A Hybrid-Strategy-Improved Dragonfly Algorithm for the Parameter Identification of an SDM

Abstract

As primary components of solar power applications, photovoltaic cells have promising development prospects. Due to the characteristics of PV cells, the identification of parameters for circuit models has become a research focus. Among the various methods of parameter estimations, metaheuristic algorithms have attracted significant interest. In this paper, a hybrid-strategy-improved dragonfly algorithm (HIDA) is proposed to meet the demand for high parameter-identification accuracy. Tent chaotic mapping generates the initial position of individual dragonflies and aids in increasing the population diversity. Individual dragonflies can adapt their updated positions to various scenarios using the adjacent position decision approach. The whale optimization algorithm fusion strategy incorporates the spiral bubble-net attack mechanism into the dragonfly algorithm to improve the optimization-seeking precision. Moreover, the optimal position perturbation strategy reduces the frequency of the HIDA falling into local optima from the perspective of an optimal solution. The effectiveness of the HIDA was evaluated using function test experiments and engineering application experiments. Seven unimodal and five multimodal benchmark test functions in 50, 120, and 200 dimensions were used for the function test experiments, while five CEC2013 functions and seven CEC2014 functions were also selected for the experiments. In the engineering application experiments, the HIDA was applied to the single-diode model (SDM), engineering model, double-diode model (DDM), triple-diode model (TDM), and STM-40/36 parameter identification, as well as to the solution of seven classical engineering problems. The experimental results all verify the good performance of the HIDA with high stability, a wide application range, and high accuracy.

1. Introduction

The demand for electricity is rising as a result of industrial expansion and new technology. With the frequent occurrence of extreme climate events caused by environmental pollution, traditional fossil fuels cannot satisfy people’s energy needs [1]. With the benefits of zero pollution and plentiful reserves, solar energy is gradually making its way into people’s lives as one of the main clean energy sources [2].

In recent years, the cost of utilizing solar energy has steadily decreased to equal that of using fossil fuels [3], which is inextricably linked to the research and development in respect of photovoltaic cells. A photovoltaic cell is a device that converts solar radiation into electricity. Solar cell modules are made up of photovoltaic cell packages, and photovoltaic systems are made up of solar cell modules and a range of other devices [4]. The performance of photovoltaic cells is influenced by both their internal materials and the environment [5]. Due to the unique method of receiving solar radiation, photovoltaic systems are invariably exposed to an open environment, which also creates the issue of device aging for photovoltaic cells [6]. Model simulation experiments are crucial for the rational management of PV systems. Among the common PV cell models, SDM is the model that meets most of the accuracy requirements without being unduly complex [7,8]. Furthermore, precise parameters are essential for modeling solar photovoltaic cells. However, in most cases, the supplier is unable to provide complete information [9]; therefore, the emphasis is placed on the parameter values identifying the solar cell.

The three main categories of existing photovoltaic cell parameter identification methods are analytical methods [10], deterministic methods [11], and metaheuristic methods [12]. Among these, analytical methods rely on the key PV parameters supplied by the manufacturer for parameter identification. Analytical methods employ several mathematical formulations to obtain correct current–voltage curves [13]. However, it is challenging to implement the assumptions of these mathematical formulations, which means that it is difficult to achieve high accuracy for the results obtained. Additionally, deterministic methods are founded on gradient computation [14]; one of its characteristics is its extreme sensitivity to initial conditions [15]. The above-mentioned problems are addressed at present using a variety of metaheuristic algorithms [16], including the partial swarm optimization (PSO) [17,18], symbiotic organisms search (SOS) algorithm [19], grasshopper optimization algorithm (GOA) [20,21], lion swarm optimization (LSO) [22], butterfly optimization algorithm (BOA) [23], sparrow search algorithm (SSA) [24], African vultures optimization (AVO) [25], Harris hawk optimization (HHO) [26,27], chicken swarm optimization (CSO) [28,29], artificial bee colony (ABC) algorithm [30,31], marine predators algorithm (MPA) [32,33], golden search optimization (GSO) [34], and firefly algorithm (FA) [35]. Because of their advantages of simplicity and powerful search capabilities, these population-based methods have received extensive attention. For example, Wu et al. [36] introduced a drunken walking model into the wolf pack algorithm to enhance the global search capability of the algorithm, and applied the improved algorithm to the parameter identification of SDM, DDM, and PV modules with high accuracy in the experimental results. Hussein et al. [37] used the Lambert W-function to generate an objective function that mitigates the implicit coupling relationship between voltage and current, and then used the marine predator algorithm (MPA) to solve the parameter extraction optimization problem for single- and double-diode PV models with a high degree of accuracy in the results of the study. Ibrahim et al. [38] fused a wind-driven algorithm with a fruit fly optimization algorithm to obtain a new algorithm, WDFO, which was used to identify the parameters of a two-diode PV cell model and three PV modules, and the fused algorithm converged faster than the original algorithm.

However, determining the parameters of solar cells is a complex multimodal problem. According to the previous research, not all metaheuristic algorithms perform well in determining solar cell parameters. Therefore, a swarm intelligence algorithm that can accurately identify photovoltaic cell parameters is required.

The dragonfly algorithm (DA) [39,40] simulates the foraging and natural enemy-avoidance behavior of dragonflies and was proposed by Seyedali Mirjalili in 2016. In recent years, the DA has been successfully applied to a variety of engineering problems, including feature extraction for drug databases [41], dynamic scheduling of cloud computing tasks [42], finding optimal aggregation trees in wireless sensor networks [43], achieving load balancing in cloud computing environments [44], maximizing cognitive radio throughput [45], etc. The DA has more prominent ability in respect of global exploration. Many researchers have proposed various methods to reform the DA in terms of its convergence speed and optimization-seeking accuracy. Xue et al. applied the algorithm restart concept to the dragonfly algorithm and combined disturbance observation for global PV maximum power tracking [46]. Lin et al. combined an elite strategy with the sine–cosine mechanism to ensure the robustness of the DA and the correctness of the optimization direction [47]. Zhong et al. utilized the wind-driven algorithm in the DA to speed up convergence [48]. The K-means++ clustering method was applied by Du et al. to cluster the population of the DA. For enhancing the information communication between populations of each generation, the probe guidance mechanism has also been used [49].

The above-mentioned methods are similar to the majority of improved swarm intelligence algorithms. However, there are still some problems in the improved algorithm, including insufficient local development ability, a single individual position update method, and low performance when solving multi-peak functions. To address these problems, in this paper, an improved approach is proposed based on a hybrid strategy.

The major contributions of this paper are briefly described as follows:

(i)

A new improved algorithm (HIDA) is proposed. Tent chaotic mapping is used to generate the initial positions of dragonfly individuals traversing the search space to improve the algorithm’s search capability. Nonlinear inertial weight is used to enable the algorithm’s global search and local exploitation to be balanced. The influence of neighboring individuals is considered to improve the efficiency of communication between populations. The bubble-net strategy of the whale optimization algorithm is fused to improve the local exploitation capability of the DA. Finally, to enhance the algorithm’s ability to avoid local extremes, Cauchy perturbation is applied to the optimal positions.

(ii)

Experiments are conducted using benchmark functions as well as CEC functions. The results of the rank sum test, a comparison with the chosen comparison algorithms, and a comparison with the two improved dragonfly algorithms prove that HIDA performs well in finding the best solution and is a competitive algorithm.

(iii)

The results for the SDM with five unknown parameters, the engineering model with four parameters, the DDM with seven parameters, the TDM with nine parameters, and the STM-40/36 model with five parameters demonstrate the high accuracy of HIDA at different temperatures and irradiances. Seven classical engineering applications further demonstrate the good performance of HIDA.

The structure of the paper is as follows. The related work is described in Section 2, which contains a brief description of the SDM as well as the standard DA. The detailed improvements and main processes regarding the improved DA are reflected in Section 3. Section 4 presents test results for the basic benchmark functions and CEC functions, the results for the estimation of solar cell parameters, and the test results for classical engineering problems. Finally, Section 5 summarizes the conclusions of the paper.

2. Related Work

2.1. Mathematical Diode Modeling of PV Cell and Objective Function

2.1.1. Modeling of Solar Photovoltaic System

Among many equivalent circuit models, the SDM has the advantage of being easy to implement and able to accurately simulate the characteristics of photovoltaic cells. The accuracy and complexity of PV models are proportional to each other; therefore, the desire for high-precision models inevitably leads to implementation difficulties. In addition, the accuracy is related to the number of model parameters, where a higher number means higher accuracy. There are five unknown parameters in the SDM, which helps the SDM to exchange low complexity for substantial accuracy.

The single-diode model only includes one diode. The circuit diagram of the SDM is shown in Figure 1.

The diode is used to shunt the photocurrent $I_{ph}$; thus, the load current can be determined as per Equation (1) [50].

$$I_L=I_{ph}-I_d-I_{sh}$$

(1)

According to the Schockley equation and Kirchhoff’s law, the diode reverse saturation current $I_{sd}$ and shunt resistance current $I_{sh}$ can be expressed as follows:

$$I_d=I_{sd}(e^{(\frac{q\left\{V_L+I_L{R}_S\right\}}{nKT})}-1)$$

(2)

$$I_{sh}=\frac{V_L+L_L{R}_S}{R_{sh}}$$

(3)

where $R_{sh}$ and $R_S$ represent battery depletion, while $R_L$ indicates the circuit load. The electron charge $q$ is 1.60217646 × 10⁻¹⁹ C, the Boltzmann constant $K$ is 1.38064852 × 10⁻²³ J/K, $n$ is the diode ideality factor, and $T$ is the temperature.

From the above formula, Equation (1) can be expanded as follows:

$$I_L=I_{ph}-I_{sd}(e^{(\frac{q\left\{V_L+I_L{R}_{}\right\}}{nKT})}-1)-\frac{V_L+L_L{R}_S}{R_{sh}}$$

(4)

Inside the SDM, $I_{ph}$, $I_{sd}$, $R_{sh}$, $R_S$, and $n$ need to be estimated.

2.1.2. Objective Function

The accuracy of the parameter estimation is usually evaluated using the root-mean-squared error (RMSE) as an indicator. The RMSE is used for calculating the error between the actual measured current $I_{L.\mathrm{m}}$ and the estimated current $I_{L.\mathrm{e}}$.

The expression for RMSE is as follows:

$$RMSE=\sqrt{\frac{1}{N}({{\displaystyle \sum _{\mathrm{i}=1}^N\left[I_{L.\mathrm{m}}-I_{L.\mathrm{e}}\right]}}^2)}$$

(5)

The error function equation is shown in Equation (6):

$$f(I_{ph},I_{sd},R_{sh},R_s,n)=I_{ph}-I_{sd}(e^{(\frac{q\left\{V_L+I_L{R}_S\right\}}{nKT})})-\frac{V_L+I_L{R}_S}{R_{sh}}-I_L$$

(6)

2.2. Dragonfly Algorithm

The dragonfly is one kind of flexible flying insect, which can move at high speed and stay in the air temporarily. The dragonfly algorithm mainly simulates two behaviors of the dragonfly: social and survival. Social behavior ensures that individual dragonfly follow population activities; survival behavior ensures the basic conditions for the survival of dragonflies.

In the social behavior of dragonflies, individual dragonflies not only have to move towards the population center, but also have to avoid collisions with other dragonflies, and adjust their flight speed while moving [51].

$$C_i=\frac{{\displaystyle \sum _{j=1}^N{X}_j}}{N}-X$$

(7)

$$S_i=-{\displaystyle \sum _{j=1}^{N}X}-X_j$$

(8)

$$A_i=\frac{{\displaystyle \sum _{j=1}^N{V}_j}}{N}$$

(9)

During social activities, the movement of dragonflies is mainly influenced by neighboring individuals. In the equations above, N is the number of adjacent individuals, $X_j$ denotes the position of the jth adjacent individual, and $V_j$ is the velocity of adjacent individuals.

Foraging and avoiding enemies are the basic conditions for dragonfly survival.

$$F_i=X^{+}-X$$

(10)

$$E_i=X^{-}+X$$

(11)

In survival behavior, $X^{+}$ is the location of the food source and $X^{-}$ is the location of the enemy.

The step vector ($\Delta X$) of the dragonfly movement is expressed by Equation (12):

$$\Delta X_{t+1}=(s{S}_i+a{A}_i+c{C}_i+f{F}_i+e{E}_i)+\omega \Delta X_t$$

(12)

where $s,a,c,f,e$ refer to the weights of each component of the abovementioned main behaviors, respectively.

