Accurately assessing the state of health (SOH) is paramount for ensuring the safety, reliability, and longevity of li ion battery systems in applications ranging from electric vehicles to grid storage. While extensive research focuses on point estimation of SOH, these methods inherently lack the ability to quantify the associated uncertainty in their predictions. For critical decision-making in battery management—such as maintenance scheduling, risk assessment, and end-of-life determination—interval estimates that provide a probable range for the SOH are significantly more informative and practical.
Conventional interval estimation techniques, like those based on Gaussian processes or bootstrap methods, often rely on specific assumptions about the underlying data distribution (e.g., Gaussian residuals). However, the degradation behavior of a li ion battery can be complex and non-stationary, potentially violating these assumptions and leading to unreliable intervals. The Lower Upper Bound Estimation (LUBE) framework offers a promising, distribution-free alternative. It employs a neural network with two outputs to directly predict the lower and upper bounds of the prediction interval, optimizing coverage and width without prior distributional knowledge.
Despite its appeal, the practical application of LUBE for li ion battery SOH estimation faces several challenges. Firstly, the original LUBE loss function is non-differentiable, complicating optimization with efficient gradient-based algorithms. Secondly, while differentiable approximations using sigmoid functions exist, they introduce a slope parameter that requires careful, often manual, tuning, impacting interval quality. Thirdly, most studies utilize battery capacity as the primary health indicator. Although capacity is the direct measure of SOH, its precise measurement is often impractical in real-world operations, relying on costly equipment or complete charge-discharge cycles, which limits online applicability.
To address these challenges, this paper proposes a comprehensive framework for distribution-free interval estimation of li ion battery SOH. The core contributions are: (1) Shifting from capacity-based to multiple, easily measurable health indicators derived from operational data, enhancing practical feasibility. (2) Employing Kernel Principal Component Analysis (KPCA) to perform nonlinear dimensionality reduction on these correlated health indicators, extracting robust and de-correlated features. (3) Designing a novel, fully differentiable loss function for the dual-output neural network that eliminates the need for manual slope parameter tuning, enabling stable and high-quality prediction interval generation. Experimental validation on a publicly available dataset demonstrates the effectiveness of the proposed method.
Feature Engineering for Health Representation
Instead of relying on direct capacity measurements, this method leverages indirect health indicators (HIs) that are strongly correlated with capacity fade and can be extracted from standard charge/discharge voltage and current profiles. The Incremental Capacity (IC) curve, obtained by differentiating capacity with respect to voltage ($dV$), is a powerful tool for identifying aging features. The IC curve is calculated as:
$$ \frac{dQ}{dV} \approx \frac{\Delta Q}{\Delta V} = \frac{Q_k – Q_{k-1}}{V_k – V_{k-1}} $$
where $Q_k$ and $V_k$ are the accumulated charge and voltage at sample $k$, respectively. As the li ion battery ages, characteristic peaks in the IC curve shift and diminish. From the smoothed IC curve, three intuitive health indicators are extracted:
- HI1: The maximum peak height of the IC curve.
- HI2: The voltage value at which the maximum peak occurs.
- HI3: The area under the IC curve peak.
These indicators capture different aspects of the electrochemical degradation within the li ion battery.
These raw HIs are often correlated and may contain noise. To create a more robust, low-dimensional, and decorrelated feature set for modeling, Kernel Principal Component Analysis (KPCA) is applied. KPCA performs nonlinear dimensionality reduction by mapping the original data into a high-dimensional feature space via a kernel function $\phi(\cdot)$ and then performing standard PCA in that space. The RBF kernel is commonly used:
$$ k_{\text{RBF}}(\mathbf{x}_i, \mathbf{x}_j) = \exp\left(-\frac{\|\mathbf{x}_i – \mathbf{x}_j\|^2}{2\sigma^2}\right) $$
The core computation involves solving the eigenvalue problem for the centered kernel matrix $\tilde{K}$:
$$ m \lambda \boldsymbol{\alpha} = \tilde{K} \boldsymbol{\alpha} $$
where $\boldsymbol{\alpha}$ are the eigenvectors and $\lambda$ the eigenvalues. The projection of a new sample $\mathbf{x}$ onto the $k$-th principal component in the high-dimensional space is given by:
$$ \text{PC}_k(\mathbf{x}) = \sum_{i=1}^{m} \alpha_i^k k(\mathbf{x}_i, \mathbf{x}) $$
The first few kernel principal components (KPCs), which capture the majority of the variance (e.g., >85%), are selected as the fused, de-correlated health feature vector $\mathbf{z}$ for the subsequent interval estimation model. The strength of the relationship between the fused feature and the actual capacity of the li ion battery is quantified using the Spearman’s rank correlation coefficient $\rho$:
$$ \rho = 1 – \frac{6 \sum_{i=1}^{n} d_i^2}{n(n^2 – 1)} $$
where $d_i$ is the difference between the ranks of the fused feature and capacity for the $i$-th sample, and $n$ is the number of samples. A high absolute value of $\rho$ confirms the suitability of the extracted feature for SOH representation.
