ARWLS-AFEKE: SOC Estimation and Capacity Correction of Lithium Batteries Based on a Fusion Algorithm

Abstract

Accuracy of battery charge status (SOC) estimation plays a significant role in the management of electric vehicle power batteries. However, recently, abrupt changes from SOC data often occurs in the actual operation of electric vehicles and some errors appear in the establishment of battery models and noise models, which give rise to the poorly adaptive and robust performance of traditional algorithms in the process of SOC estimation. The fusion algorithm proposed in this paper can effectively improve the accuracy of models and SOC estimation of lithium-ion batteries. Based on the second-order R-C network model, this method optimizes the accuracy of parameter identification by adopting the adaptive recursive weighted least square algorithm (ARWLS). In addition, the adaptive fading extended Kalman filter algorithm (AFEKF) is applied to estimate the SOC of lithium-ion batteries. Additionally, via introducing a fading factor, the optimal Kalman gain is updated in real-time, which can reduce the impact of data mutation on battery modeling. Compared with the offline AEKF algorithm and the EKF algorithm, the adaptive recursive weighted least square-adaptive fading extended Kalman filter (ARWLS-AFEKF) fusion algorithm had higher accuracy and adaptability, which can be adapted to the variable noise environment.

1. Introduction

As a direct energy source of pure electric vehicles, the lithium-ion battery determines the driving mileage, charging performance, and other important indicators of an electric vehicle. Battery management systems (BMSs) can monitor the real-time status of lithium-ion batteries and adjust the energy supply plan in time, thus achieving the goals of protecting battery packs, improving energy utilization, and optimizing the economy of pure electric vehicles [1]. The state of charge (SOC) is of great importance in parameter monitoring for the BMS whose accuracy of estimation directly determines the reliability and safety of the battery management system. Because of the particularity of the SOC, direct measurement becomes impossible. Only by establishing corresponding battery models and combining different algorithms, can it be estimated indirectly. The accuracy of SOC estimation depends on the choices about different estimation strategies, and the relevant research on estimation strategies mainly focuses on two aspects: the selection and identification of the battery model and the optimization of the SOC estimation method [2]. Battery equivalent models including the Rint model, the R-C network model, and the PNGV model are commonly used by scholars [3]. The second-order R-C network model, which can satisfy the accuracy of SOC estimation under ideal calculation cost, is a widely used circuit model at present. The accuracy of parameter identification relating to electrical components is strongly related to the accuracy of the model. The identification of traditional least square parameters will result in model parameters which lack real-time performance. Compared with the original algorithm, recursive least squares (RLS) are derived from the basic least squares algorithm, significantly improving the timeliness. However, in the case of a long sampling time, the problem of data saturation may occur, which will reduce the identification accuracy of the model [4,5]. Recursive least squares with a forgetting factor (FFRLS) can effectively improve the performance of algorithm tracking, but the selection of the forgetting factor needs to be selected manually [6].

Battery overcharging and over-discharging can be avoided because of the accuracy of SOC estimation, which helps to achieve battery balance and provides important data support for the calculation of electric vehicle mileage. Scholars at home and abroad have conducted in-depth studies on SOC estimation and its methods. The commonly used estimation methods are as follows: the ampere-hour integration approach (AH) [7], open-circuit voltage method (OCV) [8], and neural network method [9]. In practical applications, many algorithms or models are usually combined to solve the shortcomings of a single algorithm. Afshar et al. [10] proposed a combination algorithm based on the EKF support vector machine (EKF-SVM) to solve the problem that a single SOC estimation algorithm cannot satisfy multiple criteria simultaneously. Examples have shown that the combined algorithm combines the advantages of EKF and SVM, improving the robustness and generalization of SOC estimation and reducing its maximum estimation error by less than 1%. Based on the traditional wavelet neural network (WNN), Huang et al. [11] proposed a PCA-GA-WNN model by combining it with principal component analysis (PCA) and genetic algorithm (GA). Additionally, the simulation results show that the model having a strong global search ability and convergence ability can achieve high accuracy of SOC estimation with fewer iterations. Based on Thevenin’s equivalent circuit model, Shrivastava et al. [12] proposed a Forgotten Factor Least Squares (FFRLS) algorithm combined with an adaptive Extended Kalman Filter (AEKF) algorithm to estimate SOC. The results showed that the algorithm achieves closed-loop correction of the model system, improving the accuracy of FFRLS and making the SOC error within 1.5–2%. Based on traditional EKF, Ling and Wei [13] proposed an improved Extended Kalman Filter (I-EKF) algorithm and combined least squares to reduce the impact of model parameter changes on the algorithm. The simulation results showed that the error of the I-EKF algorithm was reduced by 1.5% at the end of battery discharge compared with traditional EKF. Wang et al. [14] proposed to identify battery parameters based on the improved particle swarm optimization for overcoming local optimum and improving convergence speed and accuracy. The algorithm reduced the relative error of the basic particle swarm by 3%. In summary, theoretical research has made significant progress in improving the accuracy of SOC estimation and fulfilling various index requirements.