The position vector ($X$) is calculated using Equations (13) and (14):

When there exist adjacent solutions,

$$X_{t+1}=X_t+\Delta X_{t+1}$$

(13)

Otherwise, we follow the $L{e}^{\prime }vy$ strategy:

$$X_{t+1}=X_t+L{e}^{\prime }vy(d)\times X_t$$

(14)

$$L{e}^{\prime }vy(x)=0.01\times \frac{r_1\times \sigma }{{\left|r_2\right|}^{1/\beta }}$$

(15)

$$\sigma ={\left(\frac{\mathrm{\Gamma }(1+\beta )+\sin (\frac{\pi \beta }{2})}{\mathrm{\Gamma }(\frac{1+\beta }{2})\times \beta \times 2^{(\beta -1)/2}}\right)}^{1/\beta }$$

(16)

where d is the dimension, $r_1$, $r_2$ are two random numbers in [0, 1], and $\beta =0.5$.

3. Improved Dragonfly Algorithm

3.1. Tent Mapping Initialization

In a metaheuristic algorithm, the initial position has a significant influence on the population diversity, and a high-quality initial position guarantees the algorithm’s performance. The basic dragonfly algorithm randomly performs initialization of the population. The generated positions may be concentrated in one place or far away from the optimal solution, which may have a negative effect on the search for the optimal solution. As a complex dynamic behavior of a nonlinear system, chaos is characterized by randomness, ergodicity, and regularity. Therefore, it is often used for the initialization process of meta-heuristic algorithms [52].

The ergodic property of the tent map is better than that of the logistic map, which means that by using the tent map, we can obtain a more uniform initial distribution in the search space [53]. Therefore, in this paper, tent mapping was applied to generate chaotic sequences for the initialization. The expression [54] is as follows:

$$x_{n+1}=\left\{\begin{array}{l}2x_n if x_n<0.5 \\ 2(1-x_n) \begin{array}{c} else x_n\ge 0.5 \end{array}\end{array}\right.$$

(17)

The expression for mapping the sequence to the search space generated using tent mapping is:

$$X=Tent(N,\dim )\cdot (ub-lb)+lb$$

(18)

where N is the number of dragonfly individuals in the population, ub and lb are the limit values of the search space, and dim is the spatial dimension.

3.2. Nonlinear Inertial Weight

The inertial weight of the basic dragonfly algorithm is linearly decreasing, which leads to a shortage of ergodicity in the algorithm search stage and insufficient exploitation ability, as well as the algorithm converging easily to local extremes.

The DA is adjusted using the inertial weight when calculating the step length, and it needs a higher inertial weight in the beginning iteration of the algorithm, so that the algorithm has a larger step length in the global search period to guarantee the algorithm completes the search for feasible solutions on a large scale faster. In the later iteration, a smaller value of inertial weight is needed, so that the algorithm can complete the local development stage with small steps. The two phases of the algorithm include a global wide-range search and local accurate localization. In order to ensure a smooth transit, inspired by [55], the inertial weight is changed to a nonlinear decreasing weight in this paper, and the expression is as follows:

$$\omega ={\omega }_{\min }+({\omega }_{\max }-{\omega }_{\min })\times (1-{(\frac{t}{\max \_iter})}^k)$$

(19)

where t is the current iteration number, $\max \_iter$ is the maximum iteration number, the maximum value of inertial weight ${\omega }_{\max }$ is 0.9, the minimum value ${\omega }_{\min }$ is 0.4, $k$ is the influence factor of the smoothness of the $\omega$ change curve, and the $\omega$ curve diagram is as follows when k takes different values:

From Figure 2, we can see that when the value of k is small, $\omega$ decreases sharply in the early iteration, which means the algorithm does not have a large step size for the global exploration. When the value is large, the inertial weight is close to linear variation, consistent with the basic DA. After many experiments, when the value is 0.6, the two main stages of the algorithm are balanced.

Considering that the dragonfly algorithm may fall into a local minimum, nonlinear weight is introduced at the optimal position.

$$X_{t+1}=\omega X^{+}+\Delta X_t$$

(20)

With the introduction of inertial weight $\omega$, the position updating of dragonfly individuals is dynamically affected by the optimal position, which avoids the aggregation of dragonfly individuals to the local optimal position.

3.3. Hybrid Strategy

3.3.1. Adjacent Position Decision Strategy

In the basic dragonfly algorithm, a radius is used to distinguish the region near the optimal solution from the more distant region. The quality of the solutions in the latter range is worse than that in the effective radius range. At this point, if there exist adjacent individuals, the current individual location is updated by the three actions of formation, aggregation, and collision avoidance. However, if the position of the adjacent individuals does not have a good influence on the current position, the aggregation and proximity of adjacent individuals that are not distinguished reduces the quality of the solution. Therefore, a judgment condition is added to the location update of adjacent individuals outside the radius range.

$$\Delta X_{t+1}=\left\{\begin{array}{l}\omega \Delta X_t+rand\left(A_i+C_i+S_i\right), \mathrm{if} fit\ge neigh\_fit\\ \omega \Delta X_t+rand\left(F_i+E_i+S_i\right), \mathrm{else} fit<neigh\_fit\end{array}\right.$$

(21)

Here, $fit$ represents the fitness value of the individual dragonfly and $neigh\_fit$ is the adjacent dragonfly fitness. When the adjacent position is better, the position update is still mainly affected by the adjacent solution. However, if the current position is better, it is necessary to consider foraging, collision avoidance, and enemy avoidance behaviors at the same time to strengthen the orientation towards the optimal individual. Thus, compared with the original single formula, the improved position update formula is more comprehensive, which prevents individual dragonfly from gathering at unnecessary positions.

3.3.2. Whale Optimization Algorithm Fusion Strategy

Different algorithms have their own strengths and weaknesses in different areas. Many studies have concluded that the DA does not perform well in local development compared to global searching. Scholars have proposed a variety of methods to improve the accuracy of the algorithm in small-scale mining. For example, the work of Zhang et al. [56] introduced the concept of the harmony search algorithm into the location update of the cuckoo algorithm. Liu et al. [57] introduced the influence component of elite individuals of the previous generation into the new solution position update, which ensures that the local development of the ABC algorithm is correctly guided. For the dragonfly algorithm, the introduction of an algorithm with strong local development ability is an admissible direction. Inspired by this, in this paper, the whale algorithm was introduced, which has strong local development ability to improve the dragonfly algorithm and adds the influence component of the elite individual to the position update formula to guide the dragonfly to the optimal position.

The direct use of the spiral bubble-net attack of the whale optimization algorithm helps the dragonfly to approach the optimal position rapidly. On the one hand, this operation is conducive to accelerating the convergence speed; on the other hand, it may lead to local extremes. Therefore, an adaptive probability threshold was introduced, and the position update method was selected within the effective range according to the probability to ensure the introduction of the spiral bubble-net attack in the local development stage. The adaptive probability threshold expression is as follows:

$$p=1-{\log }_2(1+\frac{t}{\max \_iter})$$

(22)

$p$ decreases nonlinearly, decreasing rapidly at the beginning of the algorithm, and gradually slowing down when t increases.

In the global exploration stage, the probability threshold $p$ is still high. When $p\ge 0.5$,

$$J=\left(2r_1-1\right)(2-\frac{2t}{\max \_iter})$$

(23)

$$X_{t+1}=\left\{\begin{array}{l}\omega X_t+\left[(s{S}_i+a{A}_i+c{C}_i+f{F}_i+e{E}_i)+Dist\right]\cdot J, \mathrm{if} \left|J\right|\ge 1\\ \omega X^{+}+\left[(s{S}_i+a{A}_i+c{C}_i+f{F}_i+e{E}_i)+Dist\right]\cdot J, \mathrm{else} \left|J\right|<0.5\end{array}\right.$$

(24)

where $r_1$ is a random number in [0, 1]. When parameter $\left|J\right|\ge 1$, the individual position is updated on the basis of the previous generation of individual positions, which is affected by five behaviors and the optimal solution. The algorithm searches quickly in the global exploration stage. When parameter $\left|J\right|<0.5$, the dragonfly individual searches near the current global optimal solution and is affected by five behaviors and the best solution.

Later, the algorithm reaches the local development stage. When $p<0.5$,

$$X_{t+1}=\omega X^{+}+Dist\cdot e^{bl}\cdot \cos (2\pi l)$$

(25)

The position is updated according to the spiral bubble-net attack of the whale algorithm, and the algorithm is developed near the optimal solution location.

3.3.3. Optimal Position Perturbation Strategy

In solving the problem of possible convergence to local extremes, the standard dragonfly algorithm’s method for dealing with this problem is to update the worst individual position outside the effective radius range and without adjacent solutions using $L{e}^{\prime }vy$ flight. However, the case of the optimal solution is not considered. Therefore, when the algorithm iteration stagnates and the position is no longer updated, the Cauchy mutation is introduced at the global optimal position to disturb.

$$f(x)=\frac{1}{\pi (x^2+1)}, -\infty <x<\infty$$

(26)

Compared with the common Gaussian distribution, Cauchy is more widely distributed. The slope of its curve is less steep and it has a continuous extension at both ends; so, it has a greater probability of escaping the current point [58]. Introducing the Cauchy mutation into the optimal position of the DA can enhance the ability of the algorithm to escape from local optimum constraints.

3.4. Main Steps and Process of Improved Dragonfly Algorithm

In summary, the specific descriptions of the HIDA are presented in the following pseudo-code (Algorithm 1). The flowchart of the HIDA is shown in Figure 3.

Algorithm 1. HIDA

1: Set the dragonfly population size to N, maximum number of iterations is $\max \_iter$;

2: Generate the initial location of the dragonfly individuals by tent chaotic map;

3: Obtain the fitness value fit of each dragonfly;

4: Set the position of optimal value as X+, the lowest as X−;

5: Initialize effective radius $r$ and adaptive probability threshold $p$;

6: while $t<\max \_iter$

7:  Update the weights using Equation (19);

8:  for $i=1:N$

9:   Find the adjacent solution according to r;

10:    if $dist\_to\_food>r$

11:     if the adjacent exists

12:      Update the position using Equation (21);

13:     else

14:      Update the position by $L{e}^{\prime }vy$ using Equation (14);

15:     end if

16:    else

17:      if $p\ge 0.5$

18:       Update the position using Equation (24);

19:      else

20:       Update the position by spiral bubble-net attack using Equation (25);

21:      end if

22:    end if

23:       Introduce the Cauchy perturbation to the optimal position according to Equation (26) and retain the solution with better fitness value to participate in the next iteration based on the greedy strategy.

24: end while

25: Return $fit$.

4. Experimental Results

4.1. Function Test Experiment

4.1.1. Basic Benchmark Functions

To test the validity of the HIDA, experiments were conducted using standard test functions. Among the 23 classical functions, 12 different functions were selected for experimentation.

Table 1. The 12 selected benchmark functions.

