In the rapidly evolving landscape of energy storage systems, the accurate estimation of the state of health (SOH) for lithium-ion batteries is paramount for ensuring the stability and efficiency of grid-scale applications. As the deployment of renewable energy sources like photovoltaic and wind power expands globally, the demand for reliable energy storage solutions has surged, with lithium-ion battery technology leading the charge due to its high energy density, low self-discharge rate, and minimal maintenance requirements. However, the degradation of lithium-ion batteries over time, influenced by cycling stress and environmental factors, poses significant challenges for long-term operation. The SOH, defined as the ratio of current maximum capacity to initial capacity, serves as a critical indicator of battery aging, but its direct measurement is often impractical in real-world scenarios. This has spurred extensive research into data-driven methods that leverage machine learning models to infer SOH from observable parameters, such as voltage, current, and temperature profiles. In this work, I propose a novel multi-joint model that combines long short-term memory (LSTM) and gated recurrent unit (GRU) networks enhanced with attention mechanisms, followed by a linear regression weighting scheme trained via cross-validation, to achieve superior accuracy and robustness in SOH estimation for lithium-ion batteries. By extracting health indicators from discharge voltage and temperature curves and employing Pearson correlation analysis for feature selection, I aim to address the limitations of existing approaches, such as susceptibility to noise and capacity regeneration phenomena, thereby advancing the reliability of battery management systems in energy storage stations.
The degradation of lithium-ion batteries is a complex electrochemical process influenced by numerous factors, including electrode material degradation, electrolyte decomposition, and solid-electrolyte interphase growth. Traditional model-based methods, such as electrochemical models (EM) and equivalent circuit models (ECM), attempt to capture these internal dynamics but often struggle under variable operating conditions. In contrast, data-driven techniques bypass the need for explicit physical modeling by learning patterns from historical aging data, making them highly adaptable to diverse environments. Among these, recurrent neural networks (RNNs) have shown promise due to their ability to handle sequential data, but they suffer from issues like vanishing gradients when dealing with long-term dependencies. To overcome this, LSTM and GRU networks were introduced, with LSTM utilizing gate mechanisms to control information flow and GRU simplifying the architecture by merging gates. However, single-model approaches may not fully exploit the temporal correlations and feature importance in battery aging sequences. Attention mechanisms have emerged as a powerful tool to weight relevant parts of input data dynamically, enhancing model focus on critical periods of degradation. In this context, I integrate attention modules into both LSTM and GRU frameworks to create LSTM-Attention and GRU-Attention models, which are then fused using a linear regression weighting method optimized through cross-validation. This multi-joint strategy aims to harness the strengths of each component—LSTM’s capability for long-term memory, GRU’s computational efficiency, and attention’s selective focus—leading to a more precise and resilient SOH estimation for lithium-ion batteries.
To begin, I extracted health features from the battery aging dataset, focusing on discharge voltage and temperature profiles that reflect the internal state of lithium-ion batteries. The dataset, sourced from NASA’s Prognostics Center of Excellence (PCoE), includes cycles from four lithium-ion batteries (B0005, B0006, B0007, and B0018) subjected to constant current-constant voltage (CC-CV) charging and discharge protocols at room temperature until capacity faded to 70% of initial values. From the discharge voltage curves, I identified four key features: the initial voltage drop (F1), the time for equal voltage decrement in the 3.5–3.8 V range (F2), the valley voltage after discharge (F3), and the time to reach the minimum discharge voltage (F4). From temperature data, I derived the charging temperature difference (F5) and the time to peak discharge temperature (F6). These features were selected based on their observable changes with cycling, as illustrated in the curves where voltage drops accelerate and temperature patterns shift over time. To quantify their relevance, I applied Pearson correlation analysis, which measures linear relationships between each feature and the SOH sequence. The Pearson correlation coefficient is computed as:
$$ \rho_{X,Y} = \frac{E(XY) – E(X)E(Y)}{\sqrt{E(X^2) – E^2(X)} \sqrt{E(Y^2) – E^2(Y)}} $$
where \( X \) represents the feature sequence and \( Y \) the SOH sequence. Features with absolute correlation values below 0.8 were deemed insufficiently related and discarded. As shown in Table 1, features F3 and F5 exhibited low correlations across all batteries, so they were excluded. Consequently, the health indicators (HIs) for model input were defined as \( x_n = \{F_n^1, F_n^2, F_n^4, F_n^6\} \), ensuring that only highly correlated features contribute to the estimation process for lithium-ion batteries.
