Real-Time Estimation of the State of Charge of Lithium Batteries Under a Wide Temperature Range

Abstract

The state of charge (SOC) of lithium-ion batteries is essential for their proper functioning and serves as the basis for estimating other parameters within the battery management system. To enhance the accuracy of SOC estimation in lithium-ion batteries, we propose a joint estimation method that integrates lithium-ion battery parameter identification and SOC assessment using cat swarm optimization dual Kalman filtering (CSO–DKF), which accounts for variable-temperature conditions. We adopt a second-order equivalent circuit model, utilizing the Kalman filtering (KF) algorithm as a parameter filter for dynamic parameter identification, while the extended Kalman filtering (EKF) algorithm acts as a state filter for real-time SOC estimation. These two filters operate alternately throughout the iterative process. Additionally, the cat swarm optimization (CSO) algorithm optimizes the noise covariance matrices of both filters, thereby enhancing the precision of parameter identification and SOC estimation. To support this algorithm, we establish an environmental temperature battery database and incorporate temperature variables to achieve accurate SOC estimation under variable-temperature conditions. The results indicate that creating a database that accommodates temperature variations and optimizing dual Kalman filtering through the cat swarm optimization algorithm significantly improves SOC estimation accuracy.

1. Introduction

In the context of the current advancing era and the ongoing refinement of environmental protection regulations, the development of power emission technologies for traditional fuel vehicles has encountered significant challenges. Consequently, an increasing number of manufacturers are intensifying their investments in the research, development, and production of new energy vehicles. Specifically, vehicles powered by lithium-ion batteries exhibit superior performance, offer longer driving ranges, and deliver a more favorable driving experience [1]. As a result, they are widely utilized in battery electric vehicles (BEVs), hybrid electric vehicles (HEVs), and plug-in hybrid electric vehicles (PHEVs) [2].

Over the past few decades, the continuous pursuit of high-precision real-time estimation and understanding of the chemical characteristics of lithium-ion batteries has spurred ongoing research in the field of SOC estimation. Consequently, a wide variety of methods have emerged, which can be generally classified into four categories. The first category involves calibrating the SOC through battery characterization parameters. Wang et al. proposed a strategy for updating the SOC algorithm model based on electrochemical impedance spectroscopy (EIS) [3]. The results indicated that this method effectively improved the estimation accuracy. Zheng et al. investigated the influence of different open-circuit voltage (OCV) testing methods on the online estimation of battery SOC [4], providing guidance for the engineering application of battery management systems (BMSs). Pattipati et al. found through research that the temperature and aging degree of the battery had a minimal impact on the SOC–OCV characteristics. Therefore, the normalized OCV parameter is an important means for accurately estimating the SOC [5]. The second category is the ampere-hour integration method. In essence, it accumulates the charge during battery charging and discharging to estimate the battery’s SOC. This method is simple and intuitive, and has relatively low requirements for controller hardware and storage [6]. However, it has multiple sources of error and thus cannot guarantee the accuracy of SOC estimation [7]. The third category utilizes models combined with algorithms for SOC estimation. Cheng et al. proposed a dual fuzzy-based adaptive extended Kalman filter (DFAEKF) [8]. The experiments showed that this method could effectively control the estimation error within 1%. Wang et al. put forward an SOC estimation method based on a first-order equivalent circuit model and cloud data for adaptive adjustment of the noise matrix in the extended Kalman filter [9]. The experiments demonstrated that the overall relative error of this method could be controlled within 3%. Miao et al. proposed an adaptive fractional-order unscented Kalman filter (UKF) algorithm to estimate the battery SOC without the need for parameter identification [10]. Qian et al. proposed a switching gain adaptive sliding-mode observer based on a dual-polarization (DP) equivalent circuit model to improve the SOC estimation accuracy [11]. Chen et al. presented an SOC estimation method based on a non-commensurate fractional-order (FO) observer [12]. Similarly, there are also other methods such as $H_{\infty }$ filtering [13] and particle filter algorithms [14]. The fourth category estimates the SOC through training datasets [15]. Zhang et al. optimized the multi-hidden-layer BP neural network (LMMBP) trained by the Levenberg–Marquard (L–M) algorithm using the genetic algorithm (GA) and particle swarm optimization (PSO) algorithm [16], which improved the SOC estimation accuracy and convergence speed. Zhang et al. adopted a feature extraction strategy based on variable correlation analysis and principal component analysis. The obtained compressed dataset was used as the input of the radial basis function neural network (RBFNN), and the particle swarm optimization algorithm was used to improve the RBFNN estimation model [17], thereby enhancing the SOC estimation accuracy. In addition, there are methods based on deep learning [18,19] and support vector machines [20,21].

