A novel state of health estimation model for lithium-ion batteries incorporating signal processing and optimized machine learning methods

Abstract

An accurate assessment of the state of health (SOH) is the cornerstone for guaranteeing the long-term stable operation of electrical equipment. However, the noise the data carries during cyclic aging poses a severe challenge to the accuracy of SOH estimation and the generalization ability of the model. To this end, this paper proposed a novel SOH estimation model for lithium-ion batteries that incorporates advanced signal-processing techniques and optimized machine-learning strategies. The model employs a whale optimization algorithm (WOA) to seek the optimal parameter combination (K, α) for the variational modal decomposition (VMD) method to ensure that the signals are accurately decomposed into different modes representing the SOH of batteries. Then, the excellent local feature extraction capability of the convolutional neural network (CNN) was utilized to obtain the critical features of each modal of SOH. Finally, the support vector machine (SVM) was selected as the final SOH estimation regressor based on its generalization ability and efficient performance on small sample datasets. The method proposed was validated on a two-class publicly available aging dataset of lithium-ion batteries containing different temperatures, discharge rates, and discharge depths. The results show that the WOA-VMD-based data processing technique effectively solves the interference problem of cyclic aging data noise on SOH estimation. The CNN-SVM optimized machine learning method significantly improves the accuracy of SOH estimation. Compared with traditional techniques, the fused algorithm achieves significant results in solving the interference of data noise, improving the accuracy of SOH estimation, and enhancing the generalization ability.

Graphical abstract

Keywords

state of health (SOH) estimation / optimized machine learning / signal processing / whale optimization algorithm-variational modal decomposition (WOA-VMD) / convolutional neural network-support vector machine (CNN-SVM)

1 Introduction

Against the background of the transition of the global energy consumption structure to decarbonization, the new energy industry has become an essential part of China’s strategic emerging industries. Lithium-ion batteries (LIBs) are crucial in this transition [1]. Their wide application in energy storage, electric vehicles, and other fields are driving the green transformation of the energy structure and providing a solid impetus to achieve the goal of sustainable development [2]. However, with the surge in demand for LIBs and the rapid growth of battery technology, corresponding challenges and problems have become increasingly prominent. LIBs may generate heat and gases during charging and discharging cycles due to the complexity of internal chemical reactions. If not appropriately managed, these processes can pose serious risks such as fire and explosion, especially in accelerated battery aging [3]. Therefore, it is essential to develop and optimize a comprehensive battery safety management system (BSMS) [4] to ensure stability and safety.

State of health (SOH) is a critical parameter within the BSMS system, carrying the vital mission of ensuring battery stable operation and prolonging its lifespan [5]. SOH represents the degree of degradation of the performance of a battery relative to its original, pristine state under specific conditions. SOH is a measure of the degradation of the performance of a battery compared to its mint condition under particular conditions and directly reflects the current health of the battery. It is an internal parameter that can only be inferred indirectly through theoretical estimates of centain characteristics. These characteristics mainly include directly measurable indicators such as voltage, current, temperature, and internal resistance. Currently, the mainstream SOH estimation strategy relies on the fusion of signal processing techniques (SPTs) and advanced intelligent algorithms to accurately assess the health state of the battery through finely processing these characteristic data [6].

SPT is essential in extracting and processing health factors. In this regard, Sarwar et al. [7] proposed using the total harmonic distortion (THD%) of the open-circuit voltage as an indicator of capacity attenuation and confirmed the feasibility of the nonlinear frequency response analysis method for estimating the SOH. Lin et al. [8] extracted features from various signals and evaluated the interrelationships between these features using a multimodal and multilinear fusion mechanism to obtain high-quality features. Li et al. [9] smoothed the incremental capacity (IC) curve using a Gaussian filter to accurately identify and obtain signals related to battery aging. This approach enhances the clarity of the data, facilitating better analysis of the performance of the battery over time. Fonso et al. [10] used wavelet analysis to filter the voltage and current signals to provide reliable SOC and SOH estimation data. Additionally, the SPT can be directly utilized for SOH estimation, further enhancing the accuracy of battery health assessments. Kim and Cho [11] introduced a method for lithium-ion electrochemical characterization and SOH diagnostics based on discrete wavelet variations. Li et al. [12] proposed an IC and Gaussian process regression fusion method for battery SOH prediction. Ali et al. [13] proposed a novel multi-step-ahead predictor based on the fusion of kernel recursive least squares and Gaussian process regression to accurately predict SOH and remaining useful life. Xia et al. [14] developed an online SOH diagnostic method using an adaptive Taylor Kalman filter and an improved Gaussian filter (IGF). They created an online 2D filtering framework for SOH prediction based on IC analysis and differential thermal voltammetry health factors. He et al. [15] proposed a SOH estimation method based on partial charging curve coefficients of variation, feature processing, and Gaussian process regression characteristics.