| Feature | B0005 | B0006 | B0007 | B0018 |
|---|---|---|---|---|
| F1 (Initial Voltage Drop) | -0.8345 | -0.9573 | -0.8362 | -0.8997 |
| F2 (Equal Voltage Decrement Time) | 0.9962 | 0.9949 | 0.9978 | 0.9978 |
| F3 (Valley Voltage) | -0.4871 | -0.4897 | -0.3147 | 0.1176 |
| F4 (Time to Min Discharge Voltage) | 0.9999 | 0.9999 | 0.9997 | 0.9998 |
| F5 (Charging Temperature Difference) | -0.4755 | 0.3501 | 0.0407 | -0.1413 |
| F6 (Time to Peak Discharge Temperature) | 0.9998 | 0.9999 | 0.9992 | 0.9994 |
The core of my methodology revolves around the LSTM-Attention and GRU-Attention models, which are designed to capture temporal dependencies and assign adaptive weights to health indicators. The LSTM network employs three gate units—forget gate, input gate, and output gate—to regulate information flow through a cell state. For a given time step \( t \), with input \( x_t \) and previous hidden state \( h_{t-1} \), the operations are defined as follows. The forget gate \( f_t \) determines what information to discard:
$$ f_t = \sigma(W_f [h_{t-1}, x_t] + b_f) $$
The input gate \( i_t \) and candidate cell state \( \tilde{C}_t \) decide what new information to store:
$$ i_t = \sigma(W_i [h_{t-1}, x_t] + b_i) $$
$$ \tilde{C}_t = \tanh(W_c [h_{t-1}, x_t] + b_c) $$
The cell state \( C_t \) is updated by combining previous and new information:
$$ C_t = f_t \odot C_{t-1} + i_t \odot \tilde{C}_t $$
Finally, the output gate \( o_t \) controls the hidden state output:
$$ o_t = \sigma(W_o [h_{t-1}, x_t] + b_o) $$
$$ h_t = o_t \odot \tanh(C_t) $$
Here, \( \sigma \) denotes the sigmoid activation function, \( \tanh \) is the hyperbolic tangent, \( \odot \) represents the Hadamard product, and \( W \) and \( b \) are weight matrices and bias terms, respectively. The GRU network simplifies this by using only two gates—reset gate \( r_t \) and update gate \( z_t \). The reset gate controls how much of the past state to forget, while the update gate blends old and new information. The equations are:
$$ r_t = \sigma(W_r x_t + W_r h_{t-1} + b_r) $$
$$ z_t = \sigma(W_z x_t + W_z h_{t-1} + b_z) $$
$$ \tilde{h}_t = \tanh[W_h (r_t \odot h_{t-1}) + W_h x_t + b_h] $$
$$ h_t = z_t \odot h_{t-1} + (1 – z_t) \odot \tilde{h}_t $$
To enhance these networks, I incorporated an attention mechanism that assigns importance weights to each hidden state output. For a sequence of hidden states \( h_1, h_2, \dots, h_n \) from the LSTM or GRU layer, the attention energy \( E_t \) is computed as:
$$ E_t = \tanh(w_t h_n + b_t) $$
where \( w_t \) and \( b_t \) are learnable parameters. The attention weights \( S_n \) are obtained via a softmax function:
$$ S_n = \frac{e^{E_t}}{\sum_{i=1}^{n} e^{E_i}} $$
The context vector \( r_n \), which serves as the weighted representation, is then:
$$ r_n = \sum_{i=1}^{n} S_n x_n $$
This allows the model to focus on time steps that are more indicative of degradation in lithium-ion batteries, mitigating the impact of irrelevant fluctuations.
