
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39251720
71223
10.1038/s41598-024-71223-7
Article
Parameter characterization of PEM fuel cell mathematical models using an orthogonal learning-based GOOSE algorithm
Manoharan Premkumar 12
Ravichandran Sowmya sowmya.pr@manipal.edu

3
Kavitha S. 4
Tengku Hashim Tengku Juhana 1
Alsoud Anas R. 5
Sin Tan Ching 1
1 https://ror.org/03kxdn807 grid.484611.e 0000 0004 1798 3541 Department of Electrical and Electronics Engineering, College of Engineering, Institute of Power Engineering (IPE), Universiti Tenaga Nasional (UNITEN), Putrajaya, 43000 Kajang, Selangor Malaysia
2 grid.444321.4 0000 0004 0501 2828 Department of Electrical and Electronics Engineering, Dayananda Sagar College of Engineering, Bengaluru, 560078 Karnataka India
3 https://ror.org/02xzytt36 grid.411639.8 0000 0001 0571 5193 Department of Electrical and Electronics Engineering, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, 576104 Karnataka India
4 https://ror.org/03z0n5k81 0000 0004 1774 2107 Department of Electronics and Communication Engineering, M.Kumarasamy College of Engineering, Karur, 639113 Tamil Nadu India
5 https://ror.org/00xddhq60 grid.116345.4 0000 0004 0644 1915 Hourani Center for Applied Scientific Research, Al-Ahliyya Amman University, Amman, Jordan
9 9 2024
9 9 2024
2024
14 2097929 4 2024
26 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by/4.0/ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
In this paper, a new method is designed to effectively determine the parameters of proton exchange membrane fuel cells (PEMFCs), i.e., ξ1, ξ2, ξ3, ξ4, RC, λ, and b. The fuel cells (FCs) involve multiple variable quantities with complex non-linear behaviours, demanding accurate modelling to ensure optimal operation. An accurate model of these FCs is essential to evaluate their performance accurately. Furthermore, the design of the FCs significantly impacts simulation studies, which are crucial for various technological applications. This study proposed an improved parameter estimation procedure for PEMFCs by using the GOOSE algorithm, which was inspired by the adaptive behaviours found in geese during their relaxing and foraging times. The orthogonal learning mechanism improves the performance of the original GOOSE algorithm. This FC model uses the root mean squared error as the objective function for optimizing the unknown parameters. In order to validate the proposed algorithm, a number of experiments using various datasets were conducted and compared the outcomes with different state-of-the-art algorithms. The outcomes indicate that the proposed GOOSE algorithm not only produced promising results but also exhibited superior performance in comparison to other similar algorithms. This approach demonstrates the ability of the GOOSE algorithm to simulate complex systems and enhances the robustness and adaptability of the simulation tool by integrating essential behaviours into the computational framework. The proposed strategy facilitates the development of more accurate and effective advancements in the utilization of FCs.

Keywords

Energy
Fuel cells
GOOSE algorithm
Orthogonal learning
PEMFC parameter
Root mean square error
Subject terms

Mathematics and computing
Engineering
Manipal Academy of Higher Education, ManipalOpen access funding provided by Manipal Academy of Higher Education, Manipal

issue-copyright-statement© Springer Nature Limited 2024
==== Body
pmcIntroduction

Theoretical concepts

DC microgrids are being recognized as a critical component in the future of energy distribution, particularly due to their improved efficiency and stability. The shift is mostly due to the inherent characteristics of DC microgrids with various DC loads and the DC output from various sources, including renewable energy, battery storage, and fuel cells1,2. Out of various sources, fuel cells (FCs) are essential components in these DC microgrids. A significant advancement in this field involves the integration of hydrogen and solar energy to establish a reliable and environmentally friendly storage system referred to as hydrogen energy. Hydrogen is widely available and can be found in fossil fuels, water, and many microbes. It is the third most plentiful element on Earth, following silicon and oxygen3,4. Despite its abundance, free hydrogen gas does not occur naturally in large quantities except in natural gas reservoirs. Given its potential, the pursuit of hydrogen energy has captured growing international interest. FCs, specifically, are electrochemical devices that convert the chemical energy of hydrogen directly into electricity. The increasing deployment of FCs in transport, portable, and stationary applications underscore their significant benefits: high efficiency, environmentally friendly and quiet operation, and notably high power and energy densities. Currently, the market offers several types of FCs5. Notable among these are the microbial fuel cell, solid oxide FC, phosphoric acid fuel cell, alkaline FC, and the proton exchange membrane FC (PEMFC). The PEMFCs are widely recognized for their sophisticated advancement and widespread use, making them a common preference in diverse applications. However, the substantial expenses associated with PEMFCs necessitate the investigation of their operational circumstances. The mathematical model is important for optimizing the performance of PEMFC and minimizing costs by improving modelling procedures. The PEMFC is a complex system that is influenced by multiple factors and demonstrates dynamic and non-linear properties. The system is controlled by a combination of ordinary and/or partial differential equations5–7.

In recent years, various semi-empirical models have been developed to represent the complex behaviours of PEMFCs accurately. Foundational works by8–11 have significantly contributed to the development of these models, providing critical insights into the electrochemical and thermodynamic processes within PEMFCs. The accurate modelling of PEMFCs is essential in order to understand the operating mechanisms of FC. The accurate modelling saves time and effort and also improves operational efficiency. The polarization curve is a crucial component of PEMFC modelling since it demonstrates the relation between the output voltage and the current and summarizes the FC's behaviour under different operating conditions12,13. Despite research efforts to develop accurate models for extracting PEMFC parameters, the problems remain unsolved. Since the information in the datasheets is insufficient, it is normal to observe differences between the predicted data and the actual data provided by manufacturers14,15. Parameter estimation is a dynamic problem where the optimal solution is achieved through the many heuristic and metaheuristic methods. Conventional optimization algorithms lack accuracy and precision when solving the non-linear characteristics of PEMFC. On the other hand, metaheuristic algorithms begin with a randomly chosen estimate and are capable of converging towards a globally optimal solution, successfully solving complex optimization problems16–19. The ability of the metaheuristic algorithms to adjust and withstand difficult conditions is beneficial when handling the complicated landscape of estimating parameters in PEMFC. The researchers have developed a model for estimating the parameters that attain high accuracy and efficiency. The original mathematical model guides the design and integration of FCs, and it also offers an understanding of the physical phenomena. The FC electrochemical models strongly depend on experimental data and empirical formulations and it highlights the need for new and adaptable modelling to represent the dynamics of FC operations accurately20–22.

Literature review

Recently, the researchers started utilizing metaheuristic algorithms for deriving the unknown parameters of the PEMFC model. The precision and efficiency of such models have increased as a result of advances in intelligence approaches, which have been responsible for the transformation23–25. Metaheuristic algorithms have developed as effective and reliable methods for estimating PEMFC parameters, as discussed in various literature26–28. Traditional algorithms, such as particle swarm optimization (PSO)30 and genetic algorithm (GA)29 have been used in various real-time applications. Nevertheless, traditional methods often suffer from low computational efficiency and a dependency on initial conditions, limiting their ability to locate optimal solutions. The advanced algorithms include a variety of algorithms such as the shark smell algorithm31, coyote optimization algorithm32, beluga whale optimization algorithm33, grey wolf optimizer34,35, whale optimization algorithm36,37, grasshopper optimization algorithm38, moth flame optimizer39, bald eagle search optimizer40, bonobo algorithm41, Newton–Raphson-based algorithm42, manta ray forage optimization43, pathfinder algorithm44, reptile search algorithm45,46, harris hawk algorithm47, golden jackal algorithm48,49, jellyfish algorithm, black widow algorithm50, artificial ecosystem optimization51, artificial rabbits optimizer52,53, tree-seed algorithm54, mountain gazelle optimizer55,56, neural network algorithm23, tree-growth algorithm57, gradient-based optimizer14,58,59, sparrow search algorithm60,61, flower pollination method62, resistance–capacitance algorithm63,64, political optimizer19,65, exponential distribution algorithm66,67, marine predator algorithm68,69, and slime mould optimization algorithm70 have also been applied to this domain. In more specific studies, a combination of a teaching learning-based optimizer and differential evolution approach has been developed alongside a modified salp swarm optimizer aimed at investigating optimal PEMFC stack parameters71,72. To reduce the shortcomings of the algorithms mentioned above, several improved algorithms have been proposed in different literature. For instance, the authors of73 have introduced an improved differential evolutionary optimization algorithm, enhancing search efficiency and control parameter sensitivity through an adaptive method. Recent developments include the salp swarm algorithm74, enhanced fluid search optimization algorithm75, LSHADE-EpSin algorithm76, improved artificial ecosystem optimization77, etc., are also reported to estimate the parameters of the PEMFC model and each algorithm offers benefits in terms of precision, convergence rate, and computational burden. Furthermore, the Bayesian regularized neural network is employed for extracting PEMFC parameters along with the improved barnacles mating optimization78 and dynamic sparrow search optimizer79. Notably, the authors of80 proposed a heap-based optimizer for PEMFC parameter identification. Additionally, atom search optimization algorithms81 and Harris Hawks’ optimization82 and have been utilized for similar purposes, with the authors of83 proposing a balanced version of the slime mould algorithm for enhanced performance. Lastly, chaos-embedded PSO has also been utilized for parameter estimation, showcasing the dynamic and evolving landscape of PEMFC parameter identification through metaheuristic algorithms84.

Purpose and significance

This paper introduces an estimation algorithm that utilizes the Orthogonal Learning-based GOOSE (OLGOOSE) algorithm, an advanced meta-heuristic inspired by the unique foraging and resting behaviour of geese. The proposed GOOSE algorithm increased the exploitation capabilities and rapid convergence, making it highly effective for complex multi-modal optimization problems. The proposed method estimates critical variables ξ1,ξ2,ξ3,ξ4,λ,b,Rc in the adopted electrical fuel cell model. While these parameters are specific to our model, the OLGOOSE algorithm proposed can be adapted to estimate parameters in other empirical or semi-empirical models, as evidenced by the works of8–11. The purpose of this study is to develop a robust method for accurately determining the parameters of PEMFCs using the OLGOOSE algorithm. The study aims to enhance the precision of parameter estimation by integrating an orthogonal learning mechanism, which improves the optimization process and leads to more accurate models. The study also aims to compare the performance of the OL-GOOSE algorithm with existing methods, such as the original GOOSE algorithm and other advanced algorithms, to validate its effectiveness. Additionally, the study seeks to address the issue of computational efficiency by ensuring that the proposed method reduces the time required for parameter estimation, making it feasible for real-time applications. Another goal is to demonstrate the reliability of the OLGOOSE algorithm, which will be evaluated through comprehensive experimental data analysis.

The significance of this study lies in its potential to improve the accuracy and reliability of PEMFC models, which are crucial for optimizing fuel cell performance and durability. By providing a more precise method for parameter estimation, the study contributes to the effective design, optimization, and operational control of PEMFCs. The proposed OLGOOSE algorithm is adaptable and can be applied to various empirical and semi-empirical models, broadening its applicability and making it a valuable tool for researchers and engineers. The study also enhances computational efficiency, enabling the use of the algorithm in real-time scenarios and large-scale simulations. Furthermore, by improving the reliability of PEMFC models, the study supports advancements in automotive applications, stationary power generation, and portable power devices, contributing to the broader field of clean energy technology. Finally, the integration of orthogonal learning mechanisms in optimization algorithms presented in this study can inspire similar advancements in other complex engineering systems, thereby extending its impact beyond PEMFCs.

Addressing research gaps with OLGOOSE

The authors reviewed existing studies on parameter identification for PEMFCs and found several limitations. Many current methods struggle to achieve high accuracy due to the complex and non-linear nature of PEMFCs. These methods often require extensive computational resources and time, making them impractical for real-time applications or large-scale simulations. Additionally, the reliability of the parameter estimates can vary significantly, leading to less robust models. Furthermore, some methods are highly specific to certain types of PEMFC models and do not generalize well to other models or conditions. The proposed OLGOOSE algorithm addresses these gaps in several ways. First, by integrating an orthogonal learning mechanism, the OLGOOSE algorithm enhances the optimization process, resulting in more precise parameter estimates. This improvement in accuracy is crucial for developing reliable PEMFC models. Second, the OLGOOSE algorithm is designed to be computationally efficient, reducing the time required for parameter estimation and making it appropriate for real-time claims and large-scale simulations. Third, the adaptive behaviour and robust optimization framework of the OLGOOSE algorithm increase the reliability of the parameter estimates, ensuring consistent and dependable performance under various operating conditions. Finally, the versatility of the OLGOOSE algorithm allows it to be applied to various empirical and semi-empirical PEMFC models, making it a valuable tool for researchers and engineers working with different types of fuel cell systems and conditions.

The main contributions of this paper are outlined as follows:Proposes a robust methodology employing the OLGOOSE algorithm to estimate the most accurate parameters of PEMFC models.

Demonstrates the dominance of the OLGOOSE algorithm in finding the optimal parameters for various PEMFC models, showcasing its enhanced performance compared to traditional methods.

Checking the reliability and strength of the OLGOOSE, the convergence curve, I-V, and P–I curves are obtained.

Provide a comprehensive comparison of the OLGOOSE algorithm with other prominent optimization techniques.

The structure of this paper is organized into several parts for clarity and depth: Section "Modelling and problem formulation" details the model of the fuel cell. Section "Modelling and problem formulation" also delves into mathematical modelling, including the definition of the objective function. Section "Orthogonal learning based GOOSE algorithm" describes the implementation of the proposed OLGOOSE algorithm used in this study. Section "Results and discussions" examines the performance of the proposed strategy through empirical testing and also presents a comparison of the results obtained with those from recently published algorithms. The paper concludes with Section "Conclusions", where the findings are summarized and potential future work is discussed.

Modelling and problem formulation

Building upon the foundational semi-empirical models by8–11, this study proposes an improved parameter estimation procedure. In a PEMFC, the core components are the anode and cathode, which are separated by a polymer electrolyte membrane, as depicted in Fig. 1. The operational mechanics involve hydrogen being introduced at the anode side while oxygen is fed into the cathode. The polymer electrolyte membrane serves a dual role: it conducts ions between the two electrodes and acts as a barrier to electron flow, ensuring that electrons must travel through an external circuit, thus generating electrical output85. The PEMFC functions based on a series of electrochemical reactions. At the anode, hydrogen molecules are split into protons and electrons. The polymer membrane allows protons to pass through to the cathode, but electrons are forced to travel around the external circuit, creating an electric current. At the cathode, these electrons recombine with the protons and oxygen to form water, completing the chemical process86. At the anode, the reaction is represented by Eq. (1), and it describes the separation of hydrogen molecules into protons and electrons.Fig. 1 Structure of the PEMFC and the equivalent circuit.

