Lithium-ion batteries are widely employed in battery energy storage systems due to their high energy density, long cycle life, and low self-discharge rate. However, safety issues caused by overcharging, overheating, and other factors remain significant concerns. Accurate estimation of the State of Health (SOH) of lithium-ion batteries is crucial for optimizing the operation, management, and maintenance of battery energy storage systems. Existing methods that extract health features from single-stage charging data fail to fully exploit the aging information embedded in the charging process, limiting estimation accuracy. In this paper, we propose a novel SOH estimation method for battery energy storage systems based on the fusion of two-stage charging data: the constant voltage (CV) charging stage and the relaxation stage. By combining health features from both stages, we fully utilize the battery aging information contained in the charging process, thereby improving SOH estimation accuracy. The proposed health feature combination does not require data from the constant current (CC) charging stage, making it immune to the uncertainty of charging start points and more adaptable to real-world operating conditions of battery energy storage systems. Experimental results show that the proposed feature combination significantly outperforms single-stage feature combinations, achieving an average Mean Absolute Error (MAE) of 0.66%, Mean Squared Error (MSE) of 0.85%, and an average coefficient of determination (R²) of 0.97.
To address the limitations of single-stage feature extraction, we develop a data-driven framework that integrates features from the CV charging stage and the relaxation stage. The CV charging stage provides valuable information about internal resistance and capacity fade, while the relaxation stage captures voltage recovery dynamics that are sensitive to aging. Our method extracts the duration of the last 1/6 current interval from the CV stage as a feature, along with the Shannon entropy and maximum curvature from the relaxation voltage curve. These features are then used as inputs to a Random Forest (RF) regressor, which is trained to estimate SOH. The RF model is chosen for its robustness against overfitting, ability to handle high-dimensional data, and excellent generalization performance in battery energy storage systems.
The key contributions of this work are: (1) a novel two-stage feature fusion approach that leverages both CV and relaxation data, (2) a feature extraction method that avoids dependence on CC charging data, making it suitable for practical battery energy storage systems, and (3) comprehensive experimental validation on multiple battery chemistries and operating conditions, demonstrating superior accuracy compared to state-of-the-art single-stage methods.
Figure 1 illustrates a typical charging and relaxation process in battery energy storage systems. The charging curve consists of a CC phase, a CV phase, and a relaxation phase. In many real-world scenarios, the charging start point is uncertain, and the CC phase may be incomplete. Therefore, we focus on the CV and relaxation phases, which are more stable and reproducible.
Feature Extraction and Correlation Analysis
Constant Voltage Stage Feature
During the CV charging stage, the charging current decreases exponentially. The time spent in specific current intervals is strongly correlated with battery aging. We divide the CV current range into n equal intervals and extract the duration of the last interval (the one immediately before the relaxation stage) as a health feature. For n current intervals, the dividing currents are:
$$ I_n = [I_{n1}, I_{n2}, \ldots, I_{n(i+1)}] $$
where i = 1, 2, …, n. The corresponding interval durations are:
$$ t_n = [t_{n1}, t_{n2}, \ldots, t_{ni}] $$
We select the last interval duration t_{nn} as a feature. This feature is easy to obtain because the cutoff current is preset; we only need to record the time from the moment the current reaches the lower bound of the last interval until charging ends. No full current profile storage is required, reducing data overhead.
Relaxation Stage Features
Relaxation voltage data exhibit increased randomness as the battery ages. We compute three statistical features from the relaxation voltage curve:
- Shannon Entropy (Usha): Measures the randomness of the relaxation voltage.
$$ U_{sha} = \sum_{h=1}^{q} p(w_h) \log\left(\frac{1}{p(w_h)}\right) $$
where p(wh) is the probability of voltage values.
- Maximum Curvature (Ur): Describes the maximum bending degree of the relaxation curve. First, compute the first derivative:
$$ D_1(\alpha) = \frac{U(\alpha) – U(\alpha-1)}{t(\alpha) – t(\alpha-1)} $$
Then the second derivative (curvature) is:
$$ D_2(\alpha) = \frac{D_1(\alpha) – D_1(\alpha-1)}{t(\alpha) – t(\alpha-1)} $$
and Ur = max D2(α).
