The accurate and reliable estimation of the State of Health (SOH) for lithium-ion batteries is a cornerstone for ensuring the safety, longevity, and performance of modern energy storage systems, particularly in the rapidly growing electric vehicle (EV) sector. SOH, typically defined as the ratio of current maximum capacity to its nominal capacity, quantifies the degradation level of a lithium-ion battery. Precise SOH estimation is critical for effective battery management systems (BMS) to prevent overcharge/discharge, optimize charging strategies, schedule maintenance, and ensure operational safety. As the demand for higher energy density and longer cycle life intensifies, developing robust estimation methodologies has become a focal point of research.
Traditional SOH estimation approaches can be broadly categorized into model-based and data-driven methods. Model-based methods, including electrochemical and equivalent circuit models, rely on a deep understanding of the internal physical and chemical processes within a lithium-ion battery. While they offer strong interpretability, they often involve solving complex partial differential equations and require precise knowledge of battery parameters, which are difficult to obtain in real-time and may not generalize well across different battery types and operating conditions. In contrast, data-driven methods have gained significant traction due to their ability to learn complex, non-linear relationships directly from operational data without requiring explicit physical models. These methods map measurable features, such as voltage, current, temperature, and incremental capacity (IC) curves, to the target SOH value.
Common data-driven techniques include machine learning algorithms like Support Vector Regression (SVR) and Gaussian Process Regression (GPR), and more recently, deep learning models such as Convolutional Neural Networks (CNN) and Recurrent Neural Networks (RNN), including Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU) networks. These models excel at capturing temporal dependencies and spatial features from historical cycling data of the lithium-ion battery. However, significant challenges remain. First, the phenomenon of capacity regeneration, where the measured capacity temporarily increases during cycling, can mislead prediction models that lack a strict causal understanding of the time series. Second, many deep learning architectures struggle with long-term dependency modeling and effective extraction of hidden degradation features, limiting their prediction accuracy and generalization capability across diverse datasets. Third, models that perform well on one type of lithium-ion battery or under specific cycling conditions often fail to maintain their performance on others, indicating poor generalization.
To address these limitations, we propose a novel hybrid deep learning architecture that integrates enhanced feature extraction with sophisticated sequence modeling for lithium-ion battery SOH estimation. Our model synergistically combines a Self-Attention (SA) empowered Causal Dilated Convolution (CDC) module with a Gated Recurrent Unit-Autoencoder (GRU-AE) module. The SA-CDC module is designed to extract rich temporal features while enforcing causality and expanding the receptive field to suppress the干扰 of capacity regeneration. The GRU-AE module further refines these features, capturing long-term dependencies and learning a compact, informative representation of the lithium-ion battery’s degradation trajectory. This integrated approach aims to achieve high-precision, robust, and generalizable SOH estimation.
Problem Formulation and Preliminaries
The core objective is to estimate the State of Health (SOH) of a lithium-ion battery using historical cycling data. The SOH at cycle \(k\) is commonly defined based on capacity fade:
$$SOH(k) = \frac{C_{max}(k)}{C_{nominal}} \times 100\%$$
where \(C_{max}(k)\) is the maximum available discharge capacity at cycle \(k\), and \(C_{nominal}\) is the nominal or initial capacity of the fresh lithium-ion battery. The degradation process is reflected in the gradual decline of \(C_{max}(k)\) over cycles.
We operate under a data-driven paradigm where the model learns from features extracted from charge/discharge cycles. A critical feature set is derived from the Incremental Capacity (IC) analysis. The IC curve, obtained by differentiating capacity with respect to voltage (\(dQ/dV\)), reveals characteristic peaks related to phase transformations within the electrode materials of the lithium-ion battery. The shape and evolution of these peaks are highly correlated with battery aging. From the IC curve per cycle, we extract statistical features such as the maximum IC value, mean, standard deviation, variance, and median. These features form our input sequence.
Let the input feature vector for cycle \(i\) be \(\mathbf{x}_i \in \mathbb{R}^F\), where \(F\) is the number of features (e.g., 5). Over \(N\) cycles, we have a sequence \(X = [\mathbf{x}_1, \mathbf{x}_2, …, \mathbf{x}_N]\). The corresponding target SOH values form the sequence \(Y = [SOH_1, SOH_2, …, SOH_N]\). The goal of our model \(f_{\theta}(\cdot)\), parameterized by \(\theta\), is to learn the mapping:
$$\hat{Y} = f_{\theta}(X)$$
such that the difference between the predicted SOH sequence \(\hat{Y}\) and the actual sequence \(Y\) is minimized. The performance is evaluated using standard regression metrics: Root Mean Square Error (RMSE) and Mean Absolute Error (MAE), defined as:
$$RMSE = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_i – \hat{y}_i)^2} \times 100\%$$
$$MAE = \frac{1}{n}\sum_{i=1}^{n}|y_i – \hat{y}_i| \times 100\%$$
where \(y_i\) and \(\hat{y}_i\) are the actual and predicted SOH values for the \(i\)-th sample, and \(n\) is the total number of test samples.
