## 3 Battery Mathematical Models
### KiBaM Experimental Validation  
The KiBaM model is experimentally validated for the battery stack described in Section 2. During the experiments, full charge/discharge cycles (depth of discharge = 100%) for various constant currents were implemented. More specifically, the BESS was tested for 15, 20, 25 and 30 A. In Figure 3a the charging voltage curves of the stack for the aforementioned currents are depicted.  
In the lithium-ion batteries charging curves, an exponential increment of voltage after the voltage reaches the nominal value can be noticed. Similarly, in the lithium-ion batteries discharging curves the voltage decreases exponentially from the maximum voltage to the nominal one. The KiBaM model is developed based on the operation of lead acid batteries [15], in which this phenomenon is not observed. Therefore, the KiBaM model can be applied for the charging/discharging curves of lithium ion batteries only by excluding the exponential area of the curves. Hence, the curves are trimmed in order to apply data fitting to the model (Figure 3b).  
As analyzed in Section 2 the residential BESS to be considered consists of 15 lithium iron phosphate battery cells (Model: GBS-LFP100 Ah-A) connected in series. The nominal voltage of the GBS-LFP100 Ah-A battery is 3.2-3.4 V. This implies that when the cell is fully discharged the nominal voltage is 3.2 V and when is fully charged is 3.4 V. Consequently, the corresponding nominal voltages for the whole stack are 48 V (3.2 V \(\times\) 15 cells) and 51 V (3.4 V \(\times\) 15 cells). This is verified by the experimental data (Figure 3a). It is observed at all charging curves, that the initial voltage is 48 V and the initial point of the exponential zone is 51 V.  
In order to experimentally validate the KiBaM model, the parameters \(c\) and \(k\) which are used in model's equations must be estimated. During the data fitting process, it was found that the optimal values of the aforementioned parameters are \(c\ =\ 0.045\) and \(k\ =\ 17\). The data fitting process was applied separately for each cell of the stack and for the whole stack. The performance of the estimated parameters is evaluated by calculating the mean absolute percentage deviation (MAPD) of the four charging curves (Figures 4 and 5). For the charging curve with current I \(\ =\ 15\)A (Figure 4a) the MAPD  
Figure 3: Charging voltage curves: (**a**) Whole charging curves; (**b**) Trimmed charging curves.  
is 0.31%. Additionally, at the charging curve with current I \(=\) 20 A (Figure 4b) the MAPD is 0.32%. Moreover, at the charging curves with currents I \(=\) 25 A (Figure 5a) and I \(=\) 30 A (Figure 5b) the MAPD is 0.29% and 0.34% respectively. Furthermore, by calculating the MAPD separately for the exponential part of the curve and the linear part, is proved that the fitting match is better in the linear part. This is of great significance since the linear part represents the greater part of the curve. More specifically, at the charging curve with current I \(=\) 25 A, the MAPD at the exponential part is 0.39% while at the linear part is 0.28%. Moreover, at the charging curve with current I \(=\) 30 A, the MAPD at the exponential part is 0.48% while at the linear part is 0.32%.  
Over and above, the performance of the estimated parameters is evaluated by calculating the corresponding root mean square error (RMSE) for each charging curve. The RMSE is 0.2 for the 15 A curve, 0.19 for the 20 A curve, 0.18 for the 25 A curve and 0.21 for the 30 A curve, respectively.