
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12709-0
10.1016/j.heliyon.2024.e36678
e36678
Research Article
An improved nutcracker optimization algorithm for discrete and continuous optimization problems: Design, comprehensive analysis, and engineering applications
Abdel-Basset Mohamed mohamedbasset@ieee.org
a
Mohamed Reda redamoh@zu.edu.eg
a
Hezam Ibrahim M. ialmishnanah@ksu.edu.sa
b
Sallam Karam M. KSallam@sharjah.ac.ae
cd
Hameed Ibrahim A. ibib@ntnu.no
e∗
a Faculty of Computers and Informatics, Zagazig University, Zagazig, Sharqiyah, 44519, Egypt
b Department of Statistics & Operations Research, College of Sciences, King Saud University, Riyadh, Saudi Arabia
c Department of Computer Science, College of Computing and Informatics, University of Sharjah, Sharjah, United Arab Emirates
d Faculty of Science and Technology, School of IT and Systems, University of Canberra, ACT, 2601, Australia
e Department of ICT and Natural Sciences, Norwegian University of Science and Technology (NTNU), Ålesund, Norway
∗ Corresponding author. ibib@ntnu.no
22 8 2024
15 9 2024
22 8 2024
10 17 e3667810 9 2023
1 8 2024
20 8 2024
© 2024 Published by Elsevier Ltd.
2024

https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
This study is presented to examine the performance of a newly proposed metaheuristic algorithm within discrete and continuous search spaces. Therefore, the multithresholding image segmentation problem and parameter estimation problem of both the proton exchange membrane fuel cell (PEMFC) and photovoltaic (PV) models, which have different search spaces, are used to test and verify this algorithm. The traditional techniques could not find approximate solutions for those problems in a reasonable amount of time, so researchers have used metaheuristic algorithms to overcome those shortcomings. However, the majority of metaheuristic algorithms still suffer from slow convergence speed and stagnation into local minima problems, which makes them unsuitable for tackling these optimization problems. Therefore, this study proposes an improved nutcracker optimization algorithm (INOA) for better solving those problems in an acceptable amount of time. INOA is based on improving the performance of the standard algorithm using a newly proposed convergence improvement strategy that aims to improve the convergence speed and prevent stagnation in local minima. This algorithm is first applied to estimating the unknown parameters of the single-diode, double-diode, and triple-diode models for a PV module and a solar cell. Second, four PEMFC modules are used to further observe INOA's performance for the continuous optimization challenge. Finally, the performance of INOA is investigated for solving the multi-thresholding image segmentation problem to test its effectiveness in a discrete search space. Several test images with different threshold levels were used to validate its effectiveness, stability, and scalability. Comparison to several rival optimizers using various performance indicators, such as convergence curve, standard deviation, average fitness value, and Wilcoxon rank-sum test, demonstrates that INOA is an effective alternative for solving both discrete and continuous optimization problems. Quantitively, INOA could solve those problems better than the other rival optimizers, with improvement rates for final results ranging between 0.8355 % and 3.34 % for discrete problems and 4.97 % and 99.9 % for continuous problems.

Keywords

Nutcracker optimization algorithm
Exploitation improvement strategy
Proton exchange membrane fuel cell
Photovoltaic models
Image segmentation
Kapur's entropy
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pmc1 Introduction

The optimization problem (OP) involves determining the most favorable solution within a feasible range. Various industries encounter several optimization challenges that necessitate precise optimization to achieve enhanced performance. Generally, optimization problems can be classified into two categories: discrete and continuous. Discrete problems consist of decision variables with integer or binary values, which must be meticulously optimized to obtain a near-optimal solution. In the literature, there are numerous well-established discrete optimization problems, such as the multi-threshold image segmentation problem that is extensively addressed in the literature to extract the optimal threshold values for effectively separating different objects within an image [1,2]. On the other hand, continuous problems involve decision variables that are continuous in nature, as observed in the parameter estimation problems of photovoltaic models and fuel cells, which have been extensively investigated in the literature. It is worth noting that each optimization problem may have single, multiple, or multiple objectives [3].

Over the past two decades, meta-heuristic (MH) optimization algorithms have created a great deal of interest in handling optimization problems in a variety of sectors, particularly the engineering field. This interest is due to their efficient computational cost, high accuracy, and simplicity. According to the inspiration, the MH algorithms are classified into four major groups: physics-based algorithms (i.e. Archimedes optimization algorithm [4], and multiverse optimizer [5]), evolutionary algorithms (i.e. differential evolution (DE) algorithm, evolution strategy, genetic algorithm [6] and evolutionary programming [7]), swarm-based algorithms (i.e. artificial hummingbird algorithm [8] and artificial gorilla troops optimizer [9]), and human-based algorithms (i.e. brain storm optimization [10] and past present future [11]).

The existing metaheuristic algorithms continue to have several flaws, including a lack of population variety, early convergence, and slow convergence speed, which make them unable to achieve approximate solutions for different discrete and continuous optimization problems. Therefore, this paper presents a robust metaheuristic-based optimization technique with better exploitation and exploration operators to alleviate these challenges when applied to three well-known optimization problems: multilevel image segmentation, parameter estimation of fuel cells, and parameter estimation of photovoltaic (PV) models. This algorithm is based on improving the search capability of the recently published nutcracker optimization algorithm (NOA) using a newly proposed convergence improvement strategy that could assist in boosting the convergence speed and preventing stagnation into in local minima. This improved variant of NOA is abbreviated as INOA. Both NOA and INOA are first assessed using a PV module and a solar cell under the single-diode model (SDM), double-diode model (DDM), and third-diode model (TDM) to observe their performance for estimating the unknown parameters of the PV models. Then, they are applied to find the unknown parameters of the PEMFC model for four commercial PEMFC stacks. Finally, their performance is tested and verified using several test images with various threshold levels to investigate their robustness for the image segmentation problem. To show the effectiveness of INOA and NOA for these problems, they are compared to several rival optimizers in terms of several performance metrics, such as best fitness, average fitness, worst fitness, standard deviation, and convergence curve. In addition, the Wilcoxon rank-sum test is used in comparison to reveal the difference between the INOA's outcomes and each rival optimizer's outcomes. The experimental findings show that INOA is a strong alternative for tackling discrete and continuous optimization problems. The main contributions of this study are as follows.• Proposing an improved variant of NOA, namely INOA, for solving the parameter estimation problems of both PEMFC and PV models as well as the multilevel thresholding image segmentation problem. This variant has outstanding performance in terms of convergence speed and final outcome accuracy.

• Assessing INOA using a PV module and a solar cell under the SDM, DDM, and TDM to observe its accuracy for determining the values of the unknown parameters of the PV models.

• Using four commercial PEMFC stacks to study the performance of the proposed INOA for estimating the parameters of the PEMFC model.

• Using several test images with various threshold levels to investigate its robustness for the image segmentation problem.

• In comparison to several rival optimizers, INOA is the best-performing optimizer for all optimization problems tackled in this study.

The remaining sections of the paper are consolidated as Section 3 presents the literature review. The mathematical model of the PV modules, PEMFC, and image segmentation problem are presented in section 4. Section 5 presents the details of the standard nutcracker optimization algorithm, while the detailed steps of the proposed algorithm are presented in Section 6. The results and discussions are presented in section 7. Finally, Section 8 presents the conclusion and future work of this study.

2 Literature review

2.1 Proton exchange membrane fuel cell (PEMFC)

In the last several decades, the proton exchange membrane fuel cell (PEMFC) has gained international attention as a significant renewable energy source due to its ability to constantly and immediately convert hydrogen fuel into electrical energy for the disposal of environmental pollution caused by conventional energy sources [12,13]. PEMFC modeling is necessary for simulating and designing PEMFC systems more accurately. In brief, PEMFC modeling requires correctly modeling the polarization curves, which illustrate the relationship between current and voltage. Several methods, including metaheuristic algorithms and empirical or semi-empirical-based approaches [12,[14], [15], [16], [17], [18]], have been utilized to build models of PEMFCs. For engineering applications, the semi-empirical model is preferable because it is well-known and easier to solve. However, mathematical models based on semi-empirical equations include certain unknown parameters that require accurate estimation. Consequently, this issue has been formulated as an optimization problem, referred to as the parameter estimation problem of PEMFC in the literature, and has been addressed using various optimization techniques [19]. This problem is a complex nonlinear challenge with multiple local optima, and thereby the traditional optimization techniques are not able to well handle this challengeable problem [19]. Therefore, the metaheuristic algorithms that have robust characteristics for avoiding stagnation into local minima have been applied in the literature to overcome this challenge. Some of these algorithms are reviewed in the rest of this section.

Sarhan [20] introduced an efficient adaptation of the atomic search optimizer (ASO) for this task. In a similar vein [19], integrated the Nelder-Mead simplex method (NMSM) and JAYA algorithm to boost the convergence and avoid getting stuck into local optima during parameter estimation in PEMFCs. Additionally, the enhanced cuckoo search (ECS) algorithm [21] was developed to mitigate the risk of getting trapped in local minima. This algorithm incorporated an explosive operator to enhance its performance. In another study [22], proposed a novel hybrid grey wolf optimizer (HGWO) approach for addressing the parameter estimation problem. The HGWO method combined the classical grey wolf optimizer (GWO) with crossover and mutation operators to enhance searchability and prevent the occurrence of local optima. More recently, the political optimizer (PO) and marine predators algorithm (MPA) have been applied to the parameter estimation problem, with MPA showing promising results [23]. Furthermore, Abbassi et al. [24] adapted the dandelion optimizer for estimating unknown parameters in PEMFCs and evaluated its performance under both dynamic and steady-state conditions.

Zhang et al. [25] hybridized the artificial bee colony algorithm using the crossover operator of genetic algorithms and the mutation operator of DE to enhance its exploration and exploitation capabilities for boosting the convergence speed and avoiding falling into local minima. This improved algorithm was called IABC and used in estimating the unknown parameters of the PEMFC model. Sultan et al. [26] proposed a modified variant of the bonobo optimizer, namely the quasi-opposite BO (QOBO). This variant was used to identify the unknown parameters of this problem. There are several other MH algorithms proposed recently for tackling the parameter estimation of PEMFCs, such as modified monarch butterfly optimization [27], improved fluid search optimization algorithm [28], improved barnacles mating optimization algorithm [29], ranking-based gorilla troops optimizer [30], repairable grey wolf optimization algorithm [31], improved gorilla troops technique [32], white shark optimizer [33], improved remora optimizer [34], red-tailed hawk algorithm [35], and gradient-based optimizer [36].

To prove our proposition presented in the introduction section, which says that the existing metaheuristic algorithms suffer from either falling into local minima, slow convergence speed, or time-consuming problems, in the experiments section, the performance of several metaheuristic algorithms, which have been proposed over the last few years for this problem, is analyzed in comparison to INOA. Those algorithms include the interior search algorithm (ISA) [37], gradient-based optimizer (GBO) [38], differential evolution (DE) [22], marine predators algorithm (MPA) [23], manta ray foraging optimizer (MRFO) [39], equilibrium optimizer (EO), moth-flame optimizer (MFO) [39], salp swarm algorithm (SSA) [40], and improved horse herd optimization algorithm (IHOA) [38]. This analysis shows that some algorithms, such as SSA, MPA, MFO, and DE, suffer from falling into local minima, thereby making them unsuitable for searching for the approximate solutions to this problem, and the other algorithms suffer from slow convergence speed, thereby consuming a huge number of function evaluations for achieving the desired outcomes.

2.2 Solar photovoltaic

Solar photovoltaic (PV) technology stands as a significant renewable energy source, harnessing the photovoltaic effect to generate electricity. By leveraging the natural electrical phenomenon of certain materials known as semiconductors, solar cells or diodes convert sunlight into useable energy. The absorption properties of these semiconductors dictate the extent to which they absorb light across the solar spectrum. Through this absorption process, ample energy is generated to liberate negatively charged electrons. The conversion of solar energy through photovoltaic (PV) devices is crucial in providing an economically viable and sustainable alternative to traditional power sources [41]. The successful implementation of PV systems relies heavily on the ability to design an optimal configuration that fulfills specific requirements while minimizing costs [42]. To achieve this, precise modeling, formulation, and simulation of PV systems are necessary. A PV model is employed to represent the current-voltage (I-V) and power-voltage (P-V) characteristics of the PV system under different operating conditions [43]. However, accurately estimating the parameters of a solar cell, which are not typically provided in the manufacturer's specifications, presents a challenging optimization problem. The chosen parameter values directly influence the amount of electricity generated [4]. In addressing this issue, various metaheuristic (MH) algorithms have been proposed, and the subsequent sections will discuss some of these approaches.

Qaraad et al. [44] introduced a method called QPSOL, which combines particle swarm optimization (PSO) with a local search strategy. This integration helps mitigate issues related to local minima while enhancing convergence speed. They applied QPSOL to determine the unknown parameters of three PV models. Belabbes et al. [45] proposed an improved variant of the snake optimizer (SO) specifically designed for determining the unknown parameters in both SDM and DDM. They applied this variant to effectively estimate the unknown parameters in these models. Lakshmanan [46] made modifications to the force-control parameters of the white shark optimizer (WSO) to enhance its exploration operator. Additionally, chaotic maps were employed to improve its exploitation operator. This enhanced variant, named IWSO, was utilized to identify unknown values of five and seven parameters in SDM and DDM. The black widow optimization algorithm (BWOA) was employed to tackle the parameter estimation problem in two PV models [47]. This algorithm aims to optimize the parameter values to achieve accurate estimation results. Abdel-Basset et al. [48] utilized the recently published Kepler optimization algorithm to estimate the unknown parameters of the SDM, DDM, and the three-diode model (TDM). This algorithm was specifically applied for parameter estimation in PV models.