Functions	Range	Optimum	Characteristics
$\mathrm{TF}1(x)={\displaystyle \sum _{i=1}^n{x}_i^2}$	[−100, 100]	0	Unimodal
$\mathrm{TF}2(x)={\displaystyle \sum _{i=1}^n\left|x_i\right|+{\displaystyle \prod _{i=1}^n\left|x_i\right|}}$	[−10, 10]	0	Unimodal
$\mathrm{TF}3(x)={{\displaystyle \sum _{i=1}^n\left({\displaystyle \sum _{j-1}^i{x}_j}\right)}}^2$	[−100, 100]	0	Unimodal
$\mathrm{TF}4(x)=\underset{i}{\max }\left\{\left|x_i\right|,1\le i\le n\right\}$	[−100, 100]	0	Unimodal
$\mathrm{TF}5(x)={\displaystyle \sum _{i=1}^{n-1}\left[100{(x_{i+1}-x_i^2)}^2+{(x_i-1)}^2\right]}$	[−30, 30]	0	Unimodal
$\mathrm{TF}6(x)={\displaystyle \sum _{i=1}^n{(\left[x_i+0.5\right])}^2}$	[−100, 100]	0	Unimodal
$\mathrm{TF}7(x)={\displaystyle \sum _{i=1}^{n}i{x}_i^4+random}\left[\left.0,1\right)\right.$	[−1.28, 1.28]	0	Unimodal
$\mathrm{TF}8(x)={\displaystyle \sum _{i=1}^n\left[x_i^2-10\cos \left(2\pi x_i\right)+10\right]}$	[−5.12, 5.12]	0	Multimodal
$\mathrm{TF}9(x)=-20\exp (-0.2\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n{x}_i^2}})-\exp (\frac{1}{n}{\displaystyle \sum _{i=1}^n\cos (2\pi x_i)})+20+e$	[−32, 32]	0	Multimodal
$\mathrm{TF}10(x)=\frac{1}{4000}{\displaystyle \sum _{i=1}^n{x}_i^2}-{\displaystyle \prod _{i=1}^n\cos (\frac{x_i}{\sqrt{i}})+1}$	[−600, 600]	0	Multimodal
$\begin{array}{l}\mathrm{TF}11(x)=\frac{\pi }{n}\left\{10\sin (\pi y_1)+{\displaystyle \sum _{i=1}^{n-1}{(y_i-1)}^2\left[1+10{\sin }^2(\pi y_{i+1})\right]+{(y_n-1)}^2}\right\}+{\displaystyle \sum _{i=1}^{n}u(x_i,10,100,4)}\\ y_i=1+\frac{x_i+1}{4}\\ u(x_i,a,k,m)=\left\{\begin{array}{l}k{(x_i-a)}^m x_i>a\\ 0 -a<x_i<a\\ k{(-x_i-a)}^m x_i<-a\end{array}\right.\end{array}$	[−50, 50]	0	Multimodal
$\mathrm{TF}12\left(x\right)=0.1\left\{{\sin }^2(3\pi x_1)+{\displaystyle \sum _{i=1}^n{(x_i-1)}^2\left[1+{\sin }^2(3\pi x_i+1)\right]+{(x_n-1)}^2\left[1+{\sin }^2(2\pi x_n)\right]}\right\}+{\displaystyle \sum _{i=1}^{n}u(x_i,5,100,4)}$	[−50, 50]	0	Multimodal

gives detailed information about the basic benchmark test functions, including the expression, value range, hypothetical optimal value, and the characteristics of functions. Classical functions are mainly divided into unimodal and multimodal test functions; the selected $\mathrm{TF}1~\mathrm{TF}7$ functions belong to the former category and $\mathrm{TF}8~\mathrm{TF}12$ belong to the latter. Among these, the unimodal test function only has one optimal value; it is relatively simple but can test the ability of the algorithm to rapidly find the search direction from a wide range. The multimodal test function has multiple optimal values; it is more complex, so the ability to remove local extremes can be tested.

4.1.2. Algorithm Comparison and Analysis

In the comparison experiment, in addition to the HIDA, the standard DA, the dragonfly algorithm improved by tent chaotic mapping and nonlinear inertial weight (ADA), the standard PSO, the moth–flame optimization (MFO) algorithm, and the sine–cosine algorithm (SCA) [59] were chosen. In order to avoid experimental contingency, 30 independent runs were performed using 12 standard test functions.

The experimental environment was a Windows 11 system with a 1.80 GHz CPU and 16 GB memory; the programming language was MATLAB R2020b. In the common parameters, $N$ was set to 30 and $\max \_iter$ was 1500. When the dimensions were 50, 120, and 200, respectively, the optimal results, worst results, and mean and standard deviation of the 30 runs under the 12 test functions were obtained; these are recorded in

Table 2. Test results of algorithm function under 50 dimensions.

Functions	Metric	PSO	DA	MFO	ADA	SCA	HIDA
$TF 1$	Mean	6.52 × 101	3.27 × 103	6.00 × 103	3.29	1.68 × 101	0.00
	Std	1.37 × 101	1.32 × 103	9.32 × 104	5.50	4.08 × 101	0.00
	Best	4.65 × 101	6.11 × 102	6.04 × 10−3	7.71 × 10−10	1.30 × 10−2	0.00
	Worst	9.98 × 101	5.62 × 103	3.00 × 104	1.93 × 101	1.86 × 102	0.00
$TF 2$	Mean	4.15 × 101	2.83 × 101	6.67 × 101	6.58 × 10−1	1.05 × 10−3	2.0 × 10−184
	Std	4.81	9.42	3.72 × 101	1.06	2.59 × 10−3	0.00
	Best	3.37 × 101	1.18 × 101	1.00 × 101	4.17 × 10−5	2.69 × 10−7	6.8 × 10−197
	Worst	5.64 × 101	5.43 × 101	1.80 × 102	4.16	1.21 × 10−2	2.3 × 10−183
$TF 3$	Mean	1.18 × 103	3.93 × 104	4.20 × 104	2.30 × 102	2.96 × 104	0.00
	Std	2.42 × 102	2.22 × 104	2.20 × 104	3.85 × 102	1.46 × 104	0.00
	Best	6.67 × 102	1.09 × 104	9.38 × 103	1.52 × 10−7	8.73 × 103	0.00
	Worst	1.64 × 103	1.20 × 105	1.06 × 105	1.64 × 103	6.89 × 104	0.00
$TF 4$	Mean	4.06	3.32 × 101	8.28 × 101	1.88 × 10−1	5.30 × 101	2.2 × 10−185
	Std	3.81 × 10−1	8.83	5.49	2.93 × 10−1	1.12 × 101	0.00
	Best	3.46	1.88 × 101	7.46 × 101	4.43 × 10−4	2.89 × 101	1.5 × 10−191
	Worst	5.15	5.45 × 101	9.28 × 101	1.08	7.27 × 101	2.0 × 10−184
$TF 5$	Mean	4.78 × 104	8.29 × 105	1.33 × 107	1.95 × 101	2.89 × 105	6.06 × 10−1
	Std	1.38 × 104	5.44 × 105	3.69 × 107	2.73 × 101	5.47 × 105	2.89
	Best	2.69 × 104	5.57 × 104	1.30 × 102	6.59 × 10−3	1.62 × 102	3.05 × 10−5
	Worst	8.44 × 104	2.35 × 106	1.6 × 108	9.82 × 101	2.25 × 106	1.59 × 101
$TF 6$	Mean	6.23 × 101	3.93 × 103	6.00 × 103	7.62	2.03 × 101	4.07 × 10−3
	Std	1.23 × 101	1.71 × 103	7.71 × 103	1.62 × 101	2.97 × 101	7.09 × 10−3
	Best	3.24 × 101	1.70 × 103	8.08 × 10−3	5.50 × 10−3	8.50 × 101	5.14 × 10−8
	Worst	9.56 × 101	8.17 × 103	3.01 × 104	7.26 × 101	1.69 × 102	3.72 × 10−2
$TF 7$	Mean	3.38 × 102	1.04	1.60 × 101	1.06 × 10−2	2.17 × 10−1	4.88 × 10−5
	Std	6.89 × 101	5.17 × 10−1	1.85 × 101	1.17 × 10−2	1.91 × 10−1	5.81 × 10−5
	Best	2.12 × 102	1.85 × 10−1	3.51 × 10−1	1.06 × 10−4	3.42 × 10−2	1.93 × 10−6
	Worst	4.63 × 102	2.44	7.95 × 101	4.22 × 10−2	8.01 × 10−1	2.50 × 10−4
$TF 8$	Mean	4.91 × 102	3.13 × 102	3.00 × 102	1.31 × 101	6.29 × 101	0.00
	Std	4.49 × 101	6.11 × 101	5.21 × 101	1.76 × 101	4.89 × 101	0.00
	Best	3.77 × 102	1.21 × 102	1.82 × 102	3.41 × 10−8	1.18	0.00
	Worst	5.63 × 102	4.45 × 102	3.82 × 102	5.10 × 101	1.67 × 102	0.00
$TF 9$	Mean	5.82	1.06 × 101	1.93 × 101	5.92 × 10−1	1.69 × 101	2.43 × 10−15
	Std	3.72 × 10−1	1.83	1.08	8.34 × 10−1	7.43	1.79 × 10−15
	Best	5.27	6.28	1.49 × 101	1.96 × 10−6	4.33 × 10−2	8.88 × 10−16
	Worst	6.66	1.46 × 101	1.99 × 101	2.64	2.05 × 101	4.44 × 10−15
$TF10$	Mean	8.93 × 10−1	3.45 × 101	6.65 × 101	3.82 × 10−1	8.48 × 10−1	0.00
	Std	5.53 × 10−2	1.26 × 101	7.81 × 101	4.24 × 10−1	3.43 × 10−1	0.00
	Best	7.88 × 10−1	1.80 × 101	5.45 × 10−3	3.91 × 10−11	9.73 × 10−2	0.00
	Worst	9.87 × 10−1	7.02 × 101	2.71 × 102	1.27	1.76	0.00
$TF11$	Mean	2.34	5.30 × 103	8.53 × 106	5.31 × 10−3	2.13 × 105	1.21 × 10−5
	Std	1.10	1.56 × 104	4.67 × 107	8.91 × 10−3	4.50 × 105	3.90 × 10−5
	Best	1.05	8.05	6.03 × 10−1	9.69 × 10−8	3.44	2.65 × 10−10
	Worst	6.09	7.18 × 104	2.56 × 108	3.32 × 10−2	1.85 × 106	2.09 × 10−4
$TF12$	Mean	1.28 × 101	4.30 × 105	4.10 × 107	8.63 × 10−2	1.91 × 106	5.34 × 10−4
	Std	2.56	4.88 × 105	1.25 × 108	1.57 × 10−1	4.16 × 106	1.50 × 10−3
	Best	6.89	2.14 × 101	1.69	2.37 × 10−4	9.10	3.04 × 10−8
	Worst	1.80 × 101	1.86 × 106	4.1 × 108	7.89 × 10−1	1.72 × 107	8.07 × 10−3

,

Table 3. Test results of algorithm function under 120 dimensions.

Functions	Metric	PSO	DA	MFO	ADA	SCA	HIDA
$TF 1$	Mean	5.18 × 102	2.05 × 104	3.42 × 104	1.49 × 101	6.11 × 103	0.00
	Std	5.10 × 101	7.97 × 103	1.77 × 104	3.08 × 101	4.37 × 103	0.00
	Best	4.07 × 102	6.02 × 103	6.00 × 103	2.36 × 10−9	7.54 × 101	0.00
	Worst	6.35 × 102	3.65 × 104	7.93 × 104	1.23 × 102	1.99 × 104	0.00
$TF 2$	Mean	2.31 × 1010	1.04 × 102	2.01 × 102	1.58	2.03	1.1 × 10−183
	Std	1.26 × 1011	3.83 × 101	5.55 × 101	2.24	3.56	0.00
	Best	3.07 × 102	3.86 × 101	1.19 × 102	1.04 × 10−3	1.31 × 10−2	3.7 × 10−219
	Worst	6.88 × 1011	2.19 × 102	3.55 × 102	9.64	1.85 × 101	1.1 × 10−182
$TF 3$	Mean	2.33 × 104	2.62 × 105	2.27 × 105	2.99 × 103	2.50 × 105	0.00
	Std	4.86 × 103	8.96 × 104	8.04 × 104	6.04 × 103	6.57 × 104	0.00
	Best	1.41 × 104	1.14 × 105	1.18 × 105	5.75 × 10−9	1.32 × 105	0.00
	Worst	3.51 × 104	4.84 × 105	3.82 × 105	2.65 × 104	4.21 × 105	0.00
$TF 4$	Mean	1.29 × 101	4.92 × 101	9.49 × 101	3.84 × 10−1	8.72 × 101	8.3 × 10−185
	Std	1.43	7.11	1.69	5.34 × 10−1	3.60	0.00
	Best	9.33	3.82 × 101	9.16 × 101	1.20 × 10−3	7.85 × 101	8.0 × 10−196
	Worst	1.64 × 101	6.00 × 101	9.80 × 101	2.21	9.37 × 101	1.4 × 10−183
$TF 5$	Mean	9.29 × 105	1.25 × 107	8.76 × 107	2.76 × 101	7.18 × 107	9.56 × 10−1
	Std	1.83 × 105	7.23 × 106	7.53 × 107	7.61 × 101	4.48 × 107	3.14
	Best	5.60 × 105	2.67 × 106	3.08 × 106	1.58 × 10−6	7.16 × 106	2.28 × 10−5
	Worst	1.31 × 106	2.98 × 107	2.41 × 108	3.91 × 102	2.10 × 108	1.30 × 101
$TF 6$	Mean	5.28 × 102	1.88 × 104	3.90 × 104	1.35 × 101	6.04 × 103	1.02 × 10−2
	Std	5.82 × 101	9.95 × 103	1.44 × 104	3.60 × 101	4.38 × 103	1.56 × 10−2
	Best	4.38 × 102	2.98 × 103	9.04 × 103	1.20 × 10−4	1.06 × 103	4.85 × 10−6
	Worst	6.64 × 102	3.99 × 104	7.22 × 104	1.73 × 102	1.87 × 104	7.35 × 10−2
$TF 7$	Mean	2.83 × 103	1.70 × 101	2.33 × 102	1.06 × 10−2	9.90 × 101	4.08 × 10−5
	Std	2.39 × 102	1.07 × 101	1.29 × 102	1.72 × 10−2	5.30 × 101	3.63 × 10−5
	Best	2.34 × 103	6.68 × 10−1	2.24 × 101	1.63 × 10−4	2.18 × 101	5.29 × 10−6
	Worst	3.41 × 103	4.49 × 101	4.96 × 102	9.07 × 10−2	2.45 × 102	1.79 × 10−4
$TF 8$	Mean	1.60 × 103	9.29 × 102	9.08 × 102	1.52 × 101	2.61 × 102	0.00
	Std	7.03 × 101	1.59 × 102	7.48 × 101	3.23 × 101	1.43 × 102	0.00
	Best	1.48 × 103	6.32 × 102	7.52 × 102	4.87 × 10−8	6.45 × 101	0.00
	Worst	1.76 × 103	1.23 × 103	1.05 × 103	1.20 × 102	6.72 × 102	000
$TF 9$	Mean	8.69	1.26 × 101	1.98 × 101	5.09 × 10−1	1.84 × 101	1.84 × 10−15
	Std	3.10 × 10−1	2.86	2.55 × 10−1	7.84 × 10−1	5.11	1.60 × 10−15
	Best	8.03	3.04	1.93 × 101	5.52 × 10−5	4.08	8.88 × 10−16
	Worst	9.19	1.99 × 101	1.99 × 101	2.95	2.07 × 101	4.44 × 10−15
$TF10$	Mean	1.13	1.80 × 102	3.24 × 102	3.65 × 10−1	6.37 × 101	0.00
	Std	1.19 × 10−2	7.02 × 101	1.22 × 102	4.90 × 10−1	5.05 × 101	0.00
	Best	1.11	6.23 × 101	1.57 × 102	3.17 × 10−7	1.71	0.00
	Worst	1.16	3.67 × 102	5.90 × 102	1.53	1.87 × 102	0.00
$TF11$	Mean	2.65 × 101	2.74 × 106	1.69 × 108	2.58 × 10−3	1.81 × 108	3.84 × 10−6
	Std	2.23 × 101	3.60 × 106	1.64 × 108	4.81 × 10−3	9.86 × 107	9.88 × 10−6
	Best	9.07	7.22 × 101	6.03 × 106	1.20 × 10−6	2.78 × 107	3.83 × 10−9
	Worst	1.34 × 102	1.63 × 107	4.86 × 108	2.40 × 10−2	4.08 × 108	3.97 × 10−5
$TF12$	Mean	5.40 × 103	2.15 × 107	2.78 × 108	5.53 × 10−1	3.61 × 108	1.38 × 10−3
	Std	4.36 × 103	1.81 × 107	2.88 × 108	1.05	1.89 × 108	4.23 × 10−3
	Best	7.92 × 102	2.39 × 106	1.00 × 107	1.47 × 10−4	7.99 × 107	4.57 × 10−10
	Worst	2.54 × 104	7.43 × 107	9.49 × 108	4.18	7.57 × 108	2.32 × 10−2

