In recent years, the global transition towards clean energy has accelerated, positioning electrified transportation and electrochemical energy storage as pivotal pathways for achieving decarbonization goals. While lithium-ion batteries have achieved widespread adoption, concerns regarding lithium resource scarcity and supply chain vulnerabilities persist. As an alternative, sodium-ion batteries have garnered significant attention. Owing to their abundant raw materials, lower cost profile, inherent safety characteristics, and commendable low-temperature performance, sodium-ion batteries present a compelling option for various energy storage and electric vehicle applications. However, like all electrochemical systems, sodium-ion batteries undergo performance degradation due to complex aging mechanisms occurring during charge-discharge cycling. The accurate assessment of the State of Health (SOH), a critical metric defining a battery’s current capacity relative to its initial capacity, is therefore paramount for ensuring safety, reliability, and effective lifespan management of sodium-ion battery systems.
Estimating the SOH of sodium-ion batteries presents unique challenges compared to the more mature lithium-ion technology. The fundamental aging mechanisms and capacity fade behaviors of sodium-ion batteries are not yet fully elucidated, partly due to the larger ionic radius of sodium (approximately 34% larger than lithium), which influences intercalation kinetics, phase transition behaviors, and side reactions. Traditional model-based methods, such as equivalent circuit models (ECMs) or electrochemical models, often struggle to capture these complex, multi-scale dynamic responses accurately without extensive parameterization based on comprehensive foundational data, which is still evolving for sodium-ion batteries.
Data-driven approaches offer a promising alternative, as they can learn the complex nonlinear mapping between measurable operational data and the underlying state of health without requiring an explicit physical model. Common features extracted from operational data include those from Incremental Capacity (IC) analysis and Relaxation Voltage (RV) profiles. IC analysis, which involves differentiating the capacity with respect to voltage ($IC = dQ/dU$), transforms voltage plateaus during charging into identifiable peaks. These peaks are directly related to phase transitions within the electrode materials, and their evolution (height, position, width) is strongly correlated with active material loss and impedance growth. Meanwhile, the relaxation voltage profile, recorded after charging or discharging when the current is interrupted, reflects the dissipation of polarization and the restoration of thermodynamic equilibrium within the sodium-ion battery. Features from this curve, such as voltage extrema and relaxation rates, are indicative of internal resistance and thermodynamic state changes.
While studies have explored these features individually, relying on a single feature source can limit robustness and accuracy. IC curves can be sensitive to measurement noise and require specific low-rate charging data, while RV features might be influenced by temperature fluctuations and preceding operational history. This work proposes a novel feature fusion methodology that synergistically combines health indicators derived from both Incremental Capacity analysis and Relaxation Voltage dynamics. By fusing multi-dimensional features from the charge dynamics (IC) and the post-charge equilibrium process (RV), the proposed method aims to create a more comprehensive, sensitive, and noise-resistant health indicator for sodium-ion batteries. A sophisticated machine learning model based on a Long Short-Term Memory network integrated with an Attention mechanism (LSTM-Attention) is then employed to learn the intricate mapping from this fused feature set to the SOH of the sodium-ion battery.
Fundamental Analysis Methods and Feature Engineering
The core of the proposed approach lies in the extraction and fusion of discriminative features from two distinct electrochemical signatures: the Incremental Capacity curve and the Relaxation Voltage profile. Each provides a unique lens into the internal state of the sodium-ion battery.
Incremental Capacity (IC) Analysis and Feature Extraction
Incremental Capacity Analysis is a powerful non-destructive diagnostic tool. For a constant-current (CC) charging process, the incremental capacity is computed as the derivative of the charged capacity ($Q$) with respect to the terminal voltage ($U$):
$$IC = \frac{dQ}{dU} = I \cdot \frac{dt}{dU}$$
where $I$ is the constant charging current. This transformation converts the subtle voltage plateaus in the $Q-U$ curve into distinct peaks in the $IC-U$ curve, each corresponding to major phase transition reactions within the electrode materials of the sodium-ion battery. The evolution of these peak characteristics serves as a fingerprint for degradation.
