The pursuit of carbon neutrality has catalyzed a global transition towards sustainable energy systems, placing renewable energy integration at the forefront of modern power grids. Within this paradigm, microgrids combining wind, solar, and electrochemical energy storage have emerged as a cornerstone for achieving energy sustainability and grid stability. Effective configuration of energy storage system capacity is paramount for mitigating the inherent intermittency and power fluctuations of renewable sources. Among various electrochemical storage technologies, the lithium-ion battery stands out due to its high energy density, long cycle life, and lack of memory effect, making it indispensable for applications ranging from consumer electronics and electric vehicles to large-scale grid storage. However, the performance of a lithium-ion battery inevitably degrades over time due to complex internal electrochemical processes, such as the loss of active material and depletion of electrolyte, leading to a gradual and nonlinear decline in its usable capacity. Accurate prediction of this capacity fade is crucial for predictive maintenance, optimizing battery utilization across its lifecycle, and ensuring the safety and reliability of the systems it powers.
Capacity fade in a lithium-ion battery is not a linear process. It typically exhibits a distinctive “S”-shaped trajectory comprising three stages: an initial rapid fade, a long period of gradual degradation, and a final accelerated decline leading to failure. A critical aspect complicating this trend is the “capacity regeneration” or “recovery” effect, where the apparent capacity increases slightly after periods of rest due to the rebalancing of active materials within the cell. This phenomenon introduces significant nonlinearity and noise into the capacity-time series, posing a substantial challenge for traditional long-term forecasting models. While short-term, online capacity estimation has seen significant progress, accurate long-term trend prediction remains essential for planning battery replacement, enabling second-life applications (e.g., repurposing electric vehicle batteries for grid storage), and facilitating efficient resource recycling.
Existing prediction methodologies can be broadly categorized into model-based and data-driven approaches. Model-based methods rely on constructing electrochemical or equivalent circuit models to simulate internal degradation mechanisms. Although insightful, these models are often complex, computationally intensive, and sensitive to unmodeled external factors. Conversely, data-driven methods, which learn patterns directly from historical operational data, have gained prominence for their flexibility and practicality. Techniques such as Support Vector Regression (SVR), Relevance Vector Machine (RVM), and various neural networks (e.g., Back-Propagation Neural Networks and Long Short-Term Memory networks) have been applied. However, for long-term trend prediction, many of these methods suffer from error accumulation during iterative forecasting and are particularly sensitive to the noisy fluctuations caused by capacity recovery, leading to suboptimal tracking of the underlying degradation trend, especially in the critical rapid fade phase.
To address these limitations, this work proposes a robust framework for predicting the long-term capacity fade of lithium-ion battery cells. The core innovation lies in the synergistic combination of signal processing and time series forecasting. For datasets exhibiting strong capacity recovery effects, we first employ Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) to decompose the raw capacity sequence into a finite set of oscillatory Intrinsic Mode Functions (IMFs) and a monotonic residual trend. This process effectively separates the stochastic recovery “noise” from the deterministic degradation “signal.” Subsequently, the Holt-Winters exponential smoothing method—a classical yet powerful time series model—is tailored to forecast each component. The seasonal Holt-Winters model captures the quasi-periodic oscillations in the IMFs, while the non-seasonal model tracks the smooth residual trend. The final capacity forecast is obtained by summing the predictions of all components. For capacity sequences with minimal recovery, a direct application of the non-seasonal Holt-Winters model proves sufficient. This hybrid CEEMDAN-Holt-Winters approach demonstrates superior accuracy and robustness in tracking the nonlinear capacity fade trajectory across diverse lithium-ion battery datasets.
Theoretical Foundation: CEEMDAN and Holt-Winters Models
1. Signal Decomposition with CEEMDAN
Empirical Mode Decomposition (EMD) is a adaptive algorithm for analyzing non-stationary signals like battery capacity fade. However, it suffers from mode mixing. While Ensemble EMD (EEMD) mitigates this by adding white noise, it introduces reconstruction errors. CEEMDAN provides a superior solution by adding adaptive white noise at each stage of decomposition, effectively resolving mode mixing and reducing residual noise. Given a lithium-ion battery capacity sequence \( C(t) \), the CEEMDAN algorithm proceeds as follows:
1. Create an ensemble by adding adaptive white noise \( \omega^i(t) \) to the original signal:
$$ C^i(t) = C(t) + \alpha_0 \omega^i(t), \quad i = 1, 2, …, I $$
where \( \alpha_0 \) is the noise amplitude coefficient and \( I \) is the number of ensemble members.