The Proposed SA-CDC-GRU-AE Architecture
The proposed architecture is designed to process the sequential feature data from a lithium-ion battery and output accurate SOH estimates. The overall workflow consists of: 1) Input feature preparation from raw cycling data, 2) Hierarchical feature extraction and refinement via the SA-CDC module, 3) Deep temporal dependency modeling and feature representation learning via the GRU-AE module, and 4) Final SOH regression through a fully connected layer.
1. The Self-Attention Causal Dilated Convolution (SA-CDC) Module
This module is the first stage of feature processing, specifically designed to handle the temporal sequence while addressing the issues of causality and capacity regeneration in lithium-ion battery data.
Causal Dilated Convolution (CDC): Standard convolutional operations in a temporal context can inadvertently use “future” information to predict the “past,” which violates the causality principle essential for real-time prediction. Furthermore, capturing long-range dependencies in the degradation trend of a lithium-ion battery requires a large receptive field, typically demanding many layers or large kernels. To solve both problems simultaneously, we employ Causal Dilated Convolutions.
- Causal Convolution: This is implemented by padding the input sequence with zeros only on the left side. For a 1D convolution with kernel size \(k\), the padding length is \(k-1\). This ensures that the output at time step \(t\) is only convolved from inputs at time steps \(t, t-1, …, t-k+1\).
- Dilated Convolution: This introduces a dilation rate \(d\) that spaces out the kernel weights. It effectively enlarges the receptive field exponentially without increasing the number of parameters or losing resolution. The effective receptive field size for a dilated convolution is \(k + (k-1)(d-1)\).
The combination forms a CDC layer. By stacking multiple CDC layers with increasing dilation rates (e.g., \(d=1, 2, 4, 8, …\)), the model can capture multi-scale temporal patterns from the lithium-ion battery feature sequence while strictly adhering to causal constraints. This design helps the model learn the underlying degradation trend and be less sensitive to local fluctuations like capacity regeneration, as the expanded receptive field provides a broader context.
Self-Attention (SA) Mechanism: While CDC captures multi-scale patterns, it may still focus on local correlations. To empower the model with the ability to weigh the importance of all historical time steps adaptively, we integrate a Self-Attention mechanism after the CDC layers. For a feature sequence \(H \in \mathbb{R}^{T \times D}\) output by the CDC block, the SA mechanism computes Query (Q), Key (K), and Value (V) matrices via linear projections:
$$Q = HW_Q, \quad K = HW_K, \quad V = HW_V$$
The attention scores are calculated as:
$$\text{Attention}(Q, K, V) = \text{softmax}\left(\frac{QK^T}{\sqrt{d_k}}\right)V$$
where \(d_k\) is the dimension of the key vectors. This operation allows the model to learn global dependencies across the entire sequence, identifying which past cycles are most informative for predicting the current SOH state of the lithium-ion battery. The output of the SA mechanism is a refined feature sequence that captures both local temporal patterns (via CDC) and global contextual relationships (via SA).
2. The Gated Recurrent Unit – Autoencoder (GRU-AE) Module
The features from the SA-CDC module are then passed to a GRU-AE module. This module serves a dual purpose: it acts as a powerful sequence model to capture complex long-term temporal dynamics in lithium-ion battery aging, and it learns a compressed, meaningful representation of the input features through its autoencoder structure.
Gated Recurrent Unit (GRU): GRU is a type of RNN adept at handling sequential data. It mitigates the vanishing gradient problem through gating mechanisms (reset gate \(r_t\) and update gate \(z_t\)). The equations for a GRU cell are:
$$
\begin{aligned}
z_t &= \sigma(W_z \cdot [h_{t-1}, x_t]) \\
r_t &= \sigma(W_r \cdot [h_{t-1}, x_t]) \\
\tilde{h}_t &= \tanh(W \cdot [r_t * h_{t-1}, x_t]) \\
h_t &= (1 – z_t) * h_{t-1} + z_t * \tilde{h}_t
\end{aligned}
$$
where \(x_t\) is the input, \(h_t\) is the hidden state, \(\sigma\) is the sigmoid function, and \(*\) denotes element-wise multiplication. GRU’s ability to selectively remember and forget information makes it highly suitable for modeling the long-term degradation trends of a lithium-ion battery.