Nevertheless, the previous methods have failed to consider the influence of environmental temperature variations on the real-time estimation of the SOC of lithium-ion batteries. Variations in the environmental temperature can impact the activity of the chemical materials within lithium batteries, thereby resulting in the instability of parameter estimation and consequently affecting the accuracy of real-time SOC estimation. To improve the accuracy of parameter identification as well as the precision of SOC estimation, a real-time SOC estimation method based on the dual Kalman filter optimized by the cat swarm optimization algorithm, which takes temperature variations into account, is proposed.

The main contributions of this paper are as follows:

• In light of the variations in environmental temperature, a second-order equivalent circuit model was developed, and a parameter identification approach based on the CSO algorithm for optimizing the Kalman filter (CSO-KF) was put forward;
• An environmental temperature battery database was constructed based on the parameter identification results that incorporated temperature variations. By means of mathematical relations, the correlations among the parameters at the current moment, the SOC values, and the temperature were introduced, thus enabling the acquisition of accurate parameter values within variable-temperature environments;
• A real-time SOC estimation method based on the CSO–DKF algorithm was proposed. The parameter filter and the state filter operated alternately, and the accuracy of SOC estimation was verified under three variable-temperature environments.

The specific sections of this paper are structured as follows: Section 2 elaborates on the lithium-ion battery model. Section 3 puts forward the parameter identification method for the Kalman filter optimized by the cat swarm optimization algorithm. Section 4 proposes the real-time SOC estimation method based on optimizing the dual Kalman filter with the cat swarm optimization algorithm. Section 5 presents the experimental results and corresponding analysis. Section 6 offers the conclusions

2. Battery Model

The selection of battery models is directly related to the estimation accuracy of SOC. Currently, commonly used models include electrochemical models [22,23], equivalent circuit models [24,25], and fractional-order models [26,27,28]. Taking into comprehensive consideration the model accuracy, complexity, and computational efficiency, this paper selects the second-order equivalent circuit model. Among them, $U_{oc}$ and $U_o$ are the open-circuit voltage and terminal voltage in the circuit, respectively, $R_0$ is the ohmic internal resistance, $R_1$ and $R_2$ are the polarization internal resistances, and $C_1$ and $C_2$ are the polarization capacitances. The current in the circuit is represented by I. It takes a positive value during charging and a negative value during discharging. The model is shown in Figure 1.

Based on Kirchhoff’s voltage law, the state equation of the system can be formulated as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{dSOC}{dt}}=-{\displaystyle \frac{I}{C_q}}\hfill \\ {\displaystyle \frac{d{U}_1}{dt}}=-{\displaystyle \frac{U_1}{R_1{C}_1}}+{\displaystyle \frac{1}{C_1}}I\hfill \\ {\displaystyle \frac{d{U}_2}{dt}}=-{\displaystyle \frac{U_2}{R_2{C}_2}}+{\displaystyle \frac{1}{C_2}}I\hfill \end{array}\right.\end{array}$$

(1)

Observation equation:

$$U_o=U_{OC}+I{R}_0+U_1+U_2$$

(2)

In the equation, $U_1$ denotes the voltage of the parallel branch composed of $R_1$ and $C_1$, $U_2$ represents the voltage of the parallel branch consisting of $R_2$ and $C_2$, and $C_q$ stands for the calibrated capacity of the lithium-ion battery employed in the experiment.

3. Dynamic Parameter Identification Method Based on Optimizing Kalman Filter with Cat Swarm Optimization Algorithm

3.1. Cat Swarm Optimization

The cat swarm optimization algorithm represents an optimized bionic computing approach, with its fundamental concept originating from the observation of the behaviors of felines [29]. This algorithm exhibits multiple remarkable advantages, including a straightforward algorithmic structure, a relatively low propensity to be trapped in local optima, a rapid convergence rate, and high adaptability. As a result, it has been extensively employed across a diverse range of fields. Specifically, the CSO categorizes the behavioral patterns of the cat swarm into two distinct modes, namely the seeking mode and the tracing mode.

In the seeking mode, individual cats will replicate their own positions multiple times and store the corresponding position information within the memory. Subsequently, numerous perturbations will be applied to the copies of the position information stored in the memory. Once the perturbations are completed, an evaluation of the position information will be conducted to derive the optimal position, which will then serve as the next movement direction for the cats. The specific steps of the seeking mode are detailed as follows:

• Generate M copies of its own position and store them in the memory pool, where M denotes the size of the memory pool;
• Perform random perturbations on the original positions of each individual within the memory pool in accordance with the variations in dimensions and ranges, thereby obtaining a new position to substitute the previous one;
• Compute the fitness function values of all the new positions in the memory pool and utilize these values as the criteria for optimization;
• In the memory pool, relocate the cat to the position with the highest fitness function value, thereby accomplishing the update of the cat’s position.

