In recent years, the rapid development of renewable energy and the global push for carbon peak and carbon neutrality have significantly increased the demand for new energy vehicles. Energy storage technology plays a critical role in integrating renewable energy sources and enhancing energy security. Among various energy storage devices, energy storage lithium batteries have become widely adopted due to their high output voltage, low self-discharge rate, high reliability, long life cycle, high energy density, and minimal environmental impact. However, as the number of charge-discharge cycles increases, the performance of energy storage lithium batteries gradually degrades, leading to reduced available capacity and increased internal resistance, ultimately shortening the battery’s lifespan. Therefore, accurately predicting the remaining useful life (RUL) of energy storage lithium batteries is crucial for ensuring device performance, user experience, and environmental sustainability.
The RUL of a battery refers to the number of remaining charge-discharge cycles before the battery reaches its end-of-life (EOL) threshold, typically defined as a 30% reduction in actual capacity. Accurate RUL prediction helps in timely maintenance and replacement, preventing unexpected failures. Existing RUL prediction methods can be broadly categorized into model-based approaches, data-driven approaches, and hybrid multi-model fusion approaches. Model-based methods rely on physical-chemical models to simulate internal battery processes, such as electrochemical reactions and aging, but they require extensive prior knowledge and are sensitive to parameter accuracy and experimental assumptions. Data-driven methods, including machine learning and neural networks, extract features from historical data to build predictive models. While these methods can capture complex patterns, they often struggle with noise and local fluctuations in capacity data, such as those caused by capacity regeneration phenomena, where the battery’s capacity temporarily increases during rest periods due to internal recovery mechanisms.
To address these challenges, we propose a novel multi-scale decomposition and multi-model fusion framework for energy storage lithium battery RUL prediction. Our approach leverages Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) to decompose the original capacity data into multiple intrinsic mode functions (IMFs), effectively separating noise and short-term fluctuations from the long-term degradation trend. For the high-frequency components, which contain noise and local variations, we employ a Bidirectional Long Short-Term Memory (BiLSTM) network to capture complex temporal dependencies. For the low-frequency components, which represent the main degradation trend, we use Gaussian Process Regression (GPR) for robust modeling. Additionally, we optimize the model parameters using Adaptive Particle Swarm Optimization (APSO) to enhance prediction accuracy. The predictions from each component are then fused to obtain the final RUL estimate. We validate our model on public datasets, including NASA and CALCE, through comparative, ablation, and generalization experiments. The results demonstrate that our model achieves superior performance with minimal error metrics, highlighting its robustness and high accuracy for energy storage lithium battery RUL prediction.
Methodology
Our proposed framework, termed CEEMDAN-APSO-BiLSTM-GPR, integrates multi-scale decomposition and multi-model fusion to handle the nonlinear and non-stationary nature of energy storage lithium battery capacity data. The framework consists of four main modules: data preprocessing, decomposition, inference, and prediction. Below, we detail each component and the overall workflow.
Data Preprocessing
The capacity data of energy storage lithium batteries is selected as the health indicator for RUL prediction due to its direct correlation with battery degradation. Initially, the raw data is cleaned to remove outliers and normalized to ensure consistency. The capacity time series is constructed using a sliding window approach, where a window of length L is moved stepwise over the data to preserve temporal context. For a sequence of capacity values \( X = [x_1, x_2, \ldots, x_n] \), the windowing process generates input-output pairs for model training. This step is crucial for capturing the sequential dependencies in the data.
Multi-Scale Decomposition with CEEMDAN
CEEMDAN is employed to decompose the capacity time series into multiple IMFs, which represent different frequency components. This decomposition helps in mitigating the effects of noise and local fluctuations, such as those induced by capacity regeneration. The CEEMDAN algorithm improves upon traditional EMD and EEMD by incorporating adaptive noise, leading to better mode separation and reduced mode mixing. The steps for CEEMDAN are as follows:
1. Preprocess the original signal by removing trends and means.
2. Set the number of ensemble cycles N and the noise standard deviation σ.
3. For each cycle, perform EMD on the signal to obtain IMFs:
$$ x(t) = \sum_{i=1}^{N_{\text{IMF}}} c_i(t) + r_N(t) $$
where \( c_i(t) \) are the IMFs and \( r_N(t) \) is the residue.
