A Li-Ion Battery State of Charge Estimation Strategy Based on the Suboptimal Multiple Fading Factor Extended Kalman Filter Algorithm

Abstract

The state of charge (SOC) is an important indicator for evaluating a battery management system (BMS), which is crucial for the reliability, performance, and life management of a battery. In this paper, the characteristics of a Li-ion battery are deeply studied to explore the charge/discharge curve under different environments. Meanwhile, a second-order RC equivalent circuit model is constructed. The function identification of the EMF and SOC is performed based on the least squares method. The model estimation error is verified by simulation to be less than 0.05 V. Based on the Suboptimal Multiple Fading Factor Extended Kalman Filter (SMFEKF) algorithm, the SOC under constant current and UDDS conditions are estimated. Matlab/simulink simulations illustrate that the estimated accuracy of the proposed algorithm is improved by 79.36% compared with the EKF algorithm. Finally, the validity of the algorithm is verified jointly with the BMS. The results show that the estimation error is within 4% in both constant current condition as well as UDDS conditions, and it can still be predicted quickly and accurately under the uncertainty in the initial value of the SOC.

1. Introduction

With the development of battery management systems (BMSs), the reliability of batteries has been put forward with higher requirements. The functions of BMSs mainly include the data acquisition function, battery state of charge (SOC) estimation, battery state of health (SOH) estimation, safety management, thermal management, energy management, etc. [1]. Among them, battery SOC estimation, as one of the most critical technologies, is the standard for evaluating the remaining power of a power battery and the basis for judging the service life of the battery [2,3,4,5].

Currently, the value of the SOC is mostly defined in terms of the remaining battery charge, i.e., the value of the SOC is the percentage of the remaining battery charge to its rated charge under a specific discharge multiplication condition [6], and its expression can be described as:

$$\mathrm{S}\mathrm{O}\mathrm{C}=\frac{Q_c}{Q_1}$$

(1)

where Q_c is the residual current of the battery and Q₁ is the rated capacity of the battery.

The current SOC estimation methods mainly include the ampere–time integration method, the open-circuit voltage (OCV) method, neural networks (NFs), and the Kalman filtering method. OCV is the most commonly used method for SOC estimation; it has high estimation accuracy but is unable to determine the initial value of the SOC, which leads to a gradual increase in the cumulative error of SOC estimation. The OCV method has the advantage of simplicity but has poor dynamic estimation accuracy and utility. The neural network method requires larger training data and is internally more complex. Kalman filtering is computationally fast, highly accurate, and can improve an algorithm according to different systems [7].

Kalman filtering is widely used in battery SOC estimation, which is characterized by closed-loop control and real-time performance. However, since the Kalman filter cannot be applied to nonlinear systems, more and more scholars have started to study improved Kalman filter algorithms. Wei et al. applied the untraceable Kalman filter algorithm (UKF) to estimate the battery SOC, and the estimation error was about 2%. Y. Leng proposed the capacitive Kalman filter (CKF) for lithium battery SOC estimation, and compared with the untraceable Kalman filter, the estimation error of the volumetric CKF did not exceed 2% at most, and the accuracy was 1% higher. L. Leng proposed the volumetric CKF for lithium battery SOC estimation. The estimation error of volumetric Kalman filtering did not exceed 2% at most, and the accuracy was 1% higher. L. Ji proposed the adaptive extended Kalman filtering (AEKF) algorithm, which estimated the mean and variance of unknown noise in real time and improved estimation accuracy compared with traditional extended Kalman filtering. Y.X. Xiong estimated the SOC of Li-ion batteries based on the dual Kalman filtering algorithm, and the estimated absolute error was less than 0.01%, while the estimated error was more than 0.01% and the absolute error was less than 0.019, which had high accuracy. Rui Xiong designed the Robust Extended Kalman Filter (REKF) method based on the DP model to estimate the SOC and determined the accuracy of the estimation by performing Federal Urban Driving Schedule (FUDS) experiments. In addition to the above algorithms, there are other improved Kalman filtering algorithms aimed at improving SOC estimation accuracy [8,9,10,11,12,13,14,15,16].

