On the Impact of Internal Cross-Linking and Connection Properties on the Current Distribution in Lithium-Ion Battery Modules

The ELM of a 4s4p system including cross-connectors is illustrated in Fig. 1. Additionally, Table I lists all components of the ELM with a short description. Based on their numbering, the components of Fig. 1 can be clearly assigned. As Fig. 1 illustrates, the numbering of the cells corresponds to the numbering of matrix elements. Therefore, the numbering of the cells c starts at the top left with c_1,1 and ends at the bottom right with c_X,Y. Both the numbering of the cell currents I and the resistances of the welding seams on the positive (R_wsp) and negative (R_wsn) tabs of the cell equal the cell numbering. The connector resistance between two serial cells R_conn gets its number from the cell above it. R_p and R_n are the string connector resistances on the top and bottom between two parallel strings and are therefore numbered according to the string number to their right. The numbering of these string connector resistances ends by Y. If one of the STs is positioned on the right side, the numbering ends by Y + 1. Note that there is no R_conn present at the link between R_wsp and R_p. This is due to the assumption that the tabs are directly welded onto the busbar leading to the ST (see Eq. 6). The same applies to R_wsn and R_n, respectively. The numbering of the cross-connector resistances R_m between two serial-connected cells and two parallel strings is separated from the cell numbering, but also corresponds to the numbering of matrix elements.

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Table I.  Descriptions, units, and initial values of the components used in the ELM.

Variable	Initial Value	Unit	Description
φ	4.2	V	Voltage of a single cell
I	1 C	A	Current through single cell
Iapp	$Y\cdot 1\,{\rm{C}}$	A	Current applied to the system
Im	0	A	Current through cross-connector m
Ri	28.7	mΩ	Internal resistance of a single cell
Rwsp	62.5	μΩ	Resistance of the welding seam at the positive tab
Rwsn	62.5	μΩ	Resistance of the welding seam at the negative tab
Rconn	30.9	μΩ	Resistance of the serial connection within a string
Rm	15.4	μΩ	Resistance of the cross-connector between two strings
Rp	15.4	μΩ	Resistance of the parallel connection at the positive tab
Rn	15.4	μΩ	Resistance of the parallel connection at the negative tab

In order to clarify the naming of the system, in addition to the number of cells connected in serial and in parallel, the location of the ST (left or cross) is also added. Fig. 1 illustrates the system for ST-cross. Furthermore, the system includes four cell packages (I–IV). One cell package includes all cells at the same level of serial connection. For example, cell package III includes the cells c_3,1, c_3,2, c_3,3, and c_3,4.

As shown in Fig. 2, Rumpf et al.¹³ coupled the ELM with a physicochemical model (PCM) and a thermal model (THM). The coupled model was then used to investigate the influence of the ST, the contact resistance and varying internal resistances, capacities and temperature gradients on the current distribution within parallel-connected lithium iron phosphate (LFP) cells. To calculate the string currents, Kirchhoff's nodal and mesh equations were used and an equation system was given.

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However, calculating systems including cross-connectors (see Fig. 1) is not possible with the developed equation system. Additionally, the position of the ST is limited to a side (left or right) and middle connection. Hence, two extensions of the ELM are necessary. First, the model should be capable of dealing with systems including cross-connectors, and second, the position of the ST should be freely chosen.

To ensure that the ST can be freely chosen, setting up the meshes to calculate the string currents has to be considered. Therefore, attention has to be paid to the node where the currents branch differently. Consequently, the sign of affected R_n and R_p has to be adjusted if necessary. The sign of all other components as well as the general calculation method for the currents remains the same as described by Rumpf et al.¹³

However, including cross-connectors in the ELM has more impact on the calculation algorithm because the mesh equations have to be defined differently. In addition to the increased number of cell currents I, the currents I_m through the cross-connector resistances have to be calculated as well. This means that every additional cross-connector adds three additional variables to the equation system. Note that cross-connectors can only be implemented when $X,Y\geqslant 2$ holds true. To achieve at least a quadratic equation system, the same number of equations has to be added. Therefore, the procedure of adding equations is explained for a 2s2p system (ST-left), shown in Fig. 3.