- Skewness of Voltage Drop Series (U’skr): Analyzes the symmetry of voltage distribution.
$$ U’_{skr} = \frac{(X_1-\mu)^3 + (X_2-\mu)^3 + \ldots + (X_j-\mu)^3}{j \cdot \sigma^3} $$
where μ is the mean, σ is the standard deviation, and j is the number of samples.
Correlation Analysis
We use Pearson correlation coefficient to evaluate linear correlation between each candidate feature and SOH. The formula is:
$$ \rho = \frac{\sum_{a=1}^{u} (F_a – \bar{F}) (SOH_a – \overline{SOH})}{\sqrt{\sum_{a=1}^{u} (F_a – \bar{F})^2 \sum_{a=1}^{u} (SOH_a – \overline{SOH})^2}} $$
where u is the number of samples. Table 1 shows the Pearson correlation coefficients for features extracted from a representative cell (NCA+NCM-CY25-0.5/4, cell #77) in the dataset.
| Feature | Pearson correlation ρ |
|---|---|
| Tcv (total CV duration) | -0.99 |
| t44 (last interval, n=4) | -0.99 |
| t66 (last interval, n=6) | -0.99 |
| t88 (last interval, n=8) | -0.99 |
| Usha (Shannon entropy) | 0.99 |
| Ur (max curvature) | 0.96 |
| U’skr (skewness) | -0.36 |
All CV features exhibit strong negative correlation (|ρ| ≈ 0.99). For relaxation features, Usha and Ur show high positive correlation, while skewness is weakly correlated. Hence, we select Usha and Ur as relaxation features. To determine the optimal CV interval division, we compare four candidate combinations with different n values (Tcv, t44, t66, t88) each combined with (Usha, Ur).
Random Forest Model and Implementation
We employ the Random Forest (RF) regressor for SOH estimation. The RF model is an ensemble of m decision trees. The final prediction is the average of individual tree outputs:
$$ y_{RF} = \frac{1}{m} \sum_{\beta=1}^{m} u_{\beta} $$
where uβ is the output of the β-th tree. RF uses bootstrap sampling and random feature selection to build diverse trees, which reduces overfitting and improves generalization. The implementation steps are:
- Given an original battery dataset containing features and SOH labels.
- Perform bootstrap sampling (with replacement) to create m bootstrapped datasets of the same size as the original.
- For each bootstrapped dataset, grow a decision tree without pruning. At each node, randomly select a subset of features and choose the best split.
- Aggregate predictions from all trees by averaging.
Hyperparameters are tuned using a random search strategy on a validation set. The key hyperparameters include the number of trees (En), maximum features per split (Fmax), maximum tree depth (Dmax), and minimum samples per leaf (Lmin).
Experimental Setup and Dataset
We use the publicly available dataset from [27], which includes cycling data of 18650 lithium-ion batteries with cathode chemistries NCA, NCM, and NCA+NCM. Five operating conditions are selected, summarized in Table 2.
| Dataset | Temperature (°C) | Charge rate (C) | Discharge rate (C) | Voltage range (V) |
|---|---|---|---|---|
| NCA | 25 | 1 | 1 | 4.2–2.65 |
| NCA | 45 | 0.5 | 1 | 4.2–2.5 |
| NCM | 35 | 0.5 | 1 | 4.2–2.5 |
| NCM | 45 | 0.5 | 1 | 4.2–2.5 |
| NCA+NCM | 25 | 0.5 | 4 | 4.2–2.5 |
For each condition, batteries are split randomly into training (50%) and testing (50%) sets. Table 3 lists the specific cell IDs used.
| Condition | Training cells | Testing cells |
|---|---|---|
| NCA-CY45-0.5/1 | #43,44,45,46,47,48,51,52,53,54,65,66 | #49,50,55,56,57,58,59,60,61,62,63,64 |
| NCA-CY25-1/1 | #1,3,9,8 | #2,4,5,7 |
| NCM-CY35-0.5/1 | #52,53 | #54,55 |
| NCM-CY45-0.5/1 | #1,2,6,7,12,13,16,17,18,19,21,23,24,27 | #3,4,5,8,9,11,10,14,15,20,22,25,26,28 |
| NCA+NCM-CY25-0.5/4 | #79 | #78 |
We use three metrics to evaluate SOH estimation accuracy: Mean Absolute Error (MAE), Mean Squared Error (MSE), and coefficient of determination (R²).
