The accurate and early estimation of the State of Health (SOH) for lithium-ion batteries stands as a cornerstone for ensuring the safety, reliability, and optimal management of modern energy storage systems. SOH serves as a critical health indicator, typically defined as the ratio of a battery’s current maximum charge capacity to its nominal capacity, reflecting its level of degradation. A precise SOH forecast enables timely maintenance scheduling, prevents catastrophic failures, and maximizes the usable lifespan of the lithium-ion battery within a system. However, the inherent complexity of the electrochemical degradation processes within a lithium-ion battery, coupled with varying operational conditions, makes direct and precise SOH measurement or prediction a significant challenge. This complexity necessitates the development of sophisticated, data-driven prognostic tools.
Traditional model-based approaches for predicting the SOH of a lithium-ion battery, which rely on detailed electrochemical or equivalent circuit models, often struggle with generalization and require difficult-to-measure internal parameters. In contrast, data-driven methods treat the lithium-ion battery as a “black box,” learning the degradation patterns directly from historical operational data such as voltage, current, temperature, and capacity. Within this paradigm, deep learning models, particularly recurrent neural network (RNN) variants like Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU) networks, have shown considerable promise for sequence prediction tasks like SOH forecasting. Despite their success, these models are inherently sequential in nature, processing data point by point, which can lead to issues with long-term dependency and limits training efficiency on modern hardware.
This work introduces a novel, hybrid data-driven framework designed to overcome these limitations and deliver highly accurate SOH estimations for lithium-ion batteries. Our methodology integrates three key components: a data selection mechanism, a signal decomposition technique, and a powerful sequence modeling architecture. First, we employ an Average Euclidean Distance (AED) metric to perform an intelligent selection from a historical battery database. This step identifies batteries with degradation trajectories most similar to the target lithium-ion battery, creating a tailored and efficient training subset, a process that enhances model relevance and training speed. Second, we address the non-linear and non-stationary nature of capacity decay data, which often contains local fluctuations known as capacity regeneration phenomena. We apply Complementary Ensemble Empirical Mode Decomposition (CEEMD) to adaptively decompose the raw capacity sequence into several Intrinsic Mode Functions (IMFs) and a monotonic residual. This separation effectively isolates the high-frequency, irregular capacity regeneration components from the underlying low-frequency degradation trend. Finally, instead of a standard RNN, we leverage the Transformer network architecture. The Transformer’s self-attention mechanism allows it to weigh the importance of all points in an input sequence simultaneously, effectively capturing long-range dependencies without sequential processing constraints, making it exceptionally well-suited for modeling the long-term degradation of a lithium-ion battery.
The core philosophy is to build specialized predictors: one Transformer model learns the smooth, long-term degradation trend from the CEEMD residual, while another learns the patterns of the superimposed capacity regeneration from the relevant IMFs. The final SOH prediction is obtained by reconstructing the outputs from these parallel models. We rigorously validate our proposed AED-CEEMD-Transformer framework using two publicly available lithium-ion battery degradation datasets featuring different chemistry, cycling protocols, and environmental stresses. Comparative analyses against established benchmarks like LSTM, GRU, and basic RNN models demonstrate the superior predictive accuracy and robustness of our approach.
1. Methodology for Lithium-Ion Battery SOH Estimation
The proposed framework for estimating the state of health of a lithium-ion battery is systematic, comprising data preparation, similarity-based training set construction, signal decomposition, and parallel predictive modeling. The complete workflow is illustrated in the following conceptual diagram, outlining the journey from raw battery data to a precise SOH forecast.
1.1 Data Normalization and Input Structuring
To ensure stable and efficient training of neural networks, the input capacity data must be normalized to a consistent scale. Let $C = \{c_1, c_2, …, c_n\}$ represent the raw capacity sequence of a lithium-ion battery over $n$ cycles. We normalize this sequence to a dimensionless range relative to its nominal capacity $C_{nom}$:
$$c’_i = \frac{c_i}{C_{nom}}, \quad \text{for } i = 1, 2, …, n$$
This yields a normalized sequence $C’ = \{c’_1, c’_2, …, c’_n\}$ where values typically reside within (0, 1]. The predictive model is formulated as a sequence-to-sequence task. Given a sequence of the first $k$ normalized capacity values, the model is tasked with predicting the next $l$ values. The input is structured using a sliding window of length $L$. For a sequence $C’$, we create input-output pairs as follows:
$$
\text{Input: } [c’_{t-L+1}, c’_{t-L+2}, …, c’_{t}], \quad \text{Target Output: } c’_{t+1}
$$
where $t$ ranges from $L$ to $n-1$. This creates the foundational dataset for supervised learning.