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In Fig. 3a, only two currents have to be calculated, since the currents through c_1,1 and c_2,1 as well as through c_1,2 and c_2,2 are equal. Therefore, two equations are generated by the node k₁ on the bottom of string 1 and the mesh m₁ in between the cells. This leads to a quadratic equation system that can be solved analytically.

By including the cross-connector in Fig. 3b, three additional currents are generated. Now, the four cell currents I_1,1, I_1,2, I_2,1, and I_2,2 as well as the current I_m,1,1 through the cross-connector have to be calculated. To do so, the meshes m₁ and m₂ between c_1,1 and c_1,2 as well as between c_2,1 and c_2,2 are determined. Additionally, the node k₁ on the bottom of string 1 and the nodes k₂ and k₃ at the connection points on the left and right of the cross-connector are considered. Consequently, three additional equations are generated and the equation system is analytically solvable. If the system is extended by another cell package in serial or string in parallel, additional resulting nodes and meshes surrounding the additional cross-connectors have to be defined to retrieve the required equations. Consequently, it can be summarized that even in a system including cross-connectors, enough equations can be determined to set up a quadratic equation system.

The PCM and THM used in this study are based on the model developed by Sturm et al.^23,24 The model development and its parametrization for a nickel-rich manganese cobalt (NMC-811), silicon-graphite (SiC) lithium-ion cell (LG Chem, INR18650-MJ1, 3.35 Ah²⁵), using a wide range of experimental measurements, are shown in detail in Refs. 23, 24. The model equations and the resulting parametrization are summarized in Tables A·I and A·II as well as in Fig. A·1. To maintain the validity of the single cell model the presented simulation studies were chosen within the experimental validated operating range. The single cell PCM shows a mean open-circuit-voltage (OCV) error of 5.7 mV and a mean error of 13.5 mV for 1 C discharge (see Ref. 23, Figs. 1b and 4f). The THM shows a mean temperature error of less than 0.1 K (see Ref. 24, Fig. 4d)) for the intended C-rates. Therefore, it is assumed that the PCM and THM do not significantly affect the results. Furthermore, the following assumptions were made: It is assumed that the subsequently calculated resistances apply to industrially produced, interconnected systems using large format cells. Currently, such systems can be found mainly in BEVs. The available Audi, Jaguar, and Porsche BEVs can be seen as examples.^26–28 Additionally, each cell of the system is assumed to have the same capacity, internal resistance, and thermal behavior. Unless stated otherwise, the initial values correspond to those of Table I and the ambient temperature is set to 25 °C. Moreover, the tabs of the cells are considered to be welded onto an aluminum connector plate with a thickness of 2 mm using two 14 mm welding seams on each tab.

Schmidt et al.²⁹ measured a contact resistance of ${R}_{\mathrm{Al}-\mathrm{Al}}\,=250\,\mu {\rm{\Omega }}$ for Al–Al connections built on a single welding seam with an area of 15.71 mm². Due to the doubled welding seams, the contact resistance decreases by a factor of 2. Moreover, it is assumed that due to industrialized welding techniques and increased welding areas, the contact resistance can be decreased by an additional factor of 2. In total, the contact resistance of R_wsp and R_wsn is set to 62.5 μΩ. Furthermore, the average distance between the positive and negative tabs l_ws of the cells connected in serial is assumed to be 3 cm. Additionally, the average distance of adjacent positive or negative tabs is set to be 0.5 · l_ws. With an electrical conductivity of aluminum of ${\sigma }_{\mathrm{Al}}=34.7\,\cdot \,{10}^{6}\,{\rm{S}}\,{{\rm{m}}}^{-1}$,²⁹ the values of R_conn and R_m can be calculated using Eqs. 1 and 2. R_n and R_p are assumed to have the same characteristics as R_m and are therefore set to the same value (Eq. 3).