Overall, the main contributions of this paper are listed as follows: (I) based on the above analysis, this paper uses the second-order R-C network model as the battery model, introducing dynamic genetic factors to optimize the accuracy of the least squares method in estimating the later stage through an adaptive weighting algorithm; (II) considering the noise variation during data collection, the noise matrix using adaptive fading extended Kalman filter (AFEKF) is updated in real-time; (III) then, the SOC of lithium batteries is estimated online by combining the adaptive recursive weighted least square (ARWLS); and (IV) finally, the adaptability and robustness of the adaptive recursive weighted least square-adaptive fading extended Kalman filter (ARWLS-AFEKF) fusion algorithm is verified under three dynamic conditions with a result showing that the ARWLS-AFEKF algorithm has high estimation accuracy and strong immunity under different dynamic conditions.

2. Mathematical Modeling of Power Batteries

2.1. Definition of SOC

For power lithium-ion batteries, the SOC is the ratio of the remaining available charge to maximize the available charge of the battery. This value is usually expressed as a percentage between 0 and 1 [15], and the method of calculation is shown in the following Equation (1):

$$SOC\left(t\right)=SOC\left(0\right)-\frac{{\displaystyle \int i_{c}dt}}{Q_n}$$

(1)

where $i_c$ is the instantaneous current, positive discharge current, and negative charge current, while $Q_n$ is the rated charge of the battery. $SOC\left(0\right)$ is the initial value of SOC and $SOC\left(t\right)$ is the SOC at time $t$. Equation (2) is discretized to satisfy the characteristics of the current data collected in the experiment:

$$SOC\left(k+1\right)=SOC\left(k\right)-\frac{{\displaystyle \sum _{j=1}^k{i}_j\cdot \Delta t}}{Q_n}$$

(2)

where $\Delta t$ is the sampling period, $k$ is the sampling time, and $SOC\left(k\right)$ is the SOC value of the battery at time $k$.

2.2. The Construction of Equivalent Battery Model

The accuracy and complexity of the model largely determine the accuracy of SOC estimation [16]. Taking the application of integrated engineering into practical consideration, an appropriate model with more complexity should be selected on the premise of accurately characterizing the dynamic performance of batteries. The Rint model which contains only one resistance does not reflect the polarization of batteries under frequent charging and discharging [17]. Serial RC circuits can attenuate the high-frequency signal in the circuit. The commonly used Thevenin and PNGV models can simulate electrochemical reactions inside batteries with poor results [18,19]. The third-order RC circuit model can simulate the internal battery reaction better. Although the third-order RC circuit model has a good application in simulating the internal reaction of batteries, it is not commonly used in practical projects because of the amount of computation. Therefore, the second-order RC model was chosen as an equivalent circuit model in this paper to accurately simulate the battery polarization effect. Additionally, it is simple and easy to implement the algorithm in engineering, as shown in Figure 1.

Where $U_{OC}$ represents open-circuit voltage, $R_0$ is battery ohm internal resistance, $C_1$, $C_2$ is polarized capacitance, $R_1$, $R_2$ is polarized resistance, $U_L\left(t\right)$ is circuit end voltage, $I$ is load current. Current end voltage $U\left(t\right)$ and circuit current $I$ can be measured by the sensor in real time. The mathematical model of the second-order R-C circuit is obtained by solving the circuit with Kirchhoff’s Law (KVL) and Current’s Law (KCL) as shown in Equation (3).

$$\{\begin{array}{c}{U}_L\left(t\right)=U_{OC}-I\left(t\right)\cdot R_0-U_1\left(t\right)-U_2\left(t\right)\\ \frac{d{U}_1\left(t\right)}{dt}=-\frac{U_1\left(t\right)}{R_1{C}_1}+\frac{I\left(t\right)}{C_1}\\ \frac{d{U}_2\left(t\right)}{dt}=-\frac{U_2\left(t\right)}{R_2{C}_2}+\frac{I\left(t\right)}{C_2}\end{array}$$

(3)

For the selected second-order equivalent model, the state variable is chosen as [$SOC$ $U_1$ $U_2$]. The definitions of the combination of Equation (3) and the definition of SOC are discretized, and their discrete state space equations are shown in Equation (4).

