From Classical to Reduced Semi-Empirical PEMFC Models: Evolution, Lambert W and g-Function Reformulation, and Applications in Optimization and Parameter Estimation

Semi-empirical models remain among the most widely used tools for analyzing proton exchange membrane fuel cells (PEMFCs) because they provide a practical balance between physical interpretability and computational efficiency. Existing reviews primarily focus on optimization techniques, system-level modeling, or data-driven approaches, with comparatively limited attention to the structural evolution of semi-empirical voltage models and their analytical reformulation. This paper presents a structured review of semi-empirical PEMFC models, tracing their development from classical formulations, including the Larminie–Dicks, Amphlett, and Mann models, to recent reduced-parameter models such as the Perez model and its variants. Emphasis is placed on the transition toward reduced formulations, highlighting their potential advantages in terms of numerical stability, reduced parameter coupling, and suitability for parameter-estimation and control-oriented applications. In addition, analytical reformulation approaches based on the Lambert W function and the g-function are reviewed and discussed, showing how these methods can enable partial analytical inversion of nonlinear model equations and support more systematic numerical solution procedures. A comparative benchmark analysis using Ballard Mark V and BCS 500 W PEMFC datasets is presented based on root mean square error, mean absolute error, voltage-error trends, and parameter-sensitivity analysis. The results show that reduced models can provide competitive accuracy and favorable robustness in the considered cases, while physically detailed models may remain advantageous when a richer physical interpretation is required. The review clarifies the trade-offs among physical fidelity, parameter identifiability, model complexity, and computational practicality, and provides guidance for model selection, parameter estimation, and optimization in PEMFC engineering applications.