There are several other metaheuristic algorithms proposed to tackle this problem, including the tree seed algorithm [49], enhanced leader particle swarm optimization [50], chaotic tunicate swarm algorithm [51], war strategy optimization algorithm [52], hybrid seagull optimization algorithm [53], artificial hummingbird optimization algorithm [54], grey wolf optimizer [55], spider wasp optimizer [56], Ali baba and forty thieves optimization technique [57], improved moth flame algorithms [58], and improved marine predators algorithm [59]. Each of these algorithms offers unique approaches and characteristics to address the parameter estimation challenge in PV models.

The metaheuristic algorithms proposed for this problem suffer from at least one of the following shortcomings: trap into local optima, slow convergence speed, and high computational cost. For example, in Ref. [60], Light Spectrum Optimizer was applied to estimate the unknown parameters of TDM, but it suffers from falling into local optima that prevents it from achieving better outcomes for this problem. This is deduced based on our experimental outcomes discussed later, where our proposed INOA manages to achieve superior outcomes in comparison to this algorithm. Likewise, in Ref. [61], the marine predators algorithm (MPA), which is inferior to the proposed algorithm according to the experimental results discussed later, was applied to tackle this problem. Also, the proposed INOA is compared to several other metaheuristic algorithms, such as the grey wolf optimizer [62], slime mould algorithm [63], and dandelion optimizer [64], which were adapted for solving this problem. The experimental outcomes show that those algorithms suffer from falling into local minima, which makes them unsuitable for this problem.

2.3 Image segmentation problem

Image segmentation is the extraction process of various objects from an image. It is required in image processing [65] and computer vision [66] to aid with the analysis and comprehension of the images. Threshold segmentation is one of the most common techniques used to segment the images, which can be classified into a bi-level threshold or a multi-level threshold. In the bi-level threshold, image objects can be divided into two categories: foreground (object) and background. The multi-level threshold divides the image objects into more than two classes. Threshold segmentation is straightforward, precise, quick, and space-efficient. Unfortunately, the multi-level threshold exponentially increases the time complexity. Utilized threshold strategies utilize parametric and non-parametric methodologies to determine the appropriate threshold values [67]. In the parametric method, the parameters for each image class are determined using a probability density function. The non-parametric method determines the threshold values by optimizing some functions such as fuzzy entropy [68], Kapur's entropy [69], and Otsu method [70,71] without utilizing statistical factors. Several MH techniques in the literature have been applied to optimize those functions for finding the near-optimal threshold values that could accurately segment the images. Some of these techniques are the crow search algorithm [72], improved firefly algorithm [73], moth swarm algorithm [74], multi-verse optimizer [5], water cycle algorithm [75], opposition-based Laplacian equilibrium optimizer [76], modified grasshopper algorithm [77], krill herd optimization algorithm [78], manta ray foraging optimizer [79], and growth optimizer [80].

In this study, we try to present a strong algorithm that could accurately find the near-optimal threshold values for various threshold levels, especially high threshold levels. The majority of the recently proposed metaheuristic algorithms were assessed and verified using low threshold levels, up to 7, which are unsuitable for evaluating the scalability and effectiveness of those algorithms. For example, in Ref. [81], the improved heap-based optimizer was proposed for solving this problem and assessed using low threshold levels, including 2, 3, 4, and 5. In Ref. [82], the modified reptile search algorithm was presented to segment the MRI images under threshold levels up to 10. Those levels are not enough to reveal its effectiveness for the images that require higher threshold levels. Yang et al. [83] proposed an improved variant of DE for accurately segmenting breast cancer images. This variant was evaluated using different threshold levels up to 6, which are too low to show its effectiveness and efficiency.

3 Mathematical formulation

In this paper, two continuous optimization problems, namely parameter estimation of the PEMFC and PV model, and another discrete optimization problem, namely multilevel thresholding image segmentation, will be solved by the proposed algorithm to observe its effectiveness.

3.1 Photovoltaic models and modules

Firstly, in this section, we introduce the mathematical model of the photovoltaic SDM, DDM, and TDM. Following that, we discuss the objective function utilized in conjunction with the proposed algorithm to direct the search during the entire optimization process.

3.1.1 Mathematical model of single-diode model

The SDM's electrical circuit is shown in Fig. 1. Eq. (1) describes how to derive the SDM output symbolized as I.(1) I=Iph−ID−Ish

where Iph stands for photo-generated current [84]. The current through a diode, denoted by the symbol ID, can be calculated using Eq. (2).(2) ID=Isd(exp(V+I*Rsn*Vt)−1)

where Isd is the diode's reverse saturation current; V is the diode's output voltage; R s is the series resistance; n is the diode's ideality factor; and Vt is calculated by Eq. (3).(3) Vt=k*Tq

where T represents the junction temperature in Kelvin, k represents the Boltzmann constant (1.3806503 × 10−23 J/K), and q represents the electron charge (1.60217646 × 10−19 C). Eq. (4) can be used to determine the shunt resistor current, Ish.(4) Ish=V+I*RsRsh

In which Rsh is the shunt resistance. By Substitution, I can be computed according to Eq. (5).(5) I=Iph−Isd(exp(q*(V+I*Rs)n*k*T)−1)−V+I*RsRsh

Fig. 1 SDM equivalent circuit.

Fig. 1

The above discussion suggests that five unknown parameters (Iph,Isd,n,Rs,andRsh) must be precisely estimated to run an accurate simulation of the SDM. Stochastic methods, such as the proposed algorithm, can be used to solve this optimization problem.

3.1.2 Mathematical model of double-diode model

The double diode (DD) model is meant to be a powerful alternative to the single diode (SDM) model in situations where the SDM is not optimal, such as when the irradiance level is low [85]. Fig. 2 depicts the DDM, which consists of two diodes, one of which acts as a rectifier and the other of which is used to compensate for the influence of non-idealities in the SC and the effect of current resulting from recombination. Eq. (6) is the formula used to determine DDM output.(6) I=Iph−ID1−ID2−Ish

Fig. 2 DDM electrical circuit.

Fig. 2

I can be substituted with Eq. (7).(7) I=Iph−Isd1(exp(V+I*Rsn1*Vt)−1)−Isd2(exp(V+I*Rsn2*Vt)−1)−V+I*RsRsh

where Isd1 is the first diode's current and Isd2 is the second diode's current. n1 and n2 are the diodes' ideality factors. In this formula, seven parameters, Iph, Isd1, Isd2, Rs, Rsh, n1, and n2 are absent from the manufacturing sheet. Without these variables, it is impossible to precisely recreate the DDM. Consequently, this paper attempts to estimate the near-optimal values for these parameters using the proposed technique.

3.1.3 Mathematical model of triple diode model (TDM)

The TDM depicted in Fig. 3 consists of a photocurrent source, Iph, three diodes in parallel, a shunt resistor Rsh, and a series of resistance Rs. According to Eq. (8), the TDM's output current could be computed.(8) I=Iph−∑i3ID,i−Ish

Where ID,i represents the current in the i-th diode, Ish represents the shunt current in Rsh, and I represent the current in Rs. ID,i and Ish can be determined using Eq. (9).(9) ID,i=Isdi(eq(V+IRs)NsaiKT−1)∀i∈1:3,Ish=V+IRsRsh

where Isdi is the reverse saturation current of the ith diode, ai is the ideality factor of the ith diode, and V is the output voltage of the solar cell. As a substitute, I can be represented according to Eq. (10).(10) I=Iph−Isd1(eq(V+IRs)Nsa1KT−1)−Isd2(eq(V+IRs)Nsa2KT−1)−Isd3(eq(V+IRs)Nsa3KT−1)−(V+IRs)Rsh

Fig. 3 TDM electrical circuit.

Fig. 3

On the contrary, this equation involves nine unknown parameters: Iph,Isd1,Isd2,Isd3,Rs,Rsh,a1,a2,anda3. Accurately and efficiently estimating these parameters is crucial to enhance the performance of the PV model. The main goal of this study is to identify and extract these parameters using the proposed algorithm, guided by the objective function (OF) which will be described in the subsequent section.

3.1.4 Objective function (OF)

The fundamental objective of addressing the parameter extraction problem is to identify the unknown parameter values that result in the smallest amount of discrepancy between the measured and simulated current data. Consequently, the root mean squared error (RMSE), referred to as the objective function (OF), is used to compute the difference between observed and estimated data based on Eq. (11).(11) RMSE=f(X→i)=1M*∑k=1M(Im−Ie(Ve,X→i))2

where Ie denotes the estimated current, and Im denotes the measured current. The measured data's length is indicated by M. The parameters of the ith solution are shown in X→i. Ie can be estimated by solving Eqs. (4), (6), (9) using the Newton-Raphson method and the extracted parameters shown in Xi, as defined in Eq. (12) [86].(12) Ie=I−II′

where I′ refers to the first derivative of I.

3.2 Proton exchange membrane fuel cell (PEMFC)

A PEMFC, as shown in Fig. 4, consists of a cathode and an anode separated by an electrolyte [87]. At the anode, H2 is split into H+ ions and e− by a catalyst layer, as described in eq. (13). Then, a circuit external to the cell supplies the cathode with e−, and the electrolyte supplies it with H+. After that, the oxygen in the cathode's catalyst layer combines with the electrons and protons to produce water and energy, as shown in eq. (14). The final formula representing all of the reactions is shown in Eq. (15) [88,89]:(13) Anode:H2→2H++2e−

(14) Cathode:12O2+2H++2e−→H2O

(15) Overallreaction:2H2+O2→2H2O+Heat+Electricity

Fig. 4 The structure of the PEMFC.

Fig. 4

To generate a substantial amount of energy, a single fuel cell's terminal voltage is typically insufficient. Therefore, Ncells of fuel cells need to be connected in series to achieve a higher voltage. Eq. (15) illustrates this relationship, which determines the total voltage produced by PEMFCs [[90], [91], [92], [93]].(16) Vs=Ncells(E−vact−vR−vc)

Eq. (17) provides a means to calculate the open-circuit voltage (E) for each cell. This voltage represents the potential difference across the terminals of the cell when no current is flowing. Eq. (21) can be utilized to determine the activation over-potential (vact). This overpotential arises from the energy barriers involved in initiating an electrochemical reaction within the cell. The voltage drop that occurs when protons move across the electrolyte barrier and electrons traverse from the anode to the cathode through the external circuit is known as the ohmic voltage (vR). This voltage can be calculated using eq. (23). Furthermore, Eq. (25) can be employed to calculate the concentration over-potential (vc). This overpotential arises from the concentration gradient of reactants or products involved in the electrochemical reaction.(17) E=1.229−0.85*10−3(Tfc−298.15)+4.3085×10−5Tfcln(PH2Po2)

(18) PH2=(RHa*PH2O)2[1/((RHa×PH2O)Pae(4.192Ifc/A)Tfc1.334)−1]

(19) PO2=(RHc*PH2O)[1/((RHa×PH2O)Pce(4.192Ifc/A)Tfc1.334)−1]

(20) PH2O=2.95*10−2Tc−9.18*10−5Tc2+1.44*10−7Tc3−2.18|Tc=Tfc−273.15

(21) vact=−[ξ1+ξ2Tfc+ξ3Tfcln(CO2)+ξ4Tfcln(Ifc)]

(22) CO2=PO25.08×106e498Tfc

(23) VR=Ifc(Rm+Rc)|Rm=ρmlA

(24) ρm=181.6[1+0.03(IfcA)+0.062(Ifc303)2(IfcA)2.5][λ−0.634−3(IfcA)]e4.18(Tfc−303Tfc)

(25) vc=−βln(Jmax−JJmax)

where Po2, PH2 and PH2O are, respectively, the partial pressures of oxygen (O2), hydrogen (H2), and water (H2O). The current gas's temperature, Tfc, is expressed in Kelvin. RHa and RHc stands for the relative humidity of vapor at the anode and cathode, respectively. Pa and Pc, respectively, stand for the inputs for the anode and cathode pressures (measured in atm). The oxygen concentration is represented by the symbol CO2 and is measured in mol/cm3. Rm,andRc, respectively, stand for the membrane resistance and the connection resistance. Ifc stands for the operating current of the FC. A parametric coefficient is β. The membrane is designated as having a thickness of l (CM). ρm stands for the membrane's resistance. Jmax stands for maximum current density, and J stands for real current density. Following a description of the mathematical model of the PEMFC, it was evident that the model would require accurate calculation of seven unknown parameters (ξ1, ξ2, ξ3, ξ4, λ, Rc, and β). Since this topic is framed as an optimization problem, these parameters can be calculated using an optimization technique.

3.2.1 PEMFC's objective function

The quality of each solution provided by the proposed technique is determined by comparing the optimal voltages estimated experimentally with the estimated voltages using the extracted parameters. Based on that, the objective function is represented by the sum of the squared error (SSE) rates between the measured and estimated data [75,94,95]. The mathematical model of this function is shown in Eq. (26) [91].(26) SSE=f(Xi→)=∑c=1LD|VS,measured(c)−VS,estimated(c)|2

where LD is the number of the measured points.

3.3 Image segmentation problem: Kapur's entropy

This section will describe Kapur's entropy, which is utilized to obtain appropriate threshold values by maximizing the entropy of segmented regions [69]. Assuming that the threshold values that segment an image into k comparable regions are t0,t1,t2,……..,andtk, then Kapur's entropy will maximize Eq. (27) until the optimal values for these thresholds are reached.(27) T(t0,t1,t2,……..,tk)=T0+T1+T2+……..+Tk

where:

(28) T0=−∑i=0t0−1XiW0*lnXiW0,Xi=NiW,W0=∑i=0t1−1Xi

(29) T1=−∑i=t0t1−1XiW1*lnXiW1,Xi=NiW,W1=∑i=t0t1−1Xi

(30) T2=−∑i=t1t2−1XiW2*lnXiW2,Xi=NiW,W2=∑i=t1t2−1Xi

(31) Tk=−∑i=tkL−1XiWk*lnXiWk,Xi=NiW,Wk=∑i=tkL−1Xi

T0,T1,T2,……..,andTk represent the entropies of similar regions, while Ni is the number of pixels with the same value as i. W0,W1,W2,…,andWk describe the probability of the various regions relative to the whole pixel W in an image. In our proposal, Eq. (26) is employed as a maximization fitness function to determine the ideal threshold values.