2. Perform EMD on each \( C^i(t) \) to obtain the first mode \( \overline{IMF}_1^i(t) \). The first CEEMDAN mode is the ensemble average:
$$ IMF_1(t) = \frac{1}{I} \sum_{i=1}^{I} \overline{IMF}_1^i(t) $$
The first residual is calculated as:
$$ r_1(t) = C(t) – IMF_1(t) $$
3. For the \( k \)-th stage (\( k \geq 2 \)), decompose the ensemble \( r_{k-1}(t) + \alpha_{k-1} E_{k-1}(\omega^i(t)) \) using EMD to get its first EMD mode, where \( E_{k}(\cdot) \) denotes the \( k \)-th EMD mode. The \( k \)-th CEEMDAN mode is:
$$ IMF_k(t) = \frac{1}{I} \sum_{i=1}^{I} E_1( r_{k-1}(t) + \alpha_{k-1} E_{k-1}(\omega^i(t)) ) $$
Update the residual:
$$ r_k(t) = r_{k-1}(t) – IMF_k(t) $$
4. Repeat step 3 until the residual \( r_K(t) \) becomes monotonic or has no more than two extrema, indicating no further oscillations can be extracted. The original signal is perfectly reconstructed as:
$$ C(t) = \sum_{k=1}^{K} IMF_k(t) + r_K(t) $$
Here, \( \{IMF_1, IMF_2, …, IMF_K\} \) represent oscillatory components of decreasing frequency, and \( r_K(t) \) represents the final monotonic trend component, effectively stripping the capacity recovery noise from the core degradation trend of the lithium-ion battery.
2. Trend and Seasonality Forecasting with Holt-Winters
The Holt-Winters method is a powerful exponential smoothing technique designed for time series data exhibiting trend and seasonality. It is highly effective for tracking and extrapolating smooth trends, making it ideal for forecasting the decomposed components of lithium-ion battery capacity data. The method has three primary variants: non-seasonal, additive seasonal, and multiplicative seasonal. The additive model is typically used when seasonal variations are constant in magnitude, while the multiplicative model is for variations proportional to the level of the series.
Holt-Winters Non-Seasonal Model:
This model is defined by two smoothing equations for level \( S_t \) and trend \( b_t \).
$$ S_t = \alpha C_t + (1-\alpha)(S_{t-1} + b_{t-1}) $$
$$ b_t = \gamma (S_t – S_{t-1}) + (1-\gamma)b_{t-1} $$
The \( m \)-step-ahead forecast is given by:
$$ \hat{C}_{t+m} = S_t + m b_t $$
where \( \alpha \) and \( \gamma \) are smoothing parameters between 0 and 1. This model is applied directly to smooth trend components or to capacity series without recovery noise.
Holt-Winters Additive Seasonal Model:
This model incorporates a seasonal component \( I_t \) with period \( L \). The smoothing equations are:
$$ S_t = \alpha (C_t – I_{t-L}) + (1-\alpha)(S_{t-1} + b_{t-1}) $$
$$ b_t = \gamma (S_t – S_{t-1}) + (1-\gamma)b_{t-1} $$
$$ I_t = \beta (C_t – S_t) + (1-\beta)I_{t-L} $$
The \( m \)-step-ahead forecast is:
$$ \hat{C}_{t+m} = S_t + m b_t + I_{t-L+1+(m-1)\mod L} $$
where \( \beta \) is the seasonal smoothing parameter. This model is well-suited for forecasting the oscillatory IMFs from the CEEMDAN decomposition, as they exhibit quasi-periodic behavior akin to seasonality.
The parameters \( (\alpha, \beta, \gamma) \) are typically optimized by minimizing the sum of squared one-step-ahead prediction errors on the training data. This framework allows the model to adaptively learn the evolving pattern of each component in the lithium-ion battery capacity sequence.
Proposed Methodology: A Hybrid CEEMDAN-Holt-Winters Framework
The proposed framework for predicting lithium-ion battery capacity fade is designed to be adaptive and robust. The workflow is illustrated in the following diagram and described in detail thereafter.
Step 1: Data Acquisition and Preprocessing. Obtain the historical discharge capacity degradation dataset \( \mathbf{C} = \{c_1, c_2, …, c_N\} \), where \( c_i \) is the capacity at the \( i \)-th cycle and \( N \) is the number of cycles up to the prediction point.
Step 2: Adaptive Decomposition. Analyze the capacity sequence for the presence of significant capacity recovery noise. If strong oscillatory behavior is observed, apply CEEMDAN to decompose \( \mathbf{C} \) into \( K \) IMFs and a residual trend \( R \):
$$ \mathbf{C} = \sum_{k=1}^{K} IMF_k + R $$
If the sequence is relatively smooth (minimal recovery), skip to Step 4 and apply the non-seasonal Holt-Winters model directly.
