The accurate estimation of the peak power state, or State of Power (SOP), for traction batteries is a cornerstone for ensuring the safe operation, optimizing energy management, and extending the driving range of new energy vehicles. As a dominant energy storage device, the lithium-ion battery is pivotal due to its high energy density, superior power density, and long cycle life. While significant progress has been made in estimating the State of Charge (SOC) and State of Health (SOH) for lithium-ion battery systems, achieving high-precision, robust SOP estimation remains a substantial challenge with direct implications for vehicle performance and safety.
Existing methods for SOP estimation, such as the Hybrid Pulse Power Characteristic (HPPC) test, data-driven approaches, and model-based strategies, often face limitations in accuracy, computational complexity, or adaptability to dynamic operating conditions. The HPPC method, while straightforward, fails to account for complex electrochemical dynamics and aging effects. Data-driven methods like Support Vector Machines (SVM) require extensive datasets and can be computationally burdensome. Model-based methods, particularly those employing Multi-Constraint Conditions (MCC), offer a more physically grounded approach by considering limits from voltage, current, and SOC. However, their accuracy is inherently tied to the precision of the underlying battery model and parameter identification, and cumulative errors from SOC estimation and model simplification can propagate, leading to significant deviations in the final SOP prediction, especially at the endpoints of discharge.
To address these challenges, we propose a novel, high-precision, and robust SOP estimation framework that synergistically combines a physics-based multi-constraint model with a data-driven error compensation mechanism. Our methodology, termed MCC-SSA-ELM, involves: 1) establishing a second-order RC equivalent circuit model for the lithium-ion battery and performing online parameter identification using a Forgetting Factor Recursive Least Squares (FFRLS) algorithm; 2) accurately estimating the battery’s SOC using an Adaptive Extended Kalman Filter (AEKF); 3) performing a preliminary SOP estimation under multiple constraints (voltage, SOC, and maximum design current) for different discharge durations (e.g., 30s, 2min, 5min); and 4) deploying a Sparrow Search Algorithm (SSA)-optimized Extreme Learning Machine (ELM) model to predict and dynamically compensate for the absolute error between the preliminary MCC-based SOP estimate and a reference “true” value. This hybrid approach leverages the interpretability of the physical model while harnessing the powerful nonlinear fitting capability of the optimized machine learning model to correct residual errors, thereby achieving superior estimation accuracy across various operating scenarios.
1. Battery Modeling and State Estimation Foundation
Accurate SOP estimation is fundamentally dependent on a reliable battery model and precise knowledge of its internal states, primarily the State of Charge (SOC). Our approach begins with these critical foundational steps.
1.1 Second-Order RC Equivalent Circuit Model
We employ a second-order RC equivalent circuit model to represent the dynamic behavior of the lithium-ion battery. This model offers a favorable balance between model fidelity and computational complexity for Battery Management System (BMS) applications. The model consists of an open-circuit voltage (OCV) source, an ohmic resistor, and two RC parallel networks representing the electrochemical polarization and concentration polarization effects, respectively.
The electrical behavior is governed by the following equations derived from Kirchhoff’s laws:
$$
\begin{align*}
I_{bat} &= \frac{U_1}{R_1} + C_1 \frac{dU_1}{dt} \\
I_{bat} &= \frac{U_2}{R_2} + C_2 \frac{dU_2}{dt} \\
U_L &= U_{oc}(SOC) – U_1 – U_2 – I_{bat}R_0
\end{align*}
$$
where \(I_{bat}\) is the battery current (positive for discharge), \(U_L\) is the terminal voltage, \(U_{oc}\) is the open-circuit voltage which is a function of SOC, \(R_0\) is the ohmic resistance, and \(R_1\), \(C_1\), \(R_2\), \(C_2\) are the polarization resistances and capacitances for the two RC networks, with corresponding voltages \(U_1\) and \(U_2\). The SOC is updated using the ampere-hour counting method:
$$
SOC(t) = SOC(t_0) – \frac{\eta}{Q_N} \int_{t_0}^{t} I_{bat}(\tau) d\tau
$$
where \(\eta\) is the Coulombic efficiency (assumed to be 1) and \(Q_N\) is the battery’s nominal capacity.