4 Nutcracker optimization algorithm (NOA)

Recently, the Nutcracker Optimization Algorithm (NOA) was introduced to address continuous optimization problems [96]. NOA mimics nutcrackers' foraging, storing, and retrieving of pine seeds. There are two basic types of nutcracker behavior, and they both happen at distinct times. First, during the warmer months, the nutcracker is likely collecting seeds to preserve for the colder months. Searching for hidden caches at different angles utilizing a variety of objects or markings as guides is another strategy that can be employed during the colder months of winter and spring. In the absence of their usual food source—the stored seeds—the nutcrackers will scavenge for food by randomly probing the search area. To formalize NOA's foraging and storage strategy into a mathematical model, we can define the following steps for the foraging stage.

4.1 Foraging and storage strategy

This strategy can be divided into its two basic phases, which are referred to as foraging and storage, and are described in greater detail below.• Foraging stage: Exploration phase 1

In this phase, the nutcrackers exhibit a randomized placement behavior within the search space. Each nutcracker initiates by evaluating the initial position of a cone filled with seeds. If viable seeds are detected, the nutcracker proceeds to relocate and bury them within a designated storage area. Conversely, if no seeds are found, the nutcracker engages in a search for an alternative cone positioned among various trees, including pine trees. To capture this behavior mathematically, Eq. (32) can be employed.(32) X→it+1={Xi,jtifτ1<τ2{Xmt+γ∙(XA,jt−XB,jt)+μ.(r2.Uj−Lj),ift≤Tmax/2.0XC,jt+μ∙(XA,jt−XB,jt)+μ.(r1<δ).(r2.Uj−Lj),Otherwiseotherwise

where X→it+1 represents the new position of the ith nutcracker at iteration t+1; Xi,jt is the jth variable of the ith nutcracker at iteration t; Uj and Lj are the upper bound and lower bound of the jth dimension, respectively; γ is a random number based on the levy-flight; Xbest,jt is the jth dimension of the best solution obtained so far; A, C, and B represent the indices of three nutcrackers chosen randomly from the current population; τ1, τ2, r, and r1 are numerical values generated randomly between 0 and 1; Xm,jt represents the mean of the jth dimensions for all solutions at iteration t; and μ is a random number generated according to Eq. (33).(33) μ={τ3ifr1<r2τ4ifr2<r3τ5ifr1<r3

where r2, r3, and τ4 are numbers generated randomly between 0 and 1. τ4 is a number generated randomly using normal distribution, and τ5 is a random number generated using the levy-flight (τ5).• Storage stage: Exploitation phase 1

Nutcrackers begin by transporting the food gathered in the previous exploration phase to temporary storage locations. During this initial phase of exploitation, the nutcrackers collect and store pine seed crops. Eq. (34) presents the mathematical model for this behavior.(34) X→it+1(new)={X→it+μ∙(X→bestt−X→it)∙|λ|+r1∙(X→At−X→Bt)ifτ1<τ2X→bestt+μ∙(X→At−X→Bt)ifτ1<τ3X→bestt∙lOtherwise

where λ is a number created randomly according to the lévy flight. l is a factor decreasing linearly from 1 to 0. During the optimization process, Eq. (35) shows how to tradeoff between the foraging phase and the cache to maintain the balance between exploration and exploitation operators.(35) X→it+1={Eq.(32),ifφ<Pa1Eq.(34),otherwise

where φ is a random number between 0 and 1, and Pa1 is a linearly decreasing probability from 1 to 0.

4.2 Cache-search and recovery strategy

This mechanism can be divided into two phases: The cache-search and recovery phases are covered in detail in the next two sections.• Cache-search stage: Exploration phase 2

In NOA, every nutcracker employs two reference points (RPs) for each cache as marks, as defined by Eq. (36).(36) RPs=[RP→1,1tRP→1,2t⋮⋮RP→i,1t⋮RP→N,1tRP→i,2t⋮RP→N,1t]

where RP→i,1t represents the first RPs for the ith nutcracker at iteration t. Two distinct equations were designed to produce the first and second RPs, respectively, to improve the nutcracker exploring process when searching for hidden caches. First RP can be created using Eq. (37), and second RP may be calculated using Eq. (38).(37) RP→i,1t={X→it+α∙cos(θ)∙((X→At−X→Bt))+α∙RP,ifθ=π/2X→it+α∙cos(θ)∙((X→At−X→Bt)),otherwise

(38) RP→i,2t={X→it+(α∙cos(θ)∙((U→−L→)∙τ3+L→)+α∙RP)∙U2→,ifθ=π/2X→it+α∙cos(θ)∙((U→−L→)∙τ3+L→)∙U2→,otherwise

(39) U1→={1r2→<Prp0otherwise

where r2→ is a vector generated randomly between 0 and 1; X→At is a randomly selected solution from the current population; θ is a numerical value generated randomly at the interval of 0 and π; and Prp is a probability to determine the exploration rate; and α is computed using Eq. (40).(40) α={(1−tTmax)2tTmax,ifr1>r2(tTmax)2t,otherwise

where Tmax represents the maximum number of generations and t is the current generation. Eq. (41) is utilized in NOA to activate the spatial memory for the first RP.(41) X→it+1={X→it,iff(X→it)<if(RP→i,1t)RP→i,1t,otherwise

• Recovery stage: Exploitation phase 2

The first scenario is that a nutcracker can recall the cache's location using the first RP. This behavior is represented by Eq. (42).(42) Xijt+1={Xijt,ifτ3<τ4Xijt+r1∙(Xbest,jt−Xijt)+r2∙(RP→i,1t−XCjt),otherwise

where r1,r2,τ3andτ4 are numbers generated randomly between 0 and 1, and C represents the index of a solution selected randomly from the current population. The second scenario is that the nutcracker forgets where he concealed his food with the first RP, so he will use the second RP to look for it. During early storage, a nutcracker memorizes many of the RPs it will take. Nutcrackers recover caches on the first attempt (with the first reference), whereas the proposed approach takes into consideration the possibility of failing on the first try. According to Eq. (43), the nutcrackers that are unable to reach their cache using the first RP will activate their spatial memory to the second RP.(43) X→it+1={X→it,iff(X→it)<if(RP→i,2t)RP→i,2t,otherwise

Eq. (42) is modified in NOA to account for the second RP on the assumption that a nutcracker will utilize it to find its cache, as defined in Eq. (44).(44) Xijt+1={Xijt,ifτ5<τ6Xijt+r1∙(Xbest,jt−Xijt)+r2∙(RP→i,2t−XCjt),otherwise

where τ5,andτ6 are randomly generated numbers between 0 and 1. Eq. (45) represents the tradeoff between the first and second RPs within the recovery behavior.(45) X→it+1={Eq.(42),ifτ7<τ8Eq.(44),otherwise

where τ7 and τ8 are random numbers created between 0 and 1. The first state in the preceding equation represents a nutcracker who recalls the secret store, whereas the second case represents a nutcracker who forgets about the hidden store. According to Eq. (46), the spatial memory between the first and second RPs, as well as the current position, is activated to search for the cache:(46) X→it+1={Eq.(41),iff(Eq.(41))<f(Eq.(43))Eq.(43),otherwise

In order to achieve a balance between exploitation and exploration, the recovery and cache-search phases are then randomly swapped as defined in Eq. (47).(47) X→it+1={Eq.(44),ifϕ<Pa2Eq.(45),otherwise

where Pa2 is a preset value between 0 and 1 indicating the probability of the exploitation stage in the optimization process. However, in the NOA algorithm, the nutcrackers remain in their current location if the quality of their solution is superior to that of the new solution. In conclusion, Eq. (48) expresses the preceding concept.(48) X→it+1={X→it+1,iff(X→it+1)<f(X→it)X→it,otherwise

5 Proposed algorithm

In this section, the NOA is integrated with an effective improvement strategy to strengthen its exploration and exploitation operator for solving some discrete and continuous optimization problems such as the parameter estimation of PV models and PEMFC, and multilevel thresholding image segmentation problem. In detail, the main steps of the proposed algorithm, namely improved NOA (INOA), will be discussed in the next subsections.

5.1 Initialization and evaluation

Similar to population-based algorithms, the proposed algorithm randomly initializes N solutions X→i(i∈N) within the search space of the optimization problem by Eq. (49), where each solution has d dimensions to be optimized.(49) Xi→=L→+(U→−L→).r→

where r→ is a randomly generated vector between 0 and 1. These initialized solutions are not applicable to the multithresholding image segmentation problem because it is discrete and these solutions are continuous. Therefore, these solutions will be converted into discrete solutions, as clarified next, to be relevant to this problem. The grey-scale level is represented in 8-bit, where the minimum values (L→) are 0 and the maximum values (U→) are 28−1 = 255. Assuming that we have an image with 10 various objects (K), which need to be separated by finding the near-optimal threshold values. Therefore, the proposed algorithm will first distribute its solutions randomly throughout the search space, as exemplified in Fig. 5. Afterwards, these solutions will be converted into integers by dropping all decimal places past the dot, as shown in Fig. 6. As indicated in Fig. 7, these converted numbers will be sorted ascendingly, and Eq. (26) will be utilized to determine the quality of these threshold values according to Kapur's entropy.Fig. 5 A solution representation to multilevel thresholding image segmentation.

Fig. 5

Fig. 6 Unsorted integer values.

Fig. 6

Fig. 7 Sorted integer values.

Fig. 7

5.2 Convergence improvement strategy (CIS)

This method is designed to expedite the convergence speed of the NOA while mitigating the issue of local minima. The approach consists of three key components. The first component involves comparing the fitness value of a randomly selected solution with that of the current solution. If the current solution is inferior, a new solution is generated by updating the current solution in the direction of the randomly selected solution. In addition, two new solutions are chosen randomly from the population, and the current solution is updated by moving towards the solution with the highest fitness value. It is plausible that the randomly selected solution may be closer to the near-optimal solution compared to the best-so-far solution. This strategy leverages the regions within the randomly selected solution that exhibit higher fitness values than the other two randomly selected solutions from the current set of solutions. The purpose is to accelerate convergence and prevent getting trapped in local minima. This particular aspect can be mathematically formulated as defined in Eq. (50).(50) G→i(t+1)=X→i(t)+(r*(v1→)+(1−r)*(v2→))

(51) v1→={Xit→−Xat→,iff(Xit→)≤f(Xat→)Xat→−Xit→,otherwise

(52) v2→={Xbt→−Xct→,iff(Xbt→)≤f(Xct→)Xct→−Xbt→,otherwise

where a,b,andc are three numbers randomly selected from the solutions such that a≠b≠c≠i. r is a number generated randomly between 0 and 1. The second and third folds are formulated in Eqs. (52), (53), respectively, as an attempt to accelerate the convergence speed in the direction of the best-so-far solution while avoiding getting stuck in local minima.(53) X→i(t+1)=(X→i(t)+X→a(t)+X→bestt)3+β*(−X→bestt+X→i(t))+r1*(Xbt→−Xct→)

(54) X→i(t+1)=(X→i(t)+X→a(t)+X→bestt)3+r*(X→bestt−X→i(t))*U→+r1*(Xbt→−Xct→)*U1→

(55) β=11+eσl

where σ is a constant value determined later within the experiment section; l is a variable including values linearly decreasing from −1 to −2. U→ and U1→ are binary vectors generated according to Eqs. (56), (57), respectively.(56) U→={1r2→≥r→30otherwise

(57) U1→={1r2→≥r0otherwise

Eq. (58) is used to calculate the tradeoff between these three folds.(58) X→i(t+1){Eq.(52)ifr<δEq.(53)ifr1<δ1Eq.(49)Else

where δ and δ1 are controlling parameters defined within the experiments to determine the probability of each fold.

5.3 Pseudo-code of the proposed INOA

The conventional NOA is effectively combined with CIS to present a new metaheuristic algorithm for both continuous and discrete optimization problems. This improved variant is known as improved NOA (INOA). In addition, the conventional NOA is used with a probability Pse in order to optimize the utilization of the number of function evaluations for accelerating convergence towards the near-optimal solution. In the experiments, the ideal value for this probability is examined. Finally, Algorithm 1 describes the steps of the proposed INOA. The main advantages of INOA are:• Easy to implement

• Having a better convergence speed

• Having a high ability to alleviate falling into local minima

Meanwhile, its main advantage is that it has four controlling parameters that have to be accurately estimated before starting the optimization to improve its effectiveness.Algorithm 1 Steps of the proposed INOAInput: population size N, L→, U→, and Tmax;	
Output: X→bestt	
1. Initialize N solutions by Eq. (49);	
2. Evaluation and extracting the best solution	
3. t=1;//the current evaluation//4. while (t<Tmax)	
5. r: A number generated randomly between 0 and 1.	
6. ifr<Pse	
7. σ and σ1: Numbers generated randomly between 0 and 1.	
8. Ifσ<σ1//* Foraging and storage strategy*//.	
9. φ is a random number between 0 and 1.	
10. for i=1:N	
11. ifφ < Pa1/* Exploration phase1*/	
12. Create X→it+1 by Eq. (32) and Eq. (48)	
13. else/*Exploitation phase1*/	
14. Create X→it+1 by Eq. (34) and Eq. (48)	
15. end if	
16. t=t+1	
17. end for	
18. else//* Cache-search and recovery strategy *//.	
19. Compute RP matrix by (36), (37), (38).	
20. : A number generated randomly between 0 and 1.	
21. for i=1:N	
22. ifϕ < Pa2/*Exploitation phase2*/	
23. Create X→it+1 by Eq. (46) and Eq. (48)	
24. else/*Exploration phase2*/	
25. Create X→it+1 by Eq. (47) and Eq. (48)	
26. end if	
27. t=t+1	
28. end for	
29. end if	
30. end if	
31.//* Applying CIS Strategy *//	
32. fori = 1:N	
33. Updating X→it+1 according to Eq. (58) and Eq. (48)	
34. t=t+1	
35. end for	
36. end while	

6 Results and discussion

In this paper, two well-known optimization problems, namely parameter estimation of both PEMFC and PV models, are used to check the efficacy of the proposed algorithm for problems with continuous decision variables. In addition, a well-known discrete problem known as the multi-thresholding image segmentation problem is used to disclose the efficiency of INOA for problems with discrete decision variables. INOA is extensively compared to several well-known state-of-the-art in terms of several pieces of statistical information extracted after 30 independent runs for each algorithm, including the best, average, worst, standard deviation (SD), rank, and p-value returned by the Wilcoxon rank-sum test. Experiments with MATLAB R2019a are performed on a machine with the following specifications: an Intel(R) Core(TM) i7-4700MQ CPU running at 2.40 GHz, 32 GB of random access memory (RAM), and a 64-bit version of Windows 10 Pro.

