1 Introduction

Modeling solar PV cells accurately plays an important role in the solar PV energy system, and it has received a great deal of attention in recent years. Two steps are involved in solar cell modelling, namely formulation of mathematical expressions and estimation of cell parameters. Several researchers have reported on a mathematical model derived from the data (Premkumar et al. 2020; Jordehi 2016). The PV cell is typically represented as a single diode (SD) (Alam et al. 2015), a double diode (DD) (Ram et al. 2017), or a three diode (TD) model (Allam et al. 2016). A PV panel's parameters, such as saturation current (Id), series resistance (Rse), shunt resistance (Rsh), ideality factor (a), and photocurrent (Ip), need to be estimated for any selected model. Models such as SD, DD, and TD have five, seven, and nine estimated parameters, respectively. In order to achieve optimal outputs equal to those obtained from the physical solar cells, it is imperative to find the optimal values for the above-mentioned parameters for the chosen PV model (Khanna et al. 2015; Mathew et al. 2017).

Solar cells' parameters have been estimated using optimization algorithms in recent years. The deterministic techniques include Lambert W-functions, least squares, and curve fitting. Because of the parameters' differentiability and convexity, deterministic methods have several restrictions. Since these techniques are highly sensitive to the initial solution, they tend to arrive at a local minimum rather than the global optimum (Ortiz-Conde et al. 2006; AlRashidi et al. 2011). A heuristic method is introduced as an alternative to the deterministic method, and it has been proven that heuristics methods can produce more accurate and robust results. There are a variety of heuristic methods that are based on population data and are derived from nature. In most engineering applications, heuristics resolve problems such as differentiability and convexity by excluding the problems such as differentiability and convexity. Heuristic methods are used to estimate solar PV cell parameters by considering various advantages. In recent years, many heuristic methods have been developed, like particle swarm optimization (PSO) (Amokrane and Haddadi 2017), genetic algorithms (Bastidas-Rodriguez et al. 2017), teaching–learning optimization (TLO) (Ramadan et al. 2020), cuckoo search (CS) (Kang et al. 2018), artificial bee colonies (ABCs) (Jamadi et al. 2016), Rao-1 (Wang et al. 2020), and Jaya algorithm (Luu and Nguyen 2020), many researchers have reported this. The literature survey on parameter estimation of different solar diode are discussed in Table 1 below:

Table 1 Literature survey
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In spite of their effectiveness and speed, heuristic algorithms have few limitations when compared to traditional techniques. Due to the exclusive searching mechanism of PSO and GA, these methods primarily focus on local minima, resulting in a high probability of premature convergence for multi-modal systems. CS and ABC perform better at the exploration stage, but they are very slow at achieving convergence. Therefore, when applied to multi-objective functions, these methods perform the worst. As the objective function is extracted from raw data, it's noisy, and most heuristic approaches don't perform as well as they could. To solve the solar PV cell parameter estimation problem, the researchers had to develop a heuristic method that balances local and global search capability (Ma et al. 2016).

It is clear from the literature survey that PV cell efficiency has uttermost importance so adding a series resistance to one diode as well as the more accurate algorithm increase the efficiency. The swarm based algorithm tends to struck in local minima so the hybridization is one of the best method to eliminate this problem which in turn increases the efficiency. This paper aims to use a new parameter estimation algorithm for three and four diode model of solar PV cell. Its major contributions are as follows:

  • The hybrid Particle Swarm Optimization and Rat Search Algorithm (PSORSA) algorithm is justified through five benchmark CEC2019 test functions, and the average and Standard Deviation (SD) are calculated for each of them.

  • Solar PV cell parameters are estimated at standard temperature condition and Root Mean Square Error (RMSE) is compared with other standalone algorithms.

  • The non-parametric test is performed i.e., Friedman Ranking Test and Wilcoxon’s Rank Sum Test.

2 Solar PV cell model mathematical modelling and problem formulation

The PV cell modeling process involves two main steps: First, the mathematical model is formulated, followed by parameter estimation. PV models such as SD, and DD are commonly reported by researchers.

