In the era of sustainable energy and carbon neutrality goals, lithium-ion batteries have emerged as pivotal components in energy storage systems and electric vehicles, owing to their long cycle life, high energy density, and environmental friendliness. Among various lithium-ion battery chemistries, the LiFePO4 battery, with its inherent safety, stability, and cost-effectiveness, has garnered significant attention for large-scale applications. Accurate state monitoring and risk assessment are crucial for ensuring the safe and reliable operation of LiFePO4 batteries. One fundamental parameter in this context is the open-circuit voltage (OCV), which exhibits a unique, temperature-dependent relationship with the state of charge (SOC). This relationship, encapsulated in the OCV-SOC curve, serves as a cornerstone for battery modeling, parameter identification, and SOC estimation. Consequently, obtaining a precise OCV-SOC curve is paramount for enhancing the overall accuracy of battery management systems.
Traditional methods for acquiring the OCV-SOC curve primarily include the low-current OCV test (often termed the incremental OCV test) and the rest-based OCV test (or static OCV test). The low-current OCV test involves charging or discharging the battery at an extremely low current rate, typically below C/20, where C represents the rated capacity. Under such low-current conditions, the internal electrochemical reactions are assumed to be quasi-static, and polarization effects are considered negligible. Thus, the terminal voltage during this slow process approximates the OCV, and the curve derived from averaging charge and discharge voltages is taken as the OCV-SOC relationship. However, this assumption is idealized; in practice, even minute polarization voltages can introduce errors, particularly for LiFePO4 batteries, which exhibit flat voltage plateaus and abrupt voltage changes in certain SOC ranges. These characteristics amplify the impact of polarization, leading to inaccuracies in the OCV-SOC curve obtained via the low-current method.
On the other hand, the rest-based OCV test involves charging or discharging the battery to specific SOC points (e.g., at 10% intervals) followed by prolonged rest periods (often exceeding one hour) to allow polarization relaxation. The stabilized terminal voltage after rest is recorded as the OCV at that SOC point. A polynomial function is then fitted to these discrete points to form the OCV-SOC curve. While this method accurately captures OCV at the measured SOC points, it fails to describe the OCV characteristics at non-measured points due to the limited sampling resolution. For LiFePO4 batteries, with their unique voltage profiles, this limitation becomes pronounced, as the polynomial fitting may not adequately represent the voltage plateaus and steep transitions between points.
To address these shortcomings, I propose a high-precision OCV-SOC curve acquisition method that synergistically combines the strengths of both the low-current and rest-based tests. The core idea is to leverage the continuous voltage trend from the low-current discharge curve while correcting for polarization effects using reference data from the rest-based test. This is achieved through an optimization framework that incorporates a first-order RC equivalent circuit model and employs a differential evolution algorithm for parameter tuning. The resulting optimized OCV-SOC curve demonstrates superior accuracy in simulating the true OCV characteristics of LiFePO4 batteries, thereby improving battery modeling and SOC estimation performance.
The proposed method unfolds in two main stages: piecewise fitting of the incremental discharge curve and optimization of this curve to mitigate polarization influences. The first stage addresses the inherent challenge of fitting the incremental discharge curve for LiFePO4 batteries. Due to the presence of voltage plateaus and sharp voltage changes, a single polynomial fit often yields poor accuracy in local SOC regions. To enhance fitting precision, I adopt a piecewise fitting approach. The incremental discharge curve is segmented based on the distinctive voltage characteristics of LiFePO4 batteries. Specifically, SOC ranges are divided to ensure each segment contains only one dominant feature, such as a plateau or a transition. For instance, segmentation at 90% SOC and 25% SOC can isolate key regions. Within each segment, a suitable fitting function is selected. Through empirical analysis, rational functions, as provided by tools like MATLAB’s Curve Fitting Toolbox, offer excellent accuracy. The general form of a rational function is:
$$ f(x) = \frac{p_{n+1} x^n + p_n x^{n-1} + \cdots + p_2 x + p_1}{x^m + q_m x^{m-1} + \cdots + q_2 x + q_1}, \quad 0 \leq n \leq 5, \quad 1 \leq m \leq 5 $$
where \( p_i \) and \( q_i \) are coefficients determined via fitting. To ensure smooth transitions between segments, each segment is extended slightly at its boundaries (e.g., by 2% SOC) during fitting. This prevents abrupt voltage jumps at connection points. The piecewise-fitted incremental discharge curve, denoted as \( U_{\text{id}}(S_{\text{oc}}) \), serves as the initial approximation for the OCV-SOC relationship and is the object of optimization.