Autoencoder (AE) Integration: A standard GRU used for sequence-to-value prediction might not fully exploit the internal structure of the feature sequence. By configuring GRU layers into an autoencoder, we enforce the model to learn a compressed latent representation \(Z\) that contains the most essential information needed to reconstruct the input sequence. This compression acts as a form of regularization and feature distillation.
- Encoder: Two GRU layers process the input sequence \(X_{in}\) and encode it into a context vector or a sequence of context vectors (the latent representation \(Z\)).
- Decoder: Two subsequent GRU layers take the latent representation \(Z\) and attempt to reconstruct the original input sequence, producing \(X_{recon}\).
The training objective for the AE component includes minimizing the reconstruction loss (e.g., Mean Squared Error between \(X_{in}\) and \(X_{recon}\)). The latent representation \(Z\), learned to efficiently encode the input features, is then used as the final processed feature for SOH prediction. This process ensures that the features passed to the final regression layer are both temporally informed and information-rich regarding the lithium-ion battery’s health state.
3. Full Architecture and Training
The complete SA-CDC-GRU-AE model pipeline is as follows:
- Input: The sequence of statistical features (max, mean, std, var, median of IC curve) for \(N\) cycles.
- SA-CDC Block 1 & 2: Two consecutive SA-CDC modules perform hierarchical causal feature extraction, outputting a globally-aware feature sequence.
- GRU-AE Block: The feature sequence is encoded by the GRU encoder into a latent representation \(Z\). This representation is then decoded (for reconstruction loss calculation during training). The final hidden states from the encoder (or the latent representation itself) are extracted.
- Fully Connected (FC) Layer: The extracted features are flattened and passed through one or more fully connected layers to produce the final SOH estimate for the target cycles.
The model is trained end-to-end using the Adam optimizer, with a combined loss function: \(\mathcal{L}_{total} = \mathcal{L}_{SOH} + \lambda \mathcal{L}_{recon}\), where \(\mathcal{L}_{SOH}\) is the RMSE or MAE loss for SOH prediction, \(\mathcal{L}_{recon}\) is the reconstruction loss from the AE, and \(\lambda\) is a weighting hyperparameter.
Experimental Validation and Results
To validate the effectiveness of the proposed model, we conducted extensive experiments on two widely used public datasets featuring lithium-ion battery aging: the CALCE dataset from the University of Maryland and the NASA Ames Prognostics Center of Excellence dataset.
Datasets and Preprocessing:
- CALCE: We used data from four lithium-ion batteries (CS2_35, CS2_36, CS2_37, CS2_38). They were cycled under a constant current-constant voltage (CC-CV) charge protocol (to 4.2V, cut-off 50mA) and a constant current (1.1A) discharge to 2.7V.
- NASA: We used data from four lithium-ion batteries (B0005, B0006, B0007, B0018). The charge protocol was CC-CV (1.5A to 4.2V, cut-off 20mA), and discharge was at a constant 2.0A to different cut-off voltages (2.7V, 2.5V, 2.3V, 2.5V respectively).
For each dataset, we adopted a leave-one-out validation strategy: three batteries were used for training, and the remaining one was used for testing. This rigorous test evaluates the model’s generalization capability to a new, unseen lithium-ion battery. Input features were extracted from the IC curves of each cycle as described.
Comparative Experiments
We compared the SA-CDC-GRU-AE model against several state-of-the-art and recent models for lithium-ion battery SOH estimation.
Results on CALCE Dataset: The comparison included Convolutional Transformer (CT), Informer, and a Transformer-based Deep Domain Adaptation Methodology (TDDAM). The performance metrics (average RMSE and MAE) are summarized below.
| Model | Avg. RMSE (%) | Avg. MAE (%) |
|---|---|---|
| CT | 1.460 | 1.160 |
| Informer | 1.380 | 0.983 |
| TDDAM | 1.043 | 0.803 |
| SA-CDC-GRU-AE (Proposed) | 1.009 | 0.780 |
The proposed model achieved the lowest average RMSE and MAE, demonstrating superior overall estimation accuracy for the lithium-ion batteries in this dataset.