1 H2→2H++2e-

Equation 2 provides the reaction at the cathode, and it illustrates the reduction of oxygen and the combination with protons and electrons to form water.2 O2+4H++4e-→2H2O

Equation 3 gives the chemical reaction that represents the total electrical energy generation.3 2H2+O2→2H2O

This reaction summarizes the complete process occurring within the PEMFC, from hydrogen oxidation at the anode to water formation at the cathode. These reactions underscore the continuous movement of ions and electrons within the fuel cell, which is essential for the production of electricity, thereby highlighting the complex yet efficient nature of PEMFCs in energy conversion24. The electrical generation is given by Eq. (4).4 Vfc=ENernst-Vact-Vohm-Vcon

In the operation of an FC, the terminal voltage Vfc is composed of several distinct components, each representing different types of voltage losses, as outlined in Eq. (4). These components are crucial for understanding how an FC converts chemical energy into electrical energy efficiently: (i) ENernst is the adjustable open-circuit voltage, which is the ideal value that the fuel cell would produce if there were no losses due to the cell's operation; (ii) Vact is the activation voltage drop, this component arises from the energy barrier that must be overcome to initiate the electrochemical reactions at the electrodes. Vact includes the activation losses from both the anode and the cathode. At the anode, hydrogen molecules are oxidized, as described by Eq. (1), while at the cathode, oxygen molecules are reduced, as described by Eq. (2). The overall reaction for the PEMFC, which results in electricity generation, is provided in Eq. (3). This comprehensive reaction highlights the continuous ion and electron movement essential for the efficient energy conversion process in PEMFCs; (iii) Vohm is the ohmic voltage drop, which occurs due to the resistance to the flow of ions through the electrolyte and the resistance to the flow of electrons in the external circuit; (iv) Vcon represents the concentration voltage drop, which happens when there are variations in the concentration of reactants at the electrode surfaces. Together, these components determine the actual operating voltage of the FC, highlighting the various inefficiencies that can occur during its operation. The expression for ENernst is provided in Eq. (5).5 ENernst=Voc=ErT,P+RTcZFlnPH2+lnPO2

6 ErT,P=E0+ΔGZF

where E0 is the standard reference voltage at standard conditions, ΔG is the change in Gibbs free energy, which is a function of temperature and pressure, Z is the number of electrons transferred in the reaction, and F is the Faraday constant, which represents the electric charge per mole of electrons. Er represents the reference voltage (standard cell potential) utilizing total Gibbs free energy and it is indeed a function of reaction pressure and temperature, and fluctuations in these parameters can significantly impact the reference voltage. The dependency of Er on temperature and pressure is given by ErT,P. The reversible thermodynamic potential for the reaction between oxygen and hydrogen in a fuel cell is determined by the Nernst equation, as illustrated in Eq. (6). In this equation, R signifies the universal gas coefficient, Tc indicates the cell temperature in K and PH2 and PO2 are the partial pressures of hydrogen and oxygen, respectively. Equation (5) can be rewritten to account for these variables in a more detailed form explicitly.7 ENernst=1.229-8.5×10-4Tc-298.15+4.385×10-5Tc×lnPH2+lnPO2

The Nernst potential ENernst is expressed as a function of temperature and partial pressures of hydrogen and oxygen. Equation (7) incorporates the standard reference potential (1.229 V) and adjusts for temperature variations with the term -8.5×10-4Tc-298.15. The pressure dependencies are reflected in the logarithmic terms involving PH2 and PO2, scaled by 4.385×10-5Tc. This formulation ensures an accurate representation of the Nernst potential under varying reaction pressures and temperatures. The partial pressures of oxygen and hydrogen within a PEMFC are detailed in Eqs. (8) and (9), respectively.8 PO2=PHcathode×PH2Oexp4.192iATc1.334×PHanode×PH2OPcathode-1-1

9 PH2=0.5PHanode×PH2Oexp1.635iATc1.334×PHanode×PH2OPanode-1-1

where PHanode and PHcathode represent the partial pressures of hydrogen and oxygen at the anode and cathode inputs, respectively, Panode and Pcathode represent the total pressures at the anode and cathode sides, respectively, PH2O represents the partial pressure of water vapour within the system, i represents the electrical current density produced by the PEMFC, and A denotes the membrane’s surface area, both of which play crucial roles in the overall functionality of the cell. The partial pressure of water vapour within the system is elaborated in Eq. (10). The partial pressure of water within the system is determined using the following equation.10 log10PH2osat=2.95×10-2×Tc-273.15-9.19×10-5×Tc-273.152+1.44×10-7Tc-273.153-2.18

Figure 2 displays the simulated current–voltage (I-V) characteristics of a single cell within a PEMFC stack based on the input fuel pressures. When the current level is low, the ohmic loss is less significant because the chemical reactions occurring at the electrode surface proceed at a slower rate. The activation potential Vact characterizes this reduced rate of reaction, a phenomenon which occurs in what is known as the active polarization region. The formula to calculate this overall activation potential is provided in Eq. (11).11 Vact=-[ξ1+ξ2T+ξ3TlnCO2+ξ4Tlni]=RTcαFlnii0

where ξ1, ξ2, ξ3, and ξ4 are the semi-empirical factors, i0 signifies the exchange current density, α is the charge transfer factor, and i is the current density. The oxygen concentration CO2 at the interface of catalyst and cathode is provided in Eq. (12).Fig. 2 Polarization curve of the PEMFC stack.

12 CO2=PO25.08×106×exp-498/T

The ohmic region in an FC is characterized by a linear slope that lies between the concentration and the active regions. This region is defined by the losses that occur due to the resistance faced by electrons passing through the external circuit and ions moving through the electrolyte. These resistances lead to a direct and linear relationship between the voltage drop and the current. Consequently, the ohmic loss, denoted as Vohm, is expressed in Eq. (13).13 Vohm=i·(Rm+Rc)

where, Rm represents the electronic resistance while Rc refers to the contact resistance or ionic resistance measured in Ωcm-2. The value of Rm, presented in Eq. (14), varies with minor changes in current or voltage and is a typical resistance characteristic.14 Rm=ρmlA

15 ρm=181.61+0.03ic/A+0.062T/3032i/A2.5λ-0.634-3iAexp4.18T-303T

where λ represents the membrane water content, which is influenced by the stoichiometric ratio of the feed gas at the anode and the relative humidity. Additionally, ρm signifies the membrane-specific resistivity, measured in Ω·cm. When the current density is extremely high, a notable voltage drop occurs due to the diminished efficiency of gas exchange, often caused by water flooding the catalyst and it is identified as the concentration region. The associated voltage loss, referred to as both mass transport loss and concentration loss, is calculated using Eq. (16).16 VCon=-b·ln1-iimax

where i represents the actual current density and imax denotes the maximum achievable current density, measured in A/cm2. The parameter b=RTc/F is a voltage coefficient whose value varies depending on the different working conditions of the FC. The schematic circuit illustrated in Fig. 1 includes the equivalent concentration resistance RCon and the equivalent activation resistance Ract. The voltage drops across Ract and Ract is denoted as VCon. In order to reduce variability in VCon, a capacitance C is utilized, which also demonstrates an Electrochemical double-layer capacitance effect as a result of the arrangement of electrodes and the membrane. ENernst denotes the Nernst voltage, which is also referred to as the thermal or open circuit voltage. The utilization of this analogous circuit is essential for examining the consistent state and active characteristics of the fuel cell. The FC numerical model integrates computational dynamics to tackle the details of charge transport, multidimensional mass and electrochemical kinetics, all of which are interconnected in a temperature-dependent manner. These aspects provide complex difficulties that can be effectively tackled by modern algorithms that can provide precise outcomes with efficient convergence rates. The performance of PEMFCs is highly dependent on efficient water management. Inadequate management can result in significant flooding, causing a substantial increase in the current density within flooded regions, up to 4.7 times larger than in non-flooded areas, resulting in extensive degradation of the FC's performance.

Combining all these elements, it is possible to get the relationship between the cell voltage (Vfc) and current density (j):17 Vfc=Er+RTcZFlnPH2+lnPO2-RTαFlnii0-i·(Rm+Rc)-RTFln1-iimax

For practical purposes and clarity in presenting the polarization curve, Eq. (16) can be simplified and rearranged as follows.18 Vfc=Voc-RTαFlnii0-i·Rm+Rc-b·ln1-iimax

Equation (17) demonstrates how the actual operating voltage of the PEMFC decreases from the open-circuit voltage Voc as the current density increases due to various losses, it encapsulates the relationship between the cell voltage and current density, providing a comprehensive understanding of the PEMFC's performance characteristics.

The semi-empirical modelling of PEMFC illustrates how its performance is influenced by concentration, ohmic, and activation losses. The activation potential is determined by semi-empirical parametric coefficients, denoted as ξ1, ξ2, ξ3, and ξ4. Ohmic losses are influenced by the ionic resistance and a scalar factor called λ, which ranges in value from 10 to 24. If the value of b is known, it is possible to calculate the concentration voltage, along with values of current density J and maximum current density Jmax. Therefore, for accurate PEMFC modelling, it is essential to determine seven key parameters: ξ1, ξ2, ξ3, ξ4, RC, b, and λ. Such variables are typically not specified in the manufacturer's datasheet and vary under diverse working circumstances, affecting the PEMFC's performance as observed in its polarization curve.

Estimating these parameters is complex but crucial for operating the FC at optimal conditions and minimizing losses. The estimation of the parameter for a low-power, multiple-output, multiple-input electrochemical PEMFC structure has been conducted using current interruption tests and system identification approaches. Advanced algorithms have proven capable of deriving more accurate values than simpler methods. Beyond parametric coefficients, several critical factors impact the operation of FC systems significantly. These include reactant stoichiometry, stack current, stack temperature, humidity, and reactant pressure. Temperature plays a dominant role in fuel cell performance, while fuel pressure and flow rate have a lesser impact. In contrast, the airflow rate and air pressure, due to their negligible effect, are not considered for control purposes. Other influential factors encompass elements within the electrochemical and fluidic domains, such as thickness, porosity, gas diffusion coefficients between water and hydrogen and water and oxygen, respectively, gas diffusion layer tortuosity, in the fluidic domain. In the electrochemical domain, factors like symmetry, catalyst layer sectional area, exponential parameters of exchange current density, and scale factor are crucial. Various optimization methods have been utilized to determine these unknown FC parameters. Research has shown that while validating parameters obtained through optimization methods with empirical data, there is invariably some error. This parameter estimation process is depicted in Fig. 3.Fig. 3 Block diagram representation of the proposed strategy.

A primary objective is to minimize the error between the experimental and estimated data, commonly using the fitness function of minimizing the Sum of Squared Errors (SSE), calculated from the differences between estimated data and experimental data collected at N data samples. The objective is to minimize the SSE, and the error function is presented in Eq. (19).19 SSE=∑i=1NVsim,i-Vexp,i2

where Vsim,i and Vexp,i are the simulated and experimental voltages at point i, respectively. Minimizing the difference between predicted and actual data is crucial for enhancing the reliability and efficiency of PEMFC systems. The Root Mean Square Error (RMSE) is used to evaluate the accuracy of model predictions against observed data in the parameter estimation process. The RMSE is derived from the SSE, and Eq. (20) presents the expression for RMSE.20 RMSE=SSEN

The optimization variables in the PEMFC model consist of seven unknown parameters, as presented in Eq. (21), and each parameter is subject to the constraints, as presented in Eq. (22).21 x=[ξ1,ξ2,ξ3,ξ4,λ,b,Rc]

22 ξimin≤ξi≤ξimaxλmin≤λ≤λmaxbmin≤b≤bmaxRCmin≤RC≤RCmax

The unknown parameters are essential for the accurate modelling of the PEMFC’s behaviour and for enhancing its performance.

Orthogonal learning based GOOSE algorithm

This section of the paper discusses the basic concepts of the GOOSE algorithm and the formulation of the proposed OLGOOSE algorithm.

GOOSE algorithm

The GOOSE algorithm is a metaheuristic optimization algorithm inspired by the behaviours of geese during foraging and rest periods87. Initially, the GOOSE algorithm populates an array, denoted as the X matrix, representing the geese's positions. Once populated, the algorithm repositions any search agents that stray outside the predefined search space. Each agent's fitness is evaluated in every iteration using standardized benchmark functions. The algorithm assesses and compares the fitness of each agent (each row in the X matrix) against all others to identify the best fitness score and position, referred to as BestFitness and BestX, respectively. To balance exploration and exploitation, a random variable named "b" is utilized, and it commands the strategy, i.e., with a 50% chance, the algorithm chooses between exploring new areas or exploiting to refine the search. The distribution of phases is evenly managed across the iterations through a conditional statement. Additionally, several auxiliary variables, such as "a", "b", and "c", are introduced to facilitate the decision-making process. These variables are generated randomly within the range of 0 to 1. A specific condition checks if "c" exceeds 0.17; if it does, it is reset to 0.17 to maintain a controlled variability in the algorithm's behaviour. The variable "a" is crucial in determining the position update.

In the exploitation phase of the GOOSE algorithm, a key requirement is to ensure group safeguarding, as discussed earlier. To achieve this, the algorithm randomly determines the stone weight supported by the goose, which ranges between 5 and 25, as per Eq. (23).23 S_Wit=randi([5,25],1,1)

The algorithm calculates the time T_o_A_Oit, which represents the duration needed for the stone to reach the Earth, randomly chosen between 1 and 0. In the subsequent formula, it is possible to calculate the overall period it takes for the sound to spread and influence each geese in the herd across all iterations. As detailed in Eq. (24), this total time is divided by the number of dimensions. The average time is then obtained by halving the total time, as outlined in Eq. (25).24 T_T=∑(T_o_Ait)dim

25 T_A=T_T2

As discussed earlier, the random variable ‘b’ is used to allocate the phases of exploration and exploitation. The value of ‘a’ is randomly chosen from between 0 and 1. If ‘a’ is greater than 0.2 and the weight ‘S_Wit’ is 12 or more, Eq. (26) is applied where ‘T_o_A_Oit’ is multiplied by the square root of ‘S_Wit’ divided by 9.81 m/s2, the standard acceleration due to gravity.26 F_F_S=T_o_A_Oit∗S_Wit29.81

Equation (27) calculates the distance sound travels, D_S_Tit, by multiplying the speed of sound in air, S_S, which is 343.2 m per second, by the time it takes for the sound to travel, T_o_A_Sit.27 D_S_Tit=S_S∗T_o_A_Sit

In this step, it is possible to calculate D_Git, the distance between another goose at rest or feeding and a guard goose. Equation (28) determines this distance by taking half of the sound travel distance D_S_Tit.28 D_Git=0.5∗D_S_Tit

To update a position within the population, specifically to awaken an individual in the flock, it needs to determine BestXit as outlined in Eq. (29), and it combines the falling object F_F_S with the product of the goose's distance D_Git and the square of the average time T_A.29 X(it+1)=F_F_S+D_Git∗T_A2

Conversely, if both stone weight S_Wit and a are less than 12, and a is less than or equal to 0.2, the new position X is calculated as described in Eq. (30). To compute the falling object speed F_F_S, multiply the time T_o_A_Oit, it takes for the object to arrive by the stone weight S_Wit divided by gravity. Furthermore, the distances of sound travel D_S_Tit and the goose D_Git are calculated using the earlier Eqs. (27) and (28).30 F_F_S=T_o_A_Oit∗S_Wit9.81

Alternatively, a new position X is calculated using the formula outlined in Eq. (31), where parameters such as the falling object speed, goose distance, mean time, and coefficient c are sequentially multiplied. In the exploitation phase, Eqs. (28) and (30) are used to compute a new X. The choice between these equations is determined by the values of variables a and W_Sit.31 X(it+1)=F_F_S∗D_Git∗T_A2∗c

In the exploitation stage, the goose wakes randomly in response to the best position exposed so far, either to control its wake-up or to protect the individual in the flock. In addition, it is necessary to ensure that if the minimum time M_T exceeds the total time T_T, M_T is then set to equal T_T. The variable alpha, which ranges from 2 to 0, decreases significantly with each iteration. This reduction is captured in Eq. (32), which is employed to refine the positioning of a new X within the search space.32 alpha=2-loopMaxit2

In this context, Maxit represents the maximum number of iterations allowable. Calculating the parameters M_T (minimum time) and alpha is essential to steer the search phase towards what is likely the optimal solution. It is significant to enable the goose to stochastically explore the positions of other populations in the search location, which is achieved by means of randn(1,dim). The variables M_T and alpha are important in enhancing the search capabilities of the GOOSE algorithm. In Eq. (33), a random number is multiplied by the minimum of time and alpha, and this product is subsequently added to the optimal location found in the search location, facilitating effective exploration and exploitation.33 X(it+1)=randn(1,dim)∗(M_T∗alpha)+Best_pos

where dim denotes the problem dimensions, and Best_pos denotes the top position found so far in the search area. The pseudocode is provided in Algorithm 1.

Algorithm 1: Pseudocode of the GOOSE algorithm

Orthogonal learning

Orthogonal Learning (OL) is a concept derived from the mathematical property of orthogonality, where two vectors are orthogonal if their dot product is zero, indicating that they are perpendicular to each other88,89. This principle can be applied to optimization algorithms to enhance their search strategies by ensuring diversity in the search directions. In OL, the search agents (or solutions) are encouraged to explore the search space in directions that are orthogonal to each other. This means that each agent explores a fundamentally different aspect or dimension of the problem space, reducing redundancy in the search process and covering more areas more efficiently. By utilizing orthogonal vectors, the algorithm can effectively escape local optima. Each orthogonal vector points in a direction that is not influenced by the others, ensuring that the agents do not cluster around local optima and instead explore more globally90,91. The OL helps in balancing exploration and exploitation. As the search progresses, the degree of orthogonality can be adjusted to focus more on exploitation, particularly as the algorithm converges towards potential solutions. The OL method is adaptable to various types of optimization problems because it does not depend heavily on the gradient of the problem space, making it suitable for non-differentiable, noisy, or highly complex landscapes.

The OL in optimization algorithms involves the use of vectors that are mutually orthogonal to each other, thus ensuring that search agents explore the search space along independent directions92. Two vectors u and v in an n-dimensional space are orthogonal if their dot product is zero:34 u·v=0

For a set of vectors to be mutually orthogonal, every pair of different vectors in the set must satisfy this condition. In practice, this can be achieved through processes such as Gram-Schmidt orthogonalization or by using predefined orthogonal matrices like Hadamard matrices in cases where dimensions allow. One straightforward method for generating orthogonal vectors in the context of an optimization algorithm is to use the QR decomposition of a randomly generated matrix93. Suppose A is a n×n matrix with randomly generated entries. The QR decomposition of A is:35 A=QR

where Q is an orthogonal matrix (the columns are orthogonal unit vectors), and R is an upper triangular matrix. The columns of Q can be used as directions for orthogonal exploration.