This paper is organized as follows. In Section 2, battery capacity influencing factors are analyzed, and the correction coefficients of discharge point multiplication and temperature on capacity are calculated to derive the SOC revised form under different influencing factors. In Section 3, the battery model and parameter identification are established, the second-order RC network equivalent circuit battery model is established, the functional relationship between the EMF and SOC is obtained based on the least squares method through experimental data of open-circuit voltage measurement, and the accuracy of the model is verified through a pulse discharge experiment. In Section 4, the SMFEKF algorithm is designed based on the second-order RC equivalent circuit model, and the accuracy and feasibility of the SMFEKF algorithm are simulated and verified in Matlab/simulink. In Section 5, the SOC estimation algorithm is further validated by building a BMS platform.

2. Battery Characterization

2.1. Charge/Discharge Experimental Platform

The experimental platform consists of a battery test system, a high- and low-temperature test chamber, an upper computer, and a lithium ternary battery. Among them, the BMS is connected to the upper computer through a CAN bus. The high- and low-temperature test chamber controls the temperature of the test environment. For the selected lithium ternary battery, the main parameters are shown in

Table 1. Parameters of the tested battery.

	Item	Parameter
Lithium-ion battery (NMC)	Electrode materials	Li(NiCoMn)O2
	Rated capacity	11 Ah
	Standard discharge current	0.2 C~1 C
	Discharge temperature	−10 °C~60 °C
	Discharge cut-off voltage	3.0 V
	Standard voltage	3.7 V
	Standard charge current	0.2 C~1 C
	Cycle life (1 C/1 C, 100% DOD)	2000 cycles
	Charge cut-off voltage	4.2 V
BTS	Used voltage range	0–100 V
	Charge and discharge current range	1–200 A
	Accuracy of current and voltage	1% (full-scale)
	Sample time	20 ms
	Temperature	10–40 °C

. Its cathode is Li(NiCoMn)O₂, a lithium nickel manganese cobalt oxide (NMC) battery. NMC is a Li-ion battery with a different type of cathode. Unlike lithium iron phosphate (LFP), which possesses good capacity and stability, NMC demonstrates an improved cycle life, thermal stability, and energy density [17]. The battery test system (BTS) is a BNT series of power battery test equipment, produced by the German Decathlon Group, with working condition simulation and battery charge/discharge test functions; the specific parameters are shown in Table 1. The software used in the upper computer is the BTS-600 (https://www.digatron.com/Portals/38/Images/documents/PRODUKTBLATT_UBT_EN.PDF?ver=2019-06-12-091045-427) battery testing system; by programming the test software, the battery testing system can be made to run the test according to the program under different control conditions. The constant temperature and humidity test chamber can control the humidity of the room and the current temperature of the experimental environment so that the battery is charged and discharged at a constant temperature.

2.2. Discharge Experiments at Various Discharge Rates

Under the external temperature of 25 °C, the discharge test of the ternary lithium battery with different discharge multiplicity is conducted, and the curves of battery capacity and discharge multiplicity are obtained, as shown in Figure 1a. The results show that with all other things being equal, the larger the discharge multiplier, the smaller the amount of power discharged by the battery. The ternary lithium-ion battery is discharged at different temperatures (10 °C, 25 °C, 40 °C) at a constant current with a 1 C discharge multiplier to obtain the terminal voltage versus time curves of the battery at different temperatures, as shown in Figure 1b.

It is assumed that the discharge multiplier is 1 C, the temperature is 25 °C, and the capacity correction factor K₁ = 1, where the capacity correction factor K_i means the ratio of the capacity of the battery when the discharge multiplication rate is “i” to the capacity of the battery when “i = 1”. The calculation results of the capacity correction coefficient when the discharge multiplication rate is 0.3 C, 0.5 C, and 1 C are shown in

Table 2. Discharge rate and the battery capacity correction factor.

Discharge rate (Ah)	3	5.5	11
Capacity correction factor (K1)	1.04	1.03	1

. Second-order fitting is carried out, which results in the curve of the capacity correction coefficient K_i, as shown in Figure 2. The expression is:

$$K_i[i(t)]=-0.0001{[i(t)]}^2-0.0035i(t)+1.053$$

(2)

2.3. Temperature Discharge Experiments at Different Temperatures

The internal chemical reaction of the battery at different temperatures is very different, and the actual working performance is also different. Discharge test experiments are carried out on the battery at different temperatures. The test results are shown in Figure 1b, from which it can be seen that under the condition of the same discharge multiplier, the higher temperature, the greater the amount of power discharged by the battery.