Results on NASA Dataset: The comparison included ResNet-LSTM, Bayesian Optimization-Sequence to Sequence (BO-Seq2Seq), and Gaussian Process Regression (GPR).
| Model | Avg. RMSE (%) | Avg. MAE (%) |
|---|---|---|
| ResNet-LSTM | 1.153 | 0.838 |
| BO-Seq2Seq | 0.655 | 0.486 |
| GPR | 0.619 | 0.499 |
| SA-CDC-GRU-AE (Proposed) | 0.488 | 0.432 |
Again, the proposed model significantly outperformed the others, achieving the best average performance. This result across two distinct datasets with different cycling protocols underscores the strong generalization ability of our architecture for lithium-ion battery SOH estimation.
Ablation Studies
To verify the contribution of each key component in the proposed model, we conducted ablation studies on both datasets. We compared the full SA-CDC-GRU-AE model against several degraded versions.
Ablation on CALCE Dataset:
| Model Variant | Description | Avg. RMSE (%) | Avg. MAE (%) |
|---|---|---|---|
| Variant 1 | GRU only (Baseline) | 1.912 | 1.489 |
| Variant 2 | SA-CNN-GRU (Standard CNN) | 1.644 | 1.345 |
| Variant 3 | SA-CDC-GRU (No AE) | 1.431 | 1.139 |
| Variant 4 | GRU-AE (No SA-CDC) | 1.447 | 1.138 |
| Full Model | SA-CDC-GRU-AE | 1.009 | 0.780 |
Ablation on NASA Dataset:
| Model Variant | Description | Avg. RMSE (%) | Avg. MAE (%) |
|---|---|---|---|
| Variant 1 | GRU only (Baseline) | 1.853 | 1.672 |
| Variant 2 | SA-CNN-GRU (Standard CNN) | 1.379 | 1.298 |
| Variant 3 | SA-CDC-GRU (No AE) | 1.172 | 1.062 |
| Variant 4 | GRU-AE (No SA-CDC) | 0.834 | 0.765 |
| Full Model | SA-CDC-GRU-AE | 0.488 | 0.432 |
The ablation results clearly demonstrate the incremental benefit of each component:
- Replacing standard CNN with the SA-CDC module (Variant 2 vs. Variant 3) improves performance, validating the effectiveness of causal dilated convolutions for lithium-ion battery sequence modeling.
- Adding the AE structure to GRU (Variant 1 vs. Variant 4) provides a substantial boost, confirming the value of representation learning.
- The combination of both the SA-CDC module and the GRU-AE module in the full model yields a synergistic effect, leading to the best performance. The full model’s RMSE on the NASA dataset is 73.7% lower than the GRU-only baseline, and on the CALCE dataset, it is 47.2% lower.
These studies conclusively prove that both the SA-CDC module (for causal, multi-scale feature extraction) and the GRU-AE module (for temporal dependency modeling and feature distillation) are crucial for achieving high-precision, generalizable SOH estimation for lithium-ion batteries.
Conclusion and Future Work
Accurate State of Health estimation is paramount for the reliable and safe operation of lithium-ion batteries in demanding applications like electric vehicles. This work presented a novel hybrid deep learning architecture, the SA-CDC-GRU-AE model, to address key challenges in SOH estimation: handling capacity regeneration, capturing long-term dependencies, and improving generalization.
The core innovation lies in the synergistic integration of two specialized modules. The SA-CDC module leverages causal dilated convolutions to enforce temporal causality and expand the receptive field, allowing the model to learn degradation trends while being robust to local anomalies like capacity regeneration in lithium-ion batteries. The Self-Attention mechanism further enhances this by enabling adaptive weighting of global historical context. Subsequently, the GRU-AE module acts as a powerful sequence modeling engine that also learns a compact, informative latent representation of the battery’s health trajectory, effectively capturing complex aging dynamics.
Comprehensive experiments on two prominent public datasets (CALCE and NASA) demonstrated the superior performance and strong generalization capability of the proposed model. It consistently achieved the lowest average RMSE and MAE compared to several contemporary models. Ablation studies confirmed the critical contribution of each architectural component to the overall performance gain.
The proposed SA-CDC-GRU-AE model provides a robust and accurate data-driven solution for lithium-ion battery SOH estimation, with significant potential for integration into next-generation Battery Management Systems. Future work will explore several promising directions. First, while purely data-driven, incorporating physical knowledge could further enhance robustness and interpretability. A promising path is the integration with Physics-Informed Neural Networks (PINN), where governing equations or constraints from degradation models could guide the learning process. Second, extending the model to perform joint estimation of SOH and other states, such as State of Charge (SOC) and Remaining Useful Life (RUL), within a unified framework would be highly valuable for comprehensive battery management. Finally, testing the model on a wider variety of lithium-ion battery chemistries, formats, and real-world driving cycle data will be essential to validate its practicality for commercial BMS applications.