A Joint Estimation Method for the SOC and SOH of Lithium-Ion Batteries Based on AR-ECM and Data-Driven Model Fusion

Abstract

Accurate estimations of State-of-Charge (SOC) and State-of-Health (SOH) are crucial for ensuring the safe and efficient operation of lithium-ion batteries in Battery Management Systems (BMSs). This paper proposes a novel joint estimation method integrating an Autoregressive Equivalent Circuit Model (AR-ECM) with a data-driven model to address the strong coupling between SOC and SOH. First, a multi-strategy improved Ivy algorithm (MSIVY) is utilized to optimize the hyperparameters of a Hybrid Kernel Extreme Learning Machine (HKELM). Key voltage interval features, including split voltage, differential capacity, and current–voltage product, are extracted and filtered using a sliding window approach to enhance SOH prediction accuracy. The estimated SOH is subsequently incorporated into the AR-ECM state-space equations, where an enhanced particle swarm optimization algorithm optimizes the model parameters. Finally, the Extended Kalman Filter (EKF) is applied to achieve collaborative SOC–SOH estimation. Experimental results demonstrate that the proposed method achieves SOH errors below 1% and SOC errors under 2% on public datasets, showcasing its robust generalization capability and real-time performance.

1. Introduction

As energy demand and pollution rise, countries urgently need sustainable development. Electric vehicles (EVs) are key solutions, with lithium-ion batteries (LIBs) as crucial components [1]. SOH and SOC are vital for safety, performance, and reliability. These parameters are hard to measure directly, making high-precision estimation methods essential for battery technology [2].

The SOC, a key battery performance metric, represents the ratio of remaining power to total capacity [3]. SOC estimation methods fall into three categories. Category 1: The Ah integration method calculates charge changes through current integration. While simple, it suffers from inaccuracies due to initial value errors, current measurement inaccuracies, and aging effects [4,5]. Consequently, it is often combined with model-based or data-driven approaches to enhance accuracy and robustness. Wang et al. [6] proposed a method based on a combination of ampere-hour integration and an extended Kalman filter, which effectively reduces cumulative errors over long-term use. Chang et al. [7] used a new approach of combining neural networks and ampere-hour integral compensation to improve the accuracy of charge-state prediction under various environmental and operating conditions. Category 2: Physical model-based methods use adaptive filtering and state-space expressions for SOC estimation, with SOC and polarization voltage as state variables derived from OCV tests. Zeng et al. [8] developed an SOC estimation method based on a fractional-order model, which better describes the dynamic characteristics of the battery and improves the estimation accuracy. Li et al. [9] proposed an SOC estimation method based on a Dual Cubature Kalman Filter, which effectively improves estimation accuracy by capturing the dynamic behavior of the battery on different time scales. While theoretically interpretable, these methods face limitations [10,11]. They rely on complex electrochemical parameters (e.g., OCV, polarization voltage, and internal resistance) requiring precise identification and increasing model complexity. Category 3: Data-driven methods. Data-driven methods use machine learning techniques to establish a nonlinear mapping relationship between battery operating parameters and the SOC [12], and commonly used algorithms include convolutional neural networks (CNNs), recurrent neural networks (RNNs) [13], gate recurrent units (GRUs) [14], long short-term memory (LSTM) [15], and support vector machines (SVMs) [16]. Ma et al. [17] proposed an SOC estimation method based on the combination of CNNs and UKFs, which took the output of a CNN as the input of a UKF and obtained high-precision SOC estimation through self-correction. El Fallah et al. [18] developed an SOC estimation model based on a deep neural networks (DNNs), which can effectively capture the dynamic characteristics of batteries. Compared to physical models, this approach minimizes reliance on battery characteristic modeling and avoids complex parameter identification. By analyzing extensive battery operation data, it achieves high-precision SOC estimation. However, it demands high-quality, large-scale data, particularly covering diverse operating conditions, as data quality directly impacts model training and prediction accuracy [19].

The SOH is used to measure the degree of battery aging, defined as the ratio of the currently available capacity to the rated capacity, and is an important metric for assessing battery performance and degradation [20]. SOH estimation methods can be classified into model-based methods and data-driven methods. Model-based methods describe battery degradation laws by building mathematical models, such as the equivalent circuit model (ECM), electrochemical model, and fractional order model [21,22]. Li et al. [23] proposed an SOH estimation method based on an improved equivalent circuit model (ECM), which reflects the degradation of battery capacity by identifying changes in model parameters. Chen et al. [24] developed an electrochemical aging model that considers the effect of temperature, improving the accuracy of SOH estimation in different temperature environments. Although these methods have high theoretical accuracy, they rely on many experimental tests and calibrations, and the models are complex and computationally expensive, making it difficult to meet the demands of real-time applications; additionally, aging models are more sensitive to parameters and environmental changes. Data-driven methods use machine learning techniques to capture the nonlinear characteristics of battery degradation and model the relationships between health indicators (e.g., incremental capacity, differential voltage, and partial energy characteristics) and SOH, and common algorithms include RNNs [25], LSTM [26], and Gaussian process regression (GPR), among others. Peng et al. [27] developed an SOH estimation model based on IGWO–LSTM, which combined with Dropout regularization to suppress model overfitting and improve generalization ability. He et al. [28] applied GPR to SOH estimation, quantifying the uncertainty of predictions through a probabilistic model and enhancing the reliability of decision-making. Data-driven methods do not require tedious parameter identification, but they have high requirements for data quality and quantity and high computational overheads.

In addition, since single estimation methods tend to ignore the strong coupling between SOC and SOH, methods for jointly estimating the two have gained much attention. Battery degradation has a significant effect on the accuracy of SOC estimation, while inaccurate SOC estimation results may in turn interfere with SOH calibration. To address this issue, researchers have developed machine learning, multi-timescale, and dual-filtering methods for the joint estimation of SOC and SOH. Zeng et al. [29] proposed two RBF-ARX models to capture the nonlinear dynamics of batteries and establish the association between SOC, SOH, and observed values. The initial state is then sampled and inferred using the MCMC method and finally combined with UKF for joint estimation. The experimental validation uses multiple data sets to demonstrate the effectiveness of the method. Yang et al. [30] proposed a complementary cooperative algorithm based on the combination of double Kalman filtering (DEKF) and pattern recognition, which achieves high-precision joint SOC–SOH estimation. Wei et al. [31] proposed an algorithm combining an adaptive central difference Kalman filter and discrete-time sliding mode observer (ACDKF-DSMO) for improving the robustness and accuracy of SOC and SOH estimation of lithium-ion batteries. In addition, Li et al. [32] showed that fractional-order models for joint SOC–SOH estimation have higher accuracy than traditional integer-order equivalent circuit-based models and can better describe the nonlinear characteristics of batteries.