$$\left[\begin{array}{c}SOC\left(k+1\right)\\ U_1\left(k+1\right)\\ U_2\left(k+1\right)\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& e^{-\frac{1}{R_1{C}_1}}& 0\\ 0& 0& e^{-\frac{1}{R_2{C}_2}}\end{array}\right]\left[\begin{array}{c}SOC\left(k\right)\\ U_1\left(k\right)\\ U_2\left(k\right)\end{array}\right]+\left[\begin{array}{c}-\frac{\Delta t}{Q_n}\\ R_1\left(1-e^{-\frac{1}{R_1{C}_1}}\right)\\ R_2\left(1-e^{-\frac{1}{R_2{C}_2}}\right)\end{array}\right]I\left(k\right)+\left[\begin{array}{c}{\omega }_1\left(k+1\right)\\ {\omega }_2\left(k+1\right)\\ {\omega }_3\left(k+1\right)\end{array}\right]$$

(4)

The noise from the process and measurement is represented by ${\omega }_i\left(i=1,2,3\right)$ and $v$ in the formulas above to characterize the uncertainty of the model in the process of estimation. Among them, the parameters to be identified in the model are open-circuit voltage $U_{OC}$; ohm internal resistance $R_0$; polarized internal resistance $R_1$, $R_2$; and polarized capacitance $C_1$, $C_2$.

2.3. ARWLS Online Parameter Identification

Lithium batteries undergo complex internal chemical reactions during charging and discharging. With the change in the working state, the parameters of each electrical element in the battery model will also change, which will affect the estimation of SOC. The online identification method analyzes the current and voltage values which are collected to correct the current model parameters and update them in real-time. Compared with the offline identification method, this method with strong adaptability and high accuracy can accurately reflect the real-time characteristics of the circuit system. Because of its simple structure and fast convergence, the least squares method has been widely used in parameter estimation. The Laplace transformation is used for Equation (3), and the observation equation in the s domain is obtained as shown in Equation (5).

$$U_L\left(s\right)-U_{OC}\left(s\right)=I\left(s\right)\left[\frac{R_1}{1+s{R}_1{C}_1}+\frac{R_2}{1+s{R}_2{C}_2}+R_0\right]$$

(5)

In Equation (5), $s$ is a Laplace operator on the $s$ domain, noting that $G(s)$ is the transfer function of the battery model in the s domain. Combined with the bilinear transformation $S$, the transfer function in the $Z$ domain can be obtained as shown in Equation (6).

$$G\left(Z^{-1}\right)=\frac{{\lambda }_3+{\lambda }_4{Z}^{-1}+{\lambda }_5{Z}^{-2}}{1-{\lambda }_1{Z}^{-1}-{\lambda }_2{Z}^{-2}}$$

(6)

where ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$, ${\lambda }_4$, and ${\lambda }_5$ are the undetermined coefficients. Differential treatment of Equation (6) yields a different formula of Equation (7).

$$y\left(k\right)={\lambda }_{1}y\left(k-1\right)+{\lambda }_{2}y\left(k-2\right)+{\lambda }_{3}I\left(k\right)+{\lambda }_{4}I\left(k-1\right)+{\lambda }_{5}I\left(k-2\right)$$

(7)

where $y(k)$ is the difference between the open-circuit voltage and the end voltage at time $k$. Set up $\phi$ and convert Equation (7) to the least squares form $Y=\phi \times \theta$, where $\phi =\left[y\left(k-1\right),y\left(k-2\right),I\left(k\right),I\left(k-1\right),I\left(k-2\right)\right]$ and $\theta ={\left[{\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5\right]}^T$.

To distinguish it from the method of traditional least squares, the method of adaptive amnesia factor recursive least squares is adopted in this paper. Considering the phenomenon of “data saturation” caused by the influence of old data on a large amount of data by a single recursive least squares method, variable genetic factors are introduced in this paper to overcome this effect so that the characteristics of the new data can react in real-time. At the same time, dynamic genetic factors modify the proportion of old and new data in the recursive model to improve the accuracy of online identification. The expression of the recursive equation for the adaptive forgetting factor is as follows:

$$\{\begin{array}{c}\lambda \left(k\right)=1-\frac{e^2\left(k\right)}{\phi \left(k-1-l\right)P\left(k-1\right){\phi }^T\left(k-1-l\right)\cdot r}\\ \widehat{\theta }\left(k\right)=\widehat{\theta }\left(k-1\right)+K\left(k\right)\cdot e\left(k\right)\\ K\left(k\right)=\frac{P\left(k-1\right){\phi }^T\left(k-1\right)}{\lambda \left(k\right)+\phi \left(k-1\right)P\left(k-1\right){\phi }^T\left(k-1\right)}\\ P\left(k\right)=\frac{1}{\lambda \left(k\right)}\left[E-K\left(k\right)\phi \left(k-1\right)\right]P\left(k-1\right)\end{array}$$

(8)

where $E$ is the unit matrix, $P$ is the error covariance matrix, $K$ is the gain matrix, $\lambda$ is the variable genetic factor, $l$ and $r$ are the weighted adjustment coefficients, and $e(k)$ is the estimation error.