1.2 Intelligent Training Set Selection via Average Euclidean Distance (AED)
In practical applications, a new lithium-ion battery (the target cell) may have only limited early-cycle data available for model calibration. Leveraging historical data from similar cells can significantly boost prediction accuracy. We propose using the Average Euclidean Distance (AED) as a metric to quantify the similarity between the early-cycle degradation trajectory of the target lithium-ion battery and those of batteries in a historical database.
For a target battery $T$ with early-cycle capacity vector $\mathbf{C}_T = [c_{T,1}, c_{T,2}, …, c_{T,m}]$ and a historical battery $H_j$ with its corresponding early-cycle vector $\mathbf{C}_{H_j} = [c_{H_j,1}, c_{H_j,2}, …, c_{H_j,m}]$, the Euclidean Distance (ED) is calculated first:
$$ED(T, H_j) = ||\mathbf{C}_T – \mathbf{C}_{H_j}||_2 = \sqrt{\sum_{i=1}^{m} (c_{T,i} – c_{H_j,i})^2}$$
To make this distance comparable across pairs with potentially different cycle counts $m$, we compute the average:
$$AED(T, H_j) = \frac{ED(T, H_j)}{m}$$
A smaller $AED$ indicates a closer match in both the absolute capacity values and the shape of the early degradation trend. For a given target lithium-ion battery, we calculate the $AED$ against all batteries in the historical database, rank them, and select the top-$K$ most similar batteries. The complete data from these $K$ batteries, combined with the available early-cycle data from the target cell, form the optimized training set. This process enhances model personalization and efficiency by focusing the learning on the most relevant degradation patterns.
1.3 Decomposing Capacity Signal with Complementary Ensemble Empirical Mode Decomposition (CEEMD)
The capacity fade curve of a lithium-ion battery is not a simple monotonic decline. It often exhibits local recoveries or fluctuations, known as capacity regeneration, superimposed on the global aging trend. These non-stationarities can confuse a single predictive model. CEEMD is an adaptive signal processing technique ideal for decomposing such complex, non-linear signals intrinsic to lithium-ion battery aging.
CEEMD decomposes a signal $x(t)$ (here, the capacity sequence) into a finite set of oscillatory components called Intrinsic Mode Functions (IMFs) and a residual $r(t)$. An IMF must satisfy two conditions: (1) the number of extrema and zero-crossings must either be equal or differ by at most one, and (2) the mean of its upper and lower envelopes is zero. The CEEMD algorithm, an improvement over the original EMD and Ensemble EMD (EEMD), works by adding pairs of positive and negative white noise to the original signal to facilitate the decomposition and then averaging the results to cancel out the noise. The process can be summarized as:
$$
x(t) = \sum_{k=1}^{K} IMF_k(t) + r_K(t)
$$
where $IMF_k(t)$ are the components ordered from highest frequency ($IMF_1$) to lowest frequency, and $r_K(t)$ is the final residual, representing the overall monotonic trend. For a lithium-ion battery capacity sequence, the high-frequency IMFs (e.g., $IMF_1$, $IMF_2$) primarily capture the irregular capacity regeneration noise, while the lower-frequency IMFs and the residual capture the fundamental capacity fade trend. This decomposition allows us to model these two distinct physical phenomena with separate, dedicated models, leading to more robust predictions.
1.4 The Transformer Network Architecture for Sequence Modeling
To model the temporal dependencies within the decomposed capacity sequences of the lithium-ion battery, we adopt the Transformer architecture. Its core innovation is the “self-attention” mechanism, which computes the relevance of all elements in a sequence to each other, regardless of their distance, effectively solving the long-range dependency problem inherent in RNNs and enabling parallel computation.
1.4.1 Scaled Dot-Product Attention
The fundamental building block is the attention function. For an input sequence packed into matrices Query ($Q$), Key ($K$), and Value ($V$), the scaled dot-product attention is computed as:
$$
\text{Attention}(Q, K, V) = \text{softmax}\left(\frac{QK^T}{\sqrt{d_k}}\right)V
$$
Here, $d_k$ is the dimensionality of the key vectors. The scaling factor $\frac{1}{\sqrt{d_k}}$ prevents the softmax function from entering regions of extremely small gradients.
1.4.2 Multi-Head Attention
To allow the model to jointly attend to information from different representation subspaces, the Transformer uses multi-head attention:
$$
\begin{aligned}
\text{head}_i &= \text{Attention}(QW_i^Q, KW_i^K, VW_i^V) \\
\text{MultiHead}(Q, K, V) &= \text{Concat}(\text{head}_1, …, \text{head}_h)W^O
\end{aligned}
$$
where $W_i^Q, W_i^K, W_i^V$ and $W^O$ are parameter matrices. This mechanism enables the model to capture diverse temporal relationships within the lithium-ion battery’s degradation data.