Different neural networks have different hyperparameters. Selecting an appropriate optimization algorithm to automatically search for the best parameter combination solves the inefficiency and subjectivity of manual parameter tuning and can significantly improve model performance. Ge et al. [33] constructed an improved IBA-ELM model for joint SOC–SOH estimation, and improved the model performance through an optimization algorithm. Wang et al. [34] proposed an improved firefly algorithm (IFA), improved the prediction performance of the GPR model from the perspective of the internal prediction process, and applied it to the joint estimation of SOH and SOC. Ghasemi et al. [35] first proposed the theoretical framework of the basic Ivy algorithm, which can efficiently search for the optimal solution in the solution space by simulating the characteristics of ivy growing towards favorable resources. Zhang et al. [36] improved the basic Ivy algorithm by introducing an adaptive perturbation factor and adaptive growth rate, significantly improving the convergence speed and accuracy of the algorithm when dealing with high-dimensional nonlinear problems. To address the deficiencies of existing SOC–SOH joint estimation methods and optimization algorithm research, this paper proposes a novel hybrid framework that combines an autoregressive equivalent circuit model (AR-ECM) and a data-driven method to achieve high-precision joint estimation of SOC and SOH. The contributions of this research are briefly described as follows:

• We propose a novel hybrid framework integrating AR-ECM with data-driven models for SOC–SOH joint estimation. The framework adaptively incorporates SOH into SOC calculations, enabling precise estimation under battery aging conditions.
• A systematic feature selection methodology is developed combining voltage segmentation and multiple correlation analysis (MCA). This approach integrates key voltage characteristics (SV, dQ/dV, and InV) to enhance SOH prediction accuracy.
• We present an improved Ivy algorithm (IVYA) with chaotic mapping and tangent flight operators for optimizing HKELM hyperparameters, significantly improving convergence speed and estimation precision.
• Comprehensive experiments across multiple datasets demonstrate the framework’s effectiveness, achieving SOH errors below 1% and SOC errors under 2% under various operating conditions and temperatures.

The organization of the paper is as follows: Section 2 describes the dataset and the data processing process in detail. The SOC and SOH co-estimation method is described in Section 3. Section 4 shows that the predictions were accurate and that the proposed approach worked. The conclusions are presented in Section 5.

2. Data Acquisition and Processing

2.1. Dataset

This paper utilizes two datasets to evaluate the efficacy of the proposed method. Dataset I is the University of Michigan Battery Laboratory (UMBL) [37] for joint SOH–SOC estimation, and Dataset II is the McMaster University in Hamilton dataset (LG_HG2) [38] for validating the universality of the SOC estimation method. Figure 1a,b displays the Dataset I SOH and the voltages of Groups I-Cell 1. Figure 1c–e displays the voltage and SOC curves for various dynamic duty cycles. This paper uses three common temperatures: −20 °C, 25 °C, and 40 °C.

2.2. Relationship Between the SOC and SOH

The relationship between the SOC and SOH is outlined in the literature [39]. Setting the current SOH to 70.5%, the results are shown in Figure 2 without considering SOH, where the predicted SOC when the battery is charged to the upper cutoff voltage does not reach 100%. When considering the effect of SOH, the impact of SOH on SOC decreases gradually as SOH approaches the actual SOH value. Therefore, it is necessary to predict SOC and SOH jointly.

2.3. Data Processing

2.3.1. Segment Voltage Selection

Since charging mode is easier to control than discharge mode, two voltage time integrations of 3.8–4.0 V and 4.0–4.2 V in the charging phase of Group I-Cell 1 in Dataset I are taken as examples, as shown in Figure 3a. Figure 3b shows the capacity degradation curves for the two voltage time integration segments. The segmented capacity degradation curve closely mirrors the overall capacity degradation curve, accurately reflecting the capacity regeneration characteristics of the battery. The Pearson correlation coefficient (PCC) is used to quantify the relationship between these two curves, as indicated in Equation (4):

$${\rho }_{x,y}={\displaystyle \frac{\mathrm{cov}(X,Y)}{{\sigma }_X{\sigma }_Y}}{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n(x_i-\overline{x})(y_i-\overline{y})}}{\sqrt{{\displaystyle {\sum }_{i=1}^n{(x_i-\overline{x})}^2}}\sqrt{{\displaystyle {\sum }_{i=1}^n{(y_i-\overline{y})}^2}}}}$$

(1)

where $x_i$ denotes the feature, $y_i$ denotes the SOH, and $\overline{x}$ and $\overline{y}$ denote the average of the data $x_i$ and $y_i$, respectively. The selection of the constant current charging segment is determined by two key factors. The voltage rise time and SOH’s PCC are the first points. The second is that 20% to 80% is the typical battery SOC. Figure 3c shows the PCCs between Group l-Cell 1 and the SOH for different up–down voltage ranges.

According to the above analyses, different voltage bands capture the full capacity degradation pattern differently. Furthermore, the voltage range has a significant effect on certain HIFs. The peak is approximately 3.7 V on the $dQ/dV$ curve, for example. To enhance the correlation between HIFs and capacity, selecting an optimal voltage band is essential. The objective of this paper is to maximize the sum of the absolute PCCs between the four HIFs and the SOH selected within the voltage interval, as expressed in the following equation:

$$\max P_{max}={\displaystyle \sum _{i=1}^4\left|p_i\right|}$$

(2)

where $P_{\max }$ is the maximum correlation coefficient sum obtained and $p$ is the correlation coefficient between the HIFs and the capacity at the $i$th time.

Real-world EV users usually start charging before their SOC reaches 0% and typically stop charging before reaching 100%, especially regarding rapid charging. This pattern is influenced primarily by concerns regarding overcharging, which is frequently exacerbated by anxiety associated with range limitations. Finding the most constrained voltage range is essential for improving this charging strategy. Adopting this method reduces computational expenses and enhances the availability of vital data in accurate capacity assessment. Taking these factors into account, this paper suggests certain limits on both the SOC and the voltage range that is allowed:

$$\left\{\begin{array}{l}10\%<SOC<80\%\\ 0.05V<V_b-V_a<0.25V\end{array}\right.$$

(3)

where $V_a$ is the left endpoint of the voltage segment and $V_b$ is the right endpoint. According to Equations (5) and (6), the problem of voltage interval optimization is a multi-constrained nonlinear programming problem. This paper uses PSO along with adaptive inertia weights to solve it. The Dataset I voltage selection intervals are described in detail in Supplementary Table S1, and the solution algorithms are described in Supporting Document S2.

