The accurate assessment of the State of Health (SOH) for lithium-ion batteries is paramount for ensuring the safety, reliability, and longevity of energy storage systems and electric vehicles. As a lithium-ion battery undergoes repeated charge-discharge cycles, its maximum available capacity gradually fades due to complex, irreversible internal electrochemical reactions. Since SOH cannot be measured directly by sensors and its degradation is influenced by multiple factors like depth of discharge, current rate, and temperature, developing robust and accurate estimation methods remains a significant challenge.
Existing SOH estimation approaches can be broadly categorized into model-based and data-driven methods. Model-based techniques, including empirical, electrochemical, and equivalent circuit models, often face limitations such as low accuracy, high complexity unsuitable for online application, or excessive dependence on precise parameter identification. In contrast, data-driven methods leverage machine learning models to establish a direct mapping between measurable battery data and SOH, offering greater flexibility and potential for online implementation. Common machine learning models employed include Support Vector Regression (SVR), Gaussian Process Regression (GPR), and various neural network architectures. However, the performance of a single machine learning model can be limited, especially when generalizing across batteries with different aging trajectories. Ensemble learning methods, such as stacking models, have shown promise in improving generalization by combining the strengths of multiple base estimators.
A critical aspect of any data-driven SOH estimation method is the selection and extraction of informative health features. While many studies focus on features extracted from the Constant Current (CC) charging phase or the relaxation voltage after charging, these approaches have inherent limitations. Features from the CC phase can be sensitive to the starting point of charging, which is often variable in real-world applications. Methods relying on relaxation voltage require the battery to be fully charged and then left idle for a considerable time, which is impractical for many operational scenarios. The Constant Voltage (CV) charging phase, which follows the CC phase in a standard charging protocol, presents a compelling alternative. It is inherently stable, its start is clearly defined by the charge termination voltage, and it does not require a subsequent long idle period. Current research on CV-based SOH estimation has primarily utilized simple statistical indicators like charging duration and current entropy, potentially leaving richer aging information within the CV current curve untapped.
To address these gaps, I propose a novel SOH estimation method for lithium-ion batteries based on a comprehensive health feature set derived solely from the CV charging stage and a sophisticated stacking ensemble model. The core innovation lies in the fusion of both statistical and geometric features extracted from the CV charging current profile, providing a more holistic description of battery aging. Furthermore, to enhance model robustness and generalization, I design a two-layer stacking model that strategically combines estimators with different learning biases.
The primary contributions of this work are twofold. First, I introduce a novel health feature set for lithium-ion battery SOH estimation, comprising CV charging time, current information entropy, current curve skewness, and maximum curvature. This combination more fully captures the degradation patterns evident in the CV phase. Second, I construct a dedicated stacking model that integrates SVR, Linear Regression (LR), and Multilayer Perceptron (MLP) models at the first level, with an eXtreme Gradient Boosting (XGBoost) meta-learner at the second level. This architecture effectively leverages the complementary strengths of diverse base learners to achieve superior estimation accuracy and stability compared to any single model.
Dataset and Health Feature Analysis
To validate the proposed method, I utilize a publicly available aging dataset for nickel-cobalt-manganese (NCM) lithium-ion batteries. The dataset comprises aging tests on 28 cells under a constant temperature of 45°C. The charging protocol is a standard CC-CV mode with a CC current of 0.5C until the voltage reaches the upper cutoff, followed by a CV hold until the current drops to a specified threshold. Discharge is performed at a 1C rate. The cells exhibit varying cycle lives from 199 to 932 cycles, representing a wide range of aging states and speeds, which is crucial for testing the generalization capability of the estimation model.
Health Feature Extraction from CV Charging Data
As a lithium-ion battery ages, its internal resistance increases and active material is lost. These changes directly influence the dynamics of the CV charging stage. Specifically, the time required to complete the CV phase lengthens, and the shape of the current decay curve evolves. To quantify these changes comprehensively, I propose extracting four distinct health features from the CV charging current data, categorized as statistical and geometric features.
