Accurate estimation of the State of Health (SOH) is a cornerstone of advanced Battery Management Systems (BMS) for lithium-ion batteries. Precise SOH knowledge is paramount for ensuring the reliable operation, safety, and longevity of energy storage systems, ranging from electric vehicles to grid-scale applications. The degradation of li ion battery cells, characterized by capacity fade and power loss, is an inevitable consequence of electrochemical and mechanical stress during charge-discharge cycling. Therefore, developing robust, data-driven methods for precise SOH estimation remains a critical research frontier.
Data-driven methodologies have emerged as powerful tools for li ion battery SOH estimation, circumventing the need for complex and often incomplete electrochemical models. These methods learn the intricate relationship between measurable battery operational data—such as voltage, current, and temperature profiles—and the underlying degradation state. The predictive performance of such models hinges on two key factors: the relevance and quality of the extracted health features (HFs) and the capacity of the predictive algorithm to capture complex, non-linear temporal dependencies within the data.
In this work, we propose a novel data-driven framework that synergizes carefully engineered statistical health features with an advanced neural network architecture, the Convolutional Fastformer, to achieve highly accurate and stable SOH estimation for li ion batteries. Our approach begins with a detailed analysis of the charging process, from which we derive statistically meaningful features. Subsequently, we employ a model that combines the local feature extraction prowess of Convolutional Neural Networks (CNNs) with the efficient global sequence modeling capability of the Fastformer, an attention-based model with linear complexity. To efficiently determine the optimal configuration for our model, we utilize an orthogonal experimental design. We validate the proposed method using a widely recognized public dataset, benchmarking its performance against several established neural network models.
Methodology
1. Charging Data Analysis and Health Feature Extraction
The foundation of an effective data-driven model lies in the selection of informative inputs. For li ion battery applications, the charging process is often more controlled and reproducible than discharging, which is highly dependent on load profiles. Consequently, we focus on extracting health indicators from the charging voltage curves. Our analysis utilizes data from a public dataset comprising 124 commercial LiFePO4/graphite cells cycled to failure. We specifically analyze cells undergoing a multi-stage Constant Current (CC) and Constant Voltage (CV) charging protocol, a common strategy to balance fast charging and battery health.
The charging protocol can be described as a sequence: CC1(Q1)-CC2(Q2)-CC3-CV, where CC1, CC2, and CC3 are distinct constant current rates, Q1 and Q2 are the State-of-Charge (SOC) points where the current rate switches, and CV is the final constant voltage phase. As the li ion battery degrades over cycles, the shapes of the voltage curves during these stages shift characteristically. For instance, the terminal voltage at the end of the CC1 stage tends to increase with cycle count, while the current during the CV phase decays at different rates. These observable changes encode information about the internal degradation mechanisms.
To quantify these morphological changes, we extract statistical features from the voltage-time series of each CC stage and the current-time series of the CV stage for every charge cycle. For a given time-series segment X (e.g., CC1 voltage), we compute six statistical descriptors:
- Mean: $$\mu_X = \frac{1}{N}\sum_{i=1}^{N} x_i$$
- Variance: $$\sigma_X^2 = \frac{1}{N}\sum_{i=1}^{N} (x_i – \mu_X)^2$$
- Skewness: $$\gamma_X = \frac{\frac{1}{N}\sum_{i=1}^{N} (x_i – \mu_X)^3}{\sigma_X^3}$$
- Kurtosis: $$\kappa_X = \frac{\frac{1}{N}\sum_{i=1}^{N} (x_i – \mu_X)^4}{\sigma_X^4}$$
- Maximum Value: $$X_{max} = \max(\{x_1, x_2, …, x_N\})$$
- Minimum Value: $$X_{min} = \min(\{x_1, x_2, …, x_N\})$$
Not all extracted features are equally informative for predicting the capacity of a li ion battery. To select the most relevant features and reduce dimensionality—thereby mitigating the risk of overfitting—we calculate the Pearson correlation coefficient between each candidate feature sequence and the measured capacity sequence. The Pearson coefficient $\rho$ for a feature $x_j$ and capacity $y$ is given by:
$$
\rho_{x_j, y} = \frac{\sum_{i=1}^{M} (x_j^{(i)} – \bar{x}_j)(y^{(i)} – \bar{y})}{\sqrt{\sum_{i=1}^{M} (x_j^{(i)} – \bar{x}_j)^2 \sum_{i=1}^{M} (y^{(i)} – \bar{y})^2}}
$$
where $M$ is the number of cycles, and $\bar{x}_j$ and $\bar{y}$ are the mean values. Features with $|\rho|$ close to 1 exhibit a strong linear relationship with capacity. Our analysis revealed that three voltage-based features demonstrated exceptionally high correlation: the maximum voltage in the CC1 stage, the minimum voltage in the CC2 stage, and the maximum voltage in the CC3 stage. These three features were selected as the final health indicator vector for our model, effectively capturing the essence of the li ion battery’s aging from the charging data.
