Lithium-Ion Battery State of Health Estimation Based on Feature Reconstruction and Transformer-GRU Parallel Architecture

Abstract

Estimating the state of health of lithium-ion batteries in energy storage systems is a key step in their subsequent safety monitoring and energy optimization management. This study proposes a method for estimating the state of health of lithium-ion batteries based on feature reconstruction and Transformer-GRU parallel architecture to solve the problems of noisy feature data and the poor applicability of a single model to different types and operating conditions of batteries. First, the incremental capacity curve was constructed based on the charging data, smoothed using Gaussian filtering, and the diverse health features were extracted in combination with the charging voltage curve. Then, this study used the CEEMDAN algorithm to reconstruct the IC curve features, which reduces noisy data due to the process of data collection and processing. Lastly, this study used the cross-attention mechanism to fuse the Transformer and GRU neural networks, which constructed a Transformer-GRU parallel model to improve its ability to mine time-dependent features and global features for state of health estimation. This study conducted experiments using three datasets from Oxford, CALCE, and NASA. The results show that the RMSE of the state of health estimation by the proposed method is 0.0071, which is an improvement of 61.41% in the accuracy of its baseline model.

1. Introduction

Lithium-ion batteries are efficient, portable, and environmentally friendly energy storage carriers. They have been widely used in consumer electronics, renewable energy storage, and electric transportation [1,2,3]. However, lithium-ion batteries will gradually degrade during long-term use due to charging and discharging cycles and environmental changes, manifested as capacity degradation and increased internal resistance. Ageing not only reduces the equipment’s endurance but may also cause overheating, short circuits, and other safety hazards, threatening the equipment’s reliability and user safety [4]. State of Health (SOH), a key indicator of battery ageing, can reflect a battery’s performance degradation by evaluating its capacity retention or internal impedance changes [5,6]. Accurate estimation of SOH not only optimizes decisions for the battery management system but also extends battery life and improves equipment operating efficiency and reliability [7]. Therefore, SOH estimation is of great significance in the research and practical application of lithium-ion batteries and is one of the core technologies for improving battery safety and economy [8].

Many scholars have carried out in-depth studies on SOH estimation, and the main approaches can be categorized into three groups: experimental, model-based, and data-driven [9,10,11]. The experimental method directly measures and analyzes the ageing characteristics of batteries and their relationship to the SOH through many experimental tests [12]. Typical experiments include life-cycle tests, static characteristic tests, battery disassembly, and analysis. This method has high precision and accuracy, can genuinely reflect the actual SOH of the battery, and is suitable for theoretical research and new battery material development. However, promoting it in practical applications is difficult because of its long testing period, high cost, and strict experimental conditions. The model-based approach relies on the theoretical study of chemical reactions or circuit behaviors within lithium-ion batteries to establish electrochemical models or equivalent circuit models to characterize the degradation process of battery performance [13,14]. It combines with state observers to study the dynamic changes of the battery’s SOH [15]. The advantage of this method is that it has a clear physical meaning and high prediction accuracy. However, simultaneously, it requires a large amount of experimental data for parameter identification, and the model complexity is high, which requires specific computational resources. On the other hand, the dynamic accuracy of the model in describing the complex working conditions is low, and it is difficult to adapt to the real-time changes in the battery’s state in the variable operating environment.

The data-driven approach relies on much historical data from battery operation and utilizes machine learning or deep learning algorithms to predict the battery’s SOH [16,17,18]. Data-driven approaches do not require precise modeling of the battery’s internal mechanism; instead, data mining is used to capture the complex nonlinear relationship between battery performance and SOH. Their advantage lies in their adaptability and easy scalability, which makes them suitable for variable application scenarios under actual working conditions. Many existing studies have constructed charging voltage curves, differential voltage curves, incremental capacity (IC) curves, etc., based on battery charge and discharge data and extracted health features for SOH estimation [19,20,21]. However, these methods face challenges in practical applications. On the one hand, battery signals are often affected by complex working conditions and the accuracy of measurement equipment during the acquisition process, leading to the inclusion of noise in the feature data and weakening the correlation between the features and the SOH. On the other hand, differential voltage and incremental capacity curves, which need to undergo secondary data transformation to extract features, will inevitably cause interference with the features and cover up the key features of the battery ageing process due to the influence of data processing algorithms and other factors. In addition to feature selection, the data-driven model is a key factor affecting estimation accuracy. Support vector regression, long- and short-term memory neural networks, gated recurrent neural networks (GRUs), and convolutional neural networks and their various variants are widely used for SOH estimation [22,23,24]. These models perform well in different application scenarios, especially in capturing nonlinear and time-series relationships. However, these single models also have obvious limitations: they tend to perform well for a particular class or some specific battery data but lack generalization and are difficult to adapt directly to different usage environments or different types of batteries. How to design data-driven models with higher adaptability and generalization capabilities is a key factor in improving the accuracy of SOH estimation.

