# Modelling Lithium-Ion Battery Ageing in Electric Vehicle Applications—Calendar and Cycling Ageing Combination Effects

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## Abstract

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## 1. Introduction

## 2. Battery Ageing

#### 2.1. Main Ageing Mechanisms in Lithium-Ion Batteries

#### 2.2. Modelling Approaches

## 3. Experiments

#### 3.1. Experimental Setup

#### 3.2. Experimental Results

## 4. Calendar Ageing Model

#### 4.1. Model Formulation

#### 4.2. Parameter Identification

## 5. Combined Ageing Model

#### 5.1. Model Formulation

- (i)
- Mass is non-negative, then capacities must be non-negative:$$\begin{array}{cc}\hfill Q\left(t\right),\phantom{\rule{3.33333pt}{0ex}}{Q}_{F,rev}\left(t\right),\phantom{\rule{3.33333pt}{0ex}}{Q}_{F}\left(t\right)& \ge 0\hfill \end{array}$$
- (ii)
- Conservation of mass: the sum of capacities is constant and equal to initial capacity ${Q}^{0}$:$$\begin{array}{cc}\hfill Q\left(t\right)+{Q}_{F,rev}\left(t\right)+{Q}_{F}\left(t\right)& ={Q}^{0}.\hfill \end{array}$$

#### 5.2. Parameter Identification

- (i)
- establish a new parameter set: $x=(\lambda ,\phantom{\rule{3.33333pt}{0ex}}{k}_{irr},\phantom{\rule{3.33333pt}{0ex}}{k}_{s})$
- (ii)
- calculate ${Q}_{F,rev}^{eq}\left(SoC\right)$ (Equation (29))
- (iii)
- run simulation on each cycling profiles to obtain ${Q}_{F,sim}$ (Equations (19) and (20))
- (iv)
- calculate the mean absolute error, $f\left(x\right)$ (Equation (30))
- (v)
- while a minimum is not found, go to step (i)

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

LFP/C | Lithium Iron Phosphate/graphite |

NMC/C | Nickel Manganese Cobalt/graphite |

ODE | Ordinary Differential Equation |

RPT | Reference Performance Tests |

SEI | Solid Electrolyte Interface |

SoC | State of Charge |

## References

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**Figure 2.**Capacity fade under different calendar/cycling ageing conditions. Relative values of capacity fade are expressed in p.u. which means “per unit”.

**Figure 3.**Calendar ageing modelling results: natural logarithm of acceleration coefficient (${C}_{a}$) versus $SoC$. The blue circles are the experimental points, the lines are the result of model 1, 2 and 3 (Equations (9), (10) and (11) respectively). For model 2: $z=5$. For model 3: $a=0.7,b=10$.

**Figure 5.**Capacity fade simulations compared to measurements under different ageing conditions. Simulations are plotted by using lines. Measurements are plotted with markers (the legend for these points is in Figure 2).

**Figure 6.**Simulation results for cycling profiles between SoC 100 and 80% with current rates C/2. All profiles have the same weekly charge throughput: 1.4 p.u., that is, seven times 0.2 p.u. Profiles 1 and 2 have the same average SoC (98%) as profiles 3 and 4 (82%).

**Table 1.**Calendar ageing model results: absolute mean errors and maximum errors of ${C}_{a}$ for different stress formulations.

${\mathit{C}}_{\mathit{a}}$ | abs. Mean Error/Max. Error (%) | Identified Parameters | Fixed Parameters | Comments |
---|---|---|---|---|

${A}^{\prime}\xb7exp(B\xb7{SoC}^{z})$ | 7.1/22.5 | ${A}^{\prime},B$ | z = 1 | model 1 (Equation (9)) |

6.2/22.1 | z = 2 | |||

5.0/17.1 | z = 3 | |||

4.1/13.1 | z = 4 | |||

4.0/11.4 | z = 5 | model 2 (Equation (10)) | ||

4.4/13.2 | z = 6 | |||

4.9/14.4 | z = 7 | |||

5.4/15.0 | z = 8 | |||

5.8/15.2 | z = 9 | |||

6.1/15.6 | z = 10 | |||

${A}^{\prime}\xb7exp({B}_{1}\xb7\sqrt{SoC}+{B}_{2}\xb7SoC)$ | 3.7/11.4 | ${A}^{\prime},{B}_{1},{B}_{2}$ | ||

${A}^{\prime}\xb7exp({B}_{1}\xb7SoC+{B}_{2}\xb7So{C}^{2})$ | 3.8/12.2 | ${A}^{\prime},{B}_{1},{B}_{2}$ | ||

${A}^{\prime}\xb7exp({B}_{1}\xb7SoC+{B}_{2}\xb7So{C}^{2}+{B}_{3}\xb7So{C}^{3})$ | 3.5/11.9 | ${A}^{\prime},{B}_{1},{B}_{2},{B}_{3}$ | ||

${A}^{\prime}\xb7exp(B\xb7{f}_{re}\left(SoC\right))$ | 3.5/11.7 | ${A}^{\prime},B$ | a = 0.7, b = 10 | model 3 (Equation (11)) |

Parameter | Initial Value (${\mathit{x}}_{0}$) | Upper Bound (${\mathit{x}}_{\mathit{max}}$) | Lower Bound (${\mathit{x}}_{\mathit{min}}$) | Obtained Value (${\mathit{x}}_{\mathit{end}}$) |
---|---|---|---|---|

