Research on Estimation Optimization of State of Charge of Lithium-Ion Batteries Based on Kalman Filter Algorithm

Abstract

Accurate prediction of the State of Charge (SOC) of lithium-ion batteries is the foundation for the stable and efficient operation of battery management systems. This paper proposes a lithium-ion battery SOC estimation method based on the Dung Beetle Optimizer (DBO), optimizing the second-order Kalman filter algorithm (DBO-DKF). Leveraging the DBO’s fast convergence speed and strong global search capability, this method optimizes the Kalman filter algorithm in the parameter identification stage and the extended Kalman filter algorithm in the SOC estimation stage to address the issue of insufficient estimation accuracy caused by noise covariance matrices of input current and voltage measurements. Through the discharge of current tests under complex conditions, as well as comparing and analyzing credibility indicators such as MAE, RMSE, and MSE as measures of estimation accuracy, it can be verified that the proposed method effectively enhances SOC estimation accuracy.

1. Introduction

With the increasing number of cars in the world, the enormous fossil energy consumed by fuel vehicles every year has become an urgent problem that every government has to consider and solve. Electric vehicles play an important role in environmental protection, the construction of new energy systems, innovation in energy storage technology, and sustainable energy development, and have received strong policy support from governments in multiple countries [1,2,3]. Lithium-ion batteries, as the power source of electric vehicles, are one of the core components of vehicles’ electrical systems, energy management systems, and thermal management systems. They are usually composed of thousands of lithium-ion batteries connected in series or parallel to form battery modules, providing the required capacity and voltage for the operation of electric vehicles. Accurate estimation of the State of Charge (SOC) of a power battery during charging and discharging is crucial for guiding the driving behavior of electric vehicle users. Due to factors such as environmental temperature, charge and discharge rates, and series and parallel connection mode of the battery pack, lithium-ion batteries undergo electrolyte decomposition, increase in internal resistance, SEI film growth, and gradual decrease in electrode active material activity [4]. This results in time-varying and nonlinear changes in the internal parameters of the battery due to complex electrochemical reactions during operation. To obtain accurate SOC of the battery, it is necessary to establish a battery equivalent circuit model that can accurately reflect the dynamic/static characteristics of the battery, as well as an estimation method with high accuracy and suitable computational complexity for engineering applications.

The estimation methods based on battery models are divided into electrochemical models (EM), electrochemical impedance models (EIM), and equivalent circuit models (ECM). The electrochemical model uses partial differential equations to calculate the potential changes of lithium-ion batteries in electrochemical reactions and obtain the SOC of the battery. This method can accurately reflect the chemical reaction principles inside the battery, but it cannot determine the trend of changes in various internal parameters, and the computational complexity is too large to be used in the engineering field [5,6,7]. The electrochemical impedance model uses the method of applying current disturbance to the battery and measuring voltage response to obtain the electrochemical impedance spectrum in the frequency domain and to get the battery SOC [8,9]. This method can acquire the internal characteristics of the battery without being affected by external factors, but in practical applications, there are difficulties in obtaining impedance spectra, and strict control of the testing environment is required. The equivalent circuit model considers the battery as a two-port network and uses circuits composed of different types of electrical components to simulate the voltage and current changes during battery operation [10]. It is a commonly used battery model for estimating the charging state of lithium-ion batteries and has the advantages of simple structure and high accuracy [11]. Therefore, the equivalent circuit model has attracted the attention of many researchers. Chen et al. [12] used voltage response data as feature points and established an RC first-order model of the battery to achieve offline identification of battery parameters; Li et al. [13] established a DP equivalent circuit model and used an unscented Kalman filter algorithm to estimate SOC; Xia et al. [14] employed the multi forgetting factor recursive least squares method to identify the parameters of the equivalent circuit model, effectively improving the identification accuracy of time-varying parameters. Although the equivalent circuit model has the characteristic of convenient calculation, the complexity of the equivalent circuit model rises with the increase of RC networks. Therefore, selecting a high-precision and uncomplicated model is an important prerequisite for parameter identification.

The Kalman filter can continuously estimate and correct unknown state parameters based on measurement values from input signals with noise and quantify and calculate the noise. It has good performance in linear parameter identification and nonlinear SOC state estimation, but the disadvantage is that the parameter setting of system noise largely determines the accuracy of identification [15,16,17]. Reference [18] proposed an estimation method for battery SOC by combining the ampere-hour integration method and extended Kalman filter method; Reference [19] proposed an enhanced extended Kalman filter algorithm based on observation equation recombination to estimate battery SOC; Reference [20] proposed an algorithm that combined extended Kalman filtering and unscented Kalman filtering to estimate battery SOC at different stages using both algorithms.

Although the above methods can achieve an online estimation of battery SOC, in practical engineering, the noise characteristics are not easily determined by fixed values, and the system state is also constantly changing. In the case of unknown noise statistical characteristics, the accuracy and robustness of estimating battery SOC using the Kalman filtering algorithm are poor.

