**1. Introduction**

The demand for clean energy has kept increasing in recent years due to global warming and depleting oil reserves [1,2]. Fuel cells have drawn significant attention in recent years due to high efficiency and no emission of greenhouse gases. In recent years, fuel cell research has grown significantly due to possible applications such as stationary power generation and automotive applications [3]. PEMFCs have particularly drawn attention for transport applications. It has many advantages such as low operating temperature, short start-up and shut-down time, high efficiency, no waste is generated as the by-product is water [4,5]. Due to compact size, low operating temperature, and quick start-up time makes PEMFCs a reliable candidate for medium power applications like smart grid, micro gird, and power electronic devices [6]. The fuel cell has three main components: anode, cathode, and electrolyte. Both anode and cathode contain a layer of catalyst, which is separated by an electrolyte membrane to perform the redox reaction. However, the voltage (1.0 V) and current density (500–1000 mA/cm<sup>2</sup> ) delivered by a single cell is too low for any practical application, so a number of stacks are connected in series to deliver sufficient power for practical application. The performance of a fuel cell depends on multiple parameters such as operating temperature, inlet pressure of fuel and reactant, and conductivity of the membrane. In order to utilize fuel cell for wide range of application evaluation of

**Citation:** Sharma, A.; Khan, R.A.; Sharma, A.; Kashyap, D.; Rajput, S. A Novel Opposition-Based Arithmetic Optimization Algorithm for Parameter Extraction of PEM Fuel Cell. *Electronics* **2021**, *10*, 2834. https://doi.org/10.3390/ electronics10222834

Academic Editor: Jung-Min Kwon

Received: 18 October 2021 Accepted: 16 November 2021 Published: 18 November 2021

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performance under various operating conditions is necessary. Moreover, development of a mathematical model to simulate the dynamic variation in operating conditions and fuel cell performance is necessary for its integration in smart grid/microgrid [7].

Both theoretical and experimental studies have been performed to optimize the factors affecting performance such as pressure, temperature, flow rate of fuel and oxidant, reaction kinetics, and membrane thickness to get the maximum power density from fuel cell [8]. As the fuel cell performance depends on multiple interdependent factors, which makes it really difficult to develop a mathematical model to evaluate the multivariable, complex, and interrelated parameters affecting the fuel cell performance [9]. In recent years, remarkable research and development has been performed to get a better understanding of the function of PEMFC characteristics via mathematical modeling. The modeling achieves great significance in the outlook of simulation, design, exploration, and progress of high-efficiency fuel cell systems [10–12]. A reliable model facilitates monitoring of fuel cell behavior for process monitoring and designing a suitable power conditioning unit for various power applications. The development of a precise parameter estimation method using the experimental data is a pre-requisite to develop a mathematical model of fuel cell and design an appropriate power control algorithm [13]. Two different approaches have been utilized to develop a mathematical model of the fuel cell systems. In the first approach, a mechanistic model is built to simulate the heat, mass transfer, reaction kinetics, membrane conductivity, and crossover of reactants through the electrolyte membrane encountered in fuel cells [14,15]. In this approach, a three-dimensional multiphase model of fuel cell system is developed, in which the gas and liquid two-phase flow in channel and porous electrodes are investigated in detail. This approach of precise estimation of model parameters is hindered by the nonlinear and complex relations of the electrochemical equations. In the second approach, a mathematical model is developed on the basis of empirical or semi-empirical equations, which are utilized to predict the effect of different input parameters on the voltage–current characteristics of the fuel cell, without examining the physical and electrochemical phenomena taking place in fuel cell system [16]. The electrical equivalent models of fuel cell are mainly divided into static and dynamic models. The static models depends on steady-state operation of fuel cell based on polarization curve [17,18] and the dynamic models rely on characteristics of electrical terminal represented by a set of passive elements [19,20]. Although mechanistic models have been developed to evaluate the optimum parameters to get the maximum output from the fuel cell system, the actual performance of fuel cell observed in experimental studies is not precisely the same as observed in theoretical studies, irrespective of models, because of assumptions and approximations are made in modelling [21]. In order to develop the precision of the models and make it reflect the actual fuel-cell performance, it is essential to improve the parameters of the models. However, a little effort has been put forward in the area of parameters optimization.

Generally, the statistics contained in any PEMFC datasheet are insufficient to determine the effective set of parameters. However, if the precise parameters are not specified, there are significant variations between the data obtained from the model and that listed in the manufacturer's datasheet. PEMFC parameter identification can be approached as an optimization challenge, and a variety of meta-heuristic techniques can be implemented to find the best solution. Over the last ten years, various meta-heuristic optimization techniques have been applied to address the issue of PEMFC parameter estimation, which utilizes two important search strategies: (a) exploration/diversification and (b) exploitation/intensification [22,23]. The first method explores the search space globally, which avoids local optima and resolving local optima entrapment, whereas the second method explores the nearby promising solutions to improve their quality locally [24]. A proper balance between these two strategies is required to get the optimum performance. The classification of meta-heuristics method is based in the evolutionary algorithms, swarm intelligence algorithms, physics-based methods, and human-based methods. However, there is no single optimized algorithm, which can solve all optimization problems. Most

of the researchers either modify an existing algorithm or propose a new algorithm to get better result [25]. Different meta-heuristic algorithms have been utilized for parameter optimization of PEMFCs such as particle swarm optimization (PSO) [26], genetic algorithm (GA) [27], artificial neural network (ANN) [28], differential evolution (DE) [29], artificial immune system (AIS) [30], artificial bee colony (ABC) [31], bird mating optimization (BMO) [32], biography-based optimization (BMO) [33], seeker optimization algorithm (SOA) [34], backtracking search algorithm (BSA) [35], improved teaching learning-based optimization (ITLBO) [36]. Slime mold algorithm (SMA) [37], moth-flame optimization (MFO) [38], Archimedes optimization algorithm [39], Jellyfish search algorithm (JSA) [40], bonobo optimizer [41], and hybrid GWO algorithm [42] have been implemented to identify the unknown parameters of PEMFC. In this article, the authors have proposed an improved opposition-based arithmetic optimization algorithm for parameter extraction of PEMFC. To the best of the authors' knowledge, arithmetic optimization algorithm (AOA) has not been explored in this field, therefore, in this article authors have examined the performance of improved AOA for parameter extraction of fuel cells.

The main contribution of this research paper is as follows:


The manuscript is organized as follows: Section 2 describes the theoretical and mathematical model of the PEMFC, Section 3 includes the formulation of OAOA technique. Section 4 discusses the results and findings. Finally, Section 5 provides the overall conclusive remarks of the proposed study.
