Koopman Spectral Reduced-Order Modeling of Spherical Diffusion in Lithium-Ion Batteries

Abstract

Physics-based battery models provide internal electrochemical states for estimation and control, but solving the partial differential equations governing solid diffusion can be computationally expensive. This paper therefore develops an analytical Koopman spectral reduced-order model for the single particle model. The Koopman eigenfunctionals and eigenvalues are derived directly from the eigenfunctions of the self-adjoint zero-flux diffusion operator. The zero mode represents the volume average concentration, the nonzero modes describe decaying radial gradients, and projection of the current dependent surface flux yields a linear state-space model that reconstructs the average, surface, and full radial concentrations. Unlike data-driven Koopman models, the proposed formulation requires neither training data nor empirical lifting functions. Compared with a 400 control volume finite volume model (FVM) during a constant 1C discharge, the 80 Koopman mode model achieves negative and positive surface concentration RMSE values of $1.26\times10^{-5}\,\mathrm{mol/m^{3}}$ and $4.10\times10^{-6}\,\mathrm{mol/m^{3}}$, a terminal voltage RMSE of $8.42\times10^{-4}\,\mathrm{V}$. The models with 5 and 80 Koopman modes were approximately 225 and 31 times faster than the FVM model, respectively. These results demonstrate a physically interpretable and computationally efficient representation of battery diffusion dynamics.