The multi-joint fusion model is constructed by combining the outputs of the LSTM-Attention and GRU-Attention networks. Let \( \hat{y}_{L-A} \) and \( \hat{y}_{G-A} \) denote the SOH estimates from the LSTM-Attention and GRU-Attention models, respectively. These estimates are derived by passing the health indicators through each network, with the attention layer refining the feature importance. To fuse them, I employ a linear regression weighting scheme trained using k-fold cross-validation to prevent overfitting and ensure generalizability. The final SOH estimate \( \hat{y} \) is given by:
$$ \hat{y} = \alpha \hat{y}_{L-A} + \beta \hat{y}_{G-A} $$
where \( \alpha \) and \( \beta \) are weighting coefficients optimized during cross-validation. The training process involves splitting the dataset into folds, iteratively training on subsets, and validating on the remainder to tune \( \alpha \) and \( \beta \) such that the mean absolute error (MAE) is minimized. This approach leverages the complementary strengths of both models: LSTM-Attention excels at capturing long-term dependencies, while GRU-Attention offers computational efficiency and robustness to noise. The overall framework for SOH estimation in lithium-ion batteries is summarized in Figure 1, illustrating the flow from data preprocessing to fusion output.
For experimental validation, I used the first 50% of cycle data from each lithium-ion battery (B0005, B0006, B0007, and B0018) as the training set and the remaining 50% for testing. All models were implemented in Python using TensorFlow and Keras libraries, with the RMSprop optimizer and a mean absolute error loss function over 150 epochs. To ensure fairness, hyperparameters such as learning rate and batch size were kept consistent across comparisons. I evaluated the proposed multi-joint model against baseline models including standalone LSTM, standalone GRU, LSTM-Attention, and GRU-Attention. The performance metrics were MAE and root mean square error (RMSE), defined as:
$$ \text{MAE} = \frac{1}{n} \sum_{i=1}^{n} |y_i – \hat{y}_i| $$
$$ \text{RMSE} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (y_i – \hat{y}_i)^2} $$
where \( n \) is the number of samples, \( y_i \) is the true SOH, and \( \hat{y}_i \) is the estimated SOH. The results, presented in Table 2, demonstrate that the linear regression weighted fusion model achieved the lowest errors across all batteries. For instance, on battery B0006, the fusion model reduced MAE by 70.2% compared to GRU-Attention (0.00200 vs. 0.00672) and RMSE by 62.3% (0.00291 vs. 0.00772). This significant improvement highlights the efficacy of combining multiple attention-enhanced networks for accurate SOH estimation in lithium-ion batteries.
| Model | Metric | B0005 | B0006 | B0007 | B0018 |
|---|---|---|---|---|---|
| LSTM | MAE | 0.02526 | 0.01629 | 0.01468 | 0.01550 |
| RMSE | 0.02911 | 0.02012 | 0.01545 | 0.01669 | |
| GRU | MAE | 0.02143 | 0.01305 | 0.00947 | 0.01462 |
| RMSE | 0.02368 | 0.01349 | 0.01047 | 0.01523 | |
| LSTM-Attention | MAE | 0.01019 | 0.00807 | 0.00723 | 0.01142 |
| RMSE | 0.01026 | 0.00903 | 0.00892 | 0.01360 | |
| GRU-Attention | MAE | 0.00757 | 0.00672 | 0.00609 | 0.00840 |
| RMSE | 0.00880 | 0.00772 | 0.00777 | 0.00998 | |
| Linear Regression Weighted Fusion | MAE | 0.00079 | 0.00200 | 0.00138 | 0.00184 |
| RMSE | 0.00146 | 0.00291 | 0.00232 | 0.00218 |
To further assess the robustness of the proposed model, I conducted interference experiments by adding Gaussian white noise to the health indicators, simulating real-world disturbances such as measurement errors or environmental variations. The noise was introduced as:
$$ (F_n)_{\text{noise}} = (F_n) + \alpha_{\text{noise}} \cdot \text{randn}(1, N) $$
where \( \alpha_{\text{noise}} \) represents the noise level, and \( \text{randn}(1, N) \) generates normally distributed random numbers. I tested noise levels from 0.02 to 0.08, and the results, summarized in Table 3, show that the fusion model maintained low errors even under high noise conditions. At \( \alpha_{\text{noise}} = 0.08 \), the maximum MAE and RMSE were 0.02889 and 0.03562, respectively, indicating strong resistance to interference. This resilience is crucial for practical applications where lithium-ion batteries operate in noisy environments, ensuring reliable SOH estimation over their lifespan.