and

Table 4. Test results of algorithm function under 200 dimensions.

Functions	Metric	PSO	DA	MFO	ADA	SCA	HIDA
$TF 1$	Mean	1.64 × 103	3.96 × 104	1.35 × 105	7.42	2.35 × 104	0.00
	Std	1.59 × 102	2.15 × 104	2.20 × 104	2.09 × 101	1.81 × 104	0.00
	Best	1.28 × 103	6.67 × 103	9.36 × 104	7.27 × 10−9	2.68 × 103	0.00
	Worst	1.95 × 103	9.35 × 104	1.77 × 105	1.03 × 102	9.69 × 104	0.00
$TF 2$	Mean	2.4 × 1038	1.67 × 102	4.46 × 102	2.73	1.25 × 101	4.5 × 10−184
	Std	9.17 × 1038	5.90 × 101	5.90 × 101	3.12	1.35 × 101	0.00
	Best	1.41 × 1016	5.33 × 101	3.36 × 102	7.96 × 10−4	8.81 × 10−1	2.9 × 10−193
	Worst	3.88 × 1039	2.94 × 102	5.51 × 102	1.08 × 101	5.65 × 101	4.4 × 10−183
$TF 3$	Mean	8.14 × 104	8.62 × 105	6.12 × 105	8.93 × 103	8.05 × 105	0.00
	Std	1.50 × 104	3.48 × 105	1.66 × 105	1.58 × 104	1.70 × 105	0.00
	Best	5.23 × 104	3.09 × 105	3.54 × 105	3.18 × 10−3	5.08 × 105	0.00
	Worst	1.11 × 105	1.58 × 106	9.25 × 105	6.17 × 104	1.10 × 106	0.00
$TF 4$	Mean	1.98 × 101	5.69 × 101	9.70 × 101	2.08 × 10−1	9.50 × 101	5.8 × 10−185
	Std	1.27	9.18	1.05	3.04 × 10−1	1.22	0.00
	Best	1.76 × 101	4.14 × 101	9.36 × 101	1.20 × 10−3	9.30 × 101	1.1 × 10−191
	Worst	2.31 × 101	7.97 × 101	9.84 × 101	1.48	9.73 × 101	8.8 × 10−184
$TF 5$	Mean	5.66 × 106	3.38 × 107	4.3 × 108	7.70 × 101	2.93 × 108	8.23 × 10−1
	Std	8.15 × 105	1.60 × 107	1.21 × 108	1.11 × 102	5.08 × 107	4.36
	Best	4.37 × 106	6.43 × 106	1.88 × 108	1.82 × 10−3	1.88 × 108	8.27 × 10−5
	Worst	7.03 × 106	7.58 × 107	7.64 × 108	4.84 × 102	3.90 × 108	2.39 × 101
$TF 6$	Mean	1.62 × 103	3.49 × 104	1.43 × 105	5.94	2.08 × 104	1.08 × 10−2
	Std	1.47 × 102	1.74 × 104	2.43 × 104	8.40	1.12 × 104	1.97 × 10−2
	Best	1.26 × 103	9.74 × 103	8.16 × 104	7.90 × 10−4	2.36 × 103	2.33 × 10−9
	Worst	1.91 × 103	7.84 × 104	1.97 × 105	3.34 × 101	4.23 × 104	9.11 × 10−2
$TF 7$	Mean	8.50 × 103	8.82 × 101	1.35 × 103	1.29 × 10−2	8.26 × 102	3.54 × 10−5
	Std	4.95 × 102	7.59 × 101	3.70 × 102	1.50 × 10−2	3.22 × 102	2.62 × 10−5
	Best	7.53 × 103	1.55 × 101	7.24 × 102	7.52 × 10−5	1.46 × 102	4.77 × 10−6
	Worst	9.15 × 103	3.06 × 102	2.15 × 103	5.21 × 10−2	1.39 × 103	1.14 × 10−4
$TF 8$	Mean	2.86 × 103	1.61 × 103	1.81 × 103	2.81 × 101	4.80 × 102	0.00
	Std	8.91 × 101	2.37 × 102	1.04 × 102	5.25 × 101	2.09 × 102	0.00
	Best	2.63 × 103	1.07 × 103	1.54 × 103	8.73 × 10−7	7.51 × 101	0.00
	Worst	3.05 × 103	1.99 × 103	2.07 × 103	2.03 × 102	9.35 × 102	0.00
$TF 9$	Mean	1.05 × 101	1.28 × 101	1.99 × 101	3.50 × 10−1	1.83 × 101	2.55 × 10−15
	Std	3.18 × 10−1	2.28	5.71 × 10−2	5.51 × 10−1	4.39	1.80 × 10−15
	Best	9.75	6.88	1.96 × 101	3.23 × 10−5	7.54	8.88 × 10−16
	Worst	1.12 × 101	1.64 × 101	1.99 × 101	1.92	2.07 × 101	4.44 × 10−15
$TF10$	Mean	1.42	2.81 × 102	1.28 × 103	5.96 × 10−1	1.92 × 102	0.00
	Std	2.85 × 10−2	1.66 × 102	1.82 × 102	5.20 × 10−1	1.09 × 102	0.00
	Best	1.36	5.62 × 101	8.51 × 102	7.72 × 10−7	4.48 × 101	0.00
	Worst	1.47	6.28 × 102	1.64 × 103	1.54	4.79 × 102	0.00
$TF11$	Mean	1.01 × 104	1.20 × 107	8.46 × 108	6.60 × 10−3	7.60 × 108	1.66 × 10−6
	Std	1.26 × 104	1.34 × 107	3.56 × 108	1.45 × 10−2	2.49 × 108	5.00 × 10−6
	Best	1.88 × 102	1.46 × 105	2.28 × 108	4.05 × 10−7	3.56 × 108	3.04 × 10−9
	Worst	4.48 × 104	5.54 × 107	1.65 × 109	6.07 × 10−2	1.42 × 109	2.74 × 10−5
$TF12$	Mean	2.70 × 105	6.22 × 107	1.73 × 109	4.05 × 10-1	1.46 × 109	1.01 × 10−3
	Std	1.29 × 105	5.28 × 107	5.44 × 108	1.03	4.86 × 108	2.00 × 10−3
	Best	5.58 × 104	5.70 × 106	2.44 × 108	7.72 × 10−6	4.98 × 108	4.95 × 10−8
	Worst	5.46 × 105	1.99 × 108	2.67 × 109	5.19	2.65 × 109	8.82 × 10−3

. The optimal search results for each algorithm were compared.

The optimal value represents the limit of the algorithm’s optimality search in that case, and the worst value roughly reflects the minimum level of the algorithm. The mean of the multiple experimental results reflects the accuracy of the algorithm, while the Std shows the distribution of the results of multiple runs, which can reflect the stability of the algorithm. If the values of the mean and Std are sufficiently low and close in value, this means the algorithm has a good performance.

From the above tables, for 50, 120, and 200 dim, under TF1, TF3, TF8, and TF10, the HIDA found the optimal value 0.00 × 10⁰, and its mean and Std were stable at 0, indicating that the HIDA had high optimization accuracy and stability. For unimodal functions TF2 and TF4, although the HIDA did not reach the ideal value, the order of magnitude of its optimization accuracy far exceeded that of the other comparison algorithms. Furthermore, the Std can still reach 0, which shows that the results of the multiple runs of the HIDA are concentrated around the average value. Although no ideal optimal solution was obtained under TF5~TF7, TF9, TF11, and TF12, the HIDA still performed the best out of the several comparison algorithms. This was because the solution complexity increased with the dimensionality; therefore, the accuracy and stability of the algorithm were challenged to find the best solution.

The dimensionality of the search space had a significant influence on the search results. The higher dimensionality of the test function meant more complex problems, which were also more challenging for the algorithm. From the perspective of the influence of the dimension, when the dimension was gradually increased to a high dimension, the algorithm was much less effective in finding the optimum. For example, from 50 to 200 dim, the mean value of the SCA under TF7 was 2.17 × 10⁻¹ at the beginning, and it was 8.26 × 10² at 200 dim; the mean of MFO under TF11 increased from 8.53 × 10⁶ to 8.46 × 10⁸. The HIDA in this paper was almost unaffected by the influence of the dimension. When solving complex high-dimensional problems, the HIDA still had a high solving accuracy and a rapid convergence speed. This demonstrated the strong stability of the HIDA when dealing with high-dimensional complex problems.

For the purpose of more intuitively comparing the optimization results of each algorithm, convergence curves were drawn.

The average convergence curves for each function in the 50-dimensional case are given in Figure 4. Among them, Figure 4a–g and Figure 4h–l present the convergence curves for TF1~TF7 and TF8~TF12, respectively.

According to Figure 4a–l, the convergence curves of the HIDA with pentagons remain below all comparison curves, indicating that the HIDA is more accurate at locating the best solutions. Additionally, the HIDA starts to converge at around 570 iterations and reaches its maximum accuracy at around 1400 iterations at TF1 and TF3; convergence slows down at TF2 and TF4 and converges to an order of magnitude of e-190 after 1500 iterations. The HIDA starts to converge effectively around the 180th iteration, with the highest accuracy results for TF5, TF7, TF8, TF11, and TF12. It converges a little faster at TF6 and a little slower at TF9 and TF10, only starting to converge rapidly at the 600th iteration. The performance of the ADA is just behind that of the HIDA. This is because the ADA also utilizes tent chaotic mapping and nonlinear inertial weight, both of which are improvement strategies of the HIDA. The HIDA has the best performance because it not only enhances the influence ability of leaders, but also uses the spiral bubble-net attack of the whale algorithm to strengthen the local development ability. In TF5 and TF7~ TF9, the HIDA converges fastest in the initial iteration and the optimization result is the best. In TF11 and TF12, although the ADA converges the earliest, the HIDA produces better results.