2.3.2. Feature Extraction

In feature extraction, it is often desirable to obtain high-quality features via simple data operations. Therefore, for the HIFs extracted from the time–voltage curves of the charging process and the $dQ/dV$ curves, on the basis of the analysis in Section 2.3.1 and taking Group I-Cell 1 of Dataset I as an example, the voltage interval with the highest correlation coefficient is [3.53–3.81], as shown in Figure 4a, and the total sum of the correlation coefficients is 2.987. Within the same voltage range, as the number of cycles increases, the charging time and charging capacity decrease, and the rate of voltage rise (slope of the curve) increases. Therefore, four parameters, the voltage integral over time ($S$), the voltage rise time ($dv/dt$), the voltage slope ($K$), and the intercept ($b$) are selected as the basic features.

Noise significantly impacts test results if the voltage measurement interval is too short. Important information could be lost if the voltage interval is set too wide. Therefore, every 0.05 V was chosen and the $dQ/dV$ still contains considerable noise. Therefore, filtering methods can further process the data and we utilize the moving average filtering method. The principle of the filter can be described as

$$y(n)={\displaystyle \frac{1}{windowsize}}(x(n)+x(n-1)+\cdots +x(n-(windowsize-1)))$$

(4)

In Figure 4b, comparison of the filtered $dQ/dV$ curve with the original curve obtained from the original $dQ/dV$ data reveals that the filtered curve is smooth and retains the features on the curve well. Figure 4c shows that the peak of the $dQ/dV$ decreases and shifts to the right with increasing numbers of cycles, and two indicators, the $dQ/dV$ peak and $dQ/dV$ peak position are intimately associated with battery aging, so they are chosen as the HIFs.

According to the literature [40], for different discharge current demands, the battery discharge voltage distribution may be different, but the product of $I^n*V$ is relatively constant. In this work, this characteristic is used to modify the voltage fragment data obtained from the analysis as follows. First, the new capacity ($Q_{InV}$) resulting from the product of the voltage data $V(t)$ and its corresponding charging current increasing to a power of 0.05 ($V(t)*I{(t)}^{0.05}$) was utilized, as shown in Figure 4f. Next, all the start times of the voltage segments were time-shifted so that each voltage segment started at time 0, yielding the $T_{InV}-Q_{InV}$ curve, as shown in Figure 4e. The $InV$ data were then normalized in the time domain. The $T_{InV}-Q_{InV}$ curve was generated by subtracting period 1 ($T_{InV1}$) from period i ($T_{InVi}$). The features of normalized $T_{InV}$ and normalized $\Delta Inv$ were obtained, as shown in Figure 4f. The features extracted in this paper are shown in

Table 1. Extracted HIFs.

Label	Name	Detail
HIF1	$S$	Voltage integral over time
HIF2	$dv/dt$	Voltage rise time
HIF3	$K$	Voltage slope
HIF4	$b$	voltage intercept
HIF5	$dQ/d{V}_{peak}$	$dQ/dV$ Peak
HIF6	$dQ/d{V}_{pos}$	$dQ/dV$ peak position
HIF7	$T_{inv}$	Poststandardization time
HIF8	$\Delta Inv$	Standardized time lag

.

2.3.3. Feature Selection in Multiple Correlation Analysis

In machine learning, feature selection is critical. It is important to remove useless or redundant features to eliminate noise from irrelevant features and make model predictions more accurate. Five types of correlation analyses were used to analyze feature data via multiple correlation analysis (MCA) [41], including maximum information coefficient (MIC) [42], Pearson correlation coefficient (PCC), Kendall correlation (KC) [43], gray correlation analysis (GRA) [44], and Spearman correlation (SC) [45]. We take Group I-Cell 1 of Dataset I as an example, and the results are shown in Figure 5.

This study comprehensively evaluated the importance of each HIF. All the extracted features are shown in Figure 5a and different correlation results were obtained according to the MCA, as shown in Figure 5b. The obtained coefficients are sorted by features according to absolute value, and the highest-ranking HIFs are assigned the highest weight of 1, decreasing by 0.125 in turn. The weighted scores for each HIF are then added together to obtain the final total, as follows:

$$\left\{\begin{array}{l}{H}_i=sort(HI{F}_{i,j})*(1-(j-1)*0.125)\\ S_j={\displaystyle {\sum }_{i=1}^5{H}_{i,j}}\end{array}\right., i=1,2,\cdots 5,j=1,2,\cdots 8$$

(5)

where $S_j$ denotes the sum of the HIF coefficients. Figure 5c shows the ranking of all the HIFs on the basis of their importance in obtaining the features that are most closely linked to the cell’s SOH. The features above the average are selected as the input to the prediction model.

3. Method

3.1. SOH Estimation Method

3.1.1. HKELM

HKELM is an improved ELM method, which combines the advantages of global and local kernel functions to improve the generalization and prediction performance of the model. In order to improve the performance of the ELM algorithm, a kernel function is introduced. The polynomial kernel has strong generalization ability but is not suitable for local prediction, and the Gaussian kernel has good robustness but limited accuracy. Therefore, the hybrid kernel function was introduced to combine the two and strengthen the ELM model.

The hybrid kernel function is formulated as follows:

$$K(x_i,x_j)=\mu ·K_{\mathrm{local}}(x_i,x_j)+(1-\mu )K_{global}(x_i,x_j)$$

(6)

Local kernel function:

$$K_{RBF}(x_i,x_j)=\mu ·\exp (-{\displaystyle \frac{{‖x_i-x_j‖}^2}{2{\sigma }^2}})$$

(7)

Global kernel function:

$$K_{ploy}(x_i,x_j)=(1-\mu )\cdot {(\alpha \cdot (x_i,x_j)+1)}^2$$

(8)

The identity matrix a and penalty coefficient S are introduced to prevent the overfitting phenomenon caused by overcomplexity of the model.

$$\beta =H^T{({\displaystyle \frac{I_0}{z}}+H{H}^T)}^{-1}L$$

(9)

$$F(x)=K{(H{H}^T+{\displaystyle \frac{I_0}{z}})}^{-1}L$$

(10)

The improved IVYA is used to optimize the (μ, σ, α, and z) parameters of HKELM.