Using 25 °C as the discharge capacity benchmark, the accuracy of SOC estimation at different temperatures was improved by using the temperature correction factor K_T, which represents the ratio of the battery discharge capacity to the benchmark value at different temperatures. The statistics of the fitting results are shown in

Table 3. Temperature and the temperature correction factor.

Temperature (T/°C)	10	25	40
Temperature correction factor (KT)	0.95	1	1.03

. The fitting results are shown in Figure 3. The fitting polynomial is:

$$K_T[T(t)]=-0.00004{[T(t)]}^2-0.0048T(t)+0.9056$$

(3)

According to the ampere–time integration method, the equation of state of the battery SOC is:

$$SOC(t)=SO{C}_0-\frac{{\int }_{t_0}^t\eta \times I(t)dt}{C_N}$$

(4)

where C_N represents the battery calibration capacity, I(t) represent the operating current of the battery, and η represents the charging and discharging efficiency.

The aging as well as self-discharge phenomena of the battery are internal properties of the battery that cannot be avoided and are not analyzed in this paper. After considering the effects of temperature and charge/discharge multiplication on the battery capacity [18], the calculation equation of SOC is corrected as:

$$SOC(t)=SO{C}_0-\frac{K_1{K}_T{\int }_{t_0}^t\eta \times I(t)dt}{C_N}$$

(5)

3. Establishment of the Battery Model and Parameter Identification

3.1. Equivalent Circuit Modeling

In order to research the external characteristics of the battery and make the electrochemical model expression easy to calculate, in this paper, the second-order RC equivalent circuit model is improved by connecting a resistive–capacitive loop in series with the Thevenin model [19,20], as shown in Figure 4. Since the capacitor has infinite resistance, the voltage applied to the capacitor is the electric potential, which is described by EMF = f (SOC) [21].

In Figure 4, EMF denotes the electric potential of the battery as a function of the SOC of the battery, which is expressed by EMF = f (SOC).

The equations for the above equivalent circuit model can be derived from Kirchhoff’s law as:

$$\begin{array}{c}{\mathrm{U}}_{\mathrm{O}\mathrm{C}}=\mathrm{E}\mathrm{M}\mathrm{F}-{\mathrm{R}}_0{\mathrm{I}}_{\mathrm{L}}-{\mathrm{U}}_{\mathrm{P}1}-{\mathrm{U}}_{\mathrm{P}2 }\\ {\mathrm{U}\prime }_{\mathrm{P}1}=-\frac{{\mathrm{U}}_{\mathrm{P}1}}{{\mathrm{R}}_{\mathrm{P}1}{\mathrm{C}}_{\mathrm{P}1}}+\frac{{\mathrm{I}}_{\mathrm{L}}}{{\mathrm{C}}_{\mathrm{P}1}} \\ {\mathrm{U}\prime }_{\mathrm{P}2}=-\frac{{\mathrm{U}}_{\mathrm{P}2}}{{\mathrm{R}}_{\mathrm{P}2}{\mathrm{C}}_{\mathrm{P}2}}+\frac{{\mathrm{I}}_{\mathrm{L}}}{{\mathrm{C}}_{\mathrm{P}2}}\end{array}$$

(6)

where R₀ denotes the ohmic internal resistance of the battery, which is used to describe the resistance generated during the chemical reactions and ionic motion inside the battery. The two resistive–capacitive networks (RC networks) mentioned above are used to describe the polarized internal resistance of the battery, in which R_P1 and C_P1 denote the electrochemical polarization of the battery; R_P2 and C_P2 denote the concentrated polarization of the battery; U_OC denotes the open-circuit voltage of the battery; U_P1 and U_P2 denote the voltage across the polarization capacitor; U’_P1 and U’_P2 denote the derivatives of U_P1 and U_P2 with respect to time, respectively; and I_L denotes the load current in the circuit.