Swarm intelligence optimization algorithms are intelligent algorithms designed by drawing inspiration from physical laws in nature, collective behaviors of animal populations, or evolutionary logic. In recent years, engineering research has focused on utilizing the rapid optimal solution-seeking capability of swarm intelligence algorithms to optimize noise-related issues in the Kalman filtering loop. Examples include the Particle Swarm Optimization (PSO) algorithm inspired by birds’ rapid food-locating behavior during foraging, the Grey Wolf Optimization (GWO) algorithm mimicking gray wolves’ predation strategies and social hierarchy, and the Cuckoo Search (CS) algorithm simulating cuckoos’ brood parasitic behavior [21,22]. Specifically, References respectively employed PSO and GWO to optimize the system noise and process noise matrices in SOC estimation through Kalman filtering, achieving significant error reduction compared to unoptimized methods. The beetle optimization algorithm is a newly proposed intelligent algorithm that can quickly optimize globally by mimicking a series of behavioral activities of beetles. Yuan Xiangyu et al. [23] incorporated the beetle optimization algorithm into the 3D path planning of unmanned aerial vehicles, effectively improving the quality of path planning; Yi Wangyuan et al. [24] combined the beetle algorithm with neural networks to predict the energy consumption of CNC machine tools. The optimized cutting parameters are more energy-efficient and save processing costs. Liu Jiaxuan et al. [25] proposed an algorithm that combined the beetle algorithm with variational mode decomposition and a bidirectional long short-term memory network to predict the building energy consumption, effectively reducing the fluctuation error caused by the diver user behaviors and potential sensor errors. By introducing the beetle algorithm to optimize the Kalman gain coefficient, the weight of adjusting correction values and prior estimation values for the battery model and other error changes can be achieved, improving the practicality of the algorithm.

Based on the above analysis, in order to verify the effectiveness of the proposed double Kalman filter algorithm based on the beetle algorithm for secondary optimization in the SOC estimation process of batteries, this paper takes lithium iron phosphate batteries as the research object, establishes an equivalent circuit model, identifies the parameters of the model and estimates SOC under various operating conditions of voltage and current. Furthermore, it makes a comparison with the improved algorithm and constant-coefficient algorithm to analyze the estimation accuracy and convergence of the proposed improved algorithm in the SOC estimation process. The primary contributions of this article are summarized as follows:

• This study designed the Dung Beetle Optimizer (DBO) secondary optimization Kalman filtering (DBO-DKF) algorithm as the SOC estimation method. The DBO algorithm targets the absolute accumulated error between the predicted terminal voltage of the power battery model under working conditions and the measured value. It optimizes the covariance matrices of system noise and observation noise in the iterative cycles of two Kalman filtering algorithms: one for model parameter identification (KF) and the other for SOC estimation (EKF).
• This ensures real-time correction of the dynamic battery model and accuracy of experimental results. The study tested the DBO-DKF estimation method for power battery SOC under stable, complex, and realistic driving conditions of electric vehicles in the experimental phase. The optimization effects of the DBO-DKF algorithm in the parameter identification and SOC estimation stages were evaluated by comparing the sampled and estimated terminal voltage values, as well as the credibility indicators of SOC true values and estimated values.
• The research focused on the equivalent model parameter identification and SOC estimation of power batteries, utilizing the fast convergence speed and strong global search capability of the DBO algorithm. It addressed the issue of insufficient estimation accuracy caused by current and voltage measurement noise errors in the identification and estimation stages of the two Kalman filtering algorithms. The proposed DBO-DKF algorithm effectively enhances SOC estimation accuracy, computational efficiency, and robustness.

2. Materials and Methods

In this section, we explain the DBO-DKF algorithm in detail and then thoroughly analyze each component of DBO-DKF to understand its contribution to the overall process of SOC estimation.

2.1. Creation of Lithium-Ion Battery Model

As shown in Figure 1, the second-order RC equivalent circuit used in this paper to simulate the dynamic variation characteristics of batteries is composed of a circuit consisting of two polarized capacitors and polarized resistors and an ohmic internal resistance.

In Figure 1, U_oc represents the open circuit voltage of the battery; I represents the battery charging and discharging current; U_t represents the terminal voltage of the battery; R₀ represents the ohmic resistance of the battery; R₁ and R₂ represent the polarization resistance of the battery; C₁ and C₂ polarized capacitors.

From this, the state equation of the battery can be obtained:

$$\left\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}{U}_1}{\mathrm{d}t}}=-{\displaystyle \frac{1}{R_1{C}_1}}{U}_1+{\displaystyle \frac{1}{C_1}}I\\ {\displaystyle \frac{\mathrm{d}{U}_2}{\mathrm{d}t}}=-{\displaystyle \frac{1}{R_2{C}_2}}{U}_2+{\displaystyle \frac{1}{C_2}}I\end{array}\right.$$

(1)

The observation equation is

$$U_{\mathrm{t}}=U_{\mathrm{oc}}+U_1+U_2+R_{0}I$$

(2)

2.2. SOC Estimation with Kalman Filter Algorithm

2.2.1. Kalman Filter Algorithm

The principle of Kalman Filtering (KF) is to use estimating and correcting these two processes. The noise generated in the measurement results and state transition process is eliminated as much as possible to achieve accurate estimation of the current state.

The time update equation (estimation) and state update equation (correction) involved in the recursive process of the system are Equations (3) and (4).

The time update equation is

$$\left\{\begin{array}{l}{\widehat{\mathit{x}}}_n^{-}=\mathit{A}{\widehat{\mathit{x}}}_{n-1}+\mathit{B}{\zeta }_{n-1}\\ {\mathit{P}}_n^{-}=\mathit{A}{\mathit{P}}_{n-1}{\mathit{A}}^T+\mathit{R}\end{array}\right.$$

(3)

where “-” and “^” represent “prior” and “estimate”; $x$ is the state variable; ${\widehat{x}}_n$ represents the prior state estimation value of step n when the state prior to step n is known; A is the $n\times n$ order state transition matrix acting on $x_{n-1}$; B is the input control matrix of order $n\times 1$; $\zeta$ is the control vector; P is the $n\times n$ order covariance matrix; and R is the covariance matrix of the $n\times n$ order process noise.

