5.3.2. Experiment Results of VFFRLS + MIUKF

(1) Current condition

The dynamic characteristics of the UUDS cycle are very strong, and the current changes almost every second. In the experimental process, the discharge is set at a positive number, the maximum discharge current is 62 A, and the maximum charging current is 41 A. The curve of the current over time is shown in Figure 7.

(2) Comparison between measured and estimated terminal voltage

The estimated terminal voltage of battery parameters is close to the measured terminal voltage, and the gap at the end of battery discharge is slightly enlarged, which is also consistent with the intuitive feeling that the internal polarization reaction of the battery tends to be intense at the end of battery discharge, resulting in the enhancement of nonlinear characteristics. The maximum error of terminal voltage is 20.3%, the minimum value is −7.87%, and the average value of the absolute error is 0.57%. The time-varying terminal voltage curve of the measured value and estimated value and the time-varying curve of the terminal voltage error are shown in Figures 8 and 9, respectively.

**Figure 7.** Discharge current over time.

**Figure 8.** Curve of measured and estimated terminal voltage values over time.

**Figure 9.** Curve of terminal voltage error over time.

(3) Identification results of battery resistance and capacitance parameters

The parameters related to the resistance and capacitance of the battery identified by the algorithm are shown in the Figure 10.

**Figure 10.** Identification results of battery resistance and capacitance parameters.: (**a**) R0 result, (**b**) R1 result, (**c**) C1 result, (**d**) R2 result, (**e**) C2 result, (**f**) T1 and T2 results.

R0 represents ohmic internal resistance, and its size depends on the activation degree of the electrode and active material which decreases with the decrease in SOC. Therefore, generally speaking, R0 shows a gradual increase trend with the passage of discharge time—that is, it gradually increases with the decrease in SOC. The identified R0 parameters conform to this physical characteristic.

R1 and C1 are represented as slow reaction polarization phenomena in the model, and R2 and C2 are represented as fast reaction polarization phenomena in the model. Their respective products are called time constants, which should conform to R1 \* C1 > R2 \* C2, and the identified parameters also conform to this physical feature.

It is worth noting that, similarly to R0, R1 and R2 generally increase gradually with the decrease in SOC. The parameters identified in this paper are inconsistent with this. Considering that this goal is not present in the model, and the inconsistency here does not cause significant deviation from the preset objectives of the model, the identification results are still successful.

#### (4) SOC estimate vs. baseline

The maximum value of SOC error is 19%, the minimum value is −0.44%, and the average value of absolute value is 0.225%. The SOC estimation value of the algorithm can quickly adjust the gap with the reference value. After 114 sampling cycles—that is, 114 s—the error of the algorithm decreases from 19% to less than 5%, and after 416 s, it decreases to less than 1%, after which it remains below 1%.

The time-varying curves of SOC estimated value and reference value and the timevarying curves of error are shown in Figures 11 and 12, respectively.

#### 5.3.3. Comparison of Algorithm Experiment Results

Based on the same data, the online parameter estimation algorithm VFFRLS + UKF and the offline parameter estimation algorithm MIUKF, the UKF and EKF, are used for the experiment. The comparison between the online and offline algorithms is based on similar cost baselines, as it is difficult to compare them in other aspects, so the same offline resistance and capacitance parameters are used as initial value for online parameters, and the same system and measurement noise covariance matrix are used as well. The comparison results show that the online parameter estimation has obvious advantages in accuracy and stability—the average value of the absolute value of the error is small, and the error curve is stable and close to zero over time. Compared with the VFFRLS + UKF algorithm of the estimation of the same online parameter, VFFRLS + MIUKF is superior in convergence speed, accuracy and stability, which shows that the error converges to zero

faster, and the average absolute value of the error and the standard deviation of the error are small.

**Figure 11.** Curve of SOC estimated value and reference value over time.

**Figure 12.** Curve of SOC estimation error over time.

The index statistics related to the stability, accuracy and convergence speed of the SOC prediction error results of each algorithm are shown in Table 4.

**Table 4.** The index statistics related to the stability, accuracy and convergence speed of the SOC prediction error results of each algorithm.


The curve of the SOC predicted value and reference value of each algorithm over time is shown in Figure 13.

**Figure 13.** The curve of the SOC predicted value and reference value of each algorithm over time.

The time-varying curve of the SOC prediction error of each algorithm is shown in Figure 14.

**Figure 14.** The time-varying curve of SOC prediction error of each algorithm.

#### **6. Conclusions**

Accurate and real-time SOC estimation is the basis and key to realizing balanced battery management, which can reduce battery internal resistance loss and the possibility of battery overcharge and discharge. Due to the complex internal chemical and physical reactions and dynamic environmental conditions, the SOC of a battery has obvious nonlinear and time-varying characteristics, which has always been the focus of and main difficulty in battery management system research.

In this paper, a joint SOC estimation algorithm based on online parameter identification and a second-order RC equivalent circuit model is proposed, which innovatively realizes the dual estimation of MIUKF + VFFRLS. The experimental results based on UDDS test data show that the algorithm has obvious advantages in stability and accuracy compared with offline parameter + EKF, offline parameter + UKF and offline parameter +MIUKF; compared with UKF + VFFRLS, it has advantages in convergence speed, accuracy and stability. The overall performance of the fused algorithm is outstanding.

Through the above research work, the SOC estimation accuracy can be effectively improved, the battery consistency management ability can be improved, and the theoretical value and practical value can be reflected, but there are still limitations and deficiencies.

The tuning of the KF is critical to the SOC estimation results, and optimization methods can be further discussed. The accuracy of RLS algorithm is very sensitive to measurement noise, and the associated noise-compensation methods can be further studied. The modelbased SOC estimation also depends on accurate estimation of the battery capacity, and data-based capacity estimation can be further studied.

The applicability of the algorithm is also related to the efficiency of the algorithm. The calculation time of the algorithm is not compared in this experiment because the calculation time is strongly related to factors such as the type of program language and the method of coding. The battery type used in the algorithm is a ternary lithium battery with a relatively strong linear relationship between the *Uocv* and the SOC curve. The duration of experimental data is short. The factors of battery capacity attenuation and temperature change are not considered.

The follow-up research can work in the following directions: making noise-compensated methods research [22,23], optimizing the tuning of the KF [24], considering the factors of battery capacity attenuation [25] and temperature change, designing an analogous algorithm efficiency model to compare the calculation time of different types of SOC estimation algorithms, performing experiments to collect data for a longer time or to seek a larger public dataset, further verifying the effectiveness of the algorithm based on larger datasets with different batteries, further studying the effectiveness of intelligent algorithms such as neural networks and the algorithm proposed by this paper in large data sets and exploring more efficient SOC estimation methods suitable for more scenarios.

**Author Contributions:** Conceptualization, H.Y. and H.P.; methodology, H.Y. and Y.H.; software, H.Y.; validation, H.Y., Y.H. and Y.Z.; formal analysis, J.D.; investigation, H.Y. and Y.H.; resources, Y.Z.; data curation, Z.C.; writing—original draft preparation, H.Y.; writing—review and editing, Y.H.; visualization, H.Y.; supervision, H.P.; project administration, H.P.; funding acquisition, J.D. All authors have read and agreed to the published version of the manuscript. H.Y. and Y.H. contributed equally to this work.

**Funding:** This research was funded by the project 2021 Natural Science Foundation of Guangdong Province, Grant No. is 2021A1515011851.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