2. The Convolutional Fastformer Model Architecture
Recurrent Neural Networks (RNNs) and their variants like Long Short-Term Memory (LSTM) networks have been widely used for sequential data modeling. However, they can struggle with long-term dependencies due to their sequential processing nature. The Transformer architecture, based solely on attention mechanisms, overcame this by enabling parallel computation and global context access. Yet, its self-attention mechanism has quadratic complexity with respect to sequence length, making it computationally expensive for long sequences.
Our proposed model addresses these limitations by integrating a one-dimensional Convolutional layer with a Fastformer block. The architecture offers two key advantages:
- Local Feature Enhancement: A 1D convolutional layer applied to the input sequence of health features (from the previous $L$ cycles) excels at extracting local patterns and dependencies within the short-term context of the li ion battery’s aging trajectory.
- Efficient Global Context Modeling: The Fastformer core efficiently models global interactions across the entire sequence of $L$ cycles with linear complexity, effectively summarizing the long-term degradation trend.
The model workflow is as follows:
- Input and Convolution: The input is a matrix $\mathbf{X} = [\mathbf{h}_1, \mathbf{h}_2, …, \mathbf{h}_L] \in \mathbb{R}^{L \times 3}$, where $\mathbf{h}_t$ is the 3-dimensional health feature vector (CC1_Vmax, CC2_Vmin, CC3_Vmax) at cycle $t$. This matrix is processed by three separate 1D convolutional layers with kernel size $k$ to generate the Query ($\mathbf{Q}$), Key ($\mathbf{K}$), and Value ($\mathbf{V}$) matrices, enriching them with local contextual information: $\mathbf{Q}, \mathbf{K}, \mathbf{V} \in \mathbb{R}^{L \times d}$.
- Fastformer Additive Attention: The Fastformer block then processes these matrices. Unlike standard self-attention which computes pairwise interactions ($\mathbf{QK}^T$), Fastformer uses an additive attention mechanism to compress global information efficiently.
- First, a global query vector $\mathbf{q} \in \mathbb{R}^d$ is computed as a weighted sum of all query vectors $\mathbf{q}_i$ in $\mathbf{Q}$.
- This global query interacts with each key vector via element-wise product to form a global context-aware key representation.
- A global key vector $\mathbf{k} \in \mathbb{R}^d$ is then computed as a weighted sum of these context-aware keys.
- Finally, $\mathbf{k}$ interacts with each value vector $\mathbf{v}_i$ via element-wise product, and the results are transformed to produce the final output $\mathbf{R} = [\mathbf{r}_1, \mathbf{r}_2, …, \mathbf{r}_L]$.
This process can be summarized by the following core operations for the $i$-th element:
$$
\mathbf{p}_i = \mathbf{q} \odot \mathbf{k}_i, \quad \mathbf{u}_i = \mathbf{k} \odot \mathbf{v}_i
$$
where $\odot$ denotes element-wise multiplication. The output $\mathbf{R}$ captures complex global dependencies across the entire sequence of health indicators for the li ion battery. - Regression Head: The output from the Fastformer block is passed through a series of feed-forward layers with activation functions and dropout for regularization, culminating in a final linear layer that outputs the predicted SOH (or capacity) for the target cycle.