After converting the denominator in Equation (5) to the same denominator as that in Equation (6), a comparison between them will be close behind. Then, the identification equations of electrical component parameters can be obtained by using the undetermined coefficient method.

$$\{\begin{array}{c}{\tau }_1+{\tau }_2=\frac{(1+{\lambda }_2)\Delta t}{1-{\lambda }_1-{\lambda }_2}\\ {\tau }_1\cdot {\tau }_2=\frac{{\Delta }^{2}t(1+{\lambda }_1-{\lambda }_2)}{4(1-{\lambda }_1-{\lambda }_2)}\\ R_0+R_1+R_2=\frac{{\lambda }_3+{\lambda }_4+{\lambda }_5}{1-{\lambda }_1-{\lambda }_2}\\ R_0{\tau }_1{\tau }_2=\frac{{\Delta }^{2}t({\lambda }_3-{\lambda }_4+{\lambda }_5)}{4(1-{\lambda }_1-{\lambda }_2)}\\ R_0{\tau }_1+R_0{\tau }_2+R_1{\tau }_2+R_2{\tau }_1=\frac{({\lambda }_3-{\lambda }_5)\Delta t}{1-{\lambda }_1-{\lambda }_2}\end{array}$$

(9)

First, use Equation (8) to estimate the optimal parameter matrix $\theta$ in real-time, then substitute the estimated parameters into Equation (9), and finally the parameters of various electrical components are obtained in the battery model. The method of adaptive recursive least square can accurately estimate the parameters of electrical components, which provides a basis for establishing the accurate state space equations below.

3. Estimation of SOC for Lithium Batteries Using Adaptive Fading Extended Kalman Filter

3.1. Principle of SOC Estimation for AFEKF

In an online system, the Kalman filter algorithm is an optimal estimation algorithm with minimum variance. In the online system, the Kalman filter algorithm has the minimum variance linear unbiased estimation, which is an optimal estimation algorithm. It makes full use of the measured data and filters out random noise recursively to obtain accurate spatial state values. However, the state space of lithium-ion batteries is nonlinear. To get an approximately linear relationship, the Taylor expansion is introduced in EKF to linearize the model and omit the higher-order terms [20,21]. Then, EKF is optimized based on Kalman algorithm, the batteries’ spatial state equation with observation equation scores which are described as follows.

$$\{\begin{array}{c}x\left(k+1\right)=f\left[x\left(k\right),i\left(k\right)\right]+\omega \left(k\right)=A\left(k\right)x\left(k\right)+B\left(k\right)I\left(k\right)+\omega \left(k\right)\\ y\left(k+1\right)=h\left[x\left(k\right),i\left(k\right)\right]+v\left(k\right)=C\left(k\right)x\left(k\right)+D\left(k\right)I\left(k\right)+v\left(k\right)\end{array}$$

(10)

where $x\left(k\right)$ is the state vector of the system at time $k$, $i\left(k\right)$ is the input current, and $y\left(k\right)$ is the observation value. $h$ and $f$ represent the observation function and the state transfer function, respectively. The white noise of a system is equipped with a mean of zero $\omega \left(k\right)$ and a covariance of $Q\left(k\right)$. White noise carries a mean of zero $v\left(k\right)$ and a covariance of $R\left(k\right)$. Moreover, $\omega \left(k\right)$ and $v\left(k\right)$ are independent of each other. $A\left(k\right)$ represents the state transition matrix, $B\left(k\right)$ is the system that controls the matrix, $C\left(k\right)$ is the observation matrix, and $D\left(k\right)$ is the observation control matrix.