2.1 Solar PV cell mathematical modelling

2.1.1 Equivalent circuit of three diode model of a PV cell

Three diode models are further improved from double diode models. As shown in Fig. 1, three diodes operate in the three-diode equivalent circuit model of solar PV cells. Leakage current and grain boundaries, denoted by Idc3, are taken into account in this model. Flow of leakage current occurs through the shunt resistance of a three-diode equivalent circuit model (Singla and Nijhawan 2021). Semi-conductors to substrate resistance represents the series resistance of the fundamental region of solar PV cells. Equation 1 illustrates the modelling of three diodes.

$$ {\text{I}}_{{\text{O}}} = {\text{I}}_{{{\text{ph}}}} - {\text{I}}_{{{\text{rsd}}1}} \left\lfloor {{\text{exp}}\left( {\frac{{{\text{q}}({\text{V}}_{{\text{O}}} + {\text{I}}_{{\text{O}}} {\text{R}}_{{{\text{se}}}} }}{{{\text{n}}_{1} {\text{KT}}}}} \right) - 1} \right\rfloor - {\text{I}}_{{{\text{rsd}}2}} \left\lfloor {{\text{exp}}\left( {\frac{{{\text{q}}({\text{V}}_{{\text{O}}} + {\text{I}}_{{\text{O}}} {\text{R}}_{{{\text{se}}}} }}{{{\text{n}}_{2} {\text{KT}}}}} \right) - 1} \right\rfloor - {\text{I}}_{{{\text{rsd}}3}} \left\lfloor {{\text{exp}}\left( {\frac{{{\text{q}}({\text{V}}_{{\text{O}}} + {\text{I}}_{{\text{O}}} {\text{R}}_{{{\text{se}}}} }}{{{\text{n}}_{3} {\text{KT}}}}} \right) - 1} \right\rfloor { } - \frac{{{\text{V}}_{{\text{O}}} + {\text{I}}_{{\text{O}}} {\text{R}}_{{{\text{se}}}} }}{{{\text{R}}_{{{\text{sh}}}} }} $$
(1)
Fig. 1
figure 1

Three diode equivalent circuit model

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The curve fitting accuracy of a three-diode solar PV cell is high, and it is possible to determine the different components of the solar PV cell's current. However, its modeling is extremely complex. The I–V characteristics of silicon solar cells of large area are simulated using this model.

2.2 Equivalent circuit of four diode model of a PV cell

Compared with single, double, and triple diode equivalent circuit models, the four diode equivalent circuit model has numerous advantages, such as being more accurate when analyzing industrial solar PV cells, having the least amount of error between experimental and calculated data, being more accurate when fitting curves and performing well in STC. However, this model has the disadvantage of having a high level of complexity. In case of big industrial applications with size over 155.2 cm2 and 17.1% efficiency, Idc1 and Idc2 aren't good enough for representing different parameters of solar PV cells. A four-diode equivalent circuit is shown in Fig. 2.

Fig. 2
figure 2

Schematic equivalent circuit model of four diode PV cell

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Modelling of four- diode model is represented in the Eq. 2:

$$ {\text{I}}_{{\text{O}}} = {\text{I}}_{{{\text{ph}}}} - {\text{I}}_{{{\text{rsd}}1}} \left\lfloor {{\text{exp}}\left( {\frac{{{\text{q}}\left( {{\text{V}}_{{\text{O}}} + {\text{I}}_{{\text{O}}} {\text{R}}_{{{\text{se}}}} } \right)}}{{{\text{n}}_{1} {\text{KT}}}}} \right) - 1} \right\rfloor - {\text{I}}_{{{\text{rsd}}2}} \left\lfloor {{\text{exp}}\left( {\frac{{{\text{q}}\left( {{\text{V}}_{{\text{O}}} + {\text{I}}_{{\text{O}}} {\text{R}}_{{{\text{se}}}} } \right)}}{{{\text{n}}_{2} {\text{KT}}}}} \right) - 1{ }} \right\rfloor - {\text{I}}_{{{\text{rsd}}3}} \left\lfloor {{\text{exp}}\left( {\frac{{{\text{q}}\left( {{\text{V}}_{{\text{O}}} + {\text{I}}_{{\text{O}}} {\text{R}}_{{{\text{se}}}} } \right)}}{{{\text{n}}_{3} {\text{KT}}}}} \right) - 1} \right\rfloor - {\text{I}}_{{{\text{rsd}}4}} \left\lfloor {{\text{exp}}\left( {\frac{{{\text{q}}\left( {{\text{V}}_{{\text{O}}} + {\text{I}}_{{\text{O}}} {\text{R}}_{{{\text{se}}}} } \right)}}{{{\text{n}}_{4} {\text{KT}}}}} \right) - 1} \right\rfloor - \frac{{{\text{V}}_{{\text{O}}} + {\text{I}}_{{\text{O}}} {\text{R}}_{{{\text{se}}}} }}{{{\text{R}}_{{{\text{sh}}}} }} $$
(2)

where Io is the output current, Vo is the output voltage, Iph is the photocurrent. Irsd1, Irsd2, Irsd3, Irsd4 are the reverse saturation currents of the four diodes, q is the absolute value of current. n1, n2, n3, n4 is the ideal factors, K is the Boltzmann’s constant, and T is the absolute temperature of P–N junction (in Kelvin).

