A Sea Horse Optimization-Based Approach for PEM Fuel Cell Model Parameter Estimation

Abstract

This study aims to determine the model parameters of proton exchange membrane fuel cells (PEMFC) by employing the Sea Horse Optimization (SHO) algorithm, a novel metaheuristic approach inspired by natural behaviors. Although conventional algorithms in the literature have achieved considerable success in parametric modeling accuracy, many of them suffer from inherent drawbacks, such as premature convergence and entrapment in local minima. The SHO algorithm, with its adaptive and dynamic nature, is designed to overcome these limitations. To further evaluate its performance, a detailed parametric sensitivity analysis is conducted on SHO-specific control parameters. In this work, experimental polarization data from a Ballard Mark V PEMFC is used as a reference to estimate the equivalent circuit model parameters ${ϵ}_1$, ${ϵ}_2$, ${ϵ}_3$, ${ϵ}_4$, $\beta$, $\lambda$, $R_c$. The SHO algorithm achieved a mean absolute error (MAE) of 0.001079 and a coefficient of determination (R²) of 0.999791, with a model-to-experiment fit ratio of 99.92%. Compared to similar studies reported in the literature, the results indicate that the SHO algorithm offers competitive performance. Moreover, the average convergence time is recorded as 1.74 s for 5000 iteration, highlighting the algorithm’s rapid convergence and low computational cost. Overall, the SHO algorithm is demonstrated to be an efficient, robust, and promising alternative to conventional methods for parameter identification in PEMFC modeling.

1. Introduction

In recent years, fuel cells (FCs) have emerged as a key component in energy systems, owing to their high energy efficiency, low emission characteristics, and quiet operation. Among the various types, proton exchange membrane fuel cells (PEMFCs) attract significant attention due to their low operating temperature, compact structure, and rapid start-up capability. These features enable their use across a broad spectrum of applications, ranging from portable electronic devices to electric vehicles [1].

For the effective deployment of PEMFC systems, an accurate and reliable modeling approach is essential during the design and control stages. However, the modeling process is inherently complex and highly multi-variable, as these systems involve a range of interrelated physical phenomena, including electrochemical reactions, gas diffusion, humidity, and heat transfer, as well as fuel supply dynamics [2,3]. These factors underline the need for models that can accurately capture the physical behavior of fuel cells and highlight the critical balance between parametric accuracy and comprehensive system representation. The identification of accurate and reliable model design parameters plays a crucial role in both the performance prediction of PEMFCs and the overall system design process [4].

In the literature, a wide range of approaches have been developed for modeling PEMFCs. These models are generally categorized as mechanistic, empirical, semi-empirical, data-driven, and equivalent circuit models [3].

Mechanistic models represent the physical and chemical processes occurring within a fuel cell through detailed mathematical formulations. However, due to their complexity and the difficulty of obtaining precise parameter values, they are primarily used in theoretical studies. Experimental models provide straightforward estimations based on directly measured data but fail to reflect the internal dynamics of the system. Semi-empirical models combine both approaches, offering simplified yet practical solutions suitable for control-oriented applications. Data-driven models operate by learning from extensive experimental datasets; however, they are often costly and time-consuming to develop. For this reason, one of the most commonly adopted methods in engineering applications is the equivalent circuit model, which represents the electrical behavior of PEMFCs using conventional electronic components [3,5]. However, the effectiveness of this model largely depends on the accurate estimation of its parameters, which is why optimization algorithms are frequently employed. One of the most critical challenges across all modeling approaches lies in the reliable identification of these parameters. Since the required parameters are often not provided by manufacturers and can be difficult to determine experimentally, the accuracy of the model may be compromised. As a result, the use of optimization techniques becomes essential to ensure model fidelity [6].

In recent years, metaheuristic algorithms—such as Genetic Algorithms (GAs) [7], Particle Swarm Optimization (PSO) [8,9], Artificial Bee Colony (ABC) [10,11], Gazelle Optimization Algorithm (GOA) [12] and Grey Wolf Optimizer (GWO) [13]—have been widely utilized in the parameter estimation processes of PEMFC models. However, these methods do not always yield sufficient accuracy due to issues such as premature convergence or entrapment in local minima. This has increased the demand for more flexible and innovative optimization strategies. In this context, the Sea Horse Optimization (SHO) algorithm has emerged as a promising alternative, offering fast convergence, the ability to avoid local optima, low computational cost, and a high level of accuracy—making it particularly well-suited for complex engineering applications [14]. A comparison between optimization methods and the models to which they are applied from existing publications in the literature is made and is given in

Table 1. Optimization methods and descriptions of the adopted models in literature.

Ref.	Optimization Methods	Description of the Adopted Model
		Ballard Mark V	BCS	Nedstack PS6	Temasek	WNs	SR-12
[15]	Flower Pollination Algorithm	Yes	Yes	Yes	Yes	Yes	Yes
[16]	Whale Optimization Algorithm	Yes	-	-	-	-	-
[10]	Bee Colony Algorithm	-	-	-	-	Yes	-
[17]	Cuckoo Search Algorithm	Yes	Yes	-	-	Yes	Yes
[18]	Monarch Butterfly Optimization Algorithm	-	-	Yes	-	-	-
[13]	Grey Wolf Optimization Algorithm	Yes	Yes	-	Yes	Yes	Yes
[19]	Salp Swarm Optimization Algorithm	-	Yes	Yes	-	-	-
[20]	Sunflower Optimization Algorithm	-	-	Yes	-	-	-
[21]	Grasshopper Optimization Algorithm	Yes	-	-	-	Yes	Yes
[22]	Slime Mould Algorithm	Yes	-	-	-	-	Yes
[6]	Spotted Hyena Optimization Algorithm	Yes	Yes	Yes	Yes	Yes	-
[23]	Neural Network Algorithm	Yes	Yes	Yes	-	-	-
[24]	Chaos Game Optimization Technique	Yes	-	Yes	-	-	Yes
SHO	Sea Horse Optimization	Yes	-	-	-	-	-

.

