In this study, the battery data used were obtained from the National Aeronautics and Space Administration (NASA). Specifically, the dataset contains 18,650 lithium-ion batteries with a cathode material of LiNi_0.8Co_0.15Al_0.05O₂ (NCA) and anode graphite material, a rated capacity is 2 Ah. All batteries were subjected to aging tests at an ambient temperature of 24°C. As shown in Fig. 2a, during the aging experiments, each battery was charged using a constant-current and constant-voltage (CC-CV) protocol with a charging current of 1.5 A, a cutoff voltage of 4.2 V, and a cutoff current of 20 mA. In the discharging process, each battery was working under a constant-current mode at 2 A with cutoff voltages of 2.7 and 2.2 V, respectively.

With the use of batteries, continuous charge-discharge cycles trigger negative electrochemical reactions that ultimately lead to capacity degradation, as shown in Fig. 2b. In repeated charging processes, the deintercalation and intercalation of lithium ions cause the cathode material to undergo repeated lattice expansion and contraction, gradually fragmenting its structure and reducing active substances. Meanwhile, a solid electrolyte interface (SEI) film forms on the anode surface, whose continuous growth not only consumes electrolyte and lithium ions but also hinders ion transport. Additionally, the electrolyte itself decomposes on the electrode surface, generating impurities that further deteriorate the internal battery environment. The continuous accumulation of these electrochemical reactions not only causes a gradual decline in the charge storage capacity, intensifying users’ range anxiety but also accompanies significant increases in internal resistance, lithium deposition on the anode, and fragmentation of cathode particles. These phenomena lead to substantial degradation of the battery’s thermal stability, significantly increasing the probability of safety accidents such as short circuits and fires. In this study, the SOH is used to evaluate battery aging, as shown in Eq. (1): (1)SOH=CrCn×100%where the C_r denotes the actual capacity of the battery with the r cycle times, C_n is the nominal capacity of the battery.

In the process of establishing a data-driven model for SOH estimation, the extraction of health factors directly affects the estimation performance of the model. In existing studies, the independent component analysis (ICA) method is a commonly used approach for health factor extraction [42]. This method monitors and analyzes the variations between voltage and capacity, enabling effective capture of data segments containing battery aging information. However, it requires recording charging data from 0% to 100% SOC degree and performing differentiation processing, which significantly increases the workload of health factor extraction and weakens the high-efficiency advantage of data-driven models. Meanwhile, in the actual operation of vehicles, the complete charging process starting from 0% SOC degree is difficult to obtain, posing challenges for the practical application of the ICA method.

With the development of battery aging, irreversible side reactions inside the battery cause changes in internal resistance, which are manifested as varying degrees of offset in the charging voltage curve, as shown in Fig. 3a. Therefore, this paper proposes using the local voltage segment time as a health factor. Taking the constant voltage duration with a 0.2 V interval as an example, as shown in Fig. 3b, with the aging of the battery, the duration of each constant voltage interval shows a significant decreasing trend.

Although the charging time of different constant voltage intervals shows the same variation trend as battery performance degradation, the degree of aging information contained in each voltage interval varies. To enhance the utilization of key information and reduce the impact of redundant information on the data-driven model, this paper employs the grey correlation degree to analyze the importance of constant voltage intervals with 0.1 and 0.2 V spacing. The calculation of the correlation degree is shown in the following equation: (2)Pi(j)=mini⁡minj⁡|SOH(j)−Ti(j)|+ρmini⁡minj⁡|SOH(j)−Ti(j)||SOH(j)−Ti(j)|+ρmini⁡minj⁡|SOH(j)−Ti(j)|where i is the number of cycles, j is the number of constant voltage interval segments, T_i(j) is the charging time of the j-th voltage interval at the i-th cycle, and SOH(j) is the state of health at the i-th cycle. The importance of the duration in each interval is shown in Fig. 4. The constant voltage interval with the highest correlation with SOH is selected as the health factor. In Fig. 4, taking a 0.2 V interval, the charging time of the interval starting at 3.9 V exhibits an extremely strong correlation. In related studies, the ICA results of NASA battery aging data are consistent, and the variation relationship between capacity and voltage is most significant near 4.0 V [43]. In this paper, the charging time in the range of 3.9–4.1 V during the charging process is selected as the health factor for model establishment.

In the construction of data-driven models, input data plays a crucial role in the estimation performance of the model. Due to the existence of random interference and capacity regeneration phenomena during the aging process, the aging trajectory of the battery is not a monotonically decreasing degradation process. These processes are intertwined, and directly constructing a model may lead to ineffective capture of aging information. The variational mode decomposition (VMD) method can adaptively decompose the original characteristic data into multiple aging information components with different central frequencies. Each component contains different aging processes, enabling fine-grained analysis of aging data.

To address the complexity and non-stationarity of aging data, in this study, it employes the VMD method to decompose the proposed features. It can effectively eliminate noise interference in the data and extract the true trend of capacity degradation. Meanwhile, by separating different frequency components, it can accurately decompose various aging information of the battery. Compared with traditional analysis methods, VMD does not require preset parameters, has stronger adaptability and robustness, and can more accurately excavate the underlying patterns in battery aging feature data, providing a more reliable basis for the establishment of data-driven models. The calculation of VMD is as follows: (3)VMD={d1(t),⋯dn(t),r(t)}where d₁(t) represents the decomposed subsequence term, r(t) is the main degradation trend term. The constrained variational model corresponding to the signal I(t) is defined as follows: (4)min{θn}{εn}{∑n=1N‖∂t[(δt+j/π)∗εn(t)]∗exp⁡[−jθnt]‖22}