The tracing mode is designed to simulate the behavioral pattern of a cat when it is tracking a target. Specifically, the globally optimal position is exploited to update the current velocity of the cat. Subsequently, a random perturbation is introduced to modify the velocity, thereby facilitating the update of the cat’s current position. Through iterative cycles of this process, the cat is able to continuously approach the optimal solution. Assume that a cat denoted as $X_i$, which is in the tracing mode, moves at a velocity $V_i$. The steps involved in the tracing mode are detailed as follows:

• The velocity of cat $X_i$ is updated via the following formula:

  $${\nu }_{i,d}\left(t+1\right)={\nu }_{i,d}\left(t\right)+rc\left(x_{best,d}-x_{i,d}\right)$$

  (3)
  In the formula, the value range of d is from 1 to the total number of dimensions. $V_{i,d}\left(t\right)$ represents the velocity of the cat $X_i$ in the d dimension at time t before the update, and $V_{i,d}(t+1)$ represents the speed of the cat in the dimension after the update. $X_{best,d}$ is the position of the cat with the optimal fitness function value in the d dimension, $X_{i,d}$ represents the position of the cat $X_i$ in the d dimension, c is a constant, and r is a random value within the range of [0, 1];
• Update the new position of the cat based on its current position and velocity:

  $$x_{i,d}\left(t+1\right)=x_{i,d}\left(t\right)+{\nu }_{i,d}\left(t+1\right)$$

  (4)
  In the formula, $X_{i,d}$ represents the position of the cat $X_i$ in the d dimension at time t prior to the update, and $X_{i,d}(t+1)$ represents the position of the cat $X_i$ in the d dimension subsequent to the update;
• To avoid out-of-bounds scenarios, if the position in a particular dimension exceeds the defined boundary, it will be adjusted to the corresponding boundary value. By implementing the aforementioned two modes, the positions of each cat are iteratively updated, gradually approaching and ultimately attaining the global optimum. Once the program satisfies the termination criteria, the algorithm concludes.

3.2. Kalman Filtering Algorithm

The Kalman filter has the ability to accomplish the optimal estimation of system state variables and is a robust algorithm with globally exponential stability [30]. The system state equation thereof is presented as follows:

$$\left\{\begin{array}{c}{x}_{k+1}=A_k{x}_k+B_k{u}_k+w_k\hfill \\ y_k=C_k{x}_k+D_k{u}_k+v_k\hfill \end{array}\right.$$

(5)

In the formula, x and y signify the system state matrix and the system output matrix, respectively. The matrices A, B, C, and D represent the system-related matrices, with u being the system input matrix. Meanwhile, w and v stand for the noise matrices.

The core concept underlying the Kalman filter algorithm is to balance the predicted values and the observed values so as to derive the optimal result. The determination of the optimal result value hinges on the magnitudes of the prediction and observation noises. The iterative process of this algorithm can be categorized into two phases, namely prediction and correction. During the prediction phase, a further prediction is conducted based on the current state. Subsequently, in the correction phase, the predicted values obtained in the prediction phase are refined. Through recursive correction commencing from the initial state, the accurate current state value can thus be obtained.

Prediction equation:

$$\left\{\begin{array}{c}{\widehat{x}}_k^{-}=A{\widehat{x}}_{k-1}+B{u}_{k-1}\hfill \\ P_k^{-}=A{P}_{k-1}{A}^T+R^w\hfill \end{array}\right.$$

(6)

In the formula, the symbol “∧” represents the estimated value, and the symbol “-” represents the prior value. k represents the time. The prior value is the predicted value of the current moment by the system based on the previous moment, and the posterior value is the value obtained by correcting the prior value. P represents the covariance matrix of the one-step predicted value, and $R^w$ represents the covariance matrix of the observation noise.

Correction equation:

$$\left\{\begin{array}{c}{K}_k=P_k^{-}{H}^T{\left(H{P}_k^{-}{H}^T+Q^w\right)}^{-1}\hfill \\ {\widehat{x}}_k={\widehat{x}}_k^{-}+K_k\left(z_k-H{\widehat{x}}_k^{-}\right)\hfill \\ P_k=\left(I-K_{k}H\right)P_k^{-}\hfill \end{array}\right.$$

(7)

In the formula, $K_k$ denotes the Kalman gain, an essential parameter in the Kalman filtering algorithm. It serves to balance the integration of predicted and observed values, thereby determining the weight assigned to the observed values. $Q^w$ stands for the process noise covariance matrix, H represents the observation model matrix, and $z_k$ signifies the measured terminal voltage at time k.

