# Estimating Degradation Costs for Non-Cyclic Usage of Lithium-Ion Batteries
## 2 Materials and Methods
### Degradation Cost
#### 2.3.2 Version 2 to Calculate Degradation Costs  
This version is based on a uniformly distributed cost over all the life of the battery. It is an approximation used by some authors, for example, [PERSON] et al. [33], which estimates that the cost associated with using a battery is the same throughout its life. In version #2, \(\epsilon\) can be estimated using Equations (22) and (23). It is necessary to calculate \(t_{cal}(C_{loss}=20\%)\) and \(C_{cyc}(C_{loss}=20\%)\), and at end of life \(CF\) is equal to 1. This is expressed in Equations (31) and (32):  
\[t_{cal}^{x}(C_{loss}\ =\ 20\%)\ =\ t_{cal}^{x}(CF\ =\ 1),x\in\{NMC,LFP\}, \tag{35}\]  
\[Q_{cyc}^{x}(C_{loss}\ =\ 20\%)\ =\ Q_{cyc}^{x}(CF\ =\ 1),x\in\{NMC,LFP\}, \tag{36}\]  
Using Equations (1) and (13), we can obtain Equation (37), which allows calculation of the time elapsed, throughout the life of the cell, until a certain CF is reached, considering only calendar degradation for NMC cells (\(t_{cal}^{NMC}(CF)\)). \[t_{cal}^{NMC}(CF)\ =\ \left(\frac{C_{loss,cell}^{NMC}}{\alpha_{cap}}\right)^{1 /0.75}\ =\ \left(\frac{(1-\mu_{CF})CF}{\alpha_{cap}}\right)^{1/0.75}, \tag{37}\]  
Using Equation (14) we can then obtain Equation (38), which allows calculation of the total Ah-throughput used by the cell until a given \(CF\) is reached, considering only cyclic degradation for NMC cells (\(Q_{cyc}^{NMC}(CF)\)). \[Q_{cyc}^{NMC}(CF)\ =\ \left(\frac{C_{loss,cyc}^{NMC}}{\beta_{cap}}\right)^{2}\ =\ \left(\frac{(1-\mu_{CF})CF}{\beta_{cap}}\right)^{2}, \tag{38}\]  
Using Equations (22), (23), (37), and (38), we obtain Equations (39) and (40), which allow the calculation of \(\epsilon_{cal}^{NMC}\) and \(\epsilon_{cyc}^{NMC}\), with \(CF\) equal to 1. \[\epsilon_{cal}^{NMC}\ =\ \left(\frac{\alpha_{cap}}{1-\mu_{CF}}\right)^{1/0.75} \cdot\Delta t, \tag{39}\]  
\[\epsilon_{cyc}^{NMC}\ =\ \left(\frac{\beta_{cap}}{1-\mu_{CF}}\right)^{2}\cdot \Delta Q, \tag{40}\]  
Figure 4a shows \(\epsilon_{c}^{NMC}\) as a function of the \(SoC\) for a range of cell temperatures from 25 to 50 \({}^{\circ}\)C, and Figure 4b shows \(\epsilon_{cyc}^{NMC}\) as a function of the \(DoD\). Figure 3: \(\epsilon_{cal}\) and \(\epsilon_{cyc}\) for lithium iron phosphate (LFP) cells. Version 1. Using Equations (1) and (17), we can obtain Equation (41), which allows calculation of the time elapsed throughout the life of the cell until a given CF is reached, considering only calendar degradation for LFP cells (\(t_{cal}^{LFP}(CF)\)). \[t_{cal}^{LFP}(CF)\ =\ \left(\frac{C_{loss,cal}^{LFP}}{\int_{cal}^{LFP}(SoC,T)} \right)^{1/\gamma}\ =\ \left(\frac{100(1-\mu_{CF})CF}{f_{cal}^{LFP}(SoC,T)}\right)^{1/\gamma}, \tag{41}\]  
where \(f_{cal}^{LFP}\) is evaluated by Equation (42). \[f_{cal}^{LFP}\ =\ A\cdot\exp(\alpha\cdot 100\cdot SoC)\cdot\exp\left(\frac{- \beta}{T}\right), \tag{42}\]  
From Equation (18) we obtain Equation (43), which allows calculation of the total Ah-throughput used by the cell until a given CF is reached, considering only cyclic degradation for LFP chemistry (\(Q_{cyc}^{LFP}(CF)\)). \[Q_{cyc}^{LFP}(CF)\ =\ \left(\frac{C_{loss,cyc}^{LFP}}{f_{cyc}^{LFP}(CR, DoD)}\right)^{1/z}\ =\ \left(\frac{100(1-\mu_{CF})CF}{f_{cyc}^{LFP}(CR, DoD)}\right)^{1/z}, \tag{43}\]  
where \(f_{cyc}^{LFP}\) is evaluated by Equation (44). \[Q_{cyc}^{x}(C_{loss}\ =\ 20\%)\ =\ Q_{cyc}^{x}(CF\ =\ 1),x\in\{NMC,LFP\}, \tag{36}\]0  
Using Equations (22), (23), (41), and (43), we can obtain Equations (45) and (46), which allow calculation of \(\epsilon_{cal}^{LFP}\) and \(\epsilon_{cyc}^{LFP}\), for a \(\Delta t\) throughout the life of the cell, where CF is equal to 1. \[Q_{cyc}^{x}(C_{loss}\ =\ 20\%)\ =\ Q_{cyc}^{x}(CF\ =\ 1),x\in\{NMC,LFP\}, \tag{36}\]1  
\[Q_{cyc}^{x}(C_{loss}\ =\ 20\%)\ =\ Q_{cyc}^{x}(CF\ =\ 1),x\in\{NMC,LFP\}, \tag{36}\]2  
Equations (45) and (46) are the expressions used by [PERSON] et al. [32].