In the era of renewable energy and electrification, lithium-ion batteries have emerged as a cornerstone technology due to their high energy density, efficiency, and environmental friendliness. They are pivotal in applications ranging from electric vehicles to grid-scale energy storage systems. However, the performance of a lithium-ion battery degrades over time through complex electrochemical processes, which can lead to reduced capacity, safety hazards, and system failures. Accurately estimating the State of Health (SOH) of a lithium-ion battery is therefore critical for ensuring reliability, prolonging lifespan, and preventing catastrophic events. SOH is typically defined as the ratio of the current maximum charge capacity to the nominal capacity, reflecting the battery’s degradation level. Traditional methods for SOH estimation, such as electrochemical models or equivalent circuit models, often require intricate knowledge of internal parameters and can be computationally intensive. In recent years, data-driven approaches have gained popularity, leveraging machine learning algorithms to correlate battery operational data with health indicators. Yet, these methods frequently suffer from being “black boxes,” with poor interpretability and a heavy reliance on large, comprehensive datasets that include full charge or discharge cycles—data that can be time-consuming to acquire in real-world scenarios. This limitation poses a significant challenge for practical applications where only partial charging data might be available. To address these issues, we propose a novel SOH estimation framework for lithium-ion batteries that combines interpretable Shapelets—discriminative subsequences from time-series data—with an optimized neural network model. Our approach aims to provide accurate SOH predictions using minimal partial charging data, enhance model transparency, and improve generalization across diverse operating conditions.
The core innovation lies in utilizing Shapelets extracted from voltage profiles during charging. Shapelets are concise, localized patterns within a time series that capture essential shape characteristics and trends. For lithium-ion batteries, the charging voltage curve evolves as the battery ages, often showing systematic shifts relative to the initial cycle. By identifying Shapelets that correlate strongly with capacity fade, we can derive health features that are both informative and interpretable. We focus on the latter part of the charging curve, typically from 3.2 V to the cutoff voltage (e.g., 3.6 V), where aging effects are pronounced. The distance between Shapelets from subsequent cycles and those from the first cycle serves as a health indicator, reflecting degradation progression. To efficiently select the most relevant Shapelets, we employ correlation analysis, specifically Spearman’s rank correlation, to evaluate the relationship between these distance metrics and SOH. This process narrows down candidate Shapelets, ensuring that the selected features are robust and representative.
Once the Shapelets are extracted, they serve as inputs to a prediction model. We employ a Backpropagation (BP) neural network due to its ability to model complex nonlinear relationships. However, standard BP networks are prone to local optima and sensitivity to initial weights, which can compromise prediction stability. To overcome this, we integrate the Subtraction-Average-Based Optimizer (SABO), a metaheuristic algorithm designed for high-dimensional optimization problems. SABO enhances the BP network by optimizing its weights and thresholds, leading to improved convergence and accuracy. The combined SABO-BP model is trained to map the Shapelet-based features to SOH values. We validate our method using the publicly available Stanford-MIT dataset, which comprises extensive cycling data for commercial lithium-ion batteries under various fast-charging protocols. Experiments demonstrate that our approach achieves high estimation precision with mean absolute error (MAE) below 0.5% and root mean square error (RMSE) below 0.6%, even when tested on batteries with charging strategies unseen during training. This underscores the method’s effectiveness and generalizability for real-world lithium-ion battery management systems.
To provide a comprehensive exposition, this article is structured as follows: First, we delve into the background of lithium-ion battery degradation and data-driven SOH estimation, highlighting the challenges and the rationale for Shapelets. Next, we detail the methodology for Shapelet extraction and selection, followed by the formulation of the SABO-BP model. We then describe the experimental setup, including dataset characteristics and evaluation metrics. Subsequently, we present results and discussions, comparing our method with baseline approaches. Finally, we conclude with insights and future research directions. Throughout, we emphasize the practical implications for lithium-ion battery health monitoring, ensuring that the keyword “lithium-ion battery” is frequently referenced to maintain focus.
