# Estimating Degradation Costs for Non-Cyclic Usage of Lithium-Ion Batteries
## 2 Materials and Methods
### Degradation Cost  
Degradation costs at a certain point in time of the cell life are obtained by multiplying the \(Cost_{bat}\) by a value extracted from the degradation model. The methods are obtained from three different visions from the literature and homogenized in the present work. The methods that have been applied, by various authors, for the determination of degradation costs are described below.  
[PERSON] et al. [33] used the [PERSON] model, which uses an aging life loss function (\(\epsilon\)) divided into a calendar aging life loss function (\(\epsilon_{cal}\)) and a cyclical aging life loss function (\(\epsilon_{cyc}\)) as the factor by which the \(Cost_{bat}\) is multiplied to obtain the \(Cost_{dg}\). It can be calculated using Equation (21) and is a non-dimensional number.  
\[Cost_{dg}\ =\ \left(\epsilon_{cal}+\epsilon_{cyc}\right)Cost_{bat}, \tag{21}\]  
Assuming that all cells in a battery are identical, \(\epsilon_{cal}\) and \(\epsilon_{cyc}\) can be obtained for a single cell, and then used as the \(\epsilon_{cal}\) and \(\epsilon_{cyc}\) of the entire battery. It is important to note that, in order to obtain a battery \(Cost_{dg}\), the degradation study of only a single cell is needed.  
[PERSON] et al. [33] used a replacement cost, uniformly distributed over the whole lifetime of the battery, for the calendar cost, and over the total Ah-throughput used in the cell lifetime for the cyclic cost. A cost proportional to the elapsed time of current use of the battery (\(\Delta t\)) is applied to obtain the calendar cost, and in order to obtain the cyclic cost, a cost proportional to the elapsed Q of the current use of the battery (\(\Delta Q\)) is applied. The end of battery life is reached when the capacity loss is 20%.  
\[\epsilon_{cal} = \frac{\Delta t}{t(C_{loss}\ =\ 20\%)}, \tag{22}\] \[\epsilon_{cyc} = \frac{\Delta Q}{Q(C_{loss}\ =\ 20\%)}, \tag{23}\]  
where \(t(C_{loss}\ =\ 20\%)\) and \(Q(C_{loss}\ =\ 20\%)\) can be obtained from the degradation model.  
[PERSON] et al. [32] applied another strategy, where \(Cost_{deg}\) is related to CF. Applying Equations (1) and (3), as the \(PF\) is considered less than the \(CF\), Equation (24) is obtained.  
\[Cost_{deg}\ =\ CF\text{-}Cost_{bat}, \tag{24}\]  
[PERSON] et al. [31] and [PERSON] et al. [10] used the derivative of capacity over time applied at a certain point in time during the battery life, using Equations (25) and (26).  
\[\Delta C_{loss,cal}(t) = \frac{\partial C_{loss,cal}(t)}{\partial t}\Delta t, \tag{25}\] \[\Delta C_{loss,cyc}(t) = \frac{\partial C_{loss,cyc}(t)}{\partial t}\Delta t, \tag{26}\]  
where \(t\) is the actual time in the life of a battery.  
In this work, we have considered the mathematical models used by other authors, but without using capacity derivatives. Instead of that, the capacity fade increment between two points in time of the cell life before and after a given use of the cell (\(\Delta CF\)) is used. Then the degradation cost due to the use of a battery can be expressed as:  
\[Cost_{deg}\ =\ \Delta CF\text{-}Cost_{bat}\ =\ \frac{C(t)-C(t+\Delta t)}{(1- \mu_{CF})\ C_{BOL}}\text{-}Cost_{bat}, \tag{27}\]  
where \(t\) is the lifetime previous to the given use of the cell, and \(t+\Delta t\) is the time after that use.  
In order to estimate the cost, three different methodologies have been considered (versions 1, 2, and 3). Each of these has different epsilon values. All of them are described below.