Inspired by the biological behavior of seahorses, the SHO algorithm adopts a two-phase strategy in which individuals explore the search space by drifting along environmental currents and exploit promising regions through spiral movements toward elite solutions. By effectively balancing exploration and exploitation, the SHO algorithm ensures a well-distributed search over the solution space. Evaluating the potential limitations of the SHO algorithm in practical applications is crucial, especially in terms of its scalability in large-scale systems and its effectiveness in real-time control scenarios. The literature indicates that nature-inspired metaheuristic algorithms face various challenges in high-dimensional search spaces, such as convergence stability, convergence time, and resource usage. Because of their nature, such algorithms attempt to optimize many decision variables simultaneously; convergence delays can occur due to the large size of the search space, which can limit the algorithm’s efficiency, especially in large-scale systems.

From a real-time control perspective, algorithmic latency becomes a critical factor when algorithms like SHO require fast and stable responses. In systems requiring real-time performance, such as fuel cell control or load frequency regulation, the computational burden of SHO can impact its ability to respond instantaneously to system dynamics. This creates some limitations for the algorithm’s integration into online control systems. In this context, there are studies that thoroughly address the limitations of metaheuristic algorithms in energy system applications. For example, a comprehensive analysis of the use of metaheuristic methods in energy production and management indicated that algorithm complexity, convergence time, and convergence dynamics are among the fundamental problems encountered in real-time applications [25]. Similarly, another study examining the fundamental inherent challenges of nature-inspired algorithms indicated that the scalability, parameter sensitivity, and mathematical modeling deficiencies of these approaches may limit their applicability to large-scale engineering problems [26]. Therefore, to increase the effectiveness of the SHO algorithm in large-scale and time-sensitive applications, more simplified structures or hybrid solutions should be developed to improve computational efficiency; furthermore, the algorithm’s applicability to problems of different sizes should be systematically tested.

2. Mathematical Modeling of PEMFC

Fuel cells are specialized electrochemical devices that convert chemical energy directly into electrical energy. These systems include a semi-permeable membrane that allows for the passage of protons while blocking electrons. Proton exchange membrane fuel cells (PEMFCs) feature a multilayered structure comprising a polymer-based membrane, anode and cathode electrodes, catalyst layers, gas diffusion layers, bipolar plates, and sealing components.

The polarization curve represents a key performance characteristic of PEMFCs, providing insight into their voltage–current behavior under various load conditions [27]. The rapid voltage drop observed at the beginning of the curve in Figure 1 is defined as the activation loss.

This phenomenon arises from the energy barrier required to initiate electrochemical reactions on the electrode surfaces. At low current densities, this effect is predominant and represents a major factor limiting cell efficiency. In the intermediate current density region, the voltage exhibits a more linear decline, which corresponds to the ohmic polarization region. This section reflects losses due to ionic and electrical resistances within the cell. Proton conduction through the membrane and the conductivity associated with the electrodes primarily influence the voltage behavior in this region. It is typically regarded as the most efficient operating range for PEMFC systems. Toward the end of the curve, a sharp voltage drop occurs at high current densities, attributed to concentration polarization. This results from insufficient transport of reactant gases (hydrogen and oxygen) to the electrode surfaces. When the gas diffusion limits are approached, the reaction rate is constrained, causing a noticeable decline in cell performance. In conclusion, the polarization curve provides a critical foundation for analyzing PEMFC performance under varying operating conditions and plays an essential role in guiding system design and control strategy development [28].

As shown in the PEMFC structure in Figure 2, hydrogen and oxygen gases undergo an electrochemical reaction, enabling them to produce electricity [29].

In the PEMFC anode (1):

$${\mathrm{H}}_2\rightleftharpoons {\mathrm{H}}^{+}+2e^{-}$$

(1)

In the PEMFC cathode (2):

$${\displaystyle \frac{1}{2}}{\mathrm{O}}_2+2{\mathrm{H}}^{+}+2e^{-}\rightleftharpoons {\mathrm{H}}_2\mathrm{O}$$

(2)

In the entire PEMFC (3):

$${\mathrm{H}}_2+{\displaystyle \frac{1}{2}}{\mathrm{O}}_2\rightleftharpoons {\mathrm{H}}_2\mathrm{O}+W_{\mathrm{el}}+Q_{\mathrm{heat}}$$

(3)

reactions occur. The electrochemical and fundamental thermodynamic equations resulting from these reactions form the basis for deriving the mathematical model of the fuel cell [30]. The operating voltage of a single cell, $v_{\mathrm{fc}}$, is presented in Equation (4). It is defined as the difference between the open-circuit voltage E and the sum of various voltage losses. Specifically, the cell voltage is obtained by subtracting the activation loss $v_{\mathrm{act}}$, ohmic loss $v_{\mathrm{ohm}}$, and concentration loss $v_{\mathrm{conc}}$ from E, as defined in Equation (5). These loss components are calculated using Equations (6)–(11).

$$v_{\mathrm{fc}}=E-v_{\mathrm{act}}-v_{\mathrm{ohm}}-v_{\mathrm{conc}}$$

(4)

$$E=1.229-8.5\times {10}^{-4}(T_{\mathrm{fc}}-298.15)+4.308\times {10}^{-5}{T}_{\mathrm{fc}}ln\left({\displaystyle \frac{P_{{\mathrm{O}}_2}}{2}}+P_{{\mathrm{H}}_2}\right)$$

(5)

$$v_a=\left[{\xi }_1+{\xi }_2{T}_{\mathrm{fc}}+{\xi }_3{T}_{\mathrm{fc}}ln\left(C_{{\mathrm{O}}_2}\right)+{\xi }_4{T}_{\mathrm{fc}}ln\left(I_{\mathrm{fc}}\right)\right]$$

(6)

$$C_{{\mathrm{O}}_2}={\displaystyle \frac{P_{{\mathrm{O}}_2}}{5.08\times {10}^6}}exp\left({\displaystyle \frac{498}{T_{\mathrm{fc}}}}\right)$$