To obtain a clean and analyzable IC curve, data preprocessing is essential. The raw voltage-capacity data from constant-current charging segments are first interpolated to a uniform voltage interval (e.g., 1 mV) to ensure consistent differential calculation. Following this, Gaussian filtering is applied to suppress high-frequency noise while preserving the essential peak shape. The Gaussian filter kernel is defined by:
$$G(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$$
where $\mu$ is the mean and $\sigma$ controls the smoothing degree. This step is crucial for reliable feature extraction from the sodium-ion battery data.
For the sodium-ion battery under study, the IC curve typically exhibits two main peaks within the voltage window of 2.5 V to 3.9 V. Four key parameters are extracted from each peak to quantify its shape and position:
- Peak Height (H): The maximum $IC$ value of the peak, related to the intensity of the phase transition reaction.
- Peak Voltage (V_p): The voltage at which the peak maximum occurs, indicating the equilibrium potential of the redox reaction.
- Full Width at Half Maximum (FWHM): The width of the peak at half of its maximum height, reflecting the kinetics and uniformity of the reaction.
- Peak Area (A): The integrated area under the peak within the FWHM bounds, representing the reversible capacity associated with that phase transition.
Thus, for a sodium-ion battery with two IC peaks, an 8-dimensional feature vector is formed: $F_{IC} = [H_1, V_{p1}, W_1, A_1, H_2, V_{p2}, W_2, A_2]$. As the sodium-ion battery ages, these features exhibit predictable trends: peak heights diminish, peak voltages often shift (usually to higher voltages due to increased polarization), widths may broaden, and areas decrease, collectively signaling capacity loss and impedance rise.
It is critical to screen for and remove anomalous IC curves resulting from measurement artifacts or failed half-cycles before training. Common anomalies include severely shifted peaks, missing secondary peaks, spike noise, or monotonically increasing curves devoid of clear peaks.
Relaxation Voltage (RV) Analysis and Feature Extraction
When the charging current is interrupted, the terminal voltage of a sodium-ion battery does not instantly stabilize but gradually relaxes toward its equilibrium open-circuit voltage (OCV). This relaxation process, governed by the dissipation of concentration gradients (diffusion) and the discharge of the double-layer capacitance, contains valuable information about the internal state. The relaxation voltage curve after a full charge is particularly informative, as it starts from a state of maximum polarization.
Key features are extracted from the post-CC-charge relaxation voltage curve recorded during a 20-minute rest period:
- Maximum Relaxation Voltage ($V_{max}$): The instantaneous voltage at the moment charging stops. This value is influenced by the ohmic voltage drop and the state of charge.
- Minimum Equilibrium Voltage ($V_{min}$): The stabilized voltage near the end of the relaxation period, approximating the OCV at a high state of charge. Its decrease over cycles is linked to thermodynamic changes in electrode materials.
- Maximum Relaxation Slope ($S_{max}$): The maximum absolute value of the voltage derivative ($|dV/dt|$) during the initial, rapid relaxation phase. This slope is predominantly related to the ohmic resistance of the sodium-ion battery.
These three parameters form a concise 3-dimensional feature vector: $F_{RV} = [V_{max}, V_{min}, S_{max}]$. Aging in a sodium-ion battery typically leads to a decrease in $V_{min}$, an increase in $S_{max}$ (due to growing internal resistance), and altered $V_{max}$ dynamics.
Feature Fusion and Health Indicator Formation
The core innovation of this study is the fusion of the $F_{IC}$ and $F_{RV}$ feature vectors. The combined feature set provides a multi-faceted view of sodium-ion battery health, capturing information from both the dynamic intercalation process (IC) and the static recovery process (RV). This fusion enhances the sensitivity and robustness of the health indicator, as the weaknesses of one feature set (e.g., IC’s sensitivity to noise) can be compensated by the strengths of the other (e.g., RV’s relative stability). The final fused health feature vector $F_{Fused}$ is an 11-dimensional vector:
$$F_{Fused} = F_{IC} \oplus F_{RV} = [H_1, V_{p1}, W_1, A_1, H_2, V_{p2}, W_2, A_2, V_{max}, V_{min}, S_{max}]$$
where $\oplus$ denotes the concatenation operation. This comprehensive feature vector serves as the input to the machine learning model for SOH estimation.
