## 3 Result and Discussion
### Long-Term State of Health Estimation  
In this subsection, the proposed techniques for the long-term battery state of health estimation are also evaluated. Note that in this procedure, all of the short-term SOH and voltage cycles are integrated to build one macro time scale concept. The charge data for 87 cycles of battery, No. 06, are employed to evaluate the proposed techniques for long-term estimation capability.  
Figure 6 shows the long-term SOH estimation of battery No. 06. The obtained MSE and RMSE for the SOH estimation are 0.714, and 0.845, respectively. It can be seen that the GMDH, trained and tuned by battery No. 05, can be used to estimate the SOH for other batteries. Nevertheless, it is observed that, despite the better performance of GMDH for short-term estimation, in long-term SOH estimation, the fluctuation of GMDH is the most noticeable. According to Figure 6, the GMDH could not estimate the 1 st, 21 st, 54 th, and 74 th accurately. It can be concluded that the GMDH technique for long-term SOH estimation is instable.  
Figure 5: Experimental and estimated results of SOH vs. battery voltage by employing ANFIS (No.06) for two cycles.  
Figure 6: Long-term SOH estimation via GMDH for the 87 discharge cycles of battery No. 06.  
The results of SOH estimation for battery No. 06 based on ANFIS, are plotted in Figure 7. As it is noted in the figure, the MSE and RMSE are 0.041 and 0.203, respectively, which shows a better stability from the GMDH for the long-term SOH estimation. It can be observed that the ANFIS has successfully learned the effect of capacity degradation on the battery terminal voltage. Therefore, overcharging and deep-discharging can be avoided, and also, the proposed techniques can be used for smart battery charging management, as ANFIS and GMDH have the capability to respond to an optimization algorithm as soon as they receive the inputs of the system.  
Table 1 presents the performance of the evaluation, comparing the proposed techniques with the recent published articles. As shown in the table, the ANFIS model obtains a much better performance over the GMDH model. For instance, the RMSE and MSE on battery No. 06 based on GMDH is 0.845 and 0.714, while the RMSE and MSE based on ANFIS is 0.203 and 0.041, respectively. Moreover, in terms of comparison, the present results and the resent published articles used same dataset from NASA, the performance of the models introduced in the literature [16; 24; 36] are compared in Table 1. As can be observed, the RMSE and MSE based on the ANFIS model are much better than the introduced models. Nevertheless, the following limitations need to be addressed in future studies:  
1. While machine learning demonstrated an acceptable self-adaptation and high non-linearity modeling capability, a large amount of experimental data is required to obtain a high accuracy.
2. Although the introduced SOH method is more predictable and accurate under charging and discharging processes, it is not a usable method for plug-in hybrid electric vehicles (PHEVs)/PEVs when they are connected to smart charging infrastructure.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Error**} & **Ref. [36]** & **Ref. [16]** & **Ref. [24]** & **Present Study** & **Present Study** \\  & **QGPFR** & **GPR-SE** & **GMDH-DGA** & **ANFIS** & **GMDH** \\ \hline RMSE (battery No. 06) & 5.12 & 1.7064 & - & 0.203 & 0.845 \\ \hline MSE (battery No. 06) & - & - & 0.360 & 0.041 & 0.714 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Root mean square error (RMSE) results of long-term capacity estimations of adaptive neuro-fuzzy inference system (ANFIS), group method of data handling (GMDH), and a recent published article. MSE—mean square error; QGPER—quadratic polynomial mean function; DGA—geometry based approach.  
Figure 7: Long-term SOH estimation via ANFIS for the 87 discharge cycles of battery No. 06.