3. Hyperparameter Optimization via Orthogonal Experiment
Deep learning models contain several hyperparameters whose optimal values are not known a priori and can significantly impact performance. A full grid search over all combinations is computationally prohibitive. We employ an efficient Taguchi orthogonal experimental design to find a near-optimal configuration for our Convolutional Fastformer model. This method systematically tests a representative subset of all possible combinations.
We identified four critical hyperparameters for optimization, each at three levels:
| Hyperparameter | Symbol | Level 1 | Level 2 | Level 3 |
|---|---|---|---|---|
| Input Sequence Length | L | 10 | 15 | 20 |
| Convolution Kernel Size | K | 3 | 4 | 5 |
| Number of Attention Heads | H | 4 | 6 | 8 |
| Network Depth (Blocks) | N | 3 | 4 | 5 |
An L9 orthogonal array was constructed, requiring only 9 experimental runs instead of the full 81 (3^4). For each run, the model was trained on one li ion battery dataset and validated on another. The Root Mean Square Error (RMSE) on the validation set was the evaluation metric. The results and analysis are shown in the table below:
| Run | L | K | H | N | Validation RMSE |
|---|---|---|---|---|---|
| 1 | 10 | 3 | 4 | 3 | 0.00304 |
| 2 | 10 | 4 | 6 | 4 | 0.00269 |
| 3 | 10 | 5 | 8 | 5 | 0.00303 |
| 4 | 15 | 3 | 6 | 5 | 0.00314 |
| 5 | 15 | 4 | 8 | 3 | 0.00291 |
| 6 | 15 | 5 | 4 | 4 | 0.00368 |
| 7 | 20 | 3 | 8 | 4 | 0.00300 |
| 8 | 20 | 4 | 4 | 5 | 0.00283 |
| 9 | 20 | 5 | 6 | 3 | 0.00265 |
Range analysis was performed on the results to determine the influence of each factor and the optimal level. The mean RMSE for each level of a factor was calculated. The range (R) for a factor, defined as the difference between its maximum and minimum mean RMSE, indicates its influence. A larger R means the factor has a greater impact on model accuracy.
| Factor | Mean RMSE (Level 1) | Mean RMSE (Level 2) | Mean RMSE (Level 3) | Range (R) | Rank |
|---|---|---|---|---|---|
| Input Length (L) | 0.00292 | 0.00324 | 0.00282 | 0.00042 | 1 |
| Kernel Size (K) | 0.00306 | 0.00281 | 0.00312 | 0.00031 | 3 |
| Heads (H) | 0.00318 | 0.00282 | 0.00298 | 0.00036 | 2 |
| Depth (N) | 0.00286 | 0.00312 | 0.00300 | 0.00026 | 4 |
The analysis revealed that the Input Sequence Length (L) had the most significant effect on the SOH estimation error for the li ion battery, followed by the Number of Attention Heads (H). The optimal combination, corresponding to the lowest mean RMSE for each factor, was determined to be: L=20, K=4, H=6, N=3. This configuration was used for all subsequent model training and testing.
Experimental Validation and Results
1. Experimental Setup and Metrics
We validated our proposed method using data from four distinct li ion battery cells (B01, B03, B04 for testing, B02 for training). The input to the model was the sequence of the three selected health features from the previous $L=20$ cycles. The target output was the SOH (calculated as current capacity divided by initial capacity) of the next cycle. Data was normalized using Min-Max scaling. The model was trained using the Adam optimizer with an initial learning rate of 0.0001 and a batch size of 50.