$\lambda $ | 10 | 20 | 0.1 | 7.41 |

${k}_{irr}$ | 0.1 | 0.5 | 0.0001 | 0.0547 |

${k}_{s}$ | 0.1 | 0.5 | 0.0001 | 0.0548 |

(a) Model equations. | ||

Main equations (ODE system): | ||

$\frac{d{Q}_{F,rev}\left(t\right)}{dt}=\lambda \xb7({Q}_{F,rev}^{eq}-{Q}_{F,rev}\left(t\right))+{k}_{s}\xb7I\left(t\right)$ | Equation (19) | |

$\frac{d{Q}_{F}\left(t\right)}{dt}=\lambda \xb7{k}_{irr}\xb7{Q}_{F,rev}\left(t\right)$ | Equation (20) | |

Auxiliary equations: | ||

$SoC\left(t\right)=So{C}^{0}+\frac{\int I\left(t\right)\xb7dt}{Q\left(t\right)}$ | t in days | |

I in p.u./day | ||

${f}_{re}\left(SoC\right)=a+\frac{(SoC-a)}{(1+{e}^{-b(SoC-a)})}$ | Equation (12) | |

${C}_{a}\left(SoC\right)={A}^{\prime}\xb7{e}^{\left(B{f}_{re}\left(SoC\right)\right)}$ | Equation (11) | |

${Q}_{F,rev}^{eq}\left(SoC\right)=\frac{{C}_{a}\left(SoC\right)}{(\lambda \xb7{k}_{irr})}$ | Equation (29) | |

$Q\left(t\right)={Q}^{0}-{Q}_{F,rev}\left(t\right)-{Q}_{F}\left(t\right)$ | from Equation (26) | |

(b) Model parameters. | ||

Parameter | Units | Value |

${A}^{\prime}$ | p.u./day | 8.8765 × 10${}^{-5}$ |

B | (no units) | 3.2162 |

a | (no units) | 0.7 |

b | (no units) | 10 |

$\lambda $ | day${}^{-1}$ | 7.41 |

${k}_{irr}$ | (no units) | 0.0547 |

${k}_{s}$ | (no units) | 0.0548 |

**Table 4.**Simulation results for different use profiles. Underlined values are showing the differences between each four profile group (profiles 5 to 8, 9 to 12 and 13 to 16) respect to the original profiles (1 to 4).

Profile Number | ${\mathit{SoC}}_{\mathit{max}}$ | ${\mathit{SoC}}_{\mathit{min}}$ | ${\mathit{SoC}}_{\mathit{avg}}$ | I | Weekly Charge Throughput | ${\mathit{Q}}_{\mathit{F}}$ after 70 Days | Comments |
---|---|---|---|---|---|---|---|

p.u. | p.u. | p.u. | C | p.u. | % | ||

1 | 1 | 0.8 | 0.98 | 0.5 | 1.4 | 19.62 | daily SOC100-80, rest 100 |

2 | 16.89 | monday SOC100-80x7, rest 100 | |||||

3 | 0.82 | 12.03 | daily SOC80-100, rest 80 | ||||

4 | 12.08 | monday SOC80-100x7, rest 80 | |||||

5 | 1 | 0.6 | 0.98 | 0.5 | 2.8 | 26.51 | daily SOC100-60, rest 100 |

6 | 23.44 | monday SOC100-60x7, rest 100 | |||||

7 | 0.62 | 11.31 | daily SOC60-100, rest 60 | ||||

8 | 11.35 | monday SOC60-100x7, rest 60 | |||||

9 | 1 | 0.8 | 0.98 | 0.2 | 1.4 | 19.36 | daily SOC100-80 (I = C/5), rest 100 |

10 | 16.54 | monday SOC100-80x7 (I = C/5), rest 100 | |||||

11 | 0.82 | 11.64 | daily SOC80-100 (I = C/5), rest 80 | ||||

12 | 11.71 | monday SOC80-100x7 (I = C/5), rest 80 | |||||

13 | 0.8 | 0.6 | 0.78 | 0.5 | 1.4 | 13.18 | daily SOC80-60, rest 80 |

14 | 10.25 | monday SOC80-60x7, rest 80 | |||||

15 | 0.62 | 10.17 | daily SOC60-80, rest 60 | ||||

16 | 10.12 | monday SOC60-80x7, rest 60 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Redondo-Iglesias, E.; Venet, P.; Pelissier, S. Modelling Lithium-Ion Battery Ageing in Electric Vehicle Applications—Calendar and Cycling Ageing Combination Effects. *Batteries* **2020**, *6*, 14.
https://doi.org/10.3390/batteries6010014

**AMA Style**

Redondo-Iglesias E, Venet P, Pelissier S. Modelling Lithium-Ion Battery Ageing in Electric Vehicle Applications—Calendar and Cycling Ageing Combination Effects. *Batteries*. 2020; 6(1):14.
https://doi.org/10.3390/batteries6010014

**Chicago/Turabian Style**

Redondo-Iglesias, Eduardo, Pascal Venet, and Serge Pelissier. 2020. "Modelling Lithium-Ion Battery Ageing in Electric Vehicle Applications—Calendar and Cycling Ageing Combination Effects" *Batteries* 6, no. 1: 14.
https://doi.org/10.3390/batteries6010014