3.1.2. Multi-Strategy Improvement for Ivy

IVYA

The IVYA is a metaheuristic based on group intelligence that was proposed in 2024 [35]. This algorithm focuses on modeling various stages of ivy life. An ivy population’s starting point in the search space is chosen at random via Equation (11). The expression is provided below:

$$I_i=I_{\min }+rand(1,D)\odot (I_{\max }-I_{\min }),i=1,\cdots ,N_{pop}$$

(11)

where $rand(1,D)$ denotes a D-dimensional vector of uniformly distributed random numbers in the interval [0, 1]. $I_{\max }$ and $I_{\min }$ are the upper and lower bounds of the search space, respectively, and $\odot$ denotes the Hadamard product of the two vectors.

The steps of researching and population searching in the proposed IVYA.

Step 1: Coordinate and order population growth.

Let $G\upsilon$ be the growth rate, $\varphi$ be the growth rate, and $\phi$ be the correction factor for deviation from growth. Equation (12) is transformed into a difference equation for the growth rate $G{\upsilon }_i(t)$ of member $I_i$ as follows:

$${\displaystyle \frac{dG\upsilon (t)}{dt}}=\psi \cdot G\upsilon (t)\cdot \phi (G\upsilon (t))$$

(12)

$$\Delta G{\upsilon }_i(t+1)=ran{d}^2\odot (N(1,D))\odot \Delta G{\upsilon }_i(t)$$

(13)

where the vectors $\Delta G{\upsilon }_i(t)$ and $\Delta G{\upsilon }_i(t+1)$ denote the discrete-time system growth rate and where $ran{d}^2$denotes that the probability density function is equal to $1/(2\sqrt{x})$. The D-dimensional random vector $N(1,D)$ is a normal distribution random number.

Step 2: Ivy plants grow to obtain sunlight.

In the following equation, the member $I_i$ uses member $I_{ii}$ to climb and move logically toward the light source:

$$I_i^{new}=I_i+\left|N(1,D)\right|\odot (I_{ii}-I_i)+N(1,D)\odot \Delta G{\upsilon }_i,i=1,2,\cdots ,N_{pop}$$

(14)

with

$$\Delta G{\upsilon }_i=\left\{\begin{array}{ll}{I}_i\varnothing (I_{\max }-I_{\min }),& Iter=1;\\ ran{d}^2\odot (N(1,D)\odot \Delta G{\upsilon }_i),& Iter>1;\end{array}\right.$$

(15)

The vector $\left|N(1,D)\right|$ has absolute values for its components $N(1,D)$ and Hadamard divisions denoted by $\varnothing$.

Step 3: Spread and evolution of the Ivy plants.

After a phase in which the member $I_i$ roams globally through the search space to the nearest and most important neighbor $I_{ii}$, a phase occurs when $I_i$ tries to directly follow the best member of the whole population $I_{Best}$, which is equivalent to searching for a better optimal solution around the member $I_{Best}$. This is denoted as

$$I_i^{new}=I_{Best}\odot (rand(1,D)+N(1,D)\odot \Delta G{\upsilon }_i)$$

(16)

Next, the new value of $\Delta G{\upsilon }_i$, the current member growth rate $I_i^{new}$, is calculated via the following formula (this formula is identical to the initialization $\Delta G{\upsilon }_i$ formula):

$$\Delta G{\upsilon }_i^{new}=I_i^{new}\varnothing (I_{\max }-I_{\min })$$

(17)

Step 4: Survivor selection

To simulate the two alternating stages of an ivy tree’s life, i.e., “climbing” and “expanding”, the following decision-making method is used in the IVYA. When the objective function value $f(I_i)$ of the member $I_i$ is less than a multiple of $f(I_{Best})$, the parameter $\beta =(2+rand)/2$ is used. Then, the ivy tree starts expanding the width of branches and leaves (given by Equation (17)). The ivy ascends and climbs, as described by Equation (19).

Improved Methods

IVYA performs well in functional testing and practical engineering problems, but it has some limitations, such as difficulty with global optimal searching and premature convergence. To this end, this study introduces three enhancement strategies to optimize the IVYA and reduce the risk of falling into local optimal searching.

Improvement 1: A chaotic sequence is generated by using the circular chaotic mapping distributed between [0, 1], a simple process generates the random sequence, and its chaotic characteristics are used to replace random initialization so that the population is more evenly distributed in the search space, which can be expressed as

$$x_{i+1}=\mathrm{mod}(x_i+b-({\displaystyle \frac{a}{2\pi }})\sin (2\pi x_k),1),a=0.5 \mathrm{and} b=0.2.$$

(18)

Improvement 2: The improved method proposed in this paper is mainly to improve the local searching by the IVYA, Cauchy distribution inverse cumulative distribution (CDICD) function, and Tangent flight operator (TFO); these operators are used as scaling factors to control the step size. The CDICD function can further enhance the search stability of the algorithm. TFO has the function of balancing exploitation and exploration, which enhances the convergence ability of the algorithm.

The Cauchy inverse cumulative distribution can be defined as

$$f(x;a,b)={\displaystyle \frac{1}{\pi }}[{\displaystyle \frac{b}{{(x-a)}^2+b^2}}]$$

(19)

When $a=0,b=1$ is a standard Cauchy distribution, the probability density function is as follows:

$$f(x;0,1)={\displaystyle \frac{1}{\pi }}[{\displaystyle \frac{1}{x^2+1^2}}]$$

(20)

The cumulative distribution function is as follows:

$$f(x;a,b)={\displaystyle \frac{1}{\pi }}\arctan ({\displaystyle \frac{x-a}{b}})+{\displaystyle \frac{1}{2}}$$

(21)

Its Cauchy distribution function can be expressed as

$$f^{-1}(p;a,b)=a+b\tan (\pi (p-{\displaystyle \frac{1}{2}}))$$

(22)

$$p=randn(1,d)$$

(23)

where $p$ is a uniformly distributed random number in the range [0, 1] and where $d$ is the function dimension $a=0,b=0.01$. Equation (16) is rewritten as

$$I_i^{new}=I_i+\left|N(1,D)\right|\odot (I_{ii}-I_i)+N(1,D)\odot \Delta G{\upsilon }_i\cdot (0.01\tan (\pi (p-{\displaystyle \frac{1}{2}}))),i=1,2,\cdots ,N_{pop}$$

(24)

It is similar to the Cauchy function in that it is a tangent function and is calculated as follows:

$$f=\tan (\upsilon \times {\displaystyle \frac{\pi }{2}})$$

(25)

$$\upsilon =randn(1,d)$$

(26)

where $\upsilon$ is a uniformly distributed random number in the range [0, 1] and where $d$ is the dimension of the function. Equation (18) is rewritten as

$$I_i^{new}=I_{Best}\odot (rand(1,D)+N(1,D)\odot \Delta G{\upsilon }_i)\cdot \tan (\upsilon \times {\displaystyle \frac{\pi }{2}})$$

(27)

To verify the effectiveness of the proposed improved method, this study compares three test functions on the CE2005 function set and the results are shown in Figure 6. The running results and fitness values are shown in

Table 2. Prediction results of capacity with different methods.