Background and Motivation
The widespread adoption of lithium-ion batteries is driven by their superior performance metrics, but their aging behavior remains a critical concern. Degradation in a lithium-ion battery is an irreversible process influenced by factors such as cycling, temperature, and charging rates. It manifests as capacity fade and power loss, ultimately affecting system efficiency and safety. SOH serves as a quantitative measure of this degradation, typically expressed as:
$$ \text{SOH} = \frac{C_{\text{current}}}{C_{\text{nominal}}} \times 100\% $$
where \( C_{\text{current}} \) is the actual maximum capacity available and \( C_{\text{nominal}} \) is the rated capacity. As a lithium-ion battery ages, SOH declines, and when it falls below a threshold (e.g., 80%), the battery is often considered unfit for demanding applications. Accurate SOH estimation enables predictive maintenance, optimal charging strategies, and second-life assessments. Existing estimation methods can be broadly categorized into model-based and data-driven approaches. Model-based methods, such as electrochemical models, rely on physical principles but require precise parameterization and are computationally expensive. Equivalent circuit models offer simplicity but may lack accuracy under dynamic conditions. Data-driven methods, in contrast, leverage historical operational data to learn patterns without explicit physical models. These include techniques like support vector machines, Gaussian process regression, and neural networks. However, they often necessitate large datasets encompassing full charge-discharge cycles, which are not always feasible to collect. Moreover, their opaque nature—often termed “black-box”—hampers interpretability, making it difficult to trust predictions in safety-critical systems. Thus, there is a pressing need for methods that balance accuracy, data efficiency, and interpretability for lithium-ion battery SOH estimation.
Time-series analysis offers a promising avenue, as battery operational data (e.g., voltage, current, temperature) are inherently sequential. Shapelets, introduced in data mining, are subsequences that maximally discriminate between classes or capture trends in time-series data. They are interpretable because they represent specific patterns or shapes within the series. For lithium-ion batteries, charging voltage curves exhibit characteristic shapes that evolve with aging. By extracting Shapelets from these curves, we can identify localized features that correlate with SOH. This approach aligns with the need for interpretability: engineers can visualize which parts of the charging profile are most indicative of health. Additionally, Shapelets can be derived from partial charging data, addressing the data scarcity issue. For instance, even if a battery is only charged from an arbitrary state of charge (SOC) to full, the voltage segment during this process can yield meaningful Shapelets. This motivates our focus on Shapelet-based feature engineering for lithium-ion battery health assessment.
Methodology: Shapelet Extraction and Selection
Our methodology begins with processing charging voltage data from a lithium-ion battery. Let \( D = \{ T_1, T_2, \dots, T_n \} \) represent a set of time-series data for \( n \) cycles, where each \( T_i = \{ t_{i1}, t_{i2}, \dots, t_{im} \} \) is the voltage sequence of length \( m \) for cycle \( i \). The goal is to extract Shapelets—subsequences of fixed length \( l \)—from these series that best indicate SOH degradation. We concentrate on the voltage range from 3.2 V to the cutoff voltage (e.g., 3.6 V), as this region consistently appears in full charges and shows marked aging effects. For each cycle \( i \), we define a Shapelet candidate \( TS_i \) as a subsequence ending at a specific voltage point \( v_{\text{end}} \). Formally, for a given end voltage \( v_{\text{end}} \), we locate the index \( k \) such that \( t_{ik} \approx v_{\text{end}} \), and then take the preceding \( l \) points:
$$ TS_i = \{ t_{i(k-l+1)}, t_{i(k-l+2)}, \dots, t_{ik} \} $$
