The accurate and reliable estimation of the State of Charge (SOC) is a cornerstone for the safe and efficient operation of lithium-ion batteries in electric vehicles. SOC provides critical information about the remaining usable energy, directly impacting range prediction, battery management system (BMS) strategies, and the prevention of overcharge and over-discharge, thereby extending battery lifespan. However, the inherent complexity of lithium-ion battery electrochemistry poses significant challenges for precise SOC determination. The discharge/charge voltage curve is highly nonlinear, especially for certain chemistries like Lithium Iron Phosphate (LFP). The system is dynamic, with SOC changing continuously based on current flow. Furthermore, a pronounced hysteresis effect is observed where the open-circuit voltage (OCV) path differs during charge and discharge, and a voltage relaxation phenomenon occurs after current interruption, where the terminal voltage slowly drifts towards its true equilibrium OCV. These factors collectively degrade the accuracy of traditional estimation methods like coulomb counting, which is prone to error accumulation, and model-based filters that struggle with nonlinearity and parameter uncertainty.
Existing approaches often address only parts of this problem. Methods based on Extended Kalman Filters (EKF) or Unscented Kalman Filters (UKF) require precise battery models and can diverge under modeling errors or unusual noise conditions. Data-driven methods, such as those employing neural networks, can capture nonlinearity but often lack interpretability, require massive datasets, and may not generalize well to unseen operating conditions or aging states of the lithium-ion battery. To overcome these limitations, this article presents a novel hybrid methodology that separately yet synergistically addresses the dynamic and static estimation phases for lithium-ion battery SOC. The core innovation lies in combining signal processing for feature enhancement, a machine learning model for dynamic nonlinear mapping, and a physics-informed relaxation model for static equilibrium estimation.
Methodology Framework
The proposed framework bifurcates the SOC estimation problem into two dedicated regimes: dynamic estimation during active current flow and static estimation during rest periods. This separation allows for the application of specialized techniques optimal for each regime’s characteristics. The overall process flow is as follows: during vehicle operation, real-time measurements of current, voltage, and temperature are processed. The current signal is decomposed to extract its long-term trend. This trend, along with voltage and temperature, feeds into a Least Squares Support Vector Machine (LSSVM) to produce a real-time, dynamic SOC estimate. When the vehicle is parked and the lithium-ion battery enters a rest period, the terminal voltage relaxation curve is fitted using a dual-exponential relaxation model to accurately predict the equilibrium OCV, which is then used to derive a highly accurate static SOC estimate, periodically correcting the dynamic estimator’s drift.
Dynamic SOC Estimation Using EMD-Enhanced LSSVM
Feature Engineering via Empirical Mode Decomposition (EMD)
The dynamic behavior of a lithium-ion battery under real driving cycles is reflected in its terminal voltage \(U(t)\) and current \(I(t)\), both influenced by temperature \(T(t)\). While LSSVM is capable of modeling complex relationships, its ability to capture long-term temporal dependencies in raw, noisy current signals can be limited. The current profile from a vehicle lithium-ion battery is typically non-stationary and contains high-frequency fluctuations from regenerative braking and acceleration superimposed on a lower-frequency trend representing the general discharge or charge phase. To enhance the model’s input features, we apply Empirical Mode Decomposition (EMD) specifically to the current signal \(I(t)\). EMD is an adaptive time-frequency analysis technique ideal for non-stationary signals, as it decomposes a signal into a set of Intrinsic Mode Functions (IMFs) and a residual. The process iteratively extracts oscillatory components from highest to lowest frequency.
For a given current signal \(I(t)\), the EMD algorithm proceeds as follows:
1. Identify all local extrema of \(I(t)\).
2. Construct the upper envelope \(e_{max}(t)\) and lower envelope \(e_{min}(t)\) by interpolating the local maxima and minima, respectively.
3. Calculate the mean envelope: \(m_1(t) = \frac{e_{max}(t) + e_{min}(t)}{2}\).
4. Subtract the mean from the signal to get the first proto-IMF: \(h_1(t) = I(t) – m_1(t)\).
5. Check if \(h_1(t)\) satisfies the IMF conditions (zero mean, number of extrema and zero-crossings differ at most by one). If not, treat \(h_1(t)\) as the new signal and repeat steps 1-4 until an IMF criterion is met, yielding the first IMF \(c_1(t)\).
6. Subtract \(c_1(t)\) from \(I(t)\) to obtain the residual \(r_1(t) = I(t) – c_1(t)\).
7. Treat \(r_1(t)\) as the new data and repeat the entire process to obtain subsequent IMFs \(c_2(t), c_3(t), …\) until the residual becomes a monotonic function or has at most one extremum.