(7)

$$v_{\mathrm{ohm}}=I_{\mathrm{fc}}\left(R_{\mathrm{m}}+R_{\mathrm{c}}\right)$$

(8)

$$R_{\mathrm{m}}={\displaystyle \frac{{\rho }_{\mathrm{m}}·l}{M_A}}$$

(9)

$${\rho }_{\mathrm{m}}={\displaystyle \frac{181.6\left[1+0.03\left(\frac{I_{\mathrm{fc}}}{M_A}\right)+0.062{\left(\frac{I_{\mathrm{fc}}}{303}\right)}^2{\left(\frac{I_{\mathrm{fc}}}{M_A}\right)}^{2.5}\right]}{\lambda -0.634-3\left(\frac{I_{\mathrm{fc}}}{M_A}\right)exp\left[4.18\left(\frac{T_{\mathrm{fc}}-303}{T_{\mathrm{fc}}}\right)\right]}}$$

(10)

$$v_{\mathrm{conc}}=-\beta ln\left(1-{\displaystyle \frac{J}{J_{\max }}}\right)$$

(11)

The theoretical output voltage obtained from this mathematical modeling differs significantly from the ideal behavior of the fuel cell when compared to performance losses encountered in practical applications. Although this framework is effective in describing the voltage–current relationship of PEMFC systems at a theoretical level, the accuracy and validity of the model largely depend on the precision of the parameters used and the extent to which the model reflects actual system behavior [31]. Accurate identification of parameters related to physical phenomena such as activation losses, ohmic resistances, and concentration effects is crucial for ensuring reliable model predictions under both static and dynamic operating conditions. However, direct measurement of these parameters is often impractical, and conventional methods frequently fall short due to high computational costs or convergence to local minima. In this context, optimization-based approaches provide an effective solution by enhancing parameter fitting, minimizing discrepancies between model predictions and experimental data, and yielding more reliable simulation outputs [32].

In this study, the SHO algorithm, inspired by the adaptive movements of seahorses in nature, is employed to determine the optimal values of the unknown parameters ${\epsilon }_1,{\epsilon }_2,{\epsilon }_3,{\epsilon }_4,b,\lambda ,R_c$ in the PEMFC model.

3. Sea Horse Optimization Algorithm (SHO)

The SHO algorithm is a nature-inspired metaheuristic developed based on the movement patterns and environmental adaptability of seahorses. The algorithm’s exploration and exploitation mechanisms are fundamentally modeled after the spiral swimming behavior of seahorses and their tendency to drift with ocean currents. The spiral motion enables detailed local search around promising areas, while the drifting behavior facilitates broader exploration of the solution space. Additionally, seahorses’ ability to anchor themselves to marine structures such as seaweed and coral reflects the algorithm’s capacity to adapt and navigate efficiently within dynamic search environments [14].

3.1. Inspiration

The life cycle and behavioral patterns of seahorses are structured around three fundamental strategies: movement, hunting, and reproduction. In the initial phase, seahorses explore their surroundings through spiral motion and passive drifting with ocean currents, which enables them to identify potential food sources and investigate new regions. In the second phase, they engage in targeted feeding by carefully selecting prey and capturing it using their tubular snouts. This stage plays a critical role in minimizing external disturbances during the hunting process. In the final phase, reproductive behavior becomes dominant; male seahorses carry and nurture the eggs in their brood pouch until hatching. Through this natural cycle comprising exploration, foraging, and reproduction, seahorses demonstrate high levels of environmental adaptability and ecological sustainability [14].

3.2. Movement Behavior

• Condition:
  In their natural habitats, seahorses demonstrate unique movement patterns to explore food sources and investigate new living areas. To mathematically simulate this natural motion, the current best solution in the search space is considered as $X_{\mathrm{elite}}$. This reference point guides the movement of the entire population toward optimal solutions within the search domain.

  $$X_{\mathrm{new}}^1(t+1)=X_i\left(t\right)+\mathrm{Levy}\left(\lambda \right)\left(\left(X_{\mathrm{elite}}\left(t\right)-X_i\left(t\right)\right)\times x\times y\times z+X_{\mathrm{elite}}\left(t\right)\right),r_1>0$$

  (12)
  This initial behavior is inspired by the spiral motion of seahorses as they create vortices while swimming in the ocean and plays a critical role in enabling local exploitation. When the standard random variable $r_1$ exceeds a predefined threshold, seahorses imitate spiral trajectories, progressing steadily toward the elite solution. This movement behavior is further enhanced by Lévy flight patterns [33], which dynamically adjust step sizes, allowing the algorithm to explore diverse regions in early iterations and avoid premature convergence to local optima. Simultaneously, the spiral turning angle is continuously adjusted to expand the area around local optima, thereby enabling more detailed exploration. Mathematically, this spiral motion is described by Equation (12) using the three-dimensional coordinates $(x,y,z)$ that define the updated position of the seahorse. The spiral length $\rho$ and turning angle $\theta$ define the geometry of the motion, while the Lévy flight mechanism introduces stochastic variation. Together, these components allow seahorses to dynamically adapt their positions in the search space, emulating purposeful swimming behavior and maintaining an effective balance between exploration and exploitation [33].
• Condition:
  Among the movement behaviors exhibited by seahorses is Brownian motion, which is characterized by random drifting along with ocean currents [34]. This motion is activated when the value of $r_1$ is less than a predefined threshold, corresponding to the exploration phase of the algorithm. In this context, the SHO algorithm employs this behavior to prevent entrapment in local extrema and to enable a more extensive search of the solution space. Brownian motion enhances the exploration capability by simulating a new movement length for the seahorse, thereby promoting broader search dynamics. Mathematically, the updated position expression for this movement is as in Equation (13):

  $$X_{\mathrm{new}}^1(t+1)=X_i\left(t\right)+\mathrm{rand}·l·{\beta }_t·\left(X_i\left(t\right)-{\beta }_t·X_{\mathrm{elite}}\right)\mathrm{for}{r}_1\le 0$$

  (13)
  • l is a constant coefficient.
  • ${\beta }_t$ is the Brownian motion random step size coefficient, which follows a standard normal distribution and represents a randomly generated value.
  In conclusion, the movement phase of the seahorse can be summarized based on two distinct behavioral models:
  • Spiral Motion: When $r_1>0$, the seahorse moves toward the elite solution through a spiral trajectory.
  • Brownian Motion: When $r_1\le 0$, it randomly drifts with ocean currents to explore new regions.
  These two behavior mechanisms simulate seahorses’ adaptive behavior under uncertain environmental conditions and effectively balance exploration and exploitation dynamics in the algorithm.