This yields a candidate set \( DS = \{ TS_1, TS_2, \dots, TS_n \} \). To quantify degradation, we compute the Euclidean distance between each candidate Shapelet and the corresponding Shapelet from the first cycle (reference cycle). For cycle \( i \), the distance \( D(TS_1, TS_i) \) is:
$$ D(TS_1, TS_i) = \sqrt{ \sum_{j=1}^{l} \left( t_{1(k-l+j)} – t_{i(k-l+j)} \right)^2 } $$
This distance metric captures how much the charging shape has deviated from the initial state, with larger distances implying more degradation. To select Shapelets that are strongly correlated with SOH, we evaluate the Spearman correlation coefficient between the distance vector \( \mathbf{D} = [D(TS_1, TS_1), D(TS_1, TS_2), \dots, D(TS_1, TS_n)] \) and the actual SOH values \( \mathbf{C}_{\text{SOH}} = [\text{SOH}_1, \text{SOH}_2, \dots, \text{SOH}_n] \). Spearman correlation is chosen for its robustness to monotonic relationships. The coefficient \( \rho \) is calculated as:
$$ \rho(\mathbf{D}, \mathbf{C}_{\text{SOH}}) = \frac{\text{cov}(\text{rank}(\mathbf{D}), \text{rank}(\mathbf{C}_{\text{SOH}}))}{\sigma_{\text{rank}(\mathbf{D})} \sigma_{\text{rank}(\mathbf{C}_{\text{SOH}})} } $$
where \( \text{cov} \) is covariance, \( \sigma \) denotes standard deviation, and \( \text{rank} \) assigns ranks to the data. We explore various combinations of \( l \) (from 10 to 100 points) and \( v_{\text{end}} \) (from 3.42 V to 3.60 V in steps of 0.02 V). For each combination, we compute \( \rho \) and select those with high absolute values (e.g., >0.9). Based on empirical analysis, we find that Shapelets with end voltages of 3.60 V, 3.58 V, and 3.56 V (with \( l \) around 90-100) and those with end voltages of 3.46 V and 3.44 V (with \( l \) around 60-70) exhibit the strongest correlations. This selection ensures that the features are both interpretable and predictive. Table 1 summarizes the correlation analysis for sample batteries from the dataset, illustrating the effectiveness of this approach.
| End Voltage (V) | Length (points) | Correlation (ρ) | Interpretation |
|---|---|---|---|
| 3.60 | 100 | -0.978 | Strong negative correlation |
| 3.58 | 90 | -0.965 | Strong negative correlation |
| 3.56 | 80 | -0.942 | Moderate negative correlation |
| 3.46 | 70 | -0.951 | Strong negative correlation |
| 3.44 | 60 | -0.932 | Moderate negative correlation |
The negative correlation indicates that as SOH decreases, the distance increases—consistent with degradation. We select the top five Shapelets based on correlation strength and diversity in voltage regions to capture comprehensive aging signatures. These Shapelets form the feature vector for SOH prediction. This process underscores the interpretability: each Shapelet corresponds to a specific voltage segment, allowing engineers to pinpoint which parts of the charge curve are most sensitive to aging in a lithium-ion battery.
SABO-Optimized BP Neural Network Model
With the Shapelet-based features extracted, we employ a BP neural network to model the relationship between these features and SOH. A standard BP network consists of an input layer, one or more hidden layers, and an output layer. The input layer receives the feature vector (e.g., distances from five Shapelets), the hidden layers perform nonlinear transformations, and the output layer produces the estimated SOH. The network learns by adjusting weights and biases through backpropagation to minimize a loss function, typically mean squared error (MSE). However, BP networks are susceptible to local minima and slow convergence, especially with limited data. To enhance performance, we integrate the Subtraction-Average-Based Optimizer (SABO), a novel metaheuristic algorithm designed for global optimization. SABO operates by iteratively updating candidate solutions based on subtraction and averaging operations, promoting exploration and exploitation balance. The algorithm is outlined as follows:
SABO Algorithm Pseudocode:
- Initialize population of solutions (weights and biases) randomly within bounds.
- Evaluate fitness (e.g., MSE) for each solution.
