In the rapidly evolving landscape of electric vehicles (EVs), the performance and safety of power batteries are paramount. China EV battery technology, particularly lithium iron phosphate (LiFePO4) batteries, is widely adopted due to its high safety, durability, and fast charging capabilities. However, accurately estimating the discharge static voltage—the voltage measured after a period of rest following discharge—remains a critical challenge. This voltage reflects internal states such as state of charge (SOC) and state of health (SOH), influencing battery management and longevity. Existing methods often suffer from issues like poor robustness and reduced identification accuracy due to factors such as polarization effects and environmental interference. In this article, we propose a novel approach for estimating the discharge static voltage of EV power batteries, combining an equivalent circuit model with advanced optimization algorithms to enhance accuracy and reliability.
Our method addresses the delayed voltage response caused by polarization capacitance and resistance during discharge. By constructing an equivalent circuit for the battery discharge and static process, we establish the relationship between discharge static voltage and time, mitigating delay issues. We employ the forgetting factor recursive least squares (FFRLS) method to identify parameters of the RC circuit model and formulate a minimization objective function for voltage estimation. The whale swarm optimization (WSO) algorithm is then applied to optimize function parameters, avoiding local optima common in gradient-based methods. Experimental results demonstrate that our approach reduces cumulative error squares to within ±0.005, improving estimation performance and providing a reliable basis for battery performance analysis in China EV battery applications.
Introduction to EV Power Battery Discharge Mechanisms
The discharge behavior of EV power batteries, especially in China EV battery systems, is governed by electrochemical processes. During acceleration, the battery undergoes continuous discharge, where the positive and negative electrodes exhibit overpotentials that affect voltage output. The discharge voltage at time \( t \), denoted as \( V_t \), can be expressed as:
$$V_t = U_{oc} – [g^+_t + g^-_t] – I_t R_o$$
Here, \( U_{oc} \) is the open-circuit voltage, \( g^+_t \) and \( g^-_t \) are the overpotentials at the positive and negative electrodes, respectively, \( I_t \) is the discharge current, and \( R_o \) is the ohmic resistance. When discharge ceases (\( I_t = 0 \)), the battery enters a static state, and the discharge static voltage \( V_o \) is given by:
$$V_o = U_{oc} – g^+_t – g^-_t$$
However, \( g^+_t \) and \( g^-_t \) are unstable during the static period, leading to voltage recovery delays. To address this, we model the battery using an equivalent circuit that captures polarization effects. This model includes components such as polarization capacitances \( C_{p1} \) and \( C_{p2} \), and polarization resistances \( R_{p1} \) and \( R_{p2} \), representing the double-layer and electrode reaction characteristics. The relationship between static voltage and time is derived as:
$$V_o = U_{oc} – V^+_o e^{-\frac{t}{R_{p1} C_{p1}}} – V^-_o e^{-\frac{t}{R_{p2} C_{p2}}}$$
where \( V^+_o \) and \( V^-_o \) are the initial polarization voltages. This exponential decay model helps in avoiding delayed responses by describing voltage recovery. For instance, the voltage during discharge or charge can be generalized as:
$$V(t) = V_{ocv} + (V_o – V_{ocv}) \cdot e^{-\frac{t}{\tau}}$$
where \( V_{ocv} \) is the final stable open-circuit voltage, and \( \tau \) is the time constant. This formulation allows for better management of distributed energy resources, such as adjusting reactive power output to support voltage during sags.
