The global transition towards renewable energy systems has created an urgent demand for large-scale, cost-effective, and safe energy storage technologies. While lithium-ion batteries have dominated the landscape due to their high energy density, concerns regarding lithium resource scarcity and associated cost volatility are becoming increasingly prominent. In this context, sodium-ion battery technology emerges as a highly promising alternative, leveraging the natural abundance, low cost, and widespread geographical distribution of sodium. The performance and longevity of any electrochemical storage system, including the sodium-ion battery, are intrinsically linked to its internal degradation over time. Therefore, accurate and reliable estimation of the State of Health (SOH) is paramount for ensuring operational safety, optimizing performance, and facilitating predictive maintenance within Battery Management Systems (BMS).
My research focuses on addressing the critical challenge of SOH estimation for sodium-ion battery systems. Unlike lithium-ion counterparts, the study of degradation modeling and state estimation for sodium-ion battery cells is still in a nascent stage. The electrochemical degradation mechanisms, while sharing some similarities, exhibit distinct characteristics due to the larger ionic radius and different reaction kinetics of sodium ions. This necessitates the development of tailored estimation methodologies. This article presents a novel data-driven framework that synergistically combines a Recurrent Neural Network (RNN) with an Extended Kalman Filter (EKF) to achieve precise and robust SOH estimation for sodium-ion battery cells, validated through experimental cycle life data.
Defining State of Health for Batteries
The State of Health is a metric that quantifies the current condition of a battery relative to its fresh state. It is a dimensionless parameter, typically expressed as a percentage, where 100% represents a pristine battery and 0% indicates end-of-life. For a sodium-ion battery, as with other chemistries, SOH can be defined through several measurable parameters that degrade with use:
- Capacity-Based SOH (SOHC): This is the most common definition, reflecting the loss of charge storage capability.
$$ SOH_C = \frac{Q_{current}}{Q_{rated}} \times 100\% $$
where $Q_{current}$ is the maximum available capacity at the current cycle and $Q_{rated}$ is the nominal or initial rated capacity. - Resistance-Based SOH (SOHR): This definition captures the increase in internal impedance, which reduces power capability.
$$ SOH_R = \frac{R_{fresh}}{R_{current}} \times 100\% $$
where $R_{fresh}$ is the internal resistance at beginning-of-life and $R_{current}$ is the resistance at the current state. - Voltage-Based SOH: This can relate to changes in open-circuit voltage characteristics under specific conditions.
In my work, I adopt the capacity-based definition ($SOH_C$) as the primary metric for the sodium-ion battery. It is a direct indicator of runtime and energy content, and it can be correlated with underlying degradation modes. Accurate estimation of this SOH is crucial for predicting remaining range in applications and scheduling battery replacement.
Proposed Methodology: An RNN-EKF Fusion Framework
The core of my approach is a fusion architecture that leverages the temporal pattern recognition strength of Recurrent Neural Networks and the optimal state estimation capability of the Extended Kalman Filter. The goal is to estimate the sequence of SOH values over the lifetime of a sodium-ion battery using readily measurable operational data.
Feature Engineering for the Sodium-Ion Battery
Effective feature selection is critical for any data-driven model. For the sodium-ion battery SOH estimation, I identify and utilize five key features that are easily accessible from standard BMS measurements during cycling and are highly indicative of aging:
- Cycle Number (N): A direct temporal index of usage and cumulative stress.
- Discharge Capacity (Qdis): The actual delivered capacity in each cycle, which inherently carries SOH information but is the target for estimation.
- Discharge Median Voltage (Vmed): The median voltage during the discharge phase. A shift in this value often correlates with changes in internal impedance and electrode polarization.
- Discharge Starting Voltage (Vstart): The voltage at the beginning of the constant-current discharge. This can be influenced by the internal resistance and the state of charge reset point.
- Discharge Ending Voltage (Vend): The voltage at the termination of discharge. Its evolution over cycles can indicate changes in the dischargeable voltage window and kinetic limitations.
These features, $X_t = [N_t, Q_{dis,t}, V_{med,t}, V_{start,t}, V_{end,t}]$, form the input vector for the model at cycle $t$. They provide a compact yet informative snapshot of the sodium-ion battery‘s operational state and its history.
