## 4 The Performance of the Four Typical Battery Models  
In order to show the performance of the four typical modeling methods in more detail, the combined model [13], two RC ECM [10], SPM [14], and SVM battery models [15] are compared in terms of accuracy of voltage response and execution time in this section. For the details of each model, please refer to [10; 13; 14; 15]. For fairness of comparison, GA [76] is used to optimize the parameters in the combined model, two RC ECM, and SPM. Our paper uses the MATLAB command \(ga\) to implement the parameters optimization by GA [77]. A LiFePO\({}_{4}\) battery is discharged by NEDC profile in the MACCOR 4000 series test bench [10]. The nominal capacity of the battery is 10 Ah, and the nominal voltage is 3.2 V. The temperature in the test chamber is set to 25 \({}^{\circ}\)C and the sampling time is 1 s for data acquisition. The step time for the simulation of the models is also set to 1 s in this paper. In order to verify the four modeling methods, the LiFePO\({}_{4}\) battery is discharging under multi-NEDC driving cycles. All the measurements from MACCOR are shown in Figure 6.  
Figure 6: _Cont._  
OCV is measured with 5% SOC interval after two hours' battery relaxation time. Therefore, the two RC ECM can be built on the basis of the measurement in Figure 7.  
With the constraint that the diminishing of OCV with the decrease of SOC, a high-order polynomial function represents the OCV-SOC relationship, as follows:  
\[ \begin{array}{l}OCV=-330.2741\cdot SOC^{8}+1507.8350\cdot SOC^{7}-2869.7023 \cdot SOC^{6}\\ +2949.8632\cdot SOC^{5}-1773.9467\cdot SOC^{4}+632.0383\cdot SOC^{3}-128.9882 \cdot SOC^{2}\\ +13.8940\cdot SOC+2.6371\end{array}  \tag{3}\]  
Equation (3) further acts as the voltage source in the two RC ECM. The modeling results of the four modeling methods are shown in Figures 8 and 9. Figure 6: Measurement during multi-NEDC driving cycles. (**a**) Current; (**b**) voltage; (**c**) SOC. Figure 7: OCV measurement. Figure 8: Voltage prediction achieved by using the four models. (**a**) Voltage; (**b**) absolute error. Figure 9: Zoom of the voltage prediction achieved by using the four models (from 7221s to 8054 s): (**a**) Voltage; (**b**) absolute error. We can see that all the methods are able to predict the battery terminal voltage in Figure 8a. SVM and SPM achieve better results than the other two methods, as shown in Figure 9a, but the results of the two RC ECM and the combined model are also acceptable. The absolute error of the four methods (Figure 8b) proves that SVM obtains the best result of the four models, and the Mean Absolute Error (MAE) of SVM is 0.0034 V. The good accuracy of SVM is because the same data is used in SVM for training and testing. For further testing the SVM battery model, an Urban Dynamometer Driving Schedule (UDDS) [78] is applied to the SVM, trained by the measurement from NEDC. The results for the SVM battery model shown in Figure 10 are much worse than those in Figure 8. The MAE of the results in Figure 10 is 0.0632 V. Therefore, the accuracy is doubtful if another driving cycle different from the training samples is applied to the SVM battery model. The details of the absolute error in Figure 9b clearly show the performance of the four models in terms of accuracy of voltage prediction, while SPM shows comparable results to SVM. MAE and the execution time of the four models are listed in Table 5.  
The MAE of SVM is much smaller than the others three models. Although GA is applied to optimize the parameters, it is still difficult to obtain true values for each parameter for SPM. Hence, the differences in MAE are not obvious for the SPM, the two RC ECM, and the combined model, since SPM contains as many as 28 parameters. The execution time in this paper is measured in MATLAB 2017b on a 64-_bit_ computer with 2.30 GHz CPU, and the mean execution time of the battery model at each step is calculated in Table 5. It can be seen from Table 5 that the execution time of SVM is 0.0018 s, which is much longer than in the other models.