- For each iteration \( t \) up to \( T \):
- For each solution \( i \), compute new position:
$$ \mathbf{p}_{\text{new}}^i = \mathbf{p}^i + \mathbf{r} \cdot \frac{1}{L} \sum_{j=1}^{L} \left( \mathbf{p}^i \ominus \nu \mathbf{p}^j \right) $$
where \( \mathbf{r} \) is a random vector, \( L \) is population size, \( \ominus \) denotes subtraction with sign based on fitness comparison, and \( \nu \) is a random vector between 1 and 2. - Update solution if fitness improves:
$$ \mathbf{p}^i = \begin{cases}
\mathbf{p}_{\text{new}}^i & \text{if } F(\mathbf{p}_{\text{new}}^i) < F(\mathbf{p}^i) \\
\mathbf{p}^i & \text{otherwise}
\end{cases} $$
- For each solution \( i \), compute new position:
- Return best solution.
In our context, the solutions are the weight matrices \( \mathbf{W}_1 \) (input-hidden), \( \mathbf{W}_2 \) (hidden-output) and bias vectors \( \mathbf{b}_1 \), \( \mathbf{b}_2 \) of the BP network. The fitness function \( F \) is the RMSE on training data:
$$ F = \sqrt{ \frac{1}{n} \sum_{i=1}^{n} (y_i – \hat{y}_i)^2 } $$
where \( y_i \) is actual SOH, \( \hat{y}_i \) is predicted SOH, and \( n \) is number of samples. SABO optimizes these parameters over iterations, leading to a robust initialization for the BP network. After optimization, the network is trained further via backpropagation to fine-tune weights. This hybrid SABO-BP model capitalizes on SABO’s global search capability and BP’s local refinement, resulting in improved accuracy and convergence speed for lithium-ion battery SOH estimation. The architecture of the model is summarized in Table 2.
| Component | Specification | Value/Range |
|---|---|---|
| Input Layer | Number of neurons | 5 (Shapelet features) |
| Hidden Layer | Number of neurons | 10 (with tanh activation) |
| Output Layer | Number of neurons | 1 (SOH, linear activation) |
| Optimizer | SABO parameters | Population size: 30, Iterations: 100 |
| Training | Backpropagation | Learning rate: 0.01, Epochs: 500 |
| Loss Function | Mean Squared Error | Minimized via gradient descent |
This structured approach ensures that the model is both efficient and effective, leveraging interpretable features from the lithium-ion battery charging data.
Experimental Setup and Dataset
To validate our method, we use the Stanford-MIT battery dataset, a comprehensive collection of cycling data for commercial lithium-ion batteries (APR18650M1A). The dataset includes 124 batteries subjected to 72 different fast-charging protocols at a controlled temperature of 30°C. Each protocol follows a multi-step constant current (CC) pattern denoted as “C1 (Q1)-C2”, where C1 and C2 are current rates, and Q1 is the SOC at which switching occurs. Charging continues until the voltage reaches 3.6 V, followed by a constant voltage (CV) phase until current drops to a threshold. Discharge is performed at a constant current to 2.0 V. Batteries are cycled until capacity degrades to 80% of nominal, providing SOH labels for each cycle. This dataset is ideal for testing generalization, as it encompasses varied charging strategies that mimic real-world scenarios. We randomly select 13 batteries: 9 for training and 4 for testing. The test batteries employ distinct charging protocols, ensuring that the model is evaluated on unseen conditions. Table 3 details the selected test batteries and their protocols.