In the standard extended Kalman filter, the Kalman gain matrix tends to be minimal when the system is stable. If a state mutation occurs at this time, the error covariance matrix will increase. Because of the offline calculation of Kalman gain, the change cannot be perceived in time, which leads to the result that the Kalman filter cannot track the real state of the system in real-time. To improve the estimation accuracy, this paper adopts an AFEKF method that can reduce system noise. In this paper, by introducing an adaptive fading factor to the adaptive fading of system noise, the error covariance matrix is modified in real-time, and the filter gain matrix is adjusted online so that the residual signals at different times remain orthogonal. The detailed theoretical analysis is as follows:

The fading factor $\lambda (k)$ is introduced into the prior estimation error covariance matrix $P(k)$, and the system noise is correlated before and after.

$$P^{-}\left(k\right)=A\left(k-1\right)P^{+}\left(k-1\right)A^T\left(k-1\right)+\lambda \left(k\right)Q\left(k-1\right)$$

(11)

Next, the adaptive fading factor is determined:

The first residual vector estimated is $V(k)$, and its covariance matrix is defined as follows:

$$Z\left(k\right)=E\left[V\left(k\right)V^T\left(k\right)\right]=C\left(k\right)P^{-}\left(k\right)C^T\left(k\right)+R\left(k\right)$$

(12)

If the EKF state estimation error $ax$ is much smaller than the true value $x(k)$, the residual vector autocorrelation function will be valid.

$$\begin{array}{l}E\left[V\left(k\right)V^T\left(k\right)\right]\approx C\left(k+j\right)A\left(k+j-1\right)\ast \left[\mathrm{I}-\mathrm{K}\left(k+j-1\right)C\left(k+j-1\right)\right]\ast \\ A\left(k+1\right)\ast \left[\mathrm{I}-\mathrm{K}\left(k+1\right)C\left(k+1\right)\right]\ast A\left(k\right)\ast \left[P^{-}\left(k\right)C^T\left(k\right)-{\rm Z}\left(k\right)\mathrm{K}\left(k\right)\right]\end{array}$$

(13)

When the filter condition is optimal, the residual sequence output from the system is a Gaussian white noise sequence with the residual vectors that are orthogonal to each other, and the auto correlation function $E\left[V\left(k+j\right)V^T\left(k\right)\right]$ equals to 0, which results in $P^{-}\left(k\right)C^T\left(k\right)-\mathrm{K}\left(k\right)Z\left(k\right)=0$. However, in the actual process of estimation, uncontrollable factors always interfere with the estimation process, so it is necessary to adjust the gain matrix $K(k)$ for making the expression $P^{-}\left(k\right)C^T\left(k\right)-\mathrm{K}\left(k\right)Z\left(k\right)=0$ valid and forcing the residual vectors to be orthogonal to each other, ensuring that Kalman’s gain is optimal, updating the prior estimation error covariance matrix and the Kalman gain matrix to the input Equation (13) and simplifying the input formula to obtain:

$$\lambda \left(k\right)C\left(k\right)Q\left(k-1\right)C^T\left(k\right)=Z\left(k\right)-C\left(k\right)A\left(k-1\right)P^{+}\left(k-1\right)A^T\left(k-1\right)C^T\left(k\right)$$

(14)

$N(k)$ and $M(k)$ are simplified to get $N\left(k\right)=\lambda \left(k\right)M\left(k\right)$. Traces are simultaneously found on both sides of the equation:

$$\lambda \left(k\right)=\{\begin{array}{c} \lambda \left(k\right), \lambda \left(k\right)>1;\\ 1, \lambda \left(k\right)\le 1;\end{array}$$

(15)

$$\lambda \left(k\right)=\frac{tr\left[N\left(k\right)\right]}{tr\left[M\left(k\right)\right]}$$

(16)

The above procedure is to find the fading factor, which requires less computation and is easier to implement than the current method for solving the fading factor. When problems such as modeling errors, data mutations, and improper selection of initial state values arise, the fading factor makes $P^{-}\left(k\right)C^T\left(k\right)-\mathrm{K}\left(k\right)Z\left(k\right)=0$ hold by adjusting the residual weights in real time. In this way, error accumulation is avoided, and the Kalman gain is close to optimal.

3.2. ARWLS-AFEKF Fusion Algorithmic Principle

In Section 2.3 and Section 3.1, the implemented process and details of the ARWLS and AFEKF are described in detail. By combining the two algorithms, the estimation of SOC for lithium batteries can be accurately achieved. The detailed flow chart is shown in Figure 2.

Using the collected current and voltage data, the real-time estimation is carried out with the method of ARWLS, and the output parameter matrix is converted to the parameters of each electrical element by Equation (9). The parameter is substituted into Equation (10) to estimate SOC based on the AFKF algorithm. The open-circuit voltage $U_{oc}$ at the estimated SOC value is obtained from the OCV–SOC curve, which allows it to participate in the next cycle. From this loop recursion, the parameter values and SOC of the circuit model can be obtained at any time. The ARWLS-AFEKF fusion algorithm identifies the parameters of the battery model online, realizing the closed-loop correction of the noise matrix and the forgetting factor through a recursive process, which improves the robustness of the whole system and reduces the sensitivity of the algorithm to noise.