The state update equation is

$$\left\{\begin{array}{l}{\mathit{K}}_n={\mathit{P}}_n^{-}{\mathit{H}}^T{(\mathit{H}{\mathit{P}}_n^{-}{\mathit{H}}^T+\mathit{Q})}^{-1}\\ {\widehat{\mathit{x}}}_n={\widehat{\mathit{x}}}_n^{-}+{\mathit{K}}_n({\mathit{Z}}_n-\mathit{H}{\widehat{\mathit{x}}}_n^{-})\\ {\mathit{P}}_n=(\mathit{I}-{\mathit{K}}_n\mathit{H}){\mathit{P}}_n^{-}\end{array}\right.$$

(4)

where H is the $m\times n$ order observation model matrix; Q is the process noise matrix of $m\times n$ order; K_n is Kalman gain matrix of $n\times m$ order; I is the identity matrix; and Z_n is the measuring value.

By slightly transforming Equation (4), we can obtain

$${\mathit{P}}_n^{-}=\mathit{A}{\mathit{P}}_{n-1}{\mathit{A}}^T+\mathit{R}$$

(5)

Due to the nonlinear characteristics of the internal parameter variation curve of batteries during use, the extended Kalman Filter (EKF) algorithm can transform it into an approximate linear filtering problem, making it very suitable for estimating the battery SOC. The KF algorithm can solve most linear filtering problems, while the EKF algorithm is more suitable for solving nonlinear filtering problems.

2.2.2. Principle of KF Algorithm for Estimating SOC

According to the characteristics of the KF algorithm and the EKF algorithm, the KF algorithm is more suitable for processing the parameter identification stage, while the EKF algorithm can play a better role in processing the SOC estimation stage. Therefore, assuming that the parameter values of the second-order RC equivalent circuit model and the terminal voltage values in the loop remain unchanged within the sampling period T, the observation Equation (2) is discretized to obtain Equation (6):

$$U_{\mathrm{oc},k}-U_{\mathrm{t},k}=p_1\left[U_{\mathrm{t},k-1}-U_{\mathrm{oc},k-1}\right]+p_2\left[U_{\mathrm{t},k-2}-U_{\mathrm{oc},k-2}\right]+p_3{I}_k+p_4{I}_{k-1}+p_5{I}_{k-2}$$

(6)

In the equation, p₁, p₂, p₃, p₄, and p₅ are intermediate variables, and their expression is Equation (7):

$$\left\{\begin{array}{l}{p}_0=T^2/(p_1+p_2+1)\\ a=p_0{p}_1\\ b=-p_0(p_1+2p_2)/T^2\\ c=p_0(p_3+p_4+p_5)/T^2\\ d=-p_0(p_4+2p_5)/T\end{array}\right.$$

(7)

The parameter Expression (8) obtained by transforming Equation (7) is

$$\left\{\begin{array}{l}{R}_0=p_5/p_2\\ R_1=({\displaystyle \frac{b+\sqrt{b^2-4a}}{2}}c+{\displaystyle \frac{b-\sqrt{b^2-4a}}{2}}{R}_0-d)/(\sqrt{b^2-4a})\\ R_2=c-R_0-R_1\\ C_1=({\displaystyle \frac{b+\sqrt{b^2-4a}}{2}})/R_1\\ C_2=({\displaystyle \frac{b+\sqrt{b^2-4a}}{2}})/R_2\end{array}\right.$$

(8)

Set:

The output value of the state variable

$$X_n={\left\{p_{1,n}{p}_{2,n}{p}_{3,n}{p}_{4,n}\right\}}^{\mathrm{T}}$$

(9)

The terminal voltage prediction value

$$C_n=\left\{U_{t,n-1}-U_{OC,n-1}{U}_{t,n-2}-U_{OC,n-2}{I}_n{I}_{n-1}{I}_{n-2}\right\}$$

(10)

The measured value

$$Z_n=U_{OC,n}-U_{t,n}$$

(11)

The Kalman filter algorithm is used to identify battery parameters in Equation (6), and a spatial expression Equation (14) is created equation of EKF in the SOC estimation process. When $T\in \left[n,n+1\right]$, the state Equation (1) and observation Equation (2) of the second-order RC equivalent circuit model are discretized to obtain the state equation:

$$\left\{\begin{array}{l}SOC(n)=SOC(n-1)-{\displaystyle \frac{1}{C_q}}I(n-1)\\ U_1(n)={\displaystyle \frac{1}{R_1{C}_1}}{e}^{-{\displaystyle \frac{1}{R_1{C}_1}}}{U}_1(n-1)+(1-e^{-{\displaystyle \frac{1}{R_1{C}_1}}}){\displaystyle \frac{1}{C}}I(n-1)\\ U_2(n)={\displaystyle \frac{1}{R_2{C}_2}}{e}^{-{\displaystyle \frac{1}{R_2{C}_2}}}{U}_2(n-1)+(1-e^{-{\displaystyle \frac{1}{R_2{C}_2}}}){\displaystyle \frac{1}{C}}I(n-1)\end{array}\right.$$

(12)

The observation equation is

$$U_t\left(n\right)=h_n\left[SOC(n),U_1(n),U_2(n)\right]+R_{0}I(n)$$

(13)