3.3. Hunting Behavior

The hunting behavior of seahorses is a biologically efficient process, and observational studies indicate prey capture success rates exceeding 90%. In the SHO algorithm, this natural strategy is mathematically modeled to regulate the balance between exploration and exploitation. A randomly generated number $r_2$ is used as a key parameter to determine the success of the hunting attempt, with a threshold value set at 0.1 in the literature [14]. The change in this value is also analyzed within the scope of the study.

• When $r_2>0.1$, the hunting behavior is considered successful. In this case, the seahorse moves towards the prey (i.e., the elite solution $X_{\mathrm{elite}}$), leading to faster and more stable convergence. This enhances the exploitation capability of the SHO algorithm by intensifying the search around the elite region.
• When $r_2\le 0.1$, the hunting attempt fails, indicating that the seahorse is in an exploratory state. Under this condition, the algorithm adjusts the movement direction based on the distance between the current individual and the elite solution, redirecting the search toward new areas. This mechanism helps prevent entrapment in local minima and facilitates a more effective global search across the solution space.

The mathematical representation of this hunting behavior is expressed in Equation (14):

$$X_{\mathrm{new}}^2(t+1)=\left\{\begin{array}{cc}\alpha \left(X_{\mathrm{elite}}-\mathrm{rand}·X_{\mathrm{new}}^1\left(t\right)\right)+(1-\alpha )X_{\mathrm{elite}},\hfill & \mathrm{if}{r}_2>0.1\hfill \\ (1-\alpha )\left(X_{\mathrm{new}}^1\left(t\right)-\mathrm{rand}·X_{\mathrm{elite}}\right)+\alpha X_{\mathrm{new}}^1\left(t\right),\hfill & \mathrm{if}{r}_2\le 0.1\hfill \end{array}\right.$$

(14)

Here, $X_{\mathrm{new}}^1\left(t\right)$ represents the updated position of the seahorse at a given iteration, and $r_2$ denotes a random number within the interval $[0,1]$. The parameter $\alpha$ linearly decreases with iterations to adjust the step size of the seahorse during prey capture. This mechanism ensures that the hunting behavior contributes to both exploitation and exploration, thereby enhancing the overall performance of the algorithm [14].

3.4. Reproductive Behavior

The reproductive behavior of seahorses constitutes a critical phase in the SHO algorithm, enabling the generation of new candidate solutions. During this process, the population is divided into two groups based on their fitness values: male and female seahorses. Considering the biological role of male seahorses in reproduction, the SHO algorithm designates the top half of the population (in terms of fitness) as fathers, while the remaining individuals are considered mothers. This selection strategy facilitates the transfer of favorable genetic traits between parents and helps prevent the offspring from being trapped in local optima. The mathematical representation of this behavior is given in Equations (15) and (16):

• The top 50% of individuals based on fitness form the father population:

  $$\mathrm{fathers}=X_{\mathrm{sort}}^2\left(1:{\displaystyle \frac{\mathrm{pop}}{2}}\right)$$

  (15)
• The remaining 50% of individuals constitute the mother population:

  $$\mathrm{mothers}=X_{\mathrm{sort}}^2\left({\displaystyle \frac{\mathrm{pop}}{2}}+1:\mathrm{pop}\right)$$

  (16)

Here, $X_{\mathrm{sort}}^2$ denotes the sorted list of all $X_{\mathrm{new}}^2$ individuals in ascending order based on their fitness values. The sets fathers and mothers represent the male and female seahorses, respectively. Each male–female pair is randomly selected to generate offspring. For simplicity in computation, each seahorse pair produces only a single offspring. The mathematical model for production of the $i^{\mathrm{th}}$ new offspring is given in Equation (17):

$$X_i^{\mathrm{offspring}}=r_3{X}_i^{\mathrm{father}}+(1-r_3)X_i^{\mathrm{mother}}$$

(17)

In Equation (17), $r_3$ is a randomly generated number in the range $[0,1]$. The index i denotes a positive integer within the range $[1,\mathrm{pop}/2]$, where $X_i^{\mathrm{father}}$ and $X_i^{\mathrm{mother}}$ represent randomly selected individuals from the male and female populations, respectively.

Through this reproductive phase, the SHO algorithm preserves genetic diversity and promotes the generation of new candidate solutions, thereby enhancing its ability to reach the global optimum.

The basic flowchart of the SHO algorithm is given in Figure 3. The algorithm consists of creating an initial population, updating individuals using spiral or Brownian motion, generating new solutions through hunting and breeding steps, and evaluating their fitness. These steps are repeated until a predefined number of iterations are reached.

4. Simulation Study and Results

4.1. Parametric Analysis of SHO

The Sea Horse Optimization (SHO) algorithm is an metaheuristic optimization method inspired by the natural swimming behavior of seahorses and models the exploration–exploitation balance based on biological movements. In this algorithm, the solution search process is guided by two fundamental behavior mechanisms: Lévy flight and Brownian motion. Both behavior types are controlled by different parameters, and the selection of these parameters directly impacts the overall performance of the algorithm. In this study, to more clearly demonstrate the effects of the SHO algorithm on convergence behavior and solution accuracy, the lambda ($\lambda$) for Lévy flight, the Brownian motion step coefficient (l), the random step size (${\beta }_t$), and the deciding parameter $r_1$ and $r_2$ are analyzed in detail. Sensitivity analyses conducted for each parameter separately provide comprehensive insights into the search performance of SHO. In the Sea Horse Optimization (SHO) algorithm, when mathematically modeling the natural motion behavior of seahorses, Lévy flight stands out as a particularly important exploration mechanism. This movement pattern allows the algorithm to scan the solution space more broadly and directionally, increasing its ability to avoid local minima. The characteristics of Lévy flight are determined by a parameter called $\lambda$. In this study, a parametric analysis for $\lambda$ is conducted to examine the effect on the solution accuracy and convergence time of the SHO algorithm. The $\lambda$ parameter is generally defined between 0 and 2. The $\lambda$ value used in analysis is determined between 0.5 and 1.9, and the SSE (sum of squared error), convergence time, and $R^2$ (coefficient of determination) corresponding to each value are summarized in

Table 2. The effect of the $\lambda$ parameter on the SHO algorithm’s performance.