2.3 Problem formulation

An optimization technique helps identify unknown parameters and experimental I–V data for the real system. Based on the four-diode model of solar PV cells, vector defines the solution of the optimization algorithm where \(x = \left[ {R_{se } R_{sh} I_{ph} I_{rsd1} I_{rsd2} I_{rsd3} I_{rsd4} n_{1} n_{2} n_{3} n_{4} } \right]\). In order to minimize the error between measured currents and calculated currents, parameters of solar PV cells are estimated. In homogeneous form, Eqs. 1 and 2 can be rewritten as Eqs. 3 and 4 to define the objective function, and for the experimental data, the value of can be calculated.

$$ {\text{f}}\left( {{\text{V}}_{{\text{O}}} ,{\text{ I}}_{{\text{O}}} ,{\text{x}}} \right) = {\text{I}}_{{\text{O}}} - {\text{I}}_{{{\text{ph}}}} + {\text{I}}_{{{\text{rsd}}1}} \left\lfloor {{\text{exp}}\left( {\frac{{{\text{q}}\left( {{\text{V}}_{{\text{O}}} + {\text{I}}_{{\text{O}}} {\text{R}}_{{{\text{se}}}} } \right)}}{{{\text{n}}_{1} {\text{KT}}}}} \right) - 1} \right\rfloor + {\text{I}}_{{{\text{rsd}}2}} \left\lfloor {{\text{exp}}\left( {\frac{{{\text{q}}\left( {{\text{V}}_{{\text{O}}} + {\text{I}}_{{\text{O}}} {\text{R}}_{{{\text{se}}}} } \right)}}{{{\text{n}}_{2} {\text{KT}}}}} \right) - 1} \right\rfloor + {\text{I}}_{{{\text{rsd}}3}} \left\lfloor {{\text{exp}}\left( {\frac{{{\text{q}}\left( {{\text{V}}_{{\text{O}}} + {\text{I}}_{{\text{O}}} {\text{R}}_{{{\text{se}}}} } \right)}}{{{\text{n}}_{3} {\text{KT}}}}} \right) - 1} \right\rfloor - \frac{{{\text{V}}_{{\text{O}}} + {\text{I}}_{{\text{O}}} {\text{R}}_{{{\text{se}}}} }}{{{\text{R}}_{{{\text{sh}}}} }} $$
(3)
$$ {\text{f}}\left( {{\text{V}}_{{\text{O}}} ,{\text{ I}}_{{\text{O}}} ,{\text{x}}} \right) = {\text{I}}_{{\text{O}}} - {\text{I}}_{{{\text{ph}}}} + {\text{I}}_{{{\text{rsd}}1}} \left\lfloor {{\text{exp}}\left( {\frac{{{\text{q}}\left( {{\text{V}}_{{\text{O}}} + {\text{I}}_{{\text{O}}} {\text{R}}_{{{\text{se}}}} } \right)}}{{{\text{n}}_{1} {\text{KT}}}}} \right) - 1} \right\rfloor + {\text{I}}_{{{\text{rsd}}2}} \left\lfloor {{\text{exp}}\left( {\frac{{{\text{q}}\left( {{\text{V}}_{{\text{O}}} + {\text{I}}_{{\text{O}}} {\text{R}}_{{{\text{se}}}} } \right)}}{{{\text{n}}_{2} {\text{KT}}}}} \right) - 1} \right\rfloor + {\text{I}}_{{{\text{rsd}}3}} \left\lfloor {{\text{exp}}\left( {\frac{{{\text{q}}\left( {{\text{V}}_{{\text{O}}} + {\text{I}}_{{\text{O}}} {\text{R}}_{{{\text{se}}}} } \right)}}{{{\text{n}}_{3} {\text{KT}}}}} \right) - 1} \right\rfloor + {\text{I}}_{{{\text{rsd}}4}} \left\lfloor {{\text{exp}}\left( {\frac{{{\text{q}}\left( {{\text{V}}_{{\text{O}}} + {\text{I}}_{{\text{O}}} {\text{R}}_{{{\text{se}}}} } \right)}}{{{\text{n}}_{4} {\text{KT}}}}} \right) - 1} \right\rfloor - \frac{{{\text{V}}_{{\text{O}}} + {\text{I}}_{{\text{O}}} {\text{R}}_{{{\text{se}}}} }}{{{\text{R}}_{{{\text{sh}}}} }} $$
(4)