Simplify Equations (12) and (13), and we can get

$$X_n={\Phi }_{n-1}{X}_{n-1}+BI(n-1)$$

(14)

$$U_t(n)=H_n{X}_n+R_{0}I(n)$$

(15)

where

$$\left\{\begin{array}{l}{X}_n=\left[\begin{array}{l}SOC(n)\\ U_1(n)\\ U_2(n)\end{array}\right]\\ {\Phi }_{n-1}=\left(\begin{array}{ccc}1& & \\ & {\displaystyle \frac{1}{R_1{C}_1}}{e}^{-{\displaystyle \frac{1}{R_1{C}_1}}}& \\ & & {\displaystyle \frac{1}{R_2{C}_2}}{e}^{-{\displaystyle \frac{1}{R_2{C}_2}}}\end{array}\right)\\ B=\left[\begin{array}{l}-{\displaystyle \frac{1}{Cq}}\\ (1-e^{-{\displaystyle \frac{1}{R_1{C}_1}}}){\displaystyle \frac{1}{C_1}}\\ (1-e^{-{\displaystyle \frac{1}{R_2{C}_2}}}){\displaystyle \frac{1}{C_2}}\end{array}\right]\\ H_n=\left[{\displaystyle \frac{\partial h_n}{\partial SOC(n)}}{\displaystyle \frac{\partial h_n}{\partial U_1(n)}}{\displaystyle \frac{\partial h_n}{\partial U_2(n)}}\right]\end{array}\right.$$

(16)

The flowchart of Kalman filtering in battery SOC estimation is shown in Figure 2:

2.3. DBO Optimized Kalman Filter Algorithm

The Dung Beetle Optimizer (DBO) is a population-based intelligence algorithm that simulates the behavior of the Dung Beetle population and divides it into four different subpopulations for position update and optimization. The distribution ratio of the four types of dung beetles is not a fixed value and can be flexibly adjusted according to the actual situation, but the total number of the four types of dung beetles should be the same as the total population [26].

2.3.1. Dung Beetle Optimizer

• Position update mode of dung rolling beetles

The dung rolling beetles explore the entire space by rolling dung balls, and their position updates can be divided into two modes, with or without obstacles.

①

When there are no obstacles in the front:

$$x_i(t+1)=x_i(t)+\alpha \times k\times x_i(t-1)+b\times \left|x_i(t)-X^w\right|$$

(17)

In the equation, $t$ represents the current number of iterations, $x_i$ represents the position of the i-th beetle at the t-th iteration, $\alpha$ is a natural coefficient indicating whether it deviates from the original direction, usually assigned a value of −1 or 1, $k$ ∈ (0,0.2) represents the deviation coefficient, $b$ ∈ (0,0.1) represents a constant, $k$ and $b$ are set to 0.1 and 0.3 respectively, representing the global worst position.

②

When there are obstacles in the front:

$$x_i(t+1)=x_i(t)+\tan (\theta )\left|x_i(t)-x_i(t-1)\right|$$

(18)

In the equation, $\theta$ ∈ [0, $\pi$] represents the deflection angle, and when $\theta$ is equal to 0, ${\displaystyle \frac{\pi }{2}}$ or $\pi$, the position of the beetles will not update.

2.

Position update mode of oviposition beetles

The movement range of oviposition beetles is limited to the safe zone and is used to achieve local search and development within the safe zone. The boundary of their spawning area is

$$\left\{\begin{array}{l}L{b}^{*}=M\mathrm{ax}(X^{*}\times (1-R),Lb)\\ U{b}^{*}=Min(X^{*}\times (1+R),Ub)\\ R=1-{\displaystyle \frac{t}{T_{\max }}}\end{array}\right.$$

(19)

where $X^{*}$ represents the current local optimal position, $L{b}^{*}$ and $U{b}^{*}$ respectively represent the lower and upper limits of the spawning area, $T_{\max }$ represent the maximum number of iterations, $Lb$ and $Ub$ respectively represent the lower and upper limits of the optimization problem.

According to Equation (19), the position of the spawning area will dynamically change over time, so the position of the egg ball will also update with the increase in iteration times. Its position is defined as

$$B_i(t+1)=X^{*}+b_1\times (B_i(t)-L{b}^{*})+b_2\times (B_i(t)-U{b}^{*})$$

(20)

where $B_i(t)$ is the position information of the i-th egg ball at the t-th iteration, $b_1$ and $b_2$ represents independent random vectors of two $D$ dimensions.

3.

Position update mode of little beetles

The little beetles will search for food in the optimal foraging area, and their location has been updated to

$$x_i(t+1)=x_i(t)+C_1\times (x_i(t)-L{b}^b)+C_2\times (x_i(t)-U{b}^b)$$

(21)

where

$$\left\{\begin{array}{l}L{b}^b=Max(X^b\times (1-R),Lb)\\ U{b}^b=Min(X^b\times (1+R),Ub)\end{array}\right.$$

(22)

In the equation, $x_i(t)$ is the position of the i-th little beetle at the t-th iteration, $C_1$ is a random number following a normal distribution, $C_2\in (0,1)$ is a random vector, $X^b$ is the global optimal position, $L{b}^b$ and $U{b}^b$ is the lower and upper limits of the optimal foraging area. Other parameters are defined in Equation (22).

4.

Position update mode of thief beetles

The thief beetles will steal physical objects from other beetles. According to Equation (22), $X^b$ is the optimal position, so their positions are updated as

$$x_i(t+1)=X^b+S\times g\times (\left|x_i(t)-X^{*}\right|+\left|x_i(t)-X^b\right|)$$

(23)

In the equation, $x_i(t)$ represents the position of the i-th thief beetle at the t-th iteration, $g$ represents a random variable with a $1\times D$ dimension following a normal distribution, and $S$ represents a constant value.

To verify the superior optimization performance of the DBO algorithm, this paper selects three other commonly used swarm intelligence algorithms for comparison: Particle Swarm Optimization (PSO), Grey Wolf Optimization (GWO), and Cuckoo Search (CS). A comparative analysis of algorithm performance was conducted on four benchmark functions with distinct characteristics. The information on the standard test functions is shown in

Table 1. Standard test function.