3.3. SOC Capacity Correction Based on the Fusion Algorithm

Given the error caused by the SOC estimation algorithm and battery capacity that was regarded as constant, a modified fusion method with capacity correction is proposed, in which the initial SOC value of the battery is determined by open-circuit voltage and then corrected by correction factors that are related to the charge/discharge current, the Coulombic efficiency, and temperature, and obtained by the charge and discharge tests of the battery pack [22,23]. To accurately estimate the SOC during charging and discharging, the capacity correction of the fusion method should be defined as follows:

$$SOC\left(t\right)=SOC\left(0\right)-\frac{{\displaystyle \int i_c\eta dt}}{Q_n{\eta }_i{\eta }_T{\eta }_H}$$

(17)

where $\eta$ defines Coulomb efficiency, ${\eta }_i$ represents the correction factor of the multiple rates to the battery capacity, ${\eta }_T$ describes the correction factor for battery capacity by temperature, and ${\eta }_H$ defines the correction factor for battery capacity for cycle life. $SOC\left(0\right)$ can be obtained by the open-circuit voltage method. Additionally, the parameters $\eta$, ${\eta }_i$, ${\eta }_T$, and ${\eta }_H$ can be obtained by charging and discharging the battery pack.

4. Results and Discussion

4.1. Parameter Settings

A platform for battery tests was set up to obtain the capacity correction factors of power, coulomb efficiency, and temperature through tests of the battery pack charge and discharge. The whole test platform includes a power battery pack, battery charge, discharge monitoring device, and control computer, as shown in Figure 3. The power battery charging and discharging can be edited by the test management software configured by the system to achieve automated testing, and the current and voltage parameters of the battery during charging and discharging are obtained and recorded in real time.

There is a strong correlation between the open-circuit voltage (OCV) and SOC of lithium-ion batteries. Determining the function relationship between the open-circuit voltage and battery SOC under static conditions plays a significant role in SOC estimation. Since the functional relationship between SOC and OCV does not change much in the temperature range of 20–40 °C, this paper choses to carry out hybrid pulse power characteristic (HPPC) experiments on lithium batteries under ambient temperature (25 °C) to obtain OCV–SOC curves. The relevant parameters of batteries used in this experiment are shown in

Table 1. Lithium-ion battery parameters.

Parameters	Value
Rated capacity	3 A · h
Rated voltage	3.6 V
Charge cut-off voltage	4.2 V
Charge cut-off current	0.15 A
Discharge cut-off voltage	2.75 V
Density of energy	166 wh · kg−1

. Because the OCV–SOC curves of lithium batteries change more sharply at both ends, to obtain more accurate OCV–SOC curves this paper records the corresponding open-circuit voltage at 5% intervals between SOC less than 10% or greater than 90% under BJDST25, DST25, and FUDS25 operating conditions.

4.2. Determination of Initial SOC Value

The SOC of the battery pack is estimated according to Equation (17), and the initial SOC of the battery pack is estimated by the open-circuit voltage method. The open-circuit voltage method obtains the OCV of the battery after it has been charged and discharged to a stable state. Then, the SOC of batteries can be calculated based on the functional relationship between OCV and SOC. The basic method is to carry out intermittent charging and discharging tests on equal-capacity batteries and get OCV values at different SOC points in the range of 0–1.0. Then, the corresponding relationship between OCV and SOC is obtained by looking up tables or fitting. When the OCV of the battery is known, the SOC value can be estimated based on the OCV–SOC relationship curve. The test steps are as follows: (1) at room temperature, the battery pack is fully charged by a method of constant current and pressure, and the battery pack is not stationary until it is in a stable state. (2) At the discharge rate of 0.5 C, 5% of the battery is discharged. The battery pack stands for 0.5 h and records its terminal voltage, which is used as the OCV value under the current SOC. (3) Repeat step 2 until the battery end voltage drops to 25 V or the battery SOC drops to 0. The test results between OCV and SOC are shown in Figure 4. When the battery management system is running, initial SOC values are obtained by looking up the table based on the measured OCV. When the battery management system is running, initial SOC values are obtained by looking up the table based on the measured OCV. The battery pack is in a stable state before the discharge starts, and the terminal voltage of the battery is approximately equal to the value of OCV.

