The proliferation of fast-charging technology for lithium-ion batteries has significantly enhanced user convenience, particularly in applications such as electric vehicles and portable electronics. However, the implementation of aggressive fast-charging protocols, especially multi-stage constant-current strategies, accelerates battery degradation mechanisms like lithium plating and solid electrolyte interphase (SEI) layer growth, critically impacting the battery’s State of Health (SOH). Accurate and robust SOH estimation under these dynamic conditions is paramount for ensuring safety, optimizing charging strategies, and prolonging the service life of lithium-ion battery systems.
Conventional SOH estimation methods often struggle in fast-charging scenarios. Model-based approaches, including electrochemical models (EM) and equivalent circuit models (ECM) coupled with filters like Kalman filters, require precise parameterization and deep physical insight, limiting their generalization under diverse and rapidly changing fast-charge conditions. Data-driven methods, while more flexible, face challenges related to computational cost for feature extraction, reliance on complete charging data, and vulnerability to missing input features. Many existing methods primarily extract health indicators from the charging phase, overlooking the valuable degradation information embedded within discharge voltage profiles.
To address these limitations, this work proposes a novel SOH estimation framework for lithium-ion batteries subjected to multi-stage constant-current fast charging. The method leverages a simple yet effective health indicator derived from both charging and discharging voltage curves. Furthermore, a Multi-Task Learning (MTL) architecture based on a shared Long Short-Term Memory (LSTM) network is employed to enhance estimation robustness, particularly when one of the health features is unavailable. Validation on a large, public fast-charging dataset demonstrates the high accuracy and resilience of the proposed approach.
Health Feature Extraction: Equal Voltage Range Sampling Count (EVRSCN)
To mitigate computational burden and reduce noise sensitivity, we extract a health feature from the constant-current segments of both fast-charging and discharging voltage curves. The chosen feature is the Equal Voltage Range Sampling Count Number (EVRSCN).
Consider a constant-current (CC) phase with current $I$. The capacity increment $\Delta Q$ within a specific voltage window $\Delta U$ can be expressed using the ampere-hour integration method:
$$
\Delta Q = \int_{t_1}^{t_2} I \, dt = I (t_2 – t_1)
$$
where $t_1$ and $t_2$ are the start and end times corresponding to the voltage increase $\Delta U$. The time elapsed is $\Delta T = t_2 – t_1$.
If the Battery Management System (BMS) samples at a fixed time interval $\Delta t$, the number of sampling points within $\Delta U$, defined as $EVRSCN_{\Delta U}$, is:
$$
EVRSCN_{\Delta U} = \frac{\Delta T}{\Delta t}
$$
Substituting $\Delta T = \Delta Q / I$ yields:
$$
EVRSCN_{\Delta U} = \frac{\Delta Q}{I \Delta t}
$$
For a given CC phase, both $I$ and $\Delta t$ are constant. Therefore, a direct proportionality exists:
$$
EVRSCN_{\Delta U} \propto \Delta Q
$$
This establishes that $EVRSCN$ within a fixed voltage window shares a mathematical essence with capacity increment, making it a sensitive health indicator for capacity fade. In fast-charging protocols, we extract two separate $EVRSCN$ features: one from the initial high-rate constant-current charging segment ($EVRSCN_{chg}$) and another from the entire constant-current discharge segment ($EVRSCN_{dis}$). The specific voltage windows ($\Delta U_{chg}$, $\Delta U_{dis}$) are selected via a sliding window search to maximize correlation with capacity fade. This dual-feature approach comprehensively captures degradation signatures from both charge and discharge dynamics of the lithium-ion battery.
Multi-Task Learning Framework with Shared LSTM
While using both $EVRSCN_{chg}$ and $EVRSCN_{dis}$ as inputs can improve estimation accuracy, it introduces a vulnerability: the model’s performance degrades if one feature is missing due to sensor failure or data acquisition issues. To build a robust estimator, we formulate the problem within a Multi-Task Learning (MTL) paradigm.
The core idea is to train a single model to perform two related tasks jointly: the primary task of SOH estimation and a secondary task of health feature prediction. The model learns shared representations that are beneficial for both tasks, improving generalization and robustness.