For evaluating the difference between measured and calculated currents, Root Mean Square Error (RMSE) is used. RMSE can be calculated using Eq. 5.

$$\text{RMSE}=\sqrt{\frac{1}{\text{N}}{\sum }_{\text{i}=1}^{\text{N}}\left({\text{f}}_{\text{i}}\left({\text{V}}_{\text{O}},{\text{I}}_{\text{O}},\text{x}\right)\right)}$$
(5)

This equation is based on an N number of measured data and an x number of solution vectors. To reduce the value of RMSE, parameter estimation of solar PV cells is therefore the key objective.

3 Algorithm

A comparison of PSO (Yousri et al. 2019), RSA (Dhiman et al. 2020), SCA (Ali et al. 2016), GWO (Mirjalili et al. 2014), and CSA (Yang and Deb 2009) based approaches demonstrates the superiority of the PSORSA hybrid algorithm. The parameters of polycrystalline PV cells were optimized using three diode and four diode equivalent models. This section briefly discusses hybrid PSORSA.

3.1 Rat search algorithm (RSA)

Various sizes and weights of rats belonging to two species are studied in this study, which provides insights into their behavior as intelligent and social rodents. Among the activities they engage in within their territorial communities are grooming, tumbling, hopping, boxing, and chasing. It is not uncommon for rats to exhibit violent behavior during competitions for prey, resulting in fatalities. Mathematically modeling the aggressive behavior of rats during chases and battles is the focus of this research (Dhiman et al. 2020).

Mathematical modeling:

Chasing:

Animals such as rats engage in social agonistic behaviors to pursue prey in groups. For math modeling, we assume the most effective search agent knows where the prey is, and other search agents adjust their positions based on the best agent. In order to explain this mechanism, Eq. (6) is introduced.

$$\overrightarrow{Q}=B.\overrightarrow{{Q}_{j}}\left(z\right)+D.\left(\overrightarrow{{Q}_{s}}\left(z\right)-\overrightarrow{{Q}_{j}}\left(z\right)\right)$$
(6)

The rat's position is denoted by \(\overrightarrow{{Q}_{j}}\left(z\right)\), while the best optimal solution is denoted by\(\overrightarrow{{Q}_{s}}\left(z\right)\). The values of B and D are specified by Eqs. (7–8).

$$B=S-z\times \left(\frac{S}{{Max}_{iter}}\right)\text{Where},\text{ z}=\text{0,1},2,{Max}_{iter}.$$
(7)
$$ {\text{D }} = { 2}.{\text{ rand }}() $$
(8)

A parameter S represents a random number, while a parameter D represents a random number, with values [0, 2] and (Premkumar et al. 2020; Allam et al. 2016), respectively. In the iterative process, B and D are responsible for optimizing the performance of these parameters during exploration and exploitation.

Fighting:

Rats engage in battles with their prey using Eq. (9) as a mathematical representation.

$$\overrightarrow{{Q}_{j}}\left(z+1\right)=\left|\overrightarrow{{Q}_{s}}\left(z\right)-\overrightarrow{Q}\right|$$
(9)

The rat search algorithm updates the next position of the rat by utilizing equation \(\overrightarrow{{Q}_{j}}\left(z+1\right)\). As a result, the optimal solution is maintained and the locations of other search agents are adjusted in relation to it. In order to facilitate exploration and exploitation, the parameters B and D have been adjusted. RSA proposes a solution that can be obtained with a minimum amount of operators. The pseudo-code of the rat search algorithm is presented in Algorithm 1, and the flow chart of that algorithm is shown in Fig. 3.

Fig. 3
figure 3

Flow chart of RSA

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figure a

Algorithm 1 Rat search algorithm pseudo-code.