Function		Ivy	IIVY	E_WOA	IGWO	PSO
F1	Computing time (s)	0.234	0.153	0.29	0.547	0.037
	Fitness value	0	0	0	0	0.280
F4	Computing time (s)	0.234	0.153	0.297	0.559	0.040
	Fitness value	0	0	0.379	0	1.556
F9	Computing time (s)	0.232	0.15	0.307	0.571	0.55
	Fitness value	0	0	90.8932	16.664	111.613

. F1 and F4 are unimodal functions, while F9 is a multimodal function. The solution time and the optimal value are the average values of 30 calculations. The results show that the IIVY algorithm outperforms comparison algorithms such as Ivy, EWOA, IGWO, and PSO regarding convergence speed and solution quality. Especially in the complex multimodal function F9 test, the IIVY algorithm shows obvious performance advantages. Regarding solution time, IIVY is inferior to the PSO algorithm, but the PSO algorithm has lower accuracy in terms of solution quality. The above comparison verifies the effectiveness of the improved IVY algorithm in this paper.

Training and Testing Process

This paper predicts the SOH of LIBs by learning the mapping of feature inputs ($x$) to SOH outputs ($y$). The research employs a dataset comprising four features following MCA selection and SOH values from the preceding n cycles. Model training predicts the next cycle’s SOH value based on past cycles’ SOH values. The study set n = 30 (number of training examples) and F = 4 (number of features after selection), as shown in Figure 7.

The process is as follows:

Step 1: Select the training data via the sliding window method with a window size of n, which is immediately adjacent to the cycle to be predicted, k, to capture the battery degradation patterns and latest changes.

Step 2: Construct the training datasets, each of which includes the SOH values of the past n cycles with the four external feature variables of the previous cycle as inputs and the SOH values of the current cycle as outputs. A total of K sets of such training data were constructed.

Step 3: These data are used to train the MSIVY-HKELM model and determine the model parameters.

Step 4: Test the model in cycle k + 1 to predict new SOH values via past predicted SOH values and external feature variables.

Update cycle k and repeat Steps 2–4. The entire process selects the training data through a moving window of 30 cycles to ensure that the model can adapt to different battery operating conditions and accurately predict future SOH values.

3.2. SOC Estimation Method

3.2.1. AR Equivalent Circuit Model

In this paper, an AR-ECM is established, as shown in Figure 8. $U_{ocv}$ is the open-circuit voltage of the battery, $U_{in}$ is the internal voltage of the battery, $U_d$ is the terminal voltage, and $U_{AR}$ is the internal polarization voltage of the battery. On the basis of the equation between each voltage in Figure 8

$$U_d(k)=U_{ocv}(k)+U_{in}(k)=U_{ocv}-RI(k)-U_{AR}(k)$$

(28)

Considering that the polarization voltage of the battery is closely related to the current, using the current as input to the AR model inputs

$$U_{AR}={\displaystyle {\sum }_{i=1}^p{a}_i(k-i)}$$

(29)

where $p$ is the model order, $a_i(i=1,2, . . . , p)$ represents the AR model coefficients, and Equation (12) is rewritten as

$$U_d(k)=U_{ocv}(k)-RI(k)-{\displaystyle {\sum }_{i=1}^p{a}_i(k-i)}$$

(30)

When $i=0$, $a_0$ is the ohmic internal resistance and $a_0=R$ and $a_i(i=1,2, . . . , p)$ are the AR model coefficients for the polarization voltage drop.

3.2.2. AR-ECM State-Space Equation

On the basis of Figure 8, $U_{in}$ is represented as $-(RI(k)-{\displaystyle {\sum }_{i=1}^p{a}_i(k-i)})$ at time $k$ and the recursive expression is as follows:

$$\begin{array}{ll}{U}_{in}(k+1)& =U_{in}+R(I(k+1)-I(k))+{\displaystyle {\sum }_{i=1}^p{a}_i(I(k-i+1)-I(k-i))}\\ & =U_{in}(k)+{\displaystyle {\sum }_{i=1}^p{a}_i\Delta I(k+1-i)}\end{array}$$

(31)

where $\Delta I(k+1-i)=I(k+1-i)-I(k-i)$ when $i=0$, $a_0$ is the ohmic internal resistance, and $a_0=R$ and $a_i$ are the coefficients of the polarization voltage drop AR model.

The SOC uses the Ansatz integral method to establish the equation of state in discrete form:

$$SOC(k+1)=SOC(k)+I(k)*{\displaystyle \frac{\eta \Delta t}{SOH(K)*C_{rated}}}$$

(32)

is the sampling time and nominal capacity, a coefficient related to the temperature and charge/discharge multiplier, and can be expressed considering SOH as being in a long-term changing state, as follows:

$$SOH(k+1)=SOH(k)$$

(33)

Using Equations (34)–(36), the state-space equations are obtained as follows:

$$\left\{\begin{array}{l}SOH(k+1)=SOH(k)\\ SOC(k+1)=SOC(k)+I(k)*{\displaystyle \frac{\eta \Delta t}{SOH(k)*C_{rated}}}\\ U_{in}(k+1)=U_{in}(k)+{\displaystyle {\sum }_{i=0}^p{a}_i\Delta I(k+1-i)}\end{array}\right.$$

(34)

where the state transition function is $x(k)={[SOH(k) SOC(k) U_{in}(k)]}^T$ and the measurement function is $u(k)={[I(k+1) I(k),\cdots ,I(k+1-p)]}^T$. The terminal voltage is the observation variable and the observation equation is as follows:

$$U_d(k)=Gx(k)+U_{ocv}(SOC(k))+\upsilon (k)$$

(35)

According to the literature [46], $U_{OCV}$ can be expressed as a function of the battery $SOC(k)$ at time $k$, as follows:

$$U_{OCV}(k)=K_0+{\displaystyle \frac{K_1}{SOC(k)}}+K_{2}SOC(k)+K_3\ln (SOC(k))+K_4\ln (1-SOC(k))$$

(36)

3.2.3. Extended Kalman Filter SOC Estimation

For the power battery, its actual working conditions should be considered and, since the SOC cannot be measured directly and presents a nonlinear state throughout the charging and discharging process, it is more reasonable to use the extended Kalman filter [47].