Number	Name	Dimension	Scope
F1	Sphere	30	[100, 100]
F2	Schwefel	30	[−10, 10]
F3	Rastrigin	30	[5.12, 5.12]
F4	Ackley	30	[−32, 32]

, including two unimodal test functions (F1 and F2, which have only one global minimum to evaluate convergence speed and accuracy) and two multimodal test functions (F3 and F4, which feature multiple local optima and one global optimum).

The credibility criteria of mean value and standard deviation are introduced as indicators to evaluate the optimization accuracy of algorithms. The calculation methods are defined as follows: mean value (M) and standard deviation (Std):

$$\begin{array}{l}M={\displaystyle \frac{1}{P}}{\displaystyle \sum _{i=1}^P{f}_i},\\ Std=\sqrt{{\displaystyle \frac{1}{P-1}}{{\displaystyle \sum _{i=1}^P\left(f_i-M\right)}}^2,}\end{array}$$

(24)

As shown in

Table 2. Comparison of standard test function results.

Function	Evaluation	DBO	CS	PSO	GWO
F1	M Std	2.02 × 10−104 1.11 × 10−103	4.95 × 10−97 2.48 × 10−96	1.17 × 10−5 3.05 × 10−5	1.74 × 10−27 4.31 × 10−27
F2	M Std	1.37 × 10−80 7.48 × 10−79	7.91 × 10−66 4.33 × 10−65	77.11498 33.63623	8.39 × 10−6 2.85 × 10−5
F3	M Std	65.73226 26.76003	270.69723 397.69432	337.76123 685.57033	453.7054 732.1294
F4	M Std	25.73226 0.19979	179.36765 143.84788	45.11843 26.76003	27.98438 0.415927

, the functions F1 and F2, the mean fitness values of the DBO algorithm, are closer to the theoretical optimal values, indicating its superior performance in convergence and accuracy. In F3 and F4, the DBO algorithm demonstrates stronger search capability compared to the six other algorithms, particularly achieving the global optimal solution in F4. Based on the benchmark test results against commonly used swarm intelligence algorithms, the DBO algorithm exhibits exceptional accuracy and stability in both single optimal solutions and multiple local optimal optimization problems. Figure 3 shows the function curves of the four type of swarm intelligence algorithms. Therefore, optimizing the Kalman filter for SOC estimation using the DBO algorithm is demonstrated to be feasible.

It can be seen from Table 2 that the optimization accuracy of the DBO algorithm is superior to the other three intelligent optimization algorithms, and good results are obtained on all test functions. DBO showed significant performance advantages in unimodal test functions (F1, F2). For the multimodal function (F3, F4), the ability to jump out of the local optimal solution is shown very well.

Therefore, the DBO algorithm shows excellent accuracy and stability in a single optimal solution and multiple optimal solutions represented by a single peak or multiple peaks according to the test results of benchmark function compared with other commonly used group intelligence algorithms. Therefore, it is feasible to optimize Kalman filter estimation SOC through the DBO algorithm.

2.3.2. Optimizing Kalman Filter Algorithm with Dung Beetle Optimizer

Dung Beetle Optimizer (DBO) commences with random solutions and iteratively searches for the optimal solution. The continuous iteration process allows dung beetles to continually update their positions and the position of the optimal solution, thus exhibiting rapid convergence and high accuracy. By treating the key elements (in this paper, diagonal elements) of the measurement noise and system noise Q and R in Kalman filtering as the search space for the DBO algorithm:

$$Q=\left[\begin{array}{cc}{q}_1& 0\\ 0& q_2\end{array}\right],R=\left[r\right]$$

The optimization variables to be determined are (q₁, q₂, r)

State Equation:

$$X_n=f(x_{n-1},u_{n-1})+w_n w_n~N(0,Q)$$

(25)

Observation Equation:

$$Z_n=h(x_n)+v_n,v_n~N(0,R)$$

(26)

where X_n represents state variables such as SOC, and Z_n denotes the observed terminal voltage.

In the optimization process of the DBO-DKF algorithm, the fitness function serves two purposes:constraining the displacement trends of each dung beetle and acting as an effective termination criterion for updates. During each prediction step of the DKF algorithm, when determining whether to replace the optimal position at iteration t − 1 with the best position at iteration t, it is essential to minimize the influence of measurement noise and system noise. The fitness function is defined as the absolute cumulative error between the predicted value Z_n and measured value C_nX_n,n−1 of the DKF converted second-order RC circuit and the predicted terminal voltage. The optimization workflow of the DBO-DKF algorithm is as follows:

• Initialize parameters: Set the DBO population size (N), where each dung beetle corresponds to an initial set of parameters (q₁,q₂,r). These values are either randomly generated or empirically assigned within predefined boundaries.
• Define termination criteria: specify the maximum iteration count and convergence threshold to control the optimization process.
• Global search phase: Four categories of dung beetles perform randomized exploration in the parameter space to generate new solutions and identify the current best position. Calculate the fitness of each dung beetle’s position. The positional variables of the beetles act as independent variables in the fitness function, which are the elements in the covariance matrices Q (system noise) and R (measurement noise). After each iteration, compare the fitness of the updated positions of all dung beetles with the global best position. If the iteration count has not reached the predefined maximum, use the fitness values of the local best positions (individual beetles) and the global best position to determine whether convergence to the optimal solution has been achieved.
• Calculate the covariance matrix ∆ of the beetle’s position. If the matrix satisfies ∆ < ɛ (ɛ = 0.0001), it represents that the position of the t iteration is better, then update the beetle’s position variable to replace the original beetle’s position. Otherwise, keep the position unchanged until the loop position remains unchanged and the output position variable is the desired value. The absolute cumulative error between the predicted value C_nX_n,n−1 and measured value Z_n of the terminal voltage is used as the algorithm fitness value (where L is the maximum sampling point):

  $$fitness={\displaystyle \sum _{i=1}^L\left|Z_n-C_n{x}_{n,n-1}\right|}$$

  (27)
• Output the optimal solution of the DBO-KF algorithm and assign it to the system noise covariance matrix R and Q.