| Training Battery Set | Spearman’s $\rho$ vs. Capacity |
|---|---|
| CS2_36, CS2_37, CS2_38 | -0.9892 |
| CS2_35, CS2_37, CS2_38 | -0.9919 |
| CS2_35, CS2_36, CS2_38 | -0.9907 |
| CS2_35, CS2_36, CS2_37 | -0.9908 |
Distribution-Free Interval Estimation Framework
The proposed framework is built upon the LUBE concept but introduces key innovations in the loss function design. A neural network model $f_{\boldsymbol{\theta}}$ with parameters $\boldsymbol{\theta}$ is constructed with two output neurons: one predicts the lower bound $\hat{y}_l$ and the other the upper bound $\hat{y}_u$ of the SOH prediction interval. The input to the network is a sequence of the fused health features $\mathbf{z}$ from a sliding time window.
The quality of a prediction interval (PI) is primarily judged by two criteria:
- Coverage: The PI should contain the true observation with a high predefined probability $(1-\alpha)$.
- Sharpness: The width of the PI should be as narrow as possible, providing precise information.
To directly optimize for these qualities, a custom loss function $L(\boldsymbol{\theta})$ is proposed. It seamlessly combines a width loss $L_w$ and a coverage loss $L_c$:
$$ \min_{\boldsymbol{\theta}} L(\boldsymbol{\theta}) = L_w(\boldsymbol{\theta}) + L_c(\boldsymbol{\theta}) $$
The width loss $L_w$ encourages narrow intervals that are centered around the true values. It is computed only over samples where the true value $y_i$ is currently covered by the predicted interval:
$$ L_w(\boldsymbol{\theta}) = \frac{1}{c} \sum_{i=1}^{n} \left[ (y_i – \hat{y}_{l,i})^2 + (\hat{y}_{u,i} – y_i)^2 \right] \cdot k_i $$
$$ \text{where } k_i = \begin{cases} 1, & \text{if } \hat{y}_{l,i} \leq y_i \leq \hat{y}_{u,i} \\ 0, & \text{otherwise} \end{cases}, \quad c = \sum_{i=1}^{n} k_i $$
The coverage loss $L_c$ acts as a dynamic penalty to enforce the nominal confidence level. It consists of an upper penalty (UP) and a lower penalty (LP):
$$ L_c(\boldsymbol{\theta}) = \text{UP}_{\boldsymbol{\theta}} + \text{LP}_{\boldsymbol{\theta}} $$
$$ \text{UP}_{\boldsymbol{\theta}} = \sum_{i=1}^{n} \max(0, y_i – \hat{y}_{u,i}) \cdot \gamma $$
$$ \text{LP}_{\boldsymbol{\theta}} = \sum_{i=1}^{n} \max(0, \hat{y}_{l,i} – y_i) \cdot \gamma $$
The critical adaptive weight $\gamma$ is not a fixed hyperparameter. It is dynamically calculated based on the current model’s Prediction Interval Coverage Probability (PICP) shortfall from the target $(1-\alpha)$:
$$ \gamma = \lambda \cdot \max\left(0, (1-\alpha) – \text{PICP}_{\boldsymbol{\theta}}\right) $$
$$ \text{where } \text{PICP}_{\boldsymbol{\theta}} = \frac{1}{n}\sum_{i=1}^{n} k_i $$
Here, $\lambda$ is a scaling constant. This design is pivotal: when $\text{PICP}_{\boldsymbol{\theta}} < (1-\alpha)$, $\gamma > 0$, and the coverage loss dominates, forcing the network to widen intervals to improve coverage. Once the desired coverage is achieved ($\text{PICP}_{\boldsymbol{\theta}} \geq (1-\alpha)$), $\gamma$ becomes zero, the coverage loss vanishes, and the optimizer focuses solely on minimizing $L_w$ to shrink the interval width. This elegant mechanism avoids the need for the manual tuning of a sigmoid slope parameter present in earlier methods, leading to more stable and objective optimization.
Experimental Validation and Analysis
The proposed method, termed GD_LUBE (Gradient-based Differentiable LUBE), is validated using the public CALCE li ion battery dataset. A leave-one-out cross-validation scheme is employed. The primary evaluation metrics are:
- Prediction Interval Coverage Probability (PICP): The empirical coverage rate.
- Mean Prediction Interval Width (MPIW): The average width of the PIs. $$ \text{MPIW} = \frac{1}{n} \sum_{i=1}^{n} (\hat{y}_{u,i} – \hat{y}_{l,i}) $$
- Coverage Width-based Criterion (CWC): A comprehensive score that penalizes both under-coverage and excessive width. $$ \text{CWC} = \begin{cases} \text{MPIW}, & \text{if PICP} \geq \mu \\ \text{MPIW} + \exp(-\eta (\text{PICP} – \mu)), & \text{if PICP} < \mu \end{cases} $$ where $\mu = 1-\alpha$ is the nominal confidence level and $\eta$ is a penalty factor. A lower CWC indicates a better overall PI quality.
