# 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