The EKF implementation steps are as follows:

(1) Obtain the initial state SOC values and initial values of the error equation matrix, as follows:

$$x_0=SO{C}_0,P_0=\mathrm{var}(x_0)$$

(37)

where $x_0$ and $P_0$ are the initial values of the system state variables and the initial values of the state variable error covariance matrix, respectively.

(2) State variable prediction estimation:

State vector prediction

$${\widehat{x}}_{k+1|k}=f({\widehat{x}}_k,u_k)$$

(38)

$$A_{k-1}={\displaystyle \frac{\partial f(x_{k-1},u_{k-1})}{\partial x_{k-1}}}=1$$

(39)

$$C_k={\displaystyle \frac{\partial y_k}{\partial x_k}}=-{\displaystyle \frac{K_1}{x_k^2}}+K_2+{\displaystyle \frac{K_3}{x_k}}-{\displaystyle \frac{K_4}{1-x_k}}$$

(40)

$$x_k^{-}=x_{k-1}^{-}-{\displaystyle \frac{I*\Delta t}{C_n}}$$

(41)

$$y_k^{-}=K_0+{\displaystyle \frac{K_1}{x_k^{-}}}+K_2{x}_k^{-}+K_3\ln (x_k^{-})+K_4\ln (1-x_k^{-})+a_0{I}_k+a_1{I}_{k-1}+a_2{I}_{k-2}+a_3{I}_{k-3}$$

(42)

Calculate the error covariance matrix as follows:

$$P_{k+1|k}=A_k{P}_k{A}_k^T+\omega Q{\omega }^T$$

(43)

where $P_k$ is the state error covariance matrix at $k$; $Q$ is the system process noise covariance matrix; $Q$ is a fixed value of 0.0001²; $k+1|k$ is the recursive result based on the $k$ time for the $k+1$ time variable; and $\omega$ is process noise.

(3) State variable measurement update

Kalman gain calculation

$$K_{k+1}=P_{k+1|k}{C}_k^T{[C_k{P}_{k+1|k}{C}_k^T+vR{v}^T]}^{-1}$$

(44)

where $R$ is the measurement of noise covariance, using a fixed value of 0.0005², and $\nu$ is the observed noise. Calculate the SOC optimal estimates and error covariance optimal estimates, as follows:

$$\left\{\begin{array}{l}{\widehat{z}}_{k+1}=g({\widehat{x}}_{k|k},u_k)\\ {\widehat{x}}_{k+1}={\widehat{x}}_{k+1|k}+K_{k+1}(z_{k+1}-{\widehat{z}}_{k+1})\end{array}\right.$$

(45)

Error covariance matrix update:

$$P_{k+1}=(I-K_{k+1}{C}_k)P_{k+1|k}$$

(46)

3.2.4. AR Model Parameter Identification

Based on Ref. [11], the AR model order $p=3$$a_i(i=1,2, . . . , p)$ is selected. The parameters to be identified are $[K0,K1,K2,K3,K4,a_0,a_1,a_2,a_3]$, and the same algorithm as in Section 2.3.1 is used for parameter identification to reduce the solution time. In this study, we utilized the battery from Group I, specifically Cell 1, to conduct a series of tests over multiple cycles. The results obtained from these experiments are detailed in Figure 9. Throughout the testing, we observed that the error margin was consistently kept within 10 mV, which underscores the high level of accuracy achieved in our measurements. This reliable performance highlights the battery’s effectiveness under varying conditions.

3.3. Joint SOH–SOC Estimation Method

The analysis presented in Section 2.2 demonstrates that the SOH plays a crucial role in the accuracy of SOC estimation results. This paper introduces an innovative combined AR-ECM and data-driven model designed to jointly estimate both the SOC and SOH of LIBs. As illustrated in Figure 10, the approach comprises two main components: the online identification of battery parameters and the joint estimation of battery states, which includes both SOC and SOH estimations.

4. Results and Discussion

This study evaluated SOC and SOH prediction via three error matrices—MAE, MSE, and RMSE—to objectively reveal and compare method performance. During model comparison, MSE and RMSE values closer to zero indicate an acceptable model. The results for the MAE, MSE, RMSE and $R^2$, respectively, were as follows:

$$MAE={\displaystyle \frac{1}{N}}{\displaystyle \sum _{n=1}^N\left|C_n-{\widehat{C}}_n\right|}$$

(47)

$$MSE={\displaystyle \frac{1}{N}}{\displaystyle \sum _{n=1}^N(C_n-{\widehat{C}}_n}{)}^2$$

(48)

$$RMSE=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{n=1}^N(C_n-{\widehat{C}}_n}{)}^2}$$

(49)

$$R^2=1-{\displaystyle \frac{{\displaystyle \sum _{n=1}^N({\stackrel{⌢}{C}}_n-{\overline{C}}_n}{)}^2}{{\displaystyle \sum _{n=1}^N(C_n-{\overline{C}}_n}{)}^2}}$$

(50)

where $C_n$ is the actual battery capacity, ${\widehat{C}}_n$ is the predicted battery capacity, and ${\overline{C}}_n$ is the average actual battery capacity.

4.1. SOH Prediction Results

4.1.1. Prediction Results for Different Features

In this subsection, to compare the advantages of MCA feature selection versus single feature selection, following the SOH prediction process described in Section 3.1, the model parameters of the MSIVY-HKELM model are trained using the past 30 sets of training data, and then the obtained model with known parameters is used to predict the SOH values for the next cycle. For example, the SOH estimation results obtained for the Group I-Cell 1 battery in Dataset I with different eigenvalues for most inputs are shown in Figure 11.

The black curve in Figure 11a represents the real SOH recession trajectory, the red line indicates the SOH results obtained by using the features obtained from the voltage segmentation as model input, the blue line indicates the SOH results obtained by using $dQ/dV$ as model input, the green line indicates the SOH results obtained by using the InV transformed features as model input, and the red line indicates the SOH results obtained by using the MCA features after selection as model input. The red line shows the SOH results obtained after MCA feature selection is used as the model input, and the figure shows that the most accurate results are obtained after feature selection is used as the model input. All the errors in Figure 11b are less than 2%, indicating that the model has high accuracy. The prediction results are box plotted in Figure 11c, avoiding extreme outliers. Based on the different feature inputs, the median of the MCA is almost 0, and the IQR range is the smallest among all the subgroups, which indicates that the overall trend in the errors is very close to the true value, with good stability and consistency. Figure 12d shows the results of various feature errors, in which the MAE, MSE, and RMSE of $dQ/dV$ are the largest at 0.4585, 0.2975, and 0.5455, respectively, and the features selected by MCA obtain the smallest MAE, MSE and RMSE values, with values of 0.1451, 0.0251, and 0.1584, respectively. The results are shown in

Table 3. Statistical results for different characteristics.