3.2. Parameter Identification

The traditional algorithm for the parameter identification process has the problem of low accuracy, which leads to the accumulation of errors. In this paper, the least squares method is used to identify the battery model parameters. From 10% to 90%, nine SOC points are selected for experimental testing, and the least squares method is used to fit the parameters to EMF = f (SOC). The fitting error is:

$${\delta }_1=\varphi (x_1)-f(x_1)(i=1,··· , m)$$

(7)

The curve fitted by the principle of least squares should satisfy:

$$S(a_0, a_1, ···, a_n)=\sum _{i=1}^m{[a_0{\phi }_0(x_i)+a_1{\phi }_1(x_i)+···+a_n{\phi }_n(x_i)-y_i]}^2=min$$

(8)

where δ is the error of the fitted curve; ϕ(x₁) is the value of the fitted function at x_i; f(x_i) denotes the value of the discrete function at x_i; and S is a collection of the factors to determine “$a_0, a_1, ···, a_n$”.

A ternary lithium battery is selected, with a rated capacity of 11 Ah and an operating voltage range of 3.0 V–4.2 V. The relationship between the OCV and SOC during the charging and discharging of the battery is tested separately, and the test results are shown in

Table 4. OCV and SOC values.

SOC	Discharge/V	Charge/V	Average/V
0.9	4.059	4.056	4.058
0.8	4.003	3.997	4.000
0.7	3.925	3.921	3.923
0.6	3.862	3.856	3.859
0.5	3.761	3.751	3.756
0.4	3.679	3.682	3.681
0.3	3.636	3.640	3.638
0.2	3.594	3.603	3.599
0.1	3.528	3.525	3.527

. The values of EMF corresponding to different SOC values are obtained, as shown in

Table 5. EMF and SOC values.

Variable	Sampling Point
SOC	0.9	0.8	0.7	0.6	0.5	0.4	0.3	0.2	0.1
EMF	4.058	4.000	3.923	3.859	3.756	3.681	3.638	3.599	3.527

.

The experimental data from the above table are applied to fit the parameters to the coefficients to be determined in the expression. The polyfit function is used to derive the expression EMF = f (SOC) as follows:

$$\mathrm{E}\mathrm{M}\mathrm{F}=-0.572391{SOC}^3+1.05177{SOC}^2+0.124234SOC+3.51594$$

(9)

$$\begin{gathered} \mathrm{E}\mathrm{M}\mathrm{F}=14.5833{SOC}^5-39.6402{SOC}^4+39.2519{SOC}^3-16.9309{SOC}^2 \\ +3.61746SOC+3.29892 \end{gathered}$$

(10)

$$\begin{gathered} \mathrm{E}\mathrm{M}\mathrm{F}=-0.374{SOC}^7+1.3015{SOC}^6-1.8258{SOC}^5+1.3186{SOC}^4 \\ -0.5193{SOC}^3+0.1092{SOC}^2-0.0106SOC+0.0039 \end{gathered}$$

(11)

According to the fitting results in

Table 6. Error values for each order of fit.

Fitting Order	SSE	RMSE	R-Square
Third-order fitting	0.001259	0.011828	0.99550
Fourth-order fitting	0.000221	0.004956	0.99921
Fifth-order fitting	0.000052	0.002406	0.99981

, it can be seen that the higher the polynomial order, the higher the fitting accuracy, but this result is easily oscillated. The closer the R-square value is to ±1, the better the curve fit. The comprehensive analysis of this paper uses the fifth-order polynomial to fit the expression of EMF = f (SOC), where SSE denotes the sum of squares due to error, RMSE denotes root mean squared error, and R-square denotes the coefficient of determination.

3.3. Identification of Other Parameters

The HPPC is performed on the battery. The pulse discharge curve at SOC = 0.8 is used in conjunction with the least squares method to identify the parameters of the ohmic internal resistance (R₀), the two polarization internal resistances (R_P1, R_P2), and the polarization capacitance (C_P1, C_P2) in the second-order RC model. The results are shown in

Table 7. Second-order RC equivalent circuit model parameter identification results.

SOC	R0/mΩ	RP1/mΩ	CP1/F	RP2/mΩ	CP2/F
0.9	4.13	0.33	1151.52	4.00	3492.5
0.8	4.01	0.22	2182.82	3.9	3400.00
0.7	4.13	0.21	1952.38	3.5	3054.29
0.6	4.09	0.48	833.33	3.5	3231.42
0.5	4.32	0.45	555.56	3.7	3583.78
0.4	4.18	0.28	1392.86	2.9	3968.97
0.3	3.86	0. 33	1212.12	3.2	4371.88
0.2	4.63	0. 31	1193.55	3.3	4884.85

.