The statistical features describe the overall distribution and information content of the current data:
- CV Charging Time (t_cv): The total duration from the start to the end of the CV phase. This is a direct indicator of the battery’s ability to accept charge at the target voltage.
- Current Information Entropy (H_I): A measure of the uncertainty or randomness in the current sequence. It is calculated from the discretized current values and reflects the complexity of the current decay profile.
The geometric features describe the shape and curvature of the current-time curve:
- Current Curve Skewness (S_k): Measures the asymmetry of the current decay curve. A negative skewness indicates a longer tail on the left (slower initial decay), which may correlate with aging.
- Current Maximum Curvature (κ_max): Identifies the point of sharpest bend in the current curve. Curvature is computed numerically. Changes in this value can signal shifts in the dominant aging mechanisms affecting the charge acceptance kinetics.
These four features form the proposed health feature vector F used for SOH estimation:
$$
\mathbf{F} = [t_{cv}, H_I, S_k, \kappa_{max}]
$$
A summary of the proposed features is provided in Table 1.
| Category | Feature Name | Symbol |
|---|---|---|
| Statistical | CV Charging Time | $$t_{cv}$$ |
| Statistical | Current Information Entropy | $$H_I$$ |
| Geometric | Current Curve Skewness | $$S_k$$ |
| Geometric | Maximum Curvature | $$\kappa_{max}$$ |
Correlation Analysis
The relationship between each extracted feature and the reference battery SOH (defined as the ratio of current maximum capacity to initial capacity) is analyzed. To quantify the strength of these relationships, the Pearson correlation coefficient is calculated for each feature across multiple battery cells. The Pearson correlation coefficient ρ between a feature I and SOH S over k cycles is given by:
$$
\rho = \frac{\sum_{i=1}^k (I_i – \bar{I})(S_i – \bar{S})}{\sqrt{\sum_{i=1}^k (I_i – \bar{I})^2 \sum_{i=1}^k (S_i – \bar{S})^2}}
$$
where $\bar{I}$ and $\bar{S}$ are the mean values of the feature and SOH, respectively. The average absolute correlation coefficients across several cells are consistently high, as summarized in Table 2, confirming that all proposed features are strongly indicative of the lithium-ion battery’s state of health.
| Feature | Average |ρ| |
|---|---|
| $$t_{cv}$$ | 0.973 |
| $$H_I$$ | 0.980 |
| $$S_k$$ | 0.938 |
| $$\kappa_{max}$$ | 0.925 |
Proposed Stacking Ensemble Model
Given the diverse nature of the proposed features (statistical vs. geometric) and the nonlinear, complex degradation of a lithium-ion battery, a single machine learning model may not optimally capture all underlying patterns. To enhance estimation accuracy and robustness, I design a two-layer stacking ensemble model. Stacking, or stacked generalization, is an advanced ensemble technique where a meta-model is trained to combine the predictions of several base models. This approach allows the model to learn how to best leverage the different inductive biases of each base learner.
Base Learners (Level-1 Models)
The first layer of the stacking model consists of three diverse regression estimators:
- Support Vector Regression (SVR): Effective for modeling nonlinear relationships, especially in high-dimensional spaces induced by its kernel function. I employ the Radial Basis Function (RBF) kernel. The optimization objective is:
$$
\min_{\mathbf{w}, b, \xi, \xi^*} \frac{1}{2} \|\mathbf{w}\|^2 + C \sum_{i=1}^{n} (\xi_i + \xi_i^*)
$$
subject to constraints ensuring prediction errors are within a margin ε. - Linear Regression (LR): A simple model that assumes a linear relationship between features and target. It serves to capture the dominant linear trend in SOH degradation, providing a stable baseline. The model is: $y = \beta_0 + \beta_1 x_1 + … + \beta_j x_j$.
- Multilayer Perceptron (MLP): A feedforward neural network capable of approximating complex nonlinear functions. It is well-suited for learning the intricate, non-linear mapping from health features to the SOH of a lithium-ion battery.