(5)s.t.∑n=1Nεn(t)=I(t)where {ε_n} is the n-th intrinsic mode function (IMF) after decomposition, {θ_n} denotes the frequency of the n-th IMF, ∗ represents the convolution calculation, ∂_t is the derivative with respect to time, and δ_t is the Dirac delta function. Transforming the constrained problem of the above model into an unconstrained problem for solution, as shown in Eq. (6):(6)L({εn},{θn},λ)=⟨λ(t),x(t)−∑k=1Kεn(t)⟩+‖I(t)−∑k=1Kεn(t)‖22+γ{∑n=1N‖∂t[(δt+j/πt)∗εn(t)]∗exp⁡[−jθnt]‖22}where λ is the Lagrange multiplier. In the process of VMD on aging data, the SOH is employed to replace the capacity for reflecting the aging information of the battery. The decomposition results are shown in Fig. 5. Compared with the original battery aging trajectory, the aging trend term obtained after decomposition can more effectively reflect the monotonicity of battery aging, while the disturbance term reflects the phenomena of capacity regeneration and random interference in the aging process.

The gated recurrent unit (GRU), a variant of the recurrent neural network (RNN), was introduced by Wang et al. in 2014 as an optimized version of the long short-term memory (LSTM) architecture [44]. By simplifying the gating mechanism it reduces computational complexity without sacrificing sequential modeling capabilities. This improved design effectively mitigates the vanishing gradient problem inherent in traditional RNNs, enabling the model to capture long-range temporal dependencies while maintaining robust performance in sequence-to-sequence tasks. The GRU model is characterized by its simple architecture and excellent performance, widely employed in applications such as time-series data processing, natural language processing, and speech recognition. In this study, by integrating the VMD and GRU models, it proposes a VMD-GRU model for the estimation of SOH, the framework of proposed method is shown in Fig. 6.

First, VMD is employed to decompose the degradation curve of battery SOH and the proposed health indicators into a main trend term and a residual term. Then, GRU model is used to explore the potential relationships between each modal component of SOH and health factors, completing the construction of a data-driven model. Finally, the estimation results of each modal component output by the GRU model are superimposed to obtain the SOH estimation value. The GRU estimation process for each VMD component is shown in the Eq. (7).(7){zt=δ⋅(Wz⋅[ht−1,dt]+bt)rt=δ⋅(Wr⋅[ht−1,dt]+br)h∼t=tanh⁡(Wh[rtht−1,dt]+bh)ht=(1−zt)⋅ht−1+zt⋅h∼td¯=Wout⋅hend+bswhere zt denotes the update gate; dt is the input data from VMD component; rt represents the reset gate; ht and hend are the hidden states; h~t is the candidate hidden state; δ(x) and tanh⁡(x) are activation functions; Wz, Wr, Wh and Wout are weight matrices; bt, br, bh, and bs are bias terms; d¯ is the model estimation result.

In this study, to address the challenge that capacity regeneration and random interferences during battery aging make it difficult to directly obtain the SOH, it adapts the NASA battery aging dataset to verify the effectiveness of the proposed method. Meanwhile, to evaluate the generalizability of the proposed method, aging data from different batteries are respectively used as the training and testing sets. Meanwhile, to demonstrate the superiority of the model, it constructs the PSO-XGboost model and LSTM model with the same features as the comparison group. The validation results are shown in the Fig. 7.

As shown in the figures, the proposed method exhibits estimation bias in the early aging stage, demonstrating excellent consistency between the estimation results and the aging trajectory during the subsequent aging process. To further evaluate the estimation performance of the model, root mean square error (RMSE) and mean absolute error (MAE) are employed to quantify the model’s estimation accuracy. As illustrated in Fig. 8, for the aging data of two different batteries, the SOH estimation accuracy of the proposed method is remarkable, with RMSE values of 0.33% and 0.20%, and minimum MAE values of 0.94% and 1.53%, respectively. By comparison, the minimum RMSE values of the PSO-XGBoost and LSTM models using the same features are 1.23% and 0.68%, and the minimum MAE values are 2.11% and 2.48%, respectively. The validation results indicate that the proposed method not only achieves high precision but also enables accurate SOH estimation for two different batteries, demonstrating its excellent generalizability.

The above validation demonstrates the accuracy of the proposed method. To further verify the effectiveness and generalizability of the model, it uses the battery aging dataset from the University of Maryland for estimation validation. In this dataset, all batteries are charged in CCCV mode at 0.5 C with a cut-off voltage of 4.2 V, and discharged at a constant current of 1.1 A [45]. The SOH estimation results are shown in Fig. 9. It demonstrates that the estimation result of the proposed method could accurately reflect the aging trajectory of batteries.

The estimation performance of each model is shown in Fig. 10. For the SOH estimation results of two different batteries, the proposed method achieves RMSE values of 2.76% and 2.67%, with MAE values both within 3%. In contrast, the LSTM model exhibits estimation errors with RMSE values of 3.60% and 3.37%, and MAE values of 3.17% and 2.55%. Compared with the LSTM model, the proposed method demonstrates an average improvement of 22.1% in RMSE and 27.3% in MAE. In traditional SOH estimation methods, many studies have employed ICA on battery charging data, extracting local peaks and valleys as health factors. This approach typically requires obtaining complete charging data under constant current conditions. In this study, a method is proposed that enables accurate assessment of battery health status using only partial data from the battery charging process. It reduces the difficulty in data acquisition while ensuring estimation accuracy comparable to the methods using ICA-based analysis [46–48]. According to the validation results, it can be concluded that the proposed method demonstrates great accuracy across different datasets. Additionally, the excellent generalization ability of the proposed method has been validated through verification results in different batteries.