Feature Use	MAE (%)	MSE (%)	RMSE (%)	R2
SV	0.1884	0.0739	0.2719	0.998
$dQ/dV$	0.4585	0.2975	0.5455	0.994
InV	0.3051	0.1782	0.4221	0.996
MCA	0.1451	0.0251	0.1584	0.999

. This feature selection method is adaptable and accurate based on the experimental results.

4.1.2. Prediction Results for Different Kernel Functions

This section is written because, in HKELM, the choice of kernel function significantly impacts model performance. In this study, the mixed kernel function has obvious advantages compared with the single kernel function. The SOH estimation results for the Group I-Cell 1 battery in Dataset I were obtained according to Section 4.1.1, as shown in Figure 12.

The black curve in Figure 12a represents the accurate SOH decay trajectory, the red line represents the SOH result obtained by using mixed kernel functions, the blue line represents the SOH result obtained by using local RBF kernel functions, and the green line represents the SOH result obtained by global Poly kernel functions. The undesired sum and function errors in Figure 12b are all near 0 and relatively stable. This shows that the hybrid nuclei have high robustness. The results are shown in

Table 4. Prediction results of capacity with different kernel functions.

Kernel Functions	MAE (%)	MSE (%)	RMSE (%)	R2
Hybird Kernel	0.1097	0.0179	0.1304	0.999
RBF_kernel	0.3750	0.2488	0.4998	0.995
Poly_kernel	0.4667	0.3012	0.5489	0.994

. The experimental results show that the mixed kernel function has higher accuracy.

4.1.3. Prediction Results for Different Steps

Under the general definition, battery life can be over when the SOH value decays below 80% of the initial value. The battery’s SOH can be predicted more accurately by predicting the capacity value. Figure 12 shows the prediction performance with different step size prognostications and

Table 5. Prediction results of capacity with different look-ahead periods.

Look-Ahead Period	MAE (%)	MSE (%)	RMSE (%)	R2
5	0.1606	0.0019	0.1807	0.999
10	0.1953	0.0023	0.2273	0.999
20	0.2378	0.0029	0.3879	0.996
30	0.2686	0.0034	0.5923	0.992

shows the prediction errors. Training and testing are the same but we set the forward period of L cycles to vary the output SOH value. Training with data from the previous cycle yields model parameters for each cycle. A 30-cycle moving window was also used in this experiment. The steps for 5, 10, 20, and 30 cycle forward periods are shown in Figure 13.

As shown in Figure 13a, this method is highly accurate for small forward-looking intervals L, but decreases as L increases. The longer the interval is, the less accurate the forward-looking prediction. The results show that the model better predicts the capacity after 30 cycles.

The black curve in Figure 13b represents the accurate SOH recession trajectory; the red line shows the predicted SOH results for the next five steps; the blue line shows the predicted SOH results for the next ten steps; the green line shows the predicted SOH results for the next twenty steps; and the violet line shows the predicted SOH results for an additional thirty steps. The error is shown in Figure 13c. The prediction accuracy decreases progressively with increasing future prediction step length, mainly when a relatively large deviation occurs after 350 loops for the 20-step and 30-step conditions. Figure 13d shows the box plot lines of the predicted results for different step lengths, and it can be seen that the outliers are particularly prominent in the positive direction with increasing step length, which indicates that episodic errors may have a significant impact on the results. Figure 13e shows the statistical results of the indicator, which clearly shows that the longer the prediction step length is, the larger the error, and the results are shown in Table 5.

4.1.4. Comparison with Different Methods

To further prove the superiority of the proposed SOH co-estimation method, this study is compared with four recently proposed methods. These methods propose different SOC–SOH co-estimation models, including LSTM [48], GPR [49], and LS-SVM [50]. Under the same experimental conditions, Group I-Cell 1 was used in this study for comparative experiments.

Figure 14 shows the SOH estimation results of the above method. The statistical results are shown in

Table 6. Prediction results for capacity with different methods.

Method	MAE (%)	MSE (%)	RMSE (%)	R2	Ate (s)
Our proposal	0.1606	0.0019	0.1807	0.999	64.644
LSTM	0.1953	0.0023	0.2273	0.999	992.086
GPR	0.2378	0.0029	0.3879	0.996	64.159
LS-SVM	0.2686	0.0034	0.5923	0.992	63.416

. It can be seen from the results that the method proposed in this paper has significant advantages in SOH prediction. Compared with the actual value and prediction curve in Figure 14a, the prediction trajectory of HKELM is closest to the real SOH decline trend. At the same time, the initial deviation of LS-SVM and the late error of GPR/LSTM are significant. The error percentage in Figure 14b shows that the error of HKELM is strictly controlled within ±0.5%, which is significantly better than that of LS-SVM/GPR (±1%) and LSTM (±2%). The Figure 14c box plot further verifies that HKELM has the most concentrated error distribution (the IQR is only 0.4%) and no outliers, while LSTM has extreme errors. As shown in Figure 14d, the MAE (0.1606), MSE (0.0019), and RMSE (0.1807) of HKELM levels are all the highest, and their comprehensive performances are far superior to LS-SVM, GPR, and LSTM, with severe error fluctuations. The results show that HKELM is the best model choice for SOH prediction, with high accuracy and strong stability.

4.1.5. Dataset I Results for Different Batteries

This subsection explicitly analyzes different cell monomers (Group I-Cell 1, Group II-Cell 5, Group III-Cell 9, and Group IV-Cell 10) in Dataset I to evaluate the model’s prediction ability at different temperatures and charging/discharging conditions. Each subplot consists of two parts: the upper part shows the actual value of the SOH concerning the number of cycles versus the predicted value, and the lower part shows the residual curve of the prediction error. This distribution can be used to visualize model prediction performance and error variation. In the upper central figure, the black curve indicates the actual SOH, reflecting the trend in the battery health state of gradually decaying with increasing charge/discharge cycles and the overall pattern of a nonlinear decline. The red curve indicates the SOH curve predicted by the model, which is compared with the actual SOH data in green, demonstrating the expected performance of the model on different battery monomers. In addition, the Monte Carlo (MC) method yields the probability distribution of the RUL as the purple part. From Figure 15a, the RUL of Cell 1 is 292 cycles, and the RUL probability density distribution (PDD) is 288~300 cycles. From Figure 15b, the RUL of Cell 5 is 284 cycles and the PDD is 288~300 cycles. From Figure 15c, the RUL of Cell 9 is 119 cycles and the PDD is 111~123 cycles. From Figure 15d, the RUL of Cell 10 is 385 cycles and the PDD is 382~394 cycles. The predicted RUL probability density is distributed within a narrow interval close to the failure cycles and accurately covers the actual effectiveness cycles. In the error plots below, each subplot shows how the predicted values deviate from the actual values, with an expected error of ±2%. The degree of fluctuation in the residual curves reflects the accuracy and stability of the model predictions. At some stages, the residual error is small, indicating a more favorable predictive performance of the model. The results are shown in

Table 7. SOH estimation errors of the test cells.