The average convergence curve provides more evidence that the HIDA possesses a good aptitude for finding the best and quickest convergence speed.

4.1.3. Statistical Test

In addition to the basic analysis of the test results, statistical tests are required, and the Wilcoxon rank sum test is an authoritative test method. The result of the test for two algorithms is expressed as P. When P < 0.05, the former algorithm has a significant advantage over the latter algorithm [60]. To obtain more intuitive results after the comparison, the significance judgment result S was used. When S is “+”, the former algorithm is better. When S is “−”, the latter algorithm is better. When S is “=”, the two algorithms are equivalent. The dimension was set to 200 to add difficulty. The values of P and S for each function are given in

Table 5. Wilcoxon rank sum test results.

	Metric	$\mathit{T}\mathit{F}1$	$\mathit{T}\mathit{F}2$	$\mathit{T}\mathit{F}3$	$\mathit{T}\mathit{F}4$	$\mathit{T}\mathit{F}5$	$\mathit{T}\mathit{F}6$
PSO	P	1.21 × 10−12	3.02 × 10−11	1.21 × 10−12	3.02 × 10−11	3.02 × 10−11	3.02 × 10−11
	S	$+$	$+$	$+$	$+$	$+$	$+$
DA	P	1.21 × 10−12	3.02 × 10−11	1.21 × 10−12	3.02 × 10−11	3.02 × 10−11	3.02 × 10−11
	S	$+$	$+$	$+$	$+$	$+$	$+$
MFO	P	1.21 × 10−12	3.02 × 10−11	1.21 × 10−12	3.02 × 10−11	3.02 × 10−11	3.02 × 10−11
	S	$+$	$+$	$+$	$+$	$+$	$+$
ADA	P	1.21 × 10−12	3.02 × 10−11	1.21 × 10−12	2.92 × 10−11	8.48 × 10−9	8.98 × 10−10
	S	$+$	$+$	$+$	$+$	$+$	$+$
SCA	P	1.21 × 10−12	3.02 × 10−11	1.21 × 10−12	3.02 × 10−11	3.02 × 10−11	3.02 × 10−11
	S	$+$	$+$	$+$	$+$	$+$	$+$
		$\mathit{T}\mathit{F}\mathbf{7}$	$\mathit{T}\mathit{F}\mathbf{8}$	$\mathit{T}\mathit{F}\mathbf{9}$	$\mathit{T}\mathit{F}\mathbf{10}$	$\mathit{T}\mathit{F}\mathbf{11}$	$\mathit{T}\mathit{F}\mathbf{12}$
PSO	P	3.02 × 10−11	1.21 × 10−12	1.45 × 10−11	1.21 × 10−12	3.02 × 10−11	3.02 × 10−11
	S	$+$	$+$	$+$	$+$	$+$	$+$
DA	P	3.02 × 10−11	1.21 × 10−12	1.45 × 10−11	1.21 × 10−12	3.02 × 10−11	3.02 × 10−11
	S	$+$	$+$	$+$	$+$	$+$	$+$
MFO	P	3.02 × 10−11	1.21 × 10−12	1.45 × 10−11	1.21 × 10−12	3.02 × 10−11	3.02 × 10−11
	S	$+$	$+$	$+$	$+$	$+$	$+$
ADA	P	4.98 × 10−11	1.21 × 10−12	1.45 × 10−11	1.21 × 10−12	9.76 × 10−10	7.08 × 10−8
	S	$+$	$+$	$+$	$+$	$+$	$+$
SCA	P	3.02 × 10−11	1.21 × 10−12	1.45 × 10−11	1.21 × 10−12	3.02 × 10−11	3.02 × 10−11
	S	$+$	$+$	$+$	$+$	$+$	$+$
Statistical result			$+/=/-$		$12/0/0$

.

From Table 5, we can see that the values of P are significantly below the predetermined significance level of 0.05 and the S values are all “+”. The statistics show that the HIDA performs significantly better than the comparison algorithms in all 12 functions, and no other cases exist. Only the results of the ADA under several test functions are marginally smaller than the other results. This is due to some similar improvement techniques between the ADA and HIDA.

The results of significance judgment S in the table are all “+”, and the significance statistics under 12 functions all demonstrate that the HIDA has considerable advantages over the DA, ADA, PSO, MFO, and SCA.

4.1.4. Comparison with Other Improved Dragonfly Algorithms

Numerous academics have offered various enhancements to the dragonfly method since it was proposed. To demonstrate the competitiveness of the HIDA, two different improved dragonfly algorithms were introduced for comparison. Firstly, the HIDA was compared with the CGWO-DA in [61], which improved upon the dragonfly algorithm using the gray wolf optimization algorithm. According to the reference, $N$ was 50, $\max \_iter$ was 500 times, and $\dim$ was 30. The chosen comparison criteria were mean and Std. The results are recorded in

Table 6. Comparison of HIDA and CGWO-DA.

	Metric	$\mathit{T}\mathit{F}1$	$\mathit{T}\mathit{F}2$	$\mathit{T}\mathit{F}3$	$\mathit{T}\mathit{F}4$	$\mathit{T}\mathit{F}5$	$\mathit{T}\mathit{F}6$
HIDA	Mean	2.8 × 10−122	5.40 × 10−62	8.1 × 10−121	1.48 × 10−62	4.30 × 10−1	2.42 × 10−2
	Std	1.2 × 10−121	1.75 × 10−61	3.5 × 10−120	4.07 × 10−62	1.27	6.91 × 10−2
CGWO-DA	Mean	2.86 × 10−46	6.69 × 10−28	5.34 × 10−9	6.99 × 10−12	2.71 × 101	5.10 × 10−1
	Std	5.93 × 10−46	7.47 × 10−28	2.77 × 10−8	9.47 × 10−12	7.05 × 10−1	3.15 × 10−1
		$\mathit{T}\mathit{F}\mathbf{7}$	$\mathit{T}\mathit{F}\mathbf{8}$	$\mathit{T}\mathit{F}\mathbf{9}$	$\mathit{T}\mathit{F}\mathbf{10}$	$\mathit{T}\mathit{F}\mathbf{11}$	$\mathit{T}\mathit{F}\mathbf{12}$
HIDA	Mean	7.39 × 10−5	0.00	3.26 × 10−15	0.00	5.31 × 10−5	1.73 × 10−4
	Std	4.52 × 10−5	0.00	1.70 × 10−15	0.00	8.38 × 10−5	3.01 × 10−4
CGWO-DA	Mean	1.20 × 10−3	6.71 × 10−1	1.60 × 10−14	2.36 × 10−3	3.57 × 10−2	6.21 × 10−1
	Std	6.00 × 10−4	2.41	3.30 × 10−15	6.49 × 10−3	1.90 × 10−2	2.23 × 10−1

.

From Table 6, it is easy to see that the judging metrics obtained for the HIDA for all the selected functions are much lower than those of the CGWO-DA. The HIDA also achieves ideal values in the multi-peak functions TF8 and TF10. However, the best result achieved by the comparison algorithm CGWO-DA is for the TF1 function, with a mean value of 2.86 × 10⁻⁴⁶. This shows that using the whale algorithm to improve the dragonfly algorithm is better than that improved using the gray wolf algorithm.

Then, the HIDA was compared with the modified DA in [62], which improved upon the dragonfly algorithm using Brownian motion. According to the reference, the parameters were set as follows: $N$ was 40, $\max \_iter$ was 500 times, and $\dim$ was 10.

The comparison of the results in

Table 7. Comparison of HIDA and the modified DA.

	Metric	$\mathit{T}\mathit{F}1$	$\mathit{T}\mathit{F}2$	$\mathit{T}\mathit{F}3$	$\mathit{T}\mathit{F}4$	$\mathit{T}\mathit{F}5$	$\mathit{T}\mathit{F}6$
HIDA	Mean	1.3 × 10−123	1.21 × 10−62	3.2 × 10−121	2.27 × 10−62	1.05	9.69 × 10−3
	Std	3.5 × 10−123	3.61 × 10−62	1.0 × 10−120	3.92 × 10−62	2.29	1.31 × 10−2
The modified DA	Mean	4.93	8.68 × 10−1	8.40 × 101	1.78	5.74 × 102	5.58
	Std	7.16 × 10−18	3.76 × 10−5	2.10 × 10−6	2.78 × 10−3	6.79	1.32 × 10−15
		$\mathit{T}\mathit{F}\mathbf{7}$	$\mathit{T}\mathit{F}\mathbf{8}$	$\mathit{T}\mathit{F}\mathbf{9}$	$\mathit{T}\mathit{F}\mathbf{10}$	$\mathit{T}\mathit{F}\mathbf{11}$	$\mathit{T}\mathit{F}\mathbf{12}$
HIDA	Mean	7.78 × 10−5	0.00	2.78 × 10−15	0.00	1.98 × 10−4	2.14 × 10−3
	Std	5.34 × 10−5	0.00	1.80 × 10−15	0.00	2.76 × 10−4	7.05 × 10−3
The modified DA	Mean	2.22 × 10−2	2.47 × 101	2.32	4.34 × 10−1	1.29	6.71 × 10−1
	Std	4.69 × 10−3	9.48	4.87 × 10−1	7.35 × 10−2	9.83 × 10−2	4.63 × 10−3

demonstrated that the modified DA employing Brownian motion improvement only had advantages in terms of standard deviations for F6 and F12. More importantly, the modified DA had a larger mean value compared to the HIDA, implying the lower accuracy of the search. Out of the 12 functions tested in the experiment, all of the functions tested showed that the HIDA possessed smaller mean values. In the multimodal functions TF8 and TF10, the HIDA reached the ideal value. On the whole, the HIDA worked better than the comparison algorithm improved using Brownian motion. To conclude, the method used in this paper to improve the DA can be seen as a competitive method.

4.1.5. Benchmark Functions from CEC

In addition to the basic test functions, the CEC test function set is more convincing and challenging. CEC2013 and CEC2014 [60] test sets were considered in this paper.

Table 8. The 12 selected CEC test functions.

	Functions	Name	Characteristics	Range	Optimum
CEC2013	CEC04	Rotated Discus Function	Unimodal	${[-100,100]}^D$	−1100
	CEC16	Rotated Katsuura Function	Multimodal	${[-100,100]}^D$	200
	CEC17	Lunacek Bi-Rastrigin Function	Multimodal	${[-100,100]}^D$	300
	CEC20	Expanded Scaffer’s F6 Function	Composition	${[-100,100]}^D$	600
	CEC24	Composition Function 4 (n = 3, Rotated)	Composition	${[-100,100]}^D$	1000
CEC2014	CEC05	Shifted and Rotated Ackley’s Function	Multimodal	${[-100,100]}^D$	500
	CEC12	Shifted and Rotated Katsuura Function	Multimodal	${[-100,100]}^D$	1200
	CEC23	Composition Function 1 (n = 5)	Composition	${[-100,100]}^D$	2300
	CEC24	Composition Function 2 (n = 3)	Composition	${[-100,100]}^D$	2400
	CEC25	Composition Function 3 (n = 3)	Composition	${[-100,100]}^D$	2500
	CEC26	Composition Function 4 (n = 5)	Composition	${[-100,100]}^D$	2600
	CEC28	Composition Function 6 (n = 5)	Composition	${[-100,100]}^D$	2800

lists the details of the 12 functions from CEC2013 and CEC2014. The selected test function includes unimodal, multimodal, and composition functions, which are increasingly difficult to solve.

The comparison outcomes for the 30 independent runs of the HIDA, DA, PSO, MFO, whale optimization algorithm (WOA), gray wolf optimization (GWO), and HHO on the CEC functions are exhibited in

Table 9. Comparative results of different algorithms on CEC test functions.