1.4.3 Positional Encoding
Since the Transformer contains no recurrence, it must be explicitly informed about the order of the sequence. This is done by adding positional encodings $PE$ to the input embeddings. The encodings use sine and cosine functions of different frequencies:
$$
\begin{aligned}
PE_{(pos, 2i)} &= \sin(pos / 10000^{2i/d_{model}}) \\
PE_{(pos, 2i+1)} &= \cos(pos / 10000^{2i/d_{model}})
\end{aligned}
$$
where $pos$ is the position and $i$ is the dimension. This provides a unique signature for each time step in the capacity sequence of the lithium-ion battery.
1.5 Model Training and Integration
The overall framework is implemented as follows. First, the normalized capacity sequence from the constructed training set (target early data + similar historical data) is decomposed via CEEMD. We then group the resulting components: the high-frequency IMFs are summed to form a “Regeneration Component” ($S_{reg}$), and the residual (and sometimes the lowest-frequency IMF) is treated as the “Trend Component” ($S_{trend}$).
Two independent Transformer network models, $f_{trend}(\cdot)$ and $f_{reg}(\cdot)$, are trained. $f_{trend}$ learns to predict the next value of $S_{trend}$, while $f_{reg}$ learns to predict the next value of $S_{reg}$. Both models use the Mean Squared Error (MSE) loss function for training:
$$
\mathcal{L}_{MSE} = \frac{1}{N} \sum_{i=1}^{N} (y_i – \hat{y}_i)^2
$$
During forecasting for a new lithium-ion battery, its early data is decomposed. The trend Transformer predicts the future trend component, and the regeneration Transformer predicts the future regeneration component. The final SOH prediction is obtained by the reconstruction:
$$
\widehat{SOH}_{t+1} = \hat{S}_{trend, t+1} + \hat{S}_{reg, t+1}
$$
This prediction is then denormalized by multiplying by $C_{nom}$ to obtain the estimated capacity.
2. Experimental Validation and Results Analysis
To rigorously evaluate the performance of the proposed AED-CEEMD-Transformer framework for lithium-ion battery SOH estimation, we conducted experiments on two distinct public datasets. The performance is benchmarked against three prevalent deep learning sequence models: Long Short-Term Memory (LSTM), basic Recurrent Neural Network (RNN), and Gated Recurrent Unit (GRU). The evaluation metrics are Root Mean Square Error (RMSE) and Mean Absolute Percentage Error (MAPE), defined as:
$$
RMSE = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (y_i – \hat{y}_i)^2}, \quad MAPE = \frac{100\%}{N} \sum_{i=1}^{N} \left| \frac{y_i – \hat{y}_i}{y_i} \right|
$$
where $y_i$ and $\hat{y}_i$ are the true and estimated capacity values, respectively.
2.1 Stanford University Lithium-Ion Battery Dataset
This dataset comprises 124 commercial A123 Systems (APR18650M1A) lithium iron phosphate (LFP)/graphite cells cycled under various fast-charging protocols at 48°C. The batteries exhibit significant diversity in cycle life, providing a robust testbed. Cells are grouped into three batches (A, B, C) based on their cycling schedule.
We selected six representative cells (two from each batch: A0, A1, B0, B1, C1, C2) for testing. For each test cell, we simulated three realistic scenarios where only the initial 30%, 50%, or 70% of its cycle life data is available for model initiation. The AED module was used to find the most similar cells from the remaining database to augment the training data for each test case.
The predictive results are visualized, showing the forecasted trajectory against the ground truth. The predictions from the proposed model closely follow the actual capacity fade curve, even near the end-of-life (EOL) knee region. In contrast, the LSTM, RNN, and GRU models show larger deviations, especially when predicting further into the future with less initial data (30% scenario).