Through iterative loops, the absolute cumulative error between the sampled value of the actual circuit terminal voltage and the predicted value of the equivalent circuit model is used as the fitness function. The two error covariance matrices R and Q during the iteration process are corrected by the dung beetle optimizer to reduce the randomness and volatility of the results after each iteration, thereby making the predicted results closer to the actual test values.

2.3.3. SOC Estimation Based on DBO-DKF

Accurate identification of equivalent circuit model parameters is a prerequisite for improving the accuracy of SOC estimation using the EKF algorithm. Due to the complexity of the battery itself, the values of internal parameters exhibit time-varying and nonlinear characteristics with changes in SOC. Therefore, a new SOC estimation method, Double Kalman Filtering (DKF), is designed by combining the KF algorithm for model parameter identification with the EKF algorithm for SOC estimation.

The main idea of using the DKF algorithm to estimate SOC is to divide it into two Kalman filter estimation correction loops for interaction: one is to employ the KF algorithm to identify the parameters of the second-order RC equivalent circuit model, and the other is to use the EKF algorithm to estimate the charge state variables of the battery. The DKF identification process is shown in Figure 4, where the two iterative loops are not independent but alternate with each other. In the upper part of the loop, the algorithm is used to estimate the SOC of the battery. In addition to the input and output of the basic sampling parameters, the loop also requires a one-step prediction of the model parameters. In the second half of the loop, the algorithm requires a one-step prediction of SOC to advance the identification of battery model parameters.

Based on the creation of the DBO-DKF algorithm mentioned earlier, the design idea of using the DBO algorithm to optimize the noise covariance matrix $R$ and $Q$ and improve the accuracy of model parameter identification will be adopted. A similar method will be used to optimize the DKF algorithm model. According to the algorithm flow in Figure 4, the DBO algorithm is used to optimize the noise covariance matrices R^w, Q^w, R^x, and Q^x involved in the upper and lower loops of the DBO-DKF algorithm (set R^w and Q^w as the noise matrices in the EKF algorithm, R^x and Q^x as the noise matrices in the KF algorithm). The loop update process is shown in Figure 4.

3. Results

3.1. Establishment of the Experimental Platform

Through the practical experiment of the battery, the accuracy of the DBO-DKF algorithm in estimating the battery SOC can be better demonstrated. The good quality of experimental data is the basis of the estimation and optimization of battery SOC. Hence, this paper establishes an experimental platform that has a merit of high accuracy. The SOC data is collected from such a platform, which can largely reduce the non-experimental errors.

The subject of this experiment is a single-cell lithium-ion battery of a model type 18650. The technical parameters of this battery are shown in

Table 3. The technical parameters of 18,650 battery.

Parameter	Value	Parameter	Value
Rated capacity	2980 mA/h	Mass	50.0 g
Rated voltage	3.6 V	Charging temperature	+10~+45 °C
Maximum discharging current	10 A	Discharging temperature	−20~+60 °C
Maximum charging current	4 A	Charging–discharging cycles	1000
Energy density	218 Wh/g	Size	65.1 × 18.25 mm

.

The platform is shown in Figure 5, which is made up of a PC, a constant-temperature chamber, and a monitoring system. The 18,650 battery is placed into the constant-temperature chamber. The charging and discharging of this battery are performed with a pre-specified scheme which is input from a human–computer interaction interface. During the experiment, the voltage, current, discharging time, and other data are collected and stored in a PC.

3.2. The SOC Optimization Under Various Practical Scenarios

The DST test, FUDS test, BJDST test, and US60 test are conducted on the 18,650 battery. The four tests can verify the capability of the Kalman filtering algorithm to identify critical battery parameters and estimate battery SOC. Three commonly accepted performance indices, i.e., MAE, MSE, and RMSE, are employed to indicate the accuracy of such an algorithm.

During the experiment, the battery SOC is rapidly changed from fully charged and fully discharged under a complicated scenario. There exists a risk that the battery may be overheated or even exploded. Therefore, the initial battery SOC is set to 50% by a constant current discharging scheme. Then, the battery is tested under a complicated scenario. The parameters of this experiment are as follows. The temperature is set to 25 °C. The rated capacity of the battery is 50 Ah. The rated voltage of the battery is 4.2 V. The population of dung beetle N is 90. The rolling beetle, spawning beetle, juvenile beetle, and stealing beetle are 30, 20, 25, and 25, respectively. The maximum number of iteration T is 500. The state variable and covariance matrix of noise are x₀ and p₀. The Kalman filtering algorithm is influenced by the Kalman gain coefficient and presents a better convergent characteristic. So, the state variables are randomly initialized. The state variable and covariance matrix of noise are set as follows.

$$\left\{\begin{array}{l}{\widehat{w}}_0=1\\ P_{w_0}=diag\left(\left[11111\right]\right)\\ {\widehat{x}}_0={\left[0.40.10.1\right]}^T\\ P_{x_0}=diag\left(\left[111\right]\right)\end{array}\right.$$

(28)

R^w, Q^w, R^x and Q^x at a constant value, the value is

$$\left\{\begin{array}{l}{R}^w=\left(\begin{array}{ccccc}0.1& & & & \\ & 0.1& & & \\ & & 0.1& & \\ & & & 0.1& \\ & & & & 0.1\end{array}\right)\\ Q^w=0.1\\ R^x=\left[\begin{array}{ccc}0.1& & \\ & 0.1& \\ & & 0.1\end{array}\right]\\ Q^x=0.1\end{array}\right.$$

(29)

3.2.1. The Discharging Experiment of Battery Under Stable Condition

When the battery is discharging with a scheme of 0.2 C, the DBO algorithm is performed, and the global fitness curve of the dung beetle is shown in Figure 6. It can be seen that the fitness curve is decreasing rapidly with the increase in iterations. This indicates that the algorithm can find the optimal solution in an efficient manner.

