## 4 Battery Cell-Module-Pack Level Modelling
### Cell Level  
In the electrical aspects, the cell is taken as an equivalent circuit composed of an open-circuit voltage source in series with an internal resistance. The cell-level model accepts current demand \(I_{cell}\) as an input, which is solved by the module-level model. Given \(I_{cell}\), cell voltage \(V_{cell}\) is calculated as (7), where \(OCV_{cell}\) is cell OCV acquired from an LUT based on SOC and temperature and \(R_{cell}\) is cell internal resistance from an LUT based on SOC and temperature. Power delivered by the cell \(P_{cell}\) and cell power loss \(P_{loss,cell}\) are calculated as (8) and (9), respectively. Cell SOC, \(SOC_{cell}\), is calculated through coulomb counting, as (10), where \(SOC_{cell,ini}\) is the initial SOC and \(Al_{cell}\) is the ampere-hour capacity of the cell [31].  
\[V_{cell}=OCV_{cell}-R_{cell}\times I_{cell} \tag{7}\]  
Figure 7: Workflow of interactions between cell, module, and pack levels.  
\[P_{cell}=V_{cell}\times I_{cell} \tag{8}\]  
\[P_{loss,cell}={I_{cell}}^{2}\times R_{cell} \tag{9}\]  
\[SOC_{cell}=SOC_{cell,ini}-\int\frac{I_{cell}}{3600\times Ah_{cell}}dt \tag{10}\]  
Cell SOP is determined by four factors, static power capability, dynamic power capability, and 10 s and 30 s continuous discharge power capabilities, whichever is the smallest. Static power capability \(P_{max,static}\) is associated with the maximum current rate \(C\_rate_{cell}\) defined in the cell specification (\(C\_rate\) is a measure of the rate at which a battery is charged or discharged relative to its nominal capacity, and a current rate of 1C means the battery will be fully charged or discharged in one hour), and can be expressed as (11). Dynamic power capability \(P_{max,dynamic}\) considers the gap between the present cell voltage and the minimum cell voltage \(V_{cell,min}\) defined in the cell specification and can be expressed as (12) [32]. The 10 s and 30 s continuous discharge power capabilities \(P_{max,10s}\) and \(P_{max,30s}\) are measured from experiments and sorted as LUTs based on SOC and temperature. Cell SOP can then be expressed as (13). Note that the calculation of cell SOP is accomplished in the previous time step rather than in the present; by calculating cell SOP one time step behind, the occurrence of algebraic loops is avoided. Furthermore, the calculated cell SOP can be converted into the battery pack SOP for application in the motor derating control (as mentioned in Section 2).  
\[P_{max,static}=Ah_{cell}\times C\_rate_{cell}\times V_{cell} \tag{11}\]  
\[P_{max,dynamic}=\frac{V_{cell}-V_{cell,min}}{R_{cell}}\times V_{cell} \tag{12}\]  
\[SOP_{cell}=\min\Big{\{}P_{max,static},\;\;P_{max,dynamic},\;P_{max,10s},\;P_{ max,30s}\Big{\}} \tag{13}\]  
In the thermal aspects, cell heat generation \(Q_{gen}\) is assumed to be equal to cell power loss \(P_{loss,cell}\), as (14). The calculation of cell heat dissipation is similar to (3), while the difference lies in the thermal resistance, which is divided into two parts [33]: jellyroll-to-can \(r_{jell,jrold2 zon}\) and can-to-liquid \(r_{can2 coulant}\). The former is treated as a constant [34], while the latter is obtained from an LUT based on coolant flow rate, which is determined through experiments [35]. Cell heat dissipation can then be expressed as (15), and the updated temperatures of the cell and coolant can be acquired through (4) and (5), respectively.  
\[Q_{gen}=P_{loss,cell} \tag{14}\]  
\[Q_{dis}=\frac{T-T_{liq,\;in}}{r_{jelllyroll2 zon}+r_{can2 coulant}} \tag{15}\]