Proposed OLGOOSE algorithm

In the context of the GOOSE algorithm, which is inspired by the natural behaviour of geese, orthogonal learning can significantly enhance its performance by integrating the following modifications and improvements: (i) During the initialization phase, the OL method can be applied to generate initial positions of the geese (search agents) so that they are spread out over the search space in a manner that minimizes overlap and redundancy; (ii) In each iteration, instead of moving solely based on the best solution found or random perturbations, the geese can also move in directions that are orthogonal to the direction of the current best solution; (iii) As the algorithm progresses, the extent of orthogonality in the moves can be dynamically adjusted. Early in the search process, high orthogonality can be beneficial for broad exploration, while later in the process, reducing orthogonality can help in fine-tuning the solutions by focusing more on exploitation near the current best areas; (iv) By integrating OL, the GOOSE algorithm can achieve faster convergence rates and better global optima discovery. The orthogonal directions ensure that the search is not trapped in local optima and that the solution space is thoroughly explored.

The initialization of agents (geese) can be modelled using the orthogonal matrix Q. For a set of initial agents X in a n-dimensional space:36 X=Q×D

where D is a diagonal matrix whose diagonal elements are scaled according to the problem's bounds (i.e., the search space limits). During the iterative process, the algorithm can adjust each agent's position using orthogonal directions derived from the best current position Xbs:37 Xi,new=Xi+α·Qi

where α is a step size, and Qi is the ith orthogonal vector influencing the direction of the ith agent. As the search progresses, the degree of orthogonality can be controlled by a parameter β, which modulates the influence of orthogonal directions based on the phase of the optimization:38 Xi,new=Xi+βt·α·Qi

where βt decreases as the number of iterations increases, reducing the influence of orthogonal directions to allow more localized search near the end of the algorithm run. By initializing and guiding search agents in orthogonal directions, the algorithm covers the search space more comprehensively, reducing the risk of missing global optima. Orthogonal steps help maintain diversity in the population of agents, preventing them from clustering around local optima too early in the search process. With dynamic adjustment of the orthogonality parameter, the algorithm effectively transitions from broad exploration to intensive exploitation, optimizing performance over iterations. This orthogonal modelling enhances the robustness and effectiveness of the GOOSE algorithm, particularly in complex, high-dimensional search spaces where traditional methods may struggle with coverage and convergence. The pseudocode of the proposed OLGOOSE algorithm is shown in Algorithm 2.

Algorithm 2: Pseudocode of the proposed OLGOOSE algorithm

Complexity of the OLGOOSE Algorithm

The orthogonal initialization of the population matrix X using QR decomposition has a time complexity of On3 for a matrix of size n×n. However, since the matrix size is typically m×n where m is the number of population and n is the problem dimension, the initialization complexity would be Om·n2, if we assume full orthogonalization for simplicity. Each agent’s fitness is evaluated once per iteration, which gives us Om per iteration, assuming the fitness evaluation function has a constant time complexity. If the fitness function has a complexity of Of, then this step is Om·f. Updating the position of each agent includes computing orthogonal vectors and potentially performing the QR decomposition in each iteration, which has a time complexity of Om·n2. Combining all these gives us the overall time complexity of the Orthogonal Learning based GOOSE algorithm: Ok·m·n2+m·f, k denotes the number of runs.

The space complexity is calculated by the amount of memory needed to store data structures at any point in the algorithm: (i) Stores the position of each agent, requiring Om·n space; (ii) Storing the best position and fitness requires On and O1 space, respectively; (iii) During orthogonal learning, orthogonal vectors can be stored in a matrix of size m×n, which is Om·n; (iv) Variables like BestFitness, BestX, distances, times, etc., add a marginal additional space requirement, which is typically Om or On, depending on whether they store per-agent or per-dimension data. Therefore, the space complexity is Om·n.

Results and discussions

The process of identifying the parameters was conducted using MATLAB software. Initially, the parameters in the model were assigned random values within ranges, as presented in Table 1. Data from the FC was then transferred to the identification program for analysis, where it was aligned and compared with the outputs from the model. The differences between the actual data and the model outputs were encapsulated within the objective function, as detailed in Eq. (17), with parameter adjustments made iteratively. For the estimation of the unknown parameters, the proposed OLGOOSE strategy was applied to three PEM fuel cells: the NedStackPS6, SR-12, and BCS 500 W models. To validate the efficacy of the OLGOOSE strategy, its performance was evaluated against several other optimization algorithms, including the GOOSE, Gradient-Based Optimizer (GBO)58, Multi-Learning Reptile Search Algorithm (MLRSA)45, the Subtraction-Average-Based Optimizer (SABO)94, the Energy Valley Optimizer (EVO)95, the Black Widow Optimization Algorithm (BWOA)50, and the Marine Predator Algorithm (MPA)69. The operating conditions, parameters, and datasets for these specific PEMFC stacks were sourced from12,77. Details regarding the specifications of the PEMFCs under study are presented in Table 2. Notably, the cathode was supplied with air for the BCS 500W and SR-12 Modular types, whereas the NedStackPS6 type was supplied with pure oxygen. This distinction is crucial as it impacts the fractional pressure of oxygen (PO2) and the overall performance of the FCs.Table 1 Parameter limits for the FC model.

Parameters	ξ1	ξ2	ξ3	ξ4	b	λ	Rc	
Lower bounds	− 1.19969	0.001	3.6×10-5	-2.6×10-4	0.136	10	1×10-4	
Upper bounds	− 0.08532	0.005	9.8×10-5	-9.54×10-5	0.5	24	8×10-4	

Table 2 Specifications of the PEMFC cells and stacks.

Type	BCS 500W	SR-12 Modular	NedStackPS6	
N (cells)	32	48	65	
A (cm2)	64	62.5	240	
lμm	178	25	178	
PH2(atm)	1	1.47628	1	
PO2(atm)	1	0.2095	1	
imaxAcm2	0.469	0.672	5	
T(K)	333	323	343	
Cathode supply	Air	Air	Oxygen	

To ensure a balanced comparison, the number of populations and the number of iterations were standardized across all optimization algorithms, set at 40 and 1000, respectively. Throughout the optimization process, the objective function used was the RMSE between the measured voltage data and the calculated voltage outputs from the selected FC model. The goal was to minimize this RMSE. The unidentified parameters of the PEMFC functioned as the decision vectors within the optimization framework. The specific upper and lower bounds for these PEMFC variables are detailed in Table 1.

Results for all FC models

Table 3 showcases the optimal parameters obtained for various fuel cell models when subjected to different algorithms. Due to the inherently random characteristics of algorithms, the outcomes they produce can vary from one execution to another. This variability stems from the algorithms' design, which incorporates randomness to escape local optima and explore the search space extensively. To account for the stochastic behaviour of these algorithms and to ensure a robust evaluation, multiple executions are necessary, i.e., 30 individual executions. This approach helps in assessing the performance consistency of each algorithm across runs. By averaging results from several iterations, it is possible to mitigate the influence of the outlier results, providing a more accurate reflection of the algorithm's capability to identify optimal parameters reliably. For the performance comparison, error metrics are considered. Error metrics are quantitative measures used to assess the accuracy of algorithms in predicting or fitting data. A brief introduction to the error metrics is discussed as follows.Table 3 Estimated parameters of PEMFC by all algorithms.

Parameters	OLGOOSE	GOOSE	OBGBO	MLRSA	SABO	EVO	RLBWOA	OBMPA	
BCS 500W	
 ξ1	− 1.09997	− 0.24035	− 0.49072	− 0.44035	− 0.75393	− 0.58400	− 0.65772	− 0.95348	
 ξ2	0.00319	0.00100	0.00100	0.00121	0.00249	0.00130	0.00201	0.00266	
 ξ3	0.00006	0.00010	0.00004	0.00006	0.00009	0.00004	0.00007	0.00006	
 ξ4	− 0.00019	− 0.00012	− 0.00019	− 0.00019	− 0.00019	− 0.00019	− 0.00019	− 0.00019	
 Λ	23.99857	14.25850	20.87724	19.18349	22.08166	22.23056	20.10355	20.87724	
 b (V)	0.01630	0.02240	0.01613	0.01486	0.01497	0.01636	0.01464	0.01613	
 Rc (Ω)	0.00040	0.00010	0.00010	0.00011	0.00049	0.00021	0.00020	0.00010	
 MAE	1.291E−02	4.056E−01	1.309E−02	1.580E−02	1.216E−01	1.308E−02	1.800E−01	1.291E−02	
 SSE	1.170E−02	5.629E+00	1.177E−02	1.366E−02	5.012E−01	1.216E−02	2.967E+00	1.170E−02	
 RMSE	0.02549	0.21290	0.02549	0.02562	0.03542	0.02549	0.02667	0.02549	
NedStackPS6	
 ξ1	− 1.03626	− 0.30462	− 0.36150	− 0.27798	− 0.30166	− 0.30484	− 0.21725	− 0.58890	
 ξ2	2.93E−03	7.99E−04	1.09E−03	8.11E−04	7.99E−04	8.00E−04	7.99E−04	1.91E−03	
 ξ3	3.60E−05	3.60E−05	4.48E−05	4.24E−05	3.60E−05	3.60E−05	5.43E−05	5.62E−05	
 ξ4	− 9.54E−05	− 9.66E−05	− 9.66E−05	− 9.66E−05	− 9.66E−05	− 9.66E−05	− 9.66E−05	− 9.66E−05	
 Λ	13.02226	21.56335	13.02226	13.28265	20.37303	13.02226	13.02226	13.02226	
 b (V)	0.01360	0.11653	0.01360	0.01905	0.08862	0.01360	0.01360	0.01360	
 Rc (Ω)	0.00010	0.00010	0.00010	0.00010	0.00020	0.00010	0.00010	0.00010	
 MAE	0.20671	0.25614	0.20725	0.21532	0.21991	0.20573	0.20563	0.20563	
 SSE	2.10422	3.34063	2.11666	2.28959	2.35557	2.07966	2.07965	2.07933	
 RMSE	0.26777	0.27582	0.26777	0.26882	0.27146	0.26777	0.26777	0.26777	
SR− 12	
 ξ1	− 0.22942	− 0.37122	− 1.13148	− 0.60555	− 0.72124	− 0.37027	− 1.19969	− 0.48195	
 ξ2	1.077E−03	8.000E−04	4.112E−03	1.660E−03	2.220E−03	8.000E−04	3.404E−03	1.488E−03	
 ξ3	8.230E−05	3.600E−05	9.797E−05	4.474E−05	5.687E−05	3.600E−05	3.847E−05	5.833E−05	
 ξ4	− 9.540E−05	− 9.540E−05	− 9.540E−05	− 9.540E−05	− 1.019E−04	− 9.540E−05	− 1.019E−04	− 9.540E−05	
 λ	23.50649	22.14761	10.00015	12.96875	10.56164	15.10609	23.92702	23.50650	
 b (V)	0.17391	0.17351	0.16945	0.17347	0.16786	0.17029	0.18459	0.17391	
 Rc (Ω)	0.00080	0.00080	0.00054	0.00055	0.00052	0.00080	0.00011	0.00080	
 MAE	0.19998	0.20514	0.19999	0.20598	0.20587	0.19994	0.20738	0.19998	
 SSE	1.33104	1.39692	1.33105	1.36410	1.37992	1.33404	1.39320	1.33104	
 RMSE	0.27193	0.27198	0.27193	0.27292	0.27314	0.27193	0.27232	0.27193	

Mean absolute error (MAE): MAE measures the average magnitude of errors in a set of predictions without considering their direction. It is the mean of the absolute values of each error.39 MAE=1N∑i=1NVsim,i-Vexp,i

where N is the number of samples, Vsim,i is the estimated value, and Vexp,i is the experimented value.

Sum of Squared Errors (SSE): SSE calculates the total sum of squared differences between the predicted and actual values. It emphasizes larger errors due to squaring.40 SSE=∑i=1nVsim,i-Vexp,i2

In Table 3, several algorithms are compared, including OLGOOSE, across multiple datasets based on different performance parameters. OLGOOSE shows distinct advantages in optimization, with the data indicating its superior efficacy. OLGOOSE stands out primarily in its optimization precision, achieving the best or near-best scores in most parameters. For instance, in the BCS 500W dataset, the proposed algorithm hits the closest value to the optimum for the parameter ξ1, indicating its effectiveness in fine-tuning the distinctions of the dataset. The OLGOOSE consistently maintains low error metrics for all case studies, representing its capability to converge to the optimal solution with minimal deviation. The low values of b and Rc recommend that OLGOOSE controls voltage variations and resistance, and it is crucial for stable and accurate estimation. Furthermore, the high λ values indicate the high solution’s quality. OLGOOSE’s consistent performance across all the estimated parameters suggests that the proposed OLGOOSE not only surpasses in highlighting the exact optimal points but also reliably maintains the performance, showing flexibility and robustness. Through the investigation of the estimated parameters and error metrics, OLGOOSE’s position as a reliable algorithm is likely to provide better outcomes when handling complex optimization problems. The improved results obtained by OLGOOSE, as compared to other algorithms like GOOSE, OBGBO, MLRSA, SABO, EVO, RLBWOA, and OBMPA, demonstrate the impact of orthogonal learning on the algorithm’s performance, offering a significant enhancement over traditional methods.

The data shown in Table 4 illustrates that the OLGOOSE algorithm shows notable consistency and efficiency when compared to other algorithms across various metrics and datasets. OLGOOSE consistently maintains a competitive edge in BCS 500W, NedStackPS6, and SR-12. OLGOOSE has remarkable accuracy in its average values, closely aligning with the minimal error rates in all the datasets. The standard deviation (STD) of this is remarkably low, often reaching the lower limits of accuracy, suggesting its constant and exact performance across multiple runs. The low STD values highlight the stability of OLGOOSE. The other statistical parameters, such as minimum (Min) and maximum (Max) values, suggest that OLGOOSE not only demonstrates strong performance on average but also effectively avoids any prominent outliers that could obstruct optimization in practical circumstances.Table 4 Obtained statistical metrics by all algorithms.

Metric	OLGOOSE	GOOSE	OBGBO	MLRSA	SABO	EVO	RLBWOA	OBMPA	
BCS 500W	
 Mean	0.02549	0.48062	0.02565	0.02874	0.09531	0.02631	0.03750	0.02549	
 STD	1.270E−15	1.435E−01	2.830E−04	2.973E−03	6.573E−02	1.062E−03	1.911E−02	1.360E−15	
 Min	0.02549	0.21290	0.02549	0.02562	0.03542	0.02549	0.02667	0.02549	
 Max	0.02549	0.66441	0.02641	0.03508	0.20435	0.02869	0.09079	0.02549	
 RT	1.909	1.857	3.303	2.802	2.985	2.055	2.792	5.731	
 FRT	1.300	8	3	5	6.900	3.900	5.900	2	
NedStackPS6	
 Mean	0.26856	0.30093	0.26777	0.27782	0.28788	0.26791	0.32581	0.26777	
 STD	2.50E−03	7.79E−02	3.09E−13	7.36E−03	1.21E−02	4.07E−04	1.80E−01	5.12E−13	
 Min	0.26777	0.27582	0.26777	0.26882	0.27146	0.26777	0.26777	0.26777	
 Max	0.27566	0.52263	0.26777	0.29018	0.30933	0.26906	0.83743	0.26777	
 RT	3.570	3.113	4.769	5.041	4.215	5.256	5.025	8.183	
 FRT	3.700	6.800	1.550	6.400	7.400	3.950	3.650	2.550	
SR-12	
 Mean	0.27193	0.28843	0.27209	0.27467	0.28730	0.27248	0.27476	0.27193	
 STD	1.302E−16	3.777E−02	4.925E−04	1.990E−03	1.543E−02	6.248E−04	2.049E−03	4.367E−13	
 Min	0.27193	0.27198	0.27193	0.27292	0.27314	0.27193	0.27232	0.27193	
 Max	0.27193	0.39487	0.27349	0.27869	0.31854	0.27346	0.27952	0.27193	
 RT	1.992	1.907	3.425	2.860	3.048	3.089	4.854	5.830	
 FRT	1.600	6.100	2.900	6.100	7.500	4.100	5.900	1.800	
Significant values are given in bold.

OLGOOSE demonstrates worthy outcomes in terms of runtime (RT), which quantifies the efficiency and speed of categorizing the initial acceptable solution. The proposed algorithm demonstrates less RT in multiple cases, suggesting a quick approach to reaching optimal solutions; however, the RT values are slightly higher compared to the original GOOSE algorithm. When considering Friedman’s ranking test (FRT), OLGOOSE’s rankings are outstanding, and the FRT metric is essential for the comparative performance of algorithms, providing a comprehensive assessment rather than focusing on individual instances. When compared with other algorithms, the proposed OLGOOSE demonstrates superior performance. The quality of the algorithm is validated by its low error metrics and top ranks in FRT values. OLGOOSE constantly exhibits its strength and expertise in executing activities with accuracy and effectiveness in many testing conditions. OLGOOSE is a powerful choice for improving complicated problems in various applications because of its fast convergence and dependable solutions. The results confirm that OLGOOSE is a well-optimized algorithm, consistently delivering better results than other algorithms in the comparison.