The second stage focuses on optimizing \( U_{\text{id}}(S_{\text{oc}}) \) to reduce polarization-induced errors. I begin by establishing an optimization equation based on the first-order RC equivalent circuit model, which is widely used for lithium-ion battery dynamics. The model comprises an open-circuit voltage source \( U_{\text{oc}} \), an ohmic resistor \( R_0 \), and a parallel RC network (\( R_1, C_1 \)) representing polarization. When a battery transitions from rest to a constant-current discharge, the terminal voltage \( U \) relates to \( U_{\text{oc}} \) as:
$$ U = U_{\text{oc}} – R_0 I – R_1 I (1 – e^{-t/(R_1 C_1)}) $$
where \( I \) is the discharge current and \( t \) is time. For the low-current discharge test, the recorded voltage \( U_{\text{id}} \) is essentially the terminal voltage under a small current, still subject to minor polarization. Assuming the model parameters (\( R_0, R_1, C_1 \)) remain constant during the early and middle phases of discharge (which is reasonable for stable battery behavior under low currents), the equation can be rewritten as:
$$ U_{\text{op}} = U_{\text{id}} + w_2 + w_3 (1 – e^{-t/m}) $$
Here, \( U_{\text{op}} \) represents the optimized OCV-SOC curve voltage, \( w_2 = R_0 I \), \( w_3 = R_1 I \), and \( m = R_1 C_1 \). Using the ampere-hour integral method, SOC during constant-current discharge is linearly related to time: \( S_{\text{oc}} = 1 – z t \), where \( z \) is a constant. Substituting this into the above yields:
$$ U_{\text{op}}(S_{\text{oc}}) = U_{\text{id}}(S_{\text{oc}} + w_1) + w_2 + w_3 (1 – e^{-(1 – S_{\text{oc}})/w_4}) $$
where \( w_1 \) accounts for potential SOC axis offset between the low-current and rest-based tests, and \( w_4 = m z \). This equation models the polarization effect during the low-current discharge for SOC above 30%.
For SOC below 30%, battery characteristics change rapidly, and the constant parameter assumption no longer holds. Empirical analysis suggests that an additional term can capture the voltage behavior in this region. Thus, the full optimization equation for the entire SOC range becomes:
$$
U_{\text{op}}(S_{\text{oc}}) =
\begin{cases}
U_{\text{id}}(S_{\text{oc}} + w_1) + w_2 + w_3 (1 – e^{-(1 – S_{\text{oc}})/w_4}) + w_5 (e^{-w_6 S_{\text{oc}}} – e^{-0.25 w_6}), & 0\% \leq S_{\text{oc}} < 30\% \\
U_{\text{id}}(S_{\text{oc}} + w_1) + w_2 + w_3 (1 – e^{-(1 – S_{\text{oc}})/w_4}), & 30\% \leq S_{\text{oc}} \leq 100\%
\end{cases}
$$
where \( w_5 \) and \( w_6 \) are parameters for the low-SOC correction. The optimization aims to minimize the voltage difference between \( U_{\text{op}} \) and the rest-based test data at specific SOC points. The objective function is:
$$ \min G = \sum_{i=1}^{11} [U_{\text{op}}(i) – U_{\text{ss}}(i)]^2 $$
where \( U_{\text{ss}}(i) \) is the voltage from the rest-based test at the \( i \)-th SOC point (e.g., 0%, 10%, …, 100%). Constraints are set based on voltage comparisons to define reasonable bounds for the parameters \( w_1 \) to \( w_6 \).
To solve this optimization problem, I employ the differential evolution (DE) algorithm, a robust evolutionary strategy known for its convergence speed and minimal parameter tuning. DE operates by maintaining a population of candidate solutions and iteratively improving them through mutation, crossover, and selection operations. Its ability to handle non-linear, multi-modal problems makes it suitable for this application. The algorithm is configured with typical parameters: a population size of 50, crossover probability of 0.9, and differential weight of 0.5. The optimization runs until the objective function converges or a maximum iteration count is reached.
To validate the proposed method, comprehensive experiments were conducted on a fresh ANR26650 LiFePO4 battery cell. The key specifications of this LiFePO4 battery are summarized in Table 1.
| Parameter | Value |
|---|---|
| Nominal Capacity | 2.5 Ah |
| Nominal Voltage | 3.2 V |
| Charge Cut-off Voltage | 3.6 V |
| Discharge Cut-off Voltage | 2.0 V |
| Operating Temperature | -30°C to 55°C |
All tests were performed inside a temperature-controlled chamber at 25°C using a NEWARE BT4000 battery tester. Three types of tests were carried out: a low-current OCV test, a rest-based OCV test, and a dynamic driving cycle test (Federal Urban Driving Schedule, FUDS) for model validation. For the low-current test, the LiFePO4 battery was fully charged via a constant-current constant-voltage (CC-CV) protocol, rested for 4 hours, and then discharged at a C/30 rate (approximately 0.0833 A) until the cut-off voltage. The rest-based test involved discharging the battery at a C/3 rate to predetermined SOC points (100%, 90%, …, 0%), each followed by a rest period of 4 hours (6 hours for 10% and 0% SOC) to record the stabilized voltage. The FUDS test provided dynamic current profiles to assess battery model performance.
