Smoothing Methods for Numerical Differentiation to Identify Electrochemical Reactions from Open-Circuit-Potential Data

Abstract

We develop and compare methods to determine when electrochemical reactions take place within intercalation electrodes. We employ open-circuit data from the Chevrolet Bolt EV negative and positive electrodes, lithiated graphite and NMC (Ni0.6Mn0.2Co0.2O2), respectively. Electrochemical reactions correspond to peaks in plots of dx/dU vs. U, where U is the electrode potential and x is the fractional state of charge (0 ⩽ x ⩽ 1). We employ three nonparametric methods in this work to smooth and subsequently differentiate the data: a windowed cubic polynomial of fixed and variable window length and a cubic smoothing spline. The standard deviation in the measurements can be used to set appropriately the window length and the analogous smoothing parameter for the spline. The cubic polynomial with a fixed window length and the smoothing spline each have a single smoothing parameter; both methods are shown to work well and yield comparable results. The best results were achieved with the cubic polynomial incorporating a varying, adaptive window length, and the increased fidelity enables a completely automated procedure to identify reactions. The analysis clarifies the multi-reaction nature of lithiated graphite, which must be addressed in cell modeling to simulate fast-charge behavior.