A crucial step is establishing the relationship \(U_{oc}(SOC)\). This is achieved experimentally through a low-current charge-discharge test with sufficient relaxation periods, yielding a characteristic curve that is essential for subsequent SOC and SOP algorithms. The parameters for the experimental lithium-ion battery cell used in this study are summarized below.
| Battery Parameter | Value |
|---|---|
| Nominal Capacity | 31 Ah |
| Nominal Voltage | 3.7 V |
| Charge Cut-off Voltage | 4.2 V |
| Discharge Cut-off Voltage | 3.0 V |
1.2 Online Parameter Identification and SOC Estimation
The model parameters \(R_0, R_1, C_1, R_2, C_2\) are not constant and vary with SOC, temperature, and aging. Therefore, online identification is necessary. We utilize the Forgetting Factor Recursive Least Squares (FFRLS) algorithm for this purpose. The FFRLS algorithm minimizes a weighted least squares cost function and is highly effective for tracking time-varying parameters in a computationally efficient manner, making it suitable for real-time BMS implementation. The forgetting factor \(\lambda\) (typically chosen between 0.95 and 1) allows the algorithm to discount older data, enhancing its adaptability to changing battery dynamics.
With a dynamically updated battery model, we proceed to estimate the core state variable, the SOC. We employ an Adaptive Extended Kalman Filter (AEKF). The EKF linearizes the nonlinear battery system around the current operating point, while the adaptive mechanism continuously tunes the process and measurement noise covariance matrices based on innovation sequences. This significantly improves the robustness and accuracy of SOC estimation compared to a standard EKF, especially under modeling uncertainties and varying noise statistics. The accurate SOC provided by the AEKF is a critical input for the subsequent multi-constraint SOP estimation.
2. The Proposed MCC-SSA-ELM Framework for SOP Estimation
The core innovation of our work lies in the two-stage MCC-SSA-ELM framework. The first stage provides a physics-based, constrained preliminary estimate. The second stage uses an intelligent error predictor to correct the inherent inaccuracies of the first stage.
2.1 Preliminary SOP Estimation under Multi-Constraint Conditions (MCC)
The peak power a lithium-ion battery can deliver or absorb is limited by several hard constraints to ensure safety and prevent degradation. Our MCC model integrates three primary constraints: terminal voltage limits, SOC limits, and the battery’s maximum allowable current as per its design specifications.
We derive expressions for the maximum instantaneous and sustained discharge/charge currents. First, discretizing the battery model equations for a time step \(T_s\):
$$
\begin{align*}
U_{1,k} &= U_{1,k-1} e^{-T_s/\tau_1} + I_{k-1} R_1 (1 – e^{-T_s/\tau_1}) \\
U_{2,k} &= U_{2,k-2} e^{-T_s/\tau_2} + I_{k-1} R_2 (1 – e^{-T_s/\tau_2}) \\
U_{L,k} &= U_{oc}(SOC_k) – U_{1,k} – U_{2,k} – I_k R_0
\end{align*}
$$
where \(\tau_1 = R_1C_1\) and \(\tau_2 = R_2C_2\). The OCV is linearized as \(U_{oc}(SOC_k) \approx U_{oc}(SOC_{k-1}) – I_{k-1} \frac{\eta T_s}{Q_N} \left. \frac{\partial U_{oc}}{\partial SOC} \right|_{SOC_{k-1}}\).
For a sustained peak current \(I_{dis}^{peak}\) over a duration of \(L\) time steps (\(L \times T_s\) seconds), the terminal voltage at the end of the period can be projected. By enforcing the voltage limit \(U_{L, k+L} \geq U_{min}\), we solve for the voltage-constrained maximum discharge current \(I_{dis, max, k+L}^{OCV}\). Similarly, we enforce the SOC limit \(SOC_{k+L} \geq SOC_{min}\) to obtain the SOC-constrained current \(I_{dis, max, k+L}^{SOC}\) from the ampere-hour counting equation. The overall maximum allowable discharge current is then the minimum of these constrained currents and the battery’s design maximum current \(I_{dis}^{max}\):
$$
I_{dis, max, k+L}^{MCC} = \min\left( I_{dis}^{max},\ I_{dis, max, k+L}^{OCV},\ I_{dis, max, k+L}^{SOC} \right)
$$
The corresponding peak discharge power for the duration \(L\) is calculated as:
$$
P_{dis, max, k+L}^{MCC} = \min\left( P_{design}^{max},\ U_{L, k+L} \cdot I_{dis, max, k+L}^{MCC} \right)
$$
where \(U_{L, k+L}\) is the projected terminal voltage under the peak current. An analogous procedure with maximum/minimum functions reversed yields the peak charge current and power. The specific constraint values applied in our study are listed below.
| Constraint Type | Upper Limit | Lower Limit |
|---|---|---|
| State of Charge (SOC) | 1 (100%) | 0 (0%) |
| Terminal Voltage | 4.2 V | 3.0 V |
| Design Current | +320 A (Discharge) | -160 A (Charge) |
2.2 Dynamic Error Compensation using SSA-Optimized ELM
While the MCC model provides a physically meaningful estimate, its accuracy is compromised by errors from parameter identification drift, SOC estimation inaccuracies, and model simplification, particularly towards the end of discharge. We treat the absolute error between the MCC estimate \(P^{MCC}\) and a reference true SOP \(P^{true}\) (derived from experimental data under known conditions) as a learnable nonlinear function of the operating state.