3.2 Hybrid particle swarm optimization and rat search algorithm (PSORSA).

We propose a hybrid approach to combine two algorithms for increased efficiency. Our approach enhances the precision of results achieved by RSA by leveraging PSO, which is a precursor algorithm to the class of optimization algorithms known as swarm algorithms. The exploration part is done by the PSO and exploitation section is conducted by the RSA. The swarm based algorithm has tehe tendency to struck in local minima. As a result of our hybrid method, the system is not trapped in local minimums and is also enhanced in accuracy. This enhances the speed of the system and ensures the system finds the global optimum within reasonable time. The combination of RSA and PSO thus provides a powerful hybrid approach to optimize the system. The hybrid method is also extremely cost effective, since it does not require a large amount of computational power to achieve the desired result. The combination of RSA and PSO thus provides a powerful and cost-effective solution to optimize the system. The flow chart of that algorithm is shown in Fig. 4. The pseudo-code of the hybrid algorithm is presented in Algorithm 2.

Fig. 4
figure 4

Flow chart of PSORSA

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figure b

Algorithm 2 Hybrid algorithm pseudo-code.

4 Results and discussion

4.1 Benchmark test functions

To validate the effectiveness of the new algorithm, five benchmark test functions are selected from CEC 2019. The dimensions of these features are 30 and 50 as shown in Table 2, with f1 to f5 being CEC functions, as indicated in Table 2. A comparison of PSO, CSA, SCA, RSA, GWO, and PSORSA metaheuristic algorithms is carried out in this paper to evaluate the performance of the proposed metaheuristic algorithms. To make an accurate comparison between 5 benchmark test functions and the other algorithms that are being compared, a limit of 1000 feature evaluations per function is set to make a fair comparison. MATLAB 2018b was used to program the codes, and each algorithm was run 40 times independently.

Table 2 CEC 2019 benchmark test function
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Tables 3 and 4 demonstrate the average and standard deviations with dimensions of 30 and 50 for five benchmark tests achieved by the algorithms, respectively. The findings from Tables 3 and 4 show that the suggested approach is more efficient than the others. According to five benchmark test functions, the proposed algorithm offers lower mean and standard deviation values than the rest. This benchmark test function, when applied to the proposed hybrid algorithm, can be used to infer a conclusion that it is superior to all the other algorithms and offers a greater convergence rate, robustness, precision, and performance compared to any of the other algorithms. As a consequence, the hybrid algorithm yields better results than any of the other algorithms, so it should be used for optimization purposes. This algorithm can be used to solve a wide range of complex problems efficiently and reliably.

Table 3 CEC 2019 Benchmark Statistical Test with Dimension (30)
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Table 4 CEC 2019 Benchmark Statistical Test with Dimension (50)
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4.2 Engineer problem (parameter extraction of solar cells)

The proposed algorithm addresses the issue of parameter extraction for two distinct solar PV models to facilitate a more detailed analysis of their performance in this section. The parameter setting of the all the algorithm in Table 5. Two solar panels from two different manufacturers, namely, Nemy and Solar World, are being compared in terms of their specifications. The first panel is the JP270M60 model from Nemy, which is a monocrystalline cell panel and has a size rating of 270W. It has a Vm (voltage at maximum power) of 31.10V, Im (current at maximum power) of 8.68A, Voc (open circuit voltage) of 38.60V, and Isc (short circuit current) of 9.20A. The panel consists of 60 cells and is rated for a temperature of 25 °C. Secondly, we have the SW80RNA model from Solar World, which uses poly-crystallized cells to provide power to the panel. It has a Vm of 17.90V, Im of 4.49A, Voc of 21.90V, and Isc of 4.78A. Like the Nemy panel, it also consists of 60 cells and is rated for a temperature of 25 °C. A model comprising three and four diodes has been derived from these two solar panels to extract their parameters. The first parameter is Ipv, which has a lower bound of 0 and an upper bound of 1 (measured in Amperes). The next four parameters, namely Irsd1, Irsd2, Irsd3, and Irsd4, are measured in microamperes (µA) and have a lower bound of 0 and an upper bound of 1. The parameter Rse, measured in Ohms (Ω), has a lower bound of 0 and an upper bound of 0.5. The parameter Rsh, also measured in Ohms (Ω), has a lower bound of 0 and an upper bound of 100. The last set of parameters, n1, n2, n3, and n4, are unit-less and have a lower bound of 1 and an upper bound of 2. As part of this section, two different solar panel types are discussed. The first type is mono-crystalline and the second type is poly-crystalline.