In contrast, machine learning (ML) techniques can process extensive and diverse amounts of data and automatically learn intricate patterns in the data, typically providing greater accuracy and flexiblity in predictions [16]. Akbar et al. [17] developed a reliable battery SOH prediction model using ML and verified its accuracy. Li et al. [18] proposed an ML approach that combines random forest and artificial neural network methods for SOH prediction. Obregon et al. [19] explored a technique for extracting overcomplete features from electrochemical impedance spectroscopy (EIS) data using a convolutional autoencoder (CAE). They employed a deep neural network to estimate SOH. Nozarijouybari & Fathy [20] conducted a survey of the literature on ML for batteries. They found that the dataset, features, and algorithm selection greatly influenced the success of ML applications and showed the potential of machine-learning techniques in battery systems. Naresh et al. [21] examined various ML methods, such as supervised, unsupervised, and deep learning, along with appoarches to measure their effectiveness, demonstrating the ability of ML models to predict battery behavior and address the existing challenges. Qi et al. [22] explored the potential application of AI methodologies in improving battery management efficiency and enhancing the stability and reliability of battery operation, explicitly analyzing the application of ML to SOH in LIBs.

SPT and ML methods have demonstrated significant capabilities in SOH prediction, providing strong support for optimizing battery management systems [8,16]. However, each of these two approaches also faces distinct limitations. SPT has the unique advantage of directly refining critical features from the complex electrical signals generated by batteries, which often have an intuitive physical meaning, resulting in a robust physical interpretation of the prediction results. However, the predictive efficacy of this technique may be inherently limited by its underlying models and algorithms, and its prediction accuracy may suffer when confronted with the complex, variable, and unforeseen factors that arise during the battery aging process [23]. In contrast, with their powerful data-driven and nonlinear modeling capabilities, ML approaches have demonstrated outstanding potential to address the prediction of complex systems [24]. However, the success of this approach is highly dependent on a substantial amount of high-quality training data [25], and the ability of the model to generalize and be interpretable in real-world applications poses challenges [26].

The performance of a battery system, as a complex electrochemical system, is affected by the interaction of multiple factors such as temperature, charge/discharge rate, and cell internal resistance. The complexity and dynamics of these factors make accurate prediction of SOH particularly difficult [23–28]. In addition, unstable sensor data acquisition accuracy or potential failures during actual battery operation may cause the acquired data to contain noise, outliers, or missing values [29,30]. All these factors can negatively affect the model training efficiency and prediction accuracy. Given these challenges, fusing signal processing and ML techniques to optimize SOH prediction strategies is a research priority for future development. SPT is employed to optimize the data collection and processing process to improve the quality and quantity of data. Meanwhile, advanced ML techniques and methods are actively explored to improve the estimation accuracy and robustness of the battery state. To this end, this paper proposes a novel SOH estimation method that fuses data processing methods and ML methods. The method first adopts the variational modal decomposition (VMD) technique to thoroughly process the battery sampling data deeply. The optimal parameter combination (K, α) of the VMD method is sought through the whale optimization algorithm (WOA) to ensure that the signals can be accurately decomposed into different modes representing the health state of the battery. Then, a convolutional neural network (CNN) is employed to extract the high-level features of SOH. Ultimately, the health state of the battery is estimated relying on the support vector machine method, which provides solid data support and a decision-making basis for battery management and maintenance. This paper is contributive because cycle aging datasets for lithium iron phosphate batteries of two types with different capacities is analyzed. The multi-scale health factors of constant current charge and voltage change rate are constructed, and the relationship between them and capacity decline was studied. Additionally, the WOA seeks the best combination of parameters (K, α) for the VMD method. Moreover, the signal processing method WOA-VMD is used to decompose the SOH of the battery to obtain different modes of the battery health state. Furthermore, based on the modal components, a CNN-SVM ML method is used to estimate the health state of the battery and validated on two datasets.

2 Datasets and features

2.1 Datasets

The first dataset explored in this paper stems from a cyclic aging test implemented by our team on a mining-grade LiFePO₄ battery with a capacity of 228 Ah [31]. The battery is designed to be the core power source of a coal mine transport vehicle, providing stable and robust power support to ensure continuous and efficient mission performance in the harsh mine environment. Specialized equipment was used during the test, including a mainframe, a net end, a cycle test system, and a constant temperature chamber. Due to the high capacity of this type of battery, the three batteries involved in the test were labeled as BH1, BH2, and BH3. To comprehensively assess the performance of the batteries under different temperature conditions, the tests were conducted in three temperature environments, namely 20, 45, and 60 °C, to simulate the effects of various operating scenarios on the performance of the batteries. Fig.1(a) shows the variation curve of charging current versus voltage for this type of battery. More detailed information and descriptions of the tests can be found in Feng et al. [31].