Recurrent Neural Network for Temporal Modeling
RNNs are a class of neural networks designed to handle sequential data by maintaining an internal state or memory. This makes them ideal for learning the temporal degradation patterns of a sodium-ion battery. At each cycle step $t$, the network processes the input feature vector $X_t$ and updates its hidden state $h_t$, which contains compressed information from all previous cycles.
The fundamental equations for a simple RNN cell are:
$$ h_t = \tanh(W_{xh} X_t + W_{hh} h_{t-1} + b_h) $$
$$ y_t = W_{hy} h_t + b_y $$
where:
- $h_t$ is the hidden state vector at time $t$,
- $X_t$ is the input feature vector at time $t$,
- $y_t$ is the output vector at time $t$ (the preliminary SOH estimate),
- $W_{xh}$, $W_{hh}$, $W_{hy}$ are weight matrices,
- $b_h$, $b_y$ are bias vectors,
- $\tanh(\cdot)$ is the hyperbolic tangent activation function.
The recursive connection via $W_{hh} h_{t-1}$ allows the network to learn dependencies across time, effectively modeling the progressive capacity fade of the sodium-ion battery. In practice, more advanced cells like Long Short-Term Memory (LSTM) or Gated Recurrent Units (GRU) are used to mitigate the vanishing gradient problem and capture long-term dependencies, which are essential for forecasting the entire aging trajectory.
Extended Kalman Filter for Robust State Estimation
While the RNN provides a powerful non-linear mapping from features to SOH, its point estimates can be sensitive to noise and model uncertainties. To enhance robustness, I employ an Extended Kalman Filter as a post-processing smoother. The EKF is a celebrated algorithm for state estimation in non-linear dynamic systems. It operates in two steps: prediction and update, providing an optimal (in the minimum mean-square error sense) estimate of the true state by fusing a dynamic model’s prediction with noisy measurements.
I formulate the SOH estimation as a state-space problem:
- State Transition (Process Model): I model the SOH evolution as a simple linear process with noise.
$$ SOH_{k+1} = A \cdot SOH_k + w_k $$
where $SOH_k$ is the state of health at step $k$, $A$ is the state transition coefficient (often close to 1, assuming gradual degradation), and $w_k$ is the process noise, assumed to be zero-mean Gaussian with covariance $Q$. - Measurement Model: The measurement is the RNN’s preliminary SOH estimate, $\widehat{SOH}^{RNN}_k$.
$$ \widehat{SOH}^{RNN}_k = SOH_k + v_k $$
where $v_k$ is the measurement noise, assumed to be zero-mean Gaussian with covariance $R$.
The EKF algorithm then proceeds as follows for each cycle $k$:
- Prediction:
$$ \widehat{SOH}^-_{k+1} = A \cdot \widehat{SOH}_k $$
$$ P^-_{k+1} = A P_k A^T + Q $$
where $\widehat{SOH}^-_{k+1}$ is the a priori state estimate and $P^-_{k+1}$ is the a priori error covariance. - Update (Correction):
$$ K_{k+1} = P^-_{k+1} (P^-_{k+1} + R)^{-1} $$
$$ \widehat{SOH}_{k+1} = \widehat{SOH}^-_{k+1} + K_{k+1} (\widehat{SOH}^{RNN}_{k+1} – \widehat{SOH}^-_{k+1}) $$
$$ P_{k+1} = (I – K_{k+1}) P^-_{k+1} $$
where $K_{k+1}$ is the Kalman Gain, $\widehat{SOH}_{k+1}$ is the final a posteriori SOH estimate (the output of the RNN-EKF framework), and $P_{k+1}$ is the updated error covariance.
The Kalman Gain $K$ optimally balances the trust between the RNN’s “measurement” and the model’s prediction. This filtering step effectively smooths the RNN output, reducing high-frequency estimation jitter and providing a more physically plausible and stable SOH trajectory for the sodium-ion battery.
Integrated Framework Architecture
The complete estimation framework operates in a cohesive manner:
- Data Preprocessing: The raw cycling data (voltage, current, capacity) from the sodium-ion battery is processed to extract the five key feature sequences and the target SOH sequence based on capacity fade.