Table 5 Parameter setting of all algorithm
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Case 1: mono-crystalline solar panel: In this case, the solar panel will be considered to be Nemy, which is a mono-crystalline panel. The parameter extraction for a three-diode and a four-diode model is discussed in this article. Based on the error (RMSE) in Tables 6 and 7, unknown parameters are displayed for both solar models. A graph representing the root mean square error of the two models can be found in Fig. 5. Based on these two tables and figures, it is clear that the hybrid algorithm proposed is much better than the algorithm in this study. As far as convergence time, reliability, memory requirements, etc., the hybrid algorithm is superior to the standalone algorithm in terms of its performance. It was found that the hybrid algorithm is more effective than the other standalone algorithms after the Friedman ranking test (Yu et al. 2022; Chauhan and Prakash 2023; Qaraad et al. 2023; Ma et al. 2016; Singla and Nijhawan 2021) Table 8 and Wilcoxon's rank sum test (Yousri et al. 2019; Dhiman et al. 2020; Ali et al. 2016; Mirjalili et al. 2014; Yang and Deb 2009) Table 9 were performed. Based on these tests as well, it can also be concluded that the hybrid algorithm is far better than the rest of the standalone algorithms.

Table 6 Unknown parameters of three diode model
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Table 7 Unknown Parameters of Four Diode Model
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Fig. 5
figure 5

RMSE of both models

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Table 8 Friedman Ranking Test of both the Models
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Table 9 Wilcoxon’s Rank Sum Test of both the Models
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Case 2: poly-crystalline solar panel: Specifically, in this case, we are talking about a poly-crystalline solar panel that is considered as solar world. An analysis of the model parameters for three diodes and four diodes is presented here. The unknown parameters of the solar models are shown in Tables 10 and 11 with their corresponding error (RMSE). The Fig. 6 shows the RMSE error which has been calculated for both of the models. Moreover, both the table as well as the figure states that the proposed hybrid algorithm outperforms the compared algorithm by a wide margin. When it comes to convergence time, reliability and memory usage, the hybrid algorithm outperforms the standalone algorithm by a long shot when compared to the standalone algorithm. The following Friedman ranking test is conducted after the extraction of the two models according to Table 12 and Wilcoxon's rank sum test is performed according to Table 13 and from this test as well it is concluded that the proposed hybrid algorithm has a much higher ranking than the remainder of the standalone algorithms used in the comparison.

Table 10 Unknown Parameters of Three Diode Model
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Table 11 Unknown Parameters of Four Diode Model
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Fig. 6
figure 6

RMSE of both models

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Table 12 Friedman Ranking Test of both the Models
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Table 13 Wilcoxon’s Rank Sum Test of both the Models
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5 Conclusion

A new hybrid algorithm, called PSORSA, is proposed in this work to tackle global optimization issues and extract solar cell parameters under varying temperatures. This algorithm combines two meta-heuristic algorithms, PSO and RSA, to balance exploration and exploitation. Additionally, it incorporates the opposition-based learning approach to increase demographic diversity. The study focuses on two solar PV cell models, Nemy-JP270M60 and Solar World-SW80RNA, and uses a three and four diode mathematical equivalent model of PV cell. The results obtained from this study are summarized as follows:

  • PSORSA outperforms other algorithms in terms of precise solution and convergence speed of global optimization problems.

  • In terms of solution consistency, Friedman ranking test, and Wilcoxon’s rank-sum test results, PSORSA demonstrates better performance in terms of equivalent efficiency on both solar PV cell models than other compared techniques.

  • Statistical findings indicate that PSORSA is more effective in parameter extraction for both PV models.

The study concludes that PSORSA is a promising and effective method for extracting solar PV cell parameters. Furthermore, PSORSA could be applied to other energy optimization issues such as optimal distributed generation configuration, economic load dispatch, and energy scheduling problem of power system arena to obtain more promising results.

6 Future scope

The proposed algorithm can be tested on the other engineering applications. The comprehensive discussion presented here underscores the transformative impact of advanced optimization algorithms across diverse domains. The proposed algorithms not only offer effective solutions to specific decision problems but also open new frontiers for their application in addressing complex challenges in the ever-evolving landscape of decision-making. Researchers and practitioners alike are encouraged to explore and harness the full potential of these algorithms to pave the way for innovative and efficient solutions in their respective fields. In the future research, the proposed approach could be actually compared to more advanced optimization algorithms (Chen and Tan 2023; Dulebenets 2021, 2023; Singh and Pillay 2022).