*3.3. Validation*

In order to validate the discussed ECM, simulations are compared to measurements in Figure 8. Each of the used cells consists of 40 cell layers, whose resistances and capacities were randomly drawn from Gauss distributions. With the related standard deviations of the cell's resistance and the cell's capacity

$$\frac{\sigma\_{\mathcal{R}}}{\mu\_{\mathcal{R}}} = \omega\_{\mathcal{R}} \cdot \sqrt{L},\tag{7}$$

$$\frac{\stackrel{\cdots}{\sigma\_{\mathbb{C}}}}{\mu\_{\mathbb{C}}} = \omega\_{\mathbb{C}} \cdot \sqrt{L} \,\tag{8}$$

with the standard deviation of the cell's resistance *σR* and the cell's capacity *σC*, the mean value of the cell's resistance μ*R* and the cell's capacity μ*C* as well as the number of parallel-connected cell layers *L*. The standard deviations of the parameters were multiplied by the square root of *L* to

consider the statistical averaging due to the parallel cell layers. The related standard deviations *ωR* and *ωC* were set to *ωR* = 1% and *ωC* = 0.5%, which are typical parameter distributions caused by manufacturing tolerances as found in [3,4]. The expected values μ*R* and μ*C* were adjusted according to the parametrization results of each cell. The load cable resistances *R*Cab were set according to the values of Table 2. For the tab resistances *R*Tab, the average values of the measurements in Figure 5c were used.

The results of the simulation in Figure 8 agree well with those of the measurement with an RMSD of *ξ*RMSD = 0.083 A. The highest differences appeared at the end of discharge and at the first current rest. The simulated layer currents, Figure 8a, within the cells show a similar distribution with an shift depending on the cell parameters.

**Figure 8.** Comparison of measurement and simulation, with the simulated cell layer currents *Ii*,<sup>1</sup> and *Ii*,2, *<sup>i</sup>*{1, ... , 40} within cell one and two in (**a**) and the simulated cell currents as well as the measurement results of cable type 1 in (**b**).

### **4. Impacts of the Module Design**

Due to the cell packaging, inhomogeneities in terms of cooling and thermal connection of the cells to their neighbor cells arise within a battery module. The influence of the ambient temperature depends on the cell position in the module, whereby the cells at the edge of the module are mainly affected. This leads to inhomogeneous cell cooling resulting in temperature gradients. Existing tension mats, which thermally isolate the cells, can further intensify arising temperature gradients. Possible scenarios are presented in Figure 9.

In order to investigate the influence of the cell position and thermal connection of cells, the tension mats were modeled as heat impermeable and the module housing was assumed to be isotherm with a constant temperature at ambient temperature. For each scenario, which considers the effects of the border cell, two measurements were conducted to switch the border position of the two parallel-connected cells. The reference represents conditions at a constant cell temperature, which rather corresponds to most publications.

The current and temperature distribution of these scenarios for two parallel-connected cells are displayed in Figure 10. The cells were fully discharged with *I*Batt = 1C until the cell pack reached the lower voltage limit of *U*Batt = 2.8 V. Thereafter, the cells were further discharged at constant voltage of *U*Batt = 2.8 V until the battery current decreased to *I*Batt = 1/20. After a relaxation of *T*Break = 1 h, the parallel-connected cells were charged with *I*Batt = 1C until the upper voltage limit of *U*Batt = 4.2 V was reached. The ambient temperature and the initial temperature of the aluminum plates were set to *T* = 20 ◦C.

The scenarios show significant differences of the current and temperature distribution of the parallel-connected cells. In the following the scenarios will be separately discussed.

**Figure 9.** Impacts of the cell position and isolating tension mats on temperature gradients within a battery module (**a**). Considered scenarios: Reference (**b**), thermal-coupled border cell (**c**) and isolated border cell (**d**).

**Figure 10.** Measurements simulating different cell positions in a battery and thermal connections of cells to their neighbor cells of two parallel-connected cells. With the current (**a**) and temperature distribution (**c**) at discharge with *I*Batt = 1C as well as the current (**b**) and temperature distribution (**d**) at charge with *I*Batt = 1C.
