# Extending Battery Lifetime by Avoiding High SOC
## 3 Model Development  
By studying the shape of the capacity degradation as a function of FCE for the tested cells, it was found that a second order exponential function could represent the behaviour in an adequate way. The following equation was thus used  
\[Cap(SOC,C\text{-}rate,T)=a\cdot e^{(b\cdot FCE)}+c\cdot e^{(d\cdot FCE)}. \tag{1}\]  
The equation and its parameters are mathematical adaptations and can not be linked to a physical background. Since the initial capacity is known, the expression was simplified using \(c=26-a\). Equation (1) was fitted to all the data tested in 2C for both temperatures and the trend in the \(a\), \(b\) and \(d\) parameters were studied. From this, a linear relation for parameter \(a\) was established as a function of SOC  
\[a(SOC)=6.2\frac{SOC}{90}+0.093. \tag{2}\]  
The fitting procedure was then repeated, now using \(a\) from Equation (2). Parameter \(b\) was found to be depending on SOC and C-rate as  
\[ \begin{split} b(SOC,C\text{-}rate)&=\left(0.98 \left(\frac{C\text{-}rate}{2}\right)^{3.3}+0.01741\left(\frac{SOC}{20}\right) \right)\cdot\\ &\left(\frac{-0.6045}{SOC^{2.4}}-5.512\cdot 10^{-4}\right)\cdot \left(\frac{SOC}{20}\right)^{(0.05C\text{-}rate^{2}-0.35C\text{-}rate^{2}+1)} \end{split}  \tag{3}\]  
and the value for the \(b\) parameter as function of SOC for different C-rates is presented in Figure 4.  
Now, using the values calculated from Equations (2) and (3) for \(a\) and \(b\) in Equation (1), the data were fitted again. The behaviour for parameter \(d\) with its SOC and temperature dependency was more difficult to capture. A decent fit to the data could only be achieved with a 4 th order polynomial for the SOC dependence, not including the temperature dependency. With an \(R^{2}\) value of only 0.8 for the SOC dependency, it was decided to use a look-up table for the \(d\) parameter (see Table 2). For the SOC intervals missing in the test data, interpolation was used to estimate the \(d\) parameter. To calculate the \(d\) parameter at temperatures between 25 and 35 \({}^{\circ}\)C, a linear relationship was used. The parameter \(d\) showed also a dependency on C-rate. The difference between the \(d(2C)\), \(d(1C)\) and \(d(4C)\) values in 35 \({}^{\circ}\)C was used to estimate the \(d\) value according to:  
Figure 3: Calendar ageing in 25 \({}^{\circ}\)C and 35 \({}^{\circ}\)C at 15% and 90% State of Charge (SOC). \[d(C\text{-}rate<1,SOC<50) =d(2C,SOC)+\left(C\text{-}rate-1\right)\left(d(2C,10\text{-}20)-d(1C,10 \text{-}20)\right), \tag{4}\] \[d(C\text{-}rate<1,SOC<50) =d(2C,SOC)+\left(C\text{-}rate-1\right)\left(d(2C,60\text{-}70)-d( 1C,60\text{-}70)\right),\] \[d(C\text{-}rate<2,SOC<50) =d(2C,SOC)+\left(2-C\text{-}rate\right)\left(d(2C,10\text{-}20)-d( 1C,10\text{-}20)\right),\] \[d(C\text{-}rate<2,SOC<50) =d(2C,SOC)+\left(2-C\text{-}rate\right)\left(d(2C,60\text{-}70)-d( 1C,60\text{-}70)\right),\] \[d(C\text{-}rate>2,SOC\geq 50) =d(2C,SOC)+\frac{C\text{-}rate-2}{2}\left(d(2C,10\text{-}20)-d( 4C,10\text{-}20)\right),\] \[d(C\text{-}rate>2,SOC\geq 50) =d(2C,SOC)+\frac{C\text{-}rate-2}{2}\left(d(2C,60\text{-}70)-d( 4C,60\text{-}70)\right).\]  
The resulting performance of the degradation function after its parametrisation is shown in Figure 5a for the SOC behavior towards the experimental data for 25 \({}^{\circ}\)C, and in Figure 5b for 35 \({}^{\circ}\)C. The parametrised functions represent the measured results well. Furthermore, it can be clearly noted that the battery will last much longer if used in the lower SOC intervals even when using higher C-rates. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**SOC Interval (\%)** & **0–10** & **10–20** & **2  
For the calendar data, (1) was fitted to the data in 25 \({}^{\circ}\)C, though now as a function of days.