3.4. Accuracy Verification

In order to verify the accuracy of the least squares parameter identification results used in this paper, a simulation model is built in Matlab, and the battery model is verified under HPPC pulse discharge. The experimental results are shown in Figure 5a.

The pulse discharge curve at SOC = 0.8 is selected, as shown in Figure 5b. During the pulse discharge, the error is at 0.03 V–0.04 V, and the polarization phenomenon of the battery is ignored during the experimental measurement because of the presence of the polarization internal resistance. By the later zero-state response interval, the error is close to 0.01 V, and the overall error remains within 0.05 V. As verified above, the model is valid.

4. SOC Estimation Based on the SMFEKF Algorithm

Classical Kalman filtering is mostly used for the state estimation of linear systems; however, the SOC of a battery often behaves as a nonlinear system under the operating condition. EKF is used to filter a nonlinear system into an approximately linear system, and then the state estimation is completed by using classical Kalman filtering. However, when Taylor series expansion is performed for the first-order linearization truncation, the large estimation error of the higher-order terms is neglected, which is likely to lead to problems such as filter divergence.

Therefore, this paper proposes a Suboptimal Multiple Fading Factor Extended Kalman Filter (SMFEKF) algorithm [22]. By introducing multiple fading factors into the state prediction error covariance array, the fading factors are able to fade out different state data at different rates to improve the tracking capability of the system. The error covariance equation of the algorithm after the introduction of the fading factors is:

$$P_{k\left|k-1\right.}={{\lambda }_{k}A}_{k\left|k-1\right.}{P}_{k-1\left|k-1\right.}{A_{k\left|k-1\right.}}^T+T_{k\left|k-1\right.}{Q}_{k-1}{T_{k\left|k-1\right.}}^T$$

(12)

where ${\mathsf{\lambda }}_{\mathrm{k}}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}[{\mathsf{\lambda }}_{\mathrm{k}}^1,{\mathsf{\lambda }}_{\mathrm{k}}^2,···,{\mathsf{\lambda }}_{\mathrm{k}}^{\mathrm{n}}]$ is the suboptimal multiple fading factor diagonal matrix, ${\lambda }_k^i\ge 1(i=\mathrm{1,2},···,n)$, corresponding to n state channels.

By using U_P1 and U_P2 in the second-order RC equivalent circuit model as the state variables of the system, the input quantity as the charging and discharging currents, and the open-circuit voltage, U_OC, as the output quantity, the discretized state equations and observation equations can be derived as:

$$\left\{\begin{array}{c}SOC(k)=SOC(k-1)-\frac{k_I{k}_T\mathsf{\Delta }t}{C_N}{I}_L(k-1) \hfill \\ U_{P1}(k)=U_{P1}(k-1)e^{-\frac{\mathsf{\Delta }t}{{\tau }_1}}+R_{P1}(1-e^{-\frac{\mathsf{\Delta }t}{{\tau }_1}})I_L(k-1)\\ U_{P2}(k)=U_{P2}(k-1)e^{-\frac{\mathsf{\Delta }t}{{\tau }_2}}+R_{P2}(1-e^{-\frac{\mathsf{\Delta }t}{{\tau }_2}})I_L(k-1)\end{array}\right.$$

(13)

$$U_{OC}(k)=\mathrm{E}\mathrm{M}\mathrm{F}(SOC(k))-R_O{I}_L(k)-U_{P1}(k)-U_{P2}(k)+v_k$$

(14)

where $\mathsf{\Delta }\mathrm{t}$ is the time interval of the sampling sequence, τ is the time response constant, and ${\mathsf{\tau }}_1={\mathrm{R}}_{\mathrm{P}1}{\mathrm{C}}_{\mathrm{P}1}$.

