## 1 Introduction
Conversely, when the batteries are connected in series and sealed into a battery pack, uneven temperature distribution will accelerate aging of active materials locally among the batteries, leading to a drop in capacity [12; 20]. This will finally lead to capacity imbalance of battery units in the battery pack, causing malfunction [22]. From the above description, it can be seen that environmental temperature has a large influence on battery aging [23]. The uneven heating produced from a high rate of discharge will also lead to aging of some of the internal materials in the battery, speeding up the aging process [24]. Thus, only when these causes are considered will the batteries operate safely and efficiently. State of charge (SoC) is defined as the available capacity and expressed as a percentage of the battery's rated capacity. SoC estimation based on the ampere-hour method is calculated using  
\[SoC_{(t)}=SoC_{(t_{0})}-\frac{1}{C_{N(t)}}\int_{t_{0}}^{t}(\eta_{c}\cdot i(t)-S _{d})dt, \tag{1}\]  
where \(SoC_{(t)}\) is the SoC at time t; \(SoC_{(t_{0})}\) is the initial value; \(C_{N}\) is the rated capacity; \(\eta_{c}\) is the coulombic efficiency; \(S_{d}\) is the self-discharge rate; and \(i(t)\) is the current, which is positive during the discharging process and negative during the charging process. In Equation (1), coulombic efficiency (CE) can be used to estimate the relationship between SoC and cycle life [25; 26; 27]. For LiFePO\({}_{4}\) batteries under room temperature, CE > 0.994 and the self-discharge rate is lower than 5% per month [10]. If temperature and self-discharge are not considered, the equation can be simplified by assuming \(\eta_{c}=1\) and \(S_{d}=0\). However, temperature change strongly affects CE. When used under low temperature environments and for some high power applications such as electric vehicles, there will be safety issues. In addition, the capacity will fade as the number of cycles increase, causing considerable errors in SoC estimation. In order to correct the measurement errors of the battery, the work in [28] proposed practical state-of-charge. This definition uses a practical operational capacity, instead of the manufacturer's rated capacity, as the maximum amount of charge. It can be expressed as  
\[SoC_{N}=\frac{Q_{available}}{C_{max,p}}, \tag{2}\]  
where \(C_{max,p}\), represents the maximum practical capacity as measured from the operating battery at the current time. \(C_{max,p}\) may fade over time, due to the effect of battery aging. During the charging process of lithium batteries, a SEI is formed, leading to loss of active lithium ions and lowering of capacity. State of health (SoH) is another important method of estimation for battery health management. It is the direct indication of the health condition of the battery system. A battery's SoH normally ranges within 0-100%, but when it is new, SoH can be slightly larger than 100% due to product variations [29]. In this paper, to avoid confusion, the \(SoH_{N}\) of the battery refers to the ratio of battery discharge capacity to new battery rated capacity under certain conditions. \(SoH_{N}\) can be expressed as  
\[SoH_{N}=\frac{C_{max,p}}{C_{N}}. \tag{3}\]  
Battery models describe the charging and discharging behaviors of batteries and information such as calculated capacity, state of health, etc. They can be used for preventing overcharging or over discharging. SoC and SoH are both indispensable methods for battery models. The battery models are categorized into electrochemical models and equivalent circuit models [30]. Electrochemical models use equations of physical phenomena and micro-structure of materials to predict battery performance [31; 32; 33]. They are often used to study single specific batteries, analyze the phase changes of battery materials, study the influence from porous structure of battery electrode tabs [34], and optimize the tab porosity rate and tab thickness [35]. Simultaneously, electro-chemical models are also used in improving battery design and manufacturing.