4. For each IMF, estimate the noise level standard deviation σ_i and generate a white noise sequence \( n_i(t) \).
5. Add the noise to the corresponding IMF:
$$ c_i'(t) = c_i(t) + n_i(t) $$
6. Average the IMFs over N cycles to obtain the final IMFs:
$$ c_i”(t) = \frac{1}{N} \sum_{j=1}^{N} c_i'(t) $$
7. Reconstruct the denoised signal by summing the final IMFs:
$$ x'(t) = \sum_{i=1}^{N_{\text{IMF}}} c_i”(t) $$
The decomposition results in several IMFs, with high-frequency components (e.g., IMF1, IMF2) capturing noise and short-term variations, and low-frequency components (e.g., the last IMF) representing the long-term degradation trend. The Pearson correlation coefficient between each IMF and the original capacity is computed to quantify their relevance, as shown in Table 1 for a sample battery.
| IMF Component | B0005 Battery | CS2_35 Battery |
|---|---|---|
| IMF1 | 0.069 | 0.075 |
| IMF2 | 0.204 | -0.034 |
| IMF3 | 0.001 | 0.149 |
| IMF4 | 0.997 | -0.165 |
| IMF5 | – | 0.256 |
| IMF6 | – | 0.992 |
Modeling with BiLSTM and GPR
After decomposition, the high-frequency and low-frequency components are modeled separately using BiLSTM and GPR, respectively. This targeted approach allows each model to leverage its strengths: BiLSTM for capturing temporal patterns in noisy data and GPR for modeling smooth trends with uncertainty quantification.
Bidirectional Long Short-Term Memory (BiLSTM)
BiLSTM is a variant of recurrent neural networks that processes sequences in both forward and backward directions, enabling it to capture context from past and future states. For a given input sequence \( X = [x_1, x_2, \ldots, x_n] \), the BiLSTM computes hidden states as follows:
Forward LSTM:
$$ \overrightarrow{h}_t = \text{LSTM}_{\rightarrow}(x_t, \overrightarrow{h}_{t-1}, \overrightarrow{c}_{t-1}) $$
Backward LSTM:
$$ \overleftarrow{h}_t = \text{LSTM}_{\leftarrow}(x_t, \overleftarrow{h}_{t+1}, \overleftarrow{c}_{t+1}) $$
The final hidden state is the concatenation of both directions:
$$ h_t = [\overrightarrow{h}_t; \overleftarrow{h}_t] $$
Each LSTM unit uses gating mechanisms (input, forget, and output gates) to regulate information flow:
$$ f_t = \sigma(W_f \cdot [h_{t-1}, x_t] + b_f) $$
$$ i_t = \sigma(W_i \cdot [h_{t-1}, x_t] + b_i) $$
$$ g_t = \tanh(W_g \cdot [h_{t-1}, x_t] + b_g) $$
$$ C_t = f_t \cdot C_{t-1} + i_t \cdot g_t $$
$$ o_t = \sigma(W_o \cdot [h_{t-1}, x_t] + b_o) $$
$$ h_t = o_t \cdot \tanh(C_t) $$
where \( \sigma \) is the sigmoid function, \( W \) and \( b \) are weights and biases, and \( C_t \) is the cell state.
Gaussian Process Regression (GPR)
GPR is a Bayesian non-parametric approach that models the target variable as a Gaussian process. For input \( X \) and output \( Y \), we assume:
$$ y \sim \mathcal{N}(m(X), K) $$
where \( m(X) \) is the mean function (often set to zero or a constant), and \( K \) is the covariance matrix defined by a kernel function. We use a combination of Gaussian and linear kernels, with the Gaussian kernel given by:
$$ k(x_i, x_j) = \sigma_f^2 \exp\left(-\frac{(x_i – x_j)^2}{2l^2}\right) $$
where \( \sigma_f^2 \) is the variance and \( l \) is the length scale. For prediction at new inputs \( X^* \), the mean and variance are computed as:
$$ y^* = K_* [K + \sigma^2 I]^{-1} y $$
$$ \sigma^2_* = K_{**} – K_* [K + \sigma^2 I]^{-1} K_*^T $$
where \( K_* \) is the covariance between training and test points, and \( K_{**} \) is the covariance of test points.
Parameter Optimization with APSO
To enhance model performance, we use APSO to optimize the hyperparameters of both BiLSTM and GPR. APSO dynamically adjusts the inertia weight and acceleration coefficients based on the swarm’s fitness, leading to faster convergence and better global exploration. The algorithm steps are:
1. Initialize particle positions and velocities randomly.
2. Set parameters: maximum iterations T, velocity limits \( V_{\text{max}} \) and \( V_{\text{min}} \), initial inertia weight \( \omega_{\text{max}} \), minimum inertia weight \( \omega_{\text{min}} \), and acceleration factors \( c_1 \) and \( c_2 \).
