In the context of global energy transition and sustainable development, the optimization of energy structures and the innovation of smart grid technologies have become vital forces driving social progress. Energy storage systems, as a key link in balancing grid supply and demand and improving energy utilization efficiency, are increasingly important. Among various battery chemistries, lithium iron phosphate (LFP) energy storage cells stand out due to their excellent safety, long cycle life, and stable electrochemical performance. They are widely used in smart grids, home energy storage, and electric vehicles. However, as application scenarios expand and become more complex, safety concerns remain a critical challenge. The stress induced by structural constraints within battery modules has gradually become a key factor limiting performance and safety. During charge and discharge, internal stress can cause structural deformation and even lead to safety incidents. Therefore, in-depth study of the stress characteristics of LFP energy storage cells, exploring the intrinsic relationship between stress changes and battery states, and developing state estimation algorithms that integrate battery stress with electrical parameters are of great significance for enhancing the digital twin capability and safety management of energy storage systems.
LFP energy storage cells exhibit wide voltage plateaus for both positive and negative electrodes. When the battery operates within the voltage plateau region, the limited precision of voltage sensors in the battery management system makes it difficult to correct the state of charge (SOC) accurately. Additionally, the current in energy storage systems is relatively stable, making model parameters difficult to observe, which further complicates SOC estimation. Since SOC cannot be directly measured by sensors, it must be estimated using physical quantities such as current, voltage, temperature, and stress. Numerous studies have been conducted on SOC estimation methods for lithium-ion batteries, including ampere-hour integration, open-circuit voltage methods, model-based filtering approaches, and machine learning techniques. Among these, ampere-hour integration is the most basic method, integrating current over time to estimate capacity, but it suffers from open-loop error accumulation and requires periodic calibration. Open-circuit voltage methods rely on a stable OCV-SOC relationship, but the wide voltage plateau of LFP cells reduces reliability. Model-based methods using equivalent circuit models or electrochemical models combined with Kalman filters (e.g., extended Kalman filter, unscented Kalman filter) have been widely adopted, though they require accurate model parameters and computational resources. With the advancement of computer technology, data-driven methods using machine learning and deep learning have become a research hotspot, establishing mapping relationships between multi-source data and SOC.
In recent years, battery stress characteristics have gained attention in performance optimization and state assessment. Research shows that appropriate compressive stress can improve electrode-electrolyte contact and enhance battery performance. In battery module assembly, the clamping force between cells is often set to tens or hundreds of kilograms. The stress on the cell surface varies with the state of charge, providing a new information dimension for state estimation. Given the complex internal electrochemical reactions and the influence of multiple factors, methods based on a single physical field often fail to achieve high accuracy. Multi-parameter fusion techniques have gradually become a hotspot for SOC estimation. Digital twin technology for battery energy storage systems, by constructing virtual mapping models of physical entities, enables real-time multi-source data acquisition and provides multi-physics support for state estimation. By integrating electrical, thermal, and mechanical data, and using virtual-real interaction, estimation accuracy can be further improved.
Experimental Methodology
Analysis of Battery Stress Characteristics
During the charge and discharge process, the positive and negative electrodes of LFP energy storage cells undergo lithium deintercalation or intercalation reactions, leading to structural changes. In the absence of external structural constraints, the battery thickness changes accordingly. In an energy storage module, physical constraints convert the deformation into stress. It has been found that the positive electrode (LFP) exhibits an approximately linear decrease in volume during charging, with a thickness reduction of up to 7%. The negative electrode (graphite) undergoes non-monotonic volume changes, with a thickness increase of up to 8%. The combined thickness change of the cell shows a non-monotonic behavior. Correspondingly, when the cell is mechanically constrained externally, the surface stress during charging also exhibits a non-monotonic trend, generally increasing overall. During discharge, the stress decreases overall but also non-monotonically. The relationship between electrode thickness, stress, and SOC can be summarized in Table 1.
| SOC Range (%) | Positive Electrode Thickness Change (%) | Negative Electrode Thickness Change (%) | Overall Stress Trend |
|---|---|---|---|
| 0–30 | −2 (shrink) | +4 (expand) | Increase (non-monotonic) |
| 30–60 | −3 (shrink) | +6 (expand) | Increase (plateau region) |
| 60–100 | −7 (shrink) | +8 (expand) | Increase (faster at end) |
This stable mapping suggests that stress data can be effectively combined with other parameters for battery state estimation.