6.1 Application I: Parameter estimation of PV models

This section examines the effectiveness of INOA on the RTC France solar cell and a well-known PV module: Photowatt-PWP201 (PWP) [97]. Table 1 displays the open-circuit current-voltage (Voc), the short-circuit current-temperature factor (ki), the maximum output current (Im), maximum output power (Pm), short-circuit current point (ISC), the number of cells connected in series in PV modules (Ns), the maximum output voltage (Vm), and the temperature coefficient of open-circuit voltage (kv) for these PV models at standard conditions (STC) [98]. Table 2 displays the maximum and minimal values for each of the unknown parameters. Here we compare the proposed INOA to ten popular and recently proposed metaheuristic-based algorithms: NOA [96], MPA [98], Fick's law algorithm (FLA) [99], interior search algorithm (ISA) [100], grey wolf optimizer (GWO) [101], slime mould algorithm (SMA) [102], pelican optimization algorithm (POA) [103], dandelion optimizer (DO), and EO [104]. The controlling parameters of these algorithms are set as recommended in the cited paper except for Tmax and N, which are both set to 50000 and 35 in order to provide a fair comparison.Table 1 Properties of RTC France solar cell and PV models.

Table 1characteristics	RTC	PWP	
Pm[W]	0.31	11.5	
Vm[V]	0.459	12.649	
Im[A]	0.6755	0.912	
Voc[V]	0.5736	16.7785	
ISC[A]	0.7605	1.0317	
Ns	1	36	
Ki	0.000387	0.0008	
Kv	−0.003739	−0.0725	

Table 2 The search boundary of unknown parameters.

Table 2	Iph(A)	Isdi(A),i∈1:3	Rs(Ω)	Rsh(Ω)	a1	a2	a3	
U	1.1ISC	10μA	0.5	500	2	2	2	
L	0.9ISC	1nA	0	0	1	1.2	1.4	

6.1.1 INOA's sensitivity analysis

The optimal performance of INOA in dealing with parameter estimation of the PV models requires accurate identification of four newly proposed regulating parameters. In this case, we speculate based on data from extensive experiments performed on the PWP module, as shown in Fig. 8(a–d). Fig. 8(b) shows that the best value for δ is 0.01, Fig. 8(c) reports that the performance of the proposed is maximized when δ1=0.3, and Fig. 8(d) illustrates the best performance of the proposed when setting σ to 9. The likelihood that the standard NOA will be implemented is determined by another controlling parameter, Pse. Fig. 8(a) displays the outcomes of multiple experiments in which different values were chosen for this parameter. This figure demonstrates that a value of 0.6 is optimal for this parameter. All other parameters of classical NOA are set in accordance with the suggestions made in Ref. [96].Fig. 8 Adjusting the parameters of the proposed INOA for the parameter estimation of the SDM-based PWP module.

Fig. 8

6.1.2 RTC France solar cell

INOA is employed in this section to estimate the unknown parameters of the R.T.C France solar cell based on SDM, DDM, and TDM. The current-voltage data is measured on a commercial silicon RTC France solar cell with a 57-mm diameter and a temperature of 33 °C [97].A. Single-Diode model

After running each algorithm 30 independent times over the SDM-based RTC France, various statistical information is calculated and described in Table 3. Based on the results presented in Table 3, INOA is the best for all the performance metrics, followed by the classical NOA as the second-best one, while ISA is the worst. In addition, the Wilcoxon rank-sum test was applied to observe the difference between the outcomes of INOA and those of the other algorithms. This test returns a value, referred to as p-value, between each pair of algorithms; this value determines that the Null hypothesis, determining that there is no difference, is accepted if this value is greater than the significance level of 0.05; otherwise, the alternative one is accepted. To affirm the superiority of INOA, Fig. 9 (a) and (b) are presented to show the convergence curve and the five-number summary for each algorithm. This figure affirms that INOA is the best, followed by NOA as the second-best one, while ISA is the worst one. Fig. 9 (c) and (d) show the measured and calculated I-V, and the measured and estimated P-V, which appear that the estimated values significantly agree with the measured ones.B. Double-Diode model

Table 3 Performance evaluation using SDM-based RTC France.

Table 3	Best-obtained parameters	RMSE		
Iph(A)	Isd(μA)	Rs(Ω)	Rsh(Ω)	n	Best	Worst	Avg	SD	Rank	p-value	
INOA	0.7608	3.11E-07	0.037	52.890	1.477	7.7301E-04	7.7301E-04	7.7301E-04	1.2040E-17	1		
NOA	0.7607	3.63E-07	0.036	57.879	1.493	8.1925E-04	1.7862E-03	1.2592E-03	2.5752E-04	2	3.01E-11	
FLA	0.7635	3.13E-07	0.036	47.674	1.477	3.1516E-03	1.1387E-01	1.8722E-02	2.7635E-02	10	3.01E-11	
EO	0.7604	4.05E-07	0.035	65.354	1.504	9.0687E-04	7.3694E-02	6.5035E-03	1.8271E-02	9	3.01E-11	
ISA	0.7608	3.11E-07	0.037	52.899	1.477	7.7301E-04	7.3694E-02	3.0007E-02	3.6281E-02	11	2.94E-11	
RUN	0.7601	5.40E-07	0.034	82.564	1.535	1.2236E-03	1.9544E-02	7.7874E-03	2.9242E-03	8	3.01E-11	
GWO	0.7653	7.84E-09	0.049	18.099	1.182	5.3408E-03	7.3694E-02	1.4019E-02	1.2584E-02	4	3.01E-11	
MPA	0.7599	6.16E-07	0.034	99.606	1.549	1.4290E-03	6.3233E-03	4.8931E-03	1.2239E-03	3	3.01E-11	
SMA	0.7620	1.02E-07	0.041	32.991	1.374	1.8856E-03	8.6503E-03	6.9849E-03	1.7972E-03	7	3.01E-11	
DO	0.7605	2.76E-07	0.037	55.921	1.465	8.4908E-04	8.0755E-03	5.2948E-03	2.5368E-03	4	3.01E-11	
POA	0.7580	1.57E-07	0.040	96.135	1.411	2.4347E-03	8.6882E-03	5.3957E-03	1.8125E-03	5	3.01E-11	
Bold values indicate the best results.

Fig. 9 Various comparisons among algorithms using SDM.

Fig. 9

Various aforementioned performance metrics are computed and reported in Table 4 after each method has been run 30 times on the DDM-based RTC France. This table shows that INOA ranks first, followed by NOA as the second-best one, while ISA performs the poorest. The p-value presented in this table shows that the outcomes of INOA are significantly different from those of the others. Fig. 10(a) and (b) display the convergence curve and the five-number summary for each algorithm, respectively, to demonstrate INOA's superiority. Based on these figures, INOA is the best, followed by NOA in second place and SMA in last. The estimated values appear to significantly match with the measured ones, as shown in Fig. 10(c) and (d), which display the measured and calculated I-V and the measured and estimated P-V, respectively.C. Triple-Diode model

Table 4 Performance evaluation using DDM-based RTC France.

Table 4	Best-obtained parameters	RMSE		
Iph(A)	Isd1(A)	Isd2(A)	Rs(Ω)	Rsh(Ω)	n1	n2	Best	Worst	Avg	SD	Rank	p-value	
INOA	0.7608	9.17E-08	2.08E-06	0.038	58.065	1.377	2.000	7.3274E-04	7.4201E-04	7.3703E-04	2.1080E-06	1		
NOA	0.7606	2.03E-07	8.91E-07	0.037	61.952	1.444	1.942	8.7247E-04	2.5633E-03	1.3364E-03	4.7787E-04	2	3.02E-11	
FLA	0.7558	1.80E-08	2.92E-06	0.043	483.690	1.253	2.000	3.1454E-03	1.4847E-01	3.5400E-02	4.6868E-02	10	3.02E-11	
EO	0.7607	1.99E-07	6.63E-07	0.036	58.429	1.444	1.864	7.8147E-04	2.8516E-03	1.7097E-03	5.9515E-04	3	3.02E-11	
ISA	0.7608	8.08E-08	2.24E-06	0.038	58.599	1.367	2.000	7.3277E-04	6.9442E-02	1.4534E-02	2.7924E-02	9	8.33E-08	
RUN	0.7609	6.92E-08	1.52E-06	0.038	55.291	1.357	1.901	7.4639E-04	6.9443E-02	5.0538E-03	1.2247E-02	5	3.02E-11	
GWO	0.7615	2.29E-07	1.24E-06	0.035	53.269	1.459	1.954	1.1617E-03	6.9479E-02	1.0702E-02	1.7859E-02	8	3.02E-11	
MPA	0.7608	2.53E-09	5.00E-06	0.042	490.048	1.142	1.970	2.3234E-03	1.4455E-02	6.5017E-03	3.0484E-03	6	3.02E-11	
SMA	0.7456	2.23E-07	2.80E-06	0.209	39.133	1.604	1.941	1.2560E-01	3.8795E-01	2.7893E-01	6.6896E-02	11	3.02E-11	
DO	0.7608	2.30E-08	1.01E-06	0.040	52.288	1.281	1.764	9.3909E-04	6.9442E-02	6.9604E-03	1.7019E-02	7	3.02E-11	
POA	0.7602	1.75E-07	1.52E-06	0.037	74.511	1.433	1.981	9.1404E-04	8.5431E-03	2.7510E-03	1.5974E-03	4	3.02E-11	
Bold values indicate the best results.

Fig. 10 Various comparisons among algorithms using DDM-based RTC France.

Fig. 10

After 30 independent runs for each algorithm on the TDM-based RTC France, a variety of performance indicators are calculated and provided in Table 5. This table reveals that INOA is the best, followed by ISA as the second best one. This table also includes a p-value, which demonstrates that INOA results are statistically different compared to all the other algorithms. For further illustration of INOA's superiority, Fig. 11(a) and (b) show the convergence curve and the five-number summary for each algorithm, respectively. From this data, INOA is the best, followed by ISA in second place and POA in third. As can be seen in Fig. 11(c) and (d), which show the measured and calculated I-V and the measured and estimated P-V, respectively, the estimated values appear to considerably correspond with the measured ones.Table 5 Performance evaluation using TDM-based RTC France.

Table 5	Best-obtained parameters	RMSE		
Iph(A)	Isd1(A)	Isd2(A)	Isd3(A)	Rs(Ω)	Rsh(Ω)	n1	n2	n3	Best	Worst	Avg	SD	Rank	p-value	
INOA	0.7605	8.9E-08	1.8E-09	2.1E-06	0.038	61.327	1.376	1.408	2.000	7.51E-04	7.71E-04	7.56E-04	5.5E-06	1		
NOA	0.7601	5.2E-07	7.1E-08	4.0E-07	0.033	107.015	1.536	1.828	1.960	1.56E-03	3.52E-03	2.48E-03	5.4E-04	7	3.0E-11	
FLA	0.7605	2.4E-07	3.5E-07	2.0E-06	0.036	500.000	1.466	2.000	2.000	3.14E-03	5.25E-02	2.18E-02	1.3E-02	11	3.0E-11	
EO	0.7594	1.4E-07	1.2E-06	2.7E-07	0.035	134.779	1.753	1.975	1.477	1.50E-03	3.51E-03	2.45E-03	4.2E-04	6	3.0E-11	
ISA	0.7605	1.6E-06	1.2E-09	1.3E-07	0.037	59.509	2.000	1.652	1.406	7.54E-04	2.06E-03	1.06E-03	4.4E-04	2	3.8E-10	
RUN	0.7605	1.7E-06	1.2E-08	1.2E-07	0.037	60.841	1.997	1.730	1.400	7.55E-04	8.35E-03	4.47E-03	2.6E-03	10	9.0E-11	
GWO	0.7605	1.0E-06	1.1E-06	1.2E-07	0.037	70.005	1.998	1.981	1.400	8.36E-04	4.93E-03	2.65E-03	1.1E-03	9	3.0E-11	
MPA	0.7604	5.6E-07	3.9E-09	1.2E-08	0.039	61.122	1.588	1.204	1.995	8.11E-04	4.35E-03	2.34E-03	8.2E-04	5	3.0E-11	
SMA	0.7605	1.0E-09	6.9E-09	5.0E-06	0.040	100.236	1.464	1.200	2.000	1.09E-03	5.40E-03	2.48E-03	1.1E-03	7	3.0E-11	
DO	0.7605	4.0E-07	8.1E-09	3.9E-07	0.038	66.403	1.609	1.245	1.682	8.23E-04	3.25E-03	2.04E-03	5.9E-04	4	3.0E-11	
POA	0.7605	1.7E-06	2.1E-08	1.2E-07	0.038	60.495	2.000	2.000	1.400	7.53E-04	2.23E-03	1.48E-03	5.2E-04	3	4.6E-10	
Bold values highlight the best findings.