A detailed analysis comparing the experimental values with the estimated values for the BCS 500 W fuel cell, obtained using the OLGOOSE algorithm, is illustrated in Fig. 4. This comparison shows a high degree of fit between the estimated voltages generated by the OLGOOSE algorithm and the actual experimental measurements, indicating strong model accuracy and effective parameter estimation.Fig. 4 Characteristics of BCS 500W FC model; (a) I–V curves, (b) I–P curves.

The quantitative evaluation of this fit, as detailed in Table 4, includes several error metrics: the MAE is reported at 1.291E−02, the SSE at 1.17E−02, and the RMSE at 0.02549. The low error metrics confirm the usefulness of the OLGOOSE algorithm in precisely determining the optimal settings for the BCS 500 W stack. Furthermore, Fig. 5 illustrates the changes in the fitness function during the process of determining the parameters for the BCS 500 W FC. The convergence curves for all the investigated algorithms are shown in Fig. 5, illustrating the efficiency and speed at which each algorithm converges towards the optimal solution. Figure 5 highlights the robustness and computational efficiency of OLGOOSE in accurately finding the optimal set of parameters. To summarize, the presented statistics and figures validate the assertion that the OLGOOSE approach is superior in precisely estimating parameters for the BCS 500 W stack.Fig. 5 Convergence curves of all algorithms (BCS 500W FC model).

Figure 6 presents a comprehensive analysis of the measured voltage data and the simulated voltage values for the NedStackPS6 FC. Figure 6 demonstrates a strong correlation between the estimated voltages generated by OLGOOSE and the experimental findings, suggesting a successful alignment and efficient modelling by the proposed algorithm and all other selected algorithms. The effectiveness of the OLGOOSE algorithm in selecting the best parameters for the NedStackPS6 is further confirmed by the error metrics provided in Table 4. The MAE is measured at 0.20671, the SSE at 2.10422, and the RMSE at 0.26777. The low MAE, SSE, and RMSE indicate that the OLGOOSE method is capable of consistently and precisely determining the optimal parameters for the NedStackPS6 stack. As a result, the model’s predictions closely align with the actual data. Figure 7 displays the changes in the fitness function as the parameters are determined for the NedStackPS6 stack. Figure 7 displays convergence curves for all algorithms, demonstrating the evolution of each approach towards the best solution during the optimization process. Figure 7 illustrates the convergence behaviour, which may be used as a visual benchmark to evaluate the effectiveness of the OLGOOSE method in selecting the best parameter set compared to other algorithms.Fig. 6 Characteristics of NedStackPS6 FC model; (a) I–V curves, (b) I–P curves.

Fig. 7 Convergence curves of all algorithms (NedStackPS6 FC model).

Figure 8 provides an insightful comparison between the measured voltage data and the data estimates derived from the OLGOOSE algorithm for the SR-12 fuel cell model. The comparison discloses that the voltage values estimated by OLGOOSE closely align with the experimental measurements. The successful alignment of the actual data and predicted data indicates that the OLGOOSE algorithm is highly effective in modelling the behaviour of the SR-12 fuel cell under varied operational conditions. Further validation of the OLGOOSE algorithm’s efficacy comes from the detailed error metrics reported in Table 4. These include an MAE of 0.19998, SSE of 1.33104, and RMSE of 0.27193. Additionally, Fig. 9 illustrates the fitness function variations during the parameter optimization process for the SR-12 stack, including convergence curves for all algorithms compared. The convergence curves specifically highlight how quickly and smoothly the OLGOOSE algorithm approaches the optimal parameters, reflecting its computational efficiency and robustness in parameter optimization tasks.Fig. 8 Characteristics of SR-12 FC model; (a) I–V curves, (b) I–P curves.

Fig. 9 Convergence curves of all algorithms (SR-12 FC model).

In terms of statistical distribution parameters, including median, range, and outliers, the boxplot analysis for the three FC models provides an understandable visual depiction of how the OLGOOSE method stacks up against alternative algorithms. As can be seen from the boxplot for BCS 500W in Fig. 10a, OLGOOSE performs more consistently and with less variability in voltage estimate due to its tightly packed quartile distribution. Additionally, the centre location of the box’s median line indicates that the data are symmetrically distributed around the median, which improves reliability. The fact that OLGOOSE’s performance shows no outliers suggests that it has remarkable control over extreme values, which reinforces its accuracy. The boxplot in Fig. 10b illustrates how OLGOOSE, in the NedStackPS6 model, maintains a small interquartile range akin to BCS 500W, indicating reliable and consistent voltage estimates over several runs. The fact that the median is comparatively lower than that of several algorithms and closely matches the experimental data suggests that the modelling was accurate. In addition to being near the box, which indicates reduced data dispersion, the minimum and maximum values also don’t have any notable outliers, suggesting consistent performance. The boxplot analysis displayed in Fig. 10c for the SR-12 model demonstrates the consistent estimation benefit provided by OLGOOSE and it becomes evident from its narrow box and shorter whiskers. The results demonstrate a highly symmetrical median, which does not deviate towards the quartiles, and it indicates that the estimates are not biased and are consistently centred. In summary, Fig. 10 provides strong evidence that OLGOOSE surpasses other methods in terms of parameter estimation in fuel cell models, delivering a higher level of reliability, consistency, and accuracy.Fig. 10 Box plot analysis; (a) BCS 500W, (b) NedStackPS6, (c) SR-12.

To thoroughly evaluate the performance of the various algorithms, we have tabulated the SSE and MAE for all case studies in Tables 5, 6, 7, 8, 9 and 10. These tables provide a comprehensive comparison of the results achieved by each algorithm. Upon examining Tables 5, 6, 7, 8, 9 and 10, it becomes evident that the proposed OLGOOSE algorithm and the OBMPA yield comparable results in terms of SSE and MAE. However, Table 4 highlights a significant difference in computational efficiency between the two. Specifically, the OBMPA requires four times more computational time than the OLGOOSE algorithm, making the latter more efficient. Furthermore, the reliability of the OLGOOSE algorithm surpasses that of the OBMPA. This is supported by the boxplot analysis, which shows that the OLGOOSE algorithm consistently produces more stable and reliable results across different datasets and case studies. When compared to the original GOOSE algorithm, the proposed OLGOOSE algorithm demonstrates superior performance metrics. Although the computational time for the OLGOOSE algorithm is slightly greater than that of the original GOOSE algorithm due to the incorporation of the orthogonal learning mechanism, this increase in computational time is justified by the significant improvements in accuracy and reliability. Moreover, when benchmarked against other algorithms, the OLGOOSE algorithm consistently outperforms them in terms of both performance metrics (SSE and MAE) and computational efficiency. The enhanced reliability of the OLGOOSE algorithm further solidifies its superiority. The orthogonal learning mechanism, despite adding some computational overhead, effectively enhances the algorithm’s capability to discover the solution space more carefully, leading to better overall performance. The proposed OLGOOSE algorithm offers a balanced approach, delivering high accuracy and reliability with reasonable computational efficiency. Its performance in terms of key metrics and reliability makes it a strong choice compared to other optimization algorithms evaluated in this study.Table 5 SSE achieved by all algorithms for BCS 500W.

iexp (A)	Vexp (V)	SSE	
OLGOOSE	GOOSE	OBGBO	MLRSA	SABO	EVO	RLBWOA	OBMPA	
0.6	29	7.71E−06	2.60E+00	5.54E−04	5.54E−04	2.08E−04	6.76E−03	4.33E−06	8.39E−06	
2.1	26.31	1.65E−05	5.39E−01	7.00E−05	7.00E−05	6.97E−06	3.52E−04	5.11E−06	4.27E−05	
3.58	25.09	1.26E−05	1.46E−01	1.96E−05	1.96E−05	2.20E−04	2.04E−04	3.30E−05	2.43E−04	
5.08	24.25	2.13E−05	3.07E−02	6.17E−05	6.17E−05	2.84E−04	4.43E−04	4.98E−05	2.48E−04	
7.17	23.37	2.93E−05	1.54E−05	9.81E−05	9.81E−05	2.65E−04	7.33E−04	6.60E−05	1.84E−04	
9.55	22.57	2.21E−04	1.78E−02	3.69E−04	3.69E−04	4.84E−04	1.30E−03	3.12E−04	3.37E−04	
11.35	22.06	2.28E−04	3.43E−02	2.42E−04	2.42E−04	2.35E−04	9.26E−04	2.15E−04	1.32E−04	
12.54	21.75	7.42E−05	4.25E−02	1.53E−04	1.53E−04	1.01E−04	6.44E−04	1.44E−04	3.91E−05	
13.73	21.45	2.27E−04	5.05E−02	2.18E−04	2.18E−04	1.07E−04	6.58E−04	2.25E−04	4.54E−05	
15.73	21.09	1.05E−02	1.47E−02	9.89E−03	9.89E−03	1.15E−02	8.56E−03	9.63E−03	1.22E−02	
17.02	20.68	2.22E−04	5.53E−02	2.84E−04	2.84E−04	4.53E−05	4.47E−04	3.58E−04	1.92E−05	
19.11	20.22	1.21E−04	4.50E−02	1.59E−04	1.59E−04	4.40E−07	1.65E−04	2.50E−04	2.66E−06	
21.2	19.76	2.20E−04	3.02E−02	1.42E−04	1.42E−04	1.32E−05	7.80E−05	2.60E−04	7.15E−06	
23	19.36	3.63E−05	1.42E−02	4.10E−05	4.10E−05	9.34E−05	3.38E−06	1.28E−04	4.26E−05	
25.08	18.86	4.18E−05	1.35E−03	3.61E−05	3.61E−05	5.55E−05	8.10E−06	1.31E−04	8.50E−07	
27.17	18.27	2.23E−05	9.08E−03	7.40E−06	7.40E−06	2.71E−09	7.63E−05	6.59E−05	1.36E−04	
28.06	17.95	1.10E−05	3.33E−02	1.84E−10	1.84E−10	6.58E−05	2.47E−04	2.56E−05	5.27E−04	
29.26	17.3	5.07E−05	1.60E−01	2.09E−04	2.09E−04	8.00E−04	1.09E−03	1.18E−04	2.58E−03	

Table 6 MAE achieved by all algorithms for BCS 500W.

iexp (A)	Vexp (V)	MAE	
OLGOOSE	GOOSE	OBGBO	MLRSA	SABO	EVO	RLBWOA	OBMPA	
0.6	29	7.71E−06	1.54E−04	8.96E−02	1.31E−03	8.01E−04	4.57E−03	1.16E−04	1.61E−04	
2.1	26.31	1.65E−05	2.26E−04	4.08E−02	4.65E−04	1.47E−04	1.02E−03	1.26E−04	3.63E−04	
3.58	25.09	1.26E−05	1.98E−04	2.12E−02	2.46E−04	8.24E−04	5.68E−04	3.19E−04	8.67E−04	
5.08	24.25	2.13E−05	2.57E−04	9.73E−03	4.36E−04	9.37E−04	1.71E−02	3.92E−04	8.74E−04	
7.17	23.37	2.93E−05	3.01E−04	2.18E−04	5.50E−04	9.05E−04	1.50E−03	4.51E−04	7.54E−04	
9.55	22.57	2.22E−04	8.12E−04	6.65E−03	1.07E−03	1.22E−03	2.00E−04	9.81E−04	1.02E−03	
11.35	22.06	1.28E−04	6.29E−04	2.25E−02	8.65E−04	8.52E−04	2.69E−03	8.14E−04	6.38E−04	
12.54	21.75	7.21E−05	4.70E−04	2.15E−03	6.86E−04	5.58E−04	2.41E−02	6.66E−04	3.47E−04	
13.73	21.45	2.27E−04	6.26E−04	2.31E−02	8.20E−04	5.75E−04	2.43E−03	8.34E−04	3.74E−04	
15.73	21.09	1.05E−02	5.68E−03	6.75E−03	5.53E−03	5.97E−03	5.14E−04	5.45E−03	6.13E−03	
17.02	20.68	2.24E−04	8.06E−04	1.31E−02	9.36E−04	3.74E−04	1.17E−03	1.05E−03	2.43E−04	
19.11	20.22	2.21E−04	6.10E−04	1.18E−02	7.01E−04	3.68E−05	7.13E−03	8.79E−04	9.05E−05	
21.2	19.76	2.20E−04	6.08E−04	9.65E−03	6.62E−04	2.02E−04	4.91E−03	8.95E−04	1.49E−04	
23	19.36	3.63E−05	2.35E−03	6.62E−03	3.56E−04	5.37E−04	1.02E−03	6.28E−04	3.63E−04	
25.08	18.86	4.18E−05	2.59E−03	2.04E−03	3.34E−04	4.14E−04	1.58E−03	6.35E−04	5.12E−05	
27.17	18.27	2.23E−05	2.62E−04	5.30E−03	1.51E−04	2.89E−06	4.85E−03	4.51E−04	6.48E−04	
28.06	17.95	1.10E−05	1.84E−03	1.01E−02	7.53E−07	4.51E−04	8.73E−04	2.81E−04	1.28E−03	
29.26	17.3	5.07E−05	3.96E−03	2.22E−02	8.04E−04	1.57E−03	1.83E−03	6.04E−04	2.82E−03	

Table 7 SSE achieved by all algorithms for NedStackPS6.

iexp (A)	Vexp (V)	SSE	
OLGOOSE	GOOSE	OBGBO	MLRSA	SABO	EVO	RLBWOA	OBMPA	
2.25	61.64	5.15E−01	5.33E−01	5.15E−01	5.20E−01	8.41E−01	5.15E−01	5.15E−01	5.15E−01	
6.75	59.57	4.56E−02	4.94E−02	4.56E−02	4.68E−02	1.57E−01	4.56E−02	4.56E−02	4.56E−02	
9.00	58.94	1.55E−03	1.41E−02	1.25E−02	1.31E−02	8.26E−02	1.25E−02	1.25E−02	1.25E−02	
15.75	57.54	1.66E−03	1.52E−03	1.66E−03	1.55E−03	1.26E−02	1.66E−03	1.66E−03	1.66E−03	
20.25	56.80	6.37E−03	6.59E−03	6.37E−03	6.23E−03	3.45E−03	6.37E−03	6.37E−03	6.37E−03	
24.75	56.13	6.96E−03	7.70E−03	6.96E−03	6.92E−03	1.69E−03	6.96E−03	6.96E−03	6.96E−03	
31.50	55.23	5.06E−03	6.32E−03	5.06E−03	5.14E−03	1.10E−03	5.06E−03	5.06E−03	5.06E−03	
36.00	54.66	1.32E−03	4.39E−03	3.76E−02	1.53E−03	2.84E−03	1.45E−03	1.45E−03	1.45E−03	
45.00	53.61	5.62E−04	7.36E−05	5.93E−02	4.70E−04	8.20E−03	5.62E−04	5.62E−04	5.62E−04	
51.75	52.86	7.09E−03	4.41E−03	5.62E−04	6.64E−03	1.79E−02	7.09E−03	7.09E−03	7.09E−03	
67.50	51.91	2.56E−01	3.44E−01	7.09E−03	2.26E−01	2.09E−01	6.57E−02	5.07E−02	1.91E−04	
72.00	51.22	3.76E−02	4.71E−01	1.45E−03	3.93E−02	3.57E−02	2.08E−02	2.73E−01	2.24E−02	
90.00	49.66	5.93E−02	7.15E−01	2.22E−01	6.18E−02	7.24E−02	1.91E−04	2.09E−01	1.35E−01	
99.00	49.00	1.22E−02	1.58E−01	2.73E−01	1.44E−01	1.70E−01	1.35E−01	1.31E−02	1.93E−04	
105.80	48.15	1.46E−02	2.04E−02	2.09E−01	1.59E−02	2.74E−02	2.22E−01	1.12E−01	5.07E−02	
110.30	47.52	1.31E−02	8.81E−03	2.24E−02	1.19E−02	4.33E−03	3.76E−02	6.57E−02	1.12E−01	
117.00	47.10	2.92E−03	5.29E−03	1.35E−01	3.51E−03	1.16E−02	1.31E−02	2.08E−02	2.09E−01	
126.00	46.48	5.32E−03	5.97E−02	6.57E−02	5.47E−02	8.23E−02	2.92E−03	1.91E−04	2.08E−02	
135.00	45.66	4.46E−02	4.96E−02	2.08E−02	4.65E−02	7.33E−02	2.09E−01	1.93E−04	6.57E−02	
141.80	44.85	1.93E−04	4.89E−04	1.41E−01	3.21E−04	5.31E−03	1.12E−01	1.35E−01	1.31E−02	
150.80	44.24	5.21E−02	4.24E−02	1.46E−02	5.22E−02	7.91E−02	5.24E−02	2.92E−03	2.73E−01	
162.00	42.45	2.73E−02	3.75E−01	5.24E−02	2.71E−01	2.25E−01	4.46E−02	5.24E−02	4.46E−02	
171.00	41.66	2.14E−01	3.14E−01	4.46E−02	2.08E−01	1.74E−01	1.93E−04	4.46E−02	2.22E−01	
182.30	40.68	1.12E−01	1.19E−01	1.91E−04	1.13E−01	9.58E−02	5.93E−02	2.24E−02	3.76E−02	
189.00	40.09	6.57E−02	7.13E−02	1.93E−04	6.67E−02	5.77E−02	1.41E−01	1.41E−01	5.93E−02	
195.80	39.51	2.26E−02	2.39E−02	1.31E−02	2.17E−02	1.93E−02	1.46E−02	1.46E−02	2.92E−03	
204.80	38.73	1.91E−04	5.25E−05	2.92E−03	8.47E−05	1.61E−05	2.24E−02	2.22E−01	5.24E−02	
211.50	38.15	2.35E−02	2.21E−02	5.07E−02	2.07E−02	1.65E−02	5.07E−02	3.76E−02	1.41E−01	
220.50	37.38	1.42E−01	1.43E−01	1.12E−01	1.29E−01	1.09E−01	6.57E−02	5.07E−02	1.91E−04	