Battery Data Preprocessing
Data from different terminals and data loggers often have slight time offset, requiring precise alignment of timestamps. During preprocessing, startup, shutdown, and rest periods are removed, and only charge and discharge segments are retained. After time alignment and trimming, a unified dataset containing charge (ΔQ), current (I), voltage (V), temperature (T), stress (F), and differential stress (ΔF) is constructed. Outliers are eliminated using the 3-Sigma criterion, and moving average filtering is applied to reduce noise. The differential parameters ΔQ and ΔF are computed as:
$$ \Delta Q(t) = Q(t) – Q(t-\Delta t) $$
$$ \Delta F(t) = F(t) – F(t-\Delta t) $$
These features enhance the dynamic variation information and improve model training.
Construction of CNN-BiLSTM Model
To address the challenges of SOC estimation for LFP energy storage cells—especially the wide voltage plateau that makes it difficult to distinguish states—we designed a hybrid CNN-BiLSTM model that fuses electrical and stress parameters. The convolutional neural network (CNN) effectively extracts local morphological features from multi-dimensional data, such as the slopes and inflection points in voltage and stress curves. The bidirectional long short-term memory (BiLSTM) network captures temporal dependencies from both past and future directions, modeling the evolution of battery states. The architecture is illustrated conceptually below.
The input layer receives six features: ΔQ, ΔF, V, I, F, and T. The 1D-CNN layer applies 64 convolutional filters of size 3×1 with ReLU activation, followed by max pooling. The output is then fed into a BiLSTM layer with 128 memory units in each direction (forward and backward). The bidirectional outputs are concatenated into a 256-dimensional feature vector, which is passed through a fully connected layer with dropout (rate 0.3) to prevent overfitting, and finally a dense layer with sigmoid activation to output the estimated SOC. The loss function is mean squared error (MSE), optimized using Adam with an initial learning rate of 0.001, momentum 0.9, and gradient clipping at 5.0. The data is split into 60% training, 20% validation, and 20% testing. Batch size is 64, and training runs for 200 epochs.
The mathematical formulations of the LSTM gates are given below. For a single LSTM cell at time step t:
$$ f_t = \sigma(W_f \cdot [h_{t-1}, x_t] + b_f) $$
$$ i_t = \sigma(W_i \cdot [h_{t-1}, x_t] + b_i) $$
$$ \tilde{C}_t = \tanh(W_C \cdot [h_{t-1}, x_t] + b_C) $$
$$ C_t = f_t \odot C_{t-1} + i_t \odot \tilde{C}_t $$
$$ o_t = \sigma(W_o \cdot [h_{t-1}, x_t] + b_o) $$
$$ h_t = o_t \odot \tanh(C_t) $$
In the BiLSTM, a forward LSTM processes the sequence from left to right, while a backward LSTM processes from right to left. The hidden states from both directions are concatenated:
$$ H_t = [\overrightarrow{h}_t ; \overleftarrow{h}_t] $$
The 1D convolution operation is defined as:
$$ y_k = \sum_{j=1}^{3} w_j \cdot x_{k+j-1} + b $$
where w is the kernel weight and b is the bias. The ReLU activation is:
$$ \text{ReLU}(z) = \max(0, z) $$
State Estimation Procedure
We train separate models for charging and discharging processes. After training, the models are evaluated on full charge-discharge cycles (SOC from 0% to 100%) as well as partial cycles (e.g., SOC 60%–90% for charging and 40%–90% for discharging) to test generalization.
Experimental Setup
Test Cell and Apparatus
Commercially available LFP energy storage cells (GSP7514238, capacity 25 Ah, nominal voltage 3.2 V) were used. The dimensions are 238.0 mm × 140.0 mm × 7.6 mm. The maximum charge rate is 1C, and discharge rate is 2C. The cutoff voltages are 3.65 V (charge) and 2.5 V (discharge). The stress test fixture consists of three steel plates, with the cell placed between two plates. A preload force is applied via four corner nuts and monitored using a spoke-type pressure sensor (resolution 0.1 kg). Data acquisition is performed using a Rigol M300 multichannel digital recorder at a sampling interval of 10 s. Voltage and current are recorded by the battery cycler at 1 s intervals.
Test Procedure
| Step | Description |
|---|---|
| 1 | Apply 500 kg preload force to the cell fixture. |
| 2 | Perform a characterization test: charge at 1C to 3.65 V, rest 1 h, discharge at 1.5C to 2.5 V, rest 1 h. Repeat this cycle 40 times. |
| 3 | Record voltage, current, temperature, and stress data continuously. |
Results and Discussion
Electrical and Stress Data Analysis
Figure 1 (conceptual) shows the voltage and stress profiles over time during a single charge-discharge cycle. The voltage plateau is clearly visible in the middle SOC range (20%–90%). The stress increases from ~495 kg to ~530 kg during charging, with non-monotonic variations in the middle region. During discharge, stress decreases similarly. These characteristics indicate that stress provides additional discriminative information, especially in the voltage plateau region.
