A compact six-parameter circuit model reproduced the steady-state voltage curves of two proton exchange membrane fuel cells with very low fitting error, according to a computational study that combined the model with a new metaheuristic called Enhanced Artificial Rabbits Optimization. The proposed optimizer, EARO, was tested on experimental polarization data from a 5-watt Scribner 850e single cell and a 6-kilowatt NedStack PS6 stack comprising 65 cells. In repeated trials, EARO produced the lowest root mean square error among five optimization methods, including the original artificial rabbits algorithm, the grey wolf optimizer, the whale optimization algorithm and the zebra optimization algorithm. The results support the framework as a computationally lightweight way to estimate fuel-cell parameters, although the study did not validate transient behavior, resource-constrained hardware or real-time system latency.
The model represents a proton exchange membrane fuel cell as an electrical circuit whose output is governed by the thermodynamic potential and three principal losses: activation, ohmic and concentration. Activation loss dominates the early part of the polarization curve, reflecting the kinetic barrier associated with electrochemical reactions. Ohmic loss grows with current because electrons moving through electrodes and protons crossing the membrane encounter resistance. Concentration loss becomes more important at high current, when reactants cannot reach electrode surfaces rapidly enough. In the proposed lumped representation, these processes are expressed through a diode-like activation term, an internal resistance and a current-dependent resistance for concentration loss.
Only six parameters need to be estimated: the open-circuit or thermodynamic voltage, the diode ideality factor, the diode reverse-saturation current, internal resistance, a concentration-loss coefficient and an effective maximum-current scale. The model does not evaluate the logarithmic activation term at zero current. Instead, it is fitted over the measured current range, with a minimum fitting current of 0.0200 ampere in the reported single-cell case, while the thermodynamic term represents the open-circuit condition. For a stack, the cell voltage expression is multiplied by the number of cells in series. The researchers minimized the root mean square error between calculated and experimental voltages, with separate parameter bounds established for the small cell and the larger stack.
EARO extends the original artificial rabbits optimizer, a population-based metaheuristic inspired by the foraging and hiding behavior of rabbits. Its population of 100 candidate parameter sets is initialized through four complementary mechanisms: a Sobol-like quasi-random scheme for 30 percent of individuals, opposition-based learning for another 30 percent, chaotic initialization for 20 percent and purely random initialization for the remaining 20 percent. During the search, an adaptive energy factor governs movement between exploration and exploitation. The exploration phase includes elite-guided detour foraging and occasional Levy-flight jumps, while the exploitation phase concentrates on promising solutions through best-guided, hybrid and elite-mean strategies.
The enhanced optimizer also introduces several safeguards against premature convergence. Standard artificial-rabbit updates, differential-evolution mutation and a best-guided Gaussian walk are selected probabilistically, with differential evolution and Gaussian movement each used in 30 percent of updates. Candidates that cross parameter boundaries are reflected and then clipped to remain feasible. Greedy selection accepts a candidate only when it improves the voltage root mean square error, and elite preservation protects the best solution from being lost. If progress stalls, the worst portion of the population is either reinitialized randomly or regenerated around the best candidate. Every 50 iterations, 15 small perturbations are tested near the leading solution as an intensive local-refinement step.
After each global search, the best parameter set is polished with L-BFGS-B, a bounded gradient-based optimizer that uses limited-memory approximations to the inverse Hessian. Because the researchers did not provide an analytical Jacobian, SciPy estimated gradients with a forward finite-difference scheme. The local search used a function tolerance of 10 to the power of minus 15, a gradient tolerance of 10 to the power of minus 12 and a maximum of 10,000 iterations, but its result was accepted only if it improved upon the solution found by EARO. This makes EARO a hybrid method: the metaheuristic searches broadly for a promising region, while the final local search refines the fit. The comparison algorithms were tested in their original, unmodified forms, without the added L-BFGS-B stage.
For the Scribner 850e cell, the experimental dataset contained 23 measured current-voltage points at 353.15 kelvin. The cell had an active area of 5 square centimeters, used a Nafion 117 membrane and was tested with anode and cathode pressures of 1, 1.25 and 1.5 bar under fully humidified conditions. EARO achieved best root mean square errors of 3.010 × 10 to the power of minus 5, 2.977 × 10 to the power of minus 5 and 2.657 × 10 to the power of minus 5 at the three respective pressures. The associated convergence histories stabilized at approximately 160, 230 and 130 iterations, compared with about 280, 240 and 150 iterations for the original artificial rabbits optimizer.