The dataset obtained by Ma et al. [32] from Huazhong University of Science and Technology is also used for method validation. The dataset contains 77 battery data of model LFP/graphite A123 APR18650M1A. The nominal capacity and voltage of the batteries are 1.1 Ah, and 3.3 V. Battery degradation experiments were conducted in two thermostats at 30 °C using the same fast charging scheme and 77 different multi-stage discharge protocols. In this paper, three of these batteries are selected for the study. Due to their low capacity, they are labeled as BL1, BL2, and BL3. Fig.1(b) shows their charging and discharging curves. More detailed information and descriptions of the tests can be found in Ma et al. [32].

From Fig.1(a), it can be seen that the BH-type batteries are charged and discharged with a constant current at a single rate. However, as the temperature increases, the battery charging and discharging depth exhibits an increasing trend. From Fig.1(b), it is evident that the rapid charging of BL-type batteries is divided into three stages: first, charging at five times the normal current to 3.6 V; then, using one times the normal current to charge again to 3.6 V; and finally applying constant voltage charging to 3.6 V; and finally, applying constant voltage charging at 3.6 V until the current is reduced to 55 mA. In addition, battery BL1 has the shallowest discharging depth in the discharging process, while the maximum discharging depth is observed in BL2.

Fig.2 demonstrates the variation curves of the charging capacity of the six batteries during the cycling process. As seen in Fig.2, temperature and discharge depth have a significant effect on the decline in capacity. The increase in temperature substantially exacerbates the decline in battery capacity. Meanwhile, both excessively low and high discharge depths accelerate the degradation process of the battery.

It is evident from the analysis and comparison of the tests that the equipment requirements for cyclic aging testing of BH-type batteries are more stringent. This is primarily because large-capacity batteries face higher energy densities, more complex thermal management requirements, and longer test cycles in testing. During the testing of high-capacity batteries, the impact of high currents can increase the probability of failure during the testing process. Large-capacity batteries require stricter temperature control, ventilation and exhaust systems, safety isolation and protection measures, and emergency rescue and contingency plans due to their high energy density and high heat generation. According to the performance of the battery, the maximum discharge multiplier for the BH battery during the test is 1C while the BL battery can be charged and discharged with a current not exceeding 5C. As a result, the duration of the cyclic aging test varies significantly during the actual testing.

2.2 Features for SOH estimation

As an intuitive and easy-to-quantify indicator, constant-current charging time (CCCT) significantly reflects the trend of battery performance during the aging process. As the battery ages, the increase in internal resistance and the decrease in usable capacity become common, leading directly to a reduction in CCCT. Several studies have elaborated on this correlation, such as Feng et al. [5] and Nozarijouybari & Fathy [20]. Therefore, in this paper, CCCT is adopted as a critical factor for assessing the health state of a battery and designated as HF1. It is worth noting, however, that a complete record of the exact time for each charge and discharge is often challenging to achieve in a natural battery operating environment due to the complexity and uncertainty of the operating conditions [6]. Therefore, when utilizing HF1 for battery health assessment, the feasibility of practical operation and data validity must be considered comprehensively to ensure the accuracy and reliability of the assessment results.

The voltage rate (dv/dt) of change reflects the dynamic performance of a battery during charging and discharging. The rate of voltage change in a battery varies during aging, especially at the beginning and end of charging and discharging. These changes can reveal the rate and efficiency of chemical reactions within the battery [18]. Feng et al. [31] indicates that voltage fluctuations are more pronounced when the SOC is lower or higher. This phenomenon introduces errors in feature extraction for accurate analysis. Therefore, the authors propose to use features with a voltage change rate of less than 0.06 mV. This feature is labeled as HF2 in this paper.

Fig.3 illustrates the characteristic curves of the two types of batteries, HF1 and HF2.

To accurately evaluate the correlation between HF and battery capacity decline, first, HF1 and HF2 data are extracted from the aging tests. Then, a sliding window-based method is employed to identify outliers and perform interpolation. Finally, the Pearson correlation analysis method is selected to quantitatively evaluate the linear relationship between HF and battery capacity decline, calculated as shown in Eq. (1). The result is illustrated in Fig.4.

(1)\mathrm{P}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{n}=\frac{\mathrm{c}\mathrm{o}\mathrm{v}(X,Y)}{{\sigma }_X{\sigma }_Y}=\frac{E\left(XY\right)-E\left(X\right)E\left(Y\right)}{\sqrt{E\left(X^2\right)-E^2\left(X\right)}\sqrt{E\left(Y^2\right)-E^2\left(Y\right)}}.

From Fig.4, it evident that the decreasing trend of charging time is highly consistent with the trend of capacity decline, showing a strong correlation, with correlation coefficients significantly exceeding 0.965. Further analysis reveals that the BH-type batteries are particularly prominent in this regard, demonstrating a stronger correlation between their charging time and capacity decline. The characteristics of the rate of voltage change rate exhibit a sharp inverse relationship with capacity decline, showing significant variability across the aging states of the batteries. Specifically, the correlation between batteries BH2 and BH3 is hugely substantial, with a correlation coefficient as high as 0.98. In contrast, the correlation coefficient for BL2 batteries is lower, at 0.89.

