In the context of integrating renewable energy sources like solar and wind power into the grid, energy storage systems play a pivotal role in maintaining grid stability and power balance due to the intermittent nature of these resources. Retired power batteries, after proper screening and reconfiguration, can serve effectively in such systems to mitigate power fluctuations. The accurate modeling of lithium-ion batteries is a fundamental prerequisite for reliably assessing their operational state and ensuring the safe functioning of these energy storage systems. However, lithium-ion batteries exhibit highly complex and nonlinear internal electrochemical characteristics, presenting as a typical black-box system where only terminal voltage and current are externally measurable.
The development of precise equivalent models is therefore crucial. Among various modeling approaches, equivalent circuit models, composed of ideal circuit elements like resistors and capacitors, offer a clear representation of electrical characteristics with identifiable physical meanings for each component. They are relatively straightforward to establish and implement within simulation environments. Common variants include the Rint, Thevenin, PNGV, and second-order RC models. The second-order RC model, incorporating two RC networks, provides a more accurate description of the battery’s internal dynamic response and is widely adopted in practical engineering due to its favorable balance between accuracy and computational complexity. This forms the basis of the present research.
The accuracy of a fundamental equivalent circuit model for a lithium-ion battery is susceptible to multiple influencing factors, including state of charge (SOC), ambient temperature, charge/discharge current rate (C-rate), cycle life, and self-heating effects. Traditional offline parameter identification methods require extensive experimental testing across various operational conditions to obtain model parameters tailored to each specific factor. While feasible for newer batteries, this approach is less desirable for retired lithium-ion batteries, as repeated charge-discharge cycles can further degrade their remaining useful life, potentially compromising the efficiency and reliability of their second-life application in energy storage systems.
This research addresses the challenge of model accuracy degradation due to variations in SOC, temperature, and current rate. To minimize the additional aging impact of conventional exhaustive testing on retired lithium-ion batteries, we propose a novel model enhancement method based on limited sample data expansion. The core idea is to leverage the terminal voltage error data of the base model obtained under a restricted set of temperature and discharge rate conditions. A Backpropagation (BP) neural network is then constructed to predict the model error across a wider, continuous range of temperatures and various charge/discharge rates. This predicted error functions as a dynamically compensated voltage source added to the base model. This methodology aims to circumvent the need for repetitive, condition-specific error data collection experiments, thereby reducing the time and degradation costs associated with enhancing model precision for applications such as grid-tied renewable energy storage systems.
Fundamental Battery Model and Parameter Identification
The Second-Order RC Equivalent Circuit Model
The chosen fundamental model is the second-order RC equivalent circuit, as depicted in the conceptual diagram. This model effectively captures key dynamic behaviors of the lithium-ion battery. It comprises two parallel RC networks connected in series with an open-circuit voltage source and an ohmic resistor. The first RC network (R1, C1) represents the activation polarization associated with charge transfer impedance at the electrode interface. The second RC network (R2, C2) represents the concentration polarization resulting from the diffusion of lithium ions within the electrode materials. The open-circuit voltage, $U_{oc}$, is a nonlinear function of the battery’s State of Charge (SOC). The ohmic resistor, $R_0$, represents the immediate voltage drop due to internal resistance. The terminal current is denoted by $I$ (discharging positive), and the terminal voltage by $U_t$.
Applying Kirchhoff’s laws to this circuit yields the governing equations:
$$
\begin{aligned}
\frac{dU_1}{dt} &= -\frac{U_1}{R_1 C_1} + \frac{I}{C_1} \\
\frac{dU_2}{dt} &= -\frac{U_2}{R_2 C_2} + \frac{I}{C_2} \\
U_t &= U_{oc} – U_1 – U_2 – R_0 \cdot I
\end{aligned}
$$
Offline Parameter Identification
The model parameters requiring identification are $U_{oc}(SOC)$, $R_0(SOC)$, $R_1(SOC)$, $C_1(SOC)$, $R_2(SOC)$, and $C_2(SOC)$. It is acknowledged that these parameters are functions of SOC, temperature, and current rate. To streamline the process and avoid excessive testing, the initial identification focuses solely on their dependency on SOC. The subsequent compensation mechanism will address errors induced by temperature and C-rate variations. Experiments were conducted on a commercial 18650-type lithium-ion battery with a nominal voltage of 3.62V and a capacity of 2.15Ah.
