## 2 Some Existing Methods of Estimating the Lithium-Ion Battery State of Charge
### Model-Based Methods
* Butler-Volmer (BV) kinetic equations at the surface of the particles in the electrodes. However, the P2D model suffers from a notable drawback stemming from its intricacy, primarily because it incorporates a multitude of partial differential equations (PDEs) and boundary conditions. This heightened complexity significantly escalates the computational workload involved. Furthermore, the substantial computational demands associated with the application of state and parameter estimation algorithms to the PDAEs (partial differential algebraic equations) forming the P2D model make it challenging to implement these electrochemical models in real-time applications (as referenced in [54; 55]). State-of-charge estimation from the full P2D model using a modified Extended Kalman Filter (EKF) has been reported in [56], where the electrochemical model is solved by the Chebyshev orthogonal collocation method to reduce the computational burden of the battery P2D electrochemical model. However, parameter estimation of the model remains a significant challenge [18]. To address this challenge, previous efforts have been made to estimate the state of charge of lithium-ion batteries by using simplified PDE-based models, such as the Single-Particle Model (SPM) and the Single-Particle Model with Electrolyte Dynamics (SPME) (as mentioned in [57; 58; 59]). These endeavors aimed to reduce the computational complexity associated with the P2D electrochemical model's extensive set of PDEs. An important drawback of these simplified models is their limited suitability for scenarios involving higher C-rates. It has been observed, as highlighted in references [60], that these models begin to lose their accuracy when C-rates surpass 0.5 C. Additionally, it is worth noting that the Single-Particle Model with Electrolyte Dynamics (SPME) captures less cell dynamics for its performance assessment. However, the set of PDEs that form a P2D model requires a lot of development and computation time in online battery state-of-charge estimation [18]. In this respect, it is crucial to develop a new mathematical model that is suitable for real-time implementation of lithium-ion battery state of charge. Recently, the critical review of three mathematical models, namely, the energy reservoir model, the reservoir of charge model, and the concentration-based model for determining the evolution of the lithium-ion battery state of charge was provided to the power systems research community by [PERSON] et al. in their work [2], where they used these models to calculate the optimal timing of a battery energy storage system for a peak-shaving application. The authors highlighted the advantages and disadvantages of each model from a computational point of view but mainly looked at references outside the typical applications of grid-level battery energy storage systems. However, their models do not directly incorporate the initial state of charge into the dynamic state-of-charge equation of the battery, nor the nominal and terminal voltages of the battery. Although, as we have already stressed above, batteries are essential in modern applications because of their power and energy density, incorrect charging and discharging can impair their performance and cause breakdowns. A battery management system (BMS) is, therefore, essential for their smooth operation. SOC estimation is a key element of any battery management system and is essential to maintaining the health and efficiency of the battery. In this respect, the contribution of this paper is to provide a new mathematical model to simulate the state of charge of a lithium-ion battery, taking into account the efficiency, charge transfer coefficient, nominal voltage, and terminal voltage during the charging and discharging processes. To do so, we consider certain approximations in each modeling stage and discuss their physical meaning and characteristic units.