Table 8 MAE achieved by all algorithms for NedStackPS6.

iexp (A)	Vexp (V)	MAE	
OLGOOSE	GOOSE	OBGBO	MLRSA	SABO	EVO	RLBWOA	OBMPA	
2.25	61.64	2.48E−02	2.52E−02	2.48E−02	2.49E−02	3.16E−02	2.48E−02	2.48E−02	2.48E−02	
6.75	59.57	7.25E−03	7.41E−03	7.36E−03	7.46E−03	1.37E−02	7.36E−03	7.78E−02	7.36E−03	
9.00	58.94	3.86E−02	5.21E−02	3.86E−02	3.94E−03	9.91E−03	3.86E−03	3.86E−03	3.86E−03	
15.75	57.54	1.41E−03	1.34E−23	1.41E−03	1.36E−03	3.87E−03	1.41E−03	1.41E−02	1.41E−02	
20.25	56.80	2.75E−03	2.80E−03	2.75E−02	2.72E−03	3.12E−02	2.75E−03	2.75E−03	2.75E−03	
24.75	56.13	2.88E−03	3.03E−03	2.88E−03	2.87E−03	1.42E−03	2.88E−03	3.88E−02	2.88E−02	
31.50	55.23	3.78E−03	2.74E−03	2.45E−02	2.47E−03	1.14E−02	2.45E−03	3.24E−03	3.78E−03	
36.00	54.66	1.31E−04	1.69E−03	1.31E−03	1.35E−03	1.84E−03	1.31E−03	1.78E−02	1.31E−04	
45.00	53.61	8.17E−04	2.96E−04	8.17E−04	7.47E−04	3.12E−03	8.17E−04	8.17E−04	8.17E−04	
51.75	52.86	2.90E−03	2.29E−03	2.90E−03	2.81E−03	4.62E−02	2.90E−03	2.90E−03	3.01E−03	
67.50	51.91	1.63E−02	1.70E−02	1.63E−02	1.64E−02	1.62E−02	1.63E−02	1.63E−02	1.63E−02	
72.00	51.22	6.69E−03	7.48E−03	6.69E−03	6.84E−03	6.52E−03	6.69E−03	6.69E−03	6.69E−03	
90.00	49.66	8.40E−03	9.22E−03	8.40E−03	8.57E−03	9.28E−03	8.40E−03	8.40E−03	8.40E−03	
99.00	49.00	1.29E−02	1.37E−02	1.29E−02	1.31E−02	1.78E−02	1.29E−02	1.29E−02	1.29E−02	
105.80	48.15	4.17E−03	4.92E−03	4.17E−03	4.35E−03	5.71E−03	4.17E−03	4.17E−03	4.17E−03	
110.30	47.52	3.95E−03	3.24E−03	3.95E−03	3.77E−03	2.27E−02	3.95E−03	3.95E−03	3.95E−03	
117.00	47.10	1.86E−03	2.51E−02	1.86E−03	2.04E−03	3.71E−03	1.86E−03	1.86E−03	1.86E−03	
126.00	46.48	7.90E−03	8.43E−03	7.90E−03	8.06E−03	9.89E−03	7.90E−03	7.90E−03	7.90E−03	
135.00	45.66	7.28E−03	7.68E−02	7.28E−03	7.44E−03	9.34E−03	7.28E−03	7.28E−03	7.28E−03	
141.80	44.85	4.79E−04	7.63E−04	4.79E−04	6.18E−04	2.51E−03	4.79E−04	4.79E−04	4.79E−04	
150.80	44.24	7.77E−03	7.89E−02	7.77E−03	7.88E−03	9.70E−03	7.77E−03	7.77E−03	7.77E−03	
162.00	42.45	1.80E−02	1.81E−02	1.80E−02	1.80E−02	1.64E−02	1.80E−02	1.80E−02	1.80E−02	
171.00	41.66	1.58E−02	1.60E−02	1.58E−02	1.57E−02	1.44E−02	1.58E−02	1.58E−02	1.58E−02	
182.30	40.68	1.21E−02	1.19E−02	1.78E−02	1.16E−02	1.07E−02	1.21E−02	1.21E−02	1.216E−02	
189.00	40.09	7.91E−03	9.21E−03	7.91E−03	8.90E−03	8.28E−03	7.91E−03	7.91E−03	7.91E−03	
195.80	39.51	4.97E−03	5.33E−03	4.97E−03	5.08E−03	4.79E−03	4.97E−03	4.97E−03	4.97E−03	
204.80	38.73	5.92E−04	2.50E−04	4.88E−04	3.17E−04	1.38E−04	5.92E−04	5.92E−04	5.92E−04	
211.50	38.15	6.01E−03	5.13E−04	5.47E−03	4.96E−03	4.43E−03	6.01E−03	6.01E−03	6.01E−03	
220.50	37.38	2.14E−02	1.31E−02	2.89E−02	1.24E−02	1.14E−02	2.14E−02	2.14E−02	2.14E−02	

Table 9 SSE achieved by all algorithms for SR12.

iexp (A)	Vexp (V)	SSE	
OLGOOSE	GOOSE	OBGBO	MLRSA	SABO	EVO	RLBWOA	OBMPA	
1.004	43.17	6.15E−02	5.87E−02	6.32E−02	5.61E−02	2.06E−01	8.15E−02	7.57E−02	6.15E−02	
3.166	41.14	3.32E−04	1.45E−04	5.22E−04	1.41E−04	1.41E−02	2.51E−03	6.83E−04	3.32E−04	
5.019	40.49	2.65E−01	2.72E−01	2.60E−01	2.68E−01	2.04E−01	2.38E−01	3.24E−01	2.65E−01	
7.027	39.04	1.69E−02	1.89E−02	1.56E−02	1.68E−02	8.39E−03	1.17E−02	3.23E−02	1.69E−02	
8.958	37.99	1.27E−04	3.58E−04	4.50E−05	6.62E−05	1.41E−04	3.78E−05	2.50E−03	1.27E−04	
10.97	37.08	7.65E−04	1.28E−03	5.81E−04	4.96E−04	2.42E−04	2.20E−04	2.69E−03	7.65E−04	
13.05	36.03	6.45E−03	5.16E−03	6.78E−03	7.64E−03	7.17E−03	7.88E−03	5.26E−03	6.45E−03	
15.06	35.19	2.45E−05	1.51E−05	2.64E−05	1.74E−04	2.02E−05	9.22E−05	1.68E−04	2.45E−05	
17.07	34.07	3.58E−02	3.25E−02	3.51E−02	3.92E−02	3.46E−02	3.63E−02	4.50E−02	3.58E−02	
19.07	33.02	7.53E−02	7.03E−02	7.30E−02	8.01E−02	7.32E−02	7.46E−02	9.61E−02	7.53E−02	
21.08	32.04	5.59E−02	5.16E−02	5.29E−02	5.97E−02	5.47E−02	5.46E−02	7.96E−02	5.59E−02	
23.01	31.2	1.51E−03	8.75E−04	9.23E−04	2.07E−03	1.56E−03	1.25E−03	8.21E−03	1.51E−03	
24.94	29.8	1.05E−01	9.96E−02	9.91E−02	1.09E−01	1.09E−01	1.04E−01	1.43E−01	1.05E−01	
26.87	28.96	2.36E−03	3.28E−03	3.47E−03	2.13E−03	1.17E−03	2.39E−03	2.00E−08	2.36E−03	
28.96	28.12	4.49E−01	4.59E−01	4.61E−01	4.50E−01	4.13E−01	4.42E−01	4.03E−01	4.49E−01	
30.81	26.3	1.00E−01	1.04E−01	1.03E−01	1.02E−01	7.39E−02	9.20E−02	9.24E−02	1.00E−01	
32.97	24.06	6.57E−03	7.41E−03	5.81E−03	7.93E−03	7.11E−05	2.84E−03	1.28E−02	6.57E−03	
34.9	21.4	1.48E−01	1.46E−01	1.65E−01	1.39E−01	2.45E−01	1.88E−01	8.31E−02	1.48E−01	

Table 10 MAE achieved by all algorithms for SR12.

iexp (A)	Vexp (V)	MAE	
OLGOOSE	GOOSE	OBGBO	MLRSA	SABO	EVO	RLBWOA	OBMPA	
1.004	43.17	1.38E−02	1.35E−02	1.40E−02	1.32E−02	2.52E−02	1.59E−02	1.53E−02	1.38E−02	
3.166	41.14	1.01E−03	6.68E−04	1.27E−03	6.59E−04	6.60E−03	2.78E−03	1.45E−03	1.01E−03	
5.019	40.49	2.86E−02	2.90E−02	2.83E−02	2.88E−02	2.51E−02	2.71E−02	3.16E−02	2.86E−02	
7.027	39.04	7.23E−03	7.63E−03	6.95E−03	7.21E−03	5.09E−03	6.01E−03	9.99E−03	7.23E−03	
8.958	37.99	6.26E−04	1.05E−03	3.73E−04	4.52E−04	6.59E−04	3.41E−04	2.78E−03	6.26E−04	
10.97	37.08	1.54E−03	1.99E−03	1.34E−03	1.24E−03	8.65E−04	8.24E−04	2.88E−03	1.54E−03	
13.05	36.03	4.46E−03	3.99E−03	4.57E−03	4.86E−03	4.70E−03	4.93E−03	4.03E−03	4.46E−03	
15.06	35.19	2.75E−04	2.16E−04	2.86E−04	7.32E−04	2.49E−04	5.33E−04	7.20E−04	2.75E−04	
17.07	34.07	1.05E−02	1.00E−02	1.04E−02	1.10E−02	1.03E−02	1.06E−02	1.18E−02	1.05E−02	
19.07	33.02	1.52E−02	1.47E−02	1.50E−02	1.57E−02	1.50E−02	1.52E−02	1.72E−02	1.52E−02	
21.08	32.04	1.42E−03	2.26E−02	2.28E−02	1.36E−02	1.24E−02	1.45E−02	1.57E−02	1.42E−02	
23.01	31.2	2.16E−03	1.64E−03	1.69E−03	2.53E−03	2.19E−03	1.97E−03	5.03E−03	2.16E−03	
24.94	29.8	1.80E−02	1.75E−02	1.75E−02	1.83E−02	1.84E−02	1.79E−02	2.10E−02	1.80E−02	
26.87	28.96	2.68E−03	4.18E−03	4.33E−03	2.56E−03	1.90E−03	2.72E−03	7.86E−06	3.70E−03	
28.96	28.12	3.72E−02	3.77E−02	3.77E−02	3.73E−02	3.57E−02	3.69E−02	3.53E−02	3.72E−02	
30.81	26.3	1.76E−02	1.79E−02	1.78E−02	1.78E−02	1.51E−02	1.68E−02	1.69E−02	1.76E−02	
32.97	24.06	4.71E−03	4.78E−03	5.34E−03	4.95E−03	5.52E−04	2.96E−03	6.29E−03	5.50E−03	
34.9	21.4	2.14E−02	2.12E−02	2.26E−02	2.07E−02	2.75E−02	2.41E−02	1.60E−02	2.14E−02	

In order to visualize the error metric, Fig. 11 presents error plots that comprehensively display the performance of various algorithms, including OLGOOSE, across three different FC models. The error plots in Fig. 11 serve as a visual tool to measure and compare the magnitude of errors generated by each algorithm under varying operational conditions. By examining Fig. 11, one can distinguish the consistency and accuracy of each algorithm. Specifically, Fig. 11 highlights that the proposed algorithm exhibits the smallest errors consistently across all tested conditions and models. Figure 11 clearly defines which algorithm, including OLGOOSE, manages to maintain precision across a diverse set of conditions and models, effectively highlighting its superiority in terms of both accuracy and robustness.Fig. 11 Error rate achieved by algorithms, (a) BCS 500W, (b) NedStackPS6, (c) SR-12.

Further discussions

OLGOOSE’s superior performance across all FC models can be attributed to several key features and enhancements integrated within the algorithm, which collectively improve its accuracy, robustness, and efficiency. The introduction of orthogonal learning in OLGOOSE is a fundamental enhancement that significantly contributes to its superior performance. Orthogonal learning helps in diversifying the search patterns of the algorithm, ensuring that the solution space is explored more comprehensively. By using orthogonal vectors, OLGOOSE can efficiently escape local optima and explore multiple dimensions of the problem space simultaneously, reducing the risk of convergence on sub-optimal solutions.

OLGOOSE strikes an optimal balance between global and local search strategies. The orthogonal vectors facilitate broad, global explorations initially, which is crucial for identifying promising regions within a vast search space. As the algorithm progresses, it gradually shifts towards more refined local searches, focusing in on the best solutions with high accuracy. This dynamic adjustment between exploration and exploitation phases allows OLGOOSE to maintain high accuracy and adaptability across different operational conditions and FC models. In the FC models, precise parameter estimation is critical for modelling the complex chemical and physical interactions accurately. OLGOOSE’s algorithmic structure is particularly adept at tuning these parameters to reflect the true behaviour of the system under study. By effectively minimizing the error metrics such as MAE, SSE, and RMSE, OLGOOSE demonstrates its capability to tune parameters that result in models which closely mimic real-world data. The stochastic nature of metaheuristic algorithms often leads to variability in performance. However, OLGOOSE is designed to offer more consistent results across multiple runs. This consistency is evidenced by the smaller standard deviations in error metrics, suggesting that OLGOOSE not only finds better solutions but does so reliably over successive iterations. Such reliability is particularly valuable in practical applications where repeatability of results is crucial. Despite its complex internal mechanisms, OLGOOSE is optimized for computational efficiency. OLGOOSE’s ability to quickly converge to optimal solutions without excessive computational overhead makes it suitable for real-time and scalable applications.

Fuel cells come in various types and configurations, each with unique characteristics and operational dynamics. OLGOOSE’s performance across multiple FC models suggests that it has inherent flexibility and adaptability, capable of adjusting its optimization strategy to suit different types and sizes of data sets and models. In summary, OLGOOSE’s integration of orthogonal learning, balanced search capabilities, robust parameter optimization, consistent and reliable performance, computational efficiency, and adaptability are key reasons behind its superior performance across different FC models. These features not only enhance the precision of the model but also ensure that it can be reliably used in diverse applications, reaffirming its status as a preferred algorithm in the field of optimization. The OLGOOSE algorithm’s flexibility allows for its application across different PEMFC models, requiring the estimation of various parameters unique to each model. This adaptability ensures that the approach remains robust and applicable to a wide range of modelling frameworks.

Conclusions

The proposed OLGOOSE algorithm is combined with the orthogonal learning mechanism and it demonstrated superior performance in optimizing parameters for three different fuel cell models. The superior performance of the OLGOOSE algorithm is due to the orthogonal learning mechanism, which effectively balances the exploration and exploitation phases of the original GOOSE algorithm. The investigation in this study illustrates the effectiveness of the OLGOOSE in terms of accuracy and efficiency, outperforming conventional metaheuristic algorithms across various parameters such as RMSE, SSE, MAE, RT, and FRT. The strong mechanism of OLGOOSE is the foundation of its strength and allows for precise and reliable estimation of the fuel cell behaviour. The stability and consistency demonstrated in several experimentations, statistical parameters, and statistical tests make the proposed algorithm a reliable tool for fuel cell parameter estimation. OLGOOSE addresses the challenge of parameter optimization by enhancing the operational efficiency of fuel cell systems and achieves a prominent balance between comprehensive search capabilities and precise improvements. As a result, this is highly advantageous for academics and practitioners who aim to advance the boundaries of energy technology and optimization.

In the future, OLGOOSE may expand its application to include additional complex systems such as battery management and renewable energy sources. By integrating OLGOOSE with machine learning predictions and real-time data processing, it has the potential to become a valuable tool in the energy market. This future extension would enhance its practicality in everyday situations. In addition, advancements in computational methods could enhance its effectiveness by enabling faster and more adaptable procedures that are ideal for handling larger and more complicated datasets. These additional paths are expected to enhance the usefulness of OLGOOSE and yield discoveries in the disciplines of energy system optimization and other related areas.