4.3. Correction Factor Verification

The discharge ratio of batteries greatly affects the capacity of batteries. Therefore, discharging at different discharge rates will affect the estimation of the SOC on the battery pack due to differences in the actual discharge capacity of the battery pack. Electric vehicles have frequented current changes during driving. Based on understanding the characteristics of battery pack capacity varying with discharge rate, real-time correction of battery pack capacity can improve the estimation accuracy of SOC. To explore the capacity characteristics of batteries at different discharge rates, the battery packs are charged and discharged at multiples of 0.25 C, 0.50 C, 0.75 C, 1.00 C, 1.25 C, 1.50 C, and 2.00 C. With the increasing rate of discharge, the discharge capacity of the battery pack tends to decrease, but the decrease is not significant. The capacity values at different multiples are obtained by experiments as shown in Figure 5. Based on the capacity data of the 0.19 C discharged rate provided by the manufacturers, the capacity correction factors for 0.25 C, 0.50 C, 0.75 C, 1.00 C, 1.25 C, 1.50 C, and 2.00 C are calculated as shown in Figure 5a. The capacity correction factors at different multiples are obtained by fitting. Then, the capacity correction factor ${\eta }_i$ corresponding to other multiples can be calculated by interpolation. In addition, the Coulomb efficiency represents the ratio of the actual discharge to the actual charge when the battery is operating, and it is also known as the discharge efficiency of the battery. The Coulomb efficiency affects the remaining battery capacity, which affects the accuracy of battery SOC estimation. Coulomb efficiency is mainly related to temperature and discharge rate as shown in Figure 5b. To explore the influence of discharge rate on Coulomb efficiency, a battery charge and discharge test was designed and conducted at room temperature.

4.4. Verification of Algorithm Performance under Different Operating Conditions

The proposed ARWLS-AFEKF algorithm was validated under three operating conditions: BJDST25, DST25, and FUDS25. Because of the difficulty to obtain the initial value of SOC, this paper defines the SOC obtained by the AN-TIME integral in the laboratory as a benchmark value, which is defined as the reference value. To demonstrate accuracy and stability, the offline identification EKF algorithm, AEKF algorithm, and ARWLS-AFEKF algorithm were used to estimate SOC performance compared with the ARWLS-AFEKF fusion algorithm. The random function was used to simulate the mean and covariance of randomly generated noise during the simulation. The result is shown in Figure 6. At the beginning of the simulation, the SOC curve cannot be tracked very well through the EKF algorithm and AEKF algorithm, causing a large error fluctuation. This is because the noise means and covariance matrix of the EKF algorithm and AEKF algorithm are fixed values, which makes them unable to adapt to the random noise generated in the simulation process. However, the ARWLS-AFEKF algorithm quickly converged to the SOC reference value with no more than 2% error by updating the noise state parameters in real time. In the middle of the simulation, the rate of error in the EKF algorithm and AEKF algorithm decreased, but still with a rate of more than 2%. Compared with the EKF algorithm and AEKF algorithm, the AWRLS-AFEKF algorithm produced less fluctuation and kept errors within 2%. At the end of the simulation, the changes in battery parameters were increasing. Since the EKF algorithm and AEKF algorithm do not form closed-loop feedback between the offline parameter identification process and the EKF filter process, their self-correction abilities were weak and the identification parameters could not be adjusted in time, thus causing an increasing error rate of SOC estimation by more than 4%. By introducing a fade factor, the AFEKF method adapted to the iterative system noise covariance and updated the Kalman gain in real-time so that the residual vector in the estimation process always remained orthogonal and everywhere, improving the adaptive tracking ability for SOC. Compared with the standard EKF algorithm and AEKF algorithm, the proposed method had the advantages in terms of estimation accuracy and robustness.

As shown in Figure 7, due to the intentionally incorrect setting of the initial SOC value of the battery, EKF and AFEKF simulation curves had a large deviation from the real value at the beginning. however, based on the ARWLS-AFEKF algorithm, the estimation curve converged to the real value at about 400 s and the whole subsequent estimation process was consistent with the real value curve. In contrast, the standard EKF and AEKF algorithms slowly approached the true value at about 380 s, and in the subsequent estimation process, the fit degree with the true value was worse than the ARWLS-AFEKF algorithm.

