Embedded software aims to provide the main characteristics of a cell in real time to manage and optimize battery usage. A complete multidimensional model is not tractable with the low computational resources available in situ. A class of 0D-in-space models is derived from the P2D Doyle-Fuller-Newman system, with additional assumptions about the functions' shape in space, and strongly reduces the complexity while preserving the main characteristics over time. We propose a conservative, positivity-preserving model built from the P2D system based on the statement that the Butler-Volmer source term is constant in each subdomain (anode, separator, cathode). We develop a mathematically rigorous construction that provides the simplified model and design a scheme to solve the system. We highlight that the model guarantees the conservation property together with the positivity of the concentrations. In particular, the diffusion-limited current density that represents the maximum current that can be used while still preserving the positivity of the cation concentration is obtained explicitly.