3 Methodologies

3.1 Problem formulation

Starting from the definition of SOH of the battery, the value of SOH can be obtained by calculating Eq. (2) [33].

(2)\text{SOH}=\frac{C_{\mathrm{n}\mathrm{o}\mathrm{w}}}{C_{\mathrm{n}\mathrm{e}\mathrm{w}}}\times 100\mathrm{\%}.

where C_now and C_new denote the current and nominal capacity, respectively. However, C_now is challenging to obtain during actual usage. Therefore, researchers typically consider the battery aging process as an unknown internal mechanism and use ML techniques to establish a correlation model between the health factor and SOH [34], as detailed in Eq. (3).

(3)\text{SOH}=f(\mathrm{H}\mathrm{F}).

Battery aging is an intricate electrochemical process influenced by a confluence of multiple factors. During this process, the acquired or extracted feature signals usually contain modal components of various frequencies and amplitudes, which requires optimal raw data processing to obtain more valuable information. Additionally, there is a significant nonlinear relationship between the HF and SOH of the battery, which requires the selection of a suitable machine learning method. Therefore, this paper proposes combining SPT and ML techniques to optimize data and predict SOH.

3.2 Structural frameworks

To explore the complex dynamic behaviors embedded in the battery aging process, especially the multi-frequency and multi-amplitude modal components embedded in the cyclic data, this paper adopts the VMD method to effectively isolate and identify the critical features hidden behind the data. To further enhance the accuracy and robustness of the VMD decomposition, an intelligent optimization strategy using the WOA is employed to systematically determine the optimal combination of the number of VMD decomposition layers and the penalty factor. An SVM method optimized by CNN is adopted to accurately estimate battery SOH. Fig.5 visualizes the core structural framework of the approach taken in this paper.

The idea structure of the method proposed in this paper includes three parts, with specific tasks outlined as follows:

(1) The aging dataset was obtained through experiments and analyzed to extract HFs. The details are described in Sections 2.1 and 2.2.

(2) The WOA-VMD signal processing method is proposed. The WOA method is employed to identify the optimal combination of decomposition layers and penalty factors, ensuring the accuracy and stability of the decomposition results. The VMD method decomposes the SOH to obtain stable features at different frequencies. For details and implementation steps, please refer to Section 3.3.

(3) The CNN-SVM estimation method for SOH is proposed. The multi-layer convolution and pooling operations of the CNN capture the critically essential features in the component data. Subsequently, these critical features are fed into the SVM for regression prediction to accurately estimate the SOH of the battery. The details of this approach and the implementation steps are presented and elaborated in Section 3.4.

3.3 WOA-VMD

3.3.1 VMD method

VDM is a signal decomposition method that extracts modal components of different frequencies and amplitudes from the time domain [35]. This method decomposes the signal into multiple modal components with fixed frequency bandwidth by solving the optimization problem of minimizing the variational regularization function [36]. Each modal component corresponds to a specific frequency and amplitude in the signal.

Step 1: Constructing the variational problem.

Assuming that an input signal f is a superposition of a finite number of modal components u_k (t), the variational model with the corresponding constraints is

(4)\{\begin{array}{c}\underset{\left\{u_k\right\},\left\{w_k\right\}}{\mathrm{m}\mathrm{i}\mathrm{n}}\left\{{{\displaystyle \sum }}_{k=1}^n{\Vert {\mathrm{\partial }}_t\left[\left(\delta \left(t\right)+{\displaystyle \frac{\mathrm{j}}{\text{π}t}}\right)\times u_k\left(t\right)\right]e^{-\mathrm{j}{\omega }_{k}t}\Vert }_2^2\right\},\\ \begin{array}{cc}\mathrm{s}.\mathrm{t}.& {{\displaystyle \sum }}_{k=1}^n{u}_k\left(t\right)=f\left(t\right)\end{array},\end{array}

where n is the number of modes of the decomposition, a positive integer; u_k denotes the kth modal variable obtained from the VDM decomposition; ω_k denotes the center frequency of u_k; ‘*’ indicates the convolution operation; δ(t) is the Dirac function; {\mathrm{\partial }}_t is the mathematical operation of the gradient solution; and f(t) is the signal sequence of the input band decomposition.

Step 2: Introducing the Lagrangian.