4.5. Reliability Verification of ARWLS-AFEKF Fusion Algorithm

Both the root means square error (RMSE) value and mean absolute percentage error (MAPE) value are usually used as indicators to evaluate the accuracy of a regression model, and the smaller the value is, the higher the accuracy of the model will be. Yang et al. proposed that RMSE and MAPE could be used to characterize the accuracy and reliability of the algorithm when studying SOC estimation of lithium-ion batteries [24]. The data of each sampling point of the test voltage is regarded as a set of points in the discrete space, and the SOC curve fitted by the algorithm is regarded as a regression equation. The RMSE and MAPE values under different dynamic operating conditions are shown in

Table 2. RMSE value and MAPE value for fusion algorithm under different operating conditions.

Operating Conditions	RMSE/%			MAPE/%
	EKF	AEKF	ARWLS-AFEKF	EKF	AEKF	ARWLS-AFEKF
DST25	1.53	0.25	0.22	1.66	2.88	0.61
BJDST25	2.17	0.23	0.20	2.31	2.42	0.67
FUDS25	3.59	10.96	0.37	3.88	827.39	0.71

. In addition, the comparison of common SOC estimation methods is described in

Table 3. Comparison of common SOC estimation methods.

Algorithms of SOC Estimation	Advantages	Disadvantages	Robustness
EKF	No need to linearize the model and no model linearization error	Easier to scatter under perturbation and initial uncertainty	Poorer
AEKF	Overcoming the shortcoming of traditional method and suitable for the occasion of drastic current change	The model linearization process generates errors	Poor
ARWLS-AFEKF	Strong anti-interference ability	Computationally heavy	Good

.

4.6. Typical Application Scenario Analysis

The storage for power system energy is a national strategic emerging industry, and it also plays an important role in smart grid and high-proportion renewable energy systems [25]. As the size of the energy storage system expands, the arrangement of its internal batteries becomes more and more complex. If the batteries inside the storage system are not sorted accurately, inconsistencies in the battery pack may result, and more serious security problems, such as overcharging of the storage system, may appear. SOC is an important parameter to check the inconsistency of battery packs, and its accurate estimation is conducive to the safe and stable operation of power system storage. Considering the strong non-linearity, irregularity, and wide range of operating conditions of lithium-ion batteries for energy storage scenarios, traditional SOC estimation methods have some limitations [26]. Therefore, it is necessary to collect the related data in the multi-space-time scale of the energy storage system, and then build a comprehensive SOC estimation model that fuses the multi-source information to ensure that the estimation results can more accurately reflect the actual working status of the energy storage battery.

Methods of data fusion series are not only dependent on the internal characteristics of battery operation but are also applicable to highly non-linear systems. Therefore, they are widely used in estimating the SOC of Li-ion batteries for power system storage. Multi-source data fusion technology refers to the processing of multi-source information, which is related to the SOC of lithium-ion batteries by computer. Liang et al. proposed a SOC estimation method for Li-ion batteries. This method can continuously correct the key parameter values and reduce the estimation error according to the characteristic curves of batteries under different operating conditions [27]. The simulation results showed that the error of this method in estimating the SOC of the energy storage battery was generally less than 1%. Based on data fusion technology, Mostyn et al. designed a method of SOC estimation for energy storage batteries by investigating the actual operation data of Li-ion batteries in an energy storage system and extracting their characteristic curves [28]. The analysis results showed that the average error of the SOC values estimated by this method was less than 1%, which validates the feasibility and practicability of this method.

However, this type of method has high hardware requirements. With the progress and development of science and technology, the accuracy of computing and the speed of processors will be greatly improved. In addition, the related data can be pre-processed to effectively reduce the computing burden of hardware. Therefore, this kind of method has a strong application prospect in the field of power system energy storage. In addition, since the current database may not reflect the internal and external characteristics of lithium-ion batteries adequately, it is necessary to collect and fully excavate the operational data of lithium-ion batteries, and improve the estimation accuracy of SOC of Li-ion batteries.

5. Conclusions

To reduce the estimation error of SOC in BMSs, this paper used a lithium-ion battery as the research object, used a second-order R-C network as the battery model, and introduced the variable forgetting factor optimization parameter identification process by an adaptive weighting algorithm. At the same time, considering the change in noise, the new information matrix was used to update the noise parameters and complete the construction of the AFEKF algorithm. The ARWLS-AFEKF algorithm was used to estimate the SOC of the battery online. The results show that the ARWLS-AFEKF algorithm can realize the online identification of battery model parameters and provide reliable support for SOC estimation. In BJDST25, DST25, and FUDS25, the ARWLS-AFEKF algorithm can adapt to the changing noise environment, and the error with the real SOC was within 2%, and RMSE and MAPE values were within 1%. Compared with the offline identification used by the EKF algorithm and AEKF algorithm, the ARWLS-AFEKF algorithm has the advantages in terms of the SOC estimation, the convergence speed, and the robustness.