$$\begin{eqnarray}\begin{array}{rcl}{R}_{\mathrm{conn}} & = & \displaystyle \frac{1}{{\sigma }_{\mathrm{Al}}}\cdot \displaystyle \frac{{l}_{\mathrm{conn}}}{{l}_{\mathrm{ws}}\cdot {t}_{\mathrm{conn}}}\\ & = & \ \displaystyle \frac{1}{34.7\cdot {10}^{6}\,{\rm{S}}\,{{\rm{m}}}^{-1}}\cdot \displaystyle \frac{0.03\,{\rm{m}}}{0.014\,{\rm{m}}\cdot 0.002\,{\rm{m}}}=30.88\,\mu {\rm{\Omega }}\end{array}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}\begin{array}{rcl}{R}_{{\rm{m}}} & = & \displaystyle \frac{1}{{\sigma }_{\mathrm{Al}}}\cdot \displaystyle \frac{{W}_{{\rm{m}}}}{{h}_{{\rm{m}}}\cdot {t}_{{\rm{m}}}}\\ & = & \ \displaystyle \frac{1}{34.7\cdot {10}^{6}\,{\rm{S}}\,{{\rm{m}}}^{-1}}\cdot \displaystyle \frac{0.015\,{\rm{m}}}{0.014\,{\rm{m}}\cdot 0.002\,{\rm{m}}}=15.44\,\mu {\rm{\Omega }}\end{array}\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{R}_{{\rm{n}}}={R}_{{\rm{p}}}={R}_{{\rm{m}}}\end{eqnarray} \tag{ 3 }$$

A common feature of the BEVs mentioned here is the use of large format lithium-ion batteries with a capacity of approx. 60 Ah. In order to ensure practical relevance, a capacity of 60 Ah is therefore assumed to be the reference cell capacity in the following. However, the model developed by Sturm et al.²³ was validated for a 3.35 Ah cell. As the cell under consideration and the cell used by Sturm et al.²³ differ in capacity and resistance, the values of R_wsp, R_wsn, R_conn, R_m, R_p, and R_n from Table I have to be scaled to ensure comparability. Therefore, a scaling factor scaling factor SF has to be calculated, using the ratios of the cells' capacities or internal resistances. Since the focus of this study is not on high power but on high energy applications, the influence of the capacity should dominate the influence of the resistances. Consequently, the initial values of the resistances named here are scaled by the cells' capacities using ${{SF}}_{C}=17.9$.

Subsequently, the influence of different STs, varying R_wsp, R_m, and R_p on the current distribution is discussed. Therefore, a side (ST-left) and cross-connection (ST-cross) are investigated first. Following this, the influence of varying R_wsp are examined. The value of the homogeneous case is therefore increased by +100%. Next, the same variation is applied to different R_m. Finally, the rate capability of the extractable energy content of the system is examined. Note that only one R_wsp, R_m, or R_p is varied at a time. Table II summarizes the investigated cases. Note that Rumpf et al.¹³ showed that middle contacting would lead to a more homogeneous current distribution. Consequently, the deviations shown in the following exceed the deviations of middle contacting, which is why the focus of the following investigations is on ST-left and ST-cross.

As described above, it is assumed that every cell has the same thermal behavior. Consequently, the conditions (see Tables A·I and A·II) applied to the cells are equal. Furthermore, thermal coupling between adjacent cells as well as heat transfer through contact surfaces have not yet been implemented in the THM. Therefore, an equal current load leads to an equal temperature behavior, independently of a cell's position within the system. As shown below, the current distribution throughout the system exhibits relatively small deviations between cells over a wide depth of discharge (DoD) range. Consequently, uneven temperature development is not expected to have a significant influence on the current deviation for the investigated cases. However, the development of a thermally coupled model and the analysis of temperature gradients and cooling strategies within battery systems are crucial for cooling strategies and battery safety and may be the scope of future studies.

The applied current to the system equals Y · I_1C. That in turn means that the expected current load per cell is equal to a C-rate of $I=1\,{\rm{C}}$ (I_1C). Each discharge begins with fully charged cells (DoD = 0) at a cell voltage of 4.2 V and ends when a cell reaches the discharge cutoff voltage of 2.5 V (DoD = 1). This means that for the investigated 4s4p system the measurable voltage level at the STs is between 16.8 V and 10.0 V. Furthermore, the expected amount of energy that can be drawn from the system equals the sum of the extractable energies of the individual cells and can be calculated using the following equation:

$$\begin{eqnarray}&&{E}_{\mathrm{sys}}=\displaystyle \sum _{{\rm{s}}=1}^{X}\displaystyle \sum _{{\rm{p}}=1}^{Y}{E}_{{\rm{s}},{\rm{p}}}\end{eqnarray} \tag{ 4 }$$

In this matter, the nominal energy (E_N,sys) is summarized from the total of all cells ($16\cdot {E}_{{\rm{N}},\mathrm{cell}}$), which is used to compare the simulated cases shown in Table II.

