On the analytical solution of single particle models and semi-analytical solution of P2D model for lithium-ion batteries

The Doyle–Fuller–Newman (DFN) model serves as a fundamental electrochemical modelling framework widely used in the field of lithium-ion batteries, especially in its pseudo-two-dimensional (P2D) formulation. Despite its widespread use, the DFN model imposes significant computational demands, especially for tasks such as cell characterization and state estimation in electric vehicles' battery management systems (BMS). To mitigate these computational challenges, researchers have developed several reduced-order models. In this study, we tackle the associated computational efforts starting from the governing DFN equations. The analytical solutions for the transport equations in the electrode and the electrolyte are derived, considering generic time-varying boundary conditions under certain assumptions. The obtained exact solutions are then used to develop an analytical model for a single particle model with electrolyte dynamics (SPME) which provides higher predictive accuracy compared to a single particle model (SPM), especially in high C-rate scenarios, which are critical in many applications, including electric vehicles. Furthermore, the exact solution of lithium diffusion in active material particles is incorporated into the standard P2D model. This integration leads to a semi-analytical variant of the P2D model (SAP2D). A remarkable advantage of these analytical solutions is the significantly lower computational cost compared to corresponding numerical approaches. Spatial discretization becomes unnecessary, and the solutions can be obtained by a few explicit function evaluations. Moreover, when the time-varying boundary conditions are known, the need for time stepping is also eliminated.