To model and predict this error, we employ an Extreme Learning Machine (ELM). ELM is a single-hidden-layer feedforward neural network known for its extremely fast learning speed and good generalization performance. The output of an ELM with \(L\) hidden nodes is:
$$
f_{ELM}(\mathbf{x}) = \sum_{i=1}^{L} \boldsymbol{\beta}_i \cdot g(\mathbf{w}_i \cdot \mathbf{x} + b_i) = \mathbf{H}\boldsymbol{\beta}
$$
where \(\mathbf{x}\) is the input vector (e.g., containing SOC, current, voltage, etc.), \(\mathbf{w}_i\) and \(b_i\) are randomly assigned input weights and biases, \(g(\cdot)\) is the activation function, \(\boldsymbol{\beta}_i\) are the output weights solved via Moore-Penrose generalized inverse \(\boldsymbol{\beta} = \mathbf{H}^{\dagger}\mathbf{T}\) (where \(\mathbf{T}\) is the target error vector), and \(\mathbf{H}\) is the hidden layer output matrix.
The performance of ELM is sensitive to the number of hidden neurons \(L\). To automatically find the optimal network architecture, we integrate the Sparrow Search Algorithm (SSA). SSA is a recent metaheuristic optimization algorithm inspired by the foraging and anti-predation behaviors of sparrows. It features producers (discoverers), followers, and vigilantes, which work together to efficiently explore and exploit the search space. We configure SSA to optimize the hyperparameter \(L\). The position update rules for producers (discoverers) and followers in SSA are central to its search capability:
Producer (Discoverer) Update:
$$
X_{i,j}^{t+1} =
\begin{cases}
X_{i,j}^{t} \cdot \exp\left(-\frac{i}{\alpha \cdot iter_{max}}\right), & \text{if } R_2 < ST \\
X_{i,j}^{t} + Q \cdot \mathbf{L}, & \text{if } R_2 \ge ST
\end{cases}
$$
Follower Update:
$$
X_{i,j}^{t+1} =
\begin{cases}
Q \cdot \exp\left(\frac{X_{worst}^{t} – X_{i,j}^{t}}{i^2}\right), & \text{if } i > n/2 \\
X_{P}^{t+1} + |X_{i,j}^{t} – X_{P}^{t+1}| \cdot \mathbf{A}^{+} \cdot \mathbf{L}, & \text{otherwise}
\end{cases}
$$
Here, \(X_{i,j}\) represents the position (potential value of \(L\)) of a sparrow, \(R_2\) and \(ST\) are alarm and safety thresholds, \(Q\) is a random number, \(\mathbf{L}\) is a matrix of ones, \(X_P\) is the best producer’s position, and \(X_{worst}\) is the global worst position. The algorithm balances global exploration and local exploitation to find the optimal ELM structure.
The SSA-ELM model is trained on a dataset of absolute errors \(E^{abs} = |P^{MCC} – P^{true}|\) generated from the MCC estimator under various driving cycles. Once trained, it acts as a dynamic error compensator. The final, high-precision SOP estimate is obtained by:
$$
P_{dis, max, k+L}^{MCC-SSA-ELM} = P_{dis, max, k+L}^{MCC} – \widehat{E}_{k+L}^{abs}
$$
where \(\widehat{E}_{k+L}^{abs}\) is the SSA-ELM predicted absolute error at time step \(k+L\). This step effectively corrects the systematic and nonlinear deviations of the physics-based MCC model.
3. Experimental Validation and Performance Analysis
We validate the proposed framework using experimental data from a lithium manganese oxide (LMO) lithium-ion battery cell. Tests were conducted at 25°C under a Dynamic Stress Test (DST) profile to simulate real-world driving conditions. The SOP estimation performance is evaluated for three key discharge durations: 30 seconds (representing short acceleration), 2 minutes (representing medium-term hill climbing), and 5 minutes (representing sustained cruising).
