# A Multi-Particle Physics-Based Model of a Lithium-Ion Battery for Fast-Charging Control Application

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## Abstract

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## 1. Introduction

_{2}battery was used as the object to simulate the model. At the same time, conditions for the control of the precipitation of negative lithium in a lithium-ion battery in case of a high rate of charging are proposed. The simulation results show that a high simulation accuracy can be obtained from the SMP model when charging at a high rate.

## 2. Multi-Particle Battery Model

#### 2.1. P2D Model

#### 2.2. Battery Physical Modeling

_{i}, respectively. i = n is negative, i = p is positive and i = sep is separator. m is 0, 1/3, 2/3 and 1, respectively [8]. Next, taking the solution of negative electrode parameters in the model as an example, the order of P2D model will be gradually reduced.

#### 2.3. Quasi-Linearization Approximation

## 3. Potential Balance Equation Calculation

#### 3.1. Solid and Liquid Ohmic Potential

#### 3.2. Electromotive Force and Liquid Overpotential

_{2}battery is obtained by reference [10].

#### 3.3. Calculation of the Heterogeneous Pore Wall Flux

## 4. The Update of Ion Concentration

## 5. Battery Terminal Voltage

## 6. Model Simulation and Battery Experiment

_{2}battery (40 mAh). The negative electrode material of the battery was graphite. All the simulation experiments in this paper were programmed on MATLAB. The calculation flowchart of the SMP model is shown in Figure 3. The simulation parameters of the model are given in Table 3.

## 7. Results and Discussion

#### 7.1. Terminal Voltage Results

#### 7.2. Simulation Analysis of Lithium Precipitation

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Subramanian, V.R.; Diwakar, V.D.; Tapriyal, D. Efficient macro-micro scale coupled modeling of batteries. J. Electrochem. Soc.
**2005**, 152, A2002–A2008. [Google Scholar] [CrossRef] - Doyle, M. Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell. J. Electrochem. Soc.
**1993**, 140, 1526. [Google Scholar] [CrossRef] - Haran, B.S.; Popov, B.N.; White, R.E. Determination of the hydrogen diffusion coefficient in metal hydrides by impedance spectroscopy. J. Power Sources
**1998**, 75, 56–63. [Google Scholar] [CrossRef] - Hui, P. Lithium ion battery parameter identification strategy based on extended single particle model. Acta Phys. Sin.
**2018**, 67, 259–269. [Google Scholar] [CrossRef] - Majdabadi, M.M.; Farhad, S.; Farkhondeh, M.; Fraser, R.A.; Fowler, M. Simplified electrochemical multi-particle model for LiFePO4 cathodes in lithium-ion batteries. J. Power Sources
**2015**, 275, 633–643. [Google Scholar] [CrossRef] - Kemper, P.; Li, S.E.; Kum, D. Simplification of pseudo two dimensional battery model using dynamic profile of lithium concentration. J. Power Sources
**2015**, 286, 510–525. [Google Scholar] [CrossRef] - Jokar, A.; Rajabloo, B.; Désilets, M.; Lacroix, M. Review of simplified pseudo-two-dimensional models of lithium-ion batteries. J. Power Sources
**2016**, 327, 44–55. [Google Scholar] [CrossRef] - Li, X.Y. Physical Model Simplification and Parameter Identification for State Estimation of the Retired Lithium Ion Battery; Harbin Institute of Technology: Harbin, China, 2018. [Google Scholar]
- Forman, J.C.; Bashash, S.; Stein, J.L.; Fathy, H.K. Reduction of an electrochemistry-based li-ion battery model via quasi-linearization and padé approximation. J. Electrochem. Soc.
**2011**, 158, A93–A101. [Google Scholar] [CrossRef] - Han, X. Study on li-ion Battery Mechanism Model and State Estimation for Electric Vehicles; Tsinghua University: Beijing, China, 2014. [Google Scholar]

**Figure 4.**The terminal voltage fitting results under the four charging currents. (

**a**) Terminal voltage at 0.5C; (

**b**) Terminal voltage at 1C; (

**c**) Terminal voltage at 2C; (

**d**) Terminal voltage at 3C.

**Figure 6.**The heterogeneous side reaction overpotentials in the negative electrode. (

**a**) Overpotential at 0.5C; (

**b**) Overpotential at 1C; (

**c**) Overpotential at 2C; (

**d**) Overpotential at 3C.

Nomenclature | |||
---|---|---|---|

${\eta}_{\kappa}$ | Electrochemical reaction overpotential (V) | ${\varphi}_{s}$ | Solid phase potential (V) |

${R}_{f}$ | Gas constant (J/(mol·K)) | ${\varphi}_{e}^{ohm}$ | Liquid phase potential (V) |

$T$ | Temperature (K) | ${\varphi}_{e}^{con}$ | Liquid phase concentration |

$F$ | Faraday constant (C/mol) | ${U}_{s}^{surf}$ | Particle surface electromotive force (V) |

$j$ | Pore wall flux (mol·s^{−1}·m^{−3}) | ${\eta}_{f}$ | SEI film potential (V) |

${i}_{\kappa}$ | Exchange current density (A/m^{3}) | ${i}_{e}$ | Liquid phase current density (V) |

${j}^{avg}$ | Average hole wall flow (mol·s^{−1}·m^{−3}) | ${\sigma}^{eff}$ | Effective conductivity (S/m) |

${I}_{L}$ | Input current (A) | $\alpha $ | Lithium intercalation rate (%) |

${a}_{s}$ | Specific interfacial surface area (m^{−1}) | $C$ | Lithium ion concentration (mol/L) |