Step 3: Train-Test Split for Components. For each decomposed component (each \( IMF_k \) and \( R \)), separate the data into a training set (cycles before the prediction starting point \( ST \)) and a testing set (cycles from \( ST \) to end-of-life).
Step 4: Component-Wise Modeling and Forecasting.
- For the residual trend component \( R \), fit a Holt-Winters Non-Seasonal Model on its training data to capture the long-term degradation trend.
- For each oscillatory component \( IMF_k \), fit a Holt-Winters Additive Seasonal Model on its training data. The period \( L \) for each IMF is determined based on its dominant oscillation frequency.
- For a smooth capacity sequence (no decomposition), fit a single Holt-Winters Non-Seasonal Model on the original training data.
Each fitted model is then used to generate forecasts for its respective component over the desired prediction horizon.
Step 5: Forecast Reconstruction. For decomposed sequences, sum the forecasts of all individual components to obtain the final capacity prediction:
$$ \hat{\mathbf{C}}_{future} = \hat{R}_{future} + \sum_{k=1}^{K} \widehat{IMF_k}_{future} $$
For non-decomposed sequences, the output from the single Holt-Winters model is the final prediction.
| Source | Cell ID | Chemistry | Nominal Capacity (Ah) | Key Test Conditions | Degradation Characteristics |
|---|---|---|---|---|---|
| NASA PCoE | B0005 | Graphite/LiNiCoMnO2 | 2.0 | RT, CC-CV Charge, 2A CC Discharge to 2.7V/2.5V | Short cycle life, prominent capacity recovery. |
| B0006 | Short cycle life, prominent capacity recovery. | ||||
| CALCE | CS2_35 | Graphite/LiCoO2 | 1.1 | RT, CC-CV Charge, 1C CC Discharge to 2.7V | Long cycle life, clear “S”-shape, strong recovery. |
| CS2_36 | Long cycle life, clear “S”-shape, strong recovery. | ||||
| MIT-Stanford-Toyota | CH_19 | Graphite/LFP | 1.1 | 30°C, Fast Multi-step Charge, 4C CC Discharge to 2.0V | Long cycle life, smooth degradation, weak recovery. |
| CH_43 | Long cycle life, smooth degradation, weak recovery. |
Experimental Validation and Results
To comprehensively validate the proposed framework, we utilized three publicly available datasets from NASA, CALCE, and MIT-Stanford-Toyota Research Center, encompassing different lithium-ion battery chemistries, cycling protocols, and degradation characteristics (see Table 1). Two prediction starting points (ST1, ST2) were selected for each cell to evaluate performance under different amounts of historical data, with a focus on predicting the rapid fade phase following the second inflection point.
Evaluation Metrics
The prediction accuracy is quantified using Mean Absolute Error (MAE) and Root Mean Square Error (RMSE), defined as:
$$ MAE = \frac{1}{M} \sum_{i=1}^{M} | \hat{C}_i – C_i | $$
$$ RMSE = \sqrt{ \frac{1}{M} \sum_{i=1}^{M} ( \hat{C}_i – C_i )^2 } $$
where \( \hat{C}_i \) and \( C_i \) are the predicted and true capacity values for the \( i \)-th cycle in the test set of length \( M \). Lower values indicate better prediction performance.
Comparative Methods
The proposed CEEMDAN-Holt-Winters (CEEMDAN-HW) method is compared against three established approaches:
- Holt-Winters alone (HW): Direct application of the non-seasonal Holt-Winters model to the raw capacity sequence. This tests the necessity of the decomposition step.
- Optimized Support Vector Regression (SVR): A widely used kernel-based regression method. Parameters were optimized via grid search to provide a strong benchmark.
- CEEMDAN-LSTM (CEEMDAN-LSTM): A deep learning benchmark where CEEMDAN components are forecast using Long Short-Term Memory networks, a state-of-the-art sequence modeling technique.
For the Toyota dataset with weak recovery, CEEMDAN decomposition was not applied, and comparisons were made between HW, SVR, and LSTM.
Prediction Results and Analysis
The forecasting results for all cells are summarized in Table 2. The proposed CEEMDAN-HW method consistently achieves the lowest or near-lowest MAE and RMSE across nearly all test cases. Key observations are:
1. Effectiveness of Decomposition: For cells with strong capacity recovery (NASA, CALCE), the CEEMDAN-HW method significantly outperforms the standalone HW model. For instance, for cell CS2_35 at ST1, CEEMDAN-HW reduces the MAE by 76% compared to HW (0.023 vs. 0.096). This demonstrates that isolating the recovery noise via CEEMDAN allows the Holt-Winters model to focus on and more accurately track the underlying smooth degradation trend of the lithium-ion battery.
