Accurate estimation of the State of Charge (SOC) is fundamental to ensuring the reliable operation of energy storage lithium batteries. These batteries are widely used in electric vehicles and portable electronics due to their high energy density, high power density, and long cycle life. However, challenges such as overcharging and over-discharging pose risks during operation. The Battery Management System (BMS) plays a critical role in real-time monitoring and control, where SOC is a key parameter reflecting the remaining capacity of the energy storage lithium battery. Yet, SOC cannot be measured directly and is influenced by nonlinear battery characteristics and varying operating conditions, making accurate estimation a persistent research challenge.
Existing SOC estimation methods include direct methods, model-based approaches, and data-driven techniques. Direct methods, such as the ampere-hour integral method and open-circuit voltage (OCV) method, are simple but suffer from inaccuracies due to initial SOC errors or the need for long rest periods. Model-based methods, including electrochemical models and equivalent circuit models, combine physical principles with filtering algorithms like Kalman filters for real-time SOC estimation. However, these models are sensitive to factors like battery aging and temperature, requiring frequent parameter updates. Data-driven methods leverage historical data to map input features to SOC, avoiding complex physical modeling. For instance, machine learning algorithms like random forests and support vector regression have shown promise, but they often ignore temporal dependencies in battery data. Recurrent Neural Networks (RNNs) and their variants, such as Gated Recurrent Units (GRUs) and Bidirectional GRUs (BiGRUs), address this by capturing time-series patterns. Nevertheless, many deep learning approaches lack sufficient input features, limiting their performance under complex dynamic conditions.
To overcome these limitations, we propose a novel SOC estimation method that integrates a physical model with a deep learning algorithm. Our approach combines the local feature extraction capability of Convolutional Neural Networks (CNNs) and the bidirectional temporal processing of BiGRUs. By incorporating the terminal voltage output from a first-order Resistor-Capacitor (RC) model as an input feature, along with measured voltage and current, we enhance the modeling ability of the CNN-BiGRU network for energy storage lithium batteries under varying operational scenarios. This fusion of physical knowledge and data-driven techniques improves robustness and accuracy across different temperatures and aging levels.
The core of our method lies in the hybrid architecture that leverages domain knowledge. We use a first-order RC equivalent circuit model to simulate the dynamic behavior of the energy storage lithium battery. The model equations describe the relationship between current, voltage, and SOC, as shown below:
$$ \begin{cases} \dot{U_p}(t) = -\frac{U_p(t)}{R_p C_p} + \frac{I(t)}{C_p} \\ U_{\text{term}}(t) = U_{\text{oc}}(SOC) + U_p(t) + R_o I(t) \end{cases} $$
where \( t \) is time, \( I \) is the load current, \( R_o \) is the ohmic resistance, \( R_p \) is the polarization resistance, \( C_p \) is the polarization capacitance, \( U_p \) is the polarization voltage, \( U_{\text{oc}} \) is the open-circuit voltage, and \( U_{\text{term}} \) is the terminal voltage. The transfer function of the model is derived using Laplace transform:
$$ G(s) = \frac{U_{\text{term}}(s) – U_{\text{oc}}(s)}{I(s)} = R_o + \frac{R_p}{s R_p C_p + 1} $$
Discretization via bilinear transform yields the discrete-time model:
$$ G(z^{-1}) = \frac{a_2 + a_3 z^{-1}}{1 – a_1 z^{-1}} $$
where \( a_1, a_2, a_3 \) are parameters identified using recursive least squares with a forgetting factor. The model parameters \( R_o, R_p, C_p \) are computed as:
$$ R_o = \frac{a_2 – a_3}{1 + a_1}, \quad R_p = \frac{a_2 + a_3}{1 – a_1} – R_o, \quad C_p = \frac{\Delta t (1 + a_1)}{(2 – 2a_1) R_p} $$
The CNN component of our model employs one-dimensional convolutions to extract local patterns from the input sequence. For a sequence \( x_{t:t+k-1} \) of length \( k \), the convolution operation is defined as:
$$ h_t = \sigma(W_{\text{conv}} * x_{t:t+k-1} + b_{\text{conv}}) $$
where \( * \) denotes convolution, \( \sigma \) is the activation function, \( W_{\text{conv}} \) is the convolution kernel, and \( b_{\text{conv}} \) is the bias term. We use a kernel size of 13, stride of 1, and 64 channels, followed by pooling layers for dimensionality reduction.