Figure 4 illustrates the resulting current distribution of a 4s4p system connected on the left side (ST-left, (a)) and cross-connected (ST-cross, (b)), for the homogeneous case. Homogeneous here means that the same system components have the same initial values. By comparing Figs. 4a and 4b qualitatively, it can be seen that there is no difference between the current distribution within the systems. Note that equal line colors indicate quasi-identical current distributions of different cells within the systems. Therefore, it can be stated that within XsYp systems including cross-connectors, the choice of the ST is negligible for the homogeneous case, if $X,Y\geqslant 2$ holds true.

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Additionally, five current paths can be distinguished in both subplots. As a total of 16 current distributions have to occur for the analyzed 4s4p system, the noticeable current paths can be interpreted as group current paths. It is therefore necessary to clarify where the number of group paths originates from and how the groups are formed.

Considering Fig. 4, the number of group paths strongly depends on the number of cells connected in parallel, since a group current path occurs for each of the $Y$ parallel strings. These current paths differ from the applied current of I_1C for the first and the last cell package of the system due to uneven path resistances R_PR.^2,13 For parallel-connected systems with $X\gt 2$, an additional group current path occurs which lies directly on the applied current line. Consequently, the number of group current paths ${N}_{\mathrm{GCP}}$ within a XsYp system is given by the following rule:

$$\begin{eqnarray}\begin{array}{r}\begin{array}{r}\begin{array}{c}\begin{array}{c}{N}_{{\rm{G}}{\rm{C}}{\rm{P}}}=\left\{\begin{array}{cc}Y & \,{\rm{f}}{\rm{o}}{\rm{r}}\,X\leqslant 2\,\wedge \,Y\geqslant 2\\ Y+1 & \,{\rm{f}}{\rm{o}}{\rm{r}}\,X\gt 2\,\wedge \,Y\geqslant 2\end{array}\right.\end{array}\end{array}\end{array}\end{array}\end{eqnarray} \tag{ 5 }$$

Marking the cells from the same group current path in Fig. 1b leads to Fig. 5. Considering ST-left shown in Fig. 5a, the current distribution within the system follows the axis symmetry to the mirror axis between cell packages II and III. On the other hand, the current distribution regarding ST-cross leads to point symmetry to the central point of the system of Fig. 5b. Once again, the symmetries can be justified with the cells' R_PR. For example in the case of ST-cross R_PR of c_1,3 and c_4,2 to the nearest ST calculate to:

$$\begin{eqnarray}\begin{array}{rcl}{R}_{\mathrm{PR},\mathrm{1,3}} & = & \displaystyle \frac{{R}_{\mathrm{conn},\mathrm{1,3}}}{2}+{R}_{\mathrm{wsp},\mathrm{1,3}}+{R}_{{\rm{i}},\mathrm{1,3}}\\ & & +\ {R}_{\mathrm{wsn},\mathrm{1,3}}+{R}_{{\rm{p}},3}+{R}_{{\rm{p}},2}\end{array}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}\begin{array}{rcl}{R}_{\mathrm{PR},\mathrm{4,2}} & = & \displaystyle \frac{{R}_{\mathrm{conn},\mathrm{3,2}}}{2}+{R}_{\mathrm{wsp},\mathrm{4,2}}+{R}_{{\rm{i}},\mathrm{4,2}}\\ & & +\ {R}_{\mathrm{wsn},\mathrm{4,2}}+{R}_{{\rm{n}},3}+{R}_{{\rm{n}},4}\end{array}\end{eqnarray} \tag{ 7 }$$

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Since Eqs. 6 and 7 consist of equal system elements with equal values, the same current is applied to c_1,3 and c_4,2.

The symmetry effects found can now be used to examine the maximum deviations from the applied current at the beginning of discharge (BoD) and at the end of discharge (EoD) (see Fig. 4). Note that the choice made for BoD and EoD represents the maximum deviation during the entire discharge of the cells, as can be seen in Fig. 4. This is why the deviations from the applied current at these points in time can be assumed to be a worst-case scenario. Table III lists the deviations from the applied I_1C discharge current for cells c_1,1, c_1,2, c_1,3 and c_1,4 for the homogeneous case.

