## 5 Simulation Studies  
### Observability Conditions' Assessment  
All the battery parameters used in this paper, including the coefficients of a seventh-order polynomial OCV curve as shown in (12), were taken from [28]. They were identified through the pulsed charge and discharge tests by a pouch cell LIB. The hardware-in-the-loop validation was done in order to confirm the proposed model and parameters.  
\[\begin{split} V_{\text{OC}}(soc)&=a_{1}\cdot soc^{7} +a_{2}\cdot soc^{6}+a_{3}\cdot soc^{5}+a_{4}\cdot soc^{4}...\\...&+a_{5}\cdot soc^{3}+a_{6}\cdot soc^{2}+a_{7} \cdot soc^{1}+a_{8}\end{split} \tag{12}\]  
Firstly, as shown in Figure 8, three special periodic input currents \(I_{\text{batt}}\) were considered. In one cycle (300 s), the LIB was discharged and recharged by a dynamic current during a small time period at the beginning; then, a constant discharge current was applied to the battery; and finally, the battery was idle until the next period. Besides, in order to verify whether the current value will affect the estimation of the battery capacity, three constant discharge values were selected, namely 10 A, 15 A, and 20 A.  
Note that the 20 A discharge current was the default value. The EKF was initialized with the vector \(\mathbf{x_{0}}=[0,\,0,\,0.8,\,0.001\,\,\Omega,\,0.0035\,\,\Omega,\,17,\!000\,\,F,\,18 \,\,\text{Ah}]\) in the simulation test if there is no further indication.  
Although there are three estimated ECM parameters, only the estimation result of R\({}_{0}\) will be discussed as an example. In order to verify the observability condition for the ECM parameter estimation, a simulation test was conducted in which R\({}_{0}\) would increase slowly in the battery model.  
The result is shown in Figure 9a. The dashed line is the estimation of R\({}_{0}\) by EKF. Figure 9b is the enlargement of the estimation result in Figure 9a from 2100 s to 2450 s, and Figure 9c is the corresponding estimation error, while Figure 9d is the corresponding input current. From Figure 9c and Figure 9d, we can deduce that the observer estimated efficiently the resistance only when the input current was variable. Besides, the fluctuating characteristic of the estimated R\({}_{0}\) in Figure 9a could be found evidently. This can be explained from the perspective of observability. The fluctuating characteristic of the estimated R\({}_{0}\) was due to the input current dynamics.  
Figure 8: Three different input currents for the battery.  
Figure 9: Verification of the time varying behavior of \(I_{\rm batt}\) on the estimation of ECM parameters.  
* Battery input current level On the other hand, three previously mentioned input currents were fed to the battery ECM in order to verify whether the input current value would influence the observability condition for battery capacity C\({}_{\text{n}}\) estimation. Considering the effect of randomness in measurement noise, a simple quasi-Monte Carlo simulation test was conducted with 100 estimations. The estimation result based on (9) is shown in Figure 10. It confirmed that a higher value of the battery input current improved the convergence speed of the battery capacity C\({}_{\text{n}}\) estimation.
* Initial capacity value In order to evaluate the effect of the battery capacity initial value, the estimator was initialized with three sets of different values, namely 17 Ah and 23 Ah, 18 Ah and 22 Ah, and 19 Ah and 21 Ah. To minimize the randomness effect, the estimator was executed with 100 sets of synthetic data. The estimation result in Figure 11 shows that a smaller initial value of C\({}_{\text{n}}\) could guarantee a rapid convergence speed for tracking the reference value. This confirmed the experimental results from [24] displayed in the Introduction of this paper.