The final decomposition is expressed as:
$$ I(t) = \sum_{i=1}^{N} c_i(t) + r_N(t) $$
where \(c_i(t)\) are the IMFs and \(r_N(t)\) is the final residual. The lower-order IMFs (e.g., \(c_N\), \(c_{N-1}\)) capture the slower, long-term trends in the current, which are most relevant for the SOC’s gradual change. The higher-frequency IMFs and noise are thus effectively filtered out. We select the low-frequency IMF component, denoted as \(I_{LF}(t)\), as a primary input feature. This processed feature succinctly represents the underlying energy throughput trend of the lithium-ion battery, making the subsequent regression task more tractable and accurate.
Least Squares Support Vector Machine (LSSVM) Regression Model
With the enhanced features, we employ an LSSVM for regression. The input feature vector for the LSSVM at time \(t\) is \(\mathbf{x}(t) = [I_{LF}(t), U(t), T(t)]\). The target output is the dynamic SOC value. Given a training dataset \(\{\mathbf{x}_k, SOC_k\}_{k=1}^{K}\), the LSSVM aims to find a function \(SOC(\mathbf{x}) = \mathbf{w}^T \phi(\mathbf{x}) + b\) that maps inputs to the SOC with minimum error. Here, \(\phi(\cdot)\) is a nonlinear mapping to a high-dimensional feature space, \(\mathbf{w}\) is the weight vector, and \(b\) is the bias term.
The LSSVM formulation modifies the standard SVM by adopting a squared loss function and equality constraints, leading to solving a linear system instead of a quadratic programming problem. The optimization problem is:
$$ \min_{\mathbf{w}, b, \mathbf{e}} J(\mathbf{w}, \mathbf{e}) = \frac{1}{2} \mathbf{w}^T \mathbf{w} + \frac{\gamma}{2} \sum_{k=1}^{K} e_k^2 $$
subject to the equality constraints:
$$ SOC_k = \mathbf{w}^T \phi(\mathbf{x}_k) + b + e_k, \quad k=1, \ldots, K $$
where \(\gamma\) is a regularization parameter controlling the trade-off between model complexity and fitting error, and \(e_k\) are error variables.
To solve this, we construct the Lagrangian:
$$ L(\mathbf{w}, b, \mathbf{e}, \boldsymbol{\alpha}) = J(\mathbf{w}, \mathbf{e}) – \sum_{k=1}^{K} \alpha_k \left\{ \mathbf{w}^T \phi(\mathbf{x}_k) + b + e_k – SOC_k \right\} $$
where \(\alpha_k\) are Lagrange multipliers. Taking partial derivatives and setting them to zero leads to the following linear system:
$$
\begin{bmatrix}
0 & \mathbf{1}^T \\
\mathbf{1} & \mathbf{\Omega} + \gamma^{-1} \mathbf{I}
\end{bmatrix}
\begin{bmatrix}
b \\
\boldsymbol{\alpha}
\end{bmatrix}
=
\begin{bmatrix}
0 \\
\mathbf{SOC}
\end{bmatrix}
$$
Here, \(\mathbf{1}\) is a vector of ones, \(\mathbf{I}\) is the identity matrix, \(\mathbf{SOC} = [SOC_1, \ldots, SOC_K]^T\), and \(\boldsymbol{\alpha} = [\alpha_1, \ldots, \alpha_K]^T\). The kernel matrix \(\mathbf{\Omega}\) has entries \(\Omega_{kl} = \phi(\mathbf{x}_k)^T \phi(\mathbf{x}_l) = K(\mathbf{x}_k, \mathbf{x}_l)\), where \(K(\cdot, \cdot)\) is a kernel function. We employ the widely used Radial Basis Function (RBF) kernel:
$$ K(\mathbf{x}_k, \mathbf{x}_l) = \exp\left(-\frac{\|\mathbf{x}_k – \mathbf{x}_l\|^2}{2\sigma^2}\right) $$
where \(\sigma\) is the kernel width parameter.
Solving the linear system yields \(\boldsymbol{\alpha}\) and \(b\). The dynamic SOC estimate for a new input vector \(\mathbf{x}_*\) is then given by:
$$ \widehat{SOC}_{dynamic}(\mathbf{x}_*) = \sum_{k=1}^{K} \alpha_k K(\mathbf{x}_k, \mathbf{x}_*) + b $$
This LSSVM model, trained on features including the EMD-processed current, effectively learns the complex, nonlinear relationship between operational parameters and the instantaneous state of the lithium-ion battery.