The LSTM-Attention Estimation Model
To model the complex temporal dependencies and the relative importance of different features across cycles, a neural network architecture combining Long Short-Term Memory (LSTM) and an Attention mechanism is developed. LSTM networks are a type of Recurrent Neural Network (RNN) designed to learn long-term dependencies in sequence data, making them ideal for time-series data like battery aging cycles.
An LSTM unit employs a cell state $C_t$ and three gates to regulate information flow:
- Forget Gate ($f_t$): Decides what information to discard from the previous cell state.
$$f_t = \sigma(W_f \cdot [h_{t-1}, x_t] + b_f)$$ - Input Gate ($i_t$): Decides what new information to store in the cell state. It uses a sigmoid layer and a tanh layer to create candidate values $\tilde{C}_t$.
$$i_t = \sigma(W_i \cdot [h_{t-1}, x_t] + b_i)$$
$$\tilde{C}_t = \tanh(W_C \cdot [h_{t-1}, x_t] + b_C)$$ - Cell State Update: The cell state is updated by combining the old state (filtered by the forget gate) and the new candidate values (filtered by the input gate).
$$C_t = f_t \odot C_{t-1} + i_t \odot \tilde{C}_t$$ - Output Gate ($o_t$): Decides what part of the cell state to output as the hidden state $h_t$.
$$o_t = \sigma(W_o \cdot [h_{t-1}, x_t] + b_o)$$
$$h_t = o_t \odot \tanh(C_t)$$
Here, $\sigma$ is the sigmoid function, $\odot$ is the Hadamard product, $x_t$ is the input at time step $t$ (the fused feature vector for cycle $t$), $h_{t-1}$ is the previous hidden state, and $W$ and $b$ are learnable weights and biases.
The Attention mechanism is integrated to allow the model to dynamically focus on the most relevant cycles or features when making a prediction for the current cycle. It computes a weighted sum of all previous hidden states $h_i$, where the weights $\alpha_i$ are learned by the network. For a simplified form of additive attention, the context vector $context$ fed to the final layers is computed as:
$$e_i = v^T \tanh(W_h h_i + b_h)$$
$$\alpha_i = \frac{\exp(e_i)}{\sum_{j=1}^{t} \exp(e_j)}$$
$$context = \sum_{i=1}^{t} \alpha_i h_i$$
The model architecture consists of an input layer for the 11-dimensional fused features, one or more LSTM layers to process the sequential data, an Attention layer to weight the temporal information, and finally fully connected (dense) layers to regress the estimated SOH value. The Mean Squared Error (MSE) between the true SOH and estimated SOH is used as the loss function for training:
$$MSE = \frac{1}{n} \sum_{i=1}^{n} (y_{true}^{(i)} – y_{pred}^{(i)})^2$$
The Adam optimizer is typically employed for efficient training.
Experimental Validation and Results Analysis
To validate the proposed method, aging tests were conducted on fourteen 18650 cylindrical sodium-ion batteries. The test specifications are summarized in the following table:
| Parameter | Value |
|---|---|
| Rated Capacity | 1.2 Ah |
| Voltage Window | 1.5 V – 4.0 V |
| Charge Cut-off Voltage | 3.95 V |
| Discharge Cut-off Voltage | 1.5 V |
| Charge/Discharge Rate | 1C (Constant Current) |
| Relaxation Period | 20 min |
| Ambient Temperature | 35°C |
The batteries underwent consecutive CC-CV charge and CC discharge cycles until significant capacity fade was observed. Data (voltage, current, capacity) was recorded at 1 Hz. The capacity from each discharge cycle was used to calculate the true SOH: $SOH = C_{current} / C_{initial}$.
Three distinct LSTM-Attention models were constructed and compared to evaluate the contribution of feature fusion:
- Model I (IC-only): Trained using only the 8-dimensional IC feature vector ($F_{IC}$).
- Model II (RV-only): Trained using only the 3-dimensional RV feature vector ($F_{RV}$).
- Model III (Fused): Trained using the complete 11-dimensional fused feature vector ($F_{Fused}$).