1. Open-Circuit Voltage (OCV) vs. SOC Relationship: The OCV-SOC curve was established through a standard incremental discharge and rest procedure. The battery was fully charged, rested, and then subjected to successive discharges of 10% capacity (at 1C rate) followed by a 2-hour rest period until the cutoff voltage was reached. The OCV at each rest point was recorded. An 8th-order polynomial was fitted to this data, providing the functional relationship $U_{oc}(SOC)$.
2. Internal Parameter Identification via Hybrid Pulse Test: A sequence of constant-current discharge pulses (3 min at 1C) interspersed with rest periods (5 min) was applied from full charge to discharge cutoff. The voltage response during each pulse cycle was analyzed. The ohmic resistance $R_0$ was calculated from the instantaneous voltage jumps at the beginning and end of the current pulse:
$$ R_0 = \frac{(U_{A}-U_{B}) + (U_{D}-U_{C})}{2I} $$
where points A, B, C, D correspond to the voltage before discharge, after the instantaneous drop, before rest, and after the instantaneous recovery, respectively.
The relaxation voltage curve during the rest period (e.g., segment C-D-E) represents the zero-input response of the RC networks. Its equation is:
$$ U_t = U_{oc} – U_1(0)e^{-t/\tau_1} – U_2(0)e^{-t/\tau_2} $$
where $\tau_1$ and $\tau_2$ are the time constants. Fitting this curve yields $\tau_1$ and $\tau_2$.
The discharge voltage curve after a rest period (e.g., segment A-B-C) represents the zero-state response:
$$ U_t = U_{oc} – I R_0 – I R_1 (1 – e^{-t/\tau_1}) – I R_2 (1 – e^{-t/\tau_2}) $$
Substituting the known $\tau_1$, $\tau_2$, and $R_0$ and fitting this equation provides $R_1$ and $R_2$. The capacitances are then derived as $C_1 = \tau_1 / R_1$ and $C_2 = \tau_2 / R_2$. This process is repeated for each pulse to obtain parameter sets across the entire SOC range, creating a dynamic model lookup table.
Model Error Analysis and the Proposed Enhancement Strategy
Analysis of Model Error Under Varied Conditions
To evaluate the performance of the SOC-dependent base model, constant-current discharge tests were conducted at various temperatures (-10°C, 0°C, 25°C, 30°C) and discharge rates (0.3C, 0.5C, 1.0C, 1.2C). The model was simulated with the same current profiles, and its terminal voltage output was compared against experimental measurements. Two error metrics were calculated: the average error ($E_a$) and the average relative error ($E_r$):
$$
\begin{aligned}
E_a &= \frac{1}{N}\sum_{i=1}^{N} (V_{measured}^{(i)} – V_{model}^{(i)}) \\
E_r &= \frac{1}{N}\sum_{i=1}^{N} \left| \frac{V_{measured}^{(i)} – V_{model}^{(i)}}{V_{measured}^{(i)}} \right| \times 100\%
\end{aligned}
$$
The results, summarized in the table below, clearly indicate systematic deviations. The model error correlates positively with the discharge rate (higher rate, larger error) and negatively with temperature (lower temperature, larger error). This confirms that the base model, parameterized only for SOC under specific conditions, lacks the adaptability to accurately simulate the lithium-ion battery behavior across a broad operational envelope. The error essentially represents an unmodeled voltage component influenced by temperature and current.