Changing the original constraints by introducing the penalty factor ɑ and the Lagrange operator yields

(5)\begin{array}{rl}L=& \alpha {\sum }_k{\Vert {\mathrm{\partial }}_t\left[\left(\delta (t)+\mathrm{j}/\text{π}t\right)\times u_k(t)\right]e^{-\mathrm{j}{w}_{k}t}\Vert }_2^2+\\ & {\Vert f(t)-{\sum }_k{u}_k(t)\Vert }_2^2+⟨\lambda (t),f(t)-{\sum }_{k-1}^k{u}_k(t)⟩.\end{array}

Step 3: Solving saddle points.

The alternating multiplier operator is introduced to find the optimal solution of the constrained variational equation. When w > 0, update the spectrum for each mode û_k by Eq. (6).

(6){\hat{u}}_k^{n+1}\left(\omega \right)=\frac{f\left(\omega \right)-{\sum }_{i\ne k}\stackrel{\wedge }{u}\left(\omega \right)-\frac{\hat{\lambda }\left(\omega \right)}{2}}{1+2\alpha (\omega -{\omega }_k^n{)}^2}.

Center frequency ŵ_k update

(7){\hat{w}}_k^{n+1}\left(\omega \right)=\frac{{\int }_0^{\mathrm{\infty }}\omega {\left|{\hat{u}}_k^{n+1}\left(\omega \right)\right|}_2^2\mathrm{d}\omega }{{\int }_0^{\mathrm{\infty }}{\left|{\hat{u}}_k^{n+1}\left(\omega \right)\right|}_2^2\mathrm{d}\omega }.

The Lagrange multiplier y is updated as expressed in Eq. (8).

(8){\hat{\lambda }}_k^{n+1}\left(\omega \right)={\hat{\lambda }}_k^n\left(\omega \right)+\tau \left(\hat{f}\left(\omega \right)-{\sum }_{k=1}^n{\hat{u}}_k^{n+1}\left(\omega \right)\right).

The iterative stopping criterion is given in Eq. (9), where ε is the convergence threshold.

(9)

3.3.2 WOA optimization VMD theory

In the VMD decomposition algorithm, the number of decomposition layers K and the penalty factor ɑ are essential parameters. If the value of K is set too large, over-decomposition occurs; conversely, if it is too low, under-decomposition occurs. If ɑ is set to a large value, it can damage the band information, while a smaller value may lead to information redundancy. Therefore, determining the optimal parameter combination [K, ɑ] is critical. Generally, these two parameters are chosen empirically, leading to different results in VMD decomposition. This paper applies WOA to optimize the [K, ɑ] parameters of the VMD method.

The WOA was proposed by Mirjalili and Lewis in 2016 [37]. This method simulates the feeding behavior of humpback whales and demonstrates high competitiveness compared to traditional optimization algorithms. The details of WOA-VMD are outlined below.

Step 1: Adaptation function

In this paper, the minimum sample entropy is used as the fitness function, where the size of the entropy value is proportional to the signal complexity. The corresponding sample entropy of the signal varies across different operating states. For a given input time series {x(i), i = 1, 2,…, N}, when N is a finite value, the operational expression of the sample entropy is [38]

(10)SampEn(m,r,N)=-\mathrm{l}\mathrm{n}\left(\frac{B^{m+1}\left(r\right)}{B^m\left(r\right)}\right),

where m is the embedding dimension, r is the similarity tolerance, and B^m(r) and B^m+1(r) are the probabilities that two time series match m and m + 1 points under a fixed similarity tolerance.

Step 2: Initialization

The initialized random component equation is shown in Eq. (11).

(11)p=Y\times ({\omega }_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\omega }_{\mathrm{m}\mathrm{i}\mathrm{n}})+{\omega }_{\mathrm{m}\mathrm{i}\mathrm{n}},

where p and Y denote intervals, which are random numbers within [0, 1].

Step 3: Parameter optimization

Optimizing the parameters by the encircling prey algorithm, the spiral updating algorithm, and the search for prey algorithm. Equations (12) represents encircling prey.

(12)\{\begin{array}{c}D=\left|C\cdot X\times \left(t\right)-X\left(t\right)\right|,\\ X(t+1)=X\times \left(t\right)-A\cdot D,\end{array}

where t denotes the number of iterations, A and C are the coefficient vectors, and X represents the position vector. The spiral update is presented in Eq. (13).

(13)X(t+1)=D^{{}^{\prime }}\cdot e^{bl}\cdot \mathrm{c}\mathrm{o}\mathrm{s}(2\text{π}l)+X\times t,

where D^{{}^{\prime }} represents the distance vector parameter of the prey to the ith whale, b is the spiral constant, and l is a random number in [−1, 1]. The mathematical model is presented in Eq. (14).

(14)

Step 4: Finding the optimal solution

Searching for prey is employed to perform a global search, helping to avoid local optimization problems during VMD optimization. The mathematical model is represented in Eq. (15).

(15)\{\begin{array}{c}D=\left|C\cdot X_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}-X\right|,\\ X(t+1)=X_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}-A\cdot D,\end{array}

where X_rand is a random position vector chosen from the current population.

