A two phase differential evolution algorithm with perturbation and covariance matrix for PEMFC parameter estimation challenges

Parameter identification of Proton Exchange Membrane Fuel Cells (PEMFCs) is a key factor in improving the performance of the fuel cell and assuring the operational reliability. In this study, a novel algorithm PCM-DE, based on the Differential Evolution framework, is proposed. A perturbation mechanism along with a stagnation indicator based on a Covariance Matrix is incorporated into this algorithm. Three key innovations are introduced in the PCM-DE algorithm. A two phase approach based on fitness values is used to develop a parameter adaptation strategy, firstly. The idea here is to move the evolutionary process to more promising areas of the search space on different occasions. Second, a perturbation mechanism is incorporated that targets the archived population. This mechanism utilizes a novel weight coefficient, which is determined based on the fitness values and positional attributes of archived individuals, to improve exploration efficiency. Lastly, a stagnation indicator leveraging covariance matrix analysis is employed to evaluate the diversity within the population. This indicator identifies stagnant individuals and applies perturbations to them, promoting exploration and preventing premature convergence. The effectiveness of PCM-DE is validated against nine state-of-the-art algorithms, including TDE, PSO-sono, CS-DE, jSO, EDO, LSHADE, HSES, E-QUATRE, and EA4eig, through the parameter estimation of six PEMFC stacks—BCS 500 W, Nedstack 600 W PS6, SR-12 W, Horizon H-12, Ballard Mark V, and STD 250 W. Across all test cases, PCM-DE consistently achieved the lowest minimum SSE values, including 0.025493 for BCS 500 W, 0.275211 for Nedstack 600 W PS6, 0.242284 for SR-12 W, 0.102915 for Horizon H-12, 0.148632 for Ballard Mark V, and 0.283774 for STD 250 W. PCM-DE also demonstrated rapid convergence, superior robustness with the lowest standard deviations (e.g., 3.54E−16 for Nedstack 600 W PS6), and the highest computational efficiency, with runtimes as low as 0.191303 s. These numerical results emphasize PCM-DE’s ability to outperform existing algorithms in accuracy, convergence speed, and consistency, showcasing its potential for advancing PEMFC modeling and optimization. Future research will explore PCM-DE’s applicability to dynamic operating conditions and its adaptability to other energy systems, paving the way for efficient and sustainable fuel cell technologies. © The Author(s) 2025.