## 1 Introduction  
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. [PERSON] et al. [10] developed a semi-empirical reliability model for predicting the capacity loss of Li-ion cells. [PERSON] [11] 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 [PERSON] et al. [12]. [PERSON] et al. [13] 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 (D2 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. 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].