2. Superior Trend Tracking: The CEEMDAN-HW method shows a clear advantage over SVR and CEEMDAN-LSTM in long-term trend extrapolation. While optimized SVR can sometimes produce competitive linear fits, it often fails to capture the nonlinear curvature in the rapid fade phase. The CEEMDAN-LSTM method, despite its representational power, suffers from error accumulation in multi-step forecasting and sensitivity to training data, leading to divergent predictions in some cases (e.g., high errors for B0005 and CS2_36). The Holt-Winters model’s inherent structure for trend extrapolation proves more reliable.
3. Robustness and Adaptability: The framework adapts well to different lithium-ion battery behaviors. For the smooth Toyota dataset (CH_19, CH_43), the simple non-seasonal HW model itself performs excellently, often besting both SVR and LSTM. This shows that when the capacity signal is clean, the proposed method defaults to an effective and parsimonious model.
4. Impact of Prediction Start Point: As expected, predictions starting later in the battery’s life (ST2) generally yield lower errors than those starting earlier (ST1), as more historical data is available to train the models. The proposed method maintains its relative advantage at both stages.
| Cell ID | Start Point (Cycle) | CEEMDAN-HW | HW | Optimized SVR | CEEMDAN-LSTM |
|---|---|---|---|---|---|
| B0005 | ST1 (81) | 0.012 / 0.015 | 0.018 / 0.023 | 0.027 / 0.031 | 0.080 / 0.100 |
| ST2 (101) | 0.011 / 0.014 | 0.015 / 0.018 | 0.025 / 0.032 | 0.056 / 0.069 | |
| B0006 | ST1 (81) | 0.023 / 0.028 | 0.096 / 0.104 | 0.044 / 0.050 | 0.092 / 0.120 |
| ST2 (101) | 0.017 / 0.021 | 0.049 / 0.054 | 0.016 / 0.020 | 0.150 / 0.173 | |
| CS2_35 | ST1 (669) | 0.011 / 0.015 | 0.080 / 0.095 | 0.017 / 0.021 | 0.033 / 0.039 |
| ST2 (719) | 0.008 / 0.011 | 0.025 / 0.031 | 0.033 / 0.038 | 0.049 / 0.060 | |
| CS2_36 | ST1 (727) | 0.024 / 0.029 | 0.131 / 0.147 | 0.049 / 0.054 | 0.046 / 0.050 |
| ST2 (777) | 0.015 / 0.020 | 0.048 / 0.053 | 0.020 / 0.026 | 0.137 / 0.153 | |
| CH_19 | ST1 (443) | – | 0.003 / 0.004 | 0.007 / 0.008 | 0.024 / 0.035 |
| ST2 (464) | – | 0.009 / 0.009 | 0.020 / 0.023 | 0.056 / 0.076 | |
| CH_43 | ST1 (601) | – | 0.009 / 0.012 | 0.016 / 0.017 | 0.023 / 0.030 |
| ST2 (619) | – | 0.014 / 0.015 | 0.020 / 0.023 | 0.079 / 0.107 |
Note: For CH_19 and CH_43, CEEMDAN was not applied. HW denotes the direct non-seasonal Holt-Winters model. Best results for each cell/start point are in bold.
Conclusion and Future Work
This work presents a robust and adaptive data-driven framework for predicting the long-term capacity fade of lithium-ion battery cells. The core innovation is the strategic integration of CEEMDAN signal decomposition with the Holt-Winters exponential smoothing model. CEEMDAN effectively decouples the challenging capacity recovery noise from the underlying degradation trend, while Holt-Winters provides a stable and accurate mechanism for extrapolating both the oscillatory and trend components. Extensive validation on multiple datasets with varying chemistries, lifetimes, and degradation patterns demonstrates the framework’s superiority over standalone and competing hybrid methods. It achieves high prediction accuracy (with MAE often below 2.5% of nominal capacity) and exhibits strong robustness, reliably tracking the critical rapid fade phase essential for prognostic and health management of lithium-ion battery systems.
The proposed method offers a practical tool for battery management systems, enabling better planning for maintenance, replacement, and second-life deployment of lithium-ion battery packs. Future research will focus on enhancing the framework’s generality. A key direction is developing a methodology for early-life prediction, where only initial cycle data is available to forecast the entire capacity trajectory, including the identification of inflection points. Furthermore, integrating external stress factors such as temperature and load profiles into the Holt-Winters modeling structure could improve prediction accuracy under varying operational conditions, leading to a more comprehensive state-of-health forecasting system for lithium-ion battery applications.