## 1 Introduction  
The use of lithium-ion (Li-ion) batteries in energy storage systems (ESS) and electric mobility applications is increasing due to their attractive characteristics such as high specific energy and power density, high charge and discharge efficiency, long life, and low cost [1]. However, the impedance rise and capacity fade over the cycle life are still the focus of concern. By comparing the impedance or the capacity of the pristine cell with that of the aged or degraded cell, it is possible to quantify the degradation mechanisms since the capacity fades due to the change of some identified kinetic parameters [2]. Today, there is a growing interest in reusing electric mobility batteries for second-life applications such as ESS. Although they have reached the end of their life, second-life batteries from electric vehicle applications still have a residual capacity of about 70-80% [3,4]. However, the performance and efficiency of the reused battery system is highly dependent on the use of these batteries, which require a similar state of health (SOH), capacity, and remaining useful life (RUL). Therefore, it is of utmost importance to establish battery grading techniques, making it necessary to develop fast methods to evaluate the degradation of the batteries in terms of capacity, SOH, and RUL. Such techniques play a key role in the application of battery management strategies, while allowing to extend the battery lifetime and increase the reliability of the battery system. For example, an accurate estimation of the state of charge (SOC) allows avoiding overcharging and over-discharging of the battery, thus prolonging its lifetime [5]. However, electrochemical batteries exhibit a highly nonlinear behavior [5], making accurate estimation of SOC, SOH, and RUL a complex task. Due to the importance of the battery in electric mobility and energy storage applications, the analysis of the SOH is a priority for researchers. Therefore, numerous methods have been developed over the years to evaluate and estimate the SOH. They can be broadly classified into model-based, data-driven, direct [6], and indirect measurement methods. Model-based methods typically use the battery equivalent circuit or mathematical models such as Nernst or Shepherd to calculate the SOC. However, these models generally do not allow the capacity fade within cycles to be described. Data-driven methods handle large databases and usually predict the SOC using various mathematical techniques such as fuzzy logic [7,8] or neural networks [9]. Data-driven methods treat the battery system as a black box and still provide an accurate response but at the cost of high computational requirements that hinder on-line predictions. Direct methods are typically based on electrochemical impedance spectroscopy (EIS) since the impedance of the cell typically increases gradually with the number of charge-discharge cycles [10]. Due to the wide range of frequencies analyzed in EIS measurements, this technique can characterize internal reaction mechanisms associated with battery aging processes, such as ion concentrations, particle size, dendrite growth, or electrolyte and electrode aging [11,12]. As a result, EIS is more accurate than the conventional method based on temperature, current, and voltage readings from the battery management system (BMS) [10]. EIS provides detailed information about the SOH, and requires long rest periods to reduce the impact of the battery relaxation effect on EIS measurement [13]. Therefore, EIS is difficult to implement in real-world applications due to the lack of practical access to the battery chemistry [14] and the need for a variable frequency generator, making it more suitable for laboratory testing than for direct use in practical applications [6,15,16]. Finally, Coulomb counting is a very common and simple method to determine the capacity of the battery but its accuracy is highly dependent on the precision of the measurements, while presenting a cumulative error [17].