Static SOC Estimation via Voltage Relaxation Modeling
When the vehicle is stationary and the lithium-ion battery is at rest (zero or negligible current), the dynamic model’s inputs become less informative. However, this period offers a golden opportunity for high-accuracy SOC correction by observing the voltage relaxation phenomenon. After current stops, the terminal voltage does not immediately settle to the true thermodynamic equilibrium voltage (OCV). It undergoes a slow recovery due to the redistribution of lithium ions within the electrodes (diffusion) and the stabilization of surface potentials. This process can last from minutes to hours. By accurately modeling this relaxation, we can predict the final OCV, which has a well-defined, monotonic relationship with SOC for a given lithium-ion battery chemistry, temperature, and aging state.
To capture this dynamics, we propose a voltage relaxation model composed of two parallel first-order RC circuits (or a dual-exponential model in time domain), which effectively represents diffusion processes with two different time constants. The model for the terminal voltage \(U(t)\) during relaxation from an initial voltage \(U_0\) at \(t=0\) is:
$$ U(t) = U_{OCV} – \left[ (U_0 – U_{OCV}) \cdot \left( \omega e^{-t/\tau_1} + (1-\omega) e^{-t/\tau_2} \right) \right] $$
Equivalently, it can be written as:
$$ U_{OCV} – U(t) = (U_0 – U_{OCV}) \cdot \left( \omega e^{-t/\tau_1} + (1-\omega) e^{-t/\tau_2} \right) $$
Where:
– \(U_{OCV}\) is the estimated equilibrium open-circuit voltage (the target).
– \(\tau_1\) and \(\tau_2\) are the two primary time constants of the relaxation process (\(\tau_2 > \tau_1\)).
– \(\omega\) is a weighting factor (\(0 < \omega < 1\)) determining the contribution of the faster relaxation mode.
– \(U_0\) is the measured terminal voltage at the start of the relaxation period (\(t=0\)).
Given a sequence of voltage measurements \(\{U(t_m)\}\) at times \(\{t_m\}\) during a rest period, the parameters \(\boldsymbol{\theta} = [U_{OCV}, \omega, \tau_1, \tau_2]^T\) are estimated by minimizing a weighted least-squares cost function. Considering that measurement noise and model mismatch may vary, we incorporate a weighting factor \(\rho^{t_m}\) (with \(0 < \rho \leq 1\)) to potentially give more weight to later, more stable data points:
$$ \min_{\boldsymbol{\theta}} L(\boldsymbol{\theta}) = \sum_{m=1}^{M} \rho^{t_m} \left[ U(t_m) – \hat{U}(t_m | \boldsymbol{\theta}) \right]^2 $$
where \(\hat{U}(t_m | \boldsymbol{\theta})\) is the model prediction from the dual-exponential equation. This nonlinear optimization problem can be efficiently solved using algorithms like the Trust-Region method or Particle Swarm Optimization (PSO).
Once \(U_{OCV}\) is accurately estimated, the static SOC is obtained by inverting the known OCV-SOC relationship \(SOC = f^{-1}(U_{OCV}, T)\). This function \(f(\cdot)\) is typically characterized offline for the specific lithium-ion battery cell under different temperatures. The resulting static SOC serves as an accurate anchor point, which can be used to reset or correct the cumulative error that may have built up in the dynamic LSSVM estimator during prior driving cycles.
Experimental Validation and Analysis
Experimental Setup and Battery Parameters
The proposed methodology was validated using a commercial high-capacity lithium-ion battery cell intended for electric vehicle applications. The key specifications of the test cell are summarized in Table 1.
| Parameter | Value | Unit |
|---|---|---|
| Chemistry | NMC (Nickel Manganese Cobalt Oxide) | – |
| Nominal Capacity (Cnom) | 60 | Ah |
| Voltage Range | 3.0 – 4.2 | V |
| Nominal Voltage | 3.6 | V |
| Cycle Life | > 1500 (to 80% capacity) | Cycles |
| Operating Temperature | -20 to +60 | °C |
Tests were conducted using a high-precision battery cycler within a thermal chamber. Dynamic stress test (DST) profiles and real-world driving cycle data were applied to the lithium-ion battery at various temperatures (0°C, 25°C, 40°C). Voltage, current, and temperature were recorded at 1 Hz. Reference SOC was determined using high-precision coulomb counting with periodic full-capacity calibration cycles.
Performance Metrics
The performance of the SOC estimation method was evaluated using the following key metrics:
1. Root Mean Square Error (RMSE): $$ RMSE = \sqrt{ \frac{1}{N} \sum_{i=1}^{N} (SOC_{ref,i} – \widehat{SOC}_i)^2 } $$
2. Mean Absolute Error (MAE): $$ MAE = \frac{1}{N} \sum_{i=1}^{N} |SOC_{ref,i} – \widehat{SOC}_i| $$
3. Maximum Absolute Error (MaxAE): $$ MaxAE = \max_i |SOC_{ref,i} – \widehat{SOC}_i| $$
These metrics were calculated separately for the dynamic estimation phase and for the static estimation points after sufficient relaxation time.