$\mathit{\lambda }$	Time (s)	SSE	${\mathit{R}}^2$	Explanation
0.5	1.60	0.000054	0.998852	Exploitation dominant. The step length is very short, so the algorithm searches more around the current region. Exploration is limited; the solution is stable, but the probability of reaching the global minimum is low.
1.1	1.70	0.000031	0.999346	Near balance. Exploration ability has increased, but exploitation remains strong. Solution quality is high and time is acceptable.
1.3	1.72	0.000019	0.999498	Exploration-oriented balance. Step lengths are longer, enabling broader search and better solution quality. Slight increase in convergence time.
1.5	1.74	0.000010	0.999791	Optimal balance. Perfect harmony between exploration and exploitation. Best results in both accuracy and convergence. Most recommended value in the literature.
1.7	1.83	0.000037	0.999213	Exploration dominant. Longer jumps enable broad search but delay convergence and increase error.
1.9	1.85	0.000063	0.998640	Over-exploration. Very long jumps cause unstable roaming in the solution space. Exploitation is reduced, solution quality decreases.

:

The results show that a value of $\lambda =1.5$ provides the most balanced results in terms of both solution accuracy and convergence time. A value of $\lambda =0.5$ limited both exploration and exploitation capabilities due to excessively short jumps. $\lambda =1.1$ and $1.3$ also produced successful results, but the solution quality was slightly lower than $1.5$. On the other hand, $\lambda =1.7$ and $1.9$ caused the algorithm to perform more exploration, which increased the convergence time and decreased the solution quality. Experimental analysis of the l parameter, which controls the step length of Brownian motion in the SHO algorithm, reveals that the algorithm’s convergence behavior is highly sensitive to this parameter. A value of l = 0.05 provided optimal performance with both the lowest error value and the highest agreement coefficient; larger values of l were observed to increase exploration capability but delay convergence. These findings are given in

Table 3. Effect of Brownian step coefficient l on the SHO algorithm’s performance.

l	Time (s)	SSE	${\mathit{R}}^2$	Explanation
0.01	1.67	0.000036	0.999220	Small steps, faster but limited exploration. Good solution quality, but not optimal.
0.03	1.65	0.000015	0.999688	Optimal exploration–exploitation balance. Excellent performance in both time and SSE.
0.05	1.74	0.000010	0.999791	Lowest SSE and highest $R^2$. Ideal l value for the original SHO.
0.07	1.92	0.000017	0.999629	Increased exploration, but also longer duration. Slight SSE increase, stability maintained.
0.10	1.97	0.000027	0.999425	Over-exploration led to delayed convergence. SSE increased. Local stability may have weakened.

.

The effect of the ${\beta }_t$ coefficient—which is included in the Brownian motion of the Sea Horse Optimization (SHO) algorithm and governs the jump behavior in the solution space—on the algorithm’s performance is analyzed. The ${\beta }_t$ parameter directly determines the exploration capacity by affecting the direction and amplitude of the solution and plays a significant role in the convergence process. In the experimental evaluation, the algorithm was run under the same initial conditions using different ${\beta }_t$ values (0.2, 0.5, 0.8, 1.0, 1.2), and the results were evaluated based on three key metrics (SSE, time, $R^2$). The lowest SSE = 0.000012 and the highest $R^2$ = 0.999733 were obtained with a value of ${\beta }_t=0.8$. This indicates that medium-sized random jumps balance both the exploration capacity and the convergence stability of SHO. While solution quality decreases with lower ${\beta }_t$ values (e.g., 0.2) due to limited movement, higher ${\beta }_t$ values (1.2) result in deterioration in convergence stability and increased error values due to excessive jump growth. The findings in

Table 4. The effect of the ${\beta }_t$ parameter on the SHO performance algorithm.

${\mathit{\beta }}_{\mathit{t}}$	Time (s)	SSE	${\mathit{R}}^2$	Explanation
0.2	1.66	0.000055	0.998830	Low ${\beta }_t$, narrower movement, shorter time but moderate solution quality
0.5	1.71	0.000033	0.999302	Improvement starts with increasing ${\beta }_t$
0.8	1.82	0.000012	0.999733	Best performance: strong exploration, good  convergence
1.0	1.79	0.000040	0.999147	Slight degradation begins, not too high
1.2	1.80	0.000051	0.998914	Excessive jumps, performance degradation  observed

demonstrate that, within the framework of the exploration–exploitation balance emphasized in the literature, careful tuning of the ${\beta }_t$ parameter is critical to the performance of the SHO algorithm.

Another important parameter is the $r_1$ parameter in the algorithm. This parameter is a random number and determines which movement pattern an individual will adopt in a given situation. In this study, the effects of different distributions of the $r_1$ parameter on the SSE, $R^2$, and convergence time is investigated. The results of tests conducted with different distributions are summarized below:

• $r_1=\mathrm{randn}\left(\right)$: Spiral and Brownian motions are balanced. In this reference case, SSE = 0.00001 and $R^2=0.9998$ are achieved.
• $r_1=\mathrm{rand}\left(\right)$: Because all individuals demonstrate spiral motion, the algorithm converges quickly, but its exploration capability is somewhat reduced. Nevertheless, high accuracy and a quick solution are achieved.
• $r_1=0$: In this case, where all individuals explored only with Brownian motion, solution quality decreases significantly (SSE = 0.00409, $R^2=0.912$). This result demonstrates that excessive exploration bias can limit solution accuracy.
• $r_1\sim \mathrm{Beta}(2,5)$: In this distribution with a higher exploration bias, a slight increase in the SSE value is observed, while the convergence time is extended. This demonstrates that increasing exploration provides a more stable result despite the longer time requirements.
• $r_1\sim \mathrm{Beta}(5,2)$: When exploitation bias is dominant, higher accuracy is achieved in a shorter time; however, because exploration is limited, the algorithm’s ability to provide diversity in a large solution space is reduced.