Table III.  Maximum deviation to the applied I_1C current at the beginning (BoD) and at the end of discharge (EoD). The same deviations apply to the symmetry partners of the cells according to Fig. 5.

	${\rm{\Delta }}{I}_{\mathrm{1,1}}$	${\rm{\Delta }}{I}_{\mathrm{1,2}}$	${\rm{\Delta }}{I}_{\mathrm{1,3}}$	${\rm{\Delta }}{I}_{\mathrm{1,4}}$
BoD	$+1.9 \%$	$+0.3 \%$	$-0.8 \%$	$-1.4 \%$
EoD	$-3.4 \%$	$-0.5 \%$	$+1.5 \%$	$+2.4 \%$

Thus, at the beginning, I_1,1 and I_1,2 are increased by 1.9% and 0.3% due to their smaller R_PR as compared to the applied I_1C current. In contrast, I_1,3 and I_1,4 are decreased by 0.8% and 1.4%. Consequently, until currents cross at around 75% DoD, c_1,1 and c_1,2 are discharged with a higher current and their DoD increases faster. Subsequently, c_1,3 and c_1,4 have to compensate for this imbalance until the EoD. Therefore, I_1,3 and I_1,4 increase up to 1.5% and 2.4% compared to the applied I_1C current. Depending on the choice of the ST, these deviations correspond to the deviations of the symmetry partners marked in Fig. 5. Cells of cell packages II and III are not listed because their maximum deviation to the applied I_1C current is less than 0.02% and is therefore considered negligible. Furthermore, an extractable energy content for the homogeneous case E_hom of 193.2 Wh occurs for both systems. Since the nominal energy content of a single cell calculates to ${E}_{{\rm{N}},{\rm{cell}}}=12.18\,{\rm{W}}{\rm{h}}$, following Eq. 4 E_N,sys calculates to 194.9 Wh. Consequently, 99.13% of E_N,sys can be extracted for the homogeneous case.

The analysis by Hofmann et al.,³ performed with an equivalent circuit model using linearized OCV, showed that differences in OCV, impedance, and capacity are the main reasons for an inhomogeneous current distribution within parallel connections. Thereby, differences in the OCV dominate the influences of uneven capacity and impedance. To further analyze the origin of the current distribution, the PCM was used to perform a polarization analysis.³⁰ With respect to the complexity, the analysis was conducted for the upper cell package in Fig. 3b instead of a cell package of the 4s4p system. The parametrization of the PCM, ELM, and THM remained unchanged (see Table II). Figure 6 illustrates the resulting differences in polarization and OCV between the cells. Since R_PR,1,1 is decreased as compared to R_PR,1,2, c_1,1 is discharged with an increased current compared to c_1,2 at the BoD. Since the polarization of the single cell depends on the current, the resulting inhomogeneous current distribution causes a difference in the polarization of c_1,1 and c_1,2. In addition, as both cells are discharged from 0% DoD, no OCV shift occurs at the BoD. Hence, the current distribution is caused by the differences in R_PR and polarization during the BoD. However, with progressive discharge of the cells, the inhomogeneous current distribution leads to a difference of the cells' DoDs. As a result, the cells' OCVs are shifted against each other. Figure 6 also illustrates that as soon as the difference of the OCVs exceeds the difference of the polarizations, the shift of the cells' OCVs becomes dominant. As the OCV difference leads to different cell potentials, the current distribution is dominated by the OCV difference. For the modeling design, this means that the use of a validated equivalent circuit model could also be justified. However, since the polarization analysis is seen as an important tool for subsequent studies, a PCM model was also used for the sensitivity analysis shown in the following. Note that Fig. 6 does not include differences of the polarization at surrounding components, and thus the difference between the cells' OCVs and the internal polarizations η differs from zero.

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The influence of R_wsp on the current deviation within both systems is analyzed next. For this purpose, all R_wsp are successively increased by +100% for ST-left and ST-cross.