${A}_{cell}$ | Effective area of electrode (m^{2}) | $\beta $ | Activation coefficient |

$L$ | Electrode thickness (m) | $R$ | Particle radius (s) |

${R}_{f}$ | SEI film resistance (Ω) | ${R}_{ext}$ | Contact resistance (Ω) |

${q}^{avg}$ | Average concentration flux (mol/L) | ${D}_{s}$ | Solid diffusion coefficient(m^{2}/s) |

Equation Name | Governing Equations |
---|---|

Butler–Volmer equation | $\{\begin{array}{c}j={i}_{\kappa}\left(\mathrm{exp}\left(\frac{{\alpha}_{a}F}{RT}{\eta}_{\kappa}\right)-\mathrm{exp}\left(-\frac{{\alpha}_{c}F}{RT}{\eta}_{\kappa}\right)\right)\\ {i}_{\kappa}=\kappa \sqrt{{C}_{e}\left({C}_{s}^{\mathrm{max}}-{C}_{s}^{surf}\right){C}_{s}^{surf}}\end{array}$ |

Potential balance equation | ${\eta}_{\kappa}={\phi}_{s}-{\phi}_{e}^{ohm}-{\phi}_{e}^{con}-{U}_{s}^{surf}-{\eta}_{f}$ |

Ohmic effect | $\frac{\partial {\phi}_{s}}{\partial x}=-\frac{{i}_{s}}{{\sigma}_{s}^{eff}}$ $\frac{\partial {\phi}_{e}^{ohm}}{\partial x}=-\frac{{i}_{e}}{{\sigma}_{e}^{eff}}$ |

Liquid-phase diffusion equation | ${\phi}_{e}^{con}=\frac{2RT}{F}\left(1+\beta \right)\left(1-{t}_{+}^{0}\right)\frac{\partial \mathrm{ln}{C}_{e}}{\partial x}$ |

${\epsilon}_{e}\frac{\partial {C}_{e}}{\partial t}=\frac{\partial}{\partial x}\left({D}_{e}^{eff}\frac{\partial {C}_{e}}{\partial x}\right)+{a}_{s}\left(1-{t}_{+}^{0}\right)j$ | |

Solid-phase diffusion equation | $\frac{\partial {C}_{s}}{\partial t}=\frac{1}{{r}^{2}}\frac{\partial}{\partial r}\left({D}_{s}{r}^{2}\frac{\partial {C}_{s}}{\partial r}\right)$ |

Terminal voltage | ${V}_{cell}={\phi}_{s,p}-{\phi}_{s,n}-{R}_{ext}{I}_{L}$ |

Parameter | LiCoO_{2} Battery | ||
---|---|---|---|

PositiveElectrode | Separator | Negative Electrode | |

Electrode thickness L (m) | 55 × 10^{−6} | 20 × 10^{−6} | 70 × 10^{−6} |

Electrode plate area ${A}_{cell}$ (m^{2}) | 1.74 × 10^{−3} | / | 1.81 × 10^{−3} |

Transference number ${t}_{0}^{+}$ | 0.363 | ||

Effective conductivity in solid phase ${\sigma}_{s}^{eff}$ | 0.3055 | / | 40.7891 |

Maximum Li^{+} concentration in solid phase ${C}_{s}^{\mathrm{max}}$ | 56,000 | / | 26,390 |

Liquid volume fraction ${\epsilon}_{e}$ | 0.18 | 0.55 | 0.4 |

Specific interfacial surface area (m^{−1}) ${a}_{s}$ | 144,000 | / | 165,000 |

Particle radius Rs (m) | 15 × 10^{−6} | / | 10 × 10^{−6} |

Initial liquid Li+ concentration (mol/L) | 1200 | 1200 | 1200 |

Solid diffusion coefficient (m^{2}/s) ${D}_{s}$ | 2.2 × 10^{−14} | / | 5 × 10^{−14} |

Reaction rate constant(Am^{2.5}/m^{1.5}) $\kappa $ | 5.8 × 10^{−11} | / | 8.8 × 10^{−11} |

Liquid phase effective diffusion coefficient (m^{2}/s) ${D}_{e}^{eff}$ | 3.8184 × 10^{−11} | 2.0395 × 10^{−11} | 1.2649 × 10^{−11} |

Liquid phase effective conductivity (S/m) ${\kappa}_{e}^{eff}$ | 0.0816 | 0.0153 | 0.0506 |

Initial SOC $SO{C}_{i,0}$ | 0.85 | 0.02 | |

Temperature (K) $T$ | 298.15 | ||

Negative SEI film resistance (Ω) ${R}_{f}$ | 0.003 |

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**MDPI and ACS Style**

Li, X.; Hua, W.; Tian, J.; Tian, Y. A Multi-Particle Physics-Based Model of a Lithium-Ion Battery for Fast-Charging Control Application. *World Electr. Veh. J.* **2021**, *12*, 196.
https://doi.org/10.3390/wevj12040196

**AMA Style**

Li X, Hua W, Tian J, Tian Y. A Multi-Particle Physics-Based Model of a Lithium-Ion Battery for Fast-Charging Control Application. *World Electric Vehicle Journal*. 2021; 12(4):196.
https://doi.org/10.3390/wevj12040196

**Chicago/Turabian Style**

Li, Xiaoyu, Wen Hua, Jindong Tian, and Yong Tian. 2021. "A Multi-Particle Physics-Based Model of a Lithium-Ion Battery for Fast-Charging Control Application" *World Electric Vehicle Journal* 12, no. 4: 196.
https://doi.org/10.3390/wevj12040196