# Estimating Degradation Costs for Non-Cyclic Usage of Lithium-Ion Batteries
## 3 Results
### Interpretation of Simulation Results  
As mentioned previously, the results for \(\epsilon_{cal}^{LFP}\) and \(\epsilon_{cyc}^{LFP}\) in Figures 2-7 are very similar to the data of [PERSON] et al. [33], and the same is true for version #2. Furthermore, the values of \(\epsilon_{cal}^{NMC}\) and \(\epsilon_{cyc}^{NMC}\) agree with the results of [PERSON] et al. [19]. From Figures 2-7, it is clear that in all cases, model version #1 supplies higher values than versions #2 and #3 by several orders of magnitude. Thus, using version #1 as a simplified approximation to degradation cost results in over-estimated values. Moreover, version #2 gives very similar values to #3 with \(CF=0.5\). In one case only, \(\epsilon_{cyc}^{NMC}\) in Figure 6b, a visual comparison between the results of version #3 with several \(CF\) values shows the influence of actual capacity fade on degradation costs. It clearly shows that lower \(CF\) values result in higher degradation costs for the use of a cell. Cell degradation is higher at the beginning of life than at any other time during the cell life. Version #2 can be considered as an average value taking into consideration the whole cell life. Version #3 with \(CF=0.5\) is closer to this average value than version #3 with \(CF=0.9\) or with \(CF=0.1\). The results from several charge and discharge cycles shown in Figure 8 are compatible with the cell characteristics. For example, for \(SoC=0.5\), the \(OCV\) value is 3.7 V for the NMC cell. Moreover, the nominal voltage is 3.3 V for the LFP cell. The temperature during rest periods goes to ambient temperature, while it increases during charging and discharging, to a more or less stable value. A more detailed comparison between the versions can be carried out by looking at the RMSE. The RMSE is shown in Table 6 for NMC, and in Table 7 for LFP. After reviewing the data in these tables, the following can be stated:  
* The estimated degradations using versions #2 and #3 are very similar. All the values of RMSE (#2, #3) are very low, with the only exception of \(\epsilon_{cyc}^{NMC}\) with \(CF=1\). * Using version #1, the estimated degradations present the highest values, and applying versions #2 and #3, in all cases, the RMSE is lower than using versions #1 and #3. * The lowest RMSE between versions #2 and #3 is reached when \(CF=0.5\) in almost every case, and when it is not, the difference is not relevant. This result is congruent since version #2 is based on constant degradation over all the battery life, degradation estimated by version #3 is higher at the beginning of life, and the degradation grows at a rate that decreases with usage. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**cyc** & \(\Delta\)**Q (Ah)** & \(V_{qa}\) (V)** & **CR** \\ \hline NMC & 1 & \(OCV\) at 0.5 \(SoC\) & - \\ LFP & 1 & - & 1C \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters used in RMSE calculation for cyclic degradation. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Technology** & **V1** & **V2** & **V3** \\ \hline NMC & 3.86 & 3.87 & 3.95 \\ LFP & 14.2 & 14.3 & 14.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Computational time in seconds for 100 cycles of rest, discharge, rest, and charge obtained for the three versions. Figure 9 shows that discharging at lower \(CR\), cell temperature first increases, then becomes lower before growing again. This effect is also described by [PERSON] et al. (2018). The time consumed by calculations is listed in Table 5. There are no significant differences between versions #1,#2, and #3. In all three cases, the computational time is similar. However, the computational time for NMC is lower than that for LFP. This is due to the complexity of functions fitted for \(OCV\) and Thevenin parameters; for the NMC cell, they are quite simple, but for LFP, they are very complex, involving not only a dependency on \(SoC\) but also hysteresis and \(T\). Version #3 is more accurate than versions #1 and #2, and its computational time is only slightly higher than that for the other versions.