3.3. Parameter Identification

Throughout the iterative process of the Kalman filtering algorithm, the noise covariance matrices $R^w$ and $Q^w$ are incapable of being updated in tandem with the iteration and thus require to be set independently within each recursive procedure. Nevertheless, conventional approaches for setting typically involve the utilization of random numbers, which may induce fluctuations in the output results of the Kalman filter algorithm and consequently exert an adverse impact on its accuracy. When the value of $R^w$ is relatively large, greater weight is assigned to the predicted values. In contrast, when the value of $R^w$ is relatively small, the observed values are accorded greater weight. Consequently, in order to attain more reliable results, it is imperative to accurately specify the values of $R^w$ and $Q^w$. The optimization of the Kalman filter via the cat swarm optimization algorithm is precisely aimed at rectifying these two aforementioned parameters. The flowchart of the parameter identification algorithm is depicted in Figure 2.

The cat swarm optimization algorithm enhances the precision of parameter identification by optimizing the noise covariance matrices. The optimal solution of this algorithm is dictated by a fitness function. The absolute error between the measured and estimated terminal voltage values is utilized as the fitness function, as illustrated in Equation (8). In this equation, it denotes the number of sampling points.

$$fitness=\sum _{i=1}^L\left|z_k-C_k{x}_{k|k-1}\right|$$

(8)

Suppose T represents the sampling period. It is hypothesized that the parameters remain invariant within a unit sampling time. The observational Equation (2) can be discretized and reformulated as follows:

$$U_{\mathrm{OC},k}-U_{\mathrm{o},k}=s_1[U_{\mathrm{o},k-1}-U_{\mathrm{OC},k-1}]+s_2[U_{\mathrm{o},k-2}-U_{\mathrm{OC},k-2}]+s_3{i}_k+s_4{i}_{k-1}+s_5{i}_{k-2}$$

(9)

In the formula,

$$\left\{\begin{array}{c}{R}_0=s_5/s_2\hfill \\ R_1=(c{R}_1{C}_1+b{R}_0-R_1{C}_1{R}_0-d)/(R_1{C}_1-R_2{C}_2)\hfill \\ R2=c-R_1-R_0\hfill \\ R_1{C}_1=max\{(b+\sqrt{b^2-4a})/2,(b-\sqrt{b^2-4a})/2\}\hfill \\ R_2{C}_2=min\{(b+\sqrt{b^2-4a})/2,(b-\sqrt{b^2-4a})/2\}\hfill \\ s_0=T_2/(s_1+s_2-1)\hfill \\ a=s_2/s_1\hfill \\ b=-s_0(s_1+2s_2)/T\hfill \\ c=s_0(s_3+s_4+s_5)/T^2\hfill \\ d=-s_0(s_4+2s_5)/T\hfill \end{array}\right.$$

(10)

The hypothesis is as follows:

$$x_k={\left[s_{1,k},s_{2,k},s_{3,k},s_{4,k},s_{5,k}\right]}^T$$

(11)

$$C_k=\left[U_{o,k-1}-U_{OC,k-1}{U}_{o,k-2}-U_{OC,k-2}{i}_k{i}_{k-1}{i}_{k-2}\right]$$

(12)

$$z_k=U_{OC,k}-U_{o,k}$$

(13)

Consequently, the state-space equation for model parameter identification via the CSO-KF algorithm is presented as follows:

$$\left\{\begin{array}{c}{x}_k=x_{k-1}+{\xi }_k\hfill \\ z_k=H_k{x}_k+{\chi }_k\hfill \end{array}\right.$$

(14)

In the formula, ${\xi }_k$ and ${\chi }_k$ denote the system random interference noise and the random observation noise, respectively. The statistical characteristics thereof are presented as follows:

$$E\left\{{\xi }_k\right\}=0$$

(15)

$$E\left\{{\chi }_k\right\}=0$$

(16)

$$Cov\{{\chi }_j,{\chi }_k\}=\left\{\begin{array}{cc}0,\hfill & j\ne k\hfill \\ N=n,\hfill & j=k\hfill \end{array}\right.$$

(17)

$$Cov\{{\xi }_j,{\xi }_k\}=\left\{\begin{array}{cc}0,\hfill & j\ne k\hfill \\ M=diag\left\{[m_1,m_2,m_3,m_4,m_5]\right\},\hfill & j=k\hfill \end{array}\right.$$

(18)

Following the setting of the initial values for the state variables and the error covariance, the recursive process of the algorithm is depicted as in Equation (19). Upon substituting the optimal values yielded by the algorithm into Equation (10), the parameter values to be identified can be inversely deduced.

