# A Regression-Based Technique for Capacity Estimation of Lithium-Ion Batteries
## 1 Introduction  
The availability of new energy sources to different countries is fundamental for their economic development [1]. It is well known that petrol and diesel fuel vehicles lead to a significant amount of greenhouse gasses such as CO\({}_{2}\), which has severe and dangerous effects on the environment cause global warming [2]. Currently, lithium-ion batteries for electric vehicles (EVs) and hybrid electric vehicles (HEVs) are the most promising propulsion alternative to internal combustion engines (ICEs) due to not producing carbon dioxide and sound pollution. Thus, in the not-too-distant future, we will essentially see the replacement of many ICE vehicles with EVs and HEs and this replacement process has already started. The central part of these vehicles is a rechargeable battery that must have reasonable and acceptable efficiency [3]. Lithium-ion batteries have many benefits in comparison to other types of batteries, such as higher energy capacity, longer functional life, and less self-discharge in comparison with different kinds of rechargeable batteries, such as Ni-Cd batteries, which makes them attractive for many different applications, such as portable electronic devices, electric vehicles, and stationary storage [4, 5]. Therefore, this battery is attracting the attention of the electric vehicle industry [5]. A lithium-ion battery pack consists of a considerable number of battery cells, which are connected in series and parallel. Despite the above-mentioned advantages, there are also some challenges, such as costs, cycling stability, and safe operation. Parameters suchas charge and discharge rate, temperature, and voltage range will influence the safety of the battery. If critical limits of these parameters are exceeded, this will accelerate the degradation, and thus reduce the battery performance, and may even become dangerous due to the increased risk of a thermal runaway. Furthermore, it is crucial to have an exact measurement of lithium-ion battery total capacity and to provide the ability to predict the remaining useful life (RUL) over the entire service duration in order to ensure the trusted and safe operation of these batteries ([PERSON], 2018). This RUL prediction ability represents an essential part of the battery management system (BMS), which is needed to keep the battery operational in a safe, trusted, and reliable state under all conditions. This unit includes complex electronic circuitry and dedicated algorithms, which control the battery function ([PERSON], 2018). Batteries age through regular use and during storage and transport with (simultaneous) exposure to temperature. For example, changes in the crystal chemistry of the active materials occur due to frequent cycling combined with cracking of the particles and formation of a new solid-state electrolyte interface SEI (consumption of electrolyte and lithium). Two parameters show the ageing status of a battery. The first is the battery's total capacity, which is given by \(Q\), and the second is equivalent internal resistance \(R_{e}\). When \(Q\) has decreased to 80% of the initial (or nominal) capacity in an electric car, it is recommended to replace the battery because it cannot satisfy the high requirements for such applications. However, it could still be used in a second life application. Such an application could be, e.g., a stationary battery for peak shaving, which needs a lower depth of discharge (DOD) and less power. Similarly, if \(R_{e}\) grows to 160% of its initial value, the battery in an EV needs to be replaced as well ([PERSON], 2018; [PERSON], 2018). The state of health (SOH) is defined as the ratio of the currently measured maximum battery charge to its nominal capacity. The following equation shows how the SOH parameter is expressed as a percentage:  
\[SOH=\frac{Q_{act}}{Q_{nom}}*100 \tag{1}\]  
in which \(Q_{act}\) and \(Q_{nom}\) are the total capacity and the nominal capacity, respectively ([PERSON], 2018).