Fig. 11 Various comparisons among algorithms using TDM-based RTC France.

Fig. 11

6.1.3 Photowatt-PWP201 module

Here, we use the photowatt-PWP201, another PV module, to compare INOA's performance to that of competing optimizers in determining the PV module's as-yet-unknown parameters under a variety of PV models.A. Single-Diode model

We calculate and provide in Table 6 a wide range of performance metrics after 30 replicated runs of each method on the SDM-based PWP module. Based on this data, INOA emerges as the top one, followed by NOA as the second one. A p-value is also provided to reveal that INOA yields significantly different outcomes than the rest of the algorithms.B. Double-Diode model

Table 6 Performance evaluation using SDM-based PWP module.

Table 6	Best-obtained parameters	RMSE		
Iph(A)	Isd(μA)	Rs(Ω)	Rsh(Ω)	n	Best	Worst	Avg	SD	Rank	p-value	
INOA	1.0324	2.50E-06	3.45E-02	20.7867	1.3166	0.0020400	0.0024435	0.0020543	0.0000737	1		
NOA	1.0322	2.58E-06	3.44E-02	21.4815	1.3199	0.0020423	0.0025238	0.0021666	0.0001228	2	9.73E-10	
FLA	1.0259	6.62E-06	3.07E-02	100.0000	1.4229	0.0035667	0.1572649	0.0508148	0.0559063	10	3.01E-11	
EO	1.0322	2.59E-06	3.43E-02	21.4115	1.3204	0.0020420	0.0951542	0.0060308	0.0168403	5	5.48E-11	
ISA	1.0324	2.50E-06	3.45E-02	20.7876	1.3166	0.0020400	0.0951542	0.0486342	0.0473153	1	2.43E-10	
RUN	1.0318	2.32E-05	2.54E-02	99.6524	1.5899	0.0062080	0.0951542	0.0280828	0.0376406	8	3.01E-11	
GWO	1.0366	3.30E-06	3.31E-02	15.4049	1.3461	0.0028738	0.0951542	0.0608428	0.0416180	11	3.01E-11	
MPA	1.0266	3.82E-06	3.36E-02	71.2188	1.3606	0.0029178	0.0054980	0.0041261	0.0007022	3	3.01E-11	
SMA	1.0396	1.68E-06	3.51E-02	11.5031	1.2785	0.0032482	0.0951543	0.0103516	0.0161451	6	3.01E-11	
DO	1.0272	4.03E-06	3.33E-02	61.1112	1.3665	0.0027780	0.0951542	0.0238049	0.0363285	7	3.01E-11	
POA	1.0309	3.76E-06	3.32E-02	28.1371	1.3594	0.0022363	0.0081123	0.0047595	0.0015656	4	3.68E-11	
Bold values indicate the best results.

After running each method on the DDM-based PWP module 30 times, we calculate and give a wide variety of performance data in Table 7. From these results, INOA is the winner, with NOA coming in second. Furthermore, a p-value is presented to demonstrate that INOA yields significantly different results than the rest of the algorithms.C. Triple-Diode model

Table 7 Performance evaluation using DDM-based PWP module.

Table 7	Best-obtained parameters	RMSE		
Iph(A)	Isd1(A)	Isd2(A)	Rs(Ω)	Rsh(Ω)	n1	n2	Best	Worst	Avg	SD	Rank	p-value	
INOA	1.0324	2.31E-09	2.49E-06	0.0345	20.7867	1.3166	1.3166	0.0020400	0.0023454	0.0020711	0.0000669	1		
NOA	1.0325	1.49E-09	2.98E-06	0.0339	21.8105	1.1553	1.3354	0.0020856	0.0027508	0.0021795	0.0001296	2	3.35E-08	
FLA	1.0242	1.99E-06	5.00E-06	0.0360	500.0000	1.2969	1.8684	0.0043064	0.1533691	0.0592287	0.0544239	9	3.02E-11	
EO	1.0332	1.94E-08	2.13E-06	0.0350	18.3995	1.1801	1.3042	0.0020818	0.0939028	0.0175745	0.0347194	6	1.41E-09	
ISA	1.0324	2.25E-06	2.25E-06	0.0346	20.9321	1.3072	2.0000	0.0020653	0.0940767	0.0724833	0.0395037	10	6.12E-10	
RUN	1.0305	4.82E-07	2.97E-06	0.0349	26.6476	1.2233	1.4147	0.0022137	0.0939027	0.0061759	0.0165819	4	4.50E-11	
GWO	1.0393	1.79E-08	4.80E-07	0.0403	10.2600	1.0184	1.2000	0.0043963	0.0939200	0.0582602	0.0436114	8	3.02E-11	
MPA	1.0269	2.68E-09	2.54E-06	0.0356	56.3927	1.0562	1.3202	0.0033363	0.0115753	0.0078424	0.0028378	5	3.02E-11	
SMA	1.0697	3.55E-06	3.74E-06	0.0236	496.1064	1.3353	1.9234	0.0675634	0.2703185	0.1801100	0.0447674	11	3.02E-11	
DO	1.0256	3.93E-06	3.98E-06	0.0327	233.4422	1.3695	1.7740	0.0032496	0.0939030	0.0322190	0.0411050	7	3.02E-11	
POA	1.0310	1.35E-06	1.83E-06	0.0345	24.6590	1.2834	1.4391	0.0021189	0.0078216	0.0037715	0.0012984	3	4.98E-11	
Bold values indicate the best results.

After 30 independent runs for each algorithm on the TDM-based PWP module, a variety of performance indicators are calculated and provided in Table 8. This table reveals that INOA is the best, followed by ISA as the second best one. This table also includes a p-value, which demonstrates that INOA results are statistically different compared to all the other algorithms. For further illustration of INOA's superiority, Fig. 12(a) and (b) show the convergence curve and the five-number summary for each algorithm, respectively. From this data, INOA is the best, followed by ISA in second place and NOA in third. As can be seen in Fig. 12(c) and (d), which show the measured and calculated I-V and the measured and estimated P-V, respectively, the estimated values appear to considerably correspond with the measured ones.Table 8 Performance evaluation using TDM-based PWP module.

Table 8	Best-obtained parameters	RMSE		
Iph(A)	Isd1(A)	Isd2(A)	Isd3(A)	Rs(Ω)	Rsh(Ω)	n1	n2	n3	Best	Worst	Avg	SD	Rank	p-value	
INOA	1.032	1.0E-09	2.6E-06	1.0E-09	0.034	22.16	1.32	1.32	1.40	2.0507E-03	2.20E-03	2.06E-03	2.7E-05	1		
NOA	1.030	3.1E-06	1.3E-06	2.3E-06	0.033	37.25	1.35	1.61	1.86	2.4494E-03	3.29E-03	2.90E-03	2.5E-04	3	3.02E-11	
FLA	1.025	2.7E-07	5.4E-08	5.0E-06	0.033	146.85	1.58	1.20	1.40	3.2097E-03	2.51E-02	6.61E-03	4.1E-03	11	3.02E-11	
EO	1.031	1.1E-06	2.8E-06	2.6E-06	0.033	29.18	1.60	1.33	1.99	2.2483E-03	3.40E-03	2.92E-03	2.3E-04	4	3.02E-11	
ISA	1.032	2.6E-06	1.2E-09	1.8E-08	0.034	22.20	1.32	1.79	2.00	2.0509E-03	3.39E-03	2.43E-03	4.6E-04	2	3.02E-11	
RUN	1.032	3.3E-07	2.3E-07	3.0E-06	0.035	22.98	1.23	1.25	1.41	2.1025E-03	4.75E-03	2.97E-03	5.4E-04	6	3.02E-11	
GWO	1.031	1.9E-07	3.6E-08	4.5E-06	0.033	28.98	1.25	1.44	1.40	2.4142E-03	4.09E-03	3.05E-03	3.9E-04	7	3.02E-11	
MPA	1.030	1.0E-09	2.0E-06	8.0E-06	0.034	33.04	2.00	1.30	1.83	2.3276E-03	3.47E-03	2.95E-03	3.1E-04	5	3.02E-11	
SMA	1.029	1.0E-09	1.6E-07	4.1E-06	0.034	40.92	1.00	1.20	1.40	2.5582E-03	5.05E-03	3.35E-03	7.1E-04	10	3.02E-11	
DO	1.032	1.4E-06	6.6E-07	8.3E-07	0.036	20.59	1.46	1.23	1.40	2.2057E-03	4.36E-03	3.33E-03	4.6E-04	9	3.02E-11	
POA	1.031	2.0E-06	8.9E-06	8.5E-08	0.034	28.17	1.30	2.00	1.40	2.2247E-03	4.14E-03	3.17E-03	3.7E-04	8	3.02E-11	
Bold values highlight the best findings.

Fig. 12 Various comparisons among algorithms using TDM-based PWP module.

Fig. 12

6.2 Application II: Parameters selection of PEMFCs

As an additional engineering optimization problem, parameter estimation for PEMFCs is discussed here in order to evaluate the performance of the proposed INOA. In order to demonstrate the efficacy of the proposed INOA, it is compared to nine well-known metaheuristic algorithms and the classical NOA. These algorithms are ISA [37], GBO [38], DE [22], MPA [23], MRFO [39], equilibrium optimizer (EO), MFO [39], SSA [40], and IHOA [38]. Six popular PEMFC stacks (500W Stack [88], 250W stack [105], Ballard Mark V5 [106], and Avista SR-12 [107]) have been utilized to evaluate the effectiveness of each of the aforementioned techniques. Except for the number of function evaluations and the population size, which are both set to 5000 and 25, respectively, we have followed the recommendations of the cited studies when configuring the parameters of the methods being compared. The suggested HGTO has its parameters determined in a manner to be further upon in a subsequent section. The search boundary for these unknown factors according to several studies is as follows [108]:−1.1997<ξ1<−0.8532

0.001<ξ2<0.005

3.6e−5<ξ3<9.8e−5

−2.6e−4<ξ4<−9.54e−5

13.000<λ<23.000

1e−4<Rc<8e−4

1.36e−2<β<50e−2

6.2.1 INOA's parameter tuning for PEMFC

Accurately identifying four newly proposed regulating parameters is a prerequisite for optimizing INOA's performance when dealing with the parameter estimation of the PEMFC. Here, we make an educated guess by testing a broad variety of settings on a BCS 500W stack module, as illustrated in Fig. 13. Three of these parameters: δ, δ1 and σ are for the newly proposed CIS. According to Fig. 13(b–d), the optimal values for these parameters are determined to be 0.05, 0.3, and 9, respectively. During optimization, another parameter, Pse, decides how likely the conventional NOA will be carried out. Several trials with varying values for this parameter have been performed, and the results are shown in Fig. 13(a). As can be seen in this figure, a value of 0.2 is the best for this parameter. The other classical NOA parameters are established according to the recommendations provided in Ref. [96].Fig. 13 Adjusting the parameters of the proposed INOA for the parameter estimation of PEMFC.

Fig. 13

6.2.2 BCS 500W stack module

This commercial PEMFC module is commonly constructed with the following structural parameters and operating circumstances for evaluating optimization methods in the literature: Ncells=32, PO2(atm)=0.2095, A(cm2)=64, l(μm)=178, Tfc(K) = 333, Jmax(mAcm2)=469, and PH2(atm)=1.0 [109,110]. Table 9 displays the best, average (Avg), and worst SSE values, together with their SD collected across 30 independent times to demonstrate stability for each technique. This table shows that INOA is far superior to its competitors in terms of all the performance metrics. Results from INOA and competing optimizers are compared using the Wilcoxon rank-sum test to establish whether or not there is a statistically significant difference between them. Table 9 displays the p-values that determine whether or not there is a statistically significant difference between INOA and each of the other algorithms when comparing their results using a significance threshold of 0.05. INOA's potential for a significantly different outcome compared to competing optimizers is laid out in this table. The accuracy requirements of some algorithms may necessitate several function evaluations. However, this may greatly increase the temporal complexity of the algorithms, making them less acceptable for solving optimization issues than alternatives that reach the same degree of accuracy with fewer function evaluations. Thus, the convergence curves of various algorithms on this test case are displayed in Fig. 14 (a), revealing that INOA converged faster than all the rival optimizers. Fig. 14 (b) shows a boxplot of the five-number summary for thirty fitness values obtained by each algorithm. The five-number summary appears to support INOA according to this figure.Table 9 Performance evaluation on 500W stack.

Table 9Parameter	ξ1(V)	ξ2(V/k)	ξ3(V/k)	ξ4(V/k)	λ	Rc(Ω)	β	SSE (best)	SSE (Avg)	SSE (Worst)	SSE (SD)	p-value	
INOA	−0.8913	3.36E-03	8.78E-05	−1.93E-04	20.8772	1.00E-04	0.01613	0.01170	0.01177	0.01353	0.00034		
NOA	−0.8544	2.46E-03	4.52E-05	−1.94E-04	21.0093	2.17E-04	0.01505	0.02351	0.07006	0.23119	0.04758	3.02E-11	
MFO	−1.1854	4.43E-03	9.80E-05	−1.89E-04	18.7558	1.38E-04	0.01474	0.01700	5.89177	8.57576	1.80059	2.98E-11	
SSA	−0.9214	3.04E-03	6.62E-05	−1.57E-04	14.3364	1.34E-04	0.02493	2.16320	6.20032	9.48733	1.24187	3.02E-11	
MPA	−0.9262	2.51E-03	3.60E-05	−2.02E-04	22.9994	1.35E-04	0.01399	0.10044	2.94899	7.25196	2.18282	3.02E-11	
ISA	−0.9450	3.57E-03	9.05E-05	−1.93E-04	22.0370	2.07E-04	0.01619	0.01178	0.02007	0.06372	0.01093	2.61E-10	
MRFO	−1.1835	4.43E-03	9.80E-05	−1.91E-04	20.4146	1.95E-04	0.01530	0.01325	0.03764	0.13636	0.02553	3.34E-11	
GBO	−1.1997	4.17E-03	8.16E-05	−1.93E-04	20.8610	1.00E-04	0.01612	0.01170	0.02071	0.03248	0.00807	8.89E-10	
DE	−0.8532	2.16E-03	3.60E-05	−9.54E-05	13.0000	8.00E-04	0.02214	5.77578	5.80561	5.94820	0.03746	3.02E-11	
EO	−1.1619	4.30E-03	9.48E-05	−1.91E-04	20.2425	1.99E-04	0.01540	0.01400	0.03138	0.11614	0.02188	3.02E-11	
IHOA	−1.0823	4.01E-03	9.19E-05	−1.94E-04	22.9988	1.92E-04	0.01681	0.01261	0.10058	0.52536	0.13554	4.50E-11	

Fig. 14 Various comparisons among algorithms using the BCS 500W stack module.

