# Modelling and Evaluation of Battery Packs with Different Numbers of Paralleled Cells

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Battery Pack Model

_{0j,k}is the Direct Current (DC) inner resistance. The open circuit voltage (OCV) e

_{j,k}is generated according to the state of charge, SOC

_{j,k}. i

_{j,k}is the output current of the cell while the terminal voltage is denoted by u

_{j,k}.

_{1j,k}, C

_{1j,k}, R

_{2j,k}, C

_{2j,k}, can be identified by measuring the transient voltage when the cell is discharged by a constant current. R

_{1j,k}and C

_{1j,k}are denoted to have a smaller time constant than that of R

_{2j,k}and R

_{2j,k}, so that the polarization in the short term (several seconds) and the long term (in minutes level) can be demonstrated by R

_{1j,k}, C

_{1j,k}and R

_{2j,k}, R

_{2j,k}respectively. u

_{c1j,k}and u

_{c2j,k}are the voltage of C

_{1j,k}and C

_{2j,k}. u

_{j,k}is the terminal voltage of the cell. In this model, the resistance and capacitance are assumed to stay constant. The dependency on the SOC is neglected.

_{j,k}is the capacity of the corresponding cell. The output current of the pack is represented by i

_{pack}. The output of the model is the current and the voltage of each cell in the battery pack, when the load of the battery pack, i

_{pack}, is given.

_{j}. Using Kirchhoff’s law of current, the following group of equations can be derived in Equation (1):

_{j}and correspondingly the current of each cell can be calculated by Equation (2) if the polarization voltage and OCV are known.

## 3. Experimental Verification of the Proposed Model

_{line}are listed in Table 1.

## 4. Evaluation of Battery Packs Based on Monte-Carlo Experiments

#### 4.1. Statistic Features of Battery Cells

_{0j,k}), capacity (Q

_{j,k}), polarization resistors (R

_{1j,k}, R

_{2j,k}) and polarization capacitors (C

_{1j,k}, C

_{2j,k}) in the Monte-Carlo experiments, the statistic features and the distributions of the parameters are analysed based on the measured data of 50 Panasonic NCR18650PF cells.

_{1j,k}, C

_{1j,k}, R

_{2j,k}and C

_{2j,k}, are more difficult to measure, as there is no equipment specially designed for the automatic measurement of those parameters. Therefore, the polarization parameters of only 10 cells are identified manually to obtain their statistic features. As 10 results cannot show an obvious pattern in the histograms, the statistic indices of the four polarization parameters are calculated to identify their distributions. Results are listed in Table 2, in which the p-value of KS-test is the probability that the tested dataset complies with a normal distribution. From the p-values of the KS-test and the skewness values, it is seen that the C

_{2j,k}quite possibly follows the normal distribution while R

_{1j,k}, R

_{2j,k}and C

_{1j,k}should be generated by skew normal distributions.

_{0j,k}and C

_{2j,k}comply with normal distributions, represented by N in Equation (5), while the other four properties are generated according to the skew normal distributions, represented by SKEWN in Equation (5).

_{1}, μ

_{2}and σ

_{1}, σ

_{2}are respectively the mean values and standard deviations of the two normal distributions. ω

_{1}–ω

_{4}, α

_{1}–α

_{4}and ε

_{1}–ε

_{4}are the parameters of the skew normal distributions. In the possibility density function (PDF) of the skew normal distribution, Φ and ϕ are respectively the cumulative distribution function (CDF) and the PDF of a standard normal distribution N(0,1). The values of these parameters are selected to ensure that the distributions have the same mean values, standard deviations and skewness values as those of the measured data. The detailed values and calculation processes are not given in this paper due to the length limit.

#### 4.2. Simulative Evaluations with Monte-Carlo Experiments

_{1}, X

_{2 …}, X

_{N}are not physical variables in the battery model, but just a group of independent normal distributed variables to prove that the deviation of the sum value is much smaller than the deviation of individual variables. Connecting more cells in parallel will lower the capacity difference between different rows of paralleled cells, and thus reduces the proportion of the energy consumed by the balancing of the BMS.

#### 4.3. Discussion

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**The simulated and measured current waveforms of four paralleled cells when discharged by the 1 C constant current load. (

**a**) Output current waveforms of cell 1; (

**b**) Output current waveforms of cell 2; (

**c**) Output current waveforms of cell 3; (

**d**) Output current waveforms of cell 4.

**Figure 6.**The simulated and measured current waveforms of four paralleled cells when discharged by the fluctuating driving cycle load. (

**a**) Output current waveforms of cell 1; (

**b)**Output current waveforms of cell 2; (

**c**) Output current waveforms of cell 3; (

**d**) Output current waveforms of cell 4.

**Figure 7.**The frequency histograms of the capacity and the inner resistance of the 50 cells (

**a**) Frequency histogram of the capacity; (

**b**) Frequency histogram of the DC inner resistance.

**Figure 8.**The influence of the parallel number on the losses of battery packs; (

**a**) The influence on the Battery Management System (BMS) balancing loss; (

**b**) The influence on the total in-cell energy loss.

**Figure 9.**The aging process of the average capacity and the average inner resistance of each battery pack; (

**a**) Aging process of the average capacity; (

**b**) Aging process of the average inner resistance.

Cell No. | R_{line} | R_{0j,k} | R_{1j,k} | C_{1j,k} | R_{2j,k} | C_{2j,k} |
---|---|---|---|---|---|---|

