## 5 Filter Validation
### First Scheme  
As a result of the first scheme, Figure 8 presents the SOC error, comparing estimation accuracy across the type of load profiles and the number of cycles. Regarding the effect of cycle number, the SOC error generally increases as capacity decreases, except at cycle 1500. The test performed at this cycle is considered flawed because a similar anomaly is also observed in Figure 2, which shows changes in the rated capacity. The cycles earlier than 900 exhibit an RMSE of less than 3%, regardless of load type. However, the error exceeds 3% and rises rapidly from cycle 900 onward, where the SOH reaches nearly 80%. Regarding the effect of load type, the SOC error remains similar across each type until cycle 1200. From this cycle, except for cycle 1500, the increasing error in the City model is less severe than in other models based on more aggressive load profiles. The SOC error in this model remains within 5% throughout all cycles. Capacity reduction necessarily accompanies an increase in internal resistance. Due to this increase, higher C-rates in the Highway and High-speed profiles cause greater voltage drops, particularly in more aged cells. As a result, the inability to accurately predict fluctuations in the terminal voltage leads to poor SOC estimation, which is more severe at low SOC levels towards the end of the load profile. Figure 6: Comparison between the three schemes developed. This inference can be confirmed in Figure 9, which plots changes in the SOC error with its bounds against time, comparing estimation accuracy across the types of load profiles and the number of cycles. Due to space constraints, the estimation results from five specific cycles (0, 600, 1200, 1800, and 2100) are selectively displayed. The impact of an initial  
Figure 8: Variations in SOC estimates (error) over cycles in the first scheme. Figure 7: Comparison of the (**a**) sequential and (**b**) simultaneous estimators. SOC on tracking performance is as significant as that of the cycle number. Here, some initial values are not visible because of the unified range of SOC error (\(-5\)-\(5\)%) across 12 cases. To address this lack of prior information, Table 2 lists the initial SOC estimates for each load profile and cycle number. In this study, the initial SOC was determined using the SOC-OCV curve, assuming that the terminal voltage converges to the OCV after long enough relaxation periods. To mimic the actual operations of PHEVs, the test involves resting after charging to 90% SOC, as described in Section 2. In the earlier cycles, the initial SOC is commonly estimated to be 88%, resulting in an SOC error of about 2%. Similar to the mean value (RMSE) of SOC in Figure 8, the error in its initial value increases as capacity decreases, regardless of model type. This observation suggests that the curve established for a fresh cell loses effectiveness because of the physicochemical changes in aged cells. This inference was also made in Section 3, and the current results provide even stronger evidence. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Cycle** & 0 & 100 & 200 & 300 & 600 & 900 & 1200 & 1500 & 1800 & 2100 \\ \hline
**City** & 0.88 & 0.88 & 0.87 & 0.87 & 0.85 & 0.84 & 0.81 & 0.81 & 0.79 & 0.79 \\
**Highway** & 0.88 & 0.88 & 0.88 & 0.87 & 0.85 & 0.83 & 0.82 & 0.82 & 0.8 & 0.78 \\
**High-speed** & 0.88 & 0.88 & 0.88 & 0.87 & 0.85 & 0.83 & 0.82 & 0.83 & 0.8 & 0.78 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Initial SOC estimates. Figure 9: Variations in SOC estimates (error) over time under (**a**) City, (**b**) Highway, and (**c**) High-speed profiles. As another result of the first scheme, Figure 10 plots changes in the measured and estimated capacities, comparing estimation accuracy across the type of load profiles and the number of cycles. The measurements are consistent with those in Figure 2. The resulting estimates are consistent with our concerns about the weak observability of capacity in the DEKF, as discussed in Section 4.