## 4 Battery Cell-Module-Pack Level Modelling
### Module Level  
In the electrical aspects, the module-level model's purpose is to convert the module-level current demand \(I_{mod}\) into the cell-level current demand \(I_{cell}\), such that it can be accepted as an input by the cell-level model. A 4s10p module is adopted, as shown in Figure 8a. To facilitate the module-level calculations, the 4s10p module can be equivalently represented as a lumped, single circuit, as shown in Figure 8b. According to [PERSON]'s laws [36], the module-level resistance \(R_{mod}\) and OCV \(OCV_{mod}\) can be expressed as (16) and (17), respectively. Then, the module-level current \(I_{mod}\) can be solved as (18), where \(P_{mod}\) is the module-level power demand, which is solved by the pack-level model. The cell current demand \(I_{cell}\) can also be solved by the module-level model; for example, the current of cell 1, cell 20, cell 21, and cell 40 can be expressed as (19). If a battery module configuration different from 4s10p is employed, then Equations (16)-(18) will require modification and derivation through [PERSON]'s laws. For a larger battery module with more cells, the computational load will increase at the module level but, conversely, will reduce at the pack level. \[\frac{1}{R_{mod}}=\frac{1}{R1+R20+R21+R40}+\ldots+\frac{1}{R5+R16+R25+R36}+\frac {1}{R6+R15+R26+R35}+\ldots+\frac{1}{R10+R11+R30+R31} \tag{16}\]  
\[\frac{\text{OCV}_{mod}}{R_{mod}}-\frac{\text{OCV}_{1}+\text{OCV}20+\text{OCV}21 +\text{OCV}20}{R1+R20+R21+R40}+\ldots+\frac{\text{OCV}5+\text{OCV}16+\text{OCV}5 +\text{OCV}6}{R5+R16+R25+R36}+\frac{\text{OCV}6+\text{OCV}15+\text{OCV}26+ \text{OCV}35}{R6+R15+R26+R35}+\ldots+\frac{\text{OCV}10+\text{OCV}11+\text{OCV}30 +\text{OCV}31}{R10+R11+R30+R31} \tag{17}\]  
\[I_{mod}=\frac{\text{OCV}_{mod}-\sqrt{\text{OCV}_{mod}}^{2}-4R_{mod}\times P_ {mod}}{2R_{mod}} \tag{18}\]  
\[I_{all1}=I_{cell20}=I_{cell21}=I_{cell40}=\frac{\text{OCV}1+\text{OCV}20+\text{OCV }21+\text{OCV}40}{R1+R23+R21+R40}+\left(\frac{\text{LOV}_{mod}}{R_{mod}}+ \frac{\text{OCV}1+\text{OCV}20+\text{OCV}21}{R1+R20+R21+R40}\right)\times\frac {R_{mod}}{R1+\text{R20+R21+R40}-R_{mod}} \tag{19}\]  
In the thermal aspects, two module designs with the same electrical arrangement but different thermal arrangements are considered as options, as shown in Figure 8c,d. Figure 8c adopts a serial thermal arrangement, which uses a single, U-shape coolant flow through every cell in the module successively, and the cell-level coolant flow rate \(q_{cell}\) is equal to the module-level one \(q_{mod}\), as (20). In comparison, Figure 8d adopts four parallel,  
Figure 8: **(a)** 4s10p battery module and cell numbering. **(b)** Equivalent, single-circuit model at module level. **(c)** Serial thermal design of module. **(d)** Parallel thermal design of module. straight coolant flows, each passing by 10 cells in the module, and the cell-level coolant flow rate \(q_{cell}\) is one quarter of the module-level one \(q_{mod}\), as (21). Both designs alternate the coolant directions to ensure that every parallel circuit can be cooled alternately to reduce the thermally dependent inhomogeneities over all 10 parallel circuits. The two thermal designs of the battery module are typical of serial and parallel configurations; therefore, the comparison between them can reflect the general advantages and disadvantages of serial and parallel thermal designs. Figure 8 also defines the inlet and outlet coolant temperatures of each cell. For example, for both designs, the outlet coolant temperature of cell 5 is the inlet coolant temperature of cell 6 as no losses to ambient between cells are assumed in the model. In Figure 8c, the inlet coolant temperature of cell 1 \(T_{inlet,cell1}\) is considered equal to the module-level inlet coolant temperature \(T_{inlet,mod}\), and the module-level outlet coolant temperature \(T_{outlet,mod}\) is equal to the outlet coolant temperature of cell 40 \(T_{outlet,cell40}\), as (20).