The BiGRU module processes the features extracted by CNN, capturing bidirectional temporal dependencies. The GRU unit employs update and reset gates to manage information flow:
$$ z_t = \sigma(W_z x_t + U_z h_{t-1} + b_z) $$
$$ r_t = \sigma(W_r x_t + U_r h_{t-1} + b_r) $$
$$ \tilde{h}_t = \tanh(W_h x_t + U_h (r_t \odot h_{t-1}) + b_h) $$
$$ h_t = z_t \odot h_{t-1} + (1 – z_t) \odot \tilde{h}_t $$
where \( z_t \) is the update gate, \( r_t \) is the reset gate, \( \tilde{h}_t \) is the candidate hidden state, and \( h_t \) is the hidden state. In BiGRU, forward and backward GRUs are combined:
$$ \vec{h}_t = \text{GRU}(x_t, \vec{h}_{t-1}), \quad \overleftarrow{h}_t = \text{GRU}(x_t, \overleftarrow{h}_{t+1}) $$
$$ h_t = m \vec{h}_t + n \overleftarrow{h}_t + c_t $$
This allows the model to leverage both past and future context for SOC estimation in energy storage lithium batteries.
We evaluate our method using multiple datasets to ensure comprehensive validation. The first dataset is from the University of Maryland’s CALCE center, involving INR18650-20R lithium batteries tested under dynamic stress test (DST), federal urban driving schedule (FUDS), US06, and Beijing dynamic stress test (BJDST) profiles. The second dataset, referred to as NCM, consists of ternary lithium batteries (INP27148101A-55Ah) cycled between 2.8 V and 4.3 V. The third dataset is from McMaster University (A123), featuring lithium iron phosphate batteries subjected to worldwide harmonized light vehicles test procedure (WLTP) cycles, with data representing different aging stages up to 95,000 km equivalent.
For training and testing, we use DST, FUDS, and US06 profiles from CALCE for training, and BJDST for testing. In the NCM dataset, batteries 2 and 12 are used for training, and battery 10 for testing. The A123 dataset includes cycles from 0 to 3,000, with training on selected cycles and testing on others to assess aging robustness. We employ root mean square error (RMSE), mean absolute error (MAE), and coefficient of determination (\( R^2 \)) as evaluation metrics:
$$ \text{RMSE} = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (\text{SOC}_{\text{true}}(i) – \text{SOC}_{\text{pre}}(i))^2} $$
$$ \text{MAE} = \frac{1}{N} \sum_{i=1}^{N} | \text{SOC}_{\text{true}}(i) – \text{SOC}_{\text{pre}}(i) | $$
$$ R^2 = 1 – \frac{\sum_{i=1}^{N} (\text{SOC}_{\text{true}}(i) – \text{SOC}_{\text{pre}}(i))^2}{\sum_{i=1}^{N} (\text{SOC}_{\text{true}}(i) – \overline{\text{SOC}}_{\text{true}})^2} $$
Our experiments begin by validating the impact of incorporating physical model features. We compare two input configurations: \( F_1 = [U, I] \) (measured voltage and current) and \( F_2 = [U_{\text{term}}, U, I] \) (adding RC model output). Using the CALCE dataset at 25°C, the results demonstrate that \( F_2 \) significantly improves SOC estimation accuracy by providing complementary information that filters noise and enhances feature representation.
| Input Features | RMSE (%) | MAE (%) | \( R^2 \) |
|---|---|---|---|
| \( F_1 \) | 0.63 | 0.49 | 0.9993 |
| \( F_2 \) | 0.16 | 0.12 | 0.9999 |
Next, we compare different neural network architectures to highlight the contribution of CNN and BiGRU modules. The CNN-BiGRU model outperforms standalone CNN and BiGRU models, indicating synergistic effects between local feature extraction and bidirectional temporal modeling.
| Model | RMSE (%) | MAE (%) | \( R^2 \) |
|---|---|---|---|
| CNN | 0.29 | 0.21 | 0.9998 |
| BiGRU | 0.41 | 0.32 | 0.9997 |
| CNN-BiGRU | 0.16 | 0.12 | 0.9999 |
Hyperparameter tuning is conducted to optimize the model. We vary convolution kernel size, stride, and channels, finding that a kernel size of 13, stride of 1, and 64 channels yield the best performance for energy storage lithium battery SOC estimation.
