Adaptive PDE Observer for Battery SOC/SOH Estimation

This paper develops an adaptive PDE observer for battery state-of-charge (SOC) and state-of-health (SOH) estimation. Real-time state and parameter information enables operation near physical limits without compromising durability, thereby unlock-ing the full potential of battery energy storage. SOC/SOH esti-mation is technically challenging because battery dynamics are governed by electrochemical principles, mathematically modeled by partial differential equations (PDEs). Simultaneous state and parameter estimation is extremely challenging in PDE models. Consequently, several new theoretical ideas are developed, inte-grated together, and tested. These include a backstepping PDE state estimator, a Padé-based parameter identifier, nonlinear pa-rameter sensitivity analysis, and adaptive inversion of nonlinear output functions. The end result is the first combined SOC/SOH battery estimation algorithm that identifies physical system vari-ables via an electrochemical model, from measurements of voltage and current only. NOMENCLATURE A Cell cross sectional area [m2] a j Specific interfacial surface area [m2/m3] c0e Li concentration in electrolyte phase [mol/m 3] c js Li concentration in solid phase [mol/m3] c jss Li concentration at particle surface [mol/m3] c js,max Max Li concentration in solid phase [mol/m3] D js Diffusion coefficent in solid phase [m2/sec3] F Faraday’s constant [C/mol] I Input current [A] i j0 Exchange current density [V] j Positive (+) or negative (-) electrode k j Reaction rate [A·mol1.5/m5.5] L j Electrode thickness [m] nLi Total number of Li ions [mol] q Boundary input coefficient parameter R Universal gas constant [J/mol-K] R f Lumped current collector resistance [Ω] R js Particle radius [m] r Radial coordinate [m] or [m/m] T Cell temperature [K] t Time [sec] or [sec/sec] U j Equilibrium potential [V] V Output voltage [V] α j Anodic/cathodic transfer coefficient ε Diffusion parameter ε js Volume fraction of solid phase 1