Meta-Learner (Level-2 Model)
The predictions from the three base learners (SVR, LR, MLP) are concatenated to form a new feature vector. This vector is then used to train the meta-learner, which produces the final SOH estimate. For the meta-learner, I select the eXtreme Gradient Boosting (XGBoost) algorithm. XGBoost is a powerful tree-based ensemble method known for its high performance, speed, and built-in regularization to prevent overfitting. Its model is an additive ensemble of K Classification and Regression Trees (CART):
$$
\hat{y}_i = \sum_{k=1}^{K} f_k(\mathbf{x}_i), \quad f_k \in \mathcal{F}
$$
where $\mathcal{F}$ is the space of CARTs. The objective function includes a loss term l and a regularization term Ω:
$$
\mathcal{L}(\phi) = \sum_{i} l(\hat{y}_i, y_i) + \sum_{k} \Omega(f_k)
$$
The XGBoost meta-learner learns the optimal way to non-linearly combine the outputs of the base models, effectively deciding “which base model to trust more” under different circumstances implied by their joint predictions.
Experimental Validation and Results
The proposed method is rigorously evaluated on the aforementioned lithium-ion battery dataset. The data is first preprocessed: outliers are removed, and all features are standardized to have zero mean and unit variance to ensure stable and efficient training of the machine learning models, particularly SVR and MLP. The evaluation employs three standard metrics: Mean Absolute Error (MAE), Root Mean Square Error (RMSE), and the Coefficient of Determination (R²).
$$
\text{MAE} = \frac{1}{n}\sum_{i=1}^{n} |S_i^* – S_i|
$$
$$
\text{RMSE} = \sqrt{\frac{1}{n}\sum_{i=1}^{n} (S_i^* – S_i)^2}
$$
$$
R^2 = 1 – \frac{\sum_{i=1}^{n} (S_i^* – S_i)^2}{\sum_{i=1}^{n} (S_i – \bar{S})^2}
$$
where $S_i^*$ is the estimated SOH, $S_i$ is the reference SOH, and $\bar{S}$ is the mean reference SOH.
Experiment 1: Effectiveness of the Proposed Health Features
This experiment validates the superiority of the proposed four-feature set. The stacking model structure is fixed, and its performance is compared when fed with different feature combinations:
- Proposed: All four features ($$t_{cv}, H_I, S_k, \kappa_{max}$$).
- Comparison 1: Only CV charging time ($$t_{cv}$$).
- Comparison 2: CV charging time and current entropy ($$t_{cv}, H_I$$) – a common statistical pair.
- Comparison 3: Features from relaxation voltage (variance, max, skewness) – a competing method from literature.
The estimation results for several test batteries are summarized in Table 3. The results clearly demonstrate that the proposed feature combination achieves the best overall accuracy. For most test cells, the MAE and RMSE are below 1%, and R² values are very close to 1. The proposed features consistently outperform the single-feature and two-feature statistical sets, proving that adding geometric descriptors (skewness and curvature) provides significant complementary information about lithium-ion battery aging. Furthermore, the proposed method outperforms the relaxation-voltage-based method without requiring a long post-charge idle period, highlighting its practical advantage.
| Cell | Metric | Proposed | Comp.1 ($$t_{cv}$$ only) | Comp.2 ($$t_{cv}, H_I$$) | Comp.3 (Relaxation) |
|---|---|---|---|---|---|
| #1 | MAE (%) | 0.308 | 1.426 | 0.560 | 0.327 |
| RMSE (%) | 0.542 | 1.539 | 0.790 | 0.379 | |
| R² | 0.981 | 0.843 | 0.959 | 0.990 | |
| #11 | MAE (%) | 0.485 | 1.687 | 0.591 | 1.255 |
| RMSE (%) | 0.623 | 1.729 | 0.752 | 1.306 | |
| R² | 0.987 | 0.900 | 0.981 | 0.943 | |
| #21 | MAE (%) | 0.289 | 0.477 | 0.503 | 1.140 |
| RMSE (%) | 0.570 | 0.678 | 0.739 | 1.414 | |
| R² | 0.991 | 0.987 | 0.985 | 0.945 |
Experiment 2: Effectiveness of the Proposed Stacking Model
This experiment evaluates the performance gain from the ensemble architecture. Using the proposed four-feature set, the performance of the full stacking model is compared against each of its constituent base models (SVR, LR, MLP, XGBoost) trained individually. The results for a larger set of test batteries are shown in Table 4.