| Temperature (°C) | Discharge Rate (C) | Average Error (V) | Average Relative Error (%) |
|---|---|---|---|
| -10 | 0.3 | -0.3045 | 8.964 |
| 0.5 | -0.3561 | 10.650 | |
| 1.0 | -0.3904 | 11.987 | |
| 1.2 | -0.3888 | 12.009 | |
| 0 | 0.3 | -0.1364 | 3.925 |
| 0.5 | -0.1627 | 4.736 | |
| 1.0 | -0.1933 | 5.756 | |
| 1.2 | -0.2003 | 5.984 | |
| 25 | 0.3 | -0.0053 | 0.897 |
| 0.5 | -0.0144 | 0.916 | |
| 1.0 | -0.0120 | 1.135 | |
| 1.2 | -0.0112 | 0.880 | |
| 30 | 1.0 | 0.0113 | 0.976 |
BP Neural Network for Small-Sample Data Expansion
To address this without exhaustive testing, we propose modeling the average error $E_a$ as a function of temperature ($T$) and load current ($I$), i.e., $E_a = f(T, I)$. This function can be seen as a controlled voltage source. The key innovation is to learn this function from a limited dataset and use it for prediction.
1. Data Fitting and Neural Network Construction: The average error data from the limited tests (as in the table above) were used to fit various polynomial surfaces $f(x,y)$, where $x=T$ and $y=I$. Among them, a 3rd-order in both $x$ and $y$ polynomial showed the best agreement with the measured errors. This polynomial, $f(x_3, y_3)$, served as the “ground truth” generator for creating a comprehensive training dataset for a BP neural network. The network was designed with 2 input nodes (T, I), 5 nodes in a single hidden layer with a tansig activation function, and 1 output node with a linear activation function, trained using the Levenberg-Marquardt algorithm (trainlm). The training set spanned currents from 0.6A to 2.6A and temperatures from -10°C to 30°C.
2. Validation of Predictive Capability: The trained BP network was validated against both interpolation and extrapolation scenarios, including data within and beyond the training range, as well as for charging conditions (positive current). The results were compared against outputs from various polynomial fits.
| Condition (Temp, Rate) | Measured $E_a$ (V) | BP Network Output (V) | Best Poly. $f(x_3,y_3)$ Output (V) |
|---|---|---|---|
| 5°C, 1.0C Discharge | -0.1294 | -0.1327 | -0.1331 |
| 25°C, 0.7C Discharge | -0.0124 | -0.0134 | -0.0142 |
| 25°C, 2.5C Discharge | 0.0808 | 0.0193 | -0.5077 |
| 25°C, 4.0C Discharge | 0.1787 | 0.0646 | -3.4943 |
| 40°C, 0.5C Discharge | 0.0291 | 0.0252 | 0.0454 |
| 25°C, 0.6C Charge | 0.0908 | 0.0912 | 0.2639 |
The analysis confirms that: (1) For conditions within the training scope, the BP network performs equivalently to the best-fit polynomial. (2) For conditions extrapolating beyond the training scope (e.g., higher C-rates or temperatures), the BP network demonstrates superior generalization and predictive capability compared to polynomial fits, which can diverge significantly. (3) The network trained primarily on discharge data can also reasonably predict errors during charging. This validates the feasibility of the small-sample data expansion approach using a BP neural network.
The Enhanced Equivalent Circuit Model
The final enhanced model integrates the BP neural network’s prediction as a compensating voltage source in series with the base second-order RC model. The terminal voltage equation for the enhanced model becomes:
$$ U_t = U_{oc}(SOC) – U_1 – U_2 – R_0(SOC) \cdot I – V_{BP}(I, T) $$
where $V_{BP}(I, T)$ is the output of the BP network, dynamically adjusting based on the instantaneous operating current and temperature.