Cell1 | 81.3 mΩ | 37 mΩ | 8.6 mΩ | 204 F | 54.7 mΩ | 791 F |

Cell2 | 41.9 mΩ | 36 mΩ | 17.7 mΩ | 446 F | 64.3 mΩ | 1252 F |

Cell3 | 15.8 mΩ | 36 mΩ | 14.1 mΩ | 436 F | 72.3 mΩ | 1170 F |

Cell4 | 22.6 mΩ | 32 mΩ | 9.3 mΩ | 322 F | 61.2 mΩ | 1033 F |

Indices | R_{1j,k} | C_{1j,k} | R_{2j,k} | C_{2j,k} |
---|---|---|---|---|

Mean | 0.0122 Ω | 326.6 F | 0.06 Ω | 1020.4 F |

Standard Deviation | 0.003 Ω | 83.9 F | 0.012 Ω | 145.5 F |

Skewness | 1.0831 | 0.372 | −0.9416 | 0.034 |

p-Value of KS-test | 0.441 | 0.628 | 0.539 | 0.902 |

p-Values of Correlation Tests | Q_{j,k} | R_{0j,k} | R_{1j,k} | C_{1j,k} | R_{2j,k} | C_{2j,k} |
---|---|---|---|---|---|---|

Q_{j,k} | 1.00 | 0.04 | 0.21 | 0.22 | 0.33 | 0.10 |

R_{0j,k} | 0.04 | 1.00 | 0.08 | 0.18 | 0.40 | 0.35 |

R_{1j,k} | 0.21 | 0.08 | 1.00 | 0.00 | 0.18 | 0.00 |

C_{1j,k} | 0.22 | 0.18 | 0.00 | 1.00 | 0.23 | 0.00 |

R_{2j,k} | 0.33 | 0.40 | 0.18 | 0.23 | 1.00 | 0.31 |

C_{2j,k} | 0.10 | 0.35 | 0.00 | 0.00 | 0.31 | 1.00 |

**Table 4.**The capacity and the inner resistance of cells in each battery pack before and after the degradation test.

Configuration of Battery Packs | 1p1s | 2p1s | 4p1s | 9p1s | 18p1s | 36p1s | 72p1s | |
---|---|---|---|---|---|---|---|---|

New | E(Q_{j,k}) | 2.910 Ah | 2.915 Ah | 2.910 Ah | 2.918 Ah | 2.913 Ah | 2.913 Ah | 2.914 Ah |

SD(Q_{j,k}) | - | 0.0133 | 0.0058 | 0.0062 | 0.0073 | 0.0076 | 0.0082 | |

E(r_{0j,k}) | 38.16 mΩ | 34.51 mΩ | 34.91 mΩ | 36.42 mΩ | 35.64 mΩ | 33.94 mΩ | 34.26 mΩ | |

SD(r_{0j,k}) | - | 0.0033 | 0.00451 | 0.00263 | 0.00242 | 0.00269 | 0.00284 | |

After 500 cycles test | E(Q_{j,k}) | 2.372 Ah | 2.361 Ah | 2.362 Ah | 2.367 Ah | 2.365 Ah | 2.365 Ah | 2.365 Ah |

E(Q_{j,k}) change | −18.48% | −19.02% | −18.84% | −18.89% | −18.81% | −18.82% | −18.84% | |

SD(Q_{j,k}) | - | 0.0255 | 0.0301 | 0.0177 | 0.0113 | 0.0160 | 0.0185 | |

E(r_{0j,k}) | 53.33 mΩ | 48.90 mΩ | 49.00 mΩ | 51.33 mΩ | 50.10 mΩ | 47.75 mΩ | 48.22 mΩ | |

E(r_{0j,k}) change | 39.77% | 41.69% | 40.36% | 40.93% | 40.58% | 40.70% | 40.75% | |

SD(r_{0j,k}) | - | 0.00512 | 0.00513 | 0.00357 | 0.00308 | 0.00343 | 0.00352 |

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**MDPI and ACS Style**

Chang, F.; Roemer, F.; Baumann, M.; Lienkamp, M.
Modelling and Evaluation of Battery Packs with Different Numbers of Paralleled Cells. *World Electr. Veh. J.* **2018**, *9*, 8.
https://doi.org/10.3390/wevj9010008

**AMA Style**

Chang F, Roemer F, Baumann M, Lienkamp M.
Modelling and Evaluation of Battery Packs with Different Numbers of Paralleled Cells. *World Electric Vehicle Journal*. 2018; 9(1):8.
https://doi.org/10.3390/wevj9010008

**Chicago/Turabian Style**

Chang, Fengqi, Felix Roemer, Michael Baumann, and Markus Lienkamp.
2018. "Modelling and Evaluation of Battery Packs with Different Numbers of Paralleled Cells" *World Electric Vehicle Journal* 9, no. 1: 8.
https://doi.org/10.3390/wevj9010008