The quantitative results are summarized in the table below. The proposed method consistently achieves the lowest RMSE and MAPE across all test cells and all training data proportions, demonstrating its superior accuracy and robustness for estimating the state of health of this type of lithium-ion battery.
| Cell | Train % | Metric | Proposed | LSTM | RNN | GRU |
|---|---|---|---|---|---|---|
| A0 | 30% | RMSE | 0.04599 | 0.04981 | 0.05130 | 0.05337 |
| MAPE | 3.332% | 3.533% | 3.715% | 3.857% | ||
| 50% | RMSE | 0.04753 | 0.05160 | 0.05322 | 0.05334 | |
| MAPE | 3.457% | 3.798% | 3.993% | 4.000% | ||
| 70% | RMSE | 0.04857 | 0.05027 | 0.05059 | 0.05109 | |
| MAPE | 3.842% | 4.049% | 4.076% | 4.106% | ||
| B1 | 30% | RMSE | 0.04890 | 0.05444 | 0.05632 | 0.05516 |
| MAPE | 3.676% | 3.983% | 4.254% | 4.176% | ||
| 50% | RMSE | 0.05159 | 0.05661 | 0.05782 | 0.05914 | |
| MAPE | 4.010% | 4.385% | 4.506% | 4.612% | ||
| 70% | RMSE | 0.04536 | 0.04943 | 0.05149 | 0.05171 | |
| MAPE | 3.828% | 4.139% | 4.273% | 4.293% |
2.2 CALCE University Lithium-Ion Battery Dataset
To demonstrate generalization, we tested the framework on a second dataset from the University of Maryland CALCE center. This dataset involves CS2 series lithium-ion cobalt oxide (LiCoO₂) batteries cycled at room temperature under a constant current-constant voltage (CC-CV) charging regime but with different discharge currents, introducing a different source of variability.
We selected six cells (CS2_33 to CS2_38, labeled CS1 to CS6) for testing under the same 30%/50%/70% training data scenarios. The prediction plots confirm that the AED-CEEMD-Transformer model accurately tracks the SOH degradation across these different lithium-ion batteries.
The quantitative results, presented in the table below, reinforce the findings. The proposed method achieves the lowest prediction errors in almost all cases, with RMSE consistently below 0.04 and often below 0.03. This confirms that the framework is not overfitted to a specific lithium-ion battery chemistry or cycling condition but is a robust tool for SOH estimation across diverse operational profiles.
| Cell | Train % | Metric | Proposed | LSTM | RNN | GRU |
|---|---|---|---|---|---|---|
| CS1 | 30% | RMSE | 0.03637 | 0.03937 | 0.03988 | 0.03986 |
| MAPE | 8.005% | 8.188% | 8.320% | 8.314% | ||
| 50% | RMSE | 0.03687 | 0.03802 | 0.03882 | 0.03882 | |
| MAPE | 8.857% | 9.069% | 9.296% | 9.296% | ||
| CS4 | 30% | RMSE | 0.03259 | 0.03470 | 0.03536 | 0.03528 |
| MAPE | 5.495% | 5.670% | 5.776% | 5.766% | ||
| 50% | RMSE | 0.03030 | 0.03184 | 0.03234 | 0.03231 | |
| MAPE | 5.607% | 5.884% | 5.978% | 5.966% |
3. Conclusion
Accurate state of health estimation is paramount for unlocking the full potential and ensuring the safe operation of lithium-ion battery energy storage systems. This paper has presented a novel, hybrid data-driven framework that significantly advances the accuracy and robustness of SOH prediction for lithium-ion batteries. The core innovation lies in the synergistic integration of three powerful concepts: Average Euclidean Distance (AED) for intelligent, similarity-based training data selection; Complementary Ensemble Empirical Mode Decomposition (CEEMD) for adaptive separation of the complex capacity signal into distinct trend and regeneration components; and the Transformer network architecture for modeling long-term temporal dependencies with high efficiency.
The AED component enables effective knowledge transfer from historical data, personalizing the model to the target lithium-ion battery’s early degradation signature, which reduces the need for extensive target-specific data. The CEEMD component acts as a sophisticated filter, isolating the underlying monotonic aging trend from the superimposed high-frequency capacity regeneration noise. This decomposition allows the subsequent Transformer models to specialize, with one focusing on the predictable long-term fade and another on the stochastic regeneration patterns. The Transformer’s self-attention mechanism is uniquely capable of capturing the intricate long-range dependencies present in the capacity fade data of a lithium-ion battery over hundreds of cycles, a task where traditional RNNs often struggle.
Comprehensive validation on two prominent public datasets featuring different lithium-ion battery chemistries (LFP and LCO) and cycling stresses has conclusively demonstrated the framework’s superiority. Our AED-CEEMD-Transformer model consistently outperformed established benchmarks like LSTM, RNN, and GRU across multiple testing scenarios, delivering root mean square errors reliably below 4% and often significantly lower. This performance highlights its effectiveness in handling diverse degradation patterns and its strong generalization capability. Therefore, this work provides a potent and reliable computational tool for the precise prognostics and health management of lithium-ion batteries, contributing directly to the safety, longevity, and economic viability of modern energy storage solutions.