The absolute and cumulative difference between the sampled terminal voltage and the predicted one is employed as the fitness criterion for estimating battery SOC in the DBO-DEF algorithm. Figure 7 shows the comparison between the sampled terminal voltage and the predicted one when estimating battery SOC with the algorithm. Figure 7a,c show the comparison between sampled terminal voltage Z_n and predicted one $C_n{x}_{n,n-1}$ after optimizing the noise ${\zeta }_n$ and ${\chi }_n$ with the DBO algorithm. Figure 7b,d show the comparison of prediction errors. It can be seen from Figure 7 that the prediction errors decrease after the optimization of the algorithm. The errors drop from 1.8% to 0.5%.

The model parameters are identified and shown in Figure 8. These parameters are identified under a scenario in which the initial battery SOC is 50% and discharged under a scheme of 0.2 C constant current. The input of the algorithm is sample terminal voltage and current. It can be seen that the battery is nearly fully discharged when the SOC drops to 10%. The parameters of the battery change dramatically when the battery voltage drops. Therefore, it is normal to see a large oscillation of identified battery parameters.

The identified parameters of the battery are listed in

Table 4. The identified battery parameters under different battery SOC.

SOC (%)	R0 (Ω)	R1 (Ω)	R2 (Ω)	C1 (F)	C2 (F)
50%	0.0090	0.1920	0.0943	9.8453	4.8505
40%	0.0559	0.0394	0.036	151.2994	493.904
30%	0.0561	0.0677	0.024	130.815	342.5928
20%	0.0568	0.0568	0.0034	135.624	310.9536
10%	0.0592	0.0286	0.0042	908.3162	191.9889
0%	0.0692	0.0624	−0.925	405.897	138.5594

. It can be seen that internal resistance (R₀) and polarization resistance (R₁, R₂) increase with the decrease in battery SOC. The polarization capacitances C₁ and C₂ present an opposite variation trend with the parallel connected polarization resistance. This is due to the increase in polarization voltage drop when the battery is discharging. The identified internal resistance and polarization capacitance agree with their practical variation trend with the decreasing SOC.

The algorithm is also employed to estimate the battery SOC under stable conditions. Figure 9 shows the curve of estimation errors. It can be seen that the errors are below 1.5% under stable conditions.

3.2.2. The Battery Discharging Experiment Under Complicated Scenarios

In the previous sections, discussions on the performance of the DBO-DKF algorithm in optimizing SOC estimation have been carried out. This section demonstrates the feasibility of this algorithm in optimizing SOC estimation through the battery discharging experiment under complicated scenarios.

This algorithm employs the discharging current shown in Figure 10, where the discharging time is set as T = $2\times {10}^4$ s. The sampling interval of terminal voltage and input current are set as T = 1 s. The cut-off discharging voltage is set at 3.44 V.

Figure 11 shows the curves of estimated battery SOC by the DBO-DKF algorithm and measured battery SOC.

To demonstrate the superiority of DBO-DKF algorithm in estimating battery SOC, the experiments investigate the accuracy of battery SOC estimation under different parameters of the Kalman filter algorithm. Additionally, the EKF algorithm, DKF algorithm, and DBO-DFK algorithm are employed to estimate battery SOC, respectively. The parameters of the battery, which are estimated by the DKF algorithm, are listed in

Table 5. The parameters of battery, which are estimated by DKF algorithm.

Parameter	Value
Internal resistance R0	0.0561 Ω
Polarization resistance R1	0.0677 Ω
Polarization capacitance C1	130.815 F
Polarization resistance R2	0.024 Ω
Polarization capacitance C2	342.5928 F

.

Based on the true battery SOC in Figure 12, the RMSE, MAE, and R² of battery SOC estimation by EKF, DBO-DKF, and DKF algorithms can be obtained and listed in

Table 6. The RMSE, MAE, and R² of three algorithms.

Errors	EKF	DKF	DBO-DKF
RMSE (%)	6.0166	2.54736	1.7285
MAE (%)	3.8132	1.7389	1.7534
R2 (%)	0.88	0.91	0.95

. It can be seen that the RMSE and MAE presented by the DBO-DKF algorithm are below 2%. In comparison to other two algorithms, the DBO-DKF algorithm performs better in battery SOC estimation.

3.2.3. Comparison of Swarm Intelligence Algorithms for Optimizing the DKF Algorithm

To further validate the superiority of the DBO algorithm over the other three swarm intelligence algorithms in SOC estimation, as described in Section 2.3, this chapter employs four swarm intelligence algorithms—“PSO, CS, GWO, and DBO”—to optimize the noise matrices generated during the DKF-based SOC estimation process. The initial parameter settings for the three comparative swarm intelligence algorithms (excluding DBO) remain consistent with those specified in references [21,22,27]. Figure 13 demonstrates the SOC estimation results after implementing noise matrix optimization through these four algorithms in the DKF iterative process.

From the optimization effectiveness demonstrated in Figure 14 for PSO, CS, GWO, and DBO in enhancing the DKF-based SOC estimation process, and as evident from the RMSE and MAE metrics in

Table 7. The RMSE and MAE of four swarm intelligence algorithms in estimated SOC.

