In the field of new energy, energy storage lithium batteries serve as a critical technology for energy storage, widely applied in electric vehicles, aerospace, and portable devices. The State of Health (SOH) of these batteries directly impacts their safety, lifespan, and performance, making it a central task in battery management system research. Traditional SOH estimation methods, such as capacity fade analysis and internal resistance detection, face limitations like long-term monitoring requirements and sensitivity to measurement errors. To address these challenges, this study proposes a Particle Swarm Optimization-Back Propagation (PSO-BP) algorithm integrated with health feature parameters for accurate SOH estimation of energy storage lithium batteries. By analyzing voltage and temperature curves during charging and discharging, we extract two health features (HF1 and HF2) highly correlated with battery aging and validate their relevance using Pearson, Spearman, and Kendall correlation coefficients. The model is tested on the NASA 18650 energy storage lithium battery dataset, demonstrating high accuracy and robustness in SOH estimation.
The PSO algorithm is a swarm intelligence optimization method inspired by collective foraging behavior in nature. It explores multidimensional search spaces through particle cooperation and information exchange. Each particle represents a potential solution, initialized with random positions and velocities. The fitness of particles is evaluated using an objective function, and they update their states based on individual and global best positions. The velocity update formula is given by:
$$v_i(t+1) = w \cdot v_i(t) + c_1 \cdot r_1 \cdot (P_{b_i} – x_i(t)) + c_2 \cdot r_2 \cdot (G_b – x_i(t))$$
where \(v_i(t+1)\) is the velocity of particle \(i\) at time \(t+1\), \(w\) is the inertia weight, \(c_1\) and \(c_2\) are learning factors, \(r_1\) and \(r_2\) are random numbers in [0,1], \(P_{b_i}\) is the personal best position, \(G_b\) is the global best position, and \(x_i(t)\) is the position of particle \(i\) at time \(t\). The position update is expressed as:
$$x_i(t+1) = x_i(t) + v_i(t+1)$$
The algorithm iterates until convergence criteria, such as maximum iterations or fitness thresholds, are met. PSO is renowned for its simplicity and efficiency in global optimization, making it suitable for complex problems like neural network training.
BP neural networks are multi-layer feedforward networks that utilize error backpropagation to adjust weights and biases. They consist of an input layer, hidden layers, and an output layer. During forward propagation, input data is processed through weighted sums and nonlinear activation functions. For a hidden layer neuron \(j\), the input \(z_j\) is computed as:
$$z_j = \sum_i w_{ij} x_i + b_j$$
where \(w_{ij}\) is the weight from input neuron \(i\) to hidden neuron \(j\), \(x_i\) is the input value, and \(b_j\) is the bias. The output \(a_j\) is obtained through an activation function \(f\):
$$a_j = f(z_j)$$
In the backpropagation phase, the error between predicted and actual outputs is minimized using a loss function, typically the mean squared error (MSE):
$$L = \frac{1}{n} \sum_{i=1}^n (\hat{y}_i – y_i)^2$$
where \(L\) is the loss, \(n\) is the number of samples, \(y_i\) is the true value, and \(\hat{y}_i\) is the predicted value. Weights are updated via gradient descent:
$$w_{ij}(t+1) = w_{ij}(t) – \eta \cdot \frac{\partial L}{\partial w_{ij}(t)}$$
where \(\eta\) is the learning rate. Iterations continue until the loss converges, enabling the network to capture complex nonlinear relationships.
The dataset used in this study is the NASA 18650 energy storage lithium battery dataset, which includes charge-discharge cycle data under various temperatures. For instance, battery B5 was tested at 23°C ± 2°C, with a constant current charge of 1.5 A until 4.2 V, followed by constant voltage charging until the current dropped to 20 mA. Discharge was performed at 2 A constant current until the voltage fell to 2.7 V. Testing stopped when capacity degraded to 70% of the rated capacity. The dataset provides comprehensive parameters such as cycle number, capacity, voltage, temperature, and impedance, facilitating in-depth analysis of energy storage lithium battery performance.