The piecewise fitting of the incremental discharge curve was implemented as described. The curve was segmented at 90% SOC and 25% SOC, with each segment fitted using rational functions. The resulting piecewise function \( U_{\text{id}}(S_{\text{oc}}) \) achieved a root mean square error (RMSE) of 0.000498 V, significantly lower than the 0.004700 V RMSE from a 12th-order polynomial fit. This confirms the effectiveness of piecewise fitting in capturing the intricate voltage profile of LiFePO4 batteries.
Next, the optimization was performed using DE. The optimized parameters are listed in Table 2.
| Parameter | Optimized Value |
|---|---|
| \( w_1 \) | -0.0031535 |
| \( w_2 \) | 0.0147336 |
| \( w_3 \) | 0.0076522 |
| \( w_4 \) | 0.0273639 |
| \( w_5 \) | 0.0448491 |
| \( w_6 \) | 24.9339932 |
The resulting optimized OCV-SOC curve, denoted as \( U_{\text{op}}(S_{\text{oc}}) \), was compared with the curve from the rest-based test (fitted with a 12th-order polynomial). As shown in Figure 1 (conceptual), the optimized curve aligns closely with the rest-based data at measured SOC points while smoothly interpolating between points, faithfully reflecting the voltage trends of the LiFePO4 battery. This demonstrates the success of the optimization in correcting polarization errors.
To quantitatively evaluate the impact of the optimized OCV-SOC curve, I developed two second-order RC equivalent circuit models: a conventional model using the rest-based polynomial OCV-SOC curve, and an optimized model using \( U_{\text{op}}(S_{\text{oc}}) \). Both models were parameterized using the same procedure, and their voltage estimation accuracy was tested under the FUDS cycle. The voltage estimation errors are summarized in Table 3.
| Model | RMSE (V) | MAPE (%) | Maximum Voltage Error (V) |
|---|---|---|---|
| Conventional Model | 0.01398 | 0.2751 | 0.0787 |
| Optimized Model | 0.00813 | 0.1883 | 0.0546 |
The optimized model reduces RMSE by 41.8% and mean absolute percentage error (MAPE) by 31.6%, indicating a substantial improvement in modeling precision. This directly stems from the more accurate OCV representation provided by the optimized curve for the LiFePO4 battery.
Furthermore, the SOC estimation performance was assessed using the extended Kalman filter (EKF) algorithm, a prevalent method for nonlinear state estimation. EKF was applied to both models under the FUDS cycle, with reference SOC obtained from high-precision instrumentation. The SOC estimation errors are presented in Table 4.
| Model | RMSE (%) | MAPE (%) | Maximum SOC Error (%) |
|---|---|---|---|
| Conventional Model | 0.9474 | 3.624 | 1.417 |
| Optimized Model | 0.3945 | 1.644 | 0.814 |
The optimized model achieves an RMSE reduction of 58.3% in SOC estimation, underscoring the critical role of an accurate OCV-SOC curve in state estimation for LiFePO4 batteries.
In conclusion, I have presented a novel high-precision method for acquiring the OCV-SOC curve of LiFePO4 batteries. By integrating the continuous trend from the low-current discharge curve with the pointwise accuracy of the rest-based test, and by systematically correcting polarization effects through an optimization framework, the method yields an optimized OCV-SOC curve that closely mirrors the true open-circuit voltage characteristics of LiFePO4 batteries. The piecewise fitting strategy effectively handles the unique voltage plateaus and transitions of LiFePO4 chemistry, while the differential evolution algorithm efficiently solves the parameter optimization problem. Experimental validation on a commercial LiFePO4 battery cell demonstrates significant enhancements in both battery modeling accuracy and SOC estimation performance compared to conventional approaches. This method provides a robust foundation for advanced battery management systems, contributing to the safe and efficient utilization of LiFePO4 batteries in energy storage and electric mobility. Future work may extend this approach to other battery chemistries, incorporate temperature effects, and explore real-time implementation for adaptive OCV-SOC curve updates.
The proposed methodology underscores the importance of tailored approaches for specific battery types like the LiFePO4 battery. Its success hinges on the detailed characterization of the LiFePO4 battery’s voltage behavior and the intelligent fusion of experimental data. As the demand for reliable LiFePO4 battery systems grows, such precision-oriented techniques will become increasingly valuable in pushing the boundaries of performance and safety.