This study proposes a method for estimating the SOH of lithium-ion batteries based on feature reconstruction and Transformer-GRU parallel architecture, whose flow is shown in Figure 1. First, the IC curve is constructed based on the charging data, smoothed by Gaussian filtering, and the health features are extracted by combining the charging voltage curve. Then, the Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) algorithm is used to decompose the IC curve features and select some modal components for reconstruction. Finally, a parallel data-driven model is established by combining Tranformer and GRU neural networks for SOH prediction. The main work and innovations of this study are as follows:

(1)

It is based on incremental capacity analysis and a Gaussian filtering algorithm to extract IC curve features, combined with conventional charging features to form a multivariate health feature set;

(2)

The CEEMDAN algorithm is used to decompose and reconstruct the health features to reduce their noise due to data acquisition and secondary data processing;

(3)

A data-driven model with Transformer-GRU parallel architecture is built for SOH prediction, which enhances the model’s performance in handling complex and diverse data.

The rest of this study is structured as follows: Section 2 describes the dataset and its processing process; Section 3 introduces the principles associated with the SOH estimation method; Section 4 provides a discussion of the results; and Section 5 summarizes the work of this study and presents an outlook.

2. Dataset and Feature Extraction

2.1. Definition of State of Health

The state of health mainly reflects the battery’s actual capacity ratio to its rated capacity and is used to measure its degree of ageing and service life [25]. The formula for calculating the SOH is:

$$SOH={\displaystyle \frac{C_{actual}}{C_{rated}}}\times 100\%$$

(1)

where $C_{actual}$ indicates the current capacity of the battery; $C_{rated}$ indicates the rated capacity of the battery.

2.2. Introduction to the Dataset

Battery datasets provided by the University of Oxford [26], CALCE [27,28], and NASA [29] are used in this study. These cover various battery types and diverse operating conditions with representative degradation modes and failure mechanisms.

Table 1. Battery material information.

Dataset	Cathode	Anode	Rated Capacity
Oxford	Lithium nickel cobalt oxide	Graphite	740 mAh
CALCE	LiCoO2	Graphite	1100 mAh
NASA	LiNiCo0.15Al0.05O2	Graphite	2000 mAh

demonstrates the information, such as materials for different datasets. The introduction of datasets from multiple sources is sufficient to fully validate the suitability of the proposed method in multiple battery types and complex application scenarios.

The Oxford dataset consists of eight sets of lithium-ion batteries with a capacity of 740 mAh, and batteries numbered Cell_1 and Cell_7 were selected for this study. The dataset was tested at 40 degrees Celsius, using constant current and voltage charging, and discharged by the ARTEMIS driving cycle until the capacity drop was less than 80%. Data collected include voltage, temperature discharge, and capacity tests performed every 100 cycles.

The CALCE dataset was studied on 1100 mAh LiCoO₂ batteries, and the batteries numbered CS2_35 and CS2_36 were selected for this study. They were first charged at a constant rate of 0.5C to a cut-off voltage of 4.2 V and then charged at a constant voltage of 4.2 V to a cut-off current of 0.05 A. Finally, they were discharged at a constant current of 1C until the voltage was reduced to 2.7 V. The C-rate indicates the charge/discharge rate of a battery, which is related to the rated capacity of the battery. For a battery with a rated capacity of 1100 mAh, a 1C charge/discharge multiplier means its charge/discharge current is 1.1 A. The data collection items included voltage, current, temperature, charge, and internal resistance information.

The battery’s rated capacity in the NASA dataset is 2 Ah, and two batteries, B0005 and B0006, were used in this study. Both of these batteries were charged at a constant current of 1.5 A and discharged at a constant current of 2 A. The charging cutoff voltage is 4.2 V, and the discharging cutoff voltage is 2.7 V (B0005) and 2.5 V (B0006). The end-of-battery-life condition is a capacity decay to 70% of rated capacity, i.e., from 2 Ah to 1.4 Ah.