Equivalent Circuit Model and Parameter Identification
To accurately estimate the discharge static voltage for EV power batteries, we develop an equivalent circuit model that simulates the battery’s dynamic behavior. The model consists of two RC pairs: one for the positive electrode (\( R_{p1} \), \( C_{p1} \)) and one for the negative electrode (\( R_{p2} \), \( C_{p2} \)), along with the ohmic resistance \( R_o \). The discrete-time representation of the terminal voltage \( V_t \) is derived to facilitate parameter identification. After discretization, the voltage equation becomes:
$$V_t – V_{\text{ref}} = \eta_1 [V_{t-1} – V(t)] + \eta_2 [V_{t-1} – V(t)] + \eta_3 I_t + \eta_4 I_{t-1} + \eta_5 I_{t-2}$$
where \( \eta_1 \) to \( \eta_5 \) are discrete-time parameters, and \( I \) represents the current matrix. We apply the FFRLS method to identify the parameters \( G = [R_o, C_{p1}, C_{p2}, R_{p1}, R_{p2}] \). The recursive equations for FFRLS are:
$$
\begin{aligned}
Z_t &= Z_{t-1} + \psi_t [y_t – \chi^T_t Z_{t-1}] \\
\psi_t &= \frac{P_{t-1} \chi_t}{\lambda \chi_t^T P_{t-1} \chi_t} \\
P_t &= \frac{1}{\gamma} (I – \psi_t \chi_t^T \chi_t) P_{t-1}
\end{aligned}
$$
Here, \( Z_t \) is the coefficient vector, \( \psi_t \) is the gain matrix, \( P_t \) is the covariance matrix, \( \gamma \) is the forgetting factor (typically set between 0.95 and 0.99), and \( \lambda \) is a scaling factor. The parameters are computed as follows:
$$R_o = \frac{Z_t \eta_5}{\eta_2}$$
$$C_{p1} = \frac{t_1}{R_{p1}}, \quad C_{p2} = \frac{t_2}{R_{p2}}$$
$$R_{p1} = \frac{t_1 o + t_2 R_o – d}{t_1 – t_2}, \quad R_{p2} = o – R_{p1} – R_o$$
where \( t_1 \) and \( t_2 \) are time constants, \( o = R_o + R_{p1} + R_{p2} \), and \( d = R_o (t_1 + t_2) + R_{p1} t_2 + R_{p2} t_1 \). This identification process ensures that the model parameters accurately reflect the battery’s transient response, which is crucial for China EV battery systems operating under varying conditions.
| Parameter | Symbol | Typical Value Range | Unit |
|---|---|---|---|
| Ohmic Resistance | \( R_o \) | 4 – 10 | mΩ |
| Polarization Resistance (Positive) | \( R_{p1} \) | 5 – 15 | mΩ |
| Polarization Resistance (Negative) | \( R_{p2} \) | 5 – 15 | mΩ |
| Polarization Capacitance (Positive) | \( C_{p1} \) | 1000 – 5000 | F |
| Polarization Capacitance (Negative) | \( C_{p2} \) | 1000 – 5000 | F |
Optimization with Whale Swarm Algorithm
After identifying the RC model parameters, we optimize the discharge static voltage estimation by minimizing the cumulative error between estimated and actual voltages. The objective function is defined as:
$$\min \varepsilon(G) = \frac{1}{N} \sum (V^*_o – V_o)^2$$
where \( V^*_o \) is the true static voltage, \( V_o \) is the estimated voltage, and \( N \) is the number of samples. The parameters \( G = [R_o, C_{p1}, C_{p2}, R_{p1}, R_{p2}] \) are optimized using the whale swarm optimization (WSO) algorithm, which mimics the social behavior of whales to avoid local optima. The optimization steps are as follows:
- Initialization: Generate an initial population of parameter sets within predefined bounds, such as \( R_o \in [4, 10] \) mΩ and \( C_{p1} \in [1000, 5000] \) F.
- Chaotic Mapping: Use Tent chaotic mapping to ensure uniform distribution of parameters in the search space.
- Fitness Evaluation: Compute the fitness value for each parameter set based on the objective function \( \varepsilon(G) \).
- Update Mechanisms: Compare individual and global best fitness values. Update positions using:
- Random search: \( X_{\tau+1} = o X_{\tau}^a – w A | C X_{\tau}^a – X_{\tau} | \)
- Spiral update: \( X_{\tau+1} = \begin{cases} X_{\tau}^a – w A D, & \text{if } \|A\| \geq 1 \\ w D’ e \cos(2\pi l) + d X_{\tau}^e, & \text{if } \|A\| < 1 \end{cases} \)
Here, \( X_{\tau} \) is the current position, \( X_{\tau}^a \) is a randomly selected individual, \( X_{\tau}^e \) is the elite individual, \( w \) is an adaptive weight, \( A \) and \( C \) are coefficient vectors, \( D \) and \( D’ \) are distances, \( e \) is a direction parameter (set to 1), \( d \) controls spiral step size, and \( l \) is a random number.
- Termination: Repeat until convergence criteria are met, then output the optimized parameters.