The hyperparameters for the neural networks were kept consistent across models for a fair comparison. A typical configuration is shown below:
| Hyperparameter | Value |
|---|---|
| Learning Rate | 0.001 |
| Batch Size | 72 |
| Training Epochs | 1000 |
| LSTM Hidden Units | 128 |
| Activation Function | tanh (LSTM), ReLU (Dense) |
| Optimizer | Adam |
| Train/Val/Test Split | ~5:1:1 (per battery) |
The performance of the SOH estimation models was rigorously evaluated using four standard metrics:
- Mean Absolute Error (MAE): $MAE = \frac{1}{n}\sum_{i=1}^{n} |y_{true}^{(i)} – y_{pred}^{(i)}|$
- Root Mean Square Error (RMSE): $RMSE = \sqrt{\frac{1}{n}\sum_{i=1}^{n} (y_{true}^{(i)} – y_{pred}^{(i)})^2}$
- Mean Absolute Percentage Error (MAPE): $MAPE = \frac{100\%}{n} \sum_{i=1}^{n} \left| \frac{y_{true}^{(i)} – y_{pred}^{(i)}}{y_{true}^{(i)}} \right|$
- Coefficient of Determination ($R^2$): $R^2 = 1 – \frac{\sum_{i=1}^{n}(y_{true}^{(i)} – y_{pred}^{(i)})^2}{\sum_{i=1}^{n}(y_{true}^{(i)} – \bar{y}_{true})^2}$
The results from the independent test set, which contained data from batteries not used during the model’s training phase, provide the most telling evidence of the model’s generalization capability. The performance comparison is summarized in the table below.
| Model | Input Features | MAE (%) | RMSE (%) | MAPE (%) | $R^2$ |
|---|---|---|---|---|---|
| Model I | IC Features Only | ~0.45 | ~0.58 | ~0.48 | >0.995 |
| Model II | RV Features Only | ~0.85 | ~1.10 | ~0.90 | ~0.980 |
| Model III (Proposed) | Fused IC & RV Features | ~0.40 | ~0.51 | ~0.42 | >0.998 |
The analysis leads to several key conclusions. First, Model I (IC-only) performs significantly better than Model II (RV-only), demonstrating that IC features are generally more powerful and direct indicators of capacity fade in sodium-ion batteries. However, Model II still provides a reasonable estimate, confirming that relaxation dynamics contain valuable health information.
Critically, the proposed Model III, which leverages the fused feature set, achieves the best performance across all error metrics. It reduces the MAE by approximately 12% compared to the best single-feature model (Model I). More importantly, its prediction curve shows superior smoothness and stability, especially during the later stages of aging where capacity fade can become more nonlinear and volatile. The high $R^2$ value (exceeding 0.998) indicates an almost perfect fit to the true capacity degradation trend. This improvement validates the hypothesis that combining features from the dynamic charging process and the static relaxation process creates a more robust and informative health indicator for sodium-ion batteries. The attention mechanism within the LSTM-Attention model successfully learns to weight historical cycle information effectively, further enhancing estimation accuracy.
Conclusion
This study presents a novel and highly effective data-driven framework for estimating the State of Health of sodium-ion batteries. The method is centered on the fusion of health-sensitive features extracted from two complementary data sources: Incremental Capacity (IC) curves and Relaxation Voltage (RV) profiles. The IC features capture the electrochemical phase transition dynamics during charging, while the RV features characterize the polarization recovery and internal resistance changes during rest. By concatenating these into an 11-dimensional feature vector, the model gains a multi-dimensional perspective on battery degradation that is more resilient to noise and operational variability than models based on either feature set alone.
The fused features are processed by a sophisticated LSTM-Attention neural network, which is adept at modeling long-term temporal dependencies in the aging data and focusing on the most relevant historical information for the current prediction. Experimental validation on multiple 18650 sodium-ion battery cells demonstrated the superior performance of the proposed feature-fusion model. It achieved an estimation accuracy with MAE below 0.40% and RMSE below 0.51%, outperforming models using only IC or RV features. The method operates using standard charging cycle data without requiring special test protocols, making it practical for real-world implementation in battery management systems.
This work contributes a reliable and accurate solution for SOH estimation, addressing a critical need for the health management and longevity assurance of sodium-ion battery systems as they move towards broader commercialization in electric vehicles and stationary storage. The feature fusion strategy and the LSTM-Attention model architecture provide a robust framework that can be adapted and extended as understanding of sodium-ion battery degradation mechanisms continues to deepen.