$$ MAE = \frac{1}{\varepsilon} \sum_{\delta=1}^{\varepsilon} \left| y_{\delta} – \hat{y}_{\delta} \right| \times 100\% $$
$$ MSE = \frac{1}{\varepsilon} \sum_{\delta=1}^{\varepsilon} (y_{\delta} – \hat{y}_{\delta})^2 \times 100\% $$
$$ R^2 = 1 – \frac{\sum_{\delta=1}^{\varepsilon} (\hat{y}_{\delta} – y_{\delta})^2}{\sum_{\delta=1}^{\varepsilon} (\bar{y}_{\delta} – y_{\delta})^2} $$
where yδ is the true SOH (capacity), ŷδ is the predicted SOH, and ε is the number of test samples.
Results and Discussion
Selection of Optimal CV Feature
We compare four candidate feature combinations: [Tcv, Usha, Ur], [t44, Usha, Ur], [t66, Usha, Ur], and [t88, Usha, Ur]. Each combination is evaluated using RF with hyperparameters tuned via random search. Table 4 summarizes the optimal hyperparameters found for each combination.
| Feature combination | En | Fmax | Dmax | Lmin |
|---|---|---|---|---|
| [Tcv, Usha, Ur] | 20 | auto | 5 | 4 |
| [t44, Usha, Ur] | 10 | auto | 25 | 5 |
| [t66, Usha, Ur] | 50 | auto | 5 | 7 |
| [t88, Usha, Ur] | 10 | sqrt | 20 | 7 |
Table 5 presents the SOH estimation results for each combination across all five conditions. The combination [t66, Usha, Ur] achieves the lowest average MAE (0.66%) and MSE (0.85%), with the highest average R² (0.97). Hence, we select t66 as the CV feature and finalize the optimal feature set as [t66, Usha, Ur].
| Combination | Condition | MAE (%) | MSE (%) | R² |
|---|---|---|---|---|
| [Tcv, Usha, Ur] | 1 | 0.87 | 1.31 | 0.96 |
| 2 | 1.21 | 2.44 | 0.92 | |
| 3 | 0.85 | 1.17 | 0.95 | |
| 4 | 0.65 | 0.79 | 0.98 | |
| 5 | 0.38 | 0.29 | 0.99 | |
| Average | 0.79 | 1.18 | 0.96 | |
| [t44, Usha, Ur] | 1 | 0.73 | 0.88 | 0.97 |
| 2 | 1.14 | 2.20 | 0.93 | |
| 3 | 0.65 | 0.69 | 0.97 | |
| 4 | 0.70 | 0.98 | 0.97 | |
| 5 | 0.44 | 0.38 | 0.99 | |
| Average | 0.73 | 1.04 | 0.97 | |
| [t66, Usha, Ur] | 1 | 0.74 | 0.89 | 0.97 |
| 2 | 1.10 | 1.90 | 0.94 | |
| 3 | 0.45 | 0.42 | 0.98 | |
| 4 | 0.60 | 0.75 | 0.98 | |
| 5 | 0.39 | 0.29 | 0.99 | |
| Average | 0.66 | 0.85 | 0.97 | |
| [t88, Usha, Ur] | 1 | 0.65 | 0.68 | 0.98 |
| 2 | 1.76 | 5.43 | 0.83 | |
| 3 | 0.51 | 0.51 | 0.98 | |
| 4 | 0.59 | 0.67 | 0.98 | |
| 5 | 0.58 | 0.51 | 0.99 | |
| Average | 0.81 | 1.55 | 0.95 |
Comparison with Single-Stage Feature Combinations
We compare our proposed two-stage feature set [t66, Usha, Ur] with two single-stage combinations from literature:
- Relaxation-only features from [27]: [Uvar, U’skr, Umax] (variance, skewness, maximum of relaxation voltage).