Each optimizer was run 30 times at each operating condition. EARO showed the strongest consistency as well as the smallest best error. At 1 bar, its mean root mean square error was 3.010212 × 10 to the power of minus 5, while the original artificial rabbits optimizer had a mean of 2.35106685 × 10 to the power of minus 4. At 1.25 bar, the corresponding means were 4.6248195 × 10 to the power of minus 5 and 1.39796667 × 10 to the power of minus 4; at 1.5 bar, they were 2.6572892 × 10 to the power of minus 5 and 3.94585023 × 10 to the power of minus 4. The near-constant results across repeated EARO runs produced a standard deviation of only 2.144 × 10 to the power of minus 8 at 1 bar, 5.327593 × 10 to the power of minus 9 at 1.25 bar and 8.05 × 10 to the power of minus 13 at 1.5 bar.
The second case used 28 measured current-voltage points from the NedStack PS6 stack, which contained 65 series-connected cells with an active area of 240 square centimeters per cell. It operated at 343 kelvin and 1 bar with fully humidified reactant streams. EARO’s best root mean square error was 2.69204 × 10 to the power of minus 7, compared with 1.8338747 × 10 to the power of minus 5 for the original artificial rabbits optimizer and larger errors for the other three methods. Across 30 runs, EARO had a mean error of 6.81211 × 10 to the power of minus 7 and a median of 3.21676 × 10 to the power of minus 7. Its reconstructed voltage-current and power-current curves closely followed the experimental data across the stack’s operating range, and its voltage errors were lower than those produced by the four comparators.
Statistical analysis supported the performance differences. A Kruskal-Wallis test rejected equal performance among the five methods in all four datasets, with p-values between 2.61 × 10 to the power of minus 25 and 4.71 × 10 to the power of minus 23. Pairwise two-sided Wilcoxon rank-sum tests with Holm adjustment found EARO had significantly lower errors than every competitor in all 16 comparisons. The smallest effect occurred for the 5-watt cell at 1.25 bar, where EARO still beat the original artificial rabbits optimizer with a Holm-adjusted p-value of 2.278 × 10 to the power of minus 5 and a Vargha-Delaney A12 effect size of 0.8189. In the other 15 comparisons, A12 values ranged from 0.9978 to 1.0, indicating that a randomly selected EARO run had a lower error than a randomly selected competitor run in virtually every case.
The model itself is inexpensive to evaluate because it uses algebraic equations rather than a detailed electrochemical or computational-fluid-dynamics solution. The researchers described a single fitness evaluation as constant-time and the overall search complexity as linear in the number of iterations, population size and six decision variables, with additional overhead from the local-search stage. That may make offline parameter identification and repeated simulation practical in engineering workflows, but the study did not implement the method on embedded processors or demonstrate real-time operation. The authors also proposed that more accurate power predictions could eventually reduce uncertainty in fuel-cell electric vehicles and stationary microgrids, potentially limiting oversizing and improving cost estimates. They cautioned, however, that calculating financial savings would require dedicated system-level modeling. The demonstrated result is therefore narrower but well characterized: a six-parameter steady-state model, fitted with a statistically robust hybrid optimizer, closely reconstructed two laboratory-tested PEM fuel-cell datasets without being validated for transient dynamics or deployed control systems.
Subject of Research: Steady-state PEM fuel-cell circuit modeling and parameter estimation using a hybrid artificial-rabbits optimization algorithm.
Article Title: A novel circuit-level steady-state fuel cell model assisted by enhanced artificial rabbits optimization
Article References: Bayoumi, A., Sakr, W. S., Elbana, M. M., El-Sehiemy, R. A., & Abaza, A. (2026). A novel circuit-level steady-state fuel cell model assisted by enhanced artificial rabbits optimization. Results in Engineering, 32, Article 113499. https://doi.org/10.1016/j.rineng.2026.113499
Image Credits: AI Generated
DOI: 10.1016/j.rineng.2026.113499
Keywords: proton exchange membrane fuel cells, circuit modeling, parameter estimation, metaheuristic optimization, artificial rabbits optimization, polarization curves, voltage prediction, root mean square error, fuel-cell stacks, computational efficiency, novel, circuit-level
