## 2 Electrochemistry-Based Equivalent Circuit Model  
### Single-Particle Model  
The pseudo-two-dimensional model (P2D) are widely used to describe the electrochemical behavior of lithium-ion batteries [14; 19; 20]. However, P2D is hard to use in onboard cases because of its complexity. Therefore, a series of model simplification attempts were made to reduce the computational complexity. The single-particle model (SPM) is a simplified model that is derived by approximating the electrode by a single spherical particle, and is becoming a popular model in recent years for SOC estimation [21; 22]. The schematic of the SPM, which is illustrated in Figure 1.  
Based on Reference [15], the terminal voltage (\(U_{t}\)) can be expressed as:  
\[U_{t}=\Big{(}U_{P}\Big{(}c_{surf,r}\Big{)}-U_{N}\Big{(}c_{surf,r}\Big{)}\Big{)} -U_{0}-U_{DL} \tag{1}\]  
where \(c_{surf,r}\) is lithium concentration at electrode particle surface, \(U_{P}\Big{(}c_{surf,r}\Big{)}\) and \(U_{N}\Big{(}c_{surf,r}\Big{)}\) are the surface potential of positive electrode and negative electrode, respectively. \(U_{0}\) is the sum of the liquid phase voltage drop caused by the separator and electrolyte, and the voltage drop of the collector. \(U_{DL}\) is the voltage drop of the double layer, and it can be determined as:  
\[U_{DL}=IR_{CT}\bigg{(}1-e^{(\frac{-t}{\tau_{DL}})}\bigg{)} \tag{2}\]  
where \(R_{CT}\) is the resistance of the double layer, and \(\tau_{DL}\) is time constant.  
Moreover, Equation (1) can be rewritten as:  
\[U_{t}=\Big{(}U_{P}\Big{(}\text{SOC}_{surf,r}\Big{)}-U_{N}\Big{(}\text{SOC}_{ surf,r}\Big{)}\Big{)}-U_{0}-U_{DL} \tag{3}\]  
Figure 1: Schematic of the single-particle model (SPM) (modified from Reference [23]).  
where SOC\({}_{surf,r}\) is the SOC at particle surface. However, SOC\({}_{surf,r}\) could not be directly determined, and it can be expressed as:  
\[\text{SOC}_{surf,r}=\text{SOC}_{avg}+\Delta\text{SOC} \tag{4}\]  
where SOC\({}_{avg}\) is the average SOC, which is represented by the average concentration of Li\({}^{+}\) in the electrode particle (\(C_{mean}\)). \(\Delta\text{SOC}\) is the difference between SOC\({}_{avg}\) and SOC\({}_{surf,r}\), which is related to the difference of \(c_{surf,r}\) and \(C_{mean}\) in solid-phase diffusion process. According to Reference [15], \(\Delta\text{SOC}\) follows the Equation (5):  
\[\Delta SOC=K_{SD}I_{F}\bigg{(}1-e^{\left(\frac{-t}{t_{SD}}\right)}\bigg{)} \tag{5}\]  
where \(K_{SD}\) is concentration difference parameter in solid diffusion, \(I_{F}\) is the faradaic current, and \(\tau_{SD}\) is time constant.  
When the traditional ECM is used for SOC estimation, the open circuit voltage (OCV) of the battery \(U_{OCV}\) is looked up by the SOC\({}_{avg}\) with the OCV-SOC curve. However, SOC\({}_{avg}\) could not reflect the solid-phase diffusion. In this study, we used \(U_{OCV}\big{(}\text{SOC}_{surf}\big{)}\) instead of \(U_{OCV}\big{(}\text{SOC}_{avg}\big{)}\), and Equation (3) is hence rewritten as:  
\[U_{t}=U_{OCV}\Big{(}SOC_{surf}\Big{)}-U_{0}-U_{DL} \tag{6}\]  
It is noted that the difference between the surface concentration \(c_{surf}\) and the average concentration \(C_{mean}\) indicates the solid-phase diffusion results. \(c_{surf}\) reflects the dynamic process of lithium-ion, which indirectly reflects the process of solid-phase diffusion. \(SOC_{surf}\) is closely related to \(c_{surf}\). Therefore, our model includes the solid phase diffusion process and is a simplified electrochemical model.