The flow of WOA to optimize the VMD parameters is shown in Fig.6.

3.4 CNN-SVM

Compared to the raw data, the features processed by WOA-VMD exhibit higher predictability in terms of predictive performance. Nevertheless, each HF layer may still retain some inherent unique features. This paper designs CNN-SVM as a machine-learning predictor for SOH to further improve the estimation accuracy and generalization ability of the predictors. This approach combines the advantages of both methods to achieve more accurate and generalized prediction results.

CNN can automatically create effective filters and efficiently extract deep features from the data. Thus, CNN has great potential in time series applications [39]. However, they require a substantial number of labeled training samples and are prone to overfitting when the sample size is small.

The convolution operation can be formalized for each convolutional layer as expressed in Eq. (16).

(16)h_{ij}^k=f\left(\right(W^k\times x{)}_{ij}+b_k),

where f denotes the activation function, W^k denotes the weights of the kth feature map, and b_k denotes the bias of the kth feature map.

The SVM algorithm can handle small samples and has a more extraordinary generalization ability compared to the CNN algorithm. Unlike the traditional induction-to-deduction process, SVM directly finds the optimal decision boundary through the training samples, achieving efficiency in the transaction from training samples to forecast samples via “transductive inference.” Additionally, SVM can efficiently map the data to a high-dimensional feature space, as shown in Eq. (17).

(17)f\left(x\right)={\omega }^{\mathrm{T}}\varphi \left(x\right)+b,

where \varphi is the nonlinear mapping function, ω is the weight vector, and b is the bias. The optimization of SVM can be calculated as shown in Eqs. (18) and (19).

(18)\underset{w,b,{\xi }_i,{\xi }_i^{\ast }}{\mathrm{m}\mathrm{i}\mathrm{n}}\frac{1}{2}{\omega }^{\mathrm{T}}\omega +C{\sum }_{i=1}^n{\xi }_i+{\xi }_i^{\ast },

(19)\mathrm{s}.\mathrm{t}.\{\begin{array}{c}{y}_i-\left(⟨\omega ,x_i⟩+b\right)⩽\epsilon +{\xi }_i,\\ \left(⟨\omega ,x_i⟩+b\right)-y_i⩽\epsilon +{\xi }_i^{\ast },\\ {\xi }_i,{\xi }_i^{\ast }⩾0,\end{array}i=1,2,\cdots ,l,

where l is the number of samples; x_i and y_i are the inputs and outputs of the training data, respectively; n is the number of samples; ξ_i and ξ_i* are the upper and lower training errors; and ε and C are the insensitive loss factor and regularized constant.

By using the Lagrange multipliers ɑ_i and {\alpha }_i^{\ast }, the prediction function f can be calculated as follows:

(20)f(x,{\alpha }_i,{\alpha }_i^{\ast })={\sum }_{i=1}^n({\alpha }_i-{\alpha }_i^{\ast }))k(x,x_i)+b,

where k is the kernel function.

The mind map of CNN-SVM is shown in Fig.7, whose process is as follows:

(1) The modal components from the VMD are divided into training and test datasets. The CNN consists of two convolutional layers and two fully connected layers, with convolutional layer channels set to 13, 12, and 11. The learning rate is set to 0.001, and the learning rate dropout period is 125. Dropout and Lasso regularization are used in the multilayer architecture.

(2) The fully connected layer 1 of the CNN model serves as input features to the support vector machine. The prediction results of CNN-SVM are obtained using the prediction function f.

(3) Save the CNN-SVM training model.

(4) Reload the training model. Construct the test model by directly connecting the fully connected layer 1 to the SVM layer.

4 Results and analyses

4.1 Decomposition results of SOH

To provide an objective and comprehensive assessment of the performance of the model, three key evaluation metrics are selected in this paper: mean absolute error (MAE), root mean square error (RMSE), and mean absolute percentage error (MAPE). The specific calculations for these metrics are

(21)\mathrm{M}\mathrm{A}\mathrm{E}=\frac{1}{n}{\sum }_{i=1}^n|y_i-{\hat{y}}_i|,

(22)\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{\frac{1}{n}{\sum }_{i=1}^n(y_i-{\hat{y}}_i{)}^2},

(23)\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}=\frac{1}{n}{\sum }_{i=1}^n\left|\frac{y_i-{\hat{y}}_i}{y_i}\right|\times 100\mathrm{\%},

where n is the number of samples, and y_i and ŷ_i are the actual and predicted values of the ith sample, respectively. MAE measures the accuracy of the model by calculating the average of the absolute errors between the predicted and actual values. RMSE squares the difference between the expected and actual values, making RMSE more sensitive to outliers. MAPE is suitable for scenarios where the proportion of errors is more sensitive. The smaller the values of the three metrics, the closer the predictions of the model are to the actual values, indicating the better performance.

