox^t_e can be obtained by the following equation:

(11) $\begin{array}{}o{x}_{e,j}^t=\omega \cdot \left(d{a}_j^t+d{b}_j^t\right)-x_{e,j}^t\text{.}\end{array}$

Among them, ω is a generalized coefficient uniformly distributed in [0, 1]; da^t_j and db^t_j are dynamically changing boundaries and can be defined as follows:

(12) $\begin{array}{c}d{a}_j^t={\min }_{1\le i\le n}\left(x_{i,j}^t\right)\text{,}d{b}_j^t={\max }_{1\le i\le n}\left(x_{i,j}^t\right)\text{.}\end{array}$

However, for a given problem, a certain dimension of individual ox′_e after using elite reverse learning may jump out of the boundary [lb_j, ub_j], which will make the algorithm ineffective. To avoid this situation, the following equation resets individuals outside the variable bounds:

(13) $\begin{array}{}o{x}_{e,j}^t=\left(\begin{array}{c}l{b}_j,o{x}_{e,j}^t<l{b}_j,\\ u{b}_j,o{x}_{e,j}^t>u{b}_j.\end{array}\right)\end{array}$

In addition, the number n′ of elite individuals decreases with the increase of the number of iterations. In view of this, this paper only selects the first individual in the elite individual matrix E for L times of elite reverse learning and sets L equal to the population size n in this paper. After the elite reverse learning, it is necessary to merge the L elite reverse solutions with the individuals in the matrix, and then select n′ better individuals to replace the elite solutions in the matrix. So far, elite reverse learning has been executed. After that, it is also necessary to introduce the leader’s multilearning strategy.

The leader learns to randomly select one dimension of the globally optimal individual. For example, if the jth dimension is chosen, it can be expressed as e^t _1,j. The reason for choosing only one dimension is because the local optimum is most likely to have a better solution in a certain dimension. After selecting a dimension, leader multilearning is carried out through the following 3 formulas:

(14) $\begin{array}{}{e}_{1,j}^t=e_{1,j}^t+\left(\left(u{b}_j-l{b}_j\right)\cdot \text{rand}-u{b}_j\right)\cdot R_1\text{,}\end{array}$

(15) $\begin{array}{}{e}_{1,j}^t=e_{1,j}^t+\left(\left({\lambda }_{1,j}-{\lambda }_{2,j}\right)\cdot rand2-{\lambda }_{1,j}\right)\cdot R_2\text{,}\end{array}$

(16) $\begin{array}{}{e}_{1,j}^t=e_{1,j}^t+\left(\left({\eta }_1-{\eta }_2\right)\cdot rand3-{\eta }_1\right)\cdot R_3\text{.}\end{array}$

Among them, rand1, rand2, and rand3 are random numbers in [0, 1]; η₁ and η₂ are the maximum and minimum values of all dimensions of the globally optimal individual e^t₁ in each generation. R ₁, R ₂, and R ₃ are the search radius of the multilearning method. Obviously, formula (14) is used to improve the global exploration ability, while formulas (15) and (16) are used to improve the local development ability. It should be noted that the search radii R ₁, R ₂, and R ₃ are determined by the Markov chain γ_t (t > 0):

(17) $\begin{array}{}P\left({\gamma }^{t+1}=\left(j\right){\gamma }^t=i\right)={\alpha }_{ij}^t,i,j=\mathrm{1,2},\dots ,N\text{.}\end{array}$

Among them, ${\alpha }_{ij}^t\ge 0\left(i,j\in S\right)$ is the transition probability from i to j. The number of states set in this paper is N = 2. At the same time, this paper assumes that the values of R₁, R₂, and R₃ are the same, namely, R₁ = R₂ = R₃. Therefore, when choosing the value of the search radius, we only take R₁ as an example. When the state is γ^t = 1, R₁ is taken as ρ₁ = 1. When the state is γ^t = 2, the value of R₁ is ρ₂ = 0.1. It should be pointed out that the adopted probability transition matrix Γ^t is time-varying and can be expressed as follows:

(18) $\begin{array}{}{\Gamma }^t=\left(\begin{array}{c}{\alpha }^{t}1-{\alpha }^t\\ {\alpha }^{t}1-{\alpha }^t\end{array}\right)\text{.}\end{array}$

Among them, α_t is a time-varying probability variable whose value decreases linearly in the iterative process and can be expressed as the following formula:

(19) $\begin{array}{}{\alpha }^t=\left({\alpha }_i-{\alpha }_f\right)\cdot \frac{T-t}{T}+{\alpha }_f\text{.}\end{array}$

Among them, α_i and α_f are the initial and final values of α_t, respectively. In this paper, they are set to α_i = 1 and α_f = 0. Therefore, ρ₁ occurs more frequently than ρ₂ in the early stages. However, in the late period, ρ₁ rarely occurs and ρ₂ occurs frequently. It should be noted that in leader multilearning, if the new position is better than the previous global optimal individual, the new position is accepted. Otherwise, the previous global optimal individual remains unchanged.

4. Power TN Optimization Experiment

In order to provide high-reliability and uninterrupted communication services for power systems, power communication must meet and adapt to the features that power cannot be stored, and that production, supply, and sales are all finished at the same time. A basic architecture of a smart grid is depicted in a schematic form in Figure 2, which may be seen here.

