# Lifetime Prognosis of Lithium-Ion Batteries Through Novel Accelerated Degradation Measurements and a Combined Gamma Process and Monte Carlo Method

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## Abstract

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## Featured Applications

**(1) Prognostics and health management of Li-ion batteries; (2) Lifetime evaluation using a dual-stress accelerated test; (3) Degradation modeling by a linear stochastic process with parameter random sampling technique.**

## Abstract

_{4}batteries using a novel dual dynamic stress accelerated degradation test, called D

^{2}SADT, to simulate a situation when driving an electric vehicle in the city. The Norris and Landzberg reliability model was applied to estimate activation energy of the test batteries. The test results show that the battery capacity always decreased at each measurement time-step during D

^{2}SADT to enable the novel test method. The variation of the activation energies for the test batteries indicate that the capacity loss of the test battery operated under certain power and temperature cycling conditions, which can be accelerated when the charge–discharge cycles increase. Moreover, the modeling results show that the gamma process combined with Monte Carlo simulations provides superior accuracy for predicting the lifetimes of the test batteries compared with the baseline lifetime data (i.e., real degradation route and lifetimes). The results presented high prediction quality for the proposed model as the error rates were within 5% and were obtained for all test batteries after a certain quantity of capacity loss, and remained so for at least three predictions.

## 1. Introduction

^{2}SADT) where the test battery cells were continually processed through full charging–discharging cycles through four-type-temperature-composition cycling [14]. The acceleration was defined by the ratio of the strictly operating battery to the normal operating battery in terms of the DoD ratio in these two situations. For instance, a battery pack of a plug-in hybrid EV is charged twice daily at a 50% DoD each time, once at work and once at home. If the battery pack is charged twice daily at 100% DoD each time, then the acceleration is two. Moreover, a battery pack may be operated at various temperatures when it is charged or discharged. In this study, to simplify the testing, they were represented by a four-type composition, namely one constant high, one constant low, one ramp-up, and one ramp-down temperature.

## 2. Method

#### 2.1. Physical-Based Reliability Model

_{u}is the cycles to failure in normal use conditions (h), N

_{a}is the cycles to failure in accelerated test conditions (h), ΔT

_{a}is temperature range of TC in accelerated test conditions (K), ΔT

_{u}is temperature range of TC in normal use conditions (K), f

_{u}is cycle frequency of TC in normal use conditions, f

_{a}is cycle frequency of TC in accelerated test conditions (24 h/cycle time), n and m are exponents, and $\phi \left({T}_{max}\right)$ is a function associated with the effect of maximum temperature in TC, and it is expressed particularly by the Arrhenius equation, shown below.

_{a}is activation energy (ev or kJ/mole), k is Boltzmann constant 8.625 × 10

^{−5}eV/K, ${T}_{u}^{max}$ is the maximum temperature in normal use conditions (K), ${T}_{a}^{max}$ is the maximum temperature in accelerated test conditions (K). In this study, two temperature cycling test conditions named TC1 and TC2 were setup. They represent the difference from the maximum temperature, but the other parameters, temperature range and cycle frequency, remain the same. Equation (1) was simplified to be

#### 2.2. Gamma Process Model

- y(0) = 0
- the increments ∆y(t) = y(t + $\tau $) − y(t) are independent
- ∆y(t) has a gamma distribution G(α$\tau $, β), with the probability density function (PDF) defined by

#### 2.3. Monte Carlo Simulation

## 3. Experiment

^{2}SADT that involved a battery test subsystem, a thermal chamber for TC, a data acquisition card, and a PC with software to control the tests. The test batteries were well-housed in the chamber for a long-term reliability test. The battery test subsystem was able to program various currents and time to carry out the dynamic charging–discharging test through cycling which it pumps and draws various currents to and from the cell, as well as monitor its terminal voltage and current demands. The software was going to control the main test conditions between the thermal chamber and the battery test subsystem, enabling the batteries to be tested by various temperatures and charging–discharging currents simultaneously—the so-called dual dynamic stress. Thus, the test method was realized to simulate certain driving conditions for Li-ion batteries used in EVs.