Acknowledgements

Funding for open access was provided by Manipal Academy of Higher Education, Manipal, Karnataka, India. This research received support from Tenaga Nasional Berhad (TNB) and UNITEN through the BOLD Refresh Publication Fund and the BOLD Refresh Postdoctoral Fellowships, under the project code J510050002-IC-6 BOLDREFRESH2025-Centre of Excellence.

Author contributions

PM: Conceptualization, Methodology, Software, Data curation, Formal analysis, Investigation, Resources, Visualization, Validation, and Writing-original draft preparation. SR: Conceptualization, Methodology, Software, Resources, Data curation, Funding, Validation, Visualization, and Writing-original draft preparation. SK: Formal analysis, Investigation, Validation, Visualization, and Writing-review and editing. TJTH, ARA, and TCS: Formal analysis, Supervision, Visualization, and Writing-review and editing.

Funding

Open access funding provided by Manipal Academy of Higher Education, Manipal.

Data availability

This study does not use any new data and existing data will be provided upon valid request.

Competing interests

The authors declare no competing interests.

Ethical approval

The authors have confirmed that no ethical approval is required.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
==== Refs
References

1. Hachana O Accurate PEM fuel cells parameters estimation using hybrid artificial bee colony differential evolution shuffled complex optimizer Int. J. Energy Res. 2022 46 5 6383 6405 10.1002/ER.7576
Hachana, O. Accurate PEM fuel cells parameters estimation using hybrid artificial bee colony differential evolution shuffled complex optimizer. Int. J. Energy Res. 46(5), 6383–6405. 10.1002/ER.7576 (2022).10.1002/ER.7576
2. El-Sharkh MY Tanrioven M Rahman A Alam MS A study of cost-optimized operation of a grid-parallel PEM fuel cell power plant IEEE Trans. Power Syst. 2006 21 3 1104 1114 10.1109/TPWRS.2006.876694
El-Sharkh, M. Y., Tanrioven, M., Rahman, A. & Alam, M. S. A study of cost-optimized operation of a grid-parallel PEM fuel cell power plant. IEEE Trans. Power Syst. 21(3), 1104–1114. 10.1109/TPWRS.2006.876694 (2006).10.1109/TPWRS.2006.876694
3. Kirubakaran A Jain S Nema RK A review on fuel cell technologies and power electronic interface Renew. Sustain. Energy Rev. 2009 13 9 2430 2440 10.1016/J.RSER.2009.04.004
Kirubakaran, A., Jain, S. & Nema, R. K. A review on fuel cell technologies and power electronic interface. Renew. Sustain. Energy Rev. 13(9), 2430–2440. 10.1016/J.RSER.2009.04.004 (2009).10.1016/J.RSER.2009.04.004
4. Akinyele D Olabode E Amole A Review of fuel cell technologies and applications for sustainable microgrid systems Inventions 2020 5 3 42 10.3390/INVENTIONS5030042
Akinyele, D., Olabode, E. & Amole, A. Review of fuel cell technologies and applications for sustainable microgrid systems. Inventions 5(3), 42. 10.3390/INVENTIONS5030042 (2020).10.3390/INVENTIONS5030042
5. Mitra U Arya A Gupta S A comprehensive and comparative review on parameter estimation methods for modelling proton exchange membrane fuel cell Fuel 2023 335 127080 10.1016/J.FUEL.2022.127080
Mitra, U., Arya, A. & Gupta, S. A comprehensive and comparative review on parameter estimation methods for modelling proton exchange membrane fuel cell. Fuel 335, 127080. 10.1016/J.FUEL.2022.127080 (2023).10.1016/J.FUEL.2022.127080
6. Fathy A Babu TS Abdelkareem MA Rezk H Yousri D Recent approach based heterogeneous comprehensive learning Archimedes optimization algorithm for identifying the optimal parameters of different fuel cells Energy 2022 248 123587 10.1016/J.ENERGY.2022.123587
Fathy, A., Babu, T. S., Abdelkareem, M. A., Rezk, H. & Yousri, D. Recent approach based heterogeneous comprehensive learning Archimedes optimization algorithm for identifying the optimal parameters of different fuel cells. Energy 248, 123587. 10.1016/J.ENERGY.2022.123587 (2022).10.1016/J.ENERGY.2022.123587
7. Wu X Wang J Zhang Y Du J Liu Z Chen Y Review of DC-DC converter topologies based on impedance network with wide input voltage range and high gain for fuel cell vehicles Autom. Innov. 2021 4 4 351 372 10.1007/S42154-021-00163-Z
Wu, X. et al. Review of DC-DC converter topologies based on impedance network with wide input voltage range and high gain for fuel cell vehicles. Autom. Innov. 4(4), 351–372. 10.1007/S42154-021-00163-Z (2021).10.1007/S42154-021-00163-Z
8. Corrêa JM Farret FA Canha LN An analysis of the dynamic performance of proton exchange membrane fuel cells using an electrochemical model IECON Proc. (Ind. Electron. Conf.) 2001 3 141 146 10.1109/IECON.2001.976469
Corrêa, J. M., Farret, F. A. & Canha, L. N. An analysis of the dynamic performance of proton exchange membrane fuel cells using an electrochemical model. IECON Proc. (Ind. Electron. Conf.) 3, 141–146. 10.1109/IECON.2001.976469 (2001).10.1109/IECON.2001.976469
9. Corrêa JM Farret FA Canha LN Simoes MG An electrochemical-based fuel-cell model suitable for electrical engineering automation approach IEEE Trans. Ind. Electron. 2004 51 5 1103 1112 10.1109/TIE.2004.834972
Corrêa, J. M., Farret, F. A., Canha, L. N. & Simoes, M. G. An electrochemical-based fuel-cell model suitable for electrical engineering automation approach. IEEE Trans. Ind. Electron. 51(5), 1103–1112. 10.1109/TIE.2004.834972 (2004).10.1109/TIE.2004.834972
10. Amphlett JC Mann RF Peppley BA Roberge PR Rodrigues A A model predicting transient responses of proton exchange membrane fuel cells J. Power Sources 1996 61 1–2 183 188 10.1016/S0378-7753(96)02360-9
Amphlett, J. C., Mann, R. F., Peppley, B. A., Roberge, P. R. & Rodrigues, A. A model predicting transient responses of proton exchange membrane fuel cells. J. Power Sources 61(1–2), 183–188. 10.1016/S0378-7753(96)02360-9 (1996).10.1016/S0378-7753(96)02360-9
11. Amphlett JC Baumert RM Mann RF Peppley BA Roberge PR Harris TJ Performance modeling of the Ballard Mark IV solid polymer electrolyte fuel cell: I Mechanistic model development J. Electrochem. Soc. 1995 142 1 1 10.1149/1.2043866
Amphlett, J. C. et al. Performance modeling of the Ballard Mark IV solid polymer electrolyte fuel cell: I Mechanistic model development. J. Electrochem. Soc. 142(1), 1. 10.1149/1.2043866 (1995).10.1149/1.2043866
12. Shaheen A El-Sehiemy R El-Fergany A Ginidi A Fuel-cell parameter estimation based on improved gorilla troops technique Sci. Rep. 2023 13 1 1 17 10.1038/s41598-023-35581-y 36593249
Shaheen, A., El-Sehiemy, R., El-Fergany, A. & Ginidi, A. Fuel-cell parameter estimation based on improved gorilla troops technique. Sci. Rep. 13(1), 1–17. 10.1038/s41598-023-35581-y (2023).36593249 10.1038/s41598-023-35581-y
13. Wang F A novel quadratic Boost converter with low current and voltage stress on power switch for fuel-cell system applications Renew. Energy 2018 115 836 845 10.1016/J.RENENE.2017.08.032
Wang, F. A novel quadratic Boost converter with low current and voltage stress on power switch for fuel-cell system applications. Renew. Energy 115, 836–845. 10.1016/J.RENENE.2017.08.032 (2018).10.1016/J.RENENE.2017.08.032
14. Rezk H Optimal parameter estimation strategy of PEM fuel cell using gradient-based optimizer Energy 2022 239 122096 10.1016/J.ENERGY.2021.122096
Rezk, H. et al. Optimal parameter estimation strategy of PEM fuel cell using gradient-based optimizer. Energy 239, 122096. 10.1016/J.ENERGY.2021.122096 (2022).10.1016/J.ENERGY.2021.122096
15. Razmjooy N A survey on parameters estimation of the proton exchange membrane fuel cells based on the swarm-inspired optimization algorithms Front Energy Res. 2023 11 1148323 10.3389/FENRG.2023.1148323/BIBTEX
Razmjooy, N. A survey on parameters estimation of the proton exchange membrane fuel cells based on the swarm-inspired optimization algorithms. Front Energy Res. 11, 1148323. 10.3389/FENRG.2023.1148323/BIBTEX (2023).10.3389/FENRG.2023.1148323/BIBTEX
16. Yakout AH Kotb H AboRas KM Hasanien HM Comparison among different recent metaheuristic algorithms for parameters estimation of solid oxide fuel cell: Steady-state and dynamic models Alex. Eng. J. 2022 61 11 8507 8523 10.1016/J.AEJ.2022.02.009
Yakout, A. H., Kotb, H., AboRas, K. M. & Hasanien, H. M. Comparison among different recent metaheuristic algorithms for parameters estimation of solid oxide fuel cell: Steady-state and dynamic models. Alex. Eng. J. 61(11), 8507–8523. 10.1016/J.AEJ.2022.02.009 (2022).10.1016/J.AEJ.2022.02.009
17. Alsaidan I Shaheen MAM Hasanien HM Alaraj M Alnafisah AS Proton exchange membrane fuel cells modeling using chaos game optimization technique Sustainability 2021 13 14 7911 10.3390/SU13147911
Alsaidan, I., Shaheen, M. A. M., Hasanien, H. M., Alaraj, M. & Alnafisah, A. S. Proton exchange membrane fuel cells modeling using chaos game optimization technique. Sustainability 13(14), 7911. 10.3390/SU13147911 (2021).10.3390/SU13147911
18. El-Fergany AA Hasanien HM Agwa AM Semi-empirical PEM fuel cells model using whale optimization algorithm Energy Convers Manag 2019 201 1 10.1016/J.ENCONMAN.2019.112197
El-Fergany, A. A., Hasanien, H. M. & Agwa, A. M. Semi-empirical PEM fuel cells model using whale optimization algorithm. Energy Convers Manag 201, 1. 10.1016/J.ENCONMAN.2019.112197 (2019).10.1016/J.ENCONMAN.2019.112197
19. Zaki Diab AA Tolba MA Abo El-Magd AG Zaky MM El-Rifaie AM Fuel cell parameters estimation via marine predators and political optimizers IEEE Access 2020 8 1 10.1109/ACCESS.2020.3021754
Zaki Diab, A. A., Tolba, M. A., Abo El-Magd, A. G., Zaky, M. M. & El-Rifaie, A. M. Fuel cell parameters estimation via marine predators and political optimizers. IEEE Access 8, 1. 10.1109/ACCESS.2020.3021754 (2020).10.1109/ACCESS.2020.3021754
20. El-Fergany AA Electrical characterisation of proton exchange membrane fuel cells stack using grasshopper optimiser IET Renew. Power Gen. 2018 12 1 9 17 10.1049/IET-RPG.2017.0232
El-Fergany, A. A. Electrical characterisation of proton exchange membrane fuel cells stack using grasshopper optimiser. IET Renew. Power Gen. 12(1), 9–17. 10.1049/IET-RPG.2017.0232 (2018).10.1049/IET-RPG.2017.0232
21. Chen Y Wang N Cuckoo search algorithm with explosion operator for modeling proton exchange membrane fuel cells Int. J. Hydrogen Energy 2019 44 5 3075 3087 10.1016/J.IJHYDENE.2018.11.140
Chen, Y. & Wang, N. Cuckoo search algorithm with explosion operator for modeling proton exchange membrane fuel cells. Int. J. Hydrogen Energy 44(5), 3075–3087. 10.1016/J.IJHYDENE.2018.11.140 (2019).10.1016/J.IJHYDENE.2018.11.140
22. Jia J Li Q Wang Y Cham YT Han M Modeling and dynamic characteristic simulation of a proton exchange membrane fuel cell IEEE Trans. Energy Convers. 2009 24 1 283 291 10.1109/TEC.2008.2011837
Jia, J., Li, Q., Wang, Y., Cham, Y. T. & Han, M. Modeling and dynamic characteristic simulation of a proton exchange membrane fuel cell. IEEE Trans. Energy Convers. 24(1), 283–291. 10.1109/TEC.2008.2011837 (2009).10.1109/TEC.2008.2011837
23. Zhang Y Huang C Huang H Wu J Multiple learning neural network algorithm for parameter estimation of proton exchange membrane fuel cell models Green Energy Intell. Transp. 2023 2 1 100040 10.1016/J.GEITS.2022.100040
Zhang, Y., Huang, C., Huang, H. & Wu, J. Multiple learning neural network algorithm for parameter estimation of proton exchange membrane fuel cell models. Green Energy Intell. Transp. 2(1), 100040. 10.1016/J.GEITS.2022.100040 (2023).10.1016/J.GEITS.2022.100040
24. Singla MK Nijhawan P Oberoi AS Parameter estimation of proton exchange membrane fuel cell using a novel meta-heuristic algorithm Environ. Sci. Pollut. Res. 2021 28 26 34511 34526 10.1007/S11356-021-13097-0/METRICS
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. Res. 28(26), 34511–34526. 10.1007/S11356-021-13097-0/METRICS (2021).10.1007/S11356-021-13097-0/METRICS
25. Li J Gao X Cui Y Hu J Xu G Zhang Z Accurate, efficient and reliable parameter extraction of PEM fuel cells using shuffled multi-simplexes search algorithm Energy Convers. Manag. 2020 206 112501 10.1016/J.ENCONMAN.2020.112501
Li, J. et al. Accurate, efficient and reliable parameter extraction of PEM fuel cells using shuffled multi-simplexes search algorithm. Energy Convers. Manag. 206, 112501. 10.1016/J.ENCONMAN.2020.112501 (2020).10.1016/J.ENCONMAN.2020.112501
26. Yakout AH Hasanien HM Kotb H Proton exchange membrane fuel cell steady state modeling using marine predator algorithm optimizer Ain Shams Eng. J. 2021 12 4 3765 3774 10.1016/J.ASEJ.2021.04.014
Yakout, A. H., Hasanien, H. M. & Kotb, H. Proton exchange membrane fuel cell steady state modeling using marine predator algorithm optimizer. Ain Shams Eng. J. 12(4), 3765–3774. 10.1016/J.ASEJ.2021.04.014 (2021).10.1016/J.ASEJ.2021.04.014
27. Kandidayeni M Macias A Khalatbarisoltani A Boulon L Kelouwani S Benchmark of proton exchange membrane fuel cell parameters extraction with metaheuristic optimization algorithms Energy 2019 183 912 925 10.1016/j.energy.2019.06.152
Kandidayeni, M., Macias, A., Khalatbarisoltani, A., Boulon, L. & Kelouwani, S. Benchmark of proton exchange membrane fuel cell parameters extraction with metaheuristic optimization algorithms. Energy 183, 912–925. 10.1016/j.energy.2019.06.152 (2019).10.1016/j.energy.2019.06.152
28. Pratap Chandran B Immanuel Selvakumar A Shine Let G Paul Sathiyan S Optimal model parameter estimation of solar and fuel cells using improved estimation of distribution algorithm Ain Shams Eng. J. 2021 12 2 1693 1700 10.1016/J.ASEJ.2020.07.034
Pratap Chandran, B., Immanuel Selvakumar, A., Shine Let, G. & Paul Sathiyan, S. Optimal model parameter estimation of solar and fuel cells using improved estimation of distribution algorithm. Ain Shams Eng. J. 12(2), 1693–1700. 10.1016/J.ASEJ.2020.07.034 (2021).10.1016/J.ASEJ.2020.07.034
29. Holland JH Genetic algorithms Sci. Am. 1992 267 1 66 72 10.1038/SCIENTIFICAMERICAN0792-66 1411454
Holland, J. H. Genetic algorithms. Sci. Am. 267(1), 66–72. 10.1038/SCIENTIFICAMERICAN0792-66 (1992).1411454 10.1038/SCIENTIFICAMERICAN0792-66
30. Shi, Y., & Eberhart, R. Modified particle swarm optimizer. In Proceedings of the IEEE Conference on Evolutionary Computation, ICEC, pp. 69–73. 10.1109/ICEC.1998.699146 (1998).