- CV-only features from [31]: [t43, t61, t65] (specific interval durations from CV stage).
For fair comparison, we optimize hyperparameters for each baseline using random search. Table 6 lists the optimal hyperparameters for the single-stage combinations.
| Feature combination | En | Fmax | Dmax | Lmin |
|---|---|---|---|---|
| [Uvar, U’skr, Umax] | 40 | auto | 5 | 4 |
| [t43, t61, t65] | 60 | sqrt | 20 | 8 |
Table 7 summarizes the SOH estimation results for all three combinations on each test condition. Our proposed two-stage fusion significantly outperforms both single-stage baselines in terms of average MAE, MSE, and R². For example, the relaxation-only and CV-only combinations yield average MAEs of 1.08% and 1.23%, respectively, compared to 0.66% for the proposed method. The MSE is also notably lower (0.85% vs. 2.80% and 3.56%). The improved R² (0.97 vs. 0.92 and 0.90) confirms that the two-stage features better explain the variance in SOH.
| Condition | Metric | Proposed [t66, Usha, Ur] | Relaxation-only [27] | CV-only [31] |
|---|---|---|---|---|
| 1 (NCA, 45°C, 0.5/1) | MAE (%) | 0.74 | 1.01 | 1.52 |
| MSE (%) | 0.89 | 1.77 | 4.25 | |
| R² | 0.97 | 0.95 | 0.89 | |
| 2 (NCA, 25°C, 1/1) | MAE (%) | 1.10 | 2.44 | 2.54 |
| MSE (%) | 1.90 | 10.16 | 10.92 | |
| R² | 0.94 | 0.69 | 0.66 | |
| 3 (NCM, 35°C, 0.5/1) | MAE (%) | 0.45 | 0.47 | 0.84 |
| MSE (%) | 0.42 | 0.35 | 1.02 | |
| R² | 0.98 | 0.98 | 0.96 | |
| 4 (NCM, 45°C, 0.5/1) | MAE (%) | 0.61 | 0.62 | 0.69 |
| MSE (%) | 0.75 | 0.61 | 0.88 | |
| R² | 0.98 | 0.98 | 0.98 | |
| 5 (NCA+NCM, 25°C, 0.5/4) | MAE (%) | 0.39 | 0.89 | 0.58 |
| MSE (%) | 0.29 | 1.22 | 0.72 | |
| R² | 0.99 | 0.99 | 0.99 | |
| Average | MAE (%) | 0.66 | 1.08 | 1.23 |
| MSE (%) | 0.85 | 2.80 | 3.56 | |
| R² | 0.97 | 0.92 | 0.90 |
The superiority of the two-stage fusion is especially evident in the NCA NCA-CY45-0.5/1 and NCA-CY25-1/1 conditions, where single-stage methods exhibit large errors (MSE up to 10.92%). The proposed method maintains robust performance due to the complementary aging information provided by the CV and relaxation phases. In battery energy storage systems, where charging patterns vary and CC data may be incomplete, our method offers a practical and accurate solution.
Conclusion
In this paper, we propose a novel SOH estimation method for lithium-ion batteries in battery energy storage systems by fusing health features from the constant voltage charging stage and the relaxation stage. The key findings are:
- The last 1/6 interval duration of CV charging (t66) is a highly correlated feature with SOH and is easy to extract without requiring full current profile storage.
- Relaxation stage features, specifically Shannon entropy (Usha) and maximum curvature (Ur), capture additional aging information that is not present in CV data alone.
- The combination [t66, Usha, Ur] achieves an average MAE of 0.66%, MSE of 0.85%, and R² of 0.97, significantly outperforming single-stage feature combinations.
- The proposed method does not rely on constant current charging data, making it robust to the uncertainty of charging start points and well-suited for real-world battery energy storage systems.
Future work will explore the integration of deep learning models, such as LSTM or attention mechanisms, with the proposed feature set to further improve estimation accuracy for battery energy storage systems. Additionally, we plan to validate the method on a wider range of chemistries and dynamic load profiles to enhance its generalization capability in battery energy storage systems.