# Estimating Degradation Costs for Non-Cyclic Usage of Lithium-Ion Batteries
## 2 Materials and Methods
### Degradation Cost
#### 2.3.1 Version 1 to Calculate Degradation Costs  
Degradation at a particular point in time of cell life is approximated as if the cell were always used at the beginning of its life. The attractiveness of this version is the ease of calculation. As the majority of models predict, degradation speed is higher at the beginning of battery life; thus, an over-estimation for degradation costs is expected.  
In version #1, \(\epsilon\) is considered to be approximately equal to CF. Thus, using Equation (1) we get Equation (28).  
\[\epsilon_{x}\ =\ \frac{C_{BOL}-C_{x}}{C_{BOL}(1-\mu_{CF})},x\in\{cal},cyc\}, \tag{28}\]  
where \(C_{cal}\) is the capacity reached if only calendar degradation exists, and \(C_{cyc}\) is the capacity reached if only cyclic degradation exists.  
Using Equations (9), (10), and (24), the values of the calendar aging life loss function (\(\epsilon_{cal}^{NMC}\)) and cyclical aging life loss function (\(\epsilon_{cyc}^{NMC}\)), for NMC cells, can be expressed by Equations (29) and (30).  
\[\epsilon_{cal}^{NMC}\ =\ \frac{C_{loss,cal}^{NMC}}{1-\mu_{CF}}\ =\ \frac{\alpha_{cap}(T,V)\cdot t^{0.75}}{1-\mu_{CF}}, \tag{29}\]\[\epsilon_{cyc}^{NMC}\ =\ \frac{C_{loss,cyc}^{NMC}}{1-\mu_{CF}}\ =\ \frac{\beta_{cap} \big{(}V_{qa},DoD\big{)}\cdot\sqrt{Q}}{1-\mu_{CF}}, \tag{30}\]  
where \(V_{qa}\) and \(DoD\) can be approximated using the maximum \(SoC\) over the whole cycle for NMC (\(SoC_{max}\)) and the minimum \(SoC\) over the whole cycle for NMC (\(SoC_{min}\)) as in Equations (31) and (32).  
\[V_{qa}\ =\ OCV((SoC_{max}+SoC_{min})/2,T), \tag{31}\]  
\[DoD\ =\ SoC_{max}-SoC_{min}, \tag{32}\]  
Figure 2a shows \(\epsilon_{sol}^{NMC}\) as a function of the \(SoC\) for a range of cell temperatures from 25 to 50 \({}^{\circ}\)C, and Figure 2b shows \(\epsilon_{cyc}^{NMC}\) as a function of the \(DoD\), for mean \(SoC\) value.  
From this, the calculation of cyclic degradation cost due to a single charge or discharge can be realized in one only step. Nevertheless, estimation of calendar degradation cost, if cell parameters change, needs subdivision of time in small increments, considering the parameters as constant over time. This is valid for rest time and for charging or discharging the cell, as well for NMC and LFP cells.  
For LFG cells the procedure is similar to that applied to NMC cells, but the supplied capacity loss is expressed as a percentage.  
Using Equations (17), (18), and (28), the values of the calendar aging life loss function (\(\epsilon_{cal}^{LFP}\)) and cyclical aging life loss function (\(\epsilon_{cyc}^{LFP}\)) can be expressed by Equations (33) and (34):  
\[\epsilon_{cal}^{LFP}\ =\ \frac{C_{loss,cal}^{LFP}}{100\cdot(1-\mu_{CF})}, \tag{33}\]  
\[\epsilon_{cyc}^{LFP}\ =\ \frac{C_{loss,cyc}^{LFP}}{100\cdot(1-\mu_{CF})}, \tag{34}\]  
Figure 3a shows \(\epsilon_{cal}^{LFP}\) as a function of the \(SoC\) for a range of cell temperatures from 25 to 50 \({}^{\circ}\)C, and Figure 3b shows \(\epsilon_{cyc}^{LFP}\) as a function of the \(DoD\) for CR values of 0.77, 0.8, 1, and 2.  
Figure 2: \(\epsilon_{cal}\) and \(\epsilon_{cyc}\) for nickel cobalt manganese oxide (NMC) cells. Version 1.