| Battery ID | Charging Protocol | Nominal Capacity | Cycles to 80% SOH |
|---|---|---|---|
| #n1 | 5.3C (54%) – 4.0C | 1.1 Ah | ~1000 |
| #n2 | 5.0C (67%) – 4.0C | 1.1 Ah | ~1200 |
| #n3 | 5.6C (19%) – 4.6C | 1.1 Ah | ~800 |
| #n4 | 5.6C (36%) – 4.3C | 1.1 Ah | ~700 |
For each battery, we extract charging voltage curves from every cycle. Preprocessing involves aligning sequences to the voltage range of interest (3.2 V to 3.6 V) and normalizing data to mitigate scale effects. Shapelets are then extracted as per the methodology, focusing on the selected end voltages and lengths. The corresponding distances form the feature set. We split the data per battery into training and testing sets chronologically, with earlier cycles for training and later ones for testing to simulate real-time estimation. Evaluation metrics include Mean Absolute Error (MAE), Root Mean Square Error (RMSE), and Coefficient of Determination (R²), defined as:
$$ \text{MAE} = \frac{1}{n} \sum_{i=1}^{n} |y_i – \hat{y}_i| $$
$$ \text{RMSE} = \sqrt{ \frac{1}{n} \sum_{i=1}^{n} (y_i – \hat{y}_i)^2 } $$
$$ R^2 = 1 – \frac{\sum_{i=1}^{n} (y_i – \hat{y}_i)^2}{\sum_{i=1}^{n} (y_i – \bar{y})^2 } $$
where \( \bar{y} \) is the mean of actual SOH. Lower MAE and RMSE indicate better accuracy, while R² close to 1 signifies excellent fit. We compare our SABO-BP model with baseline methods, including standard BP, genetic algorithm-optimized BP (GA-BP), and particle swarm optimization BP (PSO-BP), to demonstrate superiority. All experiments are implemented in Python, using libraries such as NumPy for calculations and scikit-learn for basic metrics.
Results and Discussion
The experimental results affirm the efficacy of our Shapelet-based SABO-BP approach for lithium-ion battery SOH estimation. For each test battery, the model outputs SOH predictions across cycles, which are plotted against true values. Figure 1 (not shown, but described) illustrates the close alignment between predicted and actual SOH curves, indicating that the model captures degradation trends accurately. Quantitatively, the evaluation metrics for all test batteries are summarized in Table 4.
| Battery ID | MAE (%) | RMSE (%) | R² |
|---|---|---|---|
| #n1 | 0.19 | 0.26 | 0.995 |
| #n2 | 0.46 | 0.58 | 0.982 |
| #n3 | 0.40 | 0.52 | 0.988 |
| #n4 | 0.45 | 0.55 | 0.984 |
The results show that MAE remains below 0.5% and RMSE below 0.6% across all batteries, with the best case achieving MAE of 0.19% and RMSE of 0.26%. R² values exceed 0.98, reaching 0.995 for Battery #n1, which demonstrates near-perfect fit. These metrics surpass typical industry requirements for SOH estimation, often set at MAE < 2%. The consistency across different charging protocols highlights the model’s robustness and generalization capability. This is crucial for real-world applications where a lithium-ion battery may undergo varying charging patterns. The interpretability aspect is also validated: by examining the selected Shapelets, we can identify that voltage segments near the cutoff (3.6 V) and around 3.44-3.46 V are most sensitive to aging. This insight can guide battery management system designs to monitor these regions closely.
To further validate the advantage of SABO optimization, we compare our model with other BP variants on Battery #n1. Table 5 presents the comparative results.
| Model | MAE (%) | RMSE (%) | R² |
|---|---|---|---|
| Standard BP | 0.52 | 0.67 | 0.970 |
| GA-BP | 0.24 | 0.32 | 0.993 |
| PSO-BP | 0.22 | 0.29 | 0.994 |
| SABO-BP (Proposed) | 0.19 | 0.26 | 0.995 |
SABO-BP outperforms all others, reducing MAE by approximately 12% relative to PSO-BP and 63% relative to standard BP. This improvement stems from SABO’s effective global search, which avoids local optima and yields better initial weights. Moreover, the training convergence is faster, as SABO reduces the number of epochs needed to achieve low error. These benefits translate to more reliable SOH estimates for lithium-ion batteries, especially when data is limited. Another key discussion point is data efficiency. Our method requires only partial charging data—specifically, voltage segments from the selected Shapelet regions. This contrasts with many data-driven methods that need full cycles. For instance, a typical full charge might take hours, whereas our approach can operate on snippets as short as a few minutes (corresponding to 60-100 voltage points at second-level sampling). This makes it feasible for online monitoring in applications like electric vehicles, where charging may be intermittent. We also analyze the impact of Shapelet count on performance. As shown in Figure 2 (not shown), increasing the number of Shapelets from 1 to 5 improves accuracy, but beyond 5, gains diminish or metrics worsen due to overfitting. Thus, we fix the number at 5, balancing complexity and predictive power.