4.2 Decomposition results of SOH

The reference SOH is first obtained by combining Eq. (2), and then the SOH of the cell is decomposed by using the WOA-VMD method to identify the optimal parameters [K, ɑ], as shown in Tab.1. Fig.8 and Fig.9 visualize the original SOH states of two different types of batteries along with the magnitude results after VMD decomposition. The data in Tab.1 indicate that the optimal parameter K exhibits high consistency for cells under various aging pathways. In contrast, the α parameter varies significantly depending on the aging path. There is an intrinsic similarity in the number of modes required to decompose the cell signals for different aging pathways. The substantial difference in the α parameter further emphasizes the importance of tuning the VMD parameters for different aging regimes to ensure the accuracy and validity of the decomposition results.

As shown in Fig.8 and Fig.9, the maximum of the error reference in the figures is 1.2 × 10⁻³, and the average error is 4.2 × 10⁻⁵. The magnitude of the reference result, as a quantitative index for evaluating the effectiveness of the VMD decomposition, directly reflects the closeness of the decomposition result to the original signal. The results show that the WOA-VMD method effectively removes the noise from the original signal while retaining essential features, and the selected parameter combinations are applicable to the current signal.

4.3 Results of SOH estimation

A comparison experiment is designed to comprehensively verify the effectiveness and superiority of the proposed hybrid model combining signal processing and ML methods in battery SOH estimation. This experimental system incorporates raw and processed data as a benchmark. It introduces data pre-processed through VMD and WOA-VMD techniques to explore the effect of data pre-processing on model performance, with K set to a fixed value of 3 in VMD.

4.3.1 70% training sample test

The experiments compare the CNN-SVM with the SVM and CNN models. To ensure the reliability and generalization ability of the experimental results, two different ratios for dividing the training and test set are used. The first division follows a ratio of 7:3, which focuses on assessing the learning ability of the model under larger training samples. Fig.10 clearly shows the performance evaluation results of each model on the training and test sets under the 7:3 division ratio.

Fig.10 clearly reveals the complexity of the battery aging process under the intertwined effects of multiple factors, such as temperature, discharge rate, and discharge depth. However, the advanced SPT and efficient ML algorithms introduced in this paper show remarkable capacity in addressing these challenges. These methods significantly enhance the accuracy of traditional SVM and CNN when applied individually in SOH evaluation. They also achieve accurate tracking and forward-looking predictions of the dynamic curve of battery SOH, showcasing excellent performance advantages.

Fig.11 displays the bounding box line plot of the estimated error when the training set comprises 70% of the data. The plot demonstrates that the proposed method in this paper has a smooth error curve with relatively small error values, strongly validating its effectiveness. Additionally, the box line plot reveals that this method exhibits good stability and consistency in data processing, with a compact distribution and no significant outliers, further supporting the superiority of the technique in error estimation.

Tab.2 presents the specific values of the performance metrics for different batteries under each model when the training set consists of 70% of the data.

Even though BH-type batteries are affected by temperature, BL-type batteries are more sensitive to the discharge strategy. However, as shown in Tab.2, the performance metrics MAE, RMSE, and MAPE obtained through the method proposed in this paper are notably low, with specific values of only 0.06%, 0.075%, and 0.075%, respectively. This remarkable achievement underscores the efficiency and superiority of the method proposed in this paper and reflects its stable performance capability under complex environmental variables. In contrast, traditional methods such as SVM and CNN exhibit significant limitations when faced with variable temperature conditions, different discharge strategies, and the diversity of battery types. Specifically, the CNN-based method shows a considerable difference of nearly ten times in the performance index when handling data from batteries BL2 and BH1, highlighting its shortcomings in adapting to different battery characteristics. Similarly, the prediction results of the SVM method show at least threefold performance fluctuations, further demonstrating that the adaptive ability and robustness of these traditional algorithms could be improved in complex and changing practical application scenarios.

4.3.2 30% training sample test

To further examine the robustness and generalization performance of the method proposed in this paper under limited training samples, the dataset was divided in a ratio of 3:7 and performed in the same tests as above. Fig.12 visualizes the dynamic trends exhibited by the performance of each model when the training set is reduced to 30% of the total data. Fig.13 shows the bounding box plot of the error of each model. In addition, to more precisely quantify the performance of each model under limited training samples, Tab.3 is compiled, which lists the specific values of the corresponding performance metrics for each model.

As can be seen from Fig.12, the estimation performance of the CNN method decreases significantly with the reduction of the number of training samples, indicating that its effectiveness is strongly dependent on sample size. Meanwhile, for cells BH2, BH3, and BL1, the estimation performance of SVM also shows a noticeable weakening trend. However, it is noteworthy that the innovative method presented herein exhibits excellent stability in SOH estimation for all cells, and its performance is remains unaffected by changes in sample size.