3.1 Foundation Model Performance
The FFRLS algorithm successfully identified the time-varying parameters of the second-order RC model online. The AEKF algorithm provided highly accurate SOC estimation with a low Root Mean Square Error (RMSE), establishing a reliable foundation for the SOP estimation pipeline.
3.2 Performance of Multi-Stage SOP Estimation
The performance of the three estimation models—MCC (baseline), MCC-ELM (with basic error compensation), and the proposed MCC-SSA-ELM—was rigorously compared. The key metrics are Absolute Error (AE), Relative Error (RE), Mean Absolute Error (MAE), Mean Relative Error (MRE), and Root Mean Square Error (RMSE).
The results demonstrate a clear and significant improvement through each stage of our framework. The standalone MCC model, while providing a reasonable estimate, exhibits noticeable errors, particularly MREs of 0.384%, 6.228%, and 6.900% for the 30s, 2min, and 5min durations, respectively. The error is more pronounced for longer durations and towards the end of discharge due to cumulative model inaccuracies.
Introducing the ELM-based error compensation (MCC-ELM) markedly reduces these errors. However, the performance of a standard ELM is dependent on its initial random parameters. By optimizing the ELM’s hidden layer structure using SSA, the proposed MCC-SSA-ELM model achieves the highest accuracy and robustness. The SSA effectively finds the optimal network configuration, enabling the ELM to learn the error dynamics more precisely.
The quantitative superiority of the MCC-SSA-ELM model is summarized in the following table, which compares the key performance metrics across the different methods and discharge durations.
| Model / Metric | Duration | Mean Absolute Error (MAE) | Root Mean Square Error (RMSE) | Mean Relative Error (MRE) |
|---|---|---|---|---|
| MCC Model | 30 s | 1.5367 | 1.5962 | 0.384% |
| 2 min | 5.3814 | 10.4229 | 6.228% | |
| 5 min | 5.8404 | 9.9889 | 6.900% | |
| MCC-ELM Model | 30 s | 0.1203 | 0.1123 | 0.006% |
| 2 min | 2.7202 | 1.9641 | 0.192% | |
| 5 min | 1.3610 | 0.8297 | 0.111% | |
| MCC-SSA-ELM Model | 30 s | 0.0101 | 0.0639 | 0.002% |
| 2 min | 1.2682 | 1.2729 | 0.113% | |
| 5 min | 0.1860 | 0.4852 | 0.042% |
The final MREs for the proposed method are 0.002%, 0.113%, and 0.042% for the 30s, 2min, and 5min durations, respectively. This represents a dramatic reduction in error compared to the baseline MCC model—by 0.382%, 6.115%, and 6.858% in MRE for the three durations. Critically, the maximum errors are consistently constrained within a very tight bound (e.g., ~0.15% or lower), demonstrating the high precision and robustness of the MCC-SSA-ELM framework across different operational time scales for the lithium-ion battery.
4. Conclusion
In this work, we have presented a novel, high-precision, and robust framework for estimating the State of Power (SOP) of lithium-ion battery systems. The proposed MCC-SSA-ELM method effectively addresses the limitations of traditional approaches by synergistically combining a physics-based multi-constraint model with an intelligently optimized data-driven error compensator. The first stage, based on voltage, SOC, and design current constraints, provides a physically interpretable preliminary SOP estimate. The second stage employs a Sparrow Search Algorithm-optimized Extreme Learning Machine (SSA-ELM) to dynamically predict and correct the residual absolute error inherent in the first-stage model.
Experimental validation on a commercial lithium-ion battery cell under a dynamic stress profile confirms the superior performance of our method. The MCC-SSA-ELM framework achieves remarkable estimation accuracy, with mean relative errors reduced to 0.002%, 0.113%, and 0.042% for 30-second, 2-minute, and 5-minute discharge durations, respectively. This represents a significant improvement over both the baseline multi-constraint method and the version using a non-optimized ELM compensator. The framework successfully mitigates the error propagation and accumulation issues seen in conventional model-based methods, particularly at low SOC levels.
This work demonstrates the practical viability of hybrid modeling for advanced BMS functions. The MCC-SSA-ELM approach offers a promising solution for achieving the high-fidelity SOP estimation required for optimizing energy management, ensuring operational safety, and prolonging the lifespan of lithium-ion battery packs in electric vehicles and energy storage systems. Future work will focus on extending this framework to incorporate the effects of temperature and aging, and on deploying the algorithm on embedded hardware for real-time, in-vehicle validation.