Figure 2 illustrates the decreasing trend of the capacity curves for the University of Oxford, CALCE, and NASA battery datasets. The Oxford dataset shows a steady linear decline in capacity; the CALCE dataset exhibits a nonlinear and fluctuating decay trend; and the NASA dataset shows a faster decline in capacity, with capacity regeneration during the decay period. These datasets cover a wide range of battery types and operating conditions, and by analyzing the different degradation patterns, the generality, robustness, and ability to capture multiple degradation characteristics of the model can be comprehensively verified.

2.3. Feature Extraction

The key to estimating the battery’s SOH lies in accurately extracting the health features that can reflect the battery’s degradation. In this study, the incremental capacity curve is constructed by combining the charging voltage with the charging amount for transformation, which can reveal the battery’s intrinsic capacity change trend and micro-degradation characteristics. At the same time, the combination of conventional charging features and incremental capacity features describes the SOH of the battery in a multi-dimensional way, which improves the differentiation of the features and the sensitivity to the battery’s degradation.

The IC curve is a characteristic curve calculated from battery charging data. It is used to characterize the differential relationship between the change in battery capacity and charging voltage. The IC curve can more sensitively reflect the changes in the battery’s internal state, especially the capacity degradation, the degree of ageing, and the declining characteristics of the active materials [30]. The formula is shown in Equation (2):

$$IC\left(V\right)={\displaystyle \frac{\Delta Q}{\Delta V}}$$

(2)

where $IC\left(V\right)$ denotes the incremental capacity value at voltage V; $\Delta Q$ is the incremental charge in the voltage interval; and $\Delta V$ is the corresponding voltage increment.

However, the IC curves obtained only by directly calculating the above formula are usually affected by the accuracy of discrete data, resulting in significant curve volatility and making it difficult to extract stable features effectively. Therefore, in this study, Gaussian filtering is used to smooth the IC curve to reduce the noise and retain the key information of the curve. Gaussian filtering [31] filters the data by weighted averaging, where the Gaussian function determines the weights. The expression of the Gaussian function is represented in Equation (3):

$$G\left(x\right)={\displaystyle \frac{1}{\sqrt{2\pi }\sigma }}{e}^{-{\displaystyle \frac{x^2}{2{\sigma }^2}}}$$

(3)

where x denotes the offset from the centre point; $\sigma$ is the standard deviation of the Gaussian distribution, which controls the degree of smoothing of the filter; in filtering the IC curve, the output of Gaussian filtering $y\left[i\right]$ can be expressed as the weighted sum of the input data $x\left[j\right]$, as shown in Equation (4):

$$y\left[i\right]=\sum _{i=-k}^{k}G\left(j\right)x\left[i+j\right]$$

(4)

where k is the radius of the filter window, which determines the smoothing range; $G\left(j\right)$ is the Gaussian weight of the corresponding position j, which is computed by the Gaussian function; and $x\left[i+j\right]$ is the $i+j$th data point of the original data.

Figure 3 shows the IC curve after Gaussian filtering; the blue curve is the original IC curve, and the orange curve is the result after smoothing. The original curve has significant fluctuation and noise, making distinguishing the key features difficult. After Gaussian filtering, the smoothed curve effectively suppresses the noise while retaining the main characteristic patterns of the curve, especially in the peak position and the overall trend.

In this study, three health features, namely, the constant-current charging time (CCCT), peak voltage of the IC curve (ICP), and average value of the IC curve (ICAV), are extracted to characterize the evolutionary trend of the battery SOH. From Figure 4, it can be seen that both the charging voltage curve and the IC curve exhibit significant changes with increase in the number of battery cycles, which are highly correlated with the decay of the battery SOH.

CCCT is gradually shortened with increase in the number of cycles, which indicates that the battery is more likely to reach the cut-off voltage under the constant charging multiplicity, reflecting the decrease in its adequate charging capacity. This feature can sensitively reflect the capacity degradation of the battery, which is positively correlated with the capacity decay. The decrease in the ICP reflects the gradual weakening of the battery’s charge capacity under the unit voltage increment, which is closely related to the decline in the active material and the accumulation of irreversible reactions. Meanwhile, the ICAV characterizes the trend of the charging amount under the same voltage increment during the whole charging cycle. As the number of cycles increases, the ICAV gradually decreases, and this feature reveals the overall characteristics of battery capacity degradation from a global perspective. The three features extracted in this study reflect the changes in the battery SOH from local to global multi-dimensionally, which provides a scientific theoretical basis and practical support for assessing the battery SOH.