To objectively evaluate the performance of our Convolutional Fastformer model for li ion battery SOH estimation, we employed two standard metrics: Mean Absolute Error (MAE) and Root Mean Square Error (RMSE), defined as:
$$
\text{MAE} = \frac{1}{N} \sum_{i=1}^{N} |y_i – \hat{y}_i|, \quad \text{RMSE} = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (y_i – \hat{y}_i)^2}
$$
where $y_i$ is the actual SOH, $\hat{y}_i$ is the estimated SOH, and $N$ is the number of test cycles. Lower values of MAE and RMSE indicate higher estimation accuracy.
2. SOH Estimation Performance
The Convolutional Fastformer model was applied to estimate the SOH of the three test li ion batteries. The estimation results, plotted against the actual capacity fade curves, demonstrated a remarkably close fit across the entire lifespan of each cell. The relative estimation error consistently remained within a tight band of ±0.8% for all test batteries, underscoring the model’s precision. The predicted curves accurately captured both the gradual capacity fade trend and the more rapid non-linear drop often observed in the later life of a li ion battery.
To benchmark our model’s effectiveness, we compared its performance against three widely-used data-driven models trained and tested on the same features and data splits:
- RNN: A standard Recurrent Neural Network.
- GRU: A Gated Recurrent Unit network.
- CNN: A purely Convolutional Neural Network.
The quantitative comparison results are summarized in the table below. The performance metrics clearly show that the proposed Convolutional Fastformer model achieved the lowest estimation error across all three test li ion battery cells.
| Battery Cell | Metric | RNN | GRU | CNN | Convolutional Fastformer |
|---|---|---|---|---|---|
| B01 | MAE | 0.296% | 0.244% | 0.204% | 0.084% |
| RMSE | 0.362% | 0.314% | 0.271% | 0.133% | |
| B03 | MAE | 0.325% | 0.251% | 0.276% | 0.080% |
| RMSE | 0.396% | 0.362% | 0.311% | 0.102% | |
| B04 | MAE | 0.317% | 0.298% | 0.261% | 0.255% |
| RMSE | 0.360% | 0.371% | 0.409% | 0.291% | |
| Average | MAE: 0.313% RMSE: 0.373% |
MAE: 0.264% RMSE: 0.349% |
MAE: 0.247% RMSE: 0.330% |
MAE: 0.140% RMSE: 0.175% |
|
The superiority of our approach is evident. The average MAE and RMSE of the Convolutional Fastformer model were 0.140% and 0.175%, respectively. Compared to the next best model (CNN), this represents a relative reduction of approximately 43% in MAE and 47% in RMSE. This significant improvement highlights the benefit of combining local convolutional feature extraction with the efficient global context modeling of the Fastformer for capturing the complex degradation dynamics of a li ion battery.
Conclusion
In this work, we have presented a novel, high-precision framework for estimating the State of Health of li ion batteries. The core contributions of this study are threefold. First, we demonstrated that simple yet powerful statistical features extracted from specific segments of the charging voltage curve (CC1 maximum, CC2 minimum, CC3 maximum) exhibit a very strong correlation with capacity fade and serve as excellent health indicators. Second, we introduced and applied the Convolutional Fastformer neural network architecture to the problem of li ion battery SOH estimation. This model effectively marries the local pattern recognition strength of CNNs with the linear-complexity, global dependency modeling of the Fastformer, creating a powerful tool for learning from sequential health feature data. Third, we employed an orthogonal experimental design to efficiently optimize the model’s hyperparameters, ensuring robust performance without exhaustive search.
Experimental validation on a public dataset confirmed the exceptional accuracy and stability of our proposed method. The Convolutional Fastformer model achieved the lowest estimation errors (MAE and RMSE) when benchmarked against RNN, GRU, and CNN models. With the maximum MAE and RMSE across all test li ion batteries being only 0.255% and 0.291% respectively, and relative errors confined within 0.8%, the model proves to be highly reliable. This level of accuracy is crucial for practical BMS applications, enabling better-informed decisions regarding battery use, maintenance, and replacement. Future work may explore the integration of features from other operational phases, adaptation to different li ion battery chemistries, and the model’s implementation for real-time, onboard SOH estimation.