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(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.
A compositional prognostic-based assessment using the gamma process and Monte Carlo simulation was implemented to monitor the likelihood values of test Lithium-ion batteries on the failure threshold associated with capacity loss. The evaluation of capacity loss for the test LiFePO4 batteries using a novel dual dynamic stress accelerated degradation test, called D2SADT, 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 D2SADT 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.
The applications of lithium (Li)-ion batteries has continuously expanded because of their excellent energy density and long cycle lifetime, particularly with portable electronic devices (e.g., laptops, phones, camcorders, and cameras), hybrid or full electric vehicles (EVs), and satellites. The capacity loss has become the major concern for developing a long-driving-mileage and high-performance EV because degradation occurs when the cell is in charging–discharging cycles and various environmental conditions [1,2,3]. Some research indicates that the operation factors associated with depth of discharge (DoD), state of charge (SOC), temperature, and the charging–discharging rate are likely to reduce battery performance and energy degradation after hundreds or thousands of charging–discharging cycles [4,5]. The mechanism of degradation can be divided into chemical and mechanical degradation. The former is caused by the formation of solid electrolyte interface films, impeding deleterious degradation reactions within the cells; the latter is generated by the cyclic expansion, and contraction of insertion or alloy materials, leading to cell cracking, fatigue, and structural distortion [6,7,8,9].
Many studies have focused on predicting the lifetimes of Li-ion batteries. Ramadass et al.  developed a semi-empirical reliability model for predicting the capacity loss of Li-ion cells. Christensen and Newman  developed a model that is capable of simulating the effect of film resistance of anodes on the charging–discharging process for a Li-ion battery. Moreover, they combined the model into a galvanostatic charging–discharging model previously developed by Doyle et al. . Ning et al.  developed a generalized model associated with the first-principle-based charging–discharging operation to simulate the degradation of rechargeable Li-ion batteries. Although this research provides scientific-based models to predict the degradation of Li-ion batteries and some of them agree with experimental data, most of their experiments were performed under a static charging–discharging process, as well as a fixed temperature. They are not realistic situations since the battery charging–discharging currents and temperature differ at all times when the EV is in operational status. Thus, it is necessary to evaluate the capacity loss of Li-ion batteries in practical conditions in order to predict their lifetimes more precisely when used in EVs.
A novel accelerated degradation test involving dual dynamic stresses, charging–discharging currents, and temperatures was developed by the authors to simulate the real conditions of driving an EV in a metropolitan area (e.g., as a commuter car). The test was called the dual dynamic stress accelerated degradation test (D2SADT) where the test battery cells were continually processed through full charging–discharging cycles through four-type-temperature-composition cycling . 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.
The degradation modeling for highly reliable products such as Li-ion batteries and LEDs is based on two aspects: stochastic processes and general degradation path approaches [15,16,17]. The stochastic processes involving the Wiener and gamma processes which indicate that the degradation of a test sample follows a random process associated with independent increments [18,19,20]. One advantage of stochastic processes is that they can generate a continuous-and-steady state, enabling the prediction to be updated during each measurement to easily develop a data-based model for prognostics and health management (PHM) of Li-ion batteries. However, it is also to represent the limitation of lacking the physical-oriented indicator to infer the failure mechanism associated with degradation behavior of the Li-ion battery.
In 1969, Norris and Landzberg  developed a practical accelerated reliability model on the basis of temperature cycling (TC) test conditions associated with cycle frequency, temperature range, and the maximum temperature. This model is to improve the Coffin–Manson equation since it is difficult to directly measure the strain associated with the thermal effect. In this study, the N–L equation was used to compute a physical-based indicator, using activation energy to illustrate the progress of degradation for the test Li-ion battery during the dual dynamic stress accelerated degradation test period. Additionally, a steady-state gamma process was applied for modeling the degradation of batteries according to observations of monitored degradation data to date. Monte Carlo simulation with controllable uncertainty was applied to simulate all the degradation paths by using the parameters of the probability distribution associated with the gamma process. Lifetime prognoses were performed in different temperature cycles with the inclusion of an increasing number of measurements. Finally, the prediction errors were computed to evaluate the quality of the lifetime prognosis by the proposed model.
2.1. Physical-Based Reliability Model
Norris and Landzberg developed a temperature cycling-based accelerated reliability model in 1969. It is more practical than using the semi-experimental Coffin–Manson equation which has difficulties in measuring the true strain associated with the thermal effect. The N-L equation is shown below:
where AF is acceleration factor, Nu is the cycles to failure in normal use conditions (h), Na is the cycles to failure in accelerated test conditions (h), ΔTa is temperature range of TC in accelerated test conditions (K), ΔTu is temperature range of TC in normal use conditions (K), fu is cycle frequency of TC in normal use conditions, fa is cycle frequency of TC in accelerated test conditions (24 h/cycle time), n and m are exponents, and is a function associated with the effect of maximum temperature in TC, and it is expressed particularly by the Arrhenius equation, shown below.
where Ea is activation energy (ev or kJ/mole), k is Boltzmann constant 8.625 × 10−5 eV/K, is the maximum temperature in normal use conditions (K), 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