Fig. 14

6.2.3 250W stack module

Here, we evaluate the performance of our proposed INOA against that of several competing optimizers by utilizing a different PEMFC stack, namely 250W stack, which is commonly utilized to probe the efficacy of the parameter estimation algorithms with the following parameters: Ncells=24, l(μm)=127, Tfc(K) = 343.15, A(cm2)=27, A(cm2)=27, Jmax(mAcm2)=860, PO2(atm)=1.0, and PH2(atm)=1.0 [109]. Results for all aforementioned performance metrics, optimal parameter values, and Wilcoxon rank-sum test results for each method after 30 separate runs are shown in Table 10. This table clearly shows that INOA outperforms all other algorithms in the vast majority of the evaluated criteria. The p-value from the Wilcoxon rank-sum test shown in the same table demonstrates how significantly different INOA is from the other optimizers. The rates of convergence reached by each approach in this test case are shown in Fig. 15(a), which serves as an additional performance metric. This figure shows that INOA is faster than any of the other rival algorithms. A boxplot presenting a five-number summary of the results achieved by each algorithm is shown in Fig. 15(b), which affirms the superiority of INOA.Table 10 Performance evaluation on 250W stack.

Table 10Parameter	ξ1(V)	ξ2(V/k)	ξ3(V/k)	ξ4(V/k)	λ	Rc(Ω)	β	SSE (best)	SSE (Avg)	SSE (Worst)	SSE (SD)	p-value	
INOA	−1.0301	3.53E-03	7.32E-05	−1.75E-04	19.9358	1.00E-04	0.01453	0.33598	0.33606	0.33733	0.00026		
NOA	−1.1091	3.23E-03	4.17E-05	−1.77E-04	21.9517	3.98E-04	0.01549	0.33952	0.35935	0.43802	0.02519	3.02E-11	
MFO	−0.8532	2.91E-03	6.74E-05	−1.74E-04	19.3380	1.00E-04	0.01360	0.33718	1.82701	2.04445	0.40567	3.33E-11	
SSA	−1.0932	3.35E-03	5.51E-05	−1.03E-04	13.6648	3.47E-04	0.01389	1.76612	2.10430	2.97399	0.39110	3.02E-11	
MPA	−0.8535	2.38E-03	3.60E-05	−1.73E-04	22.1000	8.00E-04	0.01486	0.34576	1.29498	2.59359	0.65279	3.02E-11	
ISA	−0.9950	3.78E-03	9.45E-05	−1.75E-04	19.9358	1.00E-04	0.01453	0.33598	0.34190	0.37809	0.00935	1.07E-09	
MRFO	−1.1997	4.45E-03	9.78E-05	−1.75E-04	20.1825	1.00E-04	0.01474	0.33632	0.34820	0.40312	0.01460	4.50E-11	
ChOA	−0.9820	3.66E-03	8.95E-05	−1.75E-04	19.9383	1.00E-04	0.01453	0.33598	0.33785	0.34230	0.00166	5.46E-09	
DE	−0.9401	3.00E-03	6.11E-05	−9.54E-05	14.0492	8.00E-04	0.01360	1.91526	1.94427	2.02862	0.03166	3.02E-11	
EO	−1.1808	3.97E-03	7.31E-05	−1.74E-04	19.9934	1.59E-04	0.01462	0.33665	0.34202	0.36445	0.00551	3.69E-11	
IHOA	−0.8899	3.52E-03	9.72E-05	−1.73E-04	19.8430	2.18E-04	0.01401	0.33731	0.40675	0.59601	0.08003	3.34E-11	

Fig. 15 Various comparisons among algorithms using 250W stack module.

Fig. 15

6.2.4 Ballard Mark V stack

In this section, we will compare the performance of the proposed optimizer to that of its competitors using another PEMFC stack, namely Ballard Mark V, which is frequently used in the literature [28,87,94,[111], [112], [113], [114], [115]]. The characteristics of this stack are as follows [116]: Ncells=35, Jmax(mAcm2)=1500, A(cm2)=50.6, PO2(atm)=1.0, l(μm)=178, Tfc(K) = 343, and PH2(atm)=1.0. In Table 11, the results of 30 independent runs are presented, including the performance indicators mentioned earlier, optimal parameter values, and the Wilcoxon rank-sum test. The table demonstrates that INOA outperforms other competing algorithms across a wide range of metrics. This distinction is supported by the p-value obtained from the Wilcoxon rank-sum test, which is also reported in the table. The convergence rates achieved by each method on this specific test case are visually depicted in Fig. 16(a), providing an additional measure of performance. The figure clearly illustrates that INOA significantly outperforms its competitors in terms of speed. Furthermore, Fig. 16(a), displays a boxplot with a five-number summary of the outcomes obtained by each method, further confirming the superiority of INOA over other approaches.Table 11 Performance evaluation on Ballard Mark V stack.

Table 11Parameter	ξ1(V)	ξ2(V/k)	ξ3(V/k)	ξ4(V/k)	λ	Rc(Ω)	β	SSE (best)	SSE (Avg)	SSE (Worst)	SSE (SD)	p-value	
INOA	−0.9116	2.88E-03	3.87E-05	−1.63E-04	23.0000	1.00E-04	0.01360	0.85361	0.86342	0.98448	0.03128		
NOA	−0.9709	3.10E-03	4.36E-05	−1.48E-04	22.9419	2.58E-04	0.01433	0.96809	1.27955	1.78641	0.22339	3.69E-11	
MFO	−0.8532	2.76E-03	4.30E-05	−1.55E-04	23.0000	1.64E-04	0.01360	0.89261	1.77566	2.28188	0.21182	4.07E-11	
SSA	−0.8903	3.20E-03	7.02E-05	−9.64E-05	22.7479	7.93E-04	0.01360	1.75249	1.97583	2.31132	0.19027	3.02E-11	
MPA	−1.1995	3.73E-03	4.21E-05	−1.47E-04	23.0000	1.00E-04	0.01804	0.98041	1.95581	4.81013	0.76040	3.34E-11	
ISA	−0.8542	2.67E-03	3.62E-05	−1.63E-04	23.0000	1.00E-04	0.01360	0.85361	0.96864	1.53719	0.19570	4.98E-04	
MRFO	−1.1997	4.72E-03	9.78E-05	−1.63E-04	22.9998	1.00E-04	0.01360	0.85364	1.01387	1.54656	0.26593	2.57E-07	
GBO	−1.1997	4.62E-03	9.23E-05	−1.63E-04	23.0000	1.00E-04	0.01360	0.85361	0.93443	1.54282	0.18395	2.71E-02	
DE	−0.8532	3.21E-03	7.73E-05	−9.54E-05	22.8884	8.00E-04	0.01360	1.76560	1.77265	1.83107	0.01271	3.02E-11	
EO	−0.9250	3.29E-03	6.19E-05	−1.54E-04	22.8786	1.14E-04	0.01474	0.89492	1.02066	1.44160	0.12491	3.47E-10	
IHOA	−1.1977	3.74E-03	4.10E-05	−1.56E-04	23.0000	1.87E-04	0.01360	0.89576	1.15194	1.53166	0.21424	1.33E-10	

Fig. 16 Various comparisons among algorithms using Ballard Mark V stack module.

Fig. 16

6.2.5 SR-12 PEMFC stack

In this part, the algorithm's efficacy is further evaluated with an additional widely-used PEMFC stack, SR-12, with the following characteristics: Ncells=48, PO2(atm)=0.2095, l(μm)=25, Jmax(mAcm2)=672, Tfc(K) = 343, A(cm2)=62.5, and PH2(atm)=1.47628 [28,[116], [117], [118], [119]]. Table 12 displays the outcomes of 30 independent runs for each method, including results for the aforementioned performance metrics, optimal parameter values, and Wilcoxon rank-sum tests. As can be seen from the comparison table below, INOA performs better than its competitors for all performance metrics. In the same table, the p-value from the Wilcoxon rank-sum test reveals that INOA is significantly different from the other optimizers. A further measure of performance is provided by displaying the rates of convergence achieved by each method in this test case, as shown in Fig. 17(a). This figure demonstrates that INOA is faster than all the rival optimizers. The superiority of INOA is confirmed by a boxplot in Fig. 17(b), which displays a five-number summary of the results achieved by each algorithm.Table 12 Performance evaluation on SR-12 stack.

Table 12	ξ1(V)	ξ2(V/k)	ξ3(V/k)	ξ4(V/k)	λ	Rc(Ω)	β	SSE (best)	SSE (Avg)	SSE (Worst)	SSE (SD)	p-value	
INOA	−1.0641	3.07E-03	3.93E-05	−1.06E-04	21.5177	2.73E-04	0.15001	1.4210E-04	1.4482E-04	1.7467E-04	6.145E-06		
NOA	−1.0018	3.63E-03	7.99E-05	−1.06E-04	14.2199	1.82E-04	0.14832	2.2926E-03	3.9442E-02	1.8278E-01	4.482E-02	3.02E-11	
MFO	−1.0365	2.92E-03	3.60E-05	−9.54E-05	15.7556	8.00E-04	0.14314	1.0304E-01	1.4450E-01	4.4878E-01	8.515E-02	3.02E-11	
SSA	−1.0172	2.92E-03	3.91E-05	−9.62E-05	13.7118	7.96E-04	0.14164	9.1869E-02	1.6677E-01	2.9486E-01	6.196E-02	3.02E-11	
MPA	−0.8534	2.38E-03	3.76E-05	−1.03E-04	22.9992	4.31E-04	0.14996	2.3594E-02	1.1917E+00	5.3421E+00	1.275E+00	3.02E-11	
ISA	−1.0473	3.64E-03	7.31E-05	−1.06E-04	18.7773	2.28E-04	0.14985	1.4463E-04	5.6636E-03	3.0966E-02	8.050E-03	2.61E-10	
MRFO	−0.8532	2.35E-03	3.60E-05	−1.06E-04	20.4930	2.49E-04	0.15010	1.5950E-04	5.8870E-02	3.0440E-01	7.573E-02	3.34E-11	
GBO	−1.1997	4.44E-03	9.07E-05	−1.06E-04	22.9427	2.95E-04	0.15002	1.4589E-04	2.4672E-02	1.0271E-01	2.825E-02	5.49E-11	
DE	−1.0784	3.05E-03	3.60E-05	−9.54E-05	13.2577	8.00E-04	0.14120	1.0384E-01	2.3918E-01	4.9404E-01	1.122E-01	3.02E-11	
EO	−0.9243	3.15E-03	6.73E-05	−1.06E-04	18.7067	2.67E-04	0.14932	3.6024E-04	1.3605E-02	5.0885E-02	1.177E-02	3.02E-11	
IHOA	−0.9868	3.40E-03	6.99E-05	−1.07E-04	20.8479	2.55E-04	0.15004	1.6325E-04	2.3076E-02	6.7990E-02	2.015E-02	3.34E-11	

Fig. 17 Various comparisons among algorithms using SR-12 stack module.

Fig. 17

6.3 Application III: Multilevel thresholding image segmentation problem

In this section, an additional optimization problem, namely a multithresholding image segmentation problem, with discrete variables is employed to investigate the performance of the proposed algorithm. Twenty test images taken from the Berkeley Segmentation Dataset (BSDS500) are used to validate the performance of INOA and the rival optimizers [120]. The original images and the histogram of some of these images are shown in Fig. 18(a–p). Several robust image segmentation algorithms, such as WOA [121], SMA [122], DE [123], TLBO, GWO [124], NOA [96], EO [2], MPA [125], and IMPA [125], are used to reveal the effectiveness of the proposed INOA. The controlling parameters of those rival algorithms are set to the recommended values in the cited papers, except tmax and N, which are set to 50 and 25 to ensure a fair comparison. Table 13 shows the parameters of the rival optimizers as recommended in the cited papers.Fig. 18 Original image and its histogram.

Fig. 18

Table 13 Controlling parameter values of rival optimizers.

Table 13Algorithm	Parameter	Value	
DE [123]	Cr	0.5	
F	0.5	
WOA [121]	Constant (a)	Is linearly decreased from 2 to 0	
GWO [124]	Self-adaptive	
SMA [122]	Constant (z)	0.03	
EO [2]	a1	1	
a2	2	
IMPA [125]	P	0.5	
FADs	0.2	
perIter	3	
TLBO [126]	TF	[1,2]	
NOA [96]	Prp	0.2	
Pa2	0.4	
δ	0.05	
MPA [125]	P	0.5	
FADs	0.2	

6.3.1 Performance metrics

Several performance metrics are employed in this study to observe the quality of the segmented images compared to the original images. The first metric is the peak signal-to-noise ratio (PSNR), which is used to compute the mean square error (MSE) between the segmented and original images [127]. PSNR does not consider the image's structure, so the structural similarity index (SSIM) was designed to remedy this shortcoming by comparing segmented and original images based on similarity, contrast distortion, and brightness [127]. The last metric is the feature similarity index (FSIM) proposed to compute the feature similarity between the original image and the segmented one [128].

