Considering solid phase diffusion penetration depth to improve profile approximations: Towards accurate State estimations in lithium-ion batteries at low characteristic diffusion lengths

Abstract

Lithium-ion batteries are typically modelled using the pseudo-2-dimensional (P2D) model, where a set of partial differential equations describing transport and electrochemical reactions are solved numerically to obtain battery state information. To improve computational efficiency and enable on-board implementation for state-estimation, reduced order models are developed, where one such model reduction technique includes using a profile approximation for solving solid phase diffusion inside electrode particles. While greatly reducing model complexity, the profile approximation also leads to reduced accuracy, especially at low characteristic diffusion lengths. We address this issue by developing an improved model, which recognizes the existence of a penetration depth and the solution is obtained by dividing the particle into diffusion-free core and a shell corresponding to the penetration depth. The time-scale for the diffusion front to reach the particle centre depends on the diffusivity and particle radius. A similarity solution for this period, leads to an expression for the State of charge (SOC) in the spatio-temporal domain. The predictions are compared to that from the benchmark P2D and experimental data, revealing accuracies >99%, while adding no computational cost. This improved model is particularly useful under high C-rate or low temperature operation, where the characteristic diffusion lengths are low.