Figure 2 Schematic diagram of the basic architecture of the smart grid.

Figure 3 illustrates the design of the PCN’s two fundamental networks, which are the integrated service data network and the dispatching data network. Figure 3 also illustrates the PCN’s overall structure.

Figure 3 The dual-network power safety communication infrastructure of the dispatching data network and the integrated service network.

To validate the performance of the proposed method in power TN optimization, the experimental setup in this paper is mainly conducted from two perspectives. Firstly, it verifies the optimization performance of the proposed method, and secondly, it applies the method to optimize the power transmission network.

Firstly, to validate the performance of the proposed algorithm, this paper will conduct a comparative analysis of solution accuracy and convergence using several typical test functions. Specifically, a classic single-peak spherical function and two multipeak functions, Rosenbrock and Rastrigin, will be employed for testing. The functional forms of these test functions are as follows:

(20) $\begin{array}{c}{f}_{Spherical}\left(x\right)=\sum _{i=1}^n{x}_i^2\text{,}{f}_{Rosenbrock}\left(x\right)=\sum _{i=1}^{n-1}\left(100{\left(x_{i+1}-x_i^2\right)}^2+{\left(x_i-1\right)}^2\right)\text{,}\\ f_{Rastrigin}\left(x\right)=\sum _{i=1}^n\left(x_i^2-10\cos \left(2\pi x_i\right)+10\right)\text{.}\end{array}$

The parameter settings of each algorithm are shown in Table 1.

Table 1 Parameter settings.

Based on the parameter settings provided in Table 1, the experimental results obtained by each algorithm are shown in Table 2.

Table 2 Performance comparison of various optimization algorithms.

According to the experimental data shown in Table 2, it can be observed that the proposed BAS-RDEO algorithm outperforms other algorithms in both the average optimal value and standard deviation indicators. This indicates that the proposed algorithm has shown significant improvements in both computational accuracy and stability. Moreover, the BAS-RDEO algorithm is more adept at locating the global optimum in the solution space.

In order to further compare and analyze the performance of the model in this paper and similar algorithms, this paper selects [Reference Mohamed, Hadi and Mohamed26–Reference Duman, Kahraman and Kati29] as comparison algorithms. The parameter settings of each model are the same as in references. The effect of each algorithm in the optimization of the power TN is shown in Table 3.

Table 3 The effect of each comparison algorithm in power TN optimization.

From the experimental results shown in Table 2, it can be seen that in most cases, our method obtains the optimal effect, and the average optimization effect is 86.067. There are two sets of data in [Reference Mohamed, Hadi and Mohamed26] that exceed our method, and the average obtained was 85.368. The experimental results obtained by [Reference Duman, Kahraman, Sonmez, Guvenc, Kati and Aras28, Reference Duman, Kahraman and Kati29] are relatively close, and their mean values are 84.567 and 85.375, respectively, which are lower than the experimental data obtained by the algorithm in this paper. Our method utilizes random perturbations and self-similarity to explore the solution space effectively. It can escape local optima by introducing randomness and adapt its search strategy to different scales within the problem domain. The algorithm requires minimal problem-specific information, making it versatile for various optimization problems. In comparison, the gaining-sharing knowledge-based algorithm emphasizes knowledge exchange among individuals in a population, promoting collective intelligence and exploration. However, when focusing on the advantages of the stochastic fractal search algorithm, it stands out in its ability to efficiently explore the solution space, especially in high-dimensional problems and noisy environments. Its simplicity of implementation, robustness, and adaptability to diverse problem domains contribute to its practicality. In addition, the algorithm’s stochastic nature and self-similarity enable it to overcome local optima and discover global optima. In general, our method provides an efficient and flexible solution for optimization tasks that demand efficient exploration, adaptability, and ease of implementation. This makes our method suitable for a wide range of optimization problems. In comparison to the strategy described in [Reference Derrac, García, Molina and Herrera27], our approach demonstrates clear and convincing advantages. This demonstrates that nonparametric statistical methods function satisfactorily in general when it comes to the optimization effect of transmission networks.

5. Conclusion and Future Directions

The graphical connection relationship between each electrical component in the PN is referred to as the “PN topology,” and it is described here. The topological structure and system function of the PN are its two fundamental qualities, and the two are interrelated and affect each other in the following way: the structure places restrictions on and determines the size, type, and boundary of the power system function; the structure also sets the size of the PN. The function of the power system is the exterior manifestation of its structure, and the change in function is typically accompanied by a change in the system’s structure. Both the structure of the PN and the stability of the power system are directly impacted by the essential nodes and edges that make up the PN. In this study, an enhanced stochastic fractal search technique is used with the power TN model in order to achieve optimal results. The empirical research demonstrates that the improved stochastic fractal search method that was suggested in this paper is able to successfully increase the optimization effect of the power TN, and that it also has a beneficial effect on enhancing the efficiency of power transmission. This paper has some shortcomings that need to be addressed. To give just one illustration, the identification of the model’s ideal parameters necessitates several iterations, each of which takes a considerable amount of time and contributes to an increase in the algorithm’s level of complexity about the passage of time. The main work of this paper in the future will be carried out to address the abovementioned limitations.