6.3.2 INOA's parameter tuning for image segmentation

To optimize INOA's performance when dealing with the multithresholding picture segmentation problem, it is necessary to first precisely identify four newly proposed regulating parameters. These are estimated here by conducting extensive experiments with a wide range of values on the test image: image_1 with a threshold level of 10, as shown in Fig. 19(a–d). Three of these parameters: δ, δ1 and σ are for the newly proposed CIS. According to Fig. 19(b–d), we can deduce that the best values for these parameters are 0.05, 0.5, and 10, respectively. There is another parameter, namely Pse responsible for determining the probability of executing the classical NOA during the optimization parameters. To estimate the optimal value for this parameter, several experiments under various values have been conducted and their outcomes are reported in Fig. 19(a). This figure shows that the optimal value for this parameter is 10. Regarding the other parameters of the classical NOA, they are set as recommended in Ref. [96].Fig. 19 Adjusting the parameters of the proposed INOA for multilevel threshold image segmentation.

Fig. 19

6.3.3 The performance evaluation of INOA and other optimizers

All algorithms are executed 30 independent times and the average fitness values under different Threshold levels (k), including 5-k, 6-k, 10-n, 20-n, and 30-n, for each algorithm on all test images are presented in Fig. 20. This figure shows that INOA is competitive with the majority of the others for the small threshold levels up to 10, and superior for the threshold levels greater than that. Also, Fig. 21 shows the summation of the total average fitness values for all threshold levels. The figure demonstrates the superiority of the proposed algorithm compared with the other algorithms in fitness results. In addition, Fig. 22 displays the average PSNR values for each algorithm applied to all test images, based on the various threshold levels (k). According to this figure, INOA is comparable with the vast majority of the other algorithms for the minor threshold levels up to 10, and it is superior for the threshold levels that are higher than that. Additionally, the sum of the total average PSNR values is displayed in Fig. 23 for all threshold levels on all test images. The comparison shown in this figure between the proposed method and the other algorithm reveals that the proposed algorithm is superior. Regarding the other metrics, Fig. 24, Fig. 25, Fig. 26, Fig. 27 are presented to show the average value for these metrics obtained by each algorithm under various investigated threshold levels. Inspecting this figure reveals that INOA is on par with some of the others for the threshold smaller than or equal to 10, and superior for the threshold levels greater than that. A further comparison of performance is provided by displaying the convergence rates achieved by each method on some test images under the threshold level of 30, as shown in Fig. 28(a–c). This figure demonstrates that INOA is faster than all the rival optimizers. The superiority of INOA is confirmed by a boxplot in Fig. 29(a–c), which displays a five-number summary of the results achieved by each algorithm on some test images on the threshold level of 30.Fig. 20 Average Fitness values on each threshold level.

Fig. 20

Fig. 21 Average Fitness values for all the test images.

Fig. 21

Fig. 22 Average PSNR values on each threshold level.

Fig. 22

Fig. 23 Average PSNR values for all the test images.

Fig. 23

Fig. 24 Average SSIM values on the threshold levels: 5, 6, and 10.

Fig. 24

Fig. 25 Average SSIM values on the threshold levels: 20, and 30.

Fig. 25

Fig. 26 Average SSIM values on the threshold levels: 5, 6, and 10.

Fig. 26

Fig. 27 Average SSIM values on the threshold levels: 20, and 30.

Fig. 27

Fig. 28 Comparison in terms of convergence curve on some test images.

Fig. 28

Fig. 29 Comparison in terms of Five-number summary on some test images.

Fig. 29

7 Conclusion and future work

Solar photovoltaics (PV) and proton exchange membrane fuel cells (PEMFCs) are two well-known renewable energy sources with several benefits over other kinds of renewable energy, including no safety issues in addition to being easy to build and transport. The mathematical models of those sources contain some unknown parameters that must be accurately estimated to simulate them more accurately for generating a better amount of electricity. This problem is a complex nonlinear optimization problem with multiple local optima, thereby making it hard to solve using traditional techniques. Therefore, metaheuristic algorithms have been recently used to solve it due to their ability to solve several optimization problems. However, the existing metaheuristic algorithms for this problem still suffer from stagnation in local optima and slow convergence speed. In addition, there is another essential problem, known as the multithresholding image segmentation problem that needs to be addressed for accurately segmenting various objects within the images, thereby improving final precision in image processing and computer vision. The existing metaheuristic algorithms could not achieve accurate results for this problem, especially with increasing threshold levels. So, this paper presents a strong alternative metaheuristic algorithm based on the improved nutcracker optimization algorithm (INOA) to accurately solve the parameter estimation problem of PEMFC and PV models as well as the multilevel image segmentation problem. INOA uses an effective method that hastens the convergence rate and prevents stagnation in local minima. Four commercial PEMFC stacks, the multi-thresholding image segmentation problem, and several PV modules with single, double, and third diode models are used to test and verify the performance of INOA when applied to the discrete and continuous problems. In addition, this algorithm is extensively compared to several metaheuristic algorithms to expose its effectiveness and efficiency for each solved problem. The experimental results show that INOA is a potent substitute for solving discrete and continuous optimization issues. Future work involves applying this technique to additional optimization issues, such as the 0–1 knapsack problem and feature selection.

Ethics approval and consent to participate

The results/data/figures in this manuscript have not been published elsewhere, nor are they under consideration by another publisher. All the material is owned by the authors, and/or no permissions are required.

Consent for publication

This article does not contain any studies with human participants or animals performed by any of the authors.

Availability of data and materials

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Funding

This research is supported by the Researchers Supporting Project number (RSP2024R389 ), 10.13039/501100002383 King Saud University , Riyadh, Saudi Arabia.

CRediT authorship contribution statement

Mohamed Abdel-Basset: Writing – review & editing, Writing – original draft, Validation, Software, Methodology, Investigation, Conceptualization, Funding acquisition, Resources, Supervision, Visualization. Reda Mohamed: Writing – review & editing, Writing – original draft, Software, Methodology, Investigation, Conceptualization, Funding acquisition, Resources, Supervision, Validation, Visualization. Ibrahim M. Hezam: Writing – review & editing, Writing – original draft, Funding acquisition, Conceptualization, Investigation, Methodology, Resources, Software, Supervision, Validation, Visualization. Karam M. Sallam: Writing – review & editing, Writing – original draft, Methodology, Investigation, Funding acquisition, Validation, Visualization. Ibrahim A. Hameed: Writing – review & editing, Writing – original draft, Funding acquisition, Investigation, Methodology, Validation.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

This research is supported by the Researchers Supporting Project number (RSP2024R389 ), 10.13039/501100002383 King Saud University , Riyadh, Saudi Arabia.
==== Refs
References