$$\left\{\begin{array}{c}{\widehat{x}}_k^{-}={\widehat{x}}_{k-1}\hfill \\ P_k^{-}=P_{k-1}+R^w\hfill \\ K_k=P_k^{-}{H}_k^T{(H_k{P}_k^{-}{H}_k^T+Q^w)}^{-1}\hfill \\ P_k=(I-K_k{H}_k)P_k^{-}\hfill \\ {\widehat{x}}_k={\widehat{x}}_k^{-}+K_k(z_k-H_k{\widehat{x}}_k^{-})\hfill \end{array}\right.$$

(19)

4. Real-Time SOC Estimation Method Based on Dual Kalman Filter Optimized by Cat Swarm Optimization Algorithm

4.1. Extended Kalman Filtering Algorithm

The extended Kalman filter is a linearization approximation conducted around the estimated state through the utilization of the first-order Taylor series expansion, which builds upon the Kalman filtering algorithm [31]. It is frequently employed in nonlinear systems, and its recursive equations are shown as follows:

Prediction equation:

$${\widehat{x}}_k^{-}=f\left[{\widehat{x}}_{k-1}\right]$$

(20)

$$P_k^{-}={\mathsf{\Phi }}_{k-1}{P}_{k-1}{\mathsf{\Phi }}_{k-1}^T+R^x$$

(21)

Correction equation:

$$K_k=P_k^{-}{H}_k^T{(H_k{P}_k^{-}{H}_k^T+Q^x)}^{-1}$$

(22)

$${\widehat{x}}_k={\widehat{x}}_k^{-}+K_k\left[z_k-h\left({\widehat{x}}_k^{-}\right)\right]$$

(23)

$$P_k=(I-K_k{H}_k)P_k^{-}$$

(24)

By applying a first-order Taylor series expansion to linearize the system, we derive the discretized reformulation of state Equation (1) and observation Equation (2):

$$\left\{\begin{array}{c}SOC\left(k\right)=SOC(k-1)-{\displaystyle \frac{1}{C_q}}I(k-1)\hfill \\ U_1\left(k\right)={\displaystyle \frac{1}{R_1{C}_1}}{e}^{-{\displaystyle \frac{1}{R_1{C}_1}}}{U}_1(k-1)+(1-e^{-{\displaystyle \frac{1}{R_1{C}_1}}}){\displaystyle \frac{1}{C_1}}I(k-1)\hfill \\ U_2\left(k\right)={\displaystyle \frac{1}{R_2{C}_2}}{e}^{-{\displaystyle \frac{1}{R_2{C}_2}}}{U}_2(k-1)+(1-e^{-{\displaystyle \frac{1}{R_2{C}_2}}}){\displaystyle \frac{1}{C_2}}I(k-1)\hfill \end{array}\right.$$

(25)

$$U_o\left(k\right)=h_k[SOC\left(k\right),U_1\left(k\right),U_2\left(k\right)]+R_{0}I\left(k\right)$$

(26)

The preceding equation can be concisely expressed in simplified form as

$$X_k={\mathsf{\Phi }}_{k-1}{X}_{k-1}+BI(k-1)$$

(27)

$$U_o\left(k\right)=H_k{X}_k+R_{0}I\left(k\right)$$

(28)

In the equation,

$$X_k=\left[\begin{array}{c}SOC\left(k\right)\\ U_1\left(k\right)\\ U_2\left(k\right)\end{array}\right]$$

(29)

$${\mathsf{\Phi }}_{k-1}=\left[\begin{array}{ccc}1& & \\ & {\displaystyle \frac{1}{R_1{C}_1}}{e}^{-{\displaystyle \frac{1}{R_1{C}_1}}}& \\ & & {\displaystyle \frac{1}{R_2{C}_2}}{e}^{-{\displaystyle \frac{1}{R_2{C}_2}}}\end{array}\right]$$

(30)

$$B=\left[\begin{array}{c}-{\displaystyle \frac{1}{C_q}}\\ (1-e^{-{\displaystyle \frac{1}{R_1{C}_1}}}){\displaystyle \frac{1}{C_1}}\\ (1-e^{-{\displaystyle \frac{1}{R_2{C}_2}}}){\displaystyle \frac{1}{C_2}}\end{array}\right]$$

(31)

$$H_k=\left[\begin{array}{ccc}{\displaystyle \frac{\partial h_k}{\partial SOC\left(k\right)}}& {\displaystyle \frac{\partial h_k}{\partial U_1\left(k\right)}}& {\displaystyle \frac{\partial h_k}{\partial U_2\left(k\right)}}\end{array}\right]$$

