A Physically-Informed and Aging-Sensitive Equivalent Circuit Model for Predicting the Knee in Lithium-Ion Batteries

The growing adoption of electric vehicles and stationary energy storage systems underscores the critical need to accurately predict the lifespan of lithium-ion batteries (LIBs) under diverse operating conditions. Battery aging exhibits a path-dependent nature [1], where various degradation mechanisms interact in a highly nonlinear manner, giving rise to complex aging dynamics—such as the characteristic “ knee ” in capacity fade curves or the “ elbow ” in internal resistance increase curves [1], which pose significant predictive challenges. While physiochemical models have been utilized to characterize these aging behaviors, their limitations include high computational demands and the intricate process of parameter estimation. A novel aging-sensitive, physically-informed equivalent circuit model (ECM) has been developed by integrating key degradation modes, specifically the loss of active material (LAM) and loss of lithium inventory (LLI), with their respective rates governed by physicochemical aging kinetics [2]. A schematic representation is provided in Figure 1a. The model comprises two voltage sources, V 0,PE ( X PE ) and V 0,NE ( X NE ), with corresponding current leakage paths attributed to LLI and LAM. Additionally, it includes two R-RC components: R dyn.,PE , R stat.,PE , C dyn.,PE, and R dyn.,NE , R stat.,NE , C dyn.,NE , which capture the resistive and dynamic behavior of the positive and negative electrodes, respectively. A series resistor, R s , accounts for the resistance contributions from the electrolyte and current collector system. By employing charge balance equations for the two half-cells ( Q PE , Q NE ), the model effectively integrates degradation modes. This approach maintains computational efficiency while enabling the simulation of complex, dynamic, and realistic load profiles over extended timeframes. The developed model was parameterized for a 3350 mAh cylindrical lithium-ion cell featuring a graphite anode and an NCA cathode, operated under a dynamic stress test (DST) driving protocol, which has been extensively studied in prior research [3]. The rate expressions corresponding to each degradation mode were manually calibrated to ensure alignment between the simulated voltage response and the previously reported experimental data [3], as illustrated in Figure 1b. The ECM effectively captures the characteristic knee in the capacity fade curve (Figure 1c), which is primarily attributed to the onset of lithium plating (Figure 1d). This phenomenon arises when the increasing resistance at the negative electrode drives its potential below the threshold for lithium plating. References: [1] P.M. Attia, A. Bills, F.B. Planella, and et al., Journal of The Electrochemical Society 169 (2022) 060517 (2022) 060517. [2] P. O. Mmeka, M. Dubarry and W. G. Bessler, Journal of The Electrochemical Society , 2025, under review. [3] G. Baure, M. Dubarry, Batteries 5 (2019) 42. Figure 1