# A Regression-Based Technique for Capacity Estimation of Lithium-Ion Batteries
## 1 Introduction
The most general and common method, which is used in the laboratory to estimate SOH, is coulomb counting (CC). At first, the fully charged battery is discharged to the 0% state of charge defined by the manufacturer in particular conditions, and the whole ampere-hour, which is discharged, is being measured. Then SOH is calculated by dividing discharged ampere-hour value by the initial capacity ([PERSON], 2018). As mentioned above, this method is only used to collect data in the laboratory. Existing capacity estimation methods and SOH estimation could be categorized into three groups. The first group includes the physics-based methods that use partial differential equations to model battery electrochemical properties and thermal behavior. Even though these methods have high accuracy, they are not suitable for online use because of the high calculation load ([PERSON], 2018). The second group includes data-driven models that estimate battery capacity by analyzing such characteristics as voltage and extracting some unique features. This method is strongly correlated to battery ageing without any mathematical model. In the ICA method, for instance, the IC curve is obtained by the following equation:  
\[IC=\frac{dQ}{dV}\approx\frac{\Delta Q}{\Delta V} \tag{2}\]  
Plateaus observed in the voltage curve are transformed to characteristics as identifiable peaks. Some particular features of these peaks, such as height, area, and position, are directly related to battery capacity and make the battery state of health estimation simple [12]. Differential voltage analysis (DVA) is another method of this group. The differential voltage (DV) curve is defined as voltage gradient divided by actual capacity:  
\[\frac{dV}{dQ}=\frac{dV}{\int 1 dt}=\frac{dV}{1 dt} \tag{3}\]  
Then SOH is analytically calculated by making a relationship between DV and SO. characteristics [13]. However, despite their advantages, these methods face fundamental challenges that limit their application. For example, IC and DV curves are subjected to noise and should be smoothed before using for estimation. The smoothing process needs some computational effort. Temperature variation affects their accuracy and makes capacity prediction infected with error. In addition, charging and discharging characteristics used to extract IC and DV curves are different from the actual usage profile. Other data-driven methods exist that use algorithms such as fuzzy logic and neural network. Experimental data of complicated and nonlinear systems can be processed by a fuzzy rule set of the fuzzy logic (FL) method. There are two sets of data: crisp and fuzzy. When data are divided by definite values, they belong to a crisp group, and if they are categorized by uncertainty, they belong to the fuzzy group. Membership functions determine members of fuzzy sets. The more the appropriate membership function is defined, the better prediction of SOH will be obtained [14; 15]. In [15], a relationship between several cycles and the normalized value of battery capacity is received by an exponential curve-fitting on NASA data as follows:  
\[y_{fit}=a_{0}+a_{1}e^{-(\frac{x}{a_{1}})\beta_{1}}+a_{2}e^{-(\frac{x}{a_{2}}) \beta_{2}} \tag{4}\]  
It uses the local estimation form of \(y=ae^{-\beta x}\) to reduce the number of parameters. Then it applies the impact of parameters such as temperature, current, and DOD by three membership functions of fuzzy logic that modify \(a\) and \(\beta\) factors. A disadvantage of this model is that a significant number of data are needed to define membership functions, and thus, many laboratory tests are necessary to obtain enough data in particular operating conditions. A neural network (NN) is a powerful tool that can manage a significant amount of data in a highly complex nonlinear system. One of the advantages of a NN is that it is not necessary to have detailed knowledge about battery characteristics. A robust algorithm that can accurately predict SOH in different working conditions is another benefit of NN [16].