2.4. Feature Correlation Analysis

The correlation between the features and the battery’s SOH is an important indicator of their effectiveness, and highly correlated features can more accurately reflect the evolutionary trend of the battery’s SOH. Therefore, the Pearson correlation coefficient (PCC) [32] was used in this study to quantitatively analyze the validity and sensitivity of the health features. PCC is a statistical measure of the degree of linear correlation between two variables with values ranging from [−1,1]. By calculating the PCC between the characteristics and the SOH, the magnitude of the role of the characteristics in describing the changes in the SOH can be determined. The principle is shown in Equation (5):

$$r={\displaystyle \frac{{\sum }_{i=1}^n\left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)}{\sqrt{{\sum }_{i=1}^n{\left(x_i-\overline{x}\right)}^2}·\sqrt{{\sum }_{i=1}^n{\left(y_i-\overline{y}\right)}^2}}}$$

(5)

where r is the Pearson’s coefficient of health features versus SOH; $x_i$ and $y_i$ denote the samples of the ith cycle of features versus SOH, respectively; $\overline{x}$ and $\overline{y}$ are the means of features versus SOH, respectively; and n is the battery’s cycle numbers.

Table 2. Correlation results.

Features	Correlation Results
	Cell_1	Cell_7	CS2_35	CS2_36	B0005	B0006
CCCT	0.9988	0.9998	0.9851	0.9923	0.9981	0.9952
ICP	0.9521	0.9477	0.9688	0.9723	0.9948	0.9931
ICAV	0.9687	0.9599	0.9518	0.9741	0.9972	0.9942

demonstrates the correlation coefficients of the health features extracted in this study. All the feature correlation coefficients are greater than 0.9, indicating that the extracted features can comprehensively and stably reflect the trend of the battery SOH with high reliability and consistency.

3. Methods for Estimating State of Health

3.1. Feature Reconstruction

When the model is trained with a high correlation of health features, there is still some noise in the feature data, even though these features can well reflect the trend of the battery SOH. These noisy data will be misjudged as proper signals by the model, which will interfere with the model’s prediction results and reduce its robustness and generalization ability. Therefore, performing noise reduction on the feature data is particularly important before model training. This study uses the CEEMDAN [33] technique to reconstruct the feature sequences by noise reduction of the feature data.

Given an original health feature signal, white noise is added to it to form several different noise-enhanced signals, as shown in Equation (6):

$$x_k\left(t\right)=x\left(t\right)+ϵ·n_k\left(t\right)$$

(6)

where $x\left(t\right)$ is the original feature signal, $n_k\left(t\right)$ is the white noise added in the kth iteration, and $ϵ$ is the noise intensity.

The first intrinsic modal component (IMF) of the current signal is extracted each time, and the average value of the first IMF component is calculated by adding different noises several times, as shown in Equation (7):

$$IM{F}_1\left(t\right)={\displaystyle \frac{1}{K}}\sum _{k=1}^{K}IM{F}_{1,k}\left(t\right)$$

(7)

where K is the number of signals added to the noise; after extracting the first IMF, it is removed from the original signal to obtain a new residual signal $r_1\left(t\right)$. For subsequent IMF extraction, the weighted noise of the residuals is gradually introduced, and the decomposition process is repeated to finally decompose the signal into multiple IMFs and a residual term, as shown in Equation (9):

$$r_1\left(t\right)=x\left(t\right)-IM{F}_1\left(t\right)$$

(8)

$$x\left(t\right)=\sum _{i=1}^{n}IM{F}_i\left(t\right)+r_n\left(t\right)$$

(9)

The IMF components are screened, and the high-frequency noise components can be removed by the energy thresholding method or correlation analysis. The filtered IMF components are recombined with the residuals to obtain the noise-reduced signal $x_{\mathrm{denoised}}\left(t\right)$, which can realize the noise reduction of health features:

$$x_{\mathrm{denoised}}\left(t\right)=\sum _{i=p}^{q}IM{F}_i\left(t\right)+r_n\left(t\right)$$

(10)

where p and q are the filtered IMF component index ranges.

3.2. Transformer Encoder Network

The first layer of the Transformer-GRU parallel architecture proposed in this study is the Transformer encoder network. The core of this network consists of a self-attention mechanism and a feed-forward neural network with a structure that does not depend on the time-step order of the sequences and can capture the interactions between the features through global operations. As shown in Figure 5, its structure consists of several parts, including input embedding and location coding, a multi-head attention mechanism, a feed-forward neural network, residual connectivity, and normalization [34,35].