The stacking model achieves the lowest average MAE (0.528%) and RMSE (0.724%), and the highest average R² (0.951). It outperforms the individual LR and MLP models on every test cell. While SVR and XGBoost are strong individual performers, the stacking model matches or exceeds their accuracy on the majority of cells. This demonstrates that the meta-learner successfully synthesizes a more robust and accurate estimator than any single model could be, effectively mitigating the limitations of individual learning algorithms when applied to the complex task of lithium-ion battery SOH estimation.
| Cell | Metric | Stacking | SVR | LR | MLP | XGBoost |
|---|---|---|---|---|---|---|
| #1 | MAE (%) | 0.258 | 0.242 | 0.961 | 0.660 | 0.292 |
| RMSE (%) | 0.491 | 0.478 | 1.161 | 0.940 | 0.521 | |
| R² | 0.984 | 0.985 | 0.911 | 0.942 | 0.982 | |
| #13 | MAE (%) | 0.403 | 0.458 | 0.948 | 0.764 | 0.416 |
| RMSE (%) | 0.548 | 0.626 | 1.123 | 0.932 | 0.557 | |
| R² | 0.991 | 0.988 | 0.961 | 0.973 | 0.991 | |
| #23 | MAE (%) | 0.958 | 0.930 | 1.854 | 1.544 | 1.132 |
| RMSE (%) | 1.043 | 1.073 | 2.266 | 1.602 | 1.234 | |
| R² | 0.967 | 0.965 | 0.846 | 0.923 | 0.954 | |
| Average | MAE (%) | 0.528 | 0.549 | 1.026 | 0.788 | 0.555 |
| RMSE (%) | 0.724 | 0.742 | 1.303 | 1.026 | 0.727 | |
| R² | 0.951 | 0.945 | 0.855 | 0.906 | 0.949 |
Conclusion
This work addresses the critical challenge of accurately estimating the State of Health for lithium-ion batteries in a practical and robust manner. I have presented a novel data-driven method that leverages readily available data from the Constant Voltage charging phase. The key innovation is the extraction and fusion of a comprehensive set of health features, including both traditional statistical metrics (charging time, information entropy) and novel geometric descriptors (curve skewness, maximum curvature) from the CV current profile. This feature combination provides a richer, more informative representation of battery aging compared to existing CV-based methods.
To effectively learn from this diverse feature set and the inherent non-linearity of lithium-ion battery degradation, I designed and implemented a specialized two-layer stacking ensemble model. The model integrates SVR, Linear Regression, and MLP as base learners, whose predictions are then optimally combined by an XGBoost meta-regressor. This architecture capitalizes on the complementary strengths of different learning algorithms, resulting in superior generalization and accuracy compared to any single-model approach.
Extensive experimental validation on a public lithium-ion battery aging dataset confirms the effectiveness of the proposed method. The results demonstrate that the proposed feature set significantly improves estimation accuracy over simpler feature combinations. Furthermore, the stacking model consistently delivers more accurate and stable SOH estimates than its individual constituent models, achieving low error rates (e.g., average MAE ~0.53%) across batteries with varying aging trajectories.
The proposed method offers a practical and powerful solution for online SOH estimation of lithium-ion batteries, requiring only data from the standard CV charging stage without the need for long relaxation periods or restrictive operational conditions. Future work will focus on extending this approach to estimate the SOH of entire battery packs, where cell-to-cell inconsistencies pose an additional challenge, and on investigating model transfer techniques to enhance adaptability across different battery chemistries and formats.