2.3.1. Model Parameter Identification

The equipment employed in our experimentation included a battery testing device (BTS-5 V 100 A), an incubator (HL404C), a charging and discharging facility, and a computer. The sampling time of the battery test system was set to 0.1 s. Figure 2 shows the configuration of the battery testing system. The A123 ternary lithium-ion soft pack batteries were selected as the experimental objects, and their specifications can be seen in Table 1.

**Figure 2.** Configuration of the battery testing bench.



The OCV is an important parameter of the lithium-ion battery, and it usually has a relatively fixed corresponding relationship with the SOC. By testing the OCV at different SOC points, the SOC-OCV curve of the battery can be drawn. Generally, if the lithium-ion battery is left in the open state for a sufficiently long period, the measured battery terminal voltage can be approximately considered as the battery's OCV [37]. According to the Empirical Formula (15) and the fitted data of Table 2, Figure 3 shows the connection between OCV and SOC [11]. The fitted parameters for the OCV-SOC curve are shown in Table 3.

$$\text{CL}\_{\text{OCV}}(\text{SOC}) = \text{C}\_{0} + \text{C}\_{1}\text{SOC} + \text{C}\_{2}\frac{1}{\text{SOC}} + \text{C}\_{3}\ln(\text{SOC}) + \text{C}\_{4}\ln(1 - \text{SOC})\tag{15}$$

**Table 2.** Open-circuit voltage-state of charge (OCV-SOC) fitting curve parameter table.

**Figure 3.** Fitted curve of open-circuit voltage-state of charge (OCV-SOC).

**Table 3.** Fitted parameters for the OCV-SOC curve.


To simulate cells operating in EVs and obtain their characteristics, static capacity test (SCT), DST, FUDS, and HPPC tests were performed at an environmental temperature of 25 ◦C. Later, the MCPSO algorithm [13] was employed to identify the parameters of the above-mentioned DPM and FOM in the time domain. The results of parameter identification are shown in Tables 4 and 5.

**Table 4.** Dual-polarization model (DPM) model parameter identification results.


**Table 5.** Fractional-order model (FOM) parameter identification results.


## 2.3.2. Model Accuracy Verification

In this study, the DST test cycle conditions are used to verify the model accuracy for both the DPM and the FOM. The battery current is used as the model input variable so that the corresponding model voltage output can be measured and plotted. Figure 4 shows the current profile of DST operating condition. The comparison chart and error chart of the DPM output voltage and measured voltage are shown in Figure 5. The comparison chart and error chart of the FOM output voltage and measured voltage are shown in Figure 6.

**Figure 4.** Current profile of the dynamic stress test (DST) test cycle condition.

**Figure 5.** DST working condition verification of the dual-polarization model (DPM).

**Figure 6.** The DST working condition verification of the fractional-order model (FOM).

Figures 5 and 6 show that the output voltage curves and measured voltage curves of the DPM and the FOM are highly fitted. In addition, the voltage error of the DPM did not exceed 40 mV, and the voltage error of the FOM was kept within 20 mV. This shows that the established model yielded high accuracy under DST working conditions. However, the accuracy of the FOM was higher than that of the DPM. The quantitative results are presented for comparison in Table 6.


**Table 6.** RMSEs and relative errors for different SOC ranges.

### **3. SOC Estimation Based on HKF Algorithm**

The SOC estimation accuracy of the lithium-ion battery is essential for the BMS. It is reflected in other functions, including charging and discharging control, balance management, safety management, and fault diagnosis cannot be achieved without the high-precision SOC estimation. In this paper, an HKF algorithm based on an established DPM and the FOM is proposed for SOC estimation. The algorithm comprehensively utilizes the merits of the Ah integration method, the KF algorithm, and the EKF algorithm, with verified effectiveness under different operating conditions.

### *3.1. KF Algorithm and EKF Algorithm*

For a discrete linear system, the KF algorithm can be used to improve the estimation accuracy of the system state variables. Recursive formulas used in the algorithm are shown in Table 7. However, the KF algorithm is only effective for linear systems. The DPM and the FOM of lithium-ion batteries established in this paper both are nonlinear models, challenging the usefulness of the KF algorithm. The EKF algorithm can linearize a nonlinear system, thus introducing approximation in the process, and inducing inevitable model errors. However, the algorithm poses certain advantages, such as being simple and fast in its implementation. Hence, the EKF algorithm has been extensively used to deal with the estimation problems for nonlinear models, and its recursive formula is shown in Table 8.


**Table 7.** The Kalman filter (KF) algorithm recursive formula.


**Table 8.** The extended Kalman filter (EKF) algorithm recursive formula.