Errors	DBO	PSO	CS	GWO
RMSE	0.043	0.048	0.058	0.092
MAE	0.032	0.036	0.045	0.068

, the DBO algorithm maintains superior performance compared to the other three swarm intelligence algorithms in SOC estimation.

3.2.4. The Battery SOC Estimation Based on DBO-DKF Algorithm Under Complicated Scenarios

To verify the optimization performance of the DBO-DKF algorithm under practical scenarios, the DST, FUDS, BJDST, and US60 are employed to simulate the current output of the battery dynamically. The DBO-DKF algorithm is executed under a 50% battery SOC scenario. The estimated battery SOC is shown in the following figure.

• The optimization performance under DST scenario

Figure 14a,b show the curve of errors between true terminal voltage and measure one. Figure 14c shows the curves of true battery SOC and the estimated one by the DBO-DKF algorithm. The curve of errors is shown in Figure 14d. It can be seen that the errors are below 2% and converge to 1% when the battery SOC is within the range of 50~10%. The terminal voltage drops significantly, and the estimated battery SOC deviates from the true one, which creates an increasing error. This is due to the fact that the battery is nearly depleted when the SOC is low. The drop in voltage leads to a large variation of battery parameters. The deviation of battery SOC estimation is then created. Figure 8 shows a large oscillation of identified parameters when the SOC is moderate and low. This agrees with the previous discussions. Similar explanations can be applied to other scenarios. The dataset can be downloaded from https://uscar.org/usabc/ (accessed on 16 September 2021).

• The optimization performance under FUDS scenario

The FUDS scenario is a typical simulation of a U.S urban electric vehicle. The curve of errors between true terminal voltage and measured one is shown in Figure 15a,b. It can be seen that the estimated terminal voltage shows considerable agreement with the true terminal voltage. The true battery SOC and the estimated battery SOC by the DBO-DKF algorithm are shown in Figure 15c. Their curve of errors is shown in Figure 15d. This indicates that the errors approach 2% and converge to 1% when the battery SOC is within the range of 50~10%. The dataset can be downloaded from https://www.epa.gov/vehicle-and-fuel-emissions-testing/dynamometer-drive-schedules (accessed on 18 March 2023).

• The optimization performance under US60 scenario

Under the US60 scenario, the curve of errors between true terminal voltage and measure one is shown in Figure 16a,b. It can be seen that the predicted terminal voltage is close to the true one. The true battery SOC and the estimated battery SOC by the DBO-DKF algorithm are shown in Figure 16c. Their curve of errors is shown in Figure 16d. This indicates that the errors approach a minimum when the battery SOC reaches 30%. After this point, the errors magnify. The dataset can be downloaded from https://www.nrel.gov/transportation/secure-transportation-data (accessed on 4 April 2024).

• The optimization performance under BJDST scenario

Figure 17 shows the estimation curves under the BJDST scenario. It can be seen that the errors of estimated terminal voltage and battery SOC are significant when battery SOC is above 45% or below 10%. The fitting performance is the greatest in the middle section of the battery SOC. The dataset can be downloaded from http://www.catarc.org.cn/ (accessed on 4 April 2024).

The curves of errors under four scenarios are shown in Figure 18.

It can be seen from Figure 18 that the DBO-DKF algorithm presents the best fitting performance on the true battery SOC when the battery SOC is within the range of 45~10%. The MSE, RMSE, and MAE that measure the accuracy of terminal voltage estimation and battery SOC estimation by four algorithms are calculated and listed in

Table 8. The MSE, RMSE, and MAE of terminal voltage estimation by four algorithms.

Index	DST	FUDS	US60	BJDST
MSE (U/V)	0.0299	0.00077	0.0012	0.017
RMSE (U/V)	0.000894	0.0278	0.0353	0.0408
MAE (U/V)	0.0134	0.0149	0.0163	0.0169

and

Table 9. The MSE, RMSE, and MAE of battery SOC estimation by four complicated scenarios.

Index	DST	FUDS	US60	BJDST
MSE (%)	2.23534	2.8138	2.23534	5.4309
RMS (%)	0.015	0.0167	0.01495	0.2925
MAE (%)	0.0092	0.0122	0.0092	0.2566

. It can be seen that the DBO-DKF algorithm presents the best accuracy in fitting terminal voltage under the FUDS scenario. Additionally, the DBO-DKF algorithm presents the best accuracy in fitting battery SOC under the DST scenario.

4. Conclusions

This research employs the DBO algorithm to optimize the Kalman gain coefficient of in Kalman filter algorithm so that the problem of the noise covariance matrix can be well addressed. To address the characteristics of lithium batteries, the second-order RC equivalent circuit model is employed. The DBO-KF algorithm is employed to identify the circuit parameters dynamically. On the basis of this work, the battery SOC estimation algorithm is developed by combining the DBO-KF algorithm and the advantage of EKF in handling nonlinear problems. The two schemes of the Kalman filter algorithm for identifying parameters and estimating battery SOC are established. The state equation and observation equations are discretized. The operation procedure and algorithm flowchart of the DBO-DKF algorithm are developed.

The lithium battery SOC is measured under complicated scenarios. The DBO-DKF algorithm is compared with other algorithms for estimating battery SOC, which shows that the DBO-DKF algorithm has better accuracy. Under DST, FUDS, BJDST, and US60 scenarios of simulated electric vehicle driving behavior, the parameters identified by the DBO-DKF algorithm and battery SOC estimated by the DBO-DKF algorithm are accurate. The error is below 2% when the estimation battery SOC is 7% higher than the true battery SOC. The DBO-DKF algorithm can well fit the nonlinear characteristics of battery parameters and battery SOC when the lithium battery is discharging.