# A Strategic Pathway from Cell to Pack-Level Battery Lifetime Model Development
## 4 Performance-Based Lifetime Modeling
### Model Framework  
The lifetime model is developed using empirical representations with respective mathematical equations for estimating the cell's capacity fade resulted from cycling effects. The adapted lifetime model primarily considers the cycling capacity fade consisting of several Matlab scripts referring to the different impact factors. These impact factors are selected from the sensitivity analysis of the results explained in Section 3. The detailed construction of the framework can be found in the author's previous publications [13; 37]. As mentioned in Section 3, the calendar life imposes a minimal effect; thus, it has been excluded from the model consideration. The corresponding cycle life stress factors which  
Figure 1: The impact of the cycling conditions on aging: (**a**) full DoD cycling at 2C charge–discharge from moderate to hot temperature, (**b**) high DoD cycling in the 10–90% SoC region with 2C charge–discharge rates at various temperatures, (**c**) medium DoD cycling at various SoC regions with 2C charge–discharge rates at 25 °C, and (**d**) low DoD cycling in the 40–60% SoC region with 2C charge–discharge rates at low and 25 °C. resulted in the capacity loss representations include the temperature, depth of discharge, and mid-state of charge with their respective ratings as indicated in the aging test matrix. The multivariable model is parameterized with the sensitive impact factors and scripted to get a total and single representation. This has been achieved by using the Matlab platform obtaining the overall cycling capacity fade. The adapted modeling framework for this work is shown in Figure 2.  
The result analysis and parameter selection process make the model development smooth covering a wide range of operating conditions. First, the sensitive parameters are identified, and the corresponding data are plotted to the respective FECs. Then, the aging outputs from the sensitive parameters are fitted with mathematical equations to represent the individual influences. All the cycling stress parameters (DoD, temperature, mid-SoC, etc.) effect is scripted, separately but merged for DoD and temperature effect. The following general cycle life equation is used to fit the generated dataset. \[\text{Cycling Capacity fade }(\text{FEC, IP})=\sum_{i=0,j=0}^{\text{n,m}}(\text{Ai }(\text{FEC})^{\text{i}}+\text{Bj }(\text{IP})^{\text{j}}) \tag{2}\]  
In Equation (2), IP refers to the influential parameters for cycle life, \(\text{Aj}\) and \(\text{Bj}\) are constant coefficients, n and m are surface fitting orders, and i and j are polynomial orders to get the best fit. For instance, DoD impact on the cycle life degradation is fitted at every investigated temperature with 4 th order polynomials (i = 4, j = 2). A sample of the fitting scenario for the capacity fade in terms of DoD, and FEC is presented in Figure 3 which illustrates excellent fitting (R\({}^{2}\) = 0.98) based on Equation (2). Similarly, the mid-SoC effect is formulated in a separate script to get the quantified impact. In this way, the total aging corresponds to the necessary degradation scenarios. The Matlab curve fitting toolbox is used for this purpose. The developed model can intake dynamic or one-dimensional current signal as input together with other initial battery states such as SoC, BoL capacity, and the cycling temperature. The load signal is filtered in the initial section with a rainflow counter that is typically used for fatigue analysis. The rainflow algorithm extracts the crucial information from the dynamic profile and separates the load and no-load (zero current) conditions to be processed, accordingly, to the model scripts. The no-load situations are considered negligible due to the long calendar life of the investigated cell. The current profile is analyzed by a simple battery model (coulomb counting) that makes an SoC profiling and the rainflow counter prepares the DoD profile and cycle numbers for the whole simulated duration. Figure 2: Schematic overview of the adapted lifetime model.