(1) Input Embedding and Positional Encoding: The input received by the Transformer is a sequence of features $X=[x_1,x_2,\cdots ,x_T]$, with each time step $x_t$ representing the corresponding feature vector. In order to preserve the temporal information, the Transformer adds positional encoding after embedding the input. Positional encoding is generated using sine and cosine functions to ensure that the positional information of the sequence is embedded into the model, thus compensating for the Transformer’s ability to perceive the sequence order directly. The principle is shown in Equation (11):

$$X_{\mathrm{input}}=\mathrm{Embed}\left(X\right)+\mathrm{PosEncoding}$$

(11)

(2) Self-Attention: The self-attention mechanism is the core of the Transformer, which is used to capture the global dependency between features. Firstly, a query vector Q, a key vector K, and a value vector V are generated for the input sequence to compute the attention score by calculating the Equation (12):

$$\mathrm{Attention}(Q,K,V)=\mathrm{softmax}\left({\displaystyle \frac{Q{K}^T}{\sqrt{d_k}}}\right)V$$

(12)

where Q, K, and V are generated from the input embedding by linear transformation, respectively, and $d_k$ is the the key vector’s dimension. The attention scores are weighted and summed with the value matrix V after softmax normalization to form the feature representation.

(3) Multi-head Self-attention Mechanism: To enhance the model’s ability to learn different feature subspaces, Transformer introduces a multi-head attention mechanism based on standard self-attention. Multi-head attention divides the input matrix into multiple subspaces, performs self-attention calculation separately, and concatenates these independent representations. The principle is shown in Equations (13) and (14):

$$\mathrm{MultiHead}(Q,K,V)=\mathrm{Concat}({\mathrm{head}}_1,\cdots ,{\mathrm{head}}_h)W_O$$

(13)

$${\mathrm{head}}_i=\mathrm{Attention}(Q{W}_{Q_i},K{W}_{K_i},V{W}_{V_i})$$

(14)

where $W_{Q_i}$, $W_{K_i}$, and $W_{V_i}$ are independent projection matrices. This mechanism allows the Transformer to pay attention to the correlation of multiple features simultaneously.

(4) Feed-Forward Network: The output after the attention layer is further processed by a two-layer feed-forward neural network for feature transformation, which is used to introduce more substantial feature representation capabilities. The principle is shown in Equation (15):

$$\mathrm{FFN}\left(x\right)=\mathrm{ReLU}(x{W}_1+b_1)W_2+b_2$$

(15)

where x is the input vector; $W_1$ and $W_2$ are the weight matrices corresponding to the first and second layers of the feedforward network, respectively, which define the mapping of the linear transformation; $b_1$ and $b_2$ are the bias vectors corresponding to the two linear transformations of the feedforward network, respectively; and Relu is the activation function.

(5) Residual connection and layer normalization: Residual connection and layer normalization are used after each sublayer to stabilize the training process, as shown in Equation (16):

$$\mathrm{Output}=\mathrm{LayerNorm}(x+\mathrm{Sublayer}(x\left)\right)$$

(16)

The role of the Transformer network in this study is to model the global dependencies between battery characteristics, which can learn feature interactions across time steps and improve the accuracy of SOH estimation.

3.3. Gated Recurrent Neural Network (GRU)

The second layer of the data-driven model in this study uses a GRU, which captures the long-term dependence of features through a gating mechanism and automatically ignores irrelevant information during the learning process, reducing the impact of noise. GRU [36,37] is an improved RNN that effectively alleviates the problems of gradient vanishing and gradient explosion in traditional RNNs by introducing update gates and reset gates while reducing the number of model parameters. As shown in Figure 6, the structure of a single network unit of GRU includes the following key components:

(1) Update gate: The update gate controls the degree to which the current time step information is combined with the previous sequence information, which determines the update ratio of the hidden state. Its mathematical expression as shown in Equation (17):

$$z_t=\sigma (W_z{x}_t+U_z{h}_{t-1}+b_z)$$

(17)

where $z_t$ is the updated gate vector; $x_t$ is the input at the current time step; $h_{t-1}$ is the hidden state at the previous time step; $W_z$, $U_z$, and $b_z$ are the weight matrix and bias, respectively; and $\sigma$ is the sigmoid activation function.

(2) Reset gate: The reset gate determines the influence weight of the hidden state at the previous time step, thereby regulating the degree of forgetting of past information. Its expression as shown in Equation (18):

$$z{r}_t=\sigma (W_r{x}_t+U_r{h}_{t-1}+b_r)$$

(18)

where $r_t$ is the reset gate vector and other symbols have the same meaning as above.