Taking the discretized state equations into the nonlinear system, the state variables of the corresponding system is:

$$x_k={[SOC(k){,U}_{P1}(k),U_{P2}(k)]}^T$$

(15)

where the input is ${\mathrm{u}}_{\mathrm{k}}={\mathrm{I}}_{\mathrm{L}}(\mathrm{k})$, the corresponding nonlinear function is:

$$f(x_k,u_k)=\left[\begin{array}{ccc}1& 0& 0\\ 0& e^{-\frac{\mathsf{\Delta }t}{{\tau }_1}}& 0\\ 0& 0& e^{-\frac{\mathsf{\Delta }t}{{\tau }_2}}\end{array}\right]\times \left[\begin{array}{c}SOC(k-1)\\ U_{P1}(k-1)\\ U_{P2}(k-1)\end{array}\right]+\left[\begin{array}{c}-\frac{k_I{k}_T\mathsf{\Delta }t}{C_N}\\ R_{P1}(1-e^{-\frac{\mathsf{\Delta }t}{{\tau }_1}})\\ R_{P2}(1-e^{-\frac{\mathsf{\Delta }t}{{\tau }_2}}\end{array}\right]I_L(k-1)$$

(16)

$$h(x_k,u_k)=\mathrm{E}\mathrm{M}\mathrm{F}(SOC(k))-R_O{I}_L(k)-U_{P1}(k)-U_{P2}(k)$$

(17)

where the corresponding matrices A_k, B_k, H_k, and D_k are:

$$A_k={\left.\frac{\partial f(x_k,u_k)}{\partial x_k}\right|}_{x_k={\widehat{x}}_k}=\left[\begin{array}{ccc}1& 0& 0\\ 0& e^{-\frac{\mathsf{\Delta }t}{{\tau }_1}}& 0\\ 0& 0& e^{-\frac{\mathsf{\Delta }t}{{\tau }_2}}\end{array}\right]; B_k=\left[\begin{array}{c}-\frac{k_I{k}_T\mathsf{\Delta }t}{C_N}\\ R_{P1}(1-e^{-\frac{\mathsf{\Delta }t}{{\tau }_1}})\\ R_{P2}(1-e^{-\frac{\mathsf{\Delta }t}{{\tau }_2}}\end{array}\right]$$

(18)

$$H_k={\left.\frac{\partial h(x_k,u_k)}{\partial x_k}\right|}_{x_k={\widehat{x}}_k}=\left[\begin{array}{ccc}\frac{\partial EMF}{\partial SOC}& -1& -1\end{array}\right]; D_k=-R_0$$

(19)

The computational procedure of the specific SMFEKF algorithm is as follows:

Step 1: Initialization. Assign initial values to the system’s state variable x0, error covariance P₀, system disturbance covariance array Q₀, and observation noise covariance array R₀.

Step 2: System state vector update.

$$\left\{\begin{array}{c}{\widehat{x}}_{k\left|k-1\right.}=f({\widehat{x}}_{k-1\left|k-1\right.},u_{k-1})+w_k \hfill \\ P_{k\left|k-1\right.}={\lambda }_k{A}_{k\left|k-1\right.}{P}_{k-1\left|k-1\right.}{A_{k\left|k-1\right.}}^T+T_{k\left|k-1\right.}{Q}_{k-1}{T_{k\left|k-1\right.}}^T\hfill \\ K_k=P_{k\left|k-1\right.}{H_k}^T{(H_k{P}_{k\left|k-1\right.}{H_k}^T+R_k)}^{-1} \hfill \\ {\widehat{x}}_{k\left|k\right.}={\widehat{x}}_{k\left|k-1\right.}+H_k{{\rm Y}}_k \hfill \\ {{\rm Y}}_k=Z_k-H({\widehat{x}}_{k\left|k-1\right.},u_K) \hfill \\ P_{k\left|k\right.}=(E-{K_{k}H}_k)P_{k\left|k-1\right.} \hfill \end{array}\right.$$

(20)

5. Testing and Analysis

5.1. SOC Estimation by SMFEKF

In order to verify the effectiveness of the algorithms, the EKF and SMFEKF algorithms are used for simulation comparison. The initial value of the SOC is set to 1, the circuit is in the open circuit state, the terminal voltage of the RC link is 0, the initial value of the state variable is ${\left[\begin{array}{ccc}1& 0& 0\end{array}\right]}^{\mathrm{T}}$, and the initial mean squared error is ${\mathrm{P}}_0={10}^{-6}\mathrm{*}\left[\begin{array}{ccc}1& 0& 0;\begin{array}{ccc}0& 1& 0;\begin{array}{ccc}0& 0& 1\end{array}\end{array}\end{array}\right]$. The error covariance of process noise and the error covariance of measurement noise are taken according to empirical parameters. Q_K = 10⁻⁶, R_K = 0.05.

