## 2 Methodology
### Battery Degradation Quantification  
Probably the most common way to quantify the state of health of a cell is its capacity, expressed in Ah, which is directly related to the amount of charge it can hold. It is defined as  
\[C=\int_{t_{0}}^{t_{f}}i(t)dt\;[\text{Ah}] \tag{1}\]Note that (1) corresponds to the Coulomb counting method, which integrates the current over time over one charge cycle, where the cell is completely discharged at \(t_{0}\) and fully charged at \(t_{f}\). The degradation or fading of the cell capacity in cycle \(n\), \(\Delta C_{n}\), can be calculated as  
\[\Delta C_{n}=C_{rated}-C_{n}\ [\text{Ah}] \tag{2}\]  
where \(n\) = 1, 2, 3, \(\ldots\), \(n_{\text{max}}\) is the number of cycles and \(n\) = 1 corresponds to the initial cycle of the battery where \(C_{n\ =\ 1}\) = \(C_{\text{rated}}\).  
The SOC of a battery is related to the remaining charge \(Q_{remaining}\)[C] in the battery; so, it is often defined as the ratio of \(Q_{remaining}\) to the rated charge \(Q_{rated}\) as [5]  
\[SOC(t)=\frac{Q_{remaining}(t)}{Q_{rated}(t)}\ [-] \tag{3}\]  
The SOC can be calculated using the Coulomb counting method, which integrates the cell current over time during the usage period. The SOC at any time \(t\) of the discharging phase can be calculated as  
\[SOC(t)=SOC_{0}-\frac{1}{C_{rated}}\int_{t_{0}}^{t}i(t)dt\ [-] \tag{4}\]  
where SOC\({}_{0}\) is the state of charge at the initial time of \(t_{0}\); \(C_{rated}\) [Ah] is the rated capacity of the cell, which is obtained from the cell manufacturer; and \(i(t)\) is the instantaneous cell current. The current \(i(t)\) in (4) is positive when discharging and negative when charging.  
Although the SOH has different definitions, it is related to the ratio of the effective cell capacity to the initial capacity. One possible definition is based on the ratio of the difference between the actual cell capacity \(C_{actual}\) and the end-of-life capacity \(C_{EOL}\) (70% of \(C_{initial}\)) [31] and the difference between the initial capacity \(C_{initial}\) and the end-of-life capacity \(C_{EOL}\)[10], which can be expressed as  
\[SOH=\frac{C_{actual}-C_{EOL}}{C_{initial}-C_{EOL}}\ [-] \tag{5}\]  
However, in practical applications, it is difficult to measure the actual capacity value of the cell because it is necessary to integrate (1) from a fully discharged state to a fully charged state. In this paper, it is proposed to use an indirect method to estimate the current capacity of the battery. This method is based on the calculation of the root mean square (RMS) of the difference \(\Delta V_{RMS}\) between the first discharge cycle of the cell, which is taken as the reference cycle, and the current cycle. Note that the \(\Delta V_{RMS}\) can be used as an indicator of the state of health of the cell. The \(\Delta V_{RMS,n}\) [V] indicator is calculated at a generic cycle \(n\) in the interval [SOC\({}_{\text{min}}\), SOC\({}_{\text{max}}\)] as follows:  
\[\Delta V_{\text{RMS},n}=\sqrt{\frac{1}{SOC_{\text{max}}-SOC_{\text{min}}} \int_{SOC_{\text{min}}}^{SOC_{\text{max}}}\left[V_{ref}(SOC)-V_{n}(SOC) \right]^{2}dSOC\ [\text{V}]} \tag{6}\]  
where \(n\) = 1 corresponds to the initial or reference cycle and \(n\) is a generic cycle.  
Figure 1 explains the \(\Delta V_{RMS}\) indicator. It shows the evolution of the cell terminal voltage versus the SOC value calculated from (4) during the first discharge cycle and during the \(n\)-th cycle.  
In this paper, the SOC interval [SOC\({}_{\text{min}}\), SOC\({}_{\text{max}}\)] = [0.55, 0.75] is chosen to calculate the \(\Delta V_{RMS}\) indicator. Since the proposed \(\Delta V_{RMS}\) indicator is calculated in a narrow SOC interval, this HI is quite flexible because it does not require the cell to be fully charged and discharged. As seen in Figure 1, the linear region of the voltage vs. SOC curve falls within this interval. This interval is typically within the normal transit area for most charge and discharge cycles, regardless of their profile.