Figure 7 illustrates an example of the resulting current profile of the 4s4p system (ST-left) for a variation of R_wsp,1,1 as compared to the homogeneous case. Due to the increased contact resistance of c_1,1, I_1,1 decreases in the beginning. At the same time, I_1,2, I_1,3, and I_1,4 have to increase so that cell package I can still pass on the required current to the next cell package level (II). Consequently, the increased current through cells c_1,2, c_1,3, and c_1,4 at the BoD leads to a faster discharge of these cells. This in turn means that at the EoD, c_1,1 has to deliver a higher current as compared to the homogeneous case, whereas the currents through c_1,2, c_1,3, and c_1,4 decrease. This means that there is always an interplay of the current load within the cell package so that the overall applied current of 4 · I_1C can be passed on to the next cell package. Hence it is examined next, if a variation of R_wsp only affects cells within the same cell package and therefore, whether every cell package can be analyzed independently.

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Accordingly, Fig. 8 illustrates the deviations of the cell currents at the EoD (see Fig. 7) as compared to the applied I_1C current for a variation of R_wsp. On the left side of Fig. 8, every subplot illustrates the influence of R_wsp on the current deviation of the symmetry partners for ST-left. Consequently, the influence of R_wsp,1,1 on the currents of cell package I can be investigated by comparing the deviation of I_1,1, I_1,2, I_1,3, and I_1,4 in the subplots (a), (b), (c), and (d) (dotted lines) above index 1,1. In contrast, if a variation of R_wsp,1,1 would affect cell currents of cell packages II, III, or IV, at least one cell current of I_2,1, I_3,1, I_4,1, I_4,2, I_4,3, or I_4,4 would deviate from the homogeneous case (index h) under variation of R_wsp,1,1 (index 1,1) in Fig. 8. As this is not the case, it can be stated that a variation of R_wsp,1,1 only affects cell currents within cell package I. If one continues to analyze the effects of the variations in other R_wsp (index 1,2 − 4,4), this statement by cell package I can be extended to the entire system. Accordingly, it can generally be stated that an increased contact resistance only affects the cell package in which the increased contact resistance occurs. Thus, each cell package can be evaluated in a decoupled manner. The reason why a variation within a cell package has no influence on the surrounding cell packages is the presence of cross-connectors between the cell packages. Consequently, cross-connectors can be seen as balancing elements inside the system.

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Due to the finding that cell packages can be analyzed individually, the position of an increased R_wsp within the cell package is investigated further. Hence, a variation of R_wsp,1,1, R_wsp,1,2, R_wsp,1,3, or R_wsp,1,4 leads to a maximum increase at the EoD of the currents I_1,1, I_1,2, I_1,3 or I_1,4, of 2.94%, 2.98%, 3.01%, or 3.04% as compared to the homogeneous case. Referring to the symmetry partners shown in Fig. 5, the same deviations occur for a variation of R_wsp within cell package IV. Within cell packages II and III, a deviation of 3.01% as compared to the homogeneous case remains the same for every varied R_wsp, as indicated by the subplots (e) and (j) in Fig. 8. The maximum difference between the deviations of all investigated cases can be calculated to 0.1%. Consequently, the influence of the position of an increased R_wsp has no significant impact on the rate of current increase for the analyzed system. The reason for this is that the ratio between R_wsp to the resistance of R_m and R_p is dominated by the welding seam resistances. An increase of R_wsp reinforces this effect, which is why the named resistances are even less important.

Additionally, the impact of R_n, R_p, R_m, and R_conn on the maximum cell current load can be examined by evaluating the deviation at the EoD as compared to the applied I_1C current rate. Thus, I_1,1, I_1,2, I_1,3, and I_1,4 deviate at the EoD from the applied I_1C current in the homogeneous case by $-3.41 \%$, $-0.51 \%$, 1.46%, and 2.45%, and under variation of the corresponding R_wsp by $-0.46 \%$, 2.48%, 4.48%, and 5.50% (see Fig. 8). Therefore, it can be determined that in both cases, the maximum cell current load increases with increasing distance to the ST for cell package I. This can in turn be explained by the fact that an increased distance from the ST corresponds to an increased R_PR, which leads to a lower current load during the BoD and an increased current load during the EoD (see also Fig. 7). As described above, an increase of R_wsp reinforces this effect. Therefore, it can be summarized that the position of an increased R_wsp has no impact on the rate of current increase within the cell package, but on the maximum current deviation at the EoD.

