Deriving analytical voltage response in fractional order battery modeling

Traditional electrochemical, data-driven, and integer-order equivalent-circuit models face limitations in balancing accuracy, computational efficiency, and real-time applicability. Fractional-order equivalent circuit models have emerged as a powerful alternative, incorporating Constant Phase Elements (CPEs) to capture complex polarization effects and electrochemical dynamics, such as diffusion and charge-transfer phenomena, across a broad range of operating conditions, especially hereditary effect. This work discusses three existing fractional-order battery circuit models with constant parameters and introduces a new fourth model that includes a fractional inductor in series with a resistor in the RCPE branch. The study presents several mathematical approaches for representing the time-domain voltage responses of these models. The voltage responses are initially derived in the complex domain and subsequently obtained in the time domain using three approaches: (i) representation of solutions in terms of fractional differential equations (including multi-term cases), (ii) integral formulations derived via Laplace transform inversion using the contour-integration method, and (iii) representations based on special functions. In particular, while the first two models’ voltage responses can be expressed in terms of Mittag-Leffler functions, the third and fourth models’ are obtained as series expansions involving derivatives of these functions. The analytical time-domain voltage responses are then validated against numerical Laplace inversion and Simulink simulation results. The obtained expressions enable implementation without complex domain and require minimal data for efficient computation.