3. For each particle, evaluate the fitness function (e.g., prediction error).
4. Update personal best (\( P_{\text{best}} \)) and global best (\( G_{\text{best}} \)) positions.
5. Update velocity and position:
$$ V_i^{t+1} = \omega^t V_i^t + c_1 r_1 (P_{\text{best}_i^t – X_i^t) + c_2 r_2 (G_{\text{best}}^t – X_i^t) $$
$$ X_i^{t+1} = X_i^t + V_i^{t+1} $$
6. Adaptively update inertia weight:
$$ \omega^t = \omega_{\text{max}} – \frac{t}{T} (\omega_{\text{max}} – \omega_{\text{min}}) $$
7. Enforce velocity limits and repeat until convergence.
Fusion and RUL Calculation
The predictions from the BiLSTM (for high-frequency IMFs) and GPR (for low-frequency IMFs) are summed to reconstruct the overall capacity prediction. The RUL is then calculated as the difference between the EOL cycle and the prediction start cycle. If \( T \) is the start cycle and \( T_{\text{EOL}} \) is the actual EOL cycle, the true RUL is \( T_{\text{RUL}} = T_{\text{EOL}} – T \). The predicted RUL \( \hat{T}_{\text{RUL}} \) is derived similarly from the predicted capacity curve and the failure threshold (70% of initial capacity).
Experimental Validation
We evaluate our model on two public datasets: NASA battery aging dataset (B0005, B0006, B0007, B0018) and CALCE battery cycle test dataset (CS2_35, CS2_36). The capacity degradation curves for these energy storage lithium batteries exhibit nonlinear trends with noise, as shown in the datasets. We use time windowing with a length L to create sequential inputs. The training set comprises data from cycles 1 to T, and predictions start at cycle T.
Evaluation Metrics
We use Absolute Error (AE), Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and Coefficient of Determination (\( R^2 \)) to assess performance. AE measures the absolute difference between true and predicted RUL, while MAE and RMSE evaluate capacity prediction accuracy:
$$ \text{AE} = |T_{\text{RUL}} – \hat{T}_{\text{RUL}}| $$
$$ \text{MAE} = \frac{1}{n} \sum_{i=1}^{n} |y_i – \hat{y}_i| $$
$$ \text{RMSE} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (y_i – \hat{y}_i)^2} $$
$$ R^2 = 1 – \frac{\sum_{i=1}^{n} (y_i – \hat{y}_i)^2}{\sum_{i=1}^{n} (y_i – \bar{y})^2} $$
Results and Analysis
We conduct experiments with different prediction start points to demonstrate model robustness. For instance, on B0005, starting at cycle 80, our model achieves AE=0, MAE=0.0015 Ah, RMSE=0.0018 Ah, and \( R^2 \)=0.9994. The capacity prediction curve closely follows the true degradation, effectively mitigating noise and fluctuations. Ablation studies confirm the necessity of each component: removing CEEMDAN, APSO, BiLSTM, or GPR leads to increased errors, as summarized in Table 2.
| Components Used | AE | MAE (Ah) | RMSE (Ah) | R² |
|---|---|---|---|---|
| CEEMDAN, APSO, BiLSTM, GPR | 0 | 0.0015 | 0.0018 | 0.9994 |
| CEEMDAN, BiLSTM, GPR | 1 | 0.0035 | 0.0046 | 0.9885 |
| CEEMDAN, APSO, BiLSTM | 2 | 0.0052 | 0.0093 | 0.9324 |
| CEEMDAN, APSO, GPR | 3 | 0.0067 | 0.0113 | 0.9196 |
| BiLSTM only | 4 | 0.0265 | 0.0359 | 0.7976 |
| GPR only | 5 | 0.0385 | 0.0479 | 0.7346 |
Comparative experiments with other methods, such as LSTM, BiLSTM, EEMD-GRU-MLR, EEMD-GRU-TCN-Attention, and VMD-LSTM-GPR, show that our model consistently achieves lower errors across multiple batteries. For example, on NASA datasets, our model’s AE ranges from 0 to 1, MAE from 0.0015 to 0.0042 Ah, and RMSE from 0.0018 to 0.0050 Ah, outperforming all baselines. Generalization tests on CALCE datasets (CS2_35 and CS2_36) further confirm the model’s robustness, with AE=1, MAE≈0.003 Ah, and RMSE≈0.0045 Ah for different start points. These results highlight the model’s ability to handle diverse energy storage lithium battery data without overfitting.
Conclusion
We have presented a multi-scale decomposition and multi-model fusion framework for accurate RUL prediction of energy storage lithium batteries. By integrating CEEMDAN, BiLSTM, GPR, and APSO, our model effectively addresses noise, local fluctuations, and nonlinear degradation in capacity data. The decomposition separates high-frequency noise from low-frequency trends, enabling targeted modeling that captures both short-term variations and long-term patterns. Experimental results on NASA and CALCE datasets demonstrate that our model achieves minimal prediction errors, with AE as low as 0, MAE down to 0.15%, and RMSE down to 0.18%, along with high \( R^2 \) values. The framework exhibits strong generalization across different batteries and operating conditions, making it suitable for practical applications in energy storage lithium battery management systems. Future work could explore real-time adaptation and integration with other health indicators to further enhance prediction reliability for energy storage lithium batteries.