Varying R_wsp within a 4s4p ST-cross system leads to the results in Figs. 8f–8j. Compared to the results of ST-left, only the symmetry partners change according to Fig. 5b for ST-cross. The aforementioned conclusions apply similarly for the cross-connected system. That also holds true for the influence of the position and the rate of current increase that corresponds to the variation of R_wsp.

Furthermore, only a negligible influence of a variation of R_wsp on the extractable energy content can be observed, as the difference between the maximum and the minimum extractable energy only calculates to 0.01 Wh. This in turn means that an increased R_wsp by $+100 \%$ could not be detected by the measurable extractable energy content at the system terminals.

In addition to the influence of R_wsp, the influence of increased R_m on the current distribution is investigated next. For this reason, R_m were increased individually by $+100 \%$ for both system configurations. Table IV evaluates the maximum current deviation of I_m as compared to the homogeneous case under variation of R_m. Therefore, it can be seen that R_m,1,2 and R_m,3,2 show the highest current deviation of ${\rm{\Delta }}{I}_{\max }\,=0.024 \%$. This current is distributed to the subsequent cell package differently than in the homogeneous case. Moreover, in the worst case, ${\rm{\Delta }}{I}_{\max }$ is passed on to only a single cell of the next cell package over the entire I_1C discharge. Compared to I_1C, an additional deviation of ${\rm{\Delta }}{I}_{\max }$ corresponds to a current of 0.024% of the cells' current. Such a deviation is therefore classified as not significant. Furthermore, the extractable energy content under a variation of R_m equals the value of the homogeneous case. Accordingly, it can be said that an increase of R_m by $+100 \%$ has no significant influence on either the current distribution or the extractable energy content of the analyzed system.

Table IV.  Maximum current deviation of I_m as compared to the homogeneous case under variation of R_m by $+100 \%$. The entries marked with * have a maximum deviation of less than 3 · 10⁻⁶% and were therefore set to zero.

Rm,1,1	0.016%	Rm,1,2	0.024%	Rm,1,3	0.011%
Rm,2,1	$0{ \% }^{* }$	Rm,2,2	$0{ \% }^{* }$	Rm,2,3	$0{ \% }^{* }$
Rm,3,1	0.016%	Rm,3,2	0.024%	Rm,3,3	0.011%

Taking into account the respective symmetry partner, Table IV and the statements made apply equivalently to ST-cross.

After investigating the influence of R_wsp and R_m, the influence of varying string connectors is examined next. Therefore, R_p,2, R_p,3, and R_p,4 of cell package I are successively increased by $+100 \%$ for both systems. A variation of R_n,2, R_n,3, and R_n,4 is omitted, due to symmetry effects to cell package I.

Figure 9 illustrates the current distribution within cell package I for a variation of R_p,2 in a system with ST-left. Compared to the homogeneous case, I_1,1 is increased by 1.3% at the BoD whereas I_1,2, I_1,3, and I_1,4 are decreased by 0.5%, 0.4% and 0.4%. The reason for this deviation is that the string connectors can be seen as segmenting elements of the system. The segmenting in turn leads to increased R_PR for all strings, whose position in the current path is behind the varied resistance. Consequently, strings with higher R_PR lead to lower string currents at the BoD, and thus I_1,2, I_1,3, and I_1,4 are decreased during the BoD and only I_1,1 is increased. Consequently, if R_p,2 is increased, cell package I divides into two segments. String 1 is located in front of the increased string connector and experiences a higher current during the BoD because of the lower path resistance. However, if R_p,3 is varied, the deviation of I_1,1 and I_1,2 is smaller, since the additional current can split up to both strings. In contrast to R_p,2 and R_p,3, R_p,4 takes a special place, as it is the last string connector. Figure 1 illustrates that R_p,4 acts as an element of the string itself rather than as a string connector. Thus, a variation of R_p,4 leads to the same findings as varying R_wsp,1,4.

