*3.3. Comparison of SOC Estimation Results under Di*ff*erent Working Conditions*

The test data in this section were based on the lithium-ion power battery test platform and the related basic characteristic tests. The main parameters are shown in Table 1. Besides, the tests in this section were performed at the ambient temperature of 25 ◦C. The initial SOC values in all the experiments were set to 90%, and the commissioning and verification of the SOC estimation under the HKF for the DPM and the FOM were completed under the parameters set, as explained. The experimental scheme is shown in Table 9. The basic characteristic tests performed as required for the battery SOC estimation algorithm included the DST cycle test, the FUDS cycle test, and the HPPC cycle test. Each working condition test refers to the relevant test methods in the *USABC Electric Vehicle Battery Test Procedures Manual* and the *Freedom CAR Battery Test Manual for Power-Assist Hybrid Electric*

*Vehicles*. The specific experimental steps are given in Table 10. The sampling frequency was 10 Hz. The charging and discharging current and battery terminal voltage response of each test are shown in Figure 8.

**Table 9.** Experiments under different models and different test conditions.




Considering that the HKF needs to complete the prior estimation of the SOC by the Ah integration method, the sampling time interval is particularly important. Here we chose to collect the current and voltage signal every 0.1 s. It should be noted that, for most devices, the sampling interval cannot be completely and constantly controlled; that is, the sampling time interval fluctuates slightly around 0.1 s.

The above experimental data are battery current and voltage values. To compare the effects of DPM and FOM on SOC estimation, it was necessary to obtain reference values for SOC changes. Considering that under the experimental conditions, the voltage and current acquisition accuracy were very high, and the battery's initial SOC and maximum available capacity were known, the sufficiently accurate SOC change curve could be obtained by the Ah integration method. The curve found by integration was used as a reference value for the SOC estimation results of the two models, and the simulation results are shown in Figures 9–11.

**Figure 8.** Charging and discharging current and voltage response of battery cell A123 under different test cycle conditions.

**Figure 9.** SOC estimation results under the DST test cycle condition.

**Figure 10.** SOC estimation results under the federal urban driving schedule (FUDS) test cycle condition.

**Figure 11.** SOC estimation results under the hybrid pulse power characteristic (HPPC) test cycle condition.

As can be seen from Figure 9, the SOC estimation error based on the DPM fluctuated within a range of ±0.06 under the DST test cycle condition, and the SOC estimation error based on the FOM remained within a range of ±0.01. Figure 10 shows that the error of SOC estimation based on the DPM fluctuated within the range of ±0.06 under the FUDS test cycle condition, and the error of SOC estimation based on the FOM can fluctuate within ±0.01. It is clear from Figure 11 that under the condition of HPPC, the error of SOC estimation based on the DPM fluctuated within ±0.05 range, and the SOC estimation error based on the fractional-order model remained fluctuating within ±0.03 range. The results of SOC estimation errors under different models and different test conditions are summarized in Table 11. It can be observed that irrespective of the working conditions applied, the SOC estimation error of the FOM was always less than the estimation error of the DPM, which benefits from the more accurate simulation of the battery impedance characteristics achieved by the fractional-order model. This illustrates that the higher the accuracy of the selected model, the lower the SOC estimation error.


HPPC 0.0177 0.05 0.0098 0.0146 0.03 0.0047

**Table 11.** SOC estimation errors using different models under different test conditions.

From Figure 9b, Figure 10b, and Figure 11b, it could be seen that in the DPM, the frequency of current change in the DST mode and FUDS mode was nearly the same, and the error of SOC estimation was within ± 0.06 range; the SOC estimation error could be maintained within ±0.05 range under HPPC test cycle condition. The maximum fluctuation of the SOC estimation error between different operating conditions was 0.01. From Figure 9c, Figure 10c, and Figure 11c, the SOC estimation error of the FOM in the DST and FUDS test cycle conditions fluctuated within ±0.01 range. Under the HPPC test cycle condition, the SOC estimation result was relatively poor, but the SOC estimation error could also remain within a range of ±0.03. The maximum fluctuation of the SOC estimation error between different operating conditions was 0.02. This shows that the adaptability of the FOM was not as good as that of the DPM. As can be seen from Figures 9a and 11a, in most cases, the SOC simulation values of the two models basically fluctuated slightly around the measured values, and only fluctuated slightly larger when the voltage and current change suddenly. Moreover, the fluctuation range of the FOM was larger than that of the DPM, but it immediately converged near the theoretical value, which had better convergence than the DPM. In summary, although the SOC estimation accuracy of the FOM was higher, the adaptability of working conditions was not as good as that of the DPM.