(32)

4.2. Real-Time Estimation of SOC

The accuracy of SOC estimation via the EKF algorithm fundamentally relies on precise battery models and well-identified parameters. However, battery model parameters exhibit dynamic variations due to temperature fluctuations. To address this challenge, we propose a dual Kalman filtering framework that integrates model parameter identification with SOC estimation. Similar to the KF algorithm, the noise covariance matrices $R^x$ and $Q^x$ of the EKF algorithm remain static during iterations and must be predefined. To enhance adaptability, a swarm-intelligence-based CSO algorithm is employed to optimize the dual Kalman filter parameters dynamically. This hybrid approach significantly improves SOC estimation accuracy under time-varying parameter conditions, as validated through experimental case studies.

The dual Kalman filtering algorithm operates through two parallel Kalman filters that work in tandem during iterative cycles. Specifically, a standard KF algorithm dynamically identifies model parameters, while an EKF algorithm estimates system state variables. During initialization, the algorithm employs one-step predicted values of model parameters to update the state filter’s SOC correction term. Conversely, the one-step SOC prediction is utilized to refine the model parameter correction term, establishing a closed-loop feedback mechanism.

As illustrated in Figure 3, the CSO–DKF algorithm integrates two collaborative modules: the upper module employs an EKF algorithm for state prediction using the prior parameter ${\widehat{w}}_{k-1}$, while the lower module utilizes a standard KF algorithm for parameter correction based on ${\widehat{x}}_{k-1}$, the estimated state. Furthermore, the CSO algorithm dynamically optimizes the noise covariance matrices $R^x$, $Q^x$, $R^w$, and $Q^w$ during iterations. This optimization process generates the refined state estimate ${\widehat{x}}_k$ and parameter estimate ${\widehat{w}}_k$.

The iterative procedure of the CSO–DKF algorithm is described as follows:

• Variable initialization:

  $${\widehat{w}}_0=E\left[w_0\right]$$

  (33)

  $$P_{w_0}=E\left[\left(w-{\widehat{w}}_0\right){\left(w-{\widehat{w}}_0\right)}^T\right]$$

  (34)

  $${\widehat{x}}_0=E\left[x_0\right]$$

  (35)

  $$P_{x_0}=E\left[\left(x-{\widehat{x}}_0\right){\left(x-{\widehat{x}}_0\right)}^T\right]$$

  (36)
• The CSO algorithm yields the optimal solutions for the noise covariance matrices $R^x$, $Q^x$, $R^w$, and $Q^w$;
• Parameter prediction update:

  $${\widehat{w}}_k^{-}={\widehat{w}}_{k-1}$$

  (37)

  $$P_{w_k}^{-}=P_{w_k}+R^w$$

  (38)
• State variable prediction update:

  $${\widehat{x}}_k^{-}=f\left({\widehat{x}}_{k-1}^{-},{\widehat{w}}_k^{-}\right)$$

  (39)

  $$P_{x_k}^{-}={\mathsf{\Phi }}_{k-1}{P}_{x_{k-1}}{\mathsf{\Phi }}_{k-1}^T+R^x$$

  (40)
• State variable correction update:

  $$K_k^x=P_{x_k}^{-}{\left(H_k^x\right)}^T{\left[H_k^x{P}_{x_k}^{-}{\left(H_k^x\right)}^T+Q^x\right]}^{-1}$$

  (41)

  $${\widehat{x}}_k={\widehat{x}}_k^{-}+K_k^x\left[z_k-h\left({\widehat{x}}_k^{-}\right)\right]$$

  (42)

  $$P_{x_k}=(I-K_k^x{H}_k)P_{x_k}^{-}$$

  (43)
• Parameter correction update:

  $$K_k^w=P_{w_k}^{-}{\left(H_k^w\right)}^T{\left[H_k^w{P}_{w_k}^{-}{\left(H_k^w\right)}^T+Q^w\right]}^{-1}$$

  (44)

  $${\widehat{w}}_k={\widehat{w}}_k^{-}+K_k^w(z_k-H_k^w{\widehat{w}}_k^{-})$$

  (45)
• Output the optimal estimated value of the state variable: ${\widehat{x}}_k$.

5. Experimental Validation and Analysis

Temperature is a critical factor influencing SOC estimation in lithium-ion batteries. During battery operation, temperature fluctuates due to internal self-heating and ambient thermal variations. At lower temperatures, the electrochemical activity of battery materials diminishes, increasing internal resistance. Conversely, elevated temperatures enhance material reactivity. Both extremes degrade SOC estimation accuracy due to nonlinear electrochemical dynamics. However, most existing studies either overlook the impact of temperature variations on SOC estimation or restrict comparisons to method performance under fixed thermal conditions. To address this gap, we developed a temperature-dependent parameter identification database and integrated it with the proposed algorithm, enabling robust SOC estimation under dynamic temperature scenarios.

