## 3 Problem Analysis  
Through the above review, it is not difficult to find that lithium battery RUL prediction based on data-driven methods has made great progress in recent years. However, there is still a lack of theoretical research and need for further study. Accordingly, several problems with RUL prediction for lithium-ion batteries used in electric cars will be specifically analyzed as follows:  
1. Degradation modeling and RUL prediction methods for vehicle lithium-ion batteries with time-varying ambient temperature  
Among all the environmental factors, the temperature has a great influence on the performance of the lithium battery. The lithium battery degradation rate is greatly influenced by the ambient environmental temperature [56; 57; 58; 59]. The lithium battery may burn as the car starts in a below-freezing environment and as battery capacity rapidly fades under high-temperature conditions [60]. However, electric vehicles work outdoors, and its lithium-ion batteries were often influenced by time-varying ambient temperature. Firstly, the temperature is different between day and night, or among four seasons. Secondly, the lithium battery in the charging and discharging process generated a lot of heat, which causes the lithium battery environment temperature to change, and the difference is very big. The existing methods of degradation modeling and the prediction of RUL are mostly based on the laboratory temperature, and the ambient temperature stress level is constant. These studies [58; 61] considered the ambient temperature changes. However, they only aimed at several groups of constant temperature and assumed that a working environment temperature will remain constant. These studies ignored the variability of subsequent working environment temperature. All of these factors resulted in limitations of the RUL algorithm. Therefore, degradation modeling research and RUL prediction methods for lithium-ion batteries with time-varying ambient temperature problem are of important theoretical significance and practical value. Taking into account that the time-varying temperature, we can adopt a multi-state continuous-time Markov model process to describe a transition temperature. Considering time-varying temperature affects the rate of degradation, we can construct a class of degradation models influenced by varying random ambient temperature. 1. Degradation modeling and RUL prediction methods for vehicle lithium-ion batteries with a random variable current  
According to the vehicle behavior, the vehicle acceleration and deceleration process has randomness [4], which produces the lithium battery output current corresponding random changes. The degradation rate is different when the battery discharges at a different current [5]. For example, [PERSON] _et al._[62] tested a capacity fade of 18,650-type lithium-ion batteries cycled with different discharges and found that at 2C discharge rate, the capacity decays were 18.8% of the initial capacity after 300 cycles. And at 1C and 0.5 discharge rate, the capacity decays were only 14.2% and 10.5%, respectively. The reason is that the surface thickness of anode particles and the number of lithium-ions changes with different discharge. Now, existing research on lithium battery degradation modeling and RUL is aimed at fixed loads that build a relationship model of cycles and degradation through the constant current discharge to predict the RUL. Although studies [38; 48; 49; 51] have considered several groups of lithium battery RUL prediction problems under different constant discharge currents, they did not consider the random variation of the current under actual operation process. Random-variable current affects battery degradation rate, which results in the fact that lithium battery degradation function is a nonlinear time-varying function influenced by random effects. Therefore, random-variable current RUL prediction is an enormous challenge [59]. Searching for current rules and the RUL influence becomes the key to this problem.