- RNN Training & Prediction: The feature sequences from the early-life cycles are used to train the RNN model to predict SOH. Once trained, the RNN processes the feature sequence for the entire life or unseen data to generate a preliminary SOH estimate sequence $\{\widehat{SOH}^{RNN}_k\}$.
- EKF Smoothing: The sequence $\{\widehat{SOH}^{RNN}_k\}$ is treated as the measurement input to the EKF. The EKF then processes this sequence cycle-by-cycle according to the equations above, producing the final, refined SOH estimates $\{\widehat{SOH}_k\}$.
This fusion leverages the RNN’s ability to learn complex, non-linear aging patterns from data and the EKF’s strength in providing statistically optimal, smooth state estimates in the presence of uncertainty.
Experimental Validation on Sodium-Ion Batteries
To validate the proposed RNN-EKF framework, I conducted experiments using cycling data from laboratory-fabricated sodium-ion battery cells. Three different CR2032 coin cells were subjected to repeated charge-discharge cycles under controlled conditions until significant degradation occurred.
Test Setup and Cell Specifications
The three sodium-ion battery cells were designed with variations in cathode material and cycling rate to evaluate the method’s robustness under different conditions. All testing was performed in a constant-temperature environment.
| Cell Identifier | Cathode Material | Charge/Discharge Rate (C-rate) | Voltage Range (V) | Total Cycles Recorded |
|---|---|---|---|---|
| Cell-1 | Iron-based Compound | 5 C | 1.5 – 4.1 | 500 |
| Cell-2 | Iron-based Compound | 1 C | 1.5 – 4.1 | 500 |
| Cell-3 | Lithium Fluoride-based Compound | 5 C | 1.5 – 4.1 | 500 |
For each cell, the first 200 cycles were designated as the training dataset. The model was trained to learn the mapping from the five features to the SOH within this initial period. The subsequent 300 cycles (cycle 201 to 500) were used as the test set to evaluate the model’s prediction performance on unseen, future degradation.
Evaluation Metrics
The performance of the SOH estimation framework was quantitatively assessed using three standard metrics:
- Root Mean Square Error (RMSE): Measures the standard deviation of the estimation errors.
$$ RMSE = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (\widehat{SOH}_i – SOH_i^{true})^2} $$ - Mean Absolute Error (MAE): Measures the average magnitude of the errors.
$$ MAE = \frac{1}{n} \sum_{i=1}^{n} | \widehat{SOH}_i – SOH_i^{true} | $$ - Coefficient of Determination (R2): Represents the proportion of variance in the true SOH that is predictable from the estimate. A value closer to 1 indicates a better fit.
$$ R^2 = 1 – \frac{\sum_{i=1}^{n} (\widehat{SOH}_i – SOH_i^{true})^2}{\sum_{i=1}^{n} (SOH_i^{true} – \bar{SOH}^{true})^2} $$
where $n$ is the number of test cycles, $\widehat{SOH}_i$ is the estimated value, $SOH_i^{true}$ is the true capacity-based SOH, and $\bar{SOH}^{true}$ is the mean of the true SOH values.
Results and Comparative Analysis
The proposed RNN-EKF framework was compared against several other prominent data-driven models commonly used for time-series prediction, including Gated Recurrent Unit (GRU), Long Short-Term Memory (LSTM), and their bidirectional variants (BiGRU, BiLSTM). This comparison highlights the effectiveness of the EKF fusion step.
The estimation results for the three sodium-ion battery cells are summarized below. The tables clearly show the superior performance of the RNN-EKF method across all metrics and for all cell types.