This approach enhances the robustness of voltage estimation for China EV battery systems, as the WSO algorithm efficiently explores the parameter space without getting trapped in local minima. The optimized parameters are used to recalculate the static voltage using the exponential model, improving accuracy.
| Parameter | Symbol | Value |
|---|---|---|
| Population Size | – | 50 |
| Maximum Iterations | – | 100 |
| Forgetting Factor | \( \gamma \) | 0.98 |
| Adaptive Weight | \( w \) | 0.5 – 1.0 |
| Spiral Step Control | \( d \) | 0.1 – 0.5 |
Experimental Setup and Results Analysis
To validate our method, we conducted experiments using a LiFePO4 battery from a China EV battery manufacturer. The test platform included a battery cycler, temperature chamber, and data acquisition system. Key battery parameters are summarized in Table 3.
| Parameter | Value | Unit |
|---|---|---|
| Charging Current | 15 | A |
| Standard Cycle Life | 1700 | cycles |
| Motor Power | 5000 | W |
| Nominal Voltage | 20 | V |
| Load Voltage | 19 | V |
| Charging Voltage | 62 | V |
| Discharge Current | 170 | A |
| Charging Time | 8 | h |
| Internal Resistance | 4 | mΩ |
| Standard Capacity | 120 | Ah |
We simulated two acceleration phases in EV operation, followed by a 10-minute static period. The discharge static voltage was estimated using our method, and results were compared to measured values. The cumulative error squares, a metric for estimation accuracy, were calculated pre- and post-optimization. As shown in Table 4, the optimized parameters reduced error variation to within ±0.005, demonstrating improved performance.
| Condition | Error Range | Improvement |
|---|---|---|
| Before Optimization | -0.01 to +0.02 | Baseline |
| After Optimization | ±0.005 | ~50% reduction |
Figure 1 illustrates the voltage estimation results during different acceleration phases. The estimated voltage closely tracks the actual voltage, with minor deviations, indicating that our method effectively captures dynamic changes. This is crucial for EV power battery management, as accurate voltage estimation supports SOC and SOH analysis, enhancing battery life and safety.
Additionally, we evaluated the battery’s state of health (SOH) over time using our method and compared it to existing approaches. SOH is a critical indicator for China EV battery longevity, defined as the ratio of current capacity to initial capacity. A SOH above 80% is considered healthy, while below 60% indicates significant degradation. Our method maintained SOH above 80% throughout the test period, whereas other methods showed faster degradation, as summarized in Table 5.
| Method | SOH After 30 Days (%) | SOH After 60 Days (%) | Remarks |
|---|---|---|---|
| Proposed Method | 85 | 82 | Healthy state maintained |
| Reference Method A | 75 | 65 | Moderate degradation |
| Reference Method B | 70 | 58 | Significant degradation |
The robustness of our approach stems from the combination of FFRLS and WSO, which adapts to battery aging and environmental variations. For instance, the forgetting factor in FFRLS allows the model to prioritize recent data, reducing the impact of historical inaccuracies. Meanwhile, WSO’s global search capability ensures optimal parameter identification, even under noisy conditions common in EV power battery operations.
Conclusion
In this article, we presented a comprehensive method for estimating the discharge static voltage of EV power batteries, with a focus on China EV battery applications. By integrating an equivalent circuit model with forgetting factor recursive least squares parameter identification and whale swarm optimization, we achieved high estimation accuracy and robustness. The cumulative error squares were reduced to within ±0.005, and the method maintained battery health above 80% SOH, demonstrating its practical utility. This approach provides a reliable foundation for battery management systems, enabling better performance monitoring and lifespan extension. Future work will explore real-time implementation and adaptation to other battery chemistries, further advancing China EV battery technology.
The key formulas and algorithms discussed include the voltage model \( V_o = U_{oc} – V^+_o e^{-\frac{t}{R_{p1} C_{p1}}} – V^-_o e^{-\frac{t}{R_{p2} C_{p2}}} \), the discrete-time equation for parameter identification, and the optimization objective \( \min \varepsilon(G) = \frac{1}{N} \sum (V^*_o – V_o)^2 \). These elements collectively enhance the estimation of discharge static voltage, addressing critical challenges in EV power battery management. Through continuous refinement, our method contributes to the sustainable development of electric vehicles, aligning with global trends in clean energy and transportation.