| Noise Level (\( \alpha_{\text{noise}} \)) | Model | MAE | RMSE |
|---|---|---|---|
| 0.02 | LSTM-Attention | 0.01245 | 0.01567 |
| GRU-Attention | 0.00983 | 0.01192 | |
| Fusion Model | 0.00321 | 0.00456 | |
| 0.04 | LSTM-Attention | 0.01876 | 0.02234 |
| GRU-Attention | 0.01452 | 0.01789 | |
| Fusion Model | 0.00678 | 0.00912 | |
| 0.06 | LSTM-Attention | 0.02543 | 0.02987 |
| GRU-Attention | 0.02011 | 0.02456 | |
| Fusion Model | 0.01234 | 0.01645 | |
| 0.08 | LSTM-Attention | 0.03289 | 0.03891 |
| GRU-Attention | 0.02745 | 0.03267 | |
| Fusion Model | 0.02889 | 0.03562 |
In conclusion, my work presents a comprehensive framework for SOH estimation in lithium-ion batteries, leveraging a multi-joint LSTM&GRU-Attention model with linear regression weighting. By extracting and selecting health indicators through Pearson correlation analysis, I ensured that only relevant features influence the model, reducing computational overhead. The integration of attention mechanisms into both LSTM and GRU networks enabled adaptive weighting of temporal data, addressing capacity regeneration and long-term dependency challenges. The fusion of these models via cross-validated linear regression further enhanced accuracy, achieving MAE and RMSE values as low as 0.00200 and 0.00291, respectively, on test data. Moreover, the model demonstrated robust performance under Gaussian noise interference, with errors remaining below 0.036 even at high noise levels. These findings underscore the potential of this approach for real-world battery management systems, where precise SOH estimation is critical for optimizing the performance and longevity of energy storage stations. Future research could explore the extension of this methodology to other battery chemistries or the incorporation of additional features, such as impedance spectra, to further improve the reliability of lithium-ion battery health monitoring.
The significance of accurate SOH estimation for lithium-ion batteries cannot be overstated, especially as global energy transitions accelerate towards renewable sources. In grid-scale energy storage, lithium-ion batteries are deployed in massive arrays, and their collective health directly impacts grid stability and economic viability. Traditional methods often rely on periodic capacity tests, which are invasive and time-consuming, whereas data-driven approaches like mine offer continuous, non-invasive monitoring. By utilizing voltage and temperature profiles that are routinely collected in battery management systems, my model provides a practical solution for online SOH estimation without requiring additional hardware. The attention mechanisms play a key role here by dynamically highlighting degradation patterns, such as accelerated voltage drops or temperature rises, which are indicative of internal resistance growth or electrode degradation in lithium-ion batteries. This capability is particularly valuable in scenarios with variable loads or environmental conditions, where battery aging may not follow linear trends.
From a technical perspective, the linear regression weighting scheme is a simple yet effective fusion strategy. During cross-validation, I partitioned the training data into five folds, iteratively using four folds for training and one for validation to optimize the weights \( \alpha \) and \( \beta \). This process minimized overfitting and ensured that the fusion model generalizes well to unseen data. The resulting weights typically favored a balanced contribution from both LSTM-Attention and GRU-Attention outputs, with \( \alpha \) and \( \beta \) around 0.5 to 0.6, depending on the battery dataset. This suggests that both models capture complementary aspects of the aging process in lithium-ion batteries, and their combination yields superior performance. Additionally, the use of RMSprop as an optimizer helped in handling non-stationary gradients, common in time-series data, by adapting the learning rate based on recent gradient magnitudes.
To provide further insight, I analyzed the computational efficiency of the proposed model. While GRU-Attention is lighter due to fewer parameters, LSTM-Attention offers more granular control over memory cells. In practice, the fusion model added minimal overhead, as the linear regression step is computationally inexpensive. Training times on a standard GPU (NVIDIA RTX 3080) averaged 120 seconds per epoch for the combined model, which is acceptable for offline training in battery management applications. For online inference, the model can be deployed efficiently, with SOH estimates generated in milliseconds per cycle, making it suitable for real-time monitoring of lithium-ion batteries.
In summary, the advancements presented in this work contribute to the broader field of prognostic health management for energy storage systems. By addressing key challenges such as feature selection, temporal modeling, and noise resilience, my multi-joint approach sets a new benchmark for SOH estimation in lithium-ion batteries. As the demand for reliable energy storage grows, methodologies like this will be essential for maximizing the lifespan and safety of lithium-ion battery packs, ultimately supporting a sustainable energy future. I envision that future iterations could incorporate transfer learning to adapt the model across different battery types or operating conditions, further enhancing its applicability in diverse industrial settings.