In the rapidly evolving field of electric vehicles, lithium-ion batteries serve as a critical component due to their high energy density and efficiency. The accurate assessment of the State of Charge (SOC) is essential for optimizing battery performance, extending lifespan, and ensuring safety in electric vehicle applications, particularly in regions like China where the EV market is expanding rapidly. Traditional methods for SOC estimation often struggle with aging effects and dynamic operating conditions, leading to inaccuracies that can impact the economic viability of electric vehicles. In this study, we propose a novel approach based on an integrated Extreme Learning Machine (ELM) algorithm to evaluate the SOC of lithium-ion batteries, incorporating health characteristics to iteratively compute charging states under various conditions. This method addresses the limitations of conventional techniques by leveraging machine learning to handle nonlinearities and aging-related degradations, which are common challenges in electric vehicle batteries.
The integrated ELM model operates as a two-layer ensemble system, combining multiple ELM networks to enhance prediction accuracy and robustness. Each ELM network consists of an input layer, a hidden layer with randomly assigned weights and biases, and an output layer. The output of the integrated model is derived from the aggregated outputs of individual ELMs, reducing variance and improving generalization. For a given input vector \( x_i \) and activation function \( h(\cdot) \), the output of a single ELM can be expressed as:
$$ F_L(x) = \sum_{i=1}^{L} \beta_i \cdot h(x_i) $$
where \( L \) represents the total number of samples, \( \beta_i \) denotes the output weight matrix, and \( H \) is the hidden layer output matrix. The goal is to minimize the least squares error \( \| H\beta – T \| \), where \( T \) is the target output. In the ensemble approach, the error for each individual model output \( f_i(x) \) relative to the ensemble expectation \( f_j(x) \) is computed as:
$$ \text{Error}_j = | f_i(x_j) – f_j(x_i) | $$
This error metric guides the integration process, ensuring that the final model balances the contributions of each ELM network. The control flow of the integrated ELM algorithm involves initializing multiple ELMs, training them on subsets of data, and combining their outputs through weighted averaging or voting mechanisms. This approach is particularly beneficial for electric vehicle batteries, as it adapts to varying operational scenarios, such as different charging rates and temperatures, which are common in China EV usage.
To validate the proposed method, we utilized a lithium-ion battery cycling aging dataset, similar to those provided by NASA, which includes charge-discharge cycles under controlled conditions. The batteries were subjected to repeated cycles with a 100% depth of discharge to simulate real-world aging in electric vehicles. Health indicators (HIs), such as the time difference during constant voltage charging segments, were extracted to form features for State of Health (SOH) prediction. The relationship between SOC and SOH was analyzed to identify correlations that could enhance SOC estimation. For instance, when the terminal voltage ranges between 3.9 V and 4.0 V, SOC and SOH exhibit the highest correlation, making this voltage range ideal for monitoring in electric vehicle applications to prevent overcharging or deep discharge.
The SOC reference values under charging cutoff voltage were evaluated, considering aging effects. Without accounting for aging, SOC estimates showed significant deviations, but our integrated ELM model incorporated these effects, resulting in more accurate and stable predictions. The SOH estimation results demonstrated high consistency, with the model capturing the degradation trends over cycles. This is crucial for electric vehicles in China, where battery longevity directly impacts the total cost of ownership and environmental sustainability. The health feature analysis revealed that the voltage segment between 3.9 V and 4.0 V provides the most reliable indicators for SOC-SOH mapping, as summarized in the following table showing the correlation coefficients at different voltage intervals:
| Voltage Range (V) | Correlation Coefficient (SOC vs. SOH) |
|---|---|
| 3.7 – 3.8 | 0.75 |
| 3.8 – 3.9 | 0.82 |
| 3.9 – 4.0 | 0.95 |
Furthermore, the integrated ELM model was compared against other algorithms, such as Long Short-Term Memory (LSTM) networks and single ELM models, to evaluate its performance in SOC prediction. The Root Mean Square Error (RMSE) was used as the metric, with lower values indicating higher accuracy. The results from multiple test cycles are presented in the table below, highlighting the superiority of the integrated ELM approach for electric vehicle battery management:
| Algorithm | Cycle 1 RMSE | Cycle 2 RMSE | Cycle 3 RMSE | Cycle 4 RMSE | Cycle 5 RMSE |
|---|---|---|---|---|---|
| LSTM | 0.033 | 0.029 | 0.031 | 0.032 | 0.035 |
| Single ELM | 0.021 | 0.019 | 0.021 | 0.022 | 0.023 |
| Integrated ELM | 0.014 | 0.012 | 0.011 | 0.013 | 0.016 |
The mathematical formulation of the integrated ELM model involves optimizing the ensemble output to minimize prediction errors. For a set of \( k \) ELM models, the combined output \( F_{\text{ensemble}}(x) \) is given by:
$$ F_{\text{ensemble}}(x) = \frac{1}{k} \sum_{i=1}^{k} f_i(x) $$
where \( f_i(x) \) is the output of the \( i \)-th ELM. The variance reduction achieved through this ensemble approach enhances the model’s ability to handle uncertainties in electric vehicle battery data, such as fluctuations in temperature and load. Additionally, the health feature extraction process can be described using a linear model where the SOH is estimated from the time-based features \( t_{\text{cv}} \) (constant voltage time) and voltage \( V \):
$$ \text{SOH} = \alpha \cdot t_{\text{cv}} + \beta \cdot V + \gamma $$
where \( \alpha \), \( \beta \), and \( \gamma \) are parameters learned during training. This model aligns with the requirements for China EV applications, where fast and reliable battery diagnostics are needed to support the growing infrastructure.
In the context of electric vehicles, the integration of SOC and SOH estimation is vital for predictive maintenance and energy management. Our results indicate that the integrated ELM model not only improves SOC accuracy but also provides insights into battery aging, allowing for proactive replacements and reducing downtime. For example, the RMSE values show that the integrated ELM consistently outperforms other methods, with an average improvement of over 30% compared to LSTM. This is particularly relevant for the China EV market, where high-density urban usage demands robust battery systems. The following equation illustrates the error minimization process in the ensemble, where the total error \( E_{\text{total}} \) is minimized through iterative updates:
$$ E_{\text{total}} = \sum_{j=1}^{N} \left( F_{\text{ensemble}}(x_j) – y_j \right)^2 $$
where \( N \) is the number of data points, and \( y_j \) is the actual SOC value. By leveraging this approach, electric vehicle manufacturers in China can enhance battery management systems, leading to longer battery life and improved vehicle performance.
In conclusion, the integrated ELM algorithm offers a powerful tool for evaluating the charging state of lithium-ion batteries in electric vehicles. Its ability to incorporate health features and handle aging effects makes it suitable for the dynamic conditions of the China EV ecosystem. The empirical results demonstrate significant advancements in SOC estimation accuracy, paving the way for more efficient and sustainable electric transportation. Future work could focus on real-time implementation and integration with vehicle-to-grid systems, further solidifying the role of machine learning in advancing electric vehicle technologies.