^{2}SADT. For an entire temperature cycle, it consisted of two dwell times and two ramp times, where three DSTs operated in a dwell time period and two DSTs operated in a ramp time period so that total ten DSTs operated in one temperature cycle. The Dynamic Stress Test (DST) is a variable power discharge test which is scaled to a percentage of the maximum rated power of the test vehicles, and requires regeneration levels for the general urban driving schedule. Table 2 shows the test parameters of the temperature cycling. Two temperature cycling tests called TC1 and TC2 were planed and performed to satisfy the N-L equation for evaluating the activation energy of the test battery. Figure 3 and Table 3 presents the DST profiles and the test procedures respectively. It is noted that the total test time of one DST was 360 s, but the cycles were repeated end-to-end with no time delay between them. The test battery was discharged using the DST power profile until to the cutoff condition was reached or other criteria emerged, and then was fully charged immediately. Compared to the life cycle test in standard IEC 62660-1 [25], where the main characteristics of the test profile is charge depleting, the test cycle (DST) used in this study was more focused on the variable power discharging regime to effectively simulate dynamic discharging in city traffic and highway ones [26,27]. Figure 4 shows a timing diagram of a reference power test (RPT). It was performed using the same discharging rate currents under an ambient temperature of 25 °C each time to calculate the capacity of the test battery when five temperature cycles were completed.

## 4. Results and Discussion

^{2}SADT. The capacity loss of the test battery continuously increased under both TC1 and TC2 conditions. It can be normalized by

_{4}battery test. It is noted that the activation energy continuously decreased when the capacity loss increased overtime. The lower activation energy represented a higher reaction rate inside the battery, corresponding to speed up Lithium-ion diffusion in electrode materials. From the point of view of long-term reliability, the capacity loss of the test battery operating under certain power and temperature cycling conditions can be accelerated when the charge–discharge cycles increase.

## 5. Conclusions

_{4}) battery using the novel dual dynamic stress accelerated degradation test (D

^{2}SADT). The Norris and Landzberg reliability model was applied to estimate activation energy which indicates that the capacity loss of the test battery operated under certain power and temperature cycling conditions, which can be accelerated when the charge–discharge cycles increase. The combined gamma process and Monte Carlo simulation approach not only provides superior prediction accuracy but also achieves high quality in prognosis. In summary, the proposed model and test method could help engineers not only understand the degradation behavior according to the indicator of activation energy, but also enable monitoring of the health states of Li-ion batteries more precisely in certain real conditions. This study provides a useful and basic reference to indicate that a non-linear stochastic model can be developed in advance to predict the lifetimes of Li-ion batteries operated in more complex conditions, such as a city, highway, and off-road driving patterns.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 8.**Lifetime prediction of the last TC for two test batteries in different temperature cycles.

Cell Brand and Series # | Nominal Voltage | Geometry | Weight | Nominal Capacity | Dimensions | Operation Temperature |
---|---|---|---|---|---|---|

A123 ANR26650 | 3.3 V | Cylinder | 76 g | 2300 mAh | 25.85 mm × 65.2 mm | −30 °C ~ 55 °C |

Test Parameters | TC1 | TC2 |
---|---|---|

T_{max}, °C | 55 | 60 |

T_{min}, °C | −15 | −10 |

Temperature Range ΔT, °C | 70 | 70 |

Dwell Time, h | 35 | 35 |

Ramp Time, h | 25 | 25 |

Ramp Rate, °C/h | 2.76 | 2.76 |

Cycle Duration, day | 5 | 5 |

Frequency of Usage, cycles/day | 0.2 | 0.2 |

Step | Time (s) | % | Step | Time (s) | % |
---|---|---|---|---|---|

1 | 16 | 0 | 11 | 12 | −25 |

2 | 28 | −12.5 | 12 | 8 | 12.5 |

3 | 12 | −25 | 13 | 16 | 0 |

4 | 8 | 12.5 | 14 | 36 | −12.5 |

5 | 16 | 0 | 15 | 8 | −100 |

6 | 24 | −12.5 | 16 | 24 | −62.5 |

7 | 12 | −25 | 17 | 8 | 25 |

8 | 8 | 12.5 | 18 | 32 | −25 |

9 | 16 | 0 | 19 | 8 | 50 |

10 | 24 | −12.5 | 20 | 44 | 0 |

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**MDPI and ACS Style**

Lin, Y.-C.; Chung, K.-J.
Lifetime Prognosis of Lithium-Ion Batteries Through Novel Accelerated Degradation Measurements and a Combined Gamma Process and Monte Carlo Method. *Appl. Sci.* **2019**, *9*, 559.
https://doi.org/10.3390/app9030559

**AMA Style**

Lin Y-C, Chung K-J.
Lifetime Prognosis of Lithium-Ion Batteries Through Novel Accelerated Degradation Measurements and a Combined Gamma Process and Monte Carlo Method. *Applied Sciences*. 2019; 9(3):559.
https://doi.org/10.3390/app9030559

**Chicago/Turabian Style**

Lin, Yu-Chang, and Kuan-Jung Chung.
2019. "Lifetime Prognosis of Lithium-Ion Batteries Through Novel Accelerated Degradation Measurements and a Combined Gamma Process and Monte Carlo Method" *Applied Sciences* 9, no. 3: 559.
https://doi.org/10.3390/app9030559