The effectiveness of the KPCA preprocessing step is first verified. The table below compares interval estimation results using the original multi-dimensional HIs versus the fused single-dimensional feature from KPCA, evaluated at a 90% confidence level ($\alpha=0.1$). The results demonstrate that KPCA-based feature fusion consistently leads to superior performance, as indicated by lower CWC values, validating its role in enhancing model input quality.
| Battery Cell | With KPCA Fusion | Without KPCA Fusion (Original HIs) | ||||
|---|---|---|---|---|---|---|
| PICP | MPIW | CWC | PICP | MPIW | CWC | |
| CS2_35 | 0.9095 | 0.0765 | 0.0765 | 0.9780 | 0.0783 | 0.0783 |
| CS2_36 | 0.9060 | 0.0777 | 0.0777 | 0.9404 | 0.0798 | 0.0798 |
| CS2_37 | 0.9890 | 0.0688 | 0.0688 | 0.9681 | 0.0700 | 0.0700 |
| CS2_38 | 0.9442 | 0.0683 | 0.0683 | 0.9818 | 0.0710 | 0.0710 |
The core advantage of the proposed GD_LUBE loss function is demonstrated through a comparative study against a baseline method that uses a sigmoid-adjusted loss function (referred to as QD_LUBE). The following table summarizes the one-step-ahead prediction results for all battery cells. The proposed GD_LUBE method successfully meets or exceeds the 90% coverage target in all cases. More importantly, it consistently achieves narrower intervals (lower MPIW) and better overall scores (lower CWC) compared to the baseline. This confirms that the automatic, adaptive penalty mechanism in GD_LUBE is more effective than manually-tuned sigmoid functions in balancing coverage and sharpness for li ion battery SOH estimation.
| Battery Cell | Method | PICP | MPIW | CWC |
|---|---|---|---|---|
| CS2_35 | QD_LUBE (Baseline) | 0.9584 | 0.0850 | 0.0850 |
| GD_LUBE (Proposed) | 0.9095 | 0.0765 | 0.0765 | |
| CS2_36 | QD_LUBE (Baseline) | 0.9358 | 0.0886 | 0.0886 |
| GD_LUBE (Proposed) | 0.9060 | 0.0777 | 0.0777 | |
| CS2_37 | QD_LUBE (Baseline) | 0.9901 | 0.0761 | 0.0761 |
| GD_LUBE (Proposed) | 0.9890 | 0.0688 | 0.0688 | |
| CS2_38 | QD_LUBE (Baseline) | 0.9721 | 0.0817 | 0.0817 |
| GD_LUBE (Proposed) | 0.9442 | 0.0683 | 0.0683 |
The robustness of the method is further tested in multi-step-ahead prediction tasks. The table below shows a snapshot of the performance for up to six prediction steps ahead for one cell. The proposed GD_LUBE method maintains valid coverage (PICP near or above 0.9) across all steps while generally producing intervals with competitive or smaller width (MPIW) compared to the baseline, as reflected in the superior CWC values. This demonstrates the method’s stability and reliability for short-term prognostic forecasting of li ion battery health.
| Prediction Step | Method | PICP | MPIW | CWC |
|---|---|---|---|---|
| 1 | QD_LUBE | 0.9584 | 0.0850 | 0.0850 |
| GD_LUBE | 0.9095 | 0.0765 | 0.0765 | |
| 3 | QD_LUBE | 0.8848 | 0.0830 | 2.2208 |
| GD_LUBE | 0.9044 | 0.0754 | 0.0754 | |
| 6 | QD_LUBE | 0.7909 | 0.0778 | 234.0272 |
| GD_LUBE | 0.9079 | 0.0773 | 0.0773 |
Conclusion
This paper presented a novel, practical framework for distribution-free interval estimation of lithium-ion battery state of health. By moving away from capacity-based inputs to multiple operational health indicators and processing them with KPCA, the method gains practical applicability for online monitoring. The introduction of a novel, fully differentiable loss function with an adaptive coverage penalty mechanism overcomes key limitations of prior LUBE methods, eliminating manual parameter tuning and enabling efficient gradient-based optimization. Experimental results on real li ion battery data confirm that the method reliably produces prediction intervals that meet the specified confidence level while being sharper than those from a comparable baseline method. The framework provides a valuable tool for battery management systems, offering not only a health estimate but also a quantifiable measure of its uncertainty, which is crucial for informed and safe operational decisions. Future work will focus on enhancing the model’s generalizability across different battery chemistries and aging conditions using transfer learning techniques.