## 2 Battery Modeling Methods
### Empirical Model  
In empirical models, the battery terminal voltage is represented as a mathematical function of the SOC and the current [23]. Considered a simplified electrochemical model, an empirical model represents the essential nonlinear characteristics of a battery with reduced order polynomial or mathematical expressions.  
In Table 1, \(y_{k}\) is the terminal voltage, \(E_{0}\) is OCV when the battery is fully charged, \(R_{i}\) is the internal resistance, \(K_{1}\) is the polarization resistance, \(i_{k}\) is the instantaneous current, and \(z_{k}\) represents the battery SOC. The _Shepherd_ model [24], the _Unnewehr_ universal model [25; 26], and the _Nernst_ model [27] in Table 1 are the classical empirical models in the literature. The accuracy of the three models in predicting the terminal voltage is compared in [23]. The _Nernst_ model obtains the best accuracy, and the _Shepherd_ model performs especially well in a continuously discharging current. Generally, the three models in Table 1 can be combined for better accuracy in the following form [13]:  
\[y_{k}=E_{0}-R\cdot i_{k}-\frac{K_{1}}{z_{k}}-K_{2}\cdot z_{k}+K_{3}\cdot\ln( z_{k})+K_{4}\cdot\ln(1-z_{k}). \tag{1}\]  
There are other ways to improve the modeling accuracy of the basic empirical models. The accuracy of the empirical model can be improved by adding more parameters [28]. Since the _Shepherd_ model suffers from algebraic loop and simulation instability in real-time applications, a modified _Shepherd_ model is proposed for describing the dynamic behavior of the battery [29]. Replacing the voltage of the internal resistance with the polarization voltage term, it is validated  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Model Type** & **Model Equations** \\ \hline _Shepherd_ model [24] & \(y_{k}=E_{0}-R\cdot i_{k}-\frac{K_{1}}{z_{k}}\) \\ \hline _Unnewehr_ universal model [25; 26] & \(y_{k}=E_{0}-R\cdot i_{k}-K_{1}\cdot z_{k}\) \\ \hline _Nernst_ model [27] & \(y_{k}=E_{0}-R\cdot i_{k}-K_{2}\cdot\ln(z_{k})+K_{3}\cdot\ln(1-z_{k})\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Typical empirical models.  
under a constant current profile. In order to improve the dynamic performance of the modified _Shepherd_ model, the OCV-SOC relationship is taken into consideration in [30] and a term related to the polarization voltage is added. The error band of the modified _Shepherd_ model is within \(\pm 5\%\) in the dynamic current profile. To further solve the singularity in the _Shepherd_ model, another extension proposed in [31; 32] obtains relative error below 0.5% in FITP72 cycles and the execution time is between 2.35 \(\upmu\)s and 4.35 \(\upmu\)s. The _Nernst_ model can also be improved by adding two additional constants,\(\tau_{1}\) and \(\tau_{2}\), to have a stronger ability to describe the dynamic terminal voltage [27]. Classical empirical models have flaws during the relaxation time [33] because the hysteresis effect [34] of the battery voltage is not considered. Therefore, the term \(s\cdot M\) is added to the _Nernst_ equation to represent the hysteresis effect, where \(M\) is the correction term to be identified [35]. Moreover, an enhanced self-correcting model considering the voltage hysteresis is proposed in [36], which includes the gradual hysteresis voltage as a function of SOC and the instantaneous hysteresis voltage changes with the sign of the current. Detailed expressions of the abovementioned empirical models are illustrated in Table 2. Considering more effects inside the battery, the accuracy of the empirical model is improved but the computational burden is also increased.