The proposed network architecture is as follows:
- Input Layer: Accepts the sequence of the two health features $[EVRSCN_{chg}(k), EVRSCN_{dis}(k)]$ over cycle index $k$.
- Shared LSTM Layer: A common LSTM layer processes the input sequence to capture the temporal dynamics and interdependencies between the charging and discharging features. The LSTM cell mechanics are governed by:
$$
\begin{aligned}
f_t &= \sigma(W_f \cdot [h_{t-1}, x_t] + b_f) \\
i_t &= \sigma(W_i \cdot [h_{t-1}, x_t] + b_i) \\
\tilde{C}_t &= \tanh(W_C \cdot [h_{t-1}, x_t] + b_C) \\
C_t &= f_t \odot C_{t-1} + i_t \odot \tilde{C}_t \\
o_t &= \sigma(W_o \cdot [h_{t-1}, x_t] + b_o) \\
h_t &= o_t \odot \tanh(C_t)
\end{aligned}
$$
where $\sigma$ is the sigmoid function, $\odot$ denotes element-wise multiplication, $x_t$ is the input vector at step $t$, $h_t$ is the hidden state, $C_t$ is the cell state, and $W_*, b_*$ are weights and biases. - Task-Specific Fully Connected (FC) Layers: The shared LSTM features are fed into two separate FC layers.
- FC_SOH: Dedicated to the primary task, it outputs the estimated SOH value $\widehat{SOH}(k)$.
- FC_Feature: Dedicated to the secondary task, it outputs a predicted value for a potentially missing health feature (e.g., $\widehat{EVRSCN}_{chg}(k)$ if the charging feature is absent).
- Output Logic:
- When both health features are present, the model uses the actual features to estimate SOH directly via the FC_SOH layer.
- When one feature is missing (e.g., $EVRSCN_{chg}$), the model first uses the FC_Feature layer to predict it from the available feature ($EVRSCN_{dis}$) and the shared history. The concatenation of the predicted feature and the available feature is then passed to the FC_SOH layer for final SOH estimation.
The model is trained end-to-end by minimizing a composite loss function $\mathcal{L}_{total}$ that combines the errors from both tasks:
$$
\mathcal{L}_{total} = \alpha \cdot \mathcal{L}_{SOH} + \beta \cdot \mathcal{L}_{HI}
$$
where $\mathcal{L}_{SOH}$ is the loss for SOH estimation (e.g., Mean Squared Error), $\mathcal{L}_{HI}$ is the loss for health feature prediction, and $\alpha$, $\beta$ are weighting coefficients. This joint optimization forces the shared LSTM layer to learn representations that are generically useful for understanding the underlying degradation process of the lithium-ion battery, thereby enhancing robustness.
Experimental Validation and Results
The proposed method was validated using a public dataset (MIT-Stanford) comprising A123 APR18650M1A lithium-ion batteries cycled under various multi-stage fast-charging protocols at 30°C. Both one-step and two-step constant-current fast charges (3.6C to 7C to 80% SOC) were employed, followed by a CC-CV saturation charge. A constant 4C discharge was used. A leave-one-battery-out cross-validation scheme was adopted for evaluation.
The performance was assessed using Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and the Coefficient of Determination (R²):
$$
\begin{aligned}
RMSE &= \sqrt{\frac{1}{N} \sum_{i=1}^{N} (y_i – \hat{y}_i)^2} \\
MAE &= \frac{1}{N} \sum_{i=1}^{N} |y_i – \hat{y}_i| \\
R^2 &= 1 – \frac{\sum_{i=1}^{N} (y_i – \hat{y}_i)^2}{\sum_{i=1}^{N} (y_i – \bar{y})^2}
\end{aligned}
$$
where $y_i$ is the true SOH, $\hat{y}_i$ is the estimated SOH, $\bar{y}$ is the mean of true SOH, and $N$ is the number of samples.