(3) Candidate hidden state: The candidate hidden state is generated by resetting the hidden state at the previous time step after adjusting the door and combining it with the current input. Its expression as shown in Equation (19)

$${\tilde{h}}_t=tanh(W_h{x}_t+U_h(r_t\odot h_{t-1})+b_h)$$

(19)

where ${\tilde{h}}_t$ is the candidate hidden state and ⊙ denotes element-wise multiplication.

(4) Hidden state update: The final hidden state is calculated by the update gate based on the hidden state and candidate hidden state at the previous time step:

$$h_t=z_t\odot h_{t-1}+(1-z_t)\odot {\tilde{h}}_t$$

(20)

In this study, the GRU gradually establishes a mapping relationship between the feature data and health status by learning the temporal pattern of the input health characteristics. Its primary function is to capture the dynamic characteristics of time-series data and provide high-quality time-dependent representations for subsequent parallel fusion with the Transformer.

3.4. Transformer-GRU Parallel Architecture

As shown in Figure 7, the proposed Transformer-GRU parallel architecture consists of two branches. Branch 1 consists of two layers of Transformer encoding networks, each consisting of a position encoding layer, a multi-head attention mechanism layer, a residual connection and normalization layer, and a feed-forward layer. The feed-forward layer consists of two fully connected layers, with the activation function of the first layer being $Relu$ and the second layer not using an activation function. Branch 2 consists of a three-layer GRU neural network that receives input health feature data simultaneously with branch 1. The features extracted by branches 1 and 2 are not directly used for health status prediction. This study introduces a cross-attention mechanism to cross-fuse the Transformer and GRU output features. Finally, a fully connected layer outputs the predicted health status.

Multivariate data are first encoded using a Transformer layer based on a multi-head attention mechanism to extract global spatial features. At the same time, the data are passed through a GRU network to extract long-term dependencies, and the features are combined using a cross-attention mechanism. The model proposed in this study combines global feature dependencies with local dynamic characteristics to generate a more refined representation of the health status. This fusion mechanism can compensate for the deficiencies of a single model in its modeling ability in specific dimensions, significantly enhancing its adaptability to different types of batteries and complex application conditions.

4. Discussion of the Results

First, this chapter discusses the correlation between health characteristics before and after noise reduction. Then, this study compares the results of four methods: a single GRU and Transformer model, a Transformer-GRU parallel model, and a Transformer-GRU parallel model with feature reconstruction and noise reduction. The root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and coefficient of determination ($R^2$) are used as evaluation metrics for the SOH estimation results. In this study, the top 40% of data from each battery cell are used as the training set, and the remaining 60% are used as the test set. All experiments are performed on the torch (11.8.0) platform with an NVIDIA Geforce RTX 4060. The scientific computing libraries used in this study include numpy (1.24.3), pandas (2.0.3), scikit-learn (1.3.2), and matplotlib (3.7.5).

4.1. Feature Reconstruction Result

Compared with the directly extracted feature CCCT, the features ICP and ICAV are extracted after transformation and filtering of the charging curve. Due to missing raw data, data transformation, etc., both ICP and ICAV features are disturbed and generate noisy data, which directly leads to significantly lower feature correlation than the directly extracted features. Therefore, the CEEMDAN technique is used to reconstruct the ICP curve features to reduce the impact of noisy data on model training.

Figure 8a–c show the modal components after characteristic ICP decomposition. IMF1 is the high-frequency component, and characteristic reconstruction is performed after the high-frequency component is removed. As shown in Figure 8d, the dark red curve is the original feature curve, and the dark blue curve is the feature curve after noise reduction. The feature curve has been reconstructed using the CEEMDAN technique, and its data fluctuations have been significantly reduced. In particular, in the early stage of the cycle, the frequency and amplitude of the spike in the feature curve have been reduced, which is more consistent with the actual attenuation trend of the SOH.

Table 3. Pearson coefficient after denoising.

Features	Pearson Coefficient After Denoising
	Cell_1	Cell_7	CS2_35	CS2_36	B0005	B0006
CCCT	0.9988	0.9998	0.9851	0.9923	0.9981	0.9952
ICP	0.9768	0.9615	0.9813	0.9942	0.9940	0.9925
ICAV	0.9918	0.9824	0.9740	0.9852	0.9976	0.9939

shows the Pearson coefficients after feature denoising. Compared with Table 2, it can be seen that they have improved in most battery cells. Among them, the characteristic correlation of B0006 after denoising has slightly decreased. The main reason is that the original data have low volatility and less noise. Its high correlation makes it difficult to improve the data quality through denoising further. Overall, the CEEMDAN technique has significantly improved the signal-to-noise ratio of the data and enhanced the correlation between health characteristics and SOH.