1 Korkmaz S. An artificial algae algorithm for solving binary optimization problems 9 7 2018 1233 1247
2 Abdel-Basset M. Chang V. Mohamed R. A novel equilibrium optimization algorithm for multi-thresholding image segmentation problems Neural Comput. Appl. 2020 1 34
3 Abdel-Basset M. Mohamed R. Abouhawwash M. Balanced multi-objective optimization algorithm using improvement based reference points approach Swarm Evol. Comput. 2020 100791
4 Hashim F.A. Archimedes optimization algorithm: a new metaheuristic algorithm for solving optimization problems 51 3 2021 1531 1551
5 Mirjalili S. Multi-verse optimizer: a nature-inspired algorithm for global optimization 27 2 2016 495 513
6 Holland J.H.J.S.a. Genetic algorithms 267 1 1992 66 73
7 Yao X. Liu Y. Lin G. Evolutionary programming made faster 3 2 1999 82 102
8 Zhao W. Artificial Hummingbird Algorithm: A New Bio-Inspired Optimizer with its Engineering Applications vol. 388 2022 114194
9 Abdollahzadeh B. Soleimanian Gharehchopogh F. Mirjalili S. Artificial gorilla troops optimizer: a new nature‐inspired metaheuristic algorithm for global optimization problems 36 10 2021 5887 5958
10 Shi Y. Brain storm optimization algorithm International Conference in Swarm Intelligence 2011 Springer
11 Naik A. Satapathy S.C.J.S.C. Past present future: a new human-based algorithm for stochastic optimization 25 20 2021 12915 12976
12 Zhang W. Wang N. Yang S.J.I.j.o.h.e. Hybrid artificial bee colony algorithm for parameter estimation of proton exchange membrane fuel cell 38 14 2013 5796 5806
13 Askarzadeh A.J.I.j.o.h.e. Parameter estimation of fuel cell polarization curve using BMO algorithm 38 35 2013 15405 15413
14 Amphlett J.C. Performance modeling of the Ballard Mark IV solid polymer electrolyte fuel cell: I. Mechanistic model development 142 1 1995 1
15 Yao K. A review of mathematical models for hydrogen and direct methanol polymer electrolyte membrane fuel cells 4 1‐2 2004 3 29
16 Cheddie D. Munroe N. Review and comparison of approaches to proton exchange membrane fuel cell modeling 147 1–2 2005 72 84
17 Rajasekar N. Comparative study of PEM fuel cell parameter extraction using Genetic Algorithm Ain Shams Eng. J. 6 4 2015 1187 1194
18 Mann R.F. Development and application of a generalised steady-state electrochemical model for a PEM fuel cell 86 1–2 2000 173 180
19 Xu S. Wang Y. Wang Z. Parameter estimation of proton exchange membrane fuel cells using eagle strategy based on JAYA algorithm and Nelder-Mead simplex method Energy 173 2019 457 467
20 Sarhan G.M. Agwa A.M. El-Fergany A.A. Steady-state modeling of fuel cells based on atom search optimizer Energies 12 10 2019
21 Chen Y. Wang N. Cuckoo search algorithm with explosion operator for modeling proton exchange membrane fuel cells International Journal of Hydrogen Energy 44 5 2019 3075 3087
22 Miao D. Parameter estimation of PEM fuel cells employing the hybrid grey wolf optimization method Energy 193 2020 116616
23 Diab A.A.Z. Fuel cell parameters estimation via marine predators and political optimizers IEEE Access 8 2020 166998 167018
24 Abbassi R. Accurate key parameters estimation of PEMFCs' models based on dandelion optimization algorithm Mathematics 11 6 2023 1298
25 Zhang B. Parameter identification of proton exchange membrane fuel cell based on swarm intelligence algorithm Energy 2023 128935
26 Sultan H.M. Standard and Quasi Oppositional bonobo optimizers for parameter extraction of PEM fuel cell stacks Fuel 340 2023 127586
27 Yuan Z. Wang W. Wang H. Optimal parameter estimation for PEMFC using modified monarch butterfly optimization 44 11 2020 8427 8441
28 Qin F. Parameter estimation of PEMFC based on improved fluid search optimization algorithm Energy Rep. 6 2020 1224 1232
29 Yang Z. Model Parameter Estimation of the PEMFCs Using Improved Barnacles Mating Optimization Algorithm vol. 212 2020 118738
30 Abdel-Basset M. Developments on metaheuristic-based optimization for numerical and engineering optimization problems: analysis, design, validation, and applications Alex. Eng. J. 78 2023 175 212
31 Ebrahimi S.M. Hasanzadeh S. Khatibi S. Parameter identification of fuel cell using Repairable Grey Wolf Optimization algorithm Appl. Soft Comput. 147 2023 110791
32 Shaheen A. Fuel-cell parameter estimation based on improved gorilla troops technique Sci. Rep. 13 1 2023 8685 37248236
33 Fathy A. Alanazi A. An efficient white shark optimizer for enhancing the performance of proton exchange membrane fuel cells Sustainability 15 15 2023 11741
34 Hou M. A new optimum technique for parameter identification of the proton exchange membrane fuel cells based on improved remora optimizer Energy Sources, Part A Recovery, Util. Environ. Eff. 45 1 2023 3019 3040
35 Ferahtia S. Red-tailed hawk algorithm for numerical optimization and real-world problems Sci. Rep. 13 1 2023 12950
36 Rezk H. Optimal Parameter Estimation Strategy of PEM Fuel Cell Using Gradient-Based Optimizer vol. 239 2022 122096
37 El-Hay E. El-Hameed M. El-Fergany A. Optimized parameters of SOFC for steady state and transient simulations using interior search algorithm Energy 166 2019 451 461
38 Abdel-Basset M. Improved Meta-Metaheuristic Algorithms for Optimal Parameters Selection of Proton Exchange Membrane Fuel Cells: A Comparative Study 2023
39 Fathy A. Rezk H. Ramadan H.S.M. Recent moth-flame optimizer for enhanced solid oxide fuel cell output power via optimal parameters extraction process Energy 207 2020 118326
40 El-Fergany A.A. Extracting optimal parameters of PEM fuel cells using salp swarm optimizer Renew. Energy 119 2018 641 648
41 Wang Y. Rousis A.O. Strbac G. On microgrids and resilience: a comprehensive review on modeling and operational strategies Renew. Sustain. Energy Rev. 134 2020 110313
42 Ibrahim I.A. An improved wind driven optimization algorithm for parameters identification of a triple-diode photovoltaic cell model Energy Convers. Manag. 213 2020 112872
43 Bharadwaj P. Lehman B. Modelling flexible a-si pv for increased energy capture and improved reliability 2020 IEEE Applied Power Electronics Conference and Exposition (APEC) 2020 IEEE
44 Qaraad M. Quadratic interpolation and a new local search approach to improve particle swarm optimization: solar photovoltaic parameter estimation Expert Syst. Appl. 236 2024 121417
45 Belabbes F. Using the snake optimization metaheuristic algorithms to extract the photovoltaic cells parameters Energy Convers. Manag. 292 2023 117373
46 Lakshmanan M. Kumar C. Jasper J.S. Optimal parameter characterization of an enhanced mathematical model of solar photovoltaic cell/module using an improved white shark optimization algorithm Optim. Control Appl. Methods 44 2023 2374 2425
47 Madhiarasan M. Cotfas D.T. Cotfas P.A. Black widow optimization algorithm used to extract the parameters of photovoltaic cells and panels Mathematics 11 4 2023 967
48 Abdel-Basset M. Kepler optimization algorithm: a new metaheuristic algorithm inspired by Kepler's laws of planetary motion Knowl. Base Syst. 268 2023 110454
49 Beşkirli A. Dağ İ.J.E.R. Parameter Extraction for Photovoltaic Models with Tree Seed Algorithm vol. 9 2023 174 185
50 Jordehi A.R.J.S.E. Enhanced Leader Particle Swarm Optimisation (ELPSO): an Efficient Algorithm for Parameter Estimation of Photovoltaic (PV) Cells and Modules vol. 159 2018 78 87
51 Gupta J. Nijhawan P. Ganguli S. Parameter extraction of solar PV cell models using novel metaheuristic chaotic tunicate swarm algorithm 31 12 2021 e13244
52 Ayyarao T.S. Kumar P.P. Parameter estimation of solar PV models with a new proposed war strategy optimization algorithm 46 6 2022 7215 7238
53 Long W. Parameters Estimation of Photovoltaic Models Using a Novel Hybrid Seagull Optimization Algorithm vol. 249 2022 123760
54 Ayyarao T.S.L.V. Kishore G.I. Parameter estimation of solar PV models with artificial humming bird optimization algorithm using various objective functions Soft Comput. 2023 1 22
55 Rawat N. A new grey wolf optimization-based parameter estimation technique of solar photovoltaic. Sustainable Energy Technologies and Assessments 57 2023 103240
56 Saber S. Salem S. High-performance technique for estimating the unknown parameters of photovoltaic cells and modules based on improved spider wasp optimizer SMIJ 5 5 Oct. 2023 10.61185/SMIJ.2023.55102
57 Sharma P. Parameter estimation of solar PV using Ali baba and forty thieves optimization technique Math. Probl Eng. 2022 2022
58 Qaraad M. Photovoltaic parameter estimation using improved moth flame algorithms with local escape operators Comput. Electr. Eng. 106 2023 108603
59 Abdel-Basset M. Parameter Estimation of Photovoltaic Models Using an Improved Marine Predators Algorithm vol. 227 2021 113491
60 Saber S. Salem S. An improved light Spectrum optimizer for parameter identification of triple-diode PV model SMIJ 4 4 Sep. 2023 10.61185/SMIJ.2023.44105
61 Soliman M.A. Hasanien H.M. Alkuhayli A. Marine predators algorithm for parameters identification of triple-diode photovoltaic models IEEE Access 8 2020 155832 155842
62 Touabi C. Bentarzi H. Photovoltaic panel parameters estimation using grey wolf optimization technique Engineering Proceedings 14 1 2022 3
63 Mostafa M. A new strategy based on slime mould algorithm to extract the optimal model parameters of solar PV panel Sustain. Energy Technol. Assessments 42 2020 100849
64 Elhammoudy A. Dandelion Optimizer algorithm-based method for accurate photovoltaic model parameter identification Energy Convers. Manag. X 2023 100405
65 Kuruvilla J. A review on image processing and image segmentation 2016 International Conference on Data Mining and Advanced Computing (SAPIENCE) 2016 IEEE
66 Hu R. Utilizing Large Scale Vision and Text Datasets for Image Segmentation from Referring Expressions 2016 arXiv preprint arXiv:1608.08305
67 Dirami A. Fast multilevel thresholding for image segmentation through a multiphase level set method Signal Process. 93 1 2013 139 153
68 Oliva D. Elaziz M.A. Hinojosa S. Fuzzy entropy approaches for image segmentation Metaheuristic Algorithms for Image Segmentation: Theory and Applications 2019 Springer 141 147
69 Kapur J.N. Sahoo P.K. Wong A.K. A new method for gray-level picture thresholding using the entropy of the histogram Comput. Vis. Graph Image Process 29 3 1985 273 285
70 Otsu N. A threshold selection method from gray-level histograms IEEE transactions on systems, man, and cybernetics 9 1 1979 62 66
71 Bhandari A.K. Kumar I.V. A context sensitive energy thresholding based 3D Otsu function for image segmentation using human learning optimization Appl. Soft Comput. 82 2019 105570
72 Upadhyay P. Chhabra J.K. Kapur's Entropy Based Optimal Multilevel Image Segmentation Using Crow Search Algorithm vol. 97 2020 105522
73 Sharma A. A novel opposition based improved firefly algorithm for multilevel image segmentation 81 11 2022 15521 15544
74 Zhou Y. Meta-heuristic moth swarm algorithm for multilevel thresholding image segmentation 77 18 2018 23699 23727
75 Ben Messaoud R. Parameters determination of proton exchange membrane fuel cell stack electrical model by employing the hybrid water cycle moth‐flame optimization algorithm Int. J. Energy Res. 45 3 2021 4694 4708
76 Dinkar S.K. Opposition-based Laplacian Equilibrium Optimizer with Application in Image Segmentation Using Multilevel Thresholding vol. 174 2021 114766
77 Liang H. Modified Grasshopper Algorithm-Based Multilevel Thresholding for Color Image Segmentation vol. 7 2019 11258 11295
78 Resma K.B. Nair M.S.J.J.o.k.s.u.-c. sciences i. Multilevel thresholding for image segmentation using Krill Herd Optimization algorithm 33 5 2021 528 541
79 Ma B.J. Manta Ray Foraging Optimizer-Based Image Segmentation with a Two-Strategy Enhancement 2023 110247
80 Zhang Q. Growth Optimizer: A Powerful Metaheuristic Algorithm for Solving Continuous and Discrete Global Optimization Problems 2022 110206
81 Houssein E.H. An efficient multilevel image thresholding method based on improved heap-based optimizer Sci. Rep. 13 1 2023 9094 37277531
82 Emam M.M. Houssein E.H. Ghoniem R.M. A modified reptile search algorithm for global optimization and image segmentation: case study brain MRI images Comput. Biol. Med. 152 2023 106404
83 Yang X. Multi-level threshold segmentation framework for breast cancer images using enhanced differential evolution Biomed. Signal Process Control 80 2023 104373
84 Tan Y.T. Kirschen D.S. Jenkins N. A model of PV generation suitable for stability analysis IEEE Trans. Energy Convers. 19 4 2004 748 755
85 Askarzadeh A. Rezazadeh A.J.S.E. Parameter identification for solar cell models using harmony search-based algorithms 86 11 2012 3241 3249
86 Nunes H. A new high performance method for determining the parameters of PV cells and modules based on guaranteed convergence particle swarm optimization Appl. Energy 211 2018 774 791
87 Askarzadeh A. Rezazadeh A. A new heuristic optimization algorithm for modeling of proton exchange membrane fuel cell: bird mating optimizer Int. J. Energy Res. 37 10 2013 1196 1204
88 Corrêa J.M. An electrochemical-based fuel-cell model suitable for electrical engineering automation approach IEEE Trans. Ind. Electron. 51 5 2004 1103 1112
89 Mohammadi A. A novel approach for modeling the internal behavior of a PEMFC by using electrical circuits International Journal of Hydrogen Energy 43 25 2018 11539 11549
90 El-Fergany A.A. Electrical characterisation of proton exchange membrane fuel cells stack using grasshopper optimiser IET Renew. Power Gener. 12 1 2017 9 17
91 Mann R.F. Development and application of a generalised steady-state electrochemical model for a PEM fuel cell J. Power Sources 86 1–2 2000 173 180
92 Saleh I.M. Ali R. Zhang H. Simplified mathematical model of proton exchange membrane fuel cell based on horizon fuel cell stack Journal of Modern Power Systems and Clean Energy 4 4 2016 668 679
93 Zhang G. Jiao K. Three-dimensional multi-phase simulation of PEMFC at high current density utilizing Eulerian-Eulerian model and two-fluid model Energy Convers. Manag. 176 2018 409 421
94 Singla M.K. Nijhawan P. Oberoi A.S. Parameter estimation of proton exchange membrane fuel cell using a novel meta-heuristic algorithm Environ. Sci. Pollut. Control Ser. 2021 1 16
95 Outeiro M. A new parameter extraction method for accurate modeling of PEM fuel cells Int. J. Energy Res. 33 11 2009 978 988
96 Abdel-Basset M. Nutcracker Optimizer: A Novel Nature-Inspired Metaheuristic Algorithm for Global Optimization and Engineering Design Problems 2023 110248
97 Easwarakhanthan T. Nonlinear minimization algorithm for determining the solar cell parameters with microcomputers Int. J. Sol. Energy 4 1 1986 1 12
98 Soliman M.A. Hasanien H.M. Alkuhayli A.J.I.A. Marine predators algorithm for parameters identification of triple-diode photovoltaic models 8 2020 155832 155842
99 Hashim F.A. Fick's Law Algorithm: A Physical Law-Based Algorithm for Numerical Optimization 2022 110146
100 Gandomi A.H.J.I.t. Interior search algorithm (ISA): a novel approach for global optimization 53 4 2014 1168 1183
101 Mirjalili S. Mirjalili S.M. Lewis A.J.A.i.e.s. Grey Wolf Optimizer vol. 69 2014 46 61
102 Kumar C. A new stochastic slime mould optimization algorithm for the estimation of solar photovoltaic cell parameters Optik 223 2020 165277
103 Trojovský P. Dehghani M.J.S. Pelican optimization algorithm: a novel nature-inspired algorithm for engineering applications 22 3 2022 855
104 Faramarzi A. Equilibrium Optimizer: A Novel Optimization Algorithm vol. 191 2020 105190
105 Turgut O.E. Coban M.T. Optimal proton exchange membrane fuel cell modelling based on hybrid Teaching Learning Based Optimization–Differential Evolution algorithm Ain Shams Eng. J. 7 1 2016 347 360
106 Askarzadeh A. Rezazadeh A. A grouping-based global harmony search algorithm for modeling of proton exchange membrane fuel cell International Journal of Hydrogen Energy 36 8 2011 5047 5053
107 Agwa A.M. El-Fergany A.A. Sarhan G.M. Steady-state modeling of fuel cells based on atom search optimizer Energies 12 10 2019 1884
108 Diab A.A.Z. Coyote optimization algorithm for parameters estimation of various models of solar cells and PV modules IEEE Access 8 2020 111102 111140
109 Sultan H.M. JAYATree growth algorithm for parameter identification of proton exchange membrane fuel cell models International Journal of Interactive Multimedia & Artificial Intelligence 6 2 2020
110 Sultan H.M. Parameter identification of proton exchange membrane fuel cells using an improved salp swarm algorithm Energy Convers. Manag. 224 2020 113341
111 Zhu Y. Yousefi N. Optimal parameter identification of PEMFC stacks using adaptive sparrow search algorithm Int. J. Hydrogen Energy 46 2021 9541 9552
112 Alizadeh M. Torabi F. Precise PEM fuel cell parameter extraction based on a self-consistent model and SCCSA optimization algorithm Energy Convers. Manag. 229 2021 113777
113 Özdemir M.T. Optimal parameter estimation of polymer electrolyte membrane fuel cells model with chaos embedded particle swarm optimization Int. J. Hydrogen Energy 46 2021 16465 16480
114 Song Y. Tan X. Mizzi S. Optimal parameter extraction of the proton exchange membrane fuel cells based on a new Harris Hawks Optimization algorithm Energy Sources, Part A Recovery, Util. Environ. Eff. 2020 1 18
115 Selem S.I. Hasanien H.M. El‐Fergany A.A. Parameters extraction of PEMFC's model using manta rays foraging optimizer Int. J. Energy Res. 44 6 2020 4629 4640
116 Duan B. Cao Q. Afshar N. Optimal parameter identification for the proton exchange membrane fuel cell using Satin Bowerbird optimizer Int. J. Energy Res. 43 14 2019 8623 8632
117 Askarzadeh A. Rezazadeh A. An innovative global harmony search algorithm for parameter identification of a PEM fuel cell model IEEE Trans. Ind. Electron. 59 9 2011 3473 3480
118 Askarzadeh A. Parameter estimation of fuel cell polarization curve using BMO algorithm Int. J. Hydrogen Energy 38 35 2013 15405 15413
119 Menesy A.S. Effective parameter extraction of different polymer electrolyte membrane fuel cell stack models using a modified artificial ecosystem optimization algorithm IEEE Access 8 2020 31892 31909
120 University of California, B. https://www2.eecs.berkeley.edu/Research/Projects/CS/vision/grouping/resources.html.
121 Mirjalili S. Lewis A.J.A.i.e.s. The Whale Optimization Algorithm vol. 95 2016 51 67
122 Li S. Slime Mould Algorithm: A New Method for Stochastic Optimization vol. 111 2020 300 323
123 Storn R. Price K.J.J.o.g.o. Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces 11 4 1997 341 359
124 Wang B. Robust grey wolf optimizer for multimodal optimizations: a cross-dimensional coordination approach 92 3 2022 1 30
125 Abdel-Basset M. A hybrid COVID-19 detection model using an improved marine predators algorithm and a ranking-based diversity reduction strategy IEEE Access 8 2020 79521 79540
126 Gill H.S. Teaching-learning-based optimization algorithm to minimize cross entropy for Selecting multilevel threshold values 20 1 2019 11 25
127 Hore A. Ziou D. Image quality metrics: PSNR vs. SSIM 2010 20th International Conference on Pattern Recognition 2010 IEEE
128 Zhang L. FSIM: a feature similarity index for image quality assessment IEEE Trans. Image Process. 20 8 2011 2378 2386 21292594