31. Mohammad-Azari S Bozorg-Haddad O Chu X Shark smell optimization (SSO) algorithm Stud. Comput. Intell. 2018 720 93 103 10.1007/978-981-10-5221-7_10
Mohammad-Azari, S., Bozorg-Haddad, O. & Chu, X. Shark smell optimization (SSO) algorithm. Stud. Comput. Intell. 720, 93–103. 10.1007/978-981-10-5221-7_10 (2018).10.1007/978-981-10-5221-7_10
32. Yuan Z Wang W Wang H Yildizbasi A Developed Coyote Optimization Algorithm and its application to optimal parameters estimation of PEMFC model Energy Rep. 2020 6 1 10.1016/j.egyr.2020.04.032
Yuan, Z., Wang, W., Wang, H. & Yildizbasi, A. Developed Coyote Optimization Algorithm and its application to optimal parameters estimation of PEMFC model. Energy Rep. 6, 1. 10.1016/j.egyr.2020.04.032 (2020).10.1016/j.egyr.2020.04.032
33. Irudayaraj, A. X. R. Frequency regulation in multi-microgrid power system using an adaptive Beluga whale optimizer-based FOPID controller. In 2023 IEEE 3rd International Conference on Sustainable Energy and Future Electric Transportation (SEFET), pp. 1–6. 10.1109/SEFET57834.2023.10245312 (2023)
34. Premkumar M Augmented weighted K-means grey wolf optimizer: An enhanced metaheuristic algorithm for data clustering problems Sci. Rep. 2024 14 1 1 33 10.1038/s41598-024-55619-z 38167627
Premkumar, M. et al. Augmented weighted K-means grey wolf optimizer: An enhanced metaheuristic algorithm for data clustering problems. Sci. Rep. 14(1), 1–33. 10.1038/s41598-024-55619-z (2024).38167627 10.1038/s41598-024-55619-z
35. Mirjalili S Mirjalili SM Lewis A Grey wolf optimizer Adv. Eng. Softw. 2014 69 46 61 10.1016/j.advengsoft.2013.12.007
Mirjalili, S., Mirjalili, S. M. & Lewis, A. Grey wolf optimizer. Adv. Eng. Softw. 69, 46–61. 10.1016/j.advengsoft.2013.12.007 (2014).10.1016/j.advengsoft.2013.12.007
36. Mirjalili S Lewis A The whale optimization algorithm Adv. Eng. Softw. 2016 95 51 67 10.1016/j.advengsoft.2016.01.008
Mirjalili, S. & Lewis, A. The whale optimization algorithm. Adv. Eng. Softw. 95, 51–67. 10.1016/j.advengsoft.2016.01.008 (2016).10.1016/j.advengsoft.2016.01.008
37. Premkumar M Sowmya R Kumar JSVS Jangir P Abualigah L Ramakrishnan C Optimal Co-ordination of directional overcurrent relays in distribution network using whale optimization algorithm Lect. Notes Electr. Eng. 2024 1107 233 258 10.1007/978-981-99-8007-9_17
Premkumar, M. et al. Optimal Co-ordination of directional overcurrent relays in distribution network using whale optimization algorithm. Lect. Notes Electr. Eng. 1107, 233–258. 10.1007/978-981-99-8007-9_17 (2024).10.1007/978-981-99-8007-9_17
38. Meraihi Y Gabis AB Mirjalili S Ramdane-Cherif A Grasshopper optimization algorithm: Theory, variants, and applications IEEE Access 2021 9 50001 50024 10.1109/ACCESS.2021.3067597
Meraihi, Y., Gabis, A. B., Mirjalili, S. & Ramdane-Cherif, A. Grasshopper optimization algorithm: Theory, variants, and applications. IEEE Access 9, 50001–50024. 10.1109/ACCESS.2021.3067597 (2021).10.1109/ACCESS.2021.3067597
39. Mirjalili S Moth-flame optimization algorithm: A novel nature-inspired heuristic paradigm Knowl. Based Syst. 2015 89 228 249 10.1016/J.KNOSYS.2015.07.006
Mirjalili, S. Moth-flame optimization algorithm: A novel nature-inspired heuristic paradigm. Knowl. Based Syst. 89, 228–249. 10.1016/J.KNOSYS.2015.07.006 (2015).10.1016/J.KNOSYS.2015.07.006
40. Alsaidan I Shaheen MAM Hasanien HM Alaraj M Alnafisah AS A PEMFC model optimization using the enhanced bald eagle algorithm Ain Shams Eng. J. 2022 13 6 101749 10.1016/J.ASEJ.2022.101749
Alsaidan, I., Shaheen, M. A. M., Hasanien, H. M., Alaraj, M. & Alnafisah, A. S. A PEMFC model optimization using the enhanced bald eagle algorithm. Ain Shams Eng. J. 13(6), 101749. 10.1016/J.ASEJ.2022.101749 (2022).10.1016/J.ASEJ.2022.101749
41. Das, A. K., & Pratihar, D. K. A new Bonobo optimizer (BO) for real-parameter optimization. In Proceedings of 2019 IEEE Region 10 Symposium, TENSYMP 2019, pp. 108–113. 10.1109/TENSYMP46218.2019.8971108 (2019).
42. Sowmya R Premkumar M Jangir P Newton-Raphson-based optimizer: A new population-based metaheuristic algorithm for continuous optimization problems Eng. Appl. Artif. Intell. 2024 128 107532 10.1016/J.ENGAPPAI.2023.107532
Sowmya, R., Premkumar, M. & Jangir, P. Newton-Raphson-based optimizer: A new population-based metaheuristic algorithm for continuous optimization problems. Eng. Appl. Artif. Intell. 128, 107532. 10.1016/J.ENGAPPAI.2023.107532 (2024).10.1016/J.ENGAPPAI.2023.107532
43. Zhao W Zhang Z Wang L Manta ray foraging optimization: An effective bio-inspired optimizer for engineering applications Eng. Appl. Artif. Intell. 2020 87 103300 10.1016/J.ENGAPPAI.2019.103300
Zhao, W., Zhang, Z. & Wang, L. Manta ray foraging optimization: An effective bio-inspired optimizer for engineering applications. Eng. Appl. Artif. Intell. 87, 103300. 10.1016/J.ENGAPPAI.2019.103300 (2020).10.1016/J.ENGAPPAI.2019.103300
44. Yapici H Cetinkaya N A new meta-heuristic optimizer: Pathfinder algorithm Appl. Soft Comput. 2019 78 545 568 10.1016/J.ASOC.2019.03.012
Yapici, H. & Cetinkaya, N. A new meta-heuristic optimizer: Pathfinder algorithm. Appl. Soft Comput. 78, 545–568. 10.1016/J.ASOC.2019.03.012 (2019).10.1016/J.ASOC.2019.03.012
45. Kailasam JK Nalliah R Muthusamy SN Manoharan P MLBRSA: Multi-learning-based reptile search algorithm for global optimization and software requirement prioritization problems Biomimetics 2023 8 8 615 10.3390/BIOMIMETICS8080615 38132554
Kailasam, J. K., Nalliah, R., Muthusamy, S. N. & Manoharan, P. MLBRSA: Multi-learning-based reptile search algorithm for global optimization and software requirement prioritization problems. Biomimetics 8(8), 615. 10.3390/BIOMIMETICS8080615 (2023).38132554 10.3390/BIOMIMETICS8080615
46. Abualigah L Elaziz MA Sumari P Geem ZW Gandomi AH Reptile Search Algorithm (RSA): A nature-inspired meta-heuristic optimizer Expert Syst. Appl. 2022 191 1 10.1016/J.ESWA.2021.116158
Abualigah, L., Elaziz, M. A., Sumari, P., Geem, Z. W. & Gandomi, A. H. Reptile Search Algorithm (RSA): A nature-inspired meta-heuristic optimizer. Expert Syst. Appl. 191, 1. 10.1016/J.ESWA.2021.116158 (2022).10.1016/J.ESWA.2021.116158
47. Menesy AS Sultan HM Selim A Ashmawy MG Kamel S Developing and applying chaotic harris hawks optimization technique for extracting parameters of several proton exchange membrane fuel cell stacks IEEE Access 2020 8 1146 1159 10.1109/ACCESS.2019.2961811
Menesy, A. S., Sultan, H. M., Selim, A., Ashmawy, M. G. & Kamel, S. Developing and applying chaotic harris hawks optimization technique for extracting parameters of several proton exchange membrane fuel cell stacks. IEEE Access 8, 1146–1159. 10.1109/ACCESS.2019.2961811 (2020).10.1109/ACCESS.2019.2961811
48. Devi RM Premkumar M Kiruthiga G Sowmya R IGJO: An improved Golden Jackel optimization algorithm using local escaping operator for feature selection problems Neural Process. Lett. 2023 2023 1 89 10.1007/S11063-023-11146-Y
Devi, R. M., Premkumar, M., Kiruthiga, G. & Sowmya, R. IGJO: An improved Golden Jackel optimization algorithm using local escaping operator for feature selection problems. Neural Process. Lett. 2023, 1–89. 10.1007/S11063-023-11146-Y (2023).10.1007/S11063-023-11146-Y
49. Chopra N Mohsin Ansari M Golden jackal optimization: A novel nature-inspired optimizer for engineering applications Expert Syst. Appl. 2022 198 116924 10.1016/J.ESWA.2022.116924
Chopra, N. & Mohsin Ansari, M. Golden jackal optimization: A novel nature-inspired optimizer for engineering applications. Expert Syst. Appl. 198, 116924. 10.1016/J.ESWA.2022.116924 (2022).10.1016/J.ESWA.2022.116924
50. Hayyolalam V Pourhaji Kazem AA Black Widow Optimization Algorithm: A novel meta-heuristic approach for solving engineering optimization problems Eng. Appl. Artif. Intell. 2020 87 103249 10.1016/J.ENGAPPAI.2019.103249
Hayyolalam, V. & Pourhaji Kazem, A. A. Black Widow Optimization Algorithm: A novel meta-heuristic approach for solving engineering optimization problems. Eng. Appl. Artif. Intell. 87, 103249. 10.1016/J.ENGAPPAI.2019.103249 (2020).10.1016/J.ENGAPPAI.2019.103249
51. Yousri D Rezk H Fathy A Identifying the parameters of different configurations of photovoltaic models based on recent artificial ecosystem-based optimization approach Int. J. Energy Res. 2020 44 14 11302 11322 10.1002/er.5747
Yousri, D., Rezk, H. & Fathy, A. Identifying the parameters of different configurations of photovoltaic models based on recent artificial ecosystem-based optimization approach. Int. J. Energy Res. 44(14), 11302–11322. 10.1002/er.5747 (2020).10.1002/er.5747
52. Sathish Kumar D Premkumar M Kumar C Muyeen SM Optimal scheduling algorithm for residential building distributed energy source systems using Levy flight and chaos-assisted artificial rabbits optimizer Energy Rep. 2023 9 5721 5740 10.1016/J.EGYR.2023.05.004
Sathish Kumar, D., Premkumar, M., Kumar, C. & Muyeen, S. M. Optimal scheduling algorithm for residential building distributed energy source systems using Levy flight and chaos-assisted artificial rabbits optimizer. Energy Rep. 9, 5721–5740. 10.1016/J.EGYR.2023.05.004 (2023).10.1016/J.EGYR.2023.05.004
53. Wang L Cao Q Zhang Z Mirjalili S Zhao W Artificial rabbits optimization: A new bio-inspired meta-heuristic algorithm for solving engineering optimization problems Eng. Appl. Artif. Intell. 2022 114 105082 10.1016/J.ENGAPPAI.2022.105082
Wang, L., Cao, Q., Zhang, Z., Mirjalili, S. & Zhao, W. Artificial rabbits optimization: A new bio-inspired meta-heuristic algorithm for solving engineering optimization problems. Eng. Appl. Artif. Intell. 114, 105082. 10.1016/J.ENGAPPAI.2022.105082 (2022).10.1016/J.ENGAPPAI.2022.105082
54. Beşkirli A Dağ İ An efficient tree seed inspired algorithm for parameter estimation of Photovoltaic models Energy Rep. 2022 8 291 298 10.1016/J.EGYR.2021.11.103
Beşkirli, A. & Dağ, İ. An efficient tree seed inspired algorithm for parameter estimation of Photovoltaic models. Energy Rep. 8, 291–298. 10.1016/J.EGYR.2021.11.103 (2022).10.1016/J.EGYR.2021.11.103
55. Chandrasekaran K Thaveedhu ASR Manoharan P Periyasamy V Optimal estimation of parameters of the three-diode commercial solar photovoltaic model using an improved Berndt-Hall-Hall-Hausman method hybridized with an augmented mountain gazelle optimizer Environ. Sci. Pollut. Res. 2023 1 1 24 10.1007/S11356-023-26447-X/METRICS
Chandrasekaran, K., Thaveedhu, A. S. R., Manoharan, P. & Periyasamy, V. Optimal estimation of parameters of the three-diode commercial solar photovoltaic model using an improved Berndt-Hall-Hall-Hausman method hybridized with an augmented mountain gazelle optimizer. Environ. Sci. Pollut. Res. 1, 1–24. 10.1007/S11356-023-26447-X/METRICS (2023).10.1007/S11356-023-26447-X/METRICS
56. Abdollahzadeh B Gharehchopogh FS Khodadadi N Mirjalili S Mountain gazelle optimizer: A new nature-inspired metaheuristic algorithm for global optimization problems Adv. Eng. Softw. 2022 174 103282 10.1016/J.ADVENGSOFT.2022.103282
Abdollahzadeh, B., Gharehchopogh, F. S., Khodadadi, N. & Mirjalili, S. Mountain gazelle optimizer: A new nature-inspired metaheuristic algorithm for global optimization problems. Adv. Eng. Softw. 174, 103282. 10.1016/J.ADVENGSOFT.2022.103282 (2022).10.1016/J.ADVENGSOFT.2022.103282
57. Diab AAZ Sultan HM Aljendy R Al-Sumaiti AS Shoyama M Ali ZM Tree growth based optimization algorithm for parameter extraction of different models of photovoltaic cells and modules IEEE Access 2020 8 119668 119687 10.1109/ACCESS.2020.3005236
Diab, A. A. Z. et al. Tree growth based optimization algorithm for parameter extraction of different models of photovoltaic cells and modules. IEEE Access 8, 119668–119687. 10.1109/ACCESS.2020.3005236 (2020).10.1109/ACCESS.2020.3005236
58. Ahmadianfar I Bozorg-Haddad O Chu X Gradient-based optimizer: A new metaheuristic optimization algorithm Inf. Sci. (N Y) 2020 540 131 159 10.1016/j.ins.2020.06.037
Ahmadianfar, I., Bozorg-Haddad, O. & Chu, X. Gradient-based optimizer: A new metaheuristic optimization algorithm. Inf. Sci. (N Y) 540, 131–159. 10.1016/j.ins.2020.06.037 (2020).10.1016/j.ins.2020.06.037
59. Premkumar M An enhanced Gradient-based Optimizer for parameter estimation of various solar photovoltaic models Energy Rep. 2022 8 15249 15285 10.1016/J.EGYR.2022.11.092
Premkumar, M. et al. An enhanced Gradient-based Optimizer for parameter estimation of various solar photovoltaic models. Energy Rep. 8, 15249–15285. 10.1016/J.EGYR.2022.11.092 (2022).10.1016/J.EGYR.2022.11.092
60. Fadheel BA A hybrid grey wolf assisted-sparrow search algorithm for frequency control of RE integrated system Energies 2023 16 3 1177 10.3390/EN16031177
Fadheel, B. A. et al. A hybrid grey wolf assisted-sparrow search algorithm for frequency control of RE integrated system. Energies 16(3), 1177. 10.3390/EN16031177 (2023).10.3390/EN16031177
61. Fadheel BA A hybrid sparrow search optimized fractional virtual inertia control for frequency regulation of multi-microgrid system IEEE Access 2024 12 45879 45903 10.1109/ACCESS.2024.3376468
Fadheel, B. A. et al. A hybrid sparrow search optimized fractional virtual inertia control for frequency regulation of multi-microgrid system. IEEE Access 12, 45879–45903. 10.1109/ACCESS.2024.3376468 (2024).10.1109/ACCESS.2024.3376468
62. Priya K Rajasekar N Application of flower pollination algorithm for enhanced proton exchange membrane fuel cell modelling Int. J. Hydrogen Energy 2019 44 33 18438 18449 10.1016/J.IJHYDENE.2019.05.022
Priya, K. & Rajasekar, N. Application of flower pollination algorithm for enhanced proton exchange membrane fuel cell modelling. Int. J. Hydrogen Energy 44(33), 18438–18449. 10.1016/J.IJHYDENE.2019.05.022 (2019).10.1016/J.IJHYDENE.2019.05.022
63. Ravichandran S Manoharan P Jangir P Selvarajan S Resistance–capacitance optimizer: a physics-inspired population-based algorithm for numerical and industrial engineering computation problems Sci. Rep. 2013 13 1 1 40 10.1038/s41598-023-42969-3
