In this paper the authors study a nonlinear elliptic-parabolic system, which is motivated by mathematical models for lithium ion batteries. For the reliable and fast numerical solution of the system a reduced-order approach based on proper orthogonal decomposition (POD) is applied. The strategy is justified by a priori error analysis for the error between the solution to the coupled system and its POD approximation. The nonlinear coupling is realized by variants of the empirical interpolation introduced by Barrault et al. [C. R. Acad. Sci. Paris Ser. I, 339 (2004), pp. 667--672] and Chaturantabut and Sorensen [Application of POD and DEIM on a Dimension Reduction of Nonlinear Miscible Viscous Fingering in Porous Media, Technical Report, RICE University, 2009]. Numerical examples illustrate the efficiency of the proposed reduced-order modeling.