	Metric	$\mathit{C}\mathit{E}\mathit{C}04$	$\mathit{C}\mathit{E}\mathit{C}16$	$\mathit{C}\mathit{E}\mathit{C}17$	$\mathit{C}\mathit{E}\mathit{C}20$	$\mathit{C}\mathit{E}\mathit{C}24$	$\mathit{C}\mathit{E}\mathit{C}05$
PSO	Mean	4.88 × 104	2.03 × 102	9.73 × 102	6.15 × 102	1.39 × 103	5.21 × 102
	Std	1.11 × 104	4.29 × 10−1	1.10 × 102	2.81 × 10−2	5.91 × 101	5.64 × 10−2
DA	Mean	1.02 × 105	2.28 × 102	7.76 × 103	6.15 × 102	1.41 × 103	5.21 × 102
	Std	2.57 × 104	6.03 × 10−1	1.02 × 102	1.63 × 10−1	5.37 × 101	9.11 × 10−2
MFO	Mean	1.98 × 105	2.04 × 102	2.62 × 103	6.15 × 102	1.38 × 103	5.21 × 102
	Std	6.16 × 104	1.59	2.60 × 102	3.63 × 10−1	5.74 × 101	1.10 × 10−1
SOA	Mean	4.67 × 104	2.03 × 102	8.48 × 102	6.15 × 102	1.30 × 103	5.21 × 102
	Std	7.79 × 103	4.71 × 10−1	6.21 × 101	6.40 × 10−1	1.01 × 101	5.78 × 10−2
WOA	Mean	9.90 × 104	2.03 × 102	9.84 × 102	6.15 × 102	1.32 × 103	5.21 × 102
	Std	3.50 × 104	5.69 × 10−1	1.14 × 102	1.70 × 10−1	9.72	1.19 × 10−1
GWO	Mean	5.04 × 104	2.03 × 102	5.25 × 102	6.15 × 102	1.26 × 103	5.21 × 102
	Std	5.97 × 103	4.19 × 10−1	4.33 × 101	1.31	1.02 × 101	5.51 × 10−2
HIDA	Mean	3.87 × 104	2.03 × 102	1.24 × 103	6.15 × 102	1.33 × 103	5.21 × 102
	Std	6.61 × 102	4.08 × 10−1	6.57 × 101	4.80 × 10−4	1.36 × 101	5.42 × 10−2
	Metric	$\mathit{C}\mathit{E}\mathit{C}\mathbf{12}$	$\mathit{C}\mathit{E}\mathit{C}\mathbf{23}$	$\mathit{C}\mathit{E}\mathit{C}\mathbf{24}$	$\mathit{C}\mathit{E}\mathit{C}\mathbf{25}$	$\mathit{C}\mathit{E}\mathit{C}\mathbf{26}$	$\mathit{C}\mathit{E}\mathit{C}\mathbf{28}$
PSO	Mean	1.20 × 103	2.61 × 103	2.63 × 103	2.72 × 103	2.79 × 103	7.77 × 103
	Std	4.97 × 10−1	6.33 × 10−2	1.04 × 101	5.48	2.53 × 101	7.29 × 102
DA	Mean	1.20 × 103	2.72 × 103	2.66 × 103	2.75 × 103	2.77 × 103	6.99 × 103
	Std	7.19 × 10−1	4.51 × 101	1.23 × 101	2.18 × 101	5.08 × 101	1.31 × 103
MFO	Mean	1.20 × 103	3.00 × 103	2.91 × 103	2.77 × 103	2.71 × 103	6.11 × 103
	Std	1.15 × 101	7.27 × 101	3.10 × 101	3.69 × 101	4.11	7.87 × 102
SOA	Mean	1.20 × 103	2.68 × 103	2.60 × 103	2.71 × 103	2.70 × 103	4.25 × 103
	Std	5.34 × 10−1	3.38 × 101	1.23 × 10−2	1.05 × 101	7.78 × 10−1	2.95 × 102
WOA	Mean	1.20 × 103	2.68 × 103	2.61 × 103	2.72 × 103	2.73 × 103	5.42 × 103
	Std	7.56 × 10−1	1.59 × 101	6.87	2.00 × 101	6.51 × 101	6.69 × 102
GWO	Mean	1.20 × 103	2.64 × 103	2.60 × 103	2.71 × 103	2.75 × 103	4.01 × 103
	Std	1.34 × 101	6.50 × 101	8.34 × 10−3	5.58	6.02 × 101	2.58 × 102
HIDA	Mean	1.20 × 103	2.50 × 103	2.60 × 103	2.70 × 103	2.73 × 103	6.01 × 103
	Std	3.29 × 10−1	1.19 × 10−13	8.14 × 10−4	1.69 × 10−13	4.02 × 101	2.07 × 103

.

The priority level of the mean value is higher than the standard deviation. As can be observed, in CEC2013, the HIDA leads to the lowest values for CEC04, CEC16, and CEC20. GWO has advantages for CEC17 and CEC24. The HIDA also performs well for CEC17 only after GWO. For CEC2017, the HIDA performs exceptionally well for CEC05, CEC12, CEC23, CEC24, and CEC25. Although the mean values of the compared algorithms are nearly identical for CEC05 and CEC12, the HIDA has a smaller standard deviation. SOA and GWO perform well on CEC26 and CEC28, respectively. The HIDA also performs well on these two functions. From the abovementioned discussion, the HIDA performs better on CEC functions than other algorithms.

4.1.6. Algorithm Complexity Analysis

The complexity of the algorithm determines the efficiency of the algorithm in solving a problem. In general, the algorithmic complexity of the improved algorithm increases in order to obtain a higher accuracy in the search for the best solution. Thus, it is a challenge to balance the accuracy of the search with the complexity of the algorithm.

The population size is $N$, the maximum number of iterations is $T_{\max }$, and the dimension is $\dim$. The complexity of initializing the population in the standard DA is $O(N\times \dim )$. The complexity of all five behaviors is $O(N)$. In the location update phase, the complexity of the location update with adjacent individuals is $O(T_{\max }\times N\times \dim )$ when the distance from the optimal solution is far, and the complexity of random wandering with $L{e}^{\prime }vy$ when there are no adjacent individuals is $O(N\times \dim )$. When the optimal solution is not far away, the position update complexity is $O(T_{\max }\times N\times \dim )$. In total the algorithmic complexity of the DA is $O(T_{\max }\times N\times \dim )$.

The complexity of initializing the population in the HIDA is $O(N\times \dim )$. In the iterative phase of the algorithm, the position update complexity is $O(T_{\max }\times N\times \dim )$ when there are neighboring individuals with better positions, $O(T_{\max }\times N\times \dim )$ when there are neighboring individuals but the current position is of better quality, and $O(N\times \dim )$ when there are no neighboring solutions when the distance to the optimal solution is far. The complexity of the adaptive probability threshold $P<0.5$ is $O(T_{\max }\times N\times \dim )$ when the optimal solution is not far away and $O(T_{\max }\times N\times \dim )$ when $P\ge 0.5$; the complexity of the Cauchy perturbation for the optimal position is $O(\dim )$. The total arithmetic complexity of the HIDA is $O(T_{\max }\times N\times \dim )$.

As can be observed, the complexity of the HIDA is comparable to that of the DA, and improving the method’s optimization accuracy has no negative effects on its effectiveness.

4.2. Algorithm Application Experiment

The algorithm’s value in practical applications should take precedence over only testing the algorithm’s performance with fundamental test functions.

4.2.1. Parameter Estimation of Solar Photovoltaic Cell

In this section, the HIDA was used to solve the PV cell parameter extraction problem. The classical single-diode model was chosen for the simulation experiments. Experimental data were gathered from [63]. The actual measured data are displayed in

Table 10. Actual measurement data.

Voltage/V	Current/A	Voltage/V	Current/A
0.764	−0.2057	0.728	0.4137
0.762	−0.1291	0.7065	0.4373
0.7605	−0.0588	0.6755	0.4590
0.7605	0.0057	0.632	0.4784
0.76	0.0646	0.573	0.496
0.759	0.1185	0.499	0.5119
0.757	0.1678	0.413	0.5265
0.757	0.2132	0.3165	0.5398
0.7555	0.2545	0.212	0.5521
0.754	0.2924	0.1035	0.5633
0.7505	0.3269	−0.01	0.5736
0.7465	0.3585	−0.123	0.5833
0.7385	0.3873	−0.21	0.59

.

The range of the five parameters to be measured is displayed in

Table 11. Range of the parameters used in SDM.

PRM	UB	LB
$R_{sh}$	100	0
$I_o$	1	0
$N_s$	2	1
$I_{ph}$	1	0
$R_s$	0.5	0

. The estimated parameters are considered valid only if they are within the range.

To facilitate the discussion and analysis, the arrangement of comparative experiments was necessary. Several algorithms were added to the solution of the problem, such as the DA, WOA, GWO, MFO, and HHO. Additionally, the performance of photovoltaic cells was affected by the environment; therefore, it was more accurate to conduct experiments in a variety of settings. Since temperature as the primary factor in respect of the environmental impact [64], a control experiment was conducted, with seven temperatures covering the common temperature range.

Parameter information: $N$ was 30 and $\max \_iter$ was 1000.

Table 12. Parameters estimated by different algorithms at different temperatures.

TEMP	PRM	Method
		HIDA	DA	WOA	GWO	MFO	HHO
0 °C	$I_{ph}$	0.7616	0.7861	0.7597	0.7590	0.7608	0.7531
	$I_o$	0.5278	0.6164	0.2648	0.1715	0.2597	0.0015
	$R_{sh}$	55.8763	91.0819	68.0952	64.9598	49.0832	74.7758
	$R_s$	0.0343	0.0780	0.0374	0.0392	0.0373	0.0632
	$N_s$	1.7177	1.7402	1.6376	1.5913	1.6358	1.2162
8 °C	$I_{ph}$	0.7598	0.7592	0.7606	0.7613	0.7598	0.7589
	$I_o$	0.3616	0.4298	0.6159	0.3675	0.5913	0.2205
	$R_{sh}$	71.4923	47.6686	74.5341	56.7082	63.6258	86.3956
	$R_s$	0.0358	0.0341	0.0335	0.0363	0.0338	0.0386
	$N_s$	1.6251	1.6456	1.6869	1.6271	1.6818	1.5714
14 °C	$I_{ph}$	0.7614	0.7616	0.7599	0.7612	0.7600	0.7593
	$I_o$	0.2515	0.7161	0.3361	0.2420	0.6739	0.3665
	$R_{sh}$	41.6191	51.4359	69.9134	43.8436	75.3860	86.3543
	$R_s$	0.0373	0.0321	0.0364	0.0372	0.0331	0.0362
	$N_s$	1.5530	1.6701	1.5832	1.5488	1.6621	1.5925
20 °C	$I_{ph}$	0.7606	0.7932	0.7601	0.7619	0.7597	0.7891
	$I_o$	0.5967	0.1751	0.4076	0.3919	0.5803	0.0258
	$R_{sh}$	88.1007	52.9407	71.8980	47.1064	76.4283	63.1466
	$R_s$	0.0342	0.1085	0.0355	0.0359	0.0338	0.0441
	$N_s$	1.6141	1.4983	1.5715	1.5676	1.6750	1.3252
26 °C	$I_{ph}$	0.7612	0.7606	0.7614	0.7602	0.7609	0.7616
	$I_o$	0.9563	0.6868	0.4235	0.3758	0.2962	0.3230
	$R_{sh}$	73.2298	54.5625	49.3822	63.3955	49.3590	43.1086
	$R_s$	0.0311	0.0319	0.0348	0.0357	0.0366	0.0360
	$N_s$	1.6369	1.5986	1.5445	1.5315	1.5069	1.5161
32 °C	$I_{ph}$	0.7604	0.7666	0.7594	0.7608	0.7604	0.7599
	$I_o$	0.6612	0.6197	0.3187	0.4553	0.9514	0.4231
	$R_{sh}$	84.9927	70.1958	77.8816	63.0971	58.9697	75.5183
	$R_s$	0.0335	0.0362	0.0370	0.0352	0.0313	0.0352
	$N_s$	1.5621	1.5508	1.4844	1.5215	1.6039	1.5136
38 °C	$I_{ph}$	0.7601	0.7664	0.7616	0.7612	0.7606	0.7612
	$I_o$	0.4384	0.8570	0.2876	0.2649	0.4398	0.9741
	$R_{sh}$	73.9126	85.6477	42.6690	44.4881	61.6287	77.4674
	$R_s$	0.0351	0.0426	0.0367	0.0367	0.0349	0.0313
	$N_s$	1.4881	1.5619	1.4461	1.4381	1.4886	1.5759

shows the estimated parameter values.

Since the metaheuristic algorithms were with contingency, it was necessary to prove the credibility of the results using repetitive experiments.

Table 13. RMSE of different algorithms under different temperatures.