Performance of the Enhanced Model
The enhanced model was simulated under the same varied temperature and discharge rate conditions. A significant improvement in voltage tracking accuracy was observed. The compensation effectively reduced the systematic bias present in the base model. The quantitative error reduction is summarized below, demonstrating the enhanced adaptability and precision of the proposed model for the lithium-ion battery.
| Condition | Average Error (V) | Avg. Relative Error (%) | |||
|---|---|---|---|---|---|
| Temp (°C) | Rate (C) | Before | After | Before | After |
| -10 | 0.3 | -0.3045 | 0.0018 | 8.96 | 2.01 |
| 0.5 | -0.3561 | -0.0018 | 10.65 | 1.77 | |
| 1.0 | -0.3904 | -0.0024 | 11.99 | 1.89 | |
| 1.2 | -0.3888 | 0.0022 | 12.01 | 1.90 | |
| 0 | 0.3 | -0.1364 | -0.0033 | 3.92 | 1.61 |
| 0.5 | -0.1627 | 0.0034 | 4.74 | 1.87 | |
| 1.0 | -0.1933 | 0.0015 | 5.76 | 2.07 | |
| 1.2 | -0.2003 | -0.0029 | 5.98 | 1.94 | |
| 25 | 0.3 | -0.0053 | 0.0001 | 0.90 | 0.80 |
| 0.5 | -0.0144 | -0.0008 | 0.92 | 0.88 | |
| 1.2 | -0.0112 | -0.0003 | 0.88 | 0.83 | |
Application in Energy Storage Systems Based on the Enhanced Model
The developed high-fidelity model offers significant practical value for lithium-ion battery deployment in renewable energy storage systems. Its application can be envisioned in a multi-stage workflow:
1. Grading and Screening for Second-Life Use: The enhanced model provides a consistent and accurate framework for characterizing retired lithium-ion battery cells. By comparing model parameters (like internal resistance $R_0$ and capacity derived from OCV-SOC) or model-predicted performance under standardized loads, cells with similar characteristics can be grouped together. This facilitates the creation of homogeneous packs for cascade utilization in energy storage systems, improving overall system balance, longevity, and safety.
2. Operational State Monitoring and Safety Assessment: The model’s parameters, particularly the ohmic resistance $R_0$, are sensitive to temperature and state of health. By integrating the model with a state estimator (e.g., for SOC), the internal resistance can be monitored online. Abnormal increases can indicate rising internal temperature or degradation. Coupled with the model’s ability to predict terminal voltage accurately, this enables advanced warning for potential thermal issues, triggering battery management system interventions before critical temperatures are reached.
3. System-Level Performance Simulation: For planning and optimizing a grid-scale energy storage system, accurate cell models are essential. The enhanced model allows for realistic simulation of the entire system’s voltage response, energy throughput, and efficiency under various grid service profiles (e.g., frequency regulation, peak shaving). This supports better sizing, economic analysis, and control strategy development for the energy storage system.
Conclusion
This research presents a practical and effective method for enhancing the accuracy of equivalent circuit models for lithium-ion batteries, with particular relevance to second-life applications in energy storage. By starting from a conventional second-order RC model parameterized for SOC, we systematically analyzed its performance gaps under varying temperatures and current rates. To bridge these gaps without resorting to extensive, life-consuming testing, we proposed and validated a novel compensation strategy using a BP neural network trained on limited experimental error data.
The key findings are: First, the BP neural network demonstrated superior generalization and predictive performance compared to direct polynomial fitting, especially when extrapolating beyond the initial sample conditions for both discharge and charge scenarios. This confirms the feasibility of the small-sample data expansion concept. Second, integrating the neural network’s output as a controlled voltage source into the circuit model resulted in a significant improvement in terminal voltage prediction accuracy across a wide operational envelope. The enhanced model shows markedly reduced average and relative errors, confirming its improved adaptability. Finally, this high-precision model serves as a valuable tool for critical tasks in energy storage systems, including the efficient grading of retired lithium-ion batteries, real-time state monitoring with thermal warning capabilities, and reliable system-level performance simulation. This work contributes a model enhancement methodology that balances accuracy, practicality, and battery longevity considerations, supporting the reliable and efficient integration of lithium-ion battery technology into renewable energy infrastructure.