## 1 Introduction  
Lithium-ion battery packs used in electric vehicles (EVs) have a lifetime of 4-10 years [1]. The end of life (EoL) in EVs is reached when 20% of the pack's nominal capacity is lost [2]. The second life of lithium-ion batteries (LiBs) refers to their continued use after retirement from EVs, where \(\sim\)80% of their capacity remains available but performance is insufficient for automotive use. The state of health (SOH) of the battery is commonly monitored as the capacity drop over time. This quasi-linear reduction is mostly caused by side reactions in the anode, such as the solid electrolyte interface (SEI) growth and lithium plating (LiP) [3]. These side reactions are known to reduce the ability of lithium ions to intercalate effectively. SEI growth is the dominant side reaction during the first life of the battery, and it is primarily responsible for the quasi-linear reduction in capacity over time, but as it approaches and surpasses the EoL, a nonlinear capacity drop is observed over time due to LiP [4]. In combination, SEI growth and LiP cause further capacity reduction during the second life (after EoL), increasing cell-to-cell variation or capacity spreading among cells [5,6]. Different combinations of stress factors such as temperature, state-of-charge (SOC), depth-of-discharge (DOD), and charge/discharge rate (C-rate) during calendar (at rest) and/or cycling (when charging/discharging the battery) lead to less or more aggressive conditions favouring these side reactions, and consequently, leading to the degradation processesor battery aging, causing the reduced performance [4; 7; 8]. While it is possible to harvest cell or pack materials right after EoL, extending the operational lifetime of used batteries by giving them a second life provides a secondary revenue stream while reducing the environmental impact of the battery waste [1; 9]. In their first life (before EoL), LiBs typically work under a controlled environment, such as in EVs, where management systems regulate charging and discharging and thermal conditions. Modelling aging in first life relies on uniform and consistent data from new cells with known characterizations at the beginning of life. The primary degradation mechanisms in first life are well understood, including SEI growth, lithium plating, and electrode particle cracking, as stress factor values are known and/or recorded, such as C-rates, temperature ranges, and SOC windows, or studied through accelerated aging tests offline. In contrast, in the second life, the battery has already experienced significant degradation and has an aging history. When assembled in secondary applications, such as battery backup systems, heterogeneous aging is introduced, as the repurposed battery has been previously been exposed to varying user behaviors, environmental exposure, and variable aggressive conditions during calendar aging mode (storage). This heterogeneity introduces uncertain degradation history and, hence, variability in initial conditions across cells, modules, and battery packs, which complicates the prediction and diagnosis of future performance, including their remaining useful life. In second life, degradation modes interact, for instance, in nonlinear ways, including complex coupled degradation effects, which are exacerbated by capacity spread among cells. Therefore, the modelling of second life must also contend with data scarcity, high variability, and cell spreading, which cause cell imbalance. These challenges require diagnostic approaches that are interpretable, flexible, scalable, and robust to noise. Thus, the diagnostic focus shifts from predictive SOH to understanding the current battery condition, forecasting its remaining usability, and inferring the causes of past degradation. It is expected that modelling aging in the second life will follows similar approaches to those used in the first life, including data-based, machine learning, and electrochemically based models [4; 10].