Health feature extraction is crucial for data-driven SOH estimation. From voltage curves during charging, we observe that aging increases internal resistance, causing voltage curves to shift and time to reach charge cutoff voltage to decrease. Thus, HF1 is defined as the time to reach the charge cutoff voltage. Similarly, temperature curves show that peak temperature positions vary with aging, so HF2 is defined as the time to reach the maximum charge temperature. These features are extracted to indirectly represent the health state of energy storage lithium batteries.
Correlation analysis is performed to validate the relevance of HF1 and HF2 to SOH. Pearson, Spearman, and Kendall correlation coefficients are computed. The Pearson correlation coefficient \(\rho\) for variables X and Y is given by:
$$\rho_{X,Y} = \frac{\sum (X – \bar{X})(Y – \bar{Y})}{\sqrt{\sum (X – \bar{X})^2 \sum (Y – \bar{Y})^2}}$$
The Spearman correlation coefficient assesses monotonic relationships, while Kendall’s tau measures rank correlation. A summary of correlation coefficients for HF1 and HF2 is presented in Table 1.
| Health Feature | Pearson Coefficient | Spearman Coefficient | Kendall’s Tau |
|---|---|---|---|
| HF1 | 0.92 | 0.89 | 0.87 |
| HF2 | 0.88 | 0.85 | 0.83 |
All coefficients exceed 0.8, indicating strong correlations, thus justifying the use of HF1 and HF2 as inputs for SOH estimation in energy storage lithium batteries.
The PSO-BP model is constructed with a 2-input-1-output framework. Inputs are HF1 and HF2, and the output is SOH. The network architecture includes an input layer, hidden layers, and an output layer. PSO optimizes the initial weights and thresholds of the BP network to avoid local minima. The dataset is split into training (120 samples) and testing (48 samples) sets. After setting hyperparameters, the model is trained and evaluated.
The performance of the PSO-BP model is assessed using metrics such as root mean square error (RMSE), mean absolute error (MAE), mean bias error (MBE), mean squared error (MSE), mean absolute percentage error (MAPE), and residual prediction deviation (RPD). The formulas for these metrics are:
$$RMSE = \sqrt{\frac{1}{n} \sum_{i=1}^n (\hat{y}_i – y_i)^2}$$
$$MAE = \frac{1}{n} \sum_{i=1}^n |\hat{y}_i – y_i|$$
$$MBE = \frac{1}{n} \sum_{i=1}^n (\hat{y}_i – y_i)$$
$$MSE = \frac{1}{n} \sum_{i=1}^n (\hat{y}_i – y_i)^2$$
$$MAPE = \frac{100\%}{n} \sum_{i=1}^n \left| \frac{\hat{y}_i – y_i}{y_i} \right|$$
$$RPD = \frac{SD}{RMSE}$$
where SD is the standard deviation of the true values. The results for training and testing sets are shown in Table 2.
| Metric | Training Set | Testing Set |
|---|---|---|
| RMSE | 0.0086 | 0.0136 |
| MAE | 0.0051 | 0.0127 |
| MBE | 0.0010 | -0.0103 |
| MSE | 0.0001 | 0.0001 |
| MAPE | 0.0059 | 0.0166 |
| RPD | 8.0256 | 2.9393 |
The low error values and high RPD indicate excellent model performance. The regression analysis shows a linear relationship between predicted and true SOH values, with a fit equation of output ≈ 0.98 × target + 0.012, confirming high accuracy.
In conclusion, this research presents a PSO-BP algorithm integrated with health feature parameters for SOH estimation of energy storage lithium batteries. The extracted features HF1 and HF2 demonstrate strong correlations with SOH, and the optimized model achieves high precision in both training and testing phases. This approach enhances the reliability of battery management systems for energy storage lithium batteries, contributing to safer and more efficient energy storage solutions. Future work could explore additional health features to further improve estimation accuracy.