This paper presents a numerical calculation of the evolution of the spatially resolved solid concentration in the two electrodes of a lithium-ion cell. The microscopic solid concentration is driven by the macroscopic Butler–Volmer current density distribution, which is consequently driven by the applied current through the boundary conditions. The resulting, mostly causal, implementation of the algebraic differential equations that describe the battery electrochemical principles, even after assuming fixed electrolyte concentration, is of high order and complexity and is denoted as the full order model. The full order model is compared with the results in the works of Smith and Wang (2006, “Solid-State Diffusion Limitations on Pulse Operation of a Lithium-Ion Cell for Hybrid Electric Vehicles,” J. Power Sources, 161, pp. 628–639) and Wang et al. (2007 “Control oriented 1D Electrochemical Model of Lithium Ion Battery,” Energy Convers. Manage., 48, pp. 2565–2578) and creates our baseline model, which will be further simplified for charge estimation. We then propose a low order extended Kalman filter for the estimation of the average-electrode charge similarly to the single-particle charge estimation in the work of White and Santhanagopalan (2006, “Online Estimation of the State of Charge of a Lithium Ion Cell,” J. Power Sources, 161, pp. 1346–1355) with the following two substantial enhancements. First, we estimate the average-electrode, or single-particle, solid-electrolyte surface concentration, called critical surface charge in addition to the more traditional bulk concentration called state of charge. Moreover, we avoid the weakly observable conditions associated with estimating both electrode concentrations by recognizing that the measured cell voltage depends on the difference, and not the absolute value, of the two electrode open circuit voltages. The estimation results of the reduced, single, averaged electrode model are compared with the full order model simulation.
Electrochemical state and internal variables estimation using a reduced-order physics-based model of a lithium-ion cell and an extended Kalman filter