## 2 Battery Modeling Methods
### Electrochemical Model  
According to the electrochemical kinetics and the charge transfer process, electrochemical models are established for the purpose of describing the inner reactions inside the battery. Electrochemical models are the foundation of a series of physical laws, such as [PERSON]'s first law, [PERSON]'s law, [PERSON]'s law of diffusion, and the Butler-Volmer equation. The electrochemical model is expressed in a nonlinear Partial Differential Equations (PDEs) form. Therefore, in order to have a direct analytical solution, the prerequisite of using the electrochemical model is changing the PDEs into Ordinary Differential Equations (ODEs). Numerical methods such as integral approximation, _Pade_ approximation, the _Ritz_ method, the finite element method, and the finite difference method are often selected to discretize the nonlinear PDEs in the electrochemical models [47; 48]. In _[PERSON]_'s law of electrolysis, the pore-wall flux \(J_{1}\) is related to the divergence of current flow in the electrolyte phase. The pore-wall flux is calculated as follows:  
\[J_{n}(t)=\frac{I(t)}{F\cdot S_{n}},J_{p}(t)=-\frac{I(t)}{F\cdot S_{p}}, \tag{2}\]  
where \(n\), \(p\) denote the negative and positive electrodes, respectively. _[PERSON]_'s law indicates the distribution of potential between the electrolyte and the active material. _[PERSON]_'s law of diffusion reveals the relationship of the concentration and the diffusion flux, which is capable of describing the diffusions in both the electrolyte and the electrode. The _Butler-Volmer_ equation points out the impact of the electrode potential on the electrode current. Two kinds of electrochemical models utilized in the literature are the pseudo-2D model and the Single Particle Model (SPM), as shown in Figure 2. The pseudo-2D model [49; 50; 51] is based on the concentrated solution theory and the porous electrode theory. The structure of the porous electrode increases the area of the specific surface, which adequately facilitates the electrochemical reactions. The active material is able to sufficiently contact the electrolyte with the benefit of the porous structure. Pseudo-2D models regard the active material in the electrode as spherical particles with equal size and volume. SPM [52; 53; 54] is a simplification of a pseudo-2D model that regards the electrode as a single particle. If the liquid phase concentration and electrode potential are assumed to be constant, the reactions in the electrode are identical for different particles. Thus, the electrochemical reactions of different particles in the electrode can be considered as a single spherical particle. Compared with the pseudo-2D model, the description of the migration of the Li ions inside a solid particle is much easier for SPM. A pseudo-2D model is more suitable for analyzing the mechanism of the Li-ion battery, while SPM is possible for battery SOC estimation [54; 56; 57]. In [54], SPM is reorganized in a state-space form with only eight unknown parameters, and SOC is estimated by an Iterated Extended Kalman Filter (IEFK). Nonlinear geometric observer is designed to estimate SOC on the basis of SPM, which obtains a less than 4.5% SOC estimation error [56]. Parameters in the electrochemical models generally have their own physical explanations. However, it is normally quite difficult to obtain the real parameters in electrochemical models. Therefore, heuristic methods such as Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) are used for the parameter identification of the electrochemical model [14; 58; 59]. Despite the requirement of knowledge of the electrochemical process, the computational burden should also be considered if an electrochemical model is used in BMS. The full-order electrochemical model consists of 17 PDEs, but the number of the parameters is still as many as 30-100 after discretizing [60]. Consequently, efforts are needed to reduce the order of the electrochemical model.