The simulation verification is carried out at 25 °C and 0.5 C constant current discharge conditions, and the simulation results of the two algorithms are shown in Figure 6. From the curves in Figure 6a, it can be seen that the EKF algorithm has a strong tracking ability to the actual SOC at 1000 s and gradually deviates from the reference value later on, with the largest error between 3000 s and 4000 s. The SMFEKF algorithm estimation results are extremely close to the reference value, and the algorithm has strong tracking. From Figure 6b, it can be seen that the absolute error of SOC estimation based on the SMFEKF algorithm is closer to the zero value than that of the EKF algorithm, indicating that the SOC estimation is closer to the real value and converges faster.

In order to demonstrate more intuitively the improvement in the SOC estimation accuracy by this paper’s algorithm, the maximum absolute error\root mean square error and mean error of the SMFEKF algorithm and EKF are compared, and the results are shown in

Table 8. Comparison of estimation errors of the EKF and SMFEKF algorithms.

	εmax	RMSE	MAE
EKF	0.06327	0.03571	0.031692
SMFEKF	0.11873	0.00737	0.001295

.

As shown in Table 8, the maximum error ε_max, the root means square error RMSE, and the mean error MAE estimated by the SMFEKF algorithm are smaller than that of the EKF algorithm. The RMSE of the SOC estimation of the SMFEKF algorithm is 0.00737, which is an improvement in the estimation accuracy by 79.36% compared with the 0.03571 of the EKF algorithm. From the above analysis, it can be seen that the SMFEKF algorithm has better stability performance and higher accuracy

5.2. Algorithm Validation in Conjunction with BMS Platforms

In order to verify the feasibility and accuracy of the battery SOC estimation algorithm, a BMS platform was constructed, including a battery pack formed by a series connection of four 11 Ah Li-ion batteries, a main control board, an acquisition board, a CAN line, and a host computer. Experiments were conducted on the battery pack at constant current discharge and UDDS using a current of 1 C, and then the test results were compared and analyzed with the real value of the SOC of the battery pack. The test results are shown in Figure 7 and Figure 8.

The experimental results show that the maximum error of the algorithm estimation is within 4% under the constant current discharge condition, which is in line with the international standard within 5%. The initial value of the algorithm deviates 45% from the experimental initial value under the UDDS condition, but the initial value converges very quickly in a short period of time, which demonstrates that the algorithm proposed in this paper is still effective when the initial SOC value is unknown or incorrect.

6. Conclusions

Aiming to address the problems of existing SOC estimation methods, such as the accumulation of errors resulting from non-determinable model parameters, this paper proposes a SOC estimation method based on the SMFEKF algorithm and the least squares method. Firstly, the different factors affecting the battery capacity are analyzed by building a battery charging and discharging test platform. The correction coefficients of multiplicity and temperature on the battery capacity are derived through experiments, and the SOC calculation equation is corrected. The second-order RC network equivalent circuit model is selected as the estimation model of the SOC, combined with the least squares method to identify the model parameters, and the pulse discharge simulation test is carried out by Matlab. The test results show that at the beginning of pulse discharge, the error is in the range of 0.03 V~0.04 V, the error is close to 0.01 V in the back of the zero state response interval, and the overall error is kept within 0.05 V, which has high accuracy. The overall error remains within 0.05 V, with high accuracy.

In order to verify the estimation effect of the algorithms, the EKF and SMFEKF algorithms are simulated and analyzed in Matlab. The experimental results show that the estimation error of the SMFEKF algorithm under the constant current discharge condition improves the estimation accuracy by 79.36% compared with the EKF algorithm. Finally, the algorithm is further verified in combination with the EMS platform, indicating that the algorithm is still able to make fast and accurate predictions under the uncertainty in the initial value of the SOC. Therefore, it can be said that the algorithm proposed in this paper is an estimation method with high accuracy and good robustness.