The success of our method can be attributed to several factors. First, Shapelets provide a compact, interpretable representation of aging trends in lithium-ion battery voltage data. Second, the SABO algorithm enhances the neural network’s learning capability. Third, the use of robust correlation analysis ensures feature relevance. However, limitations exist. The method assumes availability of voltage data during charging, which may not be the case in all scenarios (e.g., during discharge-only operations). Also, the Stanford-MIT dataset is from controlled lab conditions; real-world data with noise and outliers could challenge the model. Future work could involve adapting the approach to handle such variations, perhaps by incorporating adaptive filtering or transfer learning. Additionally, extending Shapelet extraction to other signals like temperature or internal resistance could further enrich health indicators for lithium-ion batteries.
Conclusion
In this article, we have presented a novel framework for lithium-ion battery State of Health estimation that marries interpretable Shapelets with an optimized neural network. The method addresses key challenges in data-driven SOH estimation: poor interpretability and heavy data dependence. By extracting Shapelets from partial charging voltage curves, we derive health features that are both informative and explainable, allowing engineers to understand which aspects of the charging profile signify degradation. The SABO-optimized BP network then leverages these features to deliver accurate SOH predictions, with MAE consistently below 0.5% and R² above 0.98 on diverse test batteries. Experimental validation on the Stanford-MIT dataset confirms the method’s efficacy and generalization across varying charging protocols. Compared to baseline models, SABO-BP shows superior performance, underscoring the value of metaheuristic optimization in enhancing neural network training. This work contributes to the advancement of intelligent battery management systems by offering a practical, data-efficient solution for health monitoring. Looking ahead, we plan to integrate real-time adaptation mechanisms and expand the feature set to include multimodal data, further solidifying the role of interpretable machine learning in sustainable energy systems centered on lithium-ion batteries.
The implications of this research extend beyond academic interest. For industries reliant on lithium-ion batteries—such as automotive, consumer electronics, and renewable energy storage—accurate SOH estimation can optimize maintenance schedules, reduce costs, and enhance safety. By minimizing data requirements and providing transparent insights, our approach facilitates deployment in resource-constrained environments. Ultimately, as the world transitions to electrification, robust health monitoring for lithium-ion batteries will be indispensable, and methods like ours pave the way for smarter, more reliable energy storage solutions.
To recapitulate, the key contributions of this work are:
- Introduction of Shapelets as interpretable features for lithium-ion battery SOH estimation from partial charging data.
- Development of a correlation-based selection process to identify optimal Shapelets.
- Integration of the SABO algorithm with BP neural networks to improve prediction accuracy and convergence.
- Comprehensive validation on a public dataset, demonstrating high precision and generalization.
We believe that this framework sets a precedent for combining interpretability and performance in battery health diagnostics, fostering trust in machine learning models for critical infrastructure. Future endeavors will explore real-world deployment and integration with cloud-based monitoring platforms for widespread lithium-ion battery management.
In summary, lithium-ion batteries are pivotal to modern energy systems, and their health management is paramount. Our Shapelet-based SABO-BP model offers a promising tool to achieve this, balancing accuracy, efficiency, and interpretability. As research progresses, we anticipate further refinements that will unlock even greater potential for sustainable battery technologies.