The experimental data in this work are sourced from the publicly available CALCE LiFePO₄ battery dataset, which provides dynamic charge–discharge cycling profiles under various temperatures ($-10$ °C, $0$ °C, $10$ °C, $20$ °C, $25$ °C, $30$ °C, $40$ °C, and $50$ °C). Figure 4 illustrates the Dynamic Stress Test (DST) profiles across these temperature conditions.

A nonlinear correlation exists between the SOC and OCV in lithium-ion batteries, where OCV increases monotonically with SOC. Experimental data were collected across a temperature gradient from $-10$ °C to $50$ °C at $10$ °C intervals. Temperature-dependent SOC–OCV curves were derived via polynomial fitting. The OCV is influenced not only by SOC but also exhibits systematic temperature dependence. Specifically, OCV decreases by 2.9% per $10$ °C increase in temperature, as quantified at SOC = 0.5, as shown in Figure 5. This thermal sensitivity underscores the necessity of temperature compensation for accurate SOC estimation.

In this study, the Kalman filter algorithm and extended Kalman filter algorithm are employed to alternately execute online parameter identification and real-time SOC estimation. Specifically, as a parameter filter, KF utilizes the one-step predicted value of model parameters to update the SOC correction value of the state filter. Conversely, EKF acts as a state filter, leveraging the one-step predicted SOC value to renew the correction value of model parameters. Meanwhile, the CSO algorithm is introduced to optimize the noise covariance matrices of these two filters during the above process. In experimental implementation, the total number of “cats” is set to 50, with an acceleration coefficient of 2.05 and mutation operation parameter $\alpha$ of 0.9. The proportion of cats in tracking mode accounts for 0.4 of the total population, and the parameter c for updating speed in tracking mode is defined as 1.05. The algorithm termination criteria are established as follows: the variation in fitness function value within 100 iterations is less than $1\times {10}^{-8}$, or the maximum iteration number reaches 500.

Parameter identification in battery systems typically involves offline and online methods. Offline identification is suitable for fixed operational conditions, whereas this study implements online identification to dynamically update parameters using a CSO-enhanced KF. For the second-order equivalent circuit model, the proposed method dynamically identifies parameters across varying temperatures from $-10$ °C to $50$ °C at $10$ °C intervals and SOC intervals of $0.1$. The identified parameters are compiled into a temperature/SOC-dependent database. Figure 6 illustrates the 3D parametric surfaces depicting the relationships between model parameters, SOC, and temperature, demonstrating the algorithm’s capability to capture nonlinear thermal–electrochemical coupling effects.

As illustrated in Figure 6, for $R_0$, the increment of electrolyte viscosity and the decline in lithium-ion migration rate at low temperatures jointly result in the minimal fluctuation of ohmic internal resistance. $R_1$ and $R_2$ demonstrate notable susceptibility to SOC values. This originates from the two-phase coexistence (${\mathrm{LiFePO}}_4/{\mathrm{FePO}}_4$) of lithium iron phosphate (${\mathrm{LiFePO}}_4$) at high SOC. The charge transfer impedance at the phase interface fluctuates with the variation in lithium-ion deintercalation rate. Meanwhile, the abrupt change in interfacial energy barrier during phase transition induces the nonlinear variation in polarization internal resistance, thus presenting a fluctuating state in the figure. At low temperatures, the solid electrolyte interphase may experience thickening or local rupture, which destabilizes the charge–discharge characteristics of the electric double-layer capacitor. At high SOC, anode graphite approaches saturation, potentially triggering lithium metal precipitation. Such precipitation locally blocks electrode pores and alters the equivalent area of the electric double-layer capacitor. Moreover, within the equivalent circuit model, interactive effects exist among parameters. Consequently, $C_1$ exhibits significant fluctuations under low-temperature and high-SOC conditions, while the variation in $C_2$ shows irregularity. However, around $20$ °C, all parameters exhibit two peaks. Therefore, relying merely on identification results under a single ambient temperature will induce substantial deviation in SOC estimation. To enhance the real-time estimation accuracy of SOC under variable-temperature conditions, integrating Equation (46) with the database enables the acquisition of accurate parameter values corresponding to the current temperature T and SOC value.