To quantify the effect of cycling, stress vs. SOC curves for cycles 1, 10, 20, 30, and 40 are plotted. The first cycle shows a different shape due to initial settling, but from cycle 10 onward, the stress-SOC relationship stabilizes. Therefore, data from cycles 10 to 30 were used for training. At different C-rates (0.1C, 0.5C, 1C), stress curves shift: at low rates, two inflection points appear at SOC ≈ 30% and 60%, while at high rates, polarization reduces these features. This suggests that stress-based features are C-rate dependent, and training should include multiple C-rates for robustness.
Data Preprocessing Results
After preprocessing, the dataset contained features ΔQ, ΔF, V, I, F, and T. ΔF was computed over a window of 5% SOC to smooth noise. For monotonic increasing segments, ΔF = 1; for decreasing, ΔF = -1. Outliers were removed, and the final dataset had 100,000+ samples.
SOC Estimation Results
The trained CNN-BiLSTM model was tested on cycles 30–40. The SOC estimation results for charging are shown in Table 3. The root mean square error (RMSE) and mean absolute error (MAE) are computed.
| Process | RMSE (%) | MAE (%) |
|---|---|---|
| Charging (full cycle) | 1.5 | 1.1 |
| Discharging (full cycle) | 5.6 | 4.2 |
| Charging (partial 60-90% SOC) | 1.8 | 1.3 |
| Discharging (partial 40-90% SOC) | 6.1 | 4.8 |
The estimation error is larger at the beginning and end of discharge due to rapid voltage changes and polarization. For charging, the error is lower because the stress variation is more consistent.
Comparative Analysis
To validate the contribution of stress and the CNN-BiLSTM architecture, three comparison experiments were conducted:
- Input feature comparison: Using only V, I, T (3 parameters) versus the full 6 parameters (including ΔQ, F, ΔF).
- Model architecture comparison: LSTM vs. BiLSTM vs. CNN-BiLSTM, all with 6 inputs.
Table 4 summarizes the results on the charging dataset.
| Model | Input Features | RMSE (%) | MAE (%) |
|---|---|---|---|
| LSTM | V, I, T | 6.8 | 5.7 |
| LSTM | ΔQ, ΔF, V, I, F, T | 3.6 | 2.4 |
| BiLSTM | V, I, T | 6.4 | 5.1 |
| BiLSTM | ΔQ, ΔF, V, I, F, T | 2.5 | 2.1 |
| CNN-BiLSTM (proposed) | V, I, T | 5.4 | 4.5 |
| CNN-BiLSTM (proposed) | ΔQ, ΔF, V, I, F, T | 1.5 | 1.1 |
The proposed 6-input CNN-BiLSTM model achieves the lowest RMSE (1.5%) and MAE (1.1%). Compared to the 3-input CNN-BiLSTM (RMSE 5.4%), the inclusion of stress and differential features reduces error by 72%. The BiLSTM with 6 inputs (RMSE 2.5%) outperforms the plain LSTM (3.6%), demonstrating the benefit of bidirectional temporal modeling. The CNN-BiLSTM further improves by extracting local features like curve inflection points, especially in the voltage plateau region where small voltage changes are amplified by stress variations. The error distribution shows that the proposed model maintains high accuracy across the entire SOC range, with maximum error below 4% even at extreme SOC values.
Discussion on Stress Contribution
The stress parameter provides a unique signature that correlates with the state of charge. During the voltage plateau (e.g., 40%–70% SOC), the voltage change is less than 20 mV, but stress can vary by 10–20 kg. This additional dimension helps the model disambiguate SOC values that would otherwise be indistinguishable. The differential feature ΔF captures the rate of stress change, which is sensitive to phase transitions in the electrodes. For instance, the graphite staging transitions cause distinct stress slopes. The CNN layer detects these local patterns, while the BiLSTM captures the sequence of transitions. This synergy results in superior performance compared to methods relying solely on electrical signals.
Conclusion
In this work, we conducted a systematic study on the stress characteristics of lithium iron phosphate (LFP) energy storage cells and developed a novel SOC estimation method based on a CNN-BiLSTM model that fuses electrical and stress parameters. Experimental results demonstrate that the proposed model achieves an RMSE of 1.5% for charging and 5.6% for discharging on full cycles, significantly outperforming models using only electrical inputs. The inclusion of stress and its differential, along with the hybrid CNN-BiLSTM architecture, effectively addresses the challenges posed by the wide voltage plateau of LFP cells. This method can be deployed on digital twin platforms for real-time battery management, enhancing safety and efficiency in energy storage systems. Future work will explore stress-based SOH estimation and incorporate transfer learning for different cell geometries and chemistries.