4.2. State of Health Estimation Result

As can be seen from the SOH estimation results shown in Figure 9, there are significant differences in the predicted performance of different models on each cell. The single GRU and Transformer models effectively capture short-term dynamic characteristics, especially in the initial stage when the predicted results are close to the actual values. However, as the number of cycles increases, its prediction error gradually increases, especially in areas where degradation intensifies later, showing a significant deviation. The Transformer-GRU parallel model combines the Transformer’s global feature modeling capability with GRU’s long-term dependency capture capability. In most scenarios, it shows better prediction accuracy than a single model. In predicting the entire life cycle of six battery cells, the parallel model significantly reduces the error in the long-term trend prediction. For working conditions with large fluctuations in battery capacity, such as the enlarged area in Figure 9c,d, this model alleviates to some extent the shortcomings of the single GRU and Transformer models and demonstrates its ability to balance short-term dynamics and long-term trends. The Transformer-GRU parallel model based on noise reduction outperforms other models, as shown in the locally enlarged views of Figure 9a,c,f. The frequency and amplitude of sudden changes in the predicted values have improved compared to before denoising. At the same time, the SOH prediction curve, after denoising, fits the real curve more closely, showing a better prediction effect. This indicates that denoising improves the signal-to-noise ratio of the features, making the model’s predictions more robust under noise interference.

Figure 10 shows the prediction error box plot for all battery cells. Overall, the model proposed in this study shows the best error distribution characteristics for all battery cells. Compared with other models, it has the narrowest error box range, indicating that the model has a more concentrated error distribution. Meanwhile, the median error is closest to zero, reflecting less deviation and higher stability in the prediction. In contrast, the error boxes of the GRU model are generally wider, indicating greater volatility in its prediction errors. Except for the Cell_7 battery, the Transformer model has been improved on other battery cells. However, its error range is still extensive, indicating that a single model has limited adaptability to complex dynamic characteristics. The box range of the Transformer-GRU parallel model is significantly reduced on all battery cells, indicating a more concentrated error distribution and further improved prediction accuracy and stability.

Table 4. SOH estimated error.

Battery	Model	Error Index
		RMSE	MAE	MAPE	R2
Cell_1	GRU	0.0130	0.0117	1.49%	0.8149
	Transformer	0.0106	0.0096	1.22%	0.8773
	Transformer-GRU	0.0066	0.0051	0.63%	0.9517
	Proposed	0.0042	0.0039	0.42%	0.9806
Cell_7	GRU	0.0112	0.0100	1.20%	0.8455
	Transformer	0.0124	0.0114	1.38%	0.8088
	Transformer-GRU	0.0048	0.0044	0.53%	0.9711
	Proposed	0.0032	0.0029	0.35%	0.9871
CS_35	GRU	0.0219	0.0149	1.89%	0.7595
	Transformer	0.0159	0.0105	1.34%	0.8727
	Transformer-GRU	0.0076	0.0055	0.68%	0.9707
	Proposed	0.0068	0.0053	0.65%	0.9762
CS_36	GRU	0.0281	0.0206	2.73%	0.8192
	Transformer	0.0183	0.0131	1.73%	0.9230
	Transformer-GRU	0.0127	0.0097	1.25%	0.9627
	Proposed	0.0107	0.0084	1.07%	0.9736
B0005	GRU	0.0175	0.0139	1.90%	0.8970
	Transformer	0.0143	0.0114	1.60%	0.9314
	Transformer-GRU	0.0080	0.0064	0.85%	0.9783
	Proposed	0.0077	0.0047	0.73%	0.9780
B0006	GRU	0.0186	0.0149	2.30%	0.8781
	Transformer	0.0164	0.0144	2.17%	0.9046
	Transformer-GRU	0.0097	0.0074	1.11%	0.9666
	Proposed	0.0099	0.0082	1.24%	0.9651