		TEMP
		0 °C	8 °C	14 °C	20 °C	26 °C	32 °C	38 °C
RMSE	DA	4.53 × 10−2	1.02 × 10−1	5.90 × 10−2	1.26 × 10−1	5.79 × 10−2	6.52 × 10−2	7.63 × 10−2
	MFO	1.24 × 10−1	2.18 × 10−3	2.08 × 10−3	2.09 × 10−3	1.57 × 10−1	2.08 × 10−3	3.08 × 10−2
	WOA	5.64 × 10−3	2.93 × 10−2	3.18 × 10−3	4.88 × 10−2	1.99 × 10−2	1.26 × 10−3	4.26 × 10−3
	HHO	1.21 × 10−2	1.75 × 10−3	1.69 × 10−3	1.89 × 10−2	2.09 × 10−3	3.96 × 10−3	2.14 × 10−3
	GWO	3.59 × 10−3	1.04 × 10−2	3.49 × 10−3	7.76 × 10−3	2.77 × 10−3	3.52 × 10−3	5.49 × 10−3
	HIDA	1.31 × 10−3	1.01 × 10−3	1.46 × 10−3	1.45 × 10−3	2.10 × 10−3	1.51 × 10−3	3.11 × 10−3

shows the value of the evaluation standard for a single-diode model for 30 runs at 7 different temperatures. In estimation-type problems, RMSE is frequently employed as a statistic to assess the efficacy of numerical estimation. The lower its value, the higher the accuracy of the method used for photovoltaic cell parameter identification.

According to the test results at various temperatures, the HIDA leads to the lowest value of RMSE at 0 °C, 8 °C, 14 °C, and 20 °C when compared with other algorithms. When it reaches to 26 °C and 38 °C, HHO performs effectively. The WOA has a better result at 32 °C. As can be observed in Table 12 and Table 13, the parameter estimation accuracy of the HIDA is highest at slightly lower temperatures. When the temperature rises, the comparison algorithm shows an advantage; although, the accuracy of the HIDA still remains at a high level.

In addition, curves describing the current–voltage relationship and curves describing the power–voltage relationship are frequently used for direct viewing effects. The temperature was set to 20 °C. Based on the estimated parameters using the proposed HIDA method, characteristic curves are plotted in Figure 5 and Figure 6. In both figures, to facilitate differentiation, the dashed line shows the actual measured data values, and the red dots show the estimated obtained data.

As can be seen, the points estimated using the HIDA perfectly match the actual measured data. This implies that the proposed method is precise. The convergence curves of the comparison algorithms are given in Figure 7.

The comparison of the convergence curves shows that the HIDA outperforms other algorithms with a higher pace of convergence. More precisely, the HIDA converges after approximately 100 iterations, which is faster than most of the compared algorithms. About 560 iterations later, the HIDA obtains the minimal function value first. The comparison method that performs the best, i.e., the WOA, starts to converge early but progressively reaches the limit of the algorithm at 250 iterations and its search accuracy is inferior to that of the HIDA.

Investigating the numbers and the curves in the above tables and figures confirms that the HIDA is a rapid and precise method for estimating the parameters of the SDM.

4.2.2. Parameter Estimation of Photovoltaic Array Engineering Model

More frequently employed in actual engineering than the SDM is the engineering model for photovoltaic cells, which can be solved rapidly using only the technical parameters provided by the manufacture [65]. The engineering model was obtained by transforming the SDM, specifically by performing the following operations on Equation (4):

(1)

Set the photo-generated current $I_{ph}$ and the short-circuit current $I_{sc}$ to equal, because the series resistance $R_s$ is much smaller than the diode’s forward resistance

(2)

$\frac{U_L+R_s{I}_L}{R_{sh}}$ is ignored due to the high value of the parallel resistance $R_{sh}$ inside the PV cell, making $\frac{U_L+R_s{I}_L}{R_{sh}}$ much smaller than $I_{ph}$.

Thus, the I–V characteristic equation of the engineering model [66] is obtained after simplification as:

$$I_L=I_{sc}\times \left[1-C_1\times (e^{(\frac{U_L}{C_2{U}_{oc}})}-1)\right]$$

(27)

At the maximum power point, the output current $I_L$ is equal to the maximum power point current $I_m$ and the output voltage $U_L$ is equal to the maximum power point voltage $U_m$. In an open circuit, the output current value $I_L$ is 0 and the output voltage $U_L$ is equal to the open-circuit voltage $U_{oc}$. $C_1$ and $C_2$ are intermediate variables.

$$C_1=(1-\frac{I_m}{I_{sc}})e^{-\frac{U_m}{C_2{U}_{oc}}}$$

(28)

$$C_2=\frac{(\frac{U_m}{U_{oc}}-1)}{\ln (1-\frac{I_m}{I_{sc}})}$$

(29)

The characteristics of photovoltaic cells are affected mainly by irradiance and temperature; however, as it is not always possible to match the standard conditions provided by the manufacturer in real life, the specific conditions need to be adapted to the actual situation. The short-circuit current $I_{scn}$, open-circuit voltage $U_{ocn}$, maximum power point current $I_{mn}$, and maximum power point voltage $U_{mn}$ under arbitrary conditions are expressed as follows:

$$I_{scn}=I_{sc}\times \frac{S_n}{S}\times (1+a\Delta T)$$

(30)

$$U_{ocn}=U_{oc}\times (1-c\Delta T)\times \ln (e+b\Delta S)$$

(31)

$$I_{mn}=I_m\times \frac{S_n}{S}\times (1+a\Delta T)$$

(32)

$$U_{mn}=U_m\times (1-c\Delta T)\times \ln (e+b\Delta S)$$

(33)

where $\Delta T$ is the difference between the actual temperature and the standard temperature, calculated from $\Delta T=T_n-T$, and $\Delta S$ is the relative irradiance difference, calculated from $\Delta S=\frac{S_n}{S}-1$. The coefficients are set as $a=0.0025$, $b=0.5$, $c=0.00288$.

There are four parameters to be identified, namely $I_{sc}$, $I_m$, $U_{oc}$ and $U_m$; therefore, the objective function is transformed from Equations (6)–(34):

$$f(I_{sc},I_m,U_{sc},U_m)=I_L-I_{sc}\times \left[1-C_1\times (e^{(\frac{U_L}{C_2{U}_{oc}})}-1)\right]$$

(34)

The ranges of the parameters to be identified are given in

Table 14. Range of 4 parameters.

Parameter	LB	UB
$U_{oc}$(V)	30	40
$U_m$(V)	25	35
$I_{sc}$(A)	2	10
$I_m$(A)	2	8

.

Irradiance and temperature fluctuate depending on the day and weather, which can greatly affect the properties of PV cells. Therefore, in order to accurately test the effectiveness of the HIDA for parameter identification under the engineering model, 10 control groups with various irradiances and different temperatures were set up, the details of which are provided in

Table 15. Information in respect of 10 matched control groups.

Number of Test Group	$\mathit{S}$$\left({\mathbf{W}/\mathbf{m}}^2\right)$	$\mathit{T}(°C)$
No.1	1000	10
No.2	1000	15
No.3	1000	20
No.4	1000	25
No.5	1000	30
No.6	500	25
No.7	650	25
No.8	800	25
No.9	950	25
No.10	1100	25

.

The actual voltage and current data are shown in Table 10. The results of the parameter identification using the standard DA and HIDA for the set conditions and the root-mean-squared error values are given in

Table 16. Test results under different situations.

Number	Algorithm	${\mathit{U}}_{\mathit{o}\mathit{c}}$	${\mathit{U}}_{\mathit{m}}$	${\mathit{I}}_{\mathit{s}\mathit{c}}$	${\mathit{I}}_{\mathit{m}}$	RMSE
No.1	DA	30	29.2941	7.63161	5.12377	0
	HIDA	31.25	28.35	7.32	7.262	0
No.2	DA	37.4712	31.9661	2.03157	2.56828	3.14 × 10−4
	HIDA	31.25	24.4562	7.2944	7.5290	1.24 × 10−9
No.3	DA	37.6318	25	7.35477	8	6.30 × 10−3
	HIDA	31.3979	33.74	2	7.22	9.81 × 10−6
No.4	DA	30	29.2941	7.63161	5.12377	2.56 × 10−13
	HIDA	30.01	30.0653	2.0525	4.2920	0
No.5	DA	33.8981	35	2.27678	6.5647	2.40 × 10−8
	HIDA	31.25	28.8738	5.15286	5.58377	1.73 × 10−15
No.6	DA	38.8812	35	2	2.44428	2.84 × 10−7
	HIDA	39.7002	34.7647	2.17224	2.12754	2.33 × 10−14
No.7	DA	39.0699	30.7876	2.6315	2.88846	2.22 × 10−6
	HIDA	33.3171	32.2484	5.176	6.50024	1.16 × 10−14
No.8	DA	35.4175	31.5476	2	2.07339	3.12 × 10−4
	HIDA	34.8152	27.7991	3.6506	4.18334	1.62 × 10−13
No.9	DA	36.8098	30.0237	2.05152	2.84683	0.012921
	HIDA	36.4092	34.0105	3.38801	5.48152	0.019944
No.10	DA	38.9795	35	6.35218	5.13714	0.12586
	HIDA	36.1361	25.7918	4.67209	6.26081	6.4310−7

.

Observing the data in Table 16, when the irradiance constant is controlled, the RMSE of the HIDA changes from 1.24 × 10⁻⁹ to 9.81 × 10⁻⁶ for a 5 °C shift in temperature, taking the data obtained in the second and third groups as an example. If 70 °C is chosen as the maximum surface temperature, then the temperature rises by 7% and the RMSE changes by three orders of magnitude. Controlling the temperature constant, the RMSE of the HIDA changes from 1.16 × 10⁻¹⁴ to 1.62 × 10⁻¹³ with an increase in irradiance of 150, taking the data obtained in groups 7 and 8 as an example. If 1373 ${\mathrm{W}/\mathrm{m}}^2$ is chosen as the maximum solar irradiance, then the irradiance is increased by 10% and the RMSE is changed by one order of magnitude. As shown by these four sets of data, the engineering model is more sensitive to temperature in these test environments.

In addition, the RMSEs of both the DA and HIDA can reach 0 under the first test group, indicating that the parameters estimated by both the DA and HIDA have high accuracy under the test conditions in this set. A high-precision result of 0 RMSE is still attained by the HIDA at group No. 4, despite the change in experimental conditions, in contrast to the DA. In the other control group experiments, the results of the tests showed the HIDA to be an accurate and stable method of parameter identification, except in group No.9, where both DA and HIDA were less than satisfactory.

Since the accuracy of the DA identification results is not particularly low, the inaccuracies in fitting the I–V and P–V characteristic curves are difficult to observe; therefore, only the characteristic curves under standard conditions (S = 1000 ${\mathrm{W}/\mathrm{m}}^2$, T = 25 °C) are displayed. The blue dashed line in the graph shows the actual data, the green dots represent data fitted by the DA, and the red dots represent data fitted by the HIDA.

As can be seen in the curves in Figure 8, both the DA and HIDA are able to provide accurately fitted curves. Combining the data in Table 16 further confirms the accuracy of the HIDA.

4.2.3. Experiments with Classical Circuit Models

In addition to the SDM and engineering model, there are several classical equivalent circuit models, including the double-diode equivalent model (DDM) [67], the triple-diode equivalent model (TDM) [68], and the PV module model (MM) [69]. The DDM and TDM are both extended by adding diodes to the basic SDM. The MM consists of several solar cells connected in series or in parallel. In addition to being more sophisticated than the SDM, these circuit models are appropriate for applications requiring accuracy because they can mimic leakage currents, recombination current losses, and other phenomena that are not included in the SDM. As a result, they are also relatively challenging to solve.

There are seven parameters to be estimated in the DDM, namely, the photo-generated current $I_{ph}$, the diffusion current $I_{sd1}$, the saturation current $I_{sd2}$, the series resistance $R_s$, the parallel resistance $R_{sh}$, the diffusion diode ideality factor $n_1$, and the recombination diode ideality factor $n_2$.

There are nine parameters to be estimated for the TDM, namely, $I_{ph}$, the diffusion current $I_{sd3}$, the saturation current $I_{sd4}$, the third-diode saturation current $I_{sd5}$, $R_s$, $R_{sh}$, the diffusion-diode ideality factor $n_3$, the composite-diode ideality factor $n_4$, and the third-diode ideal factor $n_5$.

The parameters to be identified for the DDM and TDM are listed in

Table 17. Range of parameters for DDM and TDM.

Parameter	LB	UB
$I_{ph}$ ($\mathrm{A}$)	0	1
$I_{sdi},i\in 1:4$ ($\mathsf{\mu }\mathrm{A}$)	0	1
$R_s$ ($\mathrm{\Omega }$)	0	0.5
$R_{sh}$ ($\mathrm{\Omega }$)	0	100
$n_i,i\in 1:4$	1	2

.

The STM-40/36 model was selected for this study and required the estimation of five parameters $I_{ph}$, $I_{sd}$, $R_s$, $R_{sh}$, and the diode ideality factor $n$, the ranges of which are given in

Table 18. Range of parameters for STM-40/36.

