Control of Film Growth in Lithium Ion Battery Packs via Switches

Recent advances in lithium ion battery modeling suggest degradation may be reduced by permitting unequal but controlled charging of individual cells, delayed until immediately prior to discharge. Hence, this paper compares anode-side film formation for a standard equalization scheme versus unequal charging through switches controlled by deterministic dynamic programming (DDP) and DDP-inspired algorithms. A static model for film growth rate is derived from a first-principles battery model adopted from the electrochemical engineering literature. Coupled with this model, we consider two cells connected in parallel via relay switches. The key results we demonstrate are: (1) Enabling unequal charging and (2) delaying charging until immediately prior to discharge indeed reduces film buildup. (3) A near optimal control method can be implemented using heuristic rules designed from the DDP solution and convexity properties of film growth rate. Simulation indicates the heuristic rule achieves near optimal performance relative to the DDP solution, and over 50% reduction in film growth compared to charging both cells equally. NOMENCLATURE F Faraday’s constant [C/mol] i0 Battery pack current [A] i0,s Exchange current density [A/m2] for side reaction i1, i2 Cell current [A] hpenalty Quadratic penalty function [pm/m2] ∗Address all correspondence to this author. J Cost functional [pm/m2] J1 Intercalation current between [A/m3] solid and solution Jp Terminal state penalty function [pm/m2] Jtot Total intercalation current [A/m3] Js Current density of side reaction [A/m3] MP Molecular weight of product [mol/kg] from side reaction Q Battery cell charge capacity [A·h] q1,q2 Contactor switch position R Universal gas constant [J/K/mol] R f ilm Total film resistance at [Ω· m2] electrode/electrolyte interface Rint Battery cell internal resistance [Ω] RSEI Resistance of solid electrolyte [Ω/m2] interphase (SEI) Us,re f Equilibrium potential of [V] side reaction V Value function [pm/m2] v Battery pack voltage [V] x Spatial coordinate [m/m] z Battery cell state of charge [C/C] ∆φ Local potential difference between [V] solid and solution at anode δ f ilm Resistive film thickness [pm/m2] ηs Over potential driving side reaction [V] κP Conductivity of electrolyte [1/m/Ω] ρP Density of product from side reaction [kg/m2]