Cell No.	MAE (%)	MSE (%)	RMSE (%)	R2
1	0.1550	0.0389	0.1974	0.999
5	0.3698	0.1760	0.4495	0.995
9	0.2957	0.1240	0.3522	0.998
10	0.2251	0.0058	0.2422	0.998

.

4.2. SOC Estimation Results

4.2.1. Dataset I SOC Estimation Results

To evaluate the robustness of the SOC estimation method, in Dataset I of Group I-Cell 1, the actual initial value of the SOC is 100% when SOH = 0.9 and the initial values are set to 60% and 40%, respectively. The prediction results are shown in Figure 16. The results show that, under different initial values, the method can achieve the convergence of the estimated value and the actual value in approximately 0.2 h with high accuracy.

The final SOC estimation results for Group I-Cell 1 in Dataset I are shown in Figure 16. As shown in Figure 17a–c, joint SOC-SOH estimation can follow the actual value of the SOC of an Li-ion battery more accurately. Figure 17c depicts the errors (residuals) between the predicted and actual values, i.e., the time-domain performance of the estimation error, which is relatively small in magnitude, indicating that the EKF algorithm has good estimation accuracy. Figure 17a is a partial enlargement of Figure 17b, which compares the actual SOC and the EKF-estimated SOC in a specific period in more detail. As shown in Figure 17a, the joint estimation method is used to estimate the battery SOC, which can identify the parameters of the novel equivalent circuit online in real time according to the changes in the battery SOC and charge/discharge cycles and adjust the maximum usable capacity in real time according to the changes in the SOH. Therefore, it can follow the actual values of SOC for LIBs more accurately in the case of battery aging.

4.2.2. Dataset II SOC Estimation Results

Figure 18 shows the results of the battery SOC estimations via the EKF for Dataset II via AR-ECM under three typical operating conditions, namely, US06, UDDS, and LA92, at three temperatures, namely, −20 °C, 25 °C, and 40 °C, respectively. The statistical results are shown in

Table 8. Proposed model performance evaluation.

Model	Temperature	MAE (%)	MSE (%)	RMSE (%)	R2
US06	−20	0.5100	0.003	0.5704	0.9986
	25	0.5328	0.0045	0.6693	0.9995
	40	0.4020	0.0026	0.5051	0.9997
UDDS	−20	0.9827	0.0129	0.6788	0.9986
	25	0.9354	0.0121	0.6485	0.9986
	40	0.3590	0.0022	0.4697	0.9997
LA92	−20	0.3426	0.0021	0.4619	0.9997
	25	0.7780	0.0087	0.6302	0.999
	40	0.6065	0.0480	0.6964	0.9995
Avg.		0.6054	0.0106	0.5922	0.9992

. The black curve represents the actual SOC values, the red curve represents the SOC values estimated via the EKF, and the light blue curve represents the absolute errors between the estimated values and the actual values. The comparison shows that the EKF shows good estimation performance under different working conditions and temperatures, the estimated values are in high agreement with the actual values, and the absolute errors are always controlled within 2%. Further analysis reveals slight differences in the SOC estimation results under different operating conditions: under US06, owing to the high speed and frequent dynamic changes, the SOC decreases at a faster rate and the estimation error fluctuates slightly over time, whereas under UDDS and LA92, the SOC curves are relatively smooth and the fluctuations in the estimation errors are slight. In addition, the effect of temperature on the SOC estimation is more significant. The estimation errors are slightly greater at low temperatures (−20 °C), which may be due to the increase in internal resistance and the decrease in the electrochemical reaction rate of the battery at low temperatures and other characteristics concerning the accuracy of the model. The SOC estimation results are more stable and accurate in room temperature (25 °C) and high-temperature (40 °C) conditions.

The performance of the model was evaluated by comparative analysis with recent studies in the field, as shown in

Table 9. Performance comparison with the latest study.

Methods	MAE (%)	RMSE (%)
TCN-LSTM [51]	0.7	0.81
QTGA [52]	0.8	0.96
BiLSTM-SA [53]	0.84	1.20
Proposed	0.6054	0.5922

, focusing on RMSE and MAE indicators. The results show that the accuracy of the model is significantly improved compared to previous studies, reflecting improved performance under various evaluation criteria.

The reason for the good performance of the EKF model used in this study can be summarized as the combination with the AR-ECM equivalent circuit, which improves the accuracy of the model through the accurate parameter identification method. Overall, the EKF algorithm shows good adaptability and robustness under various complex operating conditions and temperatures. Its high-precision estimation results indicate that it can effectively cope with dynamic operating condition changes and temperature fluctuations, which provides reliable technical support for SOC estimation in practical applications. The results of this study have important theoretical value and practical significance for further improving the performance of battery management systems.

5. Conclusions

This paper proposes a joint SOH–SOC estimation method incorporating the AR-ECM and a data-driven model. First, a new equivalent circuit model that simultaneously considers the effects of battery ageing and the SOC is constructed, and its parameters are optimally solved via the PSO algorithm incorporating adaptive inertia weights. The battery AR model parameters are identified under different SOC and SOH conditions. Then, based on the features extracted from the charging process of LIBs, essential features are selected as model inputs by the MCA method, the battery SOH values are used as outputs, and the MSIVY-DLEM model is used for training, establishing the feature–SOH mapping relationship, and ultimately achieving accurate prediction of the SOH of LIBs. The predicted SOH value is subsequently multiplied by the rated capacity of the battery to calculate the actual capacity of the battery, which is used to update the AR-ECM spatial equation of state to estimate the SOC further. This paper validates the joint SOH–SOC estimation method via the University of Michigan Battery Laboratory and McMaster University Hamilton dataset. The results indicate that the process can track the SOC and SOH values of LIBs more accurately than other methods over the battery life cycle.