Estimation with Complete Health Features
When both $EVRSCN_{chg}$ and $EVRSCN_{dis}$ features were available, the proposed LSTM-MTL model achieved highly accurate SOH estimation across different fast-charging protocols. The results for selected cells are summarized below:
| Cell ID & Protocol | RMSE (%) | MAE (%) | R² |
|---|---|---|---|
| No.91 (One-step) | 0.39 | 0.35 | 0.9906 |
| No.124 (One-step) | 0.68 | 0.48 | 0.9813 |
| Cell.85 (Two-step) | 0.53 | 0.49 | 0.9857 |
| Cell.92 (Two-step) | 0.49 | 0.64 | 0.9890 |
The estimation errors (RMSE, MAE) are consistently below 1%, and the R² values exceed 0.98, indicating the model’s excellent ability to track the capacity fade trajectory of the lithium-ion battery under fast-charging stress.
Estimation with Missing Health Features
To test robustness, we simulated scenarios where either the charging or discharging health feature was completely missing for all cycles of a test battery. The model successfully predicted the missing feature using its MTL architecture and proceeded with SOH estimation.
| Cell ID & Missing Feature | RMSE (%) | MAE (%) | R² |
|---|---|---|---|
| No.91 (Missing $EVRSCN_{chg}$) | 0.43 | 0.39 | 0.9883 |
| No.124 (Missing $EVRSCN_{chg}$) | 0.78 | 0.59 | 0.9801 |
| Cell.85 (Missing $EVRSCN_{dis}$) | 0.51 | 0.67 | 0.9843 |
| Cell.92 (Missing $EVRSCN_{dis}$) | 0.68 | 0.77 | 0.9868 |
The performance degradation is minimal. RMSE and MAE remain under 1%, and R² stays above 0.98. This demonstrates the significant robustness advantage conferred by the MTL framework. The shared LSTM layer learns the intrinsic relationship between the two features and the underlying SOH, enabling reliable estimation even with partial input.
Comparative Analysis
The proposed LSTM-MTL model was compared against other popular machine learning models: Support Vector Machine (SVM), Gaussian Process Regression (GPR), and Random Forest (RF). The comparison focuses on RMSE under the challenging condition of missing health features.
| Model | No.91 (Missing Feature) RMSE (%) | No.124 (Missing Feature) RMSE (%) | Cell.85 (Missing Feature) RMSE (%) | Cell.92 (Missing Feature) RMSE (%) |
|---|---|---|---|---|
| SVM | 1.67 | 2.33 | 1.76 | 1.23 |
| GPR | 1.22 | 1.75 | 2.02 | 1.88 |
| RF | 1.64 | 2.36 | 2.46 | 1.96 |
| Proposed LSTM-MTL | 0.43 | 0.78 | 0.51 | 0.68 |
The traditional models require a separate, pre-trained model to impute the missing feature before SOH estimation, which introduces cascading errors and increases complexity. In contrast, the proposed LSTM-MTL model handles feature prediction and SOH estimation in a unified, jointly optimized framework, resulting in significantly lower error and demonstrating superior robustness for lithium-ion battery SOH monitoring.
Conclusion
This paper presents a robust and accurate SOH estimation method for lithium-ion batteries under multi-stage constant-current fast-charging conditions. The method innovatively combines a computationally efficient dual health feature ($EVRSCN$) extracted from both charge and discharge voltage curves with a Multi-Task Learning framework based on a shared LSTM network.
The key advantages are:
1. Low-Cost Feature Extraction: The $EVRSCN$ feature is simple to compute from BMS voltage data, minimizing preprocessing overhead.
2. Comprehensive Degradation Capture: Utilizing both charging and discharging profiles provides a more holistic view of battery health.
3. Enhanced Robustness: The MTL architecture enables reliable SOH estimation even when one health feature is completely missing, a critical capability for real-world applications where sensor data may be incomplete.
4. High Accuracy: The model achieves estimation errors (RMSE, MAE) below 1% and R² above 0.98 on a demanding fast-charging dataset.
The proposed approach offers a practical and effective solution for battery management systems in fast-charging applications, contributing to the safer and more durable use of lithium-ion battery technology. Future work will focus on extending the model’s capability to handle scenarios with partially available voltage data within a cycle, further increasing its applicability in real-world edge computing environments.