## 6 Simulation Setup and Results
### Simulation Results
#### 6.2.1 B036 Dataset Dual-IMM Results  
The estimation results from both algorithms are presented at two cycles to demonstrate the behaviors of the algorithms at different stages of battery life. Cycle 21 was selected to show the estimation process of the algorithms at the beginning of the experiment. At this cycle, the battery capacity was measured to be ~1.79 Ah based on the B036 dataset. Figure 6 illustrates the battery capacity estimates at cycle 21. Part (a) compares the results of the dual-KF and dual-SIF and part (b) shows the results of the dual-KF-IMM and dual-SIF-IMM. All algorithms depict a large swing in battery capacity values during the estimation process; however, the final values of the estimation process return to levels similar to the beginning of the cycle. The dual-IMM algorithms show a significantly larger swing compared to the other dual strategies, which can be attributed to their access to a faster degradation rate. Two reasons for these large swings in the experiment can be attributed to the experiment's design. One reason could be that the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model Variables** & **Value** & **IMM Variables** & **Value** \\ \hline \(V_{pa}\) & 0 & \(Q_{\theta,\ Slow}\) & \(\text{Diag}(1\times 10^{-5},3\times 10^{-5},5\times 10^{-7})\) \\ \hline \(V_{pc}\) & 0 & \(R_{\theta,\ Slow}\) & Diag(0.1,0.01,1) \\ \hline SOC & 100\% & \(Delta_{Slow}\) & Diag\(\left(5\times 10^{3},4\right)\) \\ \hline \(R_{0}\) & 0.07 & \(Q_{\theta,\ Fast}\) & Diag\(\left(5\times 10^{-7},7\times 10^{-4}\right)\) \\ \hline \(Batt_{Cap}\) & 2.00 & \(R_{\theta,\ Fast}\) & Diag(0.1,0.01,1) \\ \hline \(Q_{x}\) & Diag\(\left(1\times 10^{-5},3\times 10^{-5},5\times 10^{-7}\right)\) & \(Delta_{Fast}\) & Diag(500,0.8) \\ \hline \(R_{x}\) & Diag(0.1,0.01,1) & \(p\) & 0.99 \\ \hline \(Q_{\theta}\) & Diag\(\left(5\times 10^{-8},7\times 10^{-5}\right)\) & \(\mu\) & 0.5 \\ \hline \(R_{\theta}\) & Diag(10,0.05) & & \\ \hline \(Delta_{x}\) & Diag(7,1.5,80) & & \\ \hline \(Delta_{\theta}\) & Diag(5000,4) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Dual Filter and IMM Initial Conditions. battery model is forced to remain constant throughout the different stages of life of the battery. Another reason is that the algorithms were tuned and designed to place a high emphasis on battery capacity as one of the parameters to account for the aging process. Next, Figure 7 depicts the SOC estimation results from all algorithms at cycle 21. These SOC curves demonstrate an accurate full discharge of the battery at this stage of life. It is evident that the battery is fully depleted by the end of the discharge cycle. All algorithms show similar profiles, with slight deviations from the model's SOC curve. It is important to note that the model curve is based on the derived model from Section 5 and utilizes a constant battery capacity of 1.8 Ah. Cycle 445 was selected to show the algorithms' estimation process toward the end of battery life. At this cycle, the battery had a reported battery capacity of 1.6 Ah. Figure 8 illustrates the battery capacity estimates during this cycle. Part (a) compares the results of the dual-KF and dual-SIF and part (b) compares the dual-IMM strategy results. At this stage, all the algorithms showed strong accuracy in tracking the correct battery capacity. Figure 6: (**a**) Dual-KF and dual-SIF battery capacity estimates at cycle 21. (**b**) Dual-KF-IMM and dual-SIF-IMM battery capacity estimates at cycle 21. Figure 7: (**a**) Dual-KF and dual-SIF SOC estimates at cycle 21. (**b**) Dual-KF-IMM and dual-SIF-IMM SOC estimates at cycle 21. Figure 9 shows the SOC curves of all algorithms at cycle 445. Part (a) compares the results of the dual-KF and dual-SIF and part (b) compares the dual-IMM strategy results. At this stage, all algorithms showed a significant deviation from the derived model. Furthermore, the algorithms showed an accurate SOC curve that depleted the battery by the end of the discharging cycle.