shows the error indicators of the six battery cells. The root mean square error of the SOH estimation methods proposed in this study does not exceed 0.0107, being 0.0042, 0.0032, 0.0068, 0.0107, 0.0077 and 0.0099, respectively. Among them, the average RMSE of the GRU model is 0.0184, and the average RMSE of the Transformer model is 0.0147. Overall, the error of the Transformer is lower than that of the GRU model, but the GRU model performs better on the Cell_7 battery. This phenomenon shows that the performance of a single model is limited and often cannot be guaranteed to apply to different types of batteries under different operating conditions. The parallel model based on Transformer and GRU, which incorporates the characteristics of different models, shows better prediction performance. Its RMSE is 0.0082, a 55.43% and 44.22% reduction compared to the GRU and Transformer models. After feature reconstruction and denoising, the RMSE of SOH estimation is reduced to 0.0071, a 13.41% reduction compared to the Transformer-GRU fusion model. Using GRU as the benchmark model, the method proposed in this study improves the accuracy by 61.41%.

Table 5. Model training time analysis.

Time	GRU	Transformer	Transformer-GRU	Proposed
Training time (s)	6.5635	9.5454	15.2436	15.9055
Prediction time (s)	0.0011	0.0016	0.0026	0.0025

demonstrates the average training time of different models and the time taken to predict the SOH of a cycle. GRU is the model with the shortest training time due to its simple structure, which takes only 6.5635 s; the Transformer’s training time increases to 9.5454 s due to the higher computational complexity of the self-attention mechanism. Transformer-GRU combines Transformer and GRU; its computational overhead is more significant, and the training time rises to 15.2436 s, the longest among all models. Despite the longer training time of Transformer-GRU, its RMSE is 55.43% and 44.22% lower than GRU and Transformer, respectively, and this computational cost is reasonable and acceptable. The feature reconstruction of the method proposed in this study is performed before the model training, so its time consumption is comparable to that of the Transformer-GRU model. In terms of prediction time, the inference time of all models is at the millisecond level, which can meet the real-time prediction demand.

4.3. Segmentation of Different Training Sets

In this section, the results of the Cell_7 and CS_35 batteries are chosen to be discussed for different training set division scenarios. Since the Cell_7 battery is tested for capacity after every 100 cycles, there are fewer data samples for this battery, while CS_35 is a case with a larger number of samples. Figure 11 illustrates the SOH estimation results for both cells for 30%, 40%, and 50% training set cases. As the division of the training data changes, the prediction curve of Cell_7 deviates with the reduction in training data, while the prediction curve of CS_35 basically overlaps. As shown in

Table 6. Results for different training set divisions.

Battery	Training Set	Error Index
		RMSE	MAE	MAPE	R2
Cell_7	50%	0.0026	0.0019	0.23%	0.9880
	40%	0.0032	0.0029	0.35%	0.9871
	30%	0.0075	0.0065	0.75%	0.9528
CS_35	50%	0.0073	0.0058	0.69%	0.9714
	40%	0.0068	0.0053	0.65%	0.9762
	30%	0.0071	0.0053	0.64%	0.9787

, Cell_7 has an RMSE of 0.0026 when the training set is 50% and 0.0075 when the training data are reduced to 30%, indicating that Cell_7 has fewer data samples, and its results are more affected by the division of the training set. Nevertheless, when the training data of Cell_7 is 30%, it has only 23 effective training cycles, but its RMSE is still less than 1%, which maintains a low level. For the CS_35 battery, which has more data samples, its RMSE is only 0.0071 even though only 30% of the data are used for training, and its RMSE stays at the same level as the training data changes. In summary, the method proposed in this study demonstrates stable prediction ability under different training set divisions, verifying its feasibility for long-term reliable prediction of battery SOH.

5. Conclusions

This study proposes a lithium-ion battery state of health estimation method based on feature reconstruction and a Transformer-GRU parallel architecture. The main work and conclusions are as follows:

(1)

This study constructed an incremental capacity curve based on charging data, smoothed it using Gaussian filtering, and extracted multiple health features based on the charging voltage and IC curve. All feature correlations are more significant than 0.9, indicating a strong correlation with health status;

(2)

A Transformer-GRU parallel architecture is constructed by fusing the Transformer and GRU models using a cross-attention mechanism. The root means square error of its SOH estimation is 0.0082, with 55.43% and 44.22% reduction based on the GRU and Transformer models, respectively;

(3)

The CEEMDAN algorithm reconstructs the features of the IC curve, improving the correlation between health characteristics and health status. The root mean square error of the state of health estimation using reconstructed features is reduced by 13.41%.

In the future, we plan to conduct state of health estimations under complex working conditions and apply the results to battery management systems for safety monitoring and energy optimization management of energy storage systems.