Parameter	LB	UB
$I_{ph}$ ($\mathrm{A}$)	0	2
$I_{sd}$ ($\mathsf{\mu }\mathrm{A}$)	0	50
$R_s$ ($\mathrm{\Omega }$)	0	0.36
$R_{sh}$ ($\mathrm{\Omega }$)	0	1000
$n$	1	60

.

The actual voltages and actual currents in Table 10 were used for the DDM and TDM experiments, and the actual measured data used for the STM-40/36 PV module model [70] are given in

Table 19. Actual data for STM-40/36.

Voltage/V	Current/A	Voltage/V	Current/A
0.000	1.663	14.880	1.597
0.118	1.663	15.590	1.581
2.237	1.661	16.400	1.542
5.434	1.653	16.710	1.524
7.260	1.650	16.980	1.500
9.680	1.645	17.130	1.485
11.590	1.640	17.320	1.465
12.600	1.636	17.910	1.388
13.370	1.629	19.080	1.118
14.090	1.619	21.020	0.000

.

In order to broaden the scope of the testing, experiments were performed on the three models at respective temperatures of 5 °C, 15 °C, and 25 °C. The parameters obtained from the DA and HIDA estimates and the RMSE are recorded in

Table 20. Test results of DDM.

Algorithm	T = 5 °C
	${\mathit{I}}_{\mathit{p}\mathit{h}}$	${\mathit{I}}_{\mathit{s}\mathit{d}1}$	${\mathit{R}}_{\mathit{s}}$	${\mathit{R}}_{\mathit{s}\mathit{h}}$	${\mathit{n}}_1$	${\mathit{I}}_{\mathit{s}\mathit{d}2}$	${\mathit{n}}_2$	RMSE
DA	0.73651	3.92 × 10−7	0	43.884	1.7032	5.88 × 10−7	1.87176	0.046553
HIDA	0.762025	6.38 × 10−7	0.0301296	71.1489	2	8.43 × 10−7	1.76253	0.0031434
	T = 15 °C
DA	0.777943	7.53 × 10−7	0.0380212	63.4313	1.85498	1.00 × 10−6	1.7391	0.01737
HIDA	0.762496	7.85 × 10−7	0.0287002	54.9234	1.83636	7.33 × 10−7	1.71132	0.003856
	T = 25 °C
DA	0.79259	6.92 × 10−7	0.0639931	35.5565	1.59057	4.03 × 10−7	1.94292	0.088207
HIDA	0.761596	5.74 × 10−7	0.0336924	58.9493	1.58317	3.07 × 10−9	1.77333	0.0016405

,

Table 21. Test results of TDM.

Algorithm	T = 5 °C
	${\mathit{I}}_{\mathit{p}\mathit{h}}$	${\mathit{I}}_{\mathit{s}\mathit{d}3}$	${\mathit{R}}_{\mathit{s}}$	${\mathit{R}}_{\mathit{s}\mathit{h}}$	${\mathit{n}}_3$	${\mathit{I}}_{\mathit{s}\mathit{d}4}$	${\mathit{n}}_4$	${\mathit{I}}_{\mathit{s}\mathit{d}5}$	${\mathit{n}}_5$	RMSE
DA	0.73	8.79 × 10−7	0.025	68.18	1.93	3.42 × 10−7	1.72	1.11 × 10−7	1.72	0.023
HIDA	0.76	1.00 × 10−6	0.019	100	2	1.00 × 10−6	1.84	1.00 × 10−6	2	0.011
	T = 15 °C
DA	0.82	2.08 × 10−7	0.053	37.71	1.54	2.29 × 10−9	1.35	2.69 × 10−7	1.73	0.062
HIDA	0.76	2.83 × 10−7	0.033	71.14	1.65	2.09 × 10−7	1.93	2.13 × 10−7	1.60	0.002
	T = 25 °C
DA	0.46	5.03 × 10−7	0.001	45.81	1.96	5.19 × 10−7	1.66	5.38 × 10−7	1.94	0.246
HIDA	0.76	0	0.032	37.46	1.88	5.45 × 10−7	1.62	2.13 × 10−7	1.59	0.003

and

Table 22. Test results of STM-40/36.

Algorithm	T = 5 °C
	${\mathit{I}}_{\mathit{p}\mathit{h}}$	${\mathit{I}}_{\mathit{s}\mathit{d}}$	${\mathit{R}}_{\mathit{s}}$	${\mathit{R}}_{\mathit{s}\mathit{h}}$	$\mathit{n}$	RMSE
DA	1.725472	3.94 × 10−5	0.01927654	657.2033	2.31366	0.10595
HIDA	1.6737	8.91 × 10−7	0.005223	9.5262	1.6904	0.0046089
	T = 15 °C
DA	1.662472	7.80 × 10−6	0	750.9523	1.908739	0.021485
HIDA	1.668033	1.93 × 10−5	0	260.4144	2.071242	0.016495
	T = 25 °C
DA	1.84453	3.87 × 10−5	0.0108882	0.989867	15.3303	0.31465
HIDA	1.68633	4.37 × 10−5	0	61.0625	2.15905	0.03228

.

Among these three models, the HIDA has the highest accuracy of parameter estimation in DDM, with a root-mean-squared error of only 0.001 at 25 °C and RMSEs of 0.003 at other temperatures. The estimation of the parameters of the PV module STM-40/36 was more difficult and the accuracy of the algorithm solution decreased,; however, the RMSE obtained by the DA at 25 °C was 0.31, while the RMSE obtained by the HIDA was 0.03. It is clear from the three tables above that the RMSEs generated by the HIDA are smaller than those obtained by the DA for all three temperatures tested, whether in respect of the DDM, TDM, or STM-40/36, indicating that the HIDA is more accurate in parameter estimation and exhibits a good performance in all of these models.

For a visual comparison, the I–V and P–V curves obtained from the tests conducted on the three models are presented in Figure 9 and Figure 10, and Figure 11, respectively.

In Figure 9, the curves estimated by the HIDA fit the actual curves accurately, whether at 5 °C, 15 °C, or 25 °C, while the I–V and P–V curves estimated by the DA exhibit considerable inaccuracy, which is more pronounced at 25 °C and less so at 15 °C.

While the curves acquired by the DA could nearly imitate the fundamental shape, they did so with sizable errors that were greater than those in the DDM, and the HIDA was still able to obtain well-fitting characteristic curves in the TDM. This was a result of the TDM model being more intricate and challenging to solve than the DDM model.

The characteristic curves generated by the DA in the test results for the PV modules change significantly as the temperature rises. There is a significant error at 5 °C and a better fit at 15 °C; however, the inaccuracy is severe at 25 °C. The curve generated by the DA in the STM-40/36 model is substantially deformed at 25 °C, in contrast to the severe displacement at this temperature in the TDM. The HIDA also shows a small observable inaccuracy at 25 °C, although the I–V and P–V curves obtained have basically the same form and location.

There are two articles [71,72] applying the improved SSA and ARSO to the parameter identification of the SDM, DDM, and Photowatt-PWP201 modules. Using the DDM as an example, the RMSEs of the improved methods in these two publications can reach 9.83 × 10⁻⁴ and 9.82 × 10⁻⁴, respectively, while the improved method in this paper can reach 3.20 × 10⁻³ and the standard DA reaches 4.37 × 10⁻³. This might be a result of the algorithm’s internal logic, but combined with the test results of the benchmark functions in high dimensions, the test results of the CEC functions, and the parameter estimation results of the STM-40/36 module, the HIDA shows a significant improvement in performance compared to the DA; therefore, the HIDA is still a valid improvement method.

In conclusion, the standard DA’s ability to identify PV cell parameters was less stable and more sensitive to temperature than the HIDA. The HIDA was a reliable and efficient approach for identifying the parameters of PV cells. In contrast, the improved HIDA proposed in this paper showed a good performance in parameter estimations for the DDM, TDM, and STM-40/36 with high accuracy and high stability; therefore, the HIDA was an efficient and reliable method for PV cell parameter identification.

4.2.4. Classical Engineering Application Experiment

There are seven well-known engineering problems that are often used to verify the practicality of an algorithm, namely, the pressure vessel design problem, the gear design problem, the cantilever beam design problem [73], the three-bar truss design problem, the gas transmission compressor design problem, the car side design problem, and the piston rod optimization problem [74]. To further validate the improved effectiveness of the HIDA, it was applied to these seven engineering problems with the standard DA. The experimental results are recorded in

Table 23. Comparison of improvement effects under 7 classic projects.

Engineering Problem	Algorithm	Mean	Std
Pressure vessel design	DA	1.0069 × 106	6.6822 × 105
	HIDA	8.8013 × 105	1.6648 × 106
Gear train design	DA	1.5273 × 10−3	3.7362 × 10−3
	HIDA	2.7591 × 10−4	8.1504 × 10−4
Cantilever beam design	DA	6.5916	1.6363
	HIDA	1.5530	3.8314 × 10−2
Three bar truss design	DA	2.7106 × 102	7.0500
	HIDA	2.6976 × 102	2.9407
Gas transmission compressor design	DA	7.1438 × 107	1.8868 × 108
	HIDA	1.2387 × 106	1.6696 × 104
Car side design	DA	2.6640 × 101	1.3003
	HIDA	2.6629 × 101	1.0763
Piston rod optimization	DA	1.3009 × 102	2.6634 × 102
	HIDA	4.3767 × 101	2.4095 × 101

.

The experimental results of these seven traditional engineering applications show that the mean values obtained by the HIDA are all lower than those of the DA, demonstrating that the HIDA is more effective at discovering the optimal solutions than the DA. This shows that the practicality of the HIDA has been effectively improved compared to the DA. Combining these results with the experimental results of the PV cell parameter estimation further proves that the method proposed in this paper is a practical one.

5. Conclusions

In this research, the improved HIDA was proposed, using hybrid strategies for accurate estimations of the parameters of a solar cell model.

The HIDA exploited the characteristics of tent chaotic mapping to spread the initial population evenly around the search area. This prevented the algorithm from converging too soon due to random distribution. The inertial weight was optimized to ensure a smooth transition between the two main phases of the algorithm. The enrichment of individual dragonfly position updating methods in the hybrid strategy allowed the accuracy of the HIDA to be increased. The reference to the spiral predation mechanism of the WOA enhanced the merit-seeking accuracy of the DA. The adaptive probability threshold was used to ensure the spiral predation mechanism led the HIDA to exploit the vicinity of viable solutions. Adding the Cauchy perturbation to the elite position facilitated the possibility of the HIDA finding the correct global optimal solution.

In the function test experiments containing 12 benchmark functions and 12 CEC functions, the convergence curves and rank sum test results show that the HIDA outperforms the comparison algorithm in terms of the accuracy of the search and convergence speed, and the results of comparisons with two other improved dragonfly algorithms show that the HIDA is a competitive method. The HIDA identified parameters for five models in the PV cell parameter identification experiments: SDM, engineering model, DDM, TDM, and STM-40/36. The RMSE and the fitted PV characteristic curves show that the identification accuracy of the HIDA is greatly improved compared to the DA. The vast range of applications of the HIDA was further validated by experiments on seven traditional engineering challenges. In general, the HIDA was able to perform PV cell parameter identification with high stability and optimization-seeking accuracy. The HIDA proposed in this paper not only provided a reference idea for improving the local development capability of the algorithm, but also provided a simple and effective method for PV cell parameter identification, enriching the swarm intelligence algorithms that could be used for PV cell parameter estimation.

In the future, the following extended research can be performed. First, a dragonfly algorithm with a dynamically changing effective radius can be investigated so as to dynamically adjust the range of neighborhood solutions, foraging behavior, and enemy avoidance behavior. Alternatively, dragonfly algorithms that adaptively divide subpopulations can be studied to design different position-updating methods for different subpopulations. In addition, due to the poor generality of the existing swarm intelligence algorithms, efforts can be devoted to researching and designing a hyper-heuristic dragonfly algorithm, which can be used as a high-level heuristic strategy. Second, the HIDA was designed as an optimization method for single optimization objective problems and is not yet capable of solving multi-objective optimization problems; the performance of the HIDA in dealing with multiple objective problems will be tested in the future. Third, this study verified the feasibility and effectiveness of the HIDA to solve the parameter estimation of PV cells, and there are more application scenarios that can be further investigated, such as path planning, resource scheduling, and image segmentation. Fourth, more advanced algorithms can be utilized to solve the PV cell parameter estimation problem, such as the diffused memetic optimizer, hyper-heuristic ant colony optimization, the adaptive polyploid memetic algorithm, the multi-objective hybrid metaheuristic solution algorithm, the novel-scenario-based genetic algorithm, the self-adaptive fast fireworks algorithm, and others [75,76,77,78,79,80].