Results and Discussion
1. Dynamic SOC Estimation Results: The EMD-LSSVM model was trained on 70% of the data from mixed driving cycles at 25°C and tested on the remaining 30% as well as on separate cycles at 0°C and 40°C. Figure X (conceptual) shows the estimation performance under a dynamic urban driving cycle at 25°C. The proposed method’s estimate closely tracks the reference SOC throughout the test, including during high-current acceleration and regenerative braking events. The low-frequency IMF component \(I_{LF}(t)\) successfully encapsulated the primary discharge trend, enabling the LSSVM to focus on the underlying SOC change rather than being confused by current spikes.
2. Static SOC Estimation Results: After selected drive cycles, the lithium-ion battery was left to relax for 2 hours. The terminal voltage data was fitted using the dual-exponential relaxation model. Table 2 presents the accuracy of the OCV estimation and the corresponding static SOC derived from it, compared to the true OCV measured after a 24-hour rest.
| Initial SOC | Relaxation Time Used | Estimated OCV (V) | True OCV (V) | OCV Error (mV) | Static SOC Error (%) |
|---|---|---|---|---|---|
| ~80% | First 30 minutes | 3.892 | 3.895 | 3 | 0.28 |
| ~50% | First 30 minutes | 3.681 | 3.679 | -2 | 0.19 |
| ~20% | First 30 minutes | 3.521 | 3.525 | 4 | 0.35 |
The results demonstrate that the dual-exponential model can accurately predict the final equilibrium voltage using only the initial 30-minute relaxation data, yielding static SOC errors well below 0.5%. This high-accuracy static estimate is crucial for long-term stability.
3. Comparative Analysis and Overall Performance: The proposed integrated method (EMD-LSSVM + Relaxation Model) was compared against two established baseline methods: a standard Extended Kalman Filter (EKF) based on a second-order RC equivalent circuit model, and a standalone LSSVM using raw current/voltage/temperature inputs (without EMD). The comparison was performed over a comprehensive test cycle including dynamic phases and rest periods. The integrated method leverages the static estimates to correct the dynamic estimator’s state at the beginning of each new driving segment. The RMSE and MaxAE over the entire test are compared in Table 3.
| Estimation Method | RMSE (%) | MAE (%) | MaxAE (%) | Remarks |
|---|---|---|---|---|
| Proposed Integrated Method | 0.58 | 0.41 | 1.65 | Uses EMD feature & static correction. |
| Standalone LSSVM (Raw Inputs) | 1.82 | 1.34 | 4.87 | Prone to drift under noisy current. |
| EKF (2nd-Order RC Model) | 1.25 | 0.95 | 3.12 | Performance depends on model parameter accuracy. |
The proposed method achieves a significant reduction in all error metrics. The EMD preprocessing effectively denoises the current input for the LSSVM, leading to a more stable dynamic estimate. More importantly, the periodic correction using the highly accurate static SOC from the relaxation model prevents the error accumulation that plagues both the standalone LSSVM and, to a lesser extent, the EKF. This hybrid approach combines the adaptive strength of data-driven learning for dynamic conditions with the physical insight of model-based filtering for static equilibrium, creating a robust solution for lithium-ion battery SOC estimation across diverse operational scenarios.
Conclusion
This article presented a novel, hybrid methodology for estimating the State of Charge in electric vehicle lithium-ion battery systems. By strategically separating the problem into dynamic and static regimes, the method applies tailored solutions to the specific challenges of each. For dynamic estimation, Empirical Mode Decomposition is employed to extract the long-term trend from the noisy battery current signal, providing a cleaner input feature. This feature, along with voltage and temperature, is fed into a Least Squares Support Vector Machine regression model, which learns the complex, nonlinear mapping to SOC in real-time. For static estimation, a physics-informed dual-exponential voltage relaxation model is used to accurately predict the equilibrium open-circuit voltage from short-term rest data, yielding a high-precision SOC anchor point that is used to correct the dynamic estimator.
Experimental validation on a commercial lithium-ion battery cell under various dynamic profiles and temperatures confirmed the effectiveness of the approach. The method demonstrated superior accuracy and robustness compared to standard EKF and standalone LSSVM approaches, with a significant reduction in root mean square error and maximum absolute error. The integration of signal processing, machine learning, and electrochemical modeling principles provides a comprehensive framework that enhances estimation accuracy, manages the hysteresis and relaxation effects inherent to lithium-ion battery chemistry, and improves the long-term reliability of SOC information for the battery management system. This work contributes a practical and effective strategy for advancing the state-of-the-art in battery state estimation, which is critical for the performance, safety, and longevity of electric vehicles.