From the visual presentation in Fig.13, it is evident that the sample dataset size significantly constrains traditional SVM and CNN methods. Specifically, when the sample size is reduced to 30% of the original data, the overall error level of these two methods increases significantly, with this trend intensifying as the estimation step size increases. In contrast, the WOA-VMD method introduced in this paper demonstrates substantial improvements in estimation performance by effectively managing noise. Its superiority lies in its ability to maintain stability regardless of changes in sample size, ambient temperature, and discharge multiplicity.

From the exhaustive data analysis in Tab.3, it can be seen that the method proposed in this paper achieves the lowest values across all assessment indices. Specifically, the MAE, RMSE, and MAPE are precisely controlled at 0.014%, 0.019%, and 0.016%, respectively. This excellent performance not only highlights the high reliability of the method in terms of accuracy but also underscores its superior capability to manage fluctuations under complex environmental conditions. It is particularly noteworthy that the SOH estimation performance metrics demonstrate a high degree of consistency for both BH-type and BL-type batteries, reflecting the strong adaptability and accuracy of the method in in varying training sample conditions. In summary, the WOA-VMD + CNN-SVM method proposed in this paper represents a significant advancement in enhancing the accuracy and robustness of battery health state assessment.

In addition, by reviewing the estimation performance metrics in Tab.2 and Tab.3, it can be observed that both the VMD and the WOA-VMD data processing strategies positively impact the accuracy of battery SOH estimation. Among these, the WOA-VMD method stands out due to its unique optimization mechanism, demonstrating a more significant improvement in the estimation accuracy. Additionally, the CNN-SVM method effectively enhances the accuracy of SOH estimations with its powerful learning capability. Overall, the innovative method proposed in this paper, an ML framework that seamlessly integrates data processing and optimization techniques, shows the best performance. This approach not only combines the strengths of both fields but also achieves a significant leap in SOH estimation accuracy. It provides valuable insights and inspiration for future research and practical applications in related domains.

5 Conclusions

This paper addresses the challenges posed by noise in data acquisition during the cyclic aging process of batteries and its impact on the SOH estimation accuracy and generalization ability. Based on this analysis, a novel method is proposed and experimentally validated. The main contributions and key conclusions of this paper are as follows:

(1) Based on the detailed cycle aging data from both a 228 Ah mining lithium battery and a 1.1 Ah civil lithium battery, a comprehensive dataset was constructed by selecting battery data samples covering different temperature conditions, discharge rates, and discharge depths. A multi-scale health factor incorporating features such as constant current charging and voltage rate of change was developed by analyzing the battery aging data. The validation results indicate a strong correlation between the health factor and battery capacity, effectively supporting accurate assessement of battery health status.

(2) For the newly proposed WOA-VMD method, modal decomposition experiments were conducted on six battery samples. The experimental results show that this method effectively filters out the noise interference from the original signals while successfully retaining the critical feature information related to the state changes of the battery. This capability lays a solid foundation for subsequent battery performance analysis.

(3) Through comparative experiments, it is found that the data processing technique based on WOA-VMD effectively addresses the noise interference issues present in cyclic aging data noise, significantly enhancing the accuracy of SOH estimation. Additionally, the ML method of CNN-SVM optimization effectively improves the accuracy of SOH estimation. In contrast to traditional techniques, this fused algorithm demonstrates substantial advancements in mitigating the interference of data noise, improving the accuracy of SOH estimation, and enhancing the generalization ability across diverse conditions.

This work successfully established multiscale health factors, validating their effectiveness alongside advanced estimation algorithms across multiple batteries. However, some limitations remain. First, the current feature extraction methods have practical limitations. Future research should focus on developing more efficient data acquisition and processing strategies. This could involve leveraging advanced sensors, enhanced data collection techniques, and real-time processing methods to ensure a more robust dataset for accurate battery health assessments. By addressing these challenges, the reliability and applicability of battery monitoring systems in various operational environments can be improved. Next, the health features proposed in this paper may not fully encompass the complexity and diversity of battery health states. Further research should focus on expanding the health feature space to include a wider range of parameters that reflect different operating conditions and degradation mechanisms. This could involve investigating additional electrochemical characteristics, thermal dynamics, and mechanical stress factors. By exploring more diversified and comprehensive feature sets, a more nuanced understanding of battery health can be developed, leading to improved diagnostic and predictive capabilities in battery management systems. Finally, although the SOH estimation method proposed in this paper has been validated for two types of batteries, there is a need to develop ML models that incorporate online learning capabilities and adaptive optimization parameters. Such advancements would enable the models to continuously learn from new data and adjust to varying conditions in real-time. This flexibility is essential for addressing a broader range of application scenarios, where battery usage patterns, environmental factors, and degradation behaviors can change dynamically. By integrating these features, future models can enhance their robustness and accuracy in estimating the state of health across diverse battery systems.

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