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Comparing the maximum current deviations occurring for a variation of R_wsp (see Fig. 8) to those of a variation of R_p, the influence of varying R_wsp seems to dominate R_p's influence. However, taking into account that there is a factor of 4 between the increased values of R_p and R_wsp and only a factor of 1.7 between the maximum deviations at the EoD, it can be determined that the influence of R_p dominates the influence of R_wsp on the current distribution within the system. The same applies for R_m, since, as shown above, the influence of a variation of R_m is negligible. Consequently, it can be stated that attention has to be paid to the string connectors when assembling such a system.

The maximum deviation of the extractable energy from the homogeneous case calculates to 0.02 Wh for a variation of R_p. Thus, even an increased string connector resistance cannot be detected by means of the measurable extractable energy content. This in turn means that none of the investigated cases can be detected due to measurable quantities at the STs. The almost constant extractable energy is thus achieved by a higher load on individual system components (see deviations in Fig. 9). The impact of this effect on the durability of the components and thus of the investigated system therefore needs to be investigated in further studies.

Previous sections have shown that the influences of varying R_wsp, R_m, and R_p on the current deviation and the extractable energy content of the system is negligible for a 1 C constant-current discharge. Therefore, the influence of various C-rates is investigated to demonstrate both the rate capability and the influence of varying contact resistances on the extractable energy content of the system. For this purpose, various C-rates (1, 1.25, 1.5, 1.75, 2, 2.25, 2.5, 2.75, and 3) were applied to a 4s4p system connected on the left (ST-left). Additionally, a variation of R_p,2 by $+100 \%$ was accomplished to examine whether the extractable energy content differs significantly from the homogeneous case. As it is assumed that the analysis of the extractable energy content is more meaningful for the rate capability of the system than the maximum current deviation at a specific point in time, the following analysis focuses on the extractable energy content. Since the previous sections have shown that the variations of R_wsp and R_m have a minor influence compared to R_p, an additional analysis of the variation of R_wsp and R_m is omitted here. Furthermore, an additional analysis of ST-cross was not conducted due to the symmetry effects shown in Fig. 5.

Figure 10 illustrates the extractable energy content for the homogenous case (solid line) and for a variation of R_p,2 (dashed line) for the C-rates named above. Comparing the homogeneous and varied cases, significant deviation of the extractable energy content cannot be depicted, since the maximum deviation calculates to 0.03%. Furthermore, the rate capability of the extractable energy content can be divided into two branches. The first one is represented by the range of 1 C to 1.75 C. In this domain, the extractable energy content decreases from 99.1% (1 C) to 94.5% (1.75 C) for both cases, which equals a mean decrease of 1.53% per increase of C-rate by 0.25 C. Within the second branch, the mean gradient of energy content per increase of the C-rate by 0.25 increases to 7.54% in the remaining range between 1.75 C and 3 C. This results in an extractable energy content of 56.8% for an applied 3 C-rate.

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Consequently, the analysis clearly shows the trend of a decreasing rate capability of the extractable energy content with increasing C-rates. Furthermore, the findings show that an increased contact resistance leads to a more uneven current distribution, but that the extractable energy content remains the same. This leads to the assumption that the almost constant energy output is achieved by higher loads on individual system components, which is why the longevity has to be investigated further. In addition, for future studies the cell parametrization can be further specified in subsequent steps, as shown by Mao et al.³¹ and Rheinfeld et al.³² These extensions would take into account additional limitations of electrode kinetics, and thus increase the accuracy of the p2D model in the event of even higher C-rates, especially higher than 3 C. Table V summarizes the results of the investigated cases.

Table V.  Summary of the investigated cases.

Investigation	Initial Value	Variation	Critical Component	Result
ST	—	left, cross	cross-connectors	both ${\rm{\Delta }}{I}_{\max }=+2.40 \%$
				axis and point symmetry
Rwsp	62.5 μΩ	+100%	${R}_{\mathrm{wsp},\mathrm{1,4}}$	${\rm{\Delta }}{I}_{\max }=+5.50 \%$
Rm	15.4 μΩ	+100%	${R}_{{\rm{m}},1,2}$	${\rm{\Delta }}{I}_{\max }=+0.02 \%$
Rp	15.4 μΩ	+100%	${R}_{{\rm{p}},2}$	${\rm{\Delta }}{I}_{\max }=+3.20 \%$
iapp	1 C per cell	C-rate		${\rm{\Delta }}{E}_{\max }=-43.2 \%$ (3 C)