$$\left\{\begin{array}{c}{R}_0=F_{R_0}(SOC,T)\hfill \\ R_1=F_{R_1}(SOC,T)\hfill \\ R_2=F_{R_2}(SOC,T)\hfill \\ C_1=F_{C_1}(SOC,T)\hfill \\ C_2=F_{C_2}(SOC,T)\hfill \end{array}\right.$$

(46)

To validate the outcomes of parameter identification and the accuracy of the model, tests were conducted under DST conditions. Three sets of operating-condition currents corresponding to low temperature ($-10$ °C), room temperature ($25$ °C), and high temperature ($50$ °C) were selected as excitation inputs. The parameter identification results were incorporated into the aforementioned model to obtain the model-output voltage at the model output. Figure 7 presents the comparison curves between the model-output terminal voltage and the measured voltage, along with the corresponding error curves. As can be observed from the figures, the algorithm initially exhibits relatively large errors due to the setting of initial values. However, it can rapidly converge to values in the vicinity of the true values. The results demonstrate that, under DST conditions, the model-output voltage can closely track the measured voltage, with errors confined to a very small range. This provides an accurate battery model for SOC estimation.

The real-time estimation of SOC is achieved by optimizing the DKF algorithm with the aforementioned CSO algorithm. Subsequently, the accuracy of SOC estimation under variable-temperature conditions is validated. Specifically, three variable-temperature intervals, namely variable low-temperature, variable ambient-temperature, and variable high-temperature, are selected, as depicted in Figure 8. Figure 9 illustrates the comparison and errors between the CSO–DKF estimation algorithm proposed in this paper and SOC reference values. It can be deduced from the images that, within the three variable-temperature intervals, the black curves generated by CSO–DKF closely approximate the reference values, with errors confined to an extremely narrow range. As shown in

Table 1. SOC estimation characteristics.

	RMSE (Variable Low Temperature)	RMSE (Variable Ambient Temperature)	RMSE (Variable High Temperature)
CSO–DKF	0.30%	0.24%	0.36%
DKF	0.93%	0.88%	0.97%

, the DKF algorithm, owing to the method of introducing temperature variables proposed in this paper and its special working state of alternating iteration, also exhibits high estimation accuracy. It maintains the root-mean-square error (RMSE) of SOC estimation below 1% under the three variable-temperature conditions. However, with the optimization of the cat swarm optimization algorithm, the estimation accuracy error can be effectively controlled within 0.4%.

6. Conclusions

Lithium-ion batteries exhibit a high degree of sensitivity to temperature variations. Nonetheless, certain prior studies have been solely centered around the modeling of lithium batteries under constant temperature conditions, neglecting the influence of temperature changes. This paper, predicated on the DST operating condition data of lithium batteries across diverse temperatures ($-10$ °C, $0$ °C, $10$ °C, $20$ °C, $25$ °C, $30$ °C, $40$ °C, and $50$ °C), makes the following primary contributions:

• A second-order equivalent circuit model was constructed. Dynamic parameter identification was carried out by leveraging the CSO algorithm to optimize the KF, thereby determining the model parameters. Subsequently, the accuracy of the proposed model was verified under low-temperature ($-10$ °C), ambient-temperature ($25$ °C), and high-temperature ($50$ °C) conditions. The verification results demonstrated that the model is capable of providing favorable accuracy and exhibits robust performance with respect to temperature variations;
• An environmental temperature battery database was established based on the parameter identification results obtained at different temperatures and various SOC stages. Through mathematical expressions, the relationships among the parameters at the current moment, temperature, and SOC values were established, thus facilitating the introduction of the temperature variable during the joint estimation of dynamic parameter identification and real-time SOC;
• Building upon this foundation, the CSO algorithm was utilized to optimize the DKF for real-time SOC estimation. The state filter and the parameter filter were employed alternately. The accuracy of SOC estimation and the optimization effect of the CSO were verified within three temperature variation intervals, namely variable low temperature, variable ambient temperature, and variable high temperature. The results revealed that, under varying temperatures, this system can ensure commendable accuracy in real-time SOC estimation, thereby providing a viable approach for estimating the state of charge of lithium-ion batteries in variable-temperature environments.

Although this algorithm represents a real-time estimation scheme that strikes a balance between model accuracy and computational complexity, it still poses a challenge in engineering applications due to its relatively high requirement for chip computing power. Future research directions should focus on exploring ways to minimize the computational complexity to the greatest extent while maintaining accuracy, with the aim of applying the algorithm in practical engineering projects. Moreover, while the current framework can effectively conduct real-time SOC estimation under wide dynamic temperature ranges, the impact of capacity fade, which is induced by environmental temperature and battery aging, on SOC estimation has not been taken into account. Consequently, a real-time SOC estimation method that considers capacity fade will undoubtedly be the subject of further investigation in our future work.