## 2 Some Existing Methods of Estimating the Lithium-Ion Battery State of Charge
### Model-Based Methods  
Model-based methods attempt to model battery behavior by incorporating physical and chemical principles into mathematical equations to accurately estimate the state of charge [47]. The authors in ref. [2] reported that state-of-charge models that define battery capacity in units of energy (kWh) are classified as energy reservoir models (ERMs). Those that define it in units of charge (Ah) are classified as charge reservoir models (CRMs), and those that define it in units of concentration (mol/L) are classified as concentration-based models. The model-based approach involves the use of electrochemical, electrical, or a combination of both models to describe the battery dynamics from which the state of charge is estimated. The electrical model employs an approximate equivalent circuit to represent battery dynamics, and it incorporates adaptive filter algorithms like the Kalman Filter (KF), as referenced in [31]. In this approach, constructing the circuit models involves arranging various circuit elements in series, parallel, or combinations thereof to mimic the battery's dynamic behavior. Several equivalent circuit models have been proposed, including the Rint model, the RC model, and the Thevenin model, as discussed in [1; 2]. These models include a critical component, a voltage source, which signifies the Open Circuit Voltage (OCV) of the battery. The OCV, in turn, depends on the battery SOC. It is essential to note that the accuracy of SOC estimation is heavily impacted by the specific equivalent circuit model chosen to represent the battery. For SOC estimation within the electrical model, the Extended Kalman Filter (EKF) is utilized. The EKF is an enhanced version of the Kalman Filter (KF) designed for estimating the internal state of non-linear dynamic systems,using a state-space model, as described in [48]. Essentially, it predicts the future state of the system based on prior data. The EKF encompasses two key equations. One equation involves matrices constructed by using equivalent circuit model (ECM) parameters, along with the system SOC, measurable input matrices, and non-measurable process noise. The parameters are identified through standard testing procedures. The second equation is the measurement equation, which relates the output voltage to the system state vectors, measurable input matrices, and measurement noise. In the context of SOC estimation, the EKF leverages advanced battery cell models and demands relatively high computational resources. Furthermore, the electrical model's performance relies on capacitance and resistance values that are influenced by factors like state of charge, current, or temperature. This necessitates conducting numerous experiments to establish the relationships among these parameters, as mentioned in [49]. To enhance the accuracy of the battery equivalent circuit model, researchers, as highlighted in the papers [50], often consider temperature effects and develop thermal coupling models. However, this approach increases the complexity of the model and introduces errors in the state-of-charge estimation. The electrochemical model uses partial differential equations (PDEs) to describe battery dynamics [51]. Indeed, one of the widely used electrochemical models is the pseudo-2-D (P2D) model (refs. [52; 53]), which is a set of six partial differential equations, coupled with algebraic constraints. The P2D model takes current as its input and returns voltage as its output. The partial differential equations used in the P2D model incorporate the following parameters [53]:  
* Electrolyte lithium-ion diffusion equations in the positive electrode, negative electrode, and separator according to [PERSON]'s second law. * Solid-phase lithium-ion diffusion equations in the electrodes due to [PERSON]'s second law. * Electrolyte ohm equations in the electrodes and separator. * Solid-phase ohm equations in the positive electrode and negative electrode. * Charge conservation equations, including a positive electrode and a negative electrode, and a separator.