The findings are presented in

Table 5. The effect of the $r_1$ distribution on the SHO algorithm’s performance.

${\mathit{r}}_1$ Distribution	Best SSE	Time (s)	${\mathit{R}}^2$	Explanation
$r_1=\mathrm{randn}\left(\right)$	0.00001	1.74	0.999791	Exploration–exploitation balance. Reference case. Combination of spiral and Brownian motion. Both behaviors are active.
$r_1=\mathrm{rand}\left(\right)$	0.000018	1.64	0.999621	Only spiral motion: all individuals have $r_1>0$, leading to exploitation-only behavior. Fast convergence and high accuracy.
$r_1=0$	0.004090	1.80	0.912408	Only Brownian motion: all individuals have $r_1\le 0$, leading to exploration-only behavior. Lower solution quality, but acceptable runtime.
$r_1\sim \mathrm{Beta}(2,5)$	0.000044	1.97	0.999053	Mostly small $r_1$ values: exploration-biased. Longer runtime, moderate accuracy.
$r_1\sim \mathrm{Beta}(5,2)$	0.000032	1.77	0.999317	Mostly large $r_1$ values: exploitation-biased. Short runtime, high accuracy.

, and each distribution case is summerized.

Parametric analysis of $r_2$ is conducted to evaluate the impact. This parameter governs the effects of the predation behavior of the Sea Horse Optimization (SHO) algorithm on convergence stability and solution quality. The $r_2$ parameter plays a critical role in determining whether the solution update will be exploration–exploitation oriented. Although the literature commonly adopts $r_2\sim U(0,1)$ (uniform distribution), this study aims to systematically investigate the effect of different distribution types on algorithm performance. The findings are reported in

Table 6. Effect of $r_2$ distribution on SHO algorithm performance.

${\mathit{r}}_2$ Distribution	Best SSE	Time (s)	${\mathit{R}}^2$	Explanation
Fixed ($r_2=0.5$)	0.020477	1.61	0.562087	Low accuracy, fast runtime, but weak exploration–exploitation balance.
Beta(5,2) (exploitation-biased)	0.009850	4.15	0.789031	Better accuracy, but high exploitation increases the risk of local minima.
Beta(2,5) (exploration-biased)	0.000029	4.28	0.999375	Very high accuracy with best balance, but the longest runtime.
Uniform $U(0,1)$	0.000010	1.74	0.999791	Most balanced structure, high accuracy and short runtime, recommended default setting.

. Four different distributions were applied in the experimental evaluation:

• Fixed value: $r_2=0.5$
• Exploration-biased distribution: $r_2\sim \mathrm{Beta}(2,5)$
• Exploitation-biased distribution: $r_2\sim \mathrm{Beta}(5,2)$
• Uniform distribution: $r_2\sim U(0,1)$

The obtained results show that the performance of the SHO algorithm is directly affected by the distribution of $r_2$. When a uniform distribution is used, the algorithm achieves a very low SSE = 0.000010 in 1.74 s, and the highest $R^2=0.999791$ is observed. These results indicate that the natural balance between exploration and exploitation behaviors in the original design of SHO is positively reflected in the algorithm’s performance. The algorithm performed a more comprehensive exploration in the solution space for the Beta(2,5). In this case, the SSE value was recorded as approximately 0.000029, but the algorithm’s convergence time extended to approximately 4.28 s. Nevertheless, a high $R^2=0.999375$ was obtained. This indicates that broad search capability has a positive effect on accuracy, although it increases convergence time. On the other hand, when the exploitation-biased Beta(5,2) distribution was used, the algorithm exhibited a faster convergence process 4.15 s, but the SSE = 0.009850 and $R^2=0.789031$ remained relatively weak. These results show that the algorithm concentrated on elite regions, but the overall search efficiency decreased, and the risk of being trapped in local minima increased. In the case where a fixed $r_2=0.5$ value was used, both exploration and exploitation capabilities remained limited, resulting in the highest SSE = 0.020477 and the lowest $R^2=0.562087$. In this case, due to the algorithm’s limited adaptation capability, it is understood that the overall performance level is low.

4.2. PEM Fuel Cell Model Parameter Estimation

In this study, for PEMFC modeling, the Ballard Mark V model is employed. The model parameters and operating conditions are set as follows: $N=1$, $T=343\mathrm{K}$, $P_{{\mathrm{O}}_2}=1\mathrm{bar}$, $P_{{\mathrm{H}}_2}=1\mathrm{bar}$, $J_{\max }=1.5{\mathrm{A}/\mathrm{cm}}^2$, $l=0.0178\mathrm{cm}$, $A=50.6{\mathrm{cm}}^2$. Reference experimental measurements were obtained from the polarization curve in [28].

For the Ballard Mark V model, experimental measurements reported in the literature are used. The population size is set to 30, and the maximum number of iterations is limited to 5000. Since the performance of an optimization method is limited by parameter limits, The search area of the Ballard Mark V PEMFC model is limited, using the limit values available in the literature for the parameters to be optimized [1,2,6,12,23,35,36,37]. Boundary limits for unknown parameters in the PEMFC model are given in

Table 7. Boundary limits for unknown parameters in PEMFC model [1,2,6,12,23,35,36,37].

Model Parameter	${\mathit{\epsilon }}_1$	${\mathit{\epsilon }}_2$	${\mathit{\epsilon }}_3$	${\mathit{\epsilon }}_4$	$\mathit{\lambda }$	${\mathit{R}}_{\mathit{c}}$	$\mathit{\beta }$
Lower Bound	−1.1997	0.001	0.000036	−0.00026	10	0.0001	0.0136
Upper Bound	−0.8532	0.005	0.000098	−0.0000954	24	0.0008	0.5

.

