## 5 Simulation Studies
### Evaluation of the New Estimation Structure  
The effectiveness of the new structure will be verified in the following. Comparison with estimation results from the joint estimation frameworks composed of EKF or UKF respectively will be presented. Due to the maturity and numerous applications of KF, EKF, and UKF, the details will not be presented here. Readers can refer to [14,17] to get more information. However, key information is given as follows. Firstly, the characteristics of the noises are shown in Table 1, which were set according to the existing works and several tests. For the joint estimation frameworks based on (9), the vector \(\mathbf{x_{0}}\) was used to initialize the EKF or UKF. In the new structure, the initial value of the battery OCV was set to 3.5 V, and the other initial values were the same as in the vector \(\mathbf{x}_{0}\). First of all, as a specific term appeared in the new structure, the estimation result of OCV is shown in Figures 12 and 13. Firstly, as is shown in Figure 12, the OCV estimation converged rapidly to the reference value and could track the dynamic change of the battery OCV. Then, from the OCV estimation error in Figure 13, KF was capable of providing an accurate estimation, which was important to connect the other two estimation modules. Figure 10: Battery input current level effect on battery capacity estimation. Figure 11: Initial capacity value effect on battery capacity estimation. \begin{table}
\begin{tabular}{c c} \hline \hline \(\omega_{1}\sim\mathrm{N}(0,1\times 10^{-8})\) & \(\gamma_{1}\sim\mathrm{N}(0,1\times 10^{-7})\) \\ \hline \(\omega_{2}\sim\mathrm{N}(0,1\times 10^{-8})\) & \(\gamma_{1}\sim\mathrm{N}(0,1\times 10^{-10})\) \\ \hline \(\omega_{3}\sim\mathrm{N}(0,1\times 10^{-9})\) & \(\delta\sim\mathrm{N}(0,1\times 10^{-4})\) \\ \hline \(\omega_{4}\sim\mathrm{N}(0,1\times 10^{-12})\) & \(\varepsilon_{1}\sim\mathrm{N}(0,1\times 10^{-8})\) \\ \hline \(\omega_{5}\sim\mathrm{N}(0,1\times 10^{-10})\) & \(\varepsilon_{2}\sim\mathrm{N}(0,1\times 10^{-8})\) \\ \hline \(\omega_{6}\sim\mathrm{N}(0,1\times 10^{-10})\) & \(\varepsilon_{3}\sim\mathrm{N}(0,1\times 10^{-6})\) \\ \hline \(\omega_{7}\sim\mathrm{N}(0,1\times 10^{-10})\) & \(\psi\sim\mathrm{N}(0,1\times 10^{-6})\) \\ \hline \(v\sim\mathrm{N}(0,1\times 10^{-6})\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Characteristics of the added noises. Then, the estimation results of the other parameters are shown in Figure 14. Several improvements brought by the new structure should be noticed. Firstly, as can be seen from Figure 14a,b, compared to joint EKF or joint UKF, the main improvement with the new structure came from the estimation of the battery capacity. Not only the convergence speed was more rapid, but also the estimation was fluctuating less and more accurate. Secondly, it can be observed from Figure 14c-f that the new framework improved the estimation accuracy of the RC network. Thirdly, as can be seen from Figure 14i,j, due to the improvement of the parameter estimation, the determination of the battery SOC was also improved. Furthermore, as for the computational cost, ten simulation tests were executed for these three monitoring algorithms. As shown in Figure 14, the current cycle duration for each test was 8500 s. The simulation was performed by using the discrete time model with a sampling period equal to 1 s, with MATLAB/Simulink R2015b, on a desktop with an Intel Core i7-6700 CPU 3.4 GHz and 8 GB RAM. The average calculation times of these ten simulation tests for each monitoring algorithm are listed in Table 2.  
However, this is only a general discussion of the computation load of the newly proposed monitoring structure, because the multi-time scale can also be introduced into the new structure, which means the sampling time period for the battery capacity can be extended to a larger value according to the application. For example, the capacity estimation can even be performed quarterly or semiannually because of its slow time varying property [35].