| Kernel Size | Stride | Channels | RMSE (%) | MAE (%) |
|---|---|---|---|---|
| 13 | 1 | 64 | 0.16 | 0.12 |
| 11 | 1 | 64 | 0.16 | 0.12 |
| 15 | 1 | 64 | 0.19 | 0.14 |
| 13 | 1 | 32 | 0.17 | 0.13 |
| 13 | 1 | 128 | 0.17 | 0.12 |
| 13 | 2 | 64 | 0.20 | 0.15 |
We further assess our model on different datasets to verify its generalization. Compared to GRU, BiGRU, LSTM, and BiLSTM, the CNN-BiGRU model achieves lower errors and higher \( R^2 \) on both CALCE and NCM datasets, confirming its robustness for energy storage lithium batteries.
| Dataset | Model | RMSE (%) | MAE (%) | \( R^2 \) |
|---|---|---|---|---|
| CALCE | CNN-BiGRU | 0.16 | 0.12 | 0.9999 |
| GRU | 0.41 | 0.31 | 0.9997 | |
| BiGRU | 0.41 | 0.32 | 0.9997 | |
| LSTM | 0.27 | 0.21 | 0.9998 | |
| BiLSTM | 0.28 | 0.23 | 0.9998 | |
| NCM | CNN-BiGRU | 0.18 | 0.15 | 0.9999 |
| GRU | 0.29 | 0.21 | 0.9999 | |
| BiGRU | 0.30 | 0.22 | 0.9999 | |
| LSTM | 0.28 | 0.22 | 0.9999 | |
| BiLSTM | 0.26 | 0.20 | 0.9999 |
Temperature variations significantly affect the performance of energy storage lithium batteries. We test our model at 0°C and 45°C using the CALCE dataset. The results show that CNN-BiGRU maintains high accuracy across temperatures, though errors are slightly higher at 0°C due to reduced electrochemical activity and increased impedance.
| Temperature (°C) | Model | RMSE (%) | MAE (%) | \( R^2 \) |
|---|---|---|---|---|
| 0 | CNN-BiGRU | 0.28 | 0.19 | 0.9998 |
| GRU | 1.14 | 0.82 | 0.9972 | |
| BiGRU | 0.84 | 0.61 | 0.9984 | |
| LSTM | 1.48 | 1.12 | 0.9953 | |
| BiLSTM | 0.79 | 0.60 | 0.9986 | |
| 45 | CNN-BiGRU | 0.11 | 0.09 | 0.9999 |
| GRU | 0.27 | 0.22 | 0.9998 | |
| BiGRU | 0.26 | 0.22 | 0.9998 | |
| LSTM | 0.16 | 0.13 | 0.9999 | |
| BiLSTM | 0.21 | 0.16 | 0.9999 |
Battery aging is another critical factor. Using the A123 dataset, we train the model on data from cycles 0, 300, 600, 1200, 1500, 2400, and 2700, and test on cycles 900, 1800, 2100, and 3000. The SOC estimation errors increase gradually with aging but remain within acceptable limits, demonstrating the model’s robustness for energy storage lithium batteries over their lifespan.
| Cycle Count | RMSE (%) | MAE (%) | \( R^2 \) |
|---|---|---|---|
| 900 | 0.23 | 0.16 | 0.9984 |
| 1800 | 0.23 | 0.17 | 0.9983 |
| 2100 | 0.24 | 0.17 | 0.9983 |
| 3000 | 0.27 | 0.19 | 0.9978 |
In conclusion, our proposed method, which fuses a physical model with a CNN-BiGRU deep learning architecture, provides accurate and robust SOC estimation for energy storage lithium batteries. By integrating the RC model’s terminal voltage as an input feature, we enhance the network’s ability to capture complex dynamic behaviors. Experimental results across multiple datasets, temperatures, and aging conditions confirm the superiority of our approach over existing models. This work underscores the importance of combining domain knowledge with advanced neural networks to address challenges in battery management systems. Future research could explore adaptive learning mechanisms and multi-task learning for simultaneous SOC and state of health estimation in energy storage lithium batteries.