The following results were obtained from the optimization process: RMSE = 0.001322, MAE = 0.001079, $R^2$ = 0.999791, true/model ratio = 0.9992 (seahorse success rate), convergence time: 1.74 s, best SSE = 0.000010. Best Parameters (ObjectivePosition): ${\epsilon }_1=-0.853200$, ${\epsilon }_2=0.003250$, ${\epsilon }_3=0.000036$, ${\epsilon }_4=-0.000095$, $\lambda =12.068511$, $R_c=0.000293$, $\beta =0.013642$.

To assess the impact of varying the parameter search limits on model accuracy, sensitivity analysis was performed using limits narrowed and expanded by $\pm 10\%$ in addition to the literature limits. The results obtained with $\pm 10\%$ narrowing or expanding of the limit values are shown in

Table 8. Boundary limits for unknown parameters in PEMFC model.

Limit Range	RMSE	Best SSE
Literature values in Table 7	0.001322	0.00001
$10\%$ Narrowed	0.002709	0.000044
$10\%$ Expanded	0.002629	0.000041

.

According to the results obtained, when the limits in the literature were used, the lowest error value was obtained, RMSE was 0.001322, and the best SSE was 0.00001. In cases where the boundaries were narrowed and expanded, model accuracy remained acceptable, with RMSEs of 0.002709 and 0.002629, respectively. These findings indicate that the SHO algorithm has low sensitivity to small changes in parameter boundary values, thus achieving high solution stability and identification reliability.

The comparison between the simulated model and the experimental data and while the SHO algorithm converges over 5000 iterations is depicted in Figure 4. A gradual decrease in the error value is observed, and finally, the best SSE of 0.000010 is reached.

5. Discussion

In this study, the SHO algorithm is applied for the first time to determine the PEMFC model parameters. Unlike techniques such as genetic algorithms, particle swarm optimization (PSO), and genetic algorithms (GAs), which are frequently used in the literature, the multi-stage exploration–exploitation strategy of SHO, based on seahorse behaviors, provides more flexible and balanced navigation in the solution space. While the spiral and Brownian motion-based search mechanisms allow for extensive exploration in the early iterations, hunting and reproductive behaviors prevent the model from getting stuck in local minima. Parametric analyses demonstrate in detail how the parametric changes of the SHO algorithm affect these situations.

The parametric analyses revealed that control parameters such as $\lambda$, l, ${\beta }_t$, $r_1$, and $r_2$ used in the SHO algorithm are not only technical settings but also play a decisive role in the algorithm’s exploration–exploitation balance, convergence time, and solution quality. It was observed that $\lambda =1.5$, $l=0.05$, and ${\beta }_t=0.8$ values enable SHO to perform extensive searches and find solutions with high accuracy. Furthermore, the choice of $r_1$ and $r_2$ distribution fundamentally affects the algorithm’s behavior, as it determines whether individuals exhibit spiral or Brownian movements, or it affects the exploration–exploitation balance. These findings demonstrate that conducting a preliminary analysis of the SHO algorithm at the parameter level before applying it to different problem types is critical to its performance. This study contributes to the literature by clearly demonstrating how the structural flexibility of SHO can be optimized with correct parameter selections.

The modeling process yielded a Mean Absolute Error (MAE) of 0.001079 and a coefficient of determination $R^2$ of 0.999791, which indicates a satisfactory level of accuracy. Furthermore, the developed model, using the optimal parameter set, demonstrated a 99.92% agreement with experimental data, underscoring the high-resolution optimization capability of the SHO algorithm.

With an average convergence time of only 1.74 s, SHO exhibits low computational demand, suggesting its suitability for real-time applications. This renders it particularly promising for embedded systems or time-sensitive applications, such as in automotive systems or portable fuel cell devices.

As seen in

Table 9. Feature and algorithm comparison for SHO, PSO, GWO, and ABC.

Feature/Algorithm	SHO (Sea Horse Optimization) [38,39,40]	PSO (Particle Swarm Optimization) [8,9,41]	GWO (Grey Wolf Optimizer) [13,42]	ABC (Artificial Bee Colony) [11,12,43,44]
Biological Inspiration	Seahorse swarm behavior	Bird/fish swarm movement	Hunting and social leadership structure of gray wolves	Task-distributed search behavior within a bee colony
Exploration Mechanism	Directional and extensive search with Lévy flight and Brownian motion	Extensive search with speed and position updates	Random guidance with alpha, beta, and delta leadership	Random search for new resources with scout bees
Exploitation Mechanism	Local scanning with Brownian motion, directional position optimization	Orientation to the best position	Position updating based on $\alpha$, $\beta$, $\delta$ wolves	Concentration around the successful resource
Local Minima Avoidance	High (with Lévy flight and Brownian movement)	Medium (risk of early convergence)	Medium (fixed leader structure can be limiting)	Low–medium (high randomness may cause local stagnation)
Mathematical Structure	Position updates via motion, distance scaling, and stochastic jumps	Speed and position model with fixed parameters	weighted average of positions and nonlinear distance modulation	Stochastic position updates
Exploration, Exploration and Search Behavior Balance	Balanced (Lévy and Brownian movement)	Balance depends on user parameters	Exploration/exploitation rate generally fixed	Exploration is dominant, exploitation is limited
Adaptness	Medium–high (movement type effective)	Low–medium (parameters fixed)	Low (static leadership hierarchy)	Low (roles unchanged)
Convergence Stability	High (controlled convergence)	Medium (rapid but may be unstable)	Medium–high (may converge early)	Low (fluctuating solution quality)

, the use of Brownian motion and Lévy flights in SHO leads to a unique balance between exhaustive local search and long-distance exploratory leaps. This structure allows the algorithm to escape local optima more effectively than traditional metaheuristics that lack such movement diversity. SHO’s hybrid structure, which includes Brownian motion and Lévy flights, appears to provide a natural balance between exploration and convergence without the need for parameter tuning. This structure provides strong resistance to avoiding local minima while also supporting the algorithm’s convergence stability. While algorithms such as GWO or PSO are typically based on fixed leadership or social components, direction updates in SHO occur stochastically and adaptively. This structure is one of the main reasons for SHO’s high performance in complex solution spaces. The resulting improvement in convergence stability and solution quality is supported both theoretically and experimentally in the presented results. The experimental results are compared with other studies in the literature in

Table 10. Unknown parameter and SSE comparison for Ballard Mark V PEMFC with other methods.