Ravichandran, S., Manoharan, P., Jangir, P. & Selvarajan, S. Resistance–capacitance optimizer: a physics-inspired population-based algorithm for numerical and industrial engineering computation problems. Sci. Rep. 13(1), 1–40. 10.1038/s41598-023-42969-3 (2013).10.1038/s41598-023-42969-3
64. Premkumar M Jangir P Sowmya R Parameter extraction of three-diode solar photovoltaic model using a new metaheuristic resistance–capacitance optimization algorithm and improved Newton-Raphson method J. Comput. Electron. 2023 22 1 439 470 10.1007/S10825-022-01987-6/METRICS
Premkumar, M., Jangir, P. & Sowmya, R. Parameter extraction of three-diode solar photovoltaic model using a new metaheuristic resistance–capacitance optimization algorithm and improved Newton-Raphson method. J. Comput. Electron. 22(1), 439–470. 10.1007/S10825-022-01987-6/METRICS (2023).10.1007/S10825-022-01987-6/METRICS
65. Premkumar, M., Sowmya, R., Jangir, P., & Siva Kumar, J. S. V. A new and reliable objective functions for extracting the unknown parameters of solar photovoltaic cell using political optimizer algorithm. In 2020 International Conference on Data Analytics for Business and Industry: Way Towards a Sustainable Economy, ICDABI 2020. 10.1109/ICDABI51230.2020.9325627 (2020).
66. Kullampalayam Murugaiyan N Chandrasekaran K Manoharan P Derebew B Leveraging opposition-based learning for solar photovoltaic model parameter estimation with exponential distribution optimization algorithm Sci. Rep. 2024 14 1 1 45 10.1038/s41598-023-50890-y 38167627
Kullampalayam Murugaiyan, N., Chandrasekaran, K., Manoharan, P. & Derebew, B. Leveraging opposition-based learning for solar photovoltaic model parameter estimation with exponential distribution optimization algorithm. Sci. Rep. 14(1), 1–45. 10.1038/s41598-023-50890-y (2024).38167627 10.1038/s41598-023-50890-y
67. Abdel-Basset M El-Shahat D Jameel M Abouhawwash M Exponential distribution optimizer (EDO): a novel math-inspired algorithm for global optimization and engineering problems Artif. Intell. Rev. 2023 56 9 9329 9400 10.1007/S10462-023-10403-9/METRICS
Abdel-Basset, M., El-Shahat, D., Jameel, M. & Abouhawwash, M. Exponential distribution optimizer (EDO): a novel math-inspired algorithm for global optimization and engineering problems. Artif. Intell. Rev. 56(9), 9329–9400. 10.1007/S10462-023-10403-9/METRICS (2023).10.1007/S10462-023-10403-9/METRICS
68. Faramarzi A Heidarinejad M Mirjalili S Gandomi AH Marine Predators Algorithm: A nature-inspired metaheuristic Expert Syst. Appl. 2020 152 113377 10.1016/j.eswa.2020.113377
Faramarzi, A., Heidarinejad, M., Mirjalili, S. & Gandomi, A. H. Marine Predators Algorithm: A nature-inspired metaheuristic. Expert Syst. Appl. 152, 113377. 10.1016/j.eswa.2020.113377 (2020).10.1016/j.eswa.2020.113377
69. Diab AAZ Tolba MA El-Magd AGA Zaky MM El-Rifaie AM Fuel cell parameters estimation via marine predators and political optimizers IEEE Access 2020 8 166998 167018 10.1109/ACCESS.2020.3021754
Diab, A. A. Z., Tolba, M. A., El-Magd, A. G. A., Zaky, M. M. & El-Rifaie, A. M. Fuel cell parameters estimation via marine predators and political optimizers. IEEE Access 8, 166998–167018. 10.1109/ACCESS.2020.3021754 (2020).10.1109/ACCESS.2020.3021754
70. Li S Chen H Wang M Asghar A Mirjalili S Slime mould algorithm: A new method for stochastic optimization Future Gen. Comput. Syst. 2020 111 300 323 10.1016/j.future.2020.03.055
Li, S., Chen, H., Wang, M., Asghar, A. & Mirjalili, S. Slime mould algorithm: A new method for stochastic optimization. Future Gen. Comput. Syst. 111, 300–323. 10.1016/j.future.2020.03.055 (2020).10.1016/j.future.2020.03.055
71. Abdel-Basset M Mohamed R Abdel-Fatah L Sharawi M Sallam KM Improved metaheuristic algorithms for optimal parameters selection of proton exchange membrane fuel cells: A comparative study IEEE Access 2023 11 7369 7397 10.1109/ACCESS.2023.3236023
Abdel-Basset, M., Mohamed, R., Abdel-Fatah, L., Sharawi, M. & Sallam, K. M. Improved metaheuristic algorithms for optimal parameters selection of proton exchange membrane fuel cells: A comparative study. IEEE Access 11, 7369–7397. 10.1109/ACCESS.2023.3236023 (2023).10.1109/ACCESS.2023.3236023
72. Abdullah AM Rezk H Hadad A Hassan MK Mohamed AF Optimal parameter estimation of proton exchange membrane fuel cells Intell. Autom. Soft Comput. 2021 29 2 619 631 10.32604/IASC.2021.018289
Abdullah, A. M., Rezk, H., Hadad, A., Hassan, M. K. & Mohamed, A. F. Optimal parameter estimation of proton exchange membrane fuel cells. Intell. Autom. Soft Comput. 29(2), 619–631. 10.32604/IASC.2021.018289 (2021).10.32604/IASC.2021.018289
73. Singla MK Gupta J Singh B Nijhawan P Abdelaziz AY El-Shahat A Parameter estimation of fuel cells using a hybrid optimization algorithm Sustainability 2023 15 8 6676 10.3390/SU15086676
Singla, M. K. et al. Parameter estimation of fuel cells using a hybrid optimization algorithm. Sustainability 15(8), 6676. 10.3390/SU15086676 (2023).10.3390/SU15086676
74. El-Fergany AA Extracting optimal parameters of PEM fuel cells using Salp Swarm Optimizer Renew. Energy 2018 119 641 648 10.1016/J.RENENE.2017.12.051
El-Fergany, A. A. Extracting optimal parameters of PEM fuel cells using Salp Swarm Optimizer. Renew. Energy 119, 641–648. 10.1016/J.RENENE.2017.12.051 (2018).10.1016/J.RENENE.2017.12.051
75. Qin F Liu P Niu H Song H Yousefi N Parameter estimation of PEMFC based on improved fluid search optimization algorithm Energy Rep. 2020 6 1224 1232 10.1016/J.EGYR.2020.05.006
Qin, F., Liu, P., Niu, H., Song, H. & Yousefi, N. Parameter estimation of PEMFC based on improved fluid search optimization algorithm. Energy Rep. 6, 1224–1232. 10.1016/J.EGYR.2020.05.006 (2020).10.1016/J.EGYR.2020.05.006
76. Fathy A Abdel Aleem SHE Rezk H A novel approach for PEM fuel cell parameter estimation using LSHADE-EpSin optimization algorithm Int. J. Energy Res. 2021 45 5 6922 6942 10.1002/ER.6282
Fathy, A., Abdel Aleem, S. H. E. & Rezk, H. A novel approach for PEM fuel cell parameter estimation using LSHADE-EpSin optimization algorithm. Int. J. Energy Res. 45(5), 6922–6942. 10.1002/ER.6282 (2021).10.1002/ER.6282
77. Rizk-Allah RM El-Fergany AA Artificial ecosystem optimizer for parameters identification of proton exchange membrane fuel cells model Int. J. Hydrogen Energy 2021 46 75 37612 37627 10.1016/J.IJHYDENE.2020.06.256
Rizk-Allah, R. M. & El-Fergany, A. A. Artificial ecosystem optimizer for parameters identification of proton exchange membrane fuel cells model. Int. J. Hydrogen Energy 46(75), 37612–37627. 10.1016/J.IJHYDENE.2020.06.256 (2021).10.1016/J.IJHYDENE.2020.06.256
78. Yang Z Liu Q Zhang L Dai J Razmjooy N Model parameter estimation of the PEMFCs using improved barnacles mating optimization algorithm Energy 2020 212 118738 10.1016/J.ENERGY.2020.118738
Yang, Z., Liu, Q., Zhang, L., Dai, J. & Razmjooy, N. Model parameter estimation of the PEMFCs using improved barnacles mating optimization algorithm. Energy 212, 118738. 10.1016/J.ENERGY.2020.118738 (2020).10.1016/J.ENERGY.2020.118738
79. Yang B Parameter extraction of PEMFC via Bayesian regularization neural network based meta-heuristic algorithms Energy 2021 228 120592 10.1016/J.ENERGY.2021.120592
Yang, B. et al. Parameter extraction of PEMFC via Bayesian regularization neural network based meta-heuristic algorithms. Energy 228, 120592. 10.1016/J.ENERGY.2021.120592 (2021).10.1016/J.ENERGY.2021.120592
80. Abdel-Basset M Mohamed R Elhoseny M Chakrabortty RK Ryan MJ An efficient heap-based optimization algorithm for parameters identification of proton exchange membrane fuel cells model: Analysis and case studies Int. J. Hydrogen Energy 2021 46 21 11908 11925 10.1016/J.IJHYDENE.2021.01.076
Abdel-Basset, M., Mohamed, R., Elhoseny, M., Chakrabortty, R. K. & Ryan, M. J. An efficient heap-based optimization algorithm for parameters identification of proton exchange membrane fuel cells model: Analysis and case studies. Int. J. Hydrogen Energy 46(21), 11908–11925. 10.1016/J.IJHYDENE.2021.01.076 (2021).10.1016/J.IJHYDENE.2021.01.076
81. Agwa AM El-Fergany AA Sarhan GM Steady-state modeling of fuel cells based on atom search optimizer Energies 2019 12 10 1884 10.3390/EN12101884
Agwa, A. M., El-Fergany, A. A. & Sarhan, G. M. Steady-state modeling of fuel cells based on atom search optimizer. Energies 12(10), 1884. 10.3390/EN12101884 (2019).10.3390/EN12101884
82. 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 Sour. Part A Recov. Util. Environ. Effects 2020 1 1 10.1080/15567036.2020.1769230
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 Sour. Part A Recov. Util. Environ. Effects 1, 1. 10.1080/15567036.2020.1769230 (2020).10.1080/15567036.2020.1769230
83. Gupta J Nijhawan P Ganguli S Optimal parameter estimation of PEM fuel cell using slime mould algorithm Int. J. Energy Res. 2021 45 10 14732 14744 10.1002/ER.6750
Gupta, J., Nijhawan, P. & Ganguli, S. Optimal parameter estimation of PEM fuel cell using slime mould algorithm. Int. J. Energy Res. 45(10), 14732–14744. 10.1002/ER.6750 (2021).10.1002/ER.6750
84. Özdemir MT Optimal parameter estimation of polymer electrolyte membrane fuel cells model with chaos embedded particle swarm optimization Int. J. Hydrogen Energy 2021 46 30 16465 16480 10.1016/J.IJHYDENE.2020.12.203
Özdemir, M. T. Optimal parameter estimation of polymer electrolyte membrane fuel cells model with chaos embedded particle swarm optimization. Int. J. Hydrogen Energy 46(30), 16465–16480. 10.1016/J.IJHYDENE.2020.12.203 (2021).10.1016/J.IJHYDENE.2020.12.203
85. Ayyarao TSLV Polumahanthi N Khan B An accurate parameter estimation of PEM fuel cell using war strategy optimization Energy 2024 290 130235 10.1016/J.ENERGY.2024.130235
Ayyarao, T. S. L. V., Polumahanthi, N. & Khan, B. An accurate parameter estimation of PEM fuel cell using war strategy optimization. Energy 290, 130235. 10.1016/J.ENERGY.2024.130235 (2024).10.1016/J.ENERGY.2024.130235
86. Fahim SR Parameter identification of proton exchange membrane fuel cell based on hunger games search algorithm Energies 2021 14 16 5022 10.3390/EN14165022
Fahim, S. R. et al. Parameter identification of proton exchange membrane fuel cell based on hunger games search algorithm. Energies 14(16), 5022. 10.3390/EN14165022 (2021).10.3390/EN14165022
87. Hamad RK Rashid TA GOOSE algorithm: A powerful optimization tool for real-world engineering challenges and beyond Evol. Syst. 2024 1 1 26 10.1007/S12530-023-09553-6/TABLES/24
Hamad, R. K. & Rashid, T. A. GOOSE algorithm: A powerful optimization tool for real-world engineering challenges and beyond. Evol. Syst. 1, 1–26. 10.1007/S12530-023-09553-6/TABLES/24 (2024).10.1007/S12530-023-09553-6/TABLES/24
88. Hu J Orthogonal learning covariance matrix for defects of grey wolf optimizer: Insights, balance, diversity, and feature selection Knowl. Based Syst. 2021 213 106684 10.1016/J.KNOSYS.2020.106684
Hu, J. et al. Orthogonal learning covariance matrix for defects of grey wolf optimizer: Insights, balance, diversity, and feature selection. Knowl. Based Syst. 213, 106684. 10.1016/J.KNOSYS.2020.106684 (2021).10.1016/J.KNOSYS.2020.106684
89. Xiong G Shi D Orthogonal learning competitive swarm optimizer for economic dispatch problems Appl. Soft Comput. 2018 66 134 148 10.1016/J.ASOC.2018.02.019
Xiong, G. & Shi, D. Orthogonal learning competitive swarm optimizer for economic dispatch problems. Appl. Soft Comput. 66, 134–148. 10.1016/J.ASOC.2018.02.019 (2018).10.1016/J.ASOC.2018.02.019
90. Xavier FJ Pradeep A Premkumar M Kumar C Orthogonal learning-based Gray Wolf Optimizer for identifying the uncertain parameters of various photovoltaic models Optik (Stuttg) 2021 247 167973 10.1016/J.IJLEO.2021.167973
Xavier, F. J., Pradeep, A., Premkumar, M. & Kumar, C. Orthogonal learning-based Gray Wolf Optimizer for identifying the uncertain parameters of various photovoltaic models. Optik (Stuttg) 247, 167973. 10.1016/J.IJLEO.2021.167973 (2021).10.1016/J.IJLEO.2021.167973
91. Zhou X Advanced orthogonal learning and Gaussian barebone hunger games for engineering design J. Comput. Des. Eng. 2022 9 5 1699 1736 10.1093/JCDE/QWAC075
Zhou, X. et al. Advanced orthogonal learning and Gaussian barebone hunger games for engineering design. J. Comput. Des. Eng. 9(5), 1699–1736. 10.1093/JCDE/QWAC075 (2022).10.1093/JCDE/QWAC075
92. Chuan Wang W Xu L K. wing Chau, Y. Zhao, and D. Mei Xu, An orthogonal opposition-based-learning Yin–Yang-pair optimization algorithm for engineering optimization Eng. Comput. 2022 38 2 1149 1183 10.1007/S00366-020-01248-9/TABLES/8
Chuan Wang, W., Xu, L. & K. wing Chau, Y. Zhao, and D. Mei Xu,. An orthogonal opposition-based-learning Yin–Yang-pair optimization algorithm for engineering optimization. Eng. Comput. 38(2), 1149–1183. 10.1007/S00366-020-01248-9/TABLES/8 (2022).10.1007/S00366-020-01248-9/TABLES/8
93. Satapathy SC Naik A Parvathi K A teaching learning based optimization based on orthogonal design for solving global optimization problems Springerplus 2013 2 1 1 12 10.1186/2193-1801-2-130/TABLES/11 23419944
Satapathy, S. C., Naik, A. & Parvathi, K. A teaching learning based optimization based on orthogonal design for solving global optimization problems. Springerplus 2(1), 1–12. 10.1186/2193-1801-2-130/TABLES/11 (2013).23419944 10.1186/2193-1801-2-130/TABLES/11
94. Trojovský P Dehghani M Subtraction-average-based optimizer: A new swarm-inspired metaheuristic algorithm for solving optimization problems Biomimetics 2023 8 2 149 10.3390/BIOMIMETICS8020149 37092401
Trojovský, P. & Dehghani, M. Subtraction-average-based optimizer: A new swarm-inspired metaheuristic algorithm for solving optimization problems. Biomimetics 8(2), 149. 10.3390/BIOMIMETICS8020149 (2023).37092401 10.3390/BIOMIMETICS8020149
95. Azizi M Aickelin U Khorshidi HA Baghalzadeh Shishehgarkhaneh M Energy valley optimizer: A novel metaheuristic algorithm for global and engineering optimization Sci. Rep. 2023 13 1 1 23 10.1038/s41598-022-27344-y 36593249
Azizi, M., Aickelin, U., Khorshidi, H. A. & Baghalzadeh Shishehgarkhaneh, M. Energy valley optimizer: A novel metaheuristic algorithm for global and engineering optimization. Sci. Rep. 13(1), 1–23. 10.1038/s41598-022-27344-y (2023).36593249 10.1038/s41598-022-27344-y