| Algorithm | RMSE (%) | MAE (%) | R2 (%) |
|---|---|---|---|
| GRU | 30.41 | 26.34 | 0.00 |
| LSTM | 40.64 | 36.05 | 0.00 |
| BiGRU | 26.88 | 23.11 | 0.00 |
| BiLSTM | 26.98 | 24.26 | 0.00 |
| RNN | 2.63 | 2.29 | 87.47 |
| RNN-EKF (Proposed) | 1.80 | 1.39 | 94.13 |
| Algorithm | RMSE (%) | MAE (%) | R2 (%) |
|---|---|---|---|
| GRU | 5.88 | 5.22 | 38.06 |
| LSTM | 16.66 | 15.24 | 0.00 |
| BiGRU | 5.46 | 4.87 | 46.67 |
| BiLSTM | 11.03 | 10.29 | 0.00 |
| RNN | 1.41 | 1.16 | 96.46 |
| RNN-EKF (Proposed) | 1.31 | 1.07 | 96.91 |
| Algorithm | RMSE (%) | MAE (%) | R2 (%) |
|---|---|---|---|
| GRU | 9.43 | 15.33 | 0.00 |
| LSTM | 31.44 | 26.75 | 0.00 |
| BiGRU | 28.93 | 23.97 | 0.00 |
| BiLSTM | 37.75 | 33.86 | 0.00 |
| RNN | 2.56 | 2.15 | 77.34 |
| RNN-EKF (Proposed) | 0.80 | 0.70 | 97.77 |
Discussion of Results
The results demonstrate the outstanding efficacy of the proposed RNN-EKF framework for sodium-ion battery SOH estimation. Several key observations can be made:
- Superior Accuracy and Robustness: The RNN-EKF method consistently achieved the lowest RMSE and MAE across all three cells with different chemistries and cycling conditions. The average RMSE was approximately 1.30%, and the average MAE was about 1.05%. Most notably, it achieved an R2 fit as high as 97.77%, indicating that the estimated SOH curve explains almost all the variance in the true degradation data of the sodium-ion battery.
- Effectiveness of EKF Fusion: Comparing the standalone RNN results with the RNN-EKF results clearly shows the benefit of the Kalman filter. For instance, for Cell-1, the RNN’s RMSE was reduced from 2.63% to 1.80% after EKF smoothing, and the R2 improved from 87.47% to 94.13%. The EKF effectively acts as a denoiser and stabilizer, producing a smoother, more reliable estimate that is less susceptible to transient fluctuations in the RNN’s raw output.
- Generalizability Across Variations: The model performed well on both iron-based and lithium fluoride-based cathodes, and at both high (5C) and low (1C) discharge rates. This suggests that the selected feature set (cycle number, capacity, median, start, and end voltages) captures fundamental aging signatures that are transferable across different sodium-ion battery configurations, making the approach versatile.
- Comparison with Other Models: The poor performance of the other neural network models (GRU, LSTM, etc.) in their standalone form, as indicated by high errors and near-zero R2 in many cases, underscores the challenge of directly mapping features to SOH without careful architecture tuning or post-processing. It highlights that a simple RNN, when appropriately coupled with an optimal filter like the EKF, can form a highly effective and efficient solution for this specific sodium-ion battery estimation problem.
Conclusion and Future Perspectives
In this work, I have presented a novel and effective data-driven framework for estimating the State of Health of sodium-ion battery cells. The proposed method intelligently fuses a Recurrent Neural Network with an Extended Kalman Filter. The RNN learns the complex, non-linear temporal relationship between easily accessible operational features (cycle number, discharge capacity, median, start, and end voltages) and the battery’s capacity-based SOH. The EKF then refines the RNN’s sequential predictions, providing an optimal, smoothed estimate that is robust to measurement noise and model uncertainties.
Experimental validation on three distinct sodium-ion battery cells demonstrated the framework’s excellent performance, with an average RMSE of approximately 1.30%, an average MAE of about 1.05%, and an average R2 value exceeding 96.28%. These results confirm the method’s accuracy, robustness, and generalizability across different cell materials and cycling conditions.
This research contributes a practical and efficient tool for sodium-ion battery management, enabling more reliable assessment of battery condition, which is crucial for safety, longevity, and economic operation in applications ranging from grid storage to electric vehicles. For future work, I plan to investigate the integration of this SOH estimator with other state estimators, such as State of Charge (SOC), within a unified BMS framework. Furthermore, exploring the potential of this method for forecasting the Remaining Useful Life (RUL) of sodium-ion battery packs and adapting it to work with partial charge/discharge data from real-world operational profiles will be essential steps towards practical deployment.