Study	Best SSE	${\mathit{\epsilon }}_1$	${\mathit{\epsilon }}_2$	${\mathit{\epsilon }}_3$	${\mathit{\epsilon }}_4$	$\mathit{\lambda }$	${\mathit{R}}_{\mathit{c}}$	$\mathit{\beta }$
[15]	0.2872	−0.8775	0.0025	0.000064439	−0.00012531	12.016	0.0006369	0.0198
[16]	0.8537	−1.1978	0.0044183	0.000097214	−0.00016273	23.0	0.0001002	0.0136
[17]	0.0023765	−0.9728	0.0034480	0.000083832	−0.00011328	21.6995	0.0008	0.0136
[11]	0.002067	−1.1827	0.0037080	0.0000936	−0.00011925	11.7603	0.00078773	0.0136
[21]	0.8710	−0.8532	0.0034173	0.000098	−0.00015955	22.8458	0.0001	0.0136
[22]	0.000017729	−1.1942	0.001	0.000041234	−0.00016807	15.3311	0.0002	0.2083
[6]	0.0000104	−1.1993	0.00362	0.000036704	−0.0001784	19.6862	0.0003039	0.0665
[23]	0.85361	−0.97997	0.0036946	0.000090871	−0.00016282	23.0	0.0001	0.0136
[24]	0.85361	−1.192042	0.0036124	0.000040797	−0.0001628	23	0.0001	0.0136
SHO	0.00001	−0.8532	0.003250	0.000036	−0.000095	12.0685	0.000293	0.0136

.

The SSE metric used in this study is calculated using the polarization curve obtained under a constant temperature of 70 °C and pressure of 1 bar, conditions commonly adopted in the literature [9]. This approach, as in previous studies, is generally based on optimization only for a specific operating condition. The main reason for this is that the current–voltage characteristics of the PEMFC are highly sensitive to temperature and pressure changes. Since changes in these parameters significantly alter the current–voltage curve, direct comparison of SSE values calculated under different conditions is not meaningful in the evaluation of model performance [17]. Therefore, the vast majority of studies in the literature perform SSE calculations only using a standard polarization curve to make performance evaluations meaningful and comparable. Furthermore, even if the model is optimized for different conditions in some studies, the reported SSE values are calculated only using the polarization curve for a fixed reference condition (usually 70 °C and 1 bar) [21,24]. The same method is adopted in this study. Although the evaluation in this study is conducted under fixed conditions (70 °C and 1 bar), which are widely adopted in the literature, PEMFC model parameters, such as membrane conductance, contact resistance, and exchange current density, are known to be sensitive to environmental variations. Such dynamic variations are a fundamental challenge inherent in PEMFC modeling. The direction-adaptive and balanced search structure of the SHO algorithm stands out for its flexibility in nonlinear and time-varying optimization problems, offering advantages over fixed-structure metaheuristic methods. This feature creates promising potential for future research under different operating conditions.

6. Conclusions

In the literature, there is no study in which this algorithm is directly used in determining the equivalent circuit model parameters for proton exchange membrane fuel cells (PEMFC). In this paper, the SHO algorithm is applied to PEMFC model parameter optimization for the first time, and this gap in the literature is filled by evaluating the potential of the algorithm. In the modeling process, the experimental polarization data of the Ballard Mark V type cell are taken as reference, and the seven basic (${\epsilon }_1$, ${\epsilon }_2$, ${\epsilon }_3$, ${\epsilon }_4$, b, $\lambda$, $R_c$) parameters of the equivalent circuit model are optimized with the SHO algorithm. The SHO algorithm stands out with its high accuracy (MAE = 0.001079), strong convergence performance, and low convergence time (1.74 s); it offers a solution method that is resilient to the problems of premature convergence and getting stuck in local minimum. The high accuracy level and low convergence time offered by SHO make it a remarkable alternative in terms of PEMFC parameter estimation. Recent studies [38,39] have proposed improved versions of the original seahorse optimization algorithm by adding chaotic arrays, adaptive update rules, and improved search direction mechanisms to further improve its performance on global optimization tasks. In particular, Hashim et al. [38] introduced the Modified Seahorse Optimizer (MSHO) and demonstrated its superiority in benchmarking and engineering problems through numerous enhancements. These studies demonstrate the original SHO framework’s modular and flexible structure, which can be extended. As a result, it can be seen that the SHO-based optimization approach presented in this study is not only limited to PEMFC modeling but has also been effectively used in various applications in the energy systems literature. The SHO algorithm provides successful results in a wide range of applications, such as real-time power management optimization and range estimation in electric vehicles [44], economical load allocation and power ripple reduction in photovoltaic microgrids [45], increasing energy efficiency in hybrid systems based on solid oxide fuel cells [46], improving system stability by providing faster settlement and lower overshoot values in load–frequency control [47], and improving transient performance in PSS settings by combining it with chaotic maps [48]. In particular, thanks to its hybrid direction updates based on Brownian motion and Lévy flight, SHO offers more balanced exploration–exploitation properties and more stable convergence behavior compared to traditional metaheuristic algorithms in both nonlinear and time-sensitive systems. In this context, the SHO algorithm is considered to be an effective and flexible tool not only for fuel cell modeling but also for the optimization of larger-scale energy systems. For future research, it is recommended to explore hybrid configurations to enhance its performance in more complex systems. Moreover, extending the application of the SHO algorithm to multi-objective optimization problems and other fuel cell technologies—such as Solid Oxide Fuel Cells (SOFC) and Direct Methanol Fuel Cells (DMFC)—may further broaden its utility.