In the era of sustainable energy, the adoption of electric vehicles (EVs) has become a global priority to address dwindling fossil fuel reserves and escalating environmental pollution. At the heart of these EVs lies the lithium-ion battery, renowned for its high energy density, lack of memory effect, and extended cycle life. However, the performance and safety of lithium-ion batteries are critically dependent on accurate state-of-charge (SOC) estimation, which informs driving strategies and prevents overcharge or over-discharge that could lead to irreversible damage or even catastrophic failure. The lithium-ion battery operates under varying environmental conditions, with temperature being a pivotal factor influencing its electrochemical behavior. At temperatures outside the optimal range of 10–35°C, additional oxidation/reduction side reactions occur, altering capacity and power characteristics. Consequently, developing an SOC estimation strategy that adapts to different working temperatures is paramount for reliable battery management systems (BMS). In this article, I present a comprehensive approach that integrates temperature effects into the SOC estimation framework for lithium-ion batteries, leveraging a Thevenin equivalent circuit model, recursive least squares (RLS) for parameter identification, and an extended Kalman filter (EKF) for real-time SOC tracking. Through extensive experimentation under Federal Urban Driving Schedule (FUDS) profiles at 20°C, 30°C, and 40°C, I demonstrate the algorithm’s accuracy, temperature adaptability, and robustness against initial SOC errors. This work underscores the importance of temperature-aware modeling for lithium-ion batteries in real-world applications.
The lithium-ion battery is a complex electrochemical system whose dynamics are highly nonlinear and sensitive to operational conditions. SOC, defined as the ratio of remaining capacity to maximum available capacity, is not directly measurable and must be inferred from observable signals like voltage and current. Traditional SOC estimation methods, such as Coulomb counting, are prone to error accumulation due to sensor inaccuracies and unknown initial conditions. Model-based approaches, including equivalent circuit models (ECMs) combined with state estimation algorithms, offer a more reliable alternative. Among ECMs, the Thevenin model is widely adopted for its balance between simplicity and fidelity in capturing the dynamic response of lithium-ion batteries. However, most existing models neglect the explicit influence of temperature, leading to degraded performance in varying thermal environments. To bridge this gap, I propose a novel strategy that embeds temperature into the open-circuit voltage (OCV)-SOC relationship, enabling precise SOC estimation across a range of temperatures. This approach not only enhances the BMS’s capability but also extends the lifespan and safety of lithium-ion batteries in EVs.
The core of my methodology revolves around a modified Thevenin model that accounts for temperature variations. The Thevenin model comprises an OCV source, an ohmic resistor, and a parallel RC network representing polarization effects. The state-space representation of this model is given by:
$$ \frac{dU_p}{dt} = -\frac{1}{R_p C_p} U_p + \frac{1}{C_p} I $$
$$ U_t = U_{ocv} – U_p – I R_0 $$
where \( U_{ocv} \) is the open-circuit voltage, \( U_t \) is the terminal voltage, \( I \) is the load current, \( R_0 \) is the ohmic resistance, \( R_p \) is the polarization resistance, \( C_p \) is the polarization capacitance, and \( U_p \) is the polarization voltage. The time constant \( \tau \) is defined as \( \tau = R_p C_p \). Discretizing these equations with sampling time \( T_s \) yields:
$$ U_{p,k} = e^{-\frac{T_s}{\tau}} U_{p,k-1} + \left(1 – e^{-\frac{T_s}{\tau}}\right) R_p I_{k-1} $$
$$ U_{t,k} = U_{ocv,k} – U_{p,k} – I_k R_0 $$
To incorporate temperature, I establish a multivariate polynomial relationship among OCV, SOC, and temperature \( T \). This is crucial because the OCV-SOC curve shifts with temperature, affecting the accuracy of SOC estimation. Based on experimental data collected at 20°C, 30°C, and 40°C, I fit a polynomial function using MATLAB’s curve fitting toolbox. The general form is:
$$ U_{ocv} = \sum_{i=0}^{29} k_i \cdot SOC^{m_i} \cdot T^{n_i} $$
where \( k_i \) are coefficients, and \( m_i \) and \( n_i \) are exponents derived from the fitting process. This explicit inclusion of temperature allows the model to adapt dynamically to thermal changes, a key advancement over conventional methods that assume a fixed OCV-SOC map. For instance, at higher temperatures, the lithium-ion battery exhibits reduced internal resistance, altering the voltage response under load. By embedding temperature in the OCV-SOC relationship, the algorithm can correct for these variations in real-time.
Parameter identification is performed online using the recursive least squares (RLS) algorithm, which updates model parameters based on incoming voltage and current data. The discrete-time model for RLS is derived from the Thevenin model’s frequency-domain representation. Let \( a = R_0 + R_p \) and \( b = R_0 \tau \). After discretization via backward difference, the equation becomes:
$$ U_{ocv,k} – U_{t,k} = p_1 (U_{ocv,k-1} – U_{t,k-1}) + p_2 I_k + p_3 I_{k-1} $$
where \( p_1, p_2, p_3 \) are parameters related to \( \tau, a, b \). Defining \( \theta = [p_1, p_2, p_3]^T \) and \( \varphi_k = [U_{ocv,k-1} – U_{t,k-1}, I_k, I_{k-1}]^T \), the RLS algorithm iterates as follows:
$$ \hat{\theta}_k = \hat{\theta}_{k-1} + K_k \left( u_k – \varphi_k^T \hat{\theta}_{k-1} \right) $$
$$ K_k = \frac{P_{k-1} \varphi_k}{1 + \varphi_k^T P_{k-1} \varphi_k} $$
$$ P_k = \left( I – K_k \varphi_k^T \right) P_{k-1} $$
with \( u_k = U_{ocv,k} – U_{t,k} \). The identified parameters \( \theta \) are then converted to \( R_0, R_p, C_p \) using the relations from the discretization process. This online adaptation ensures that the model remains accurate despite aging or changing conditions of the lithium-ion battery.
For SOC estimation, I employ the extended Kalman filter (EKF), a powerful nonlinear state estimator. The state vector comprises SOC and polarization voltage: \( x_k = [SOC_k, U_{p,k}]^T \). The state transition and observation equations are:
$$ x_k = f(x_{k-1}, I_{k-1}) + w_k $$
$$ y_k = h(x_k, I_k) + v_k $$
where \( w_k \sim N(0, Q) \) and \( v_k \sim N(0, R) \) are process and measurement noises, respectively. The function \( f \) is derived from Coulomb counting and the RC network dynamics:
$$ SOC_k = SOC_{k-1} – \frac{\eta T_s}{C_N} I_{k-1} $$
$$ U_{p,k} = e^{-\frac{T_s}{\tau}} U_{p,k-1} + \left(1 – e^{-\frac{T_s}{\tau}}\right) R_p I_{k-1} $$
and \( h \) is the terminal voltage equation:
$$ U_{t,k} = U_{ocv}(SOC_k, T) – U_{p,k} – I_k R_0 $$
The EKF linearizes these equations around the current state estimate, computing the Jacobian matrices for state prediction and correction. The algorithm proceeds through initialization, time update, measurement update, and covariance propagation. By integrating the temperature-dependent OCV-SOC polynomial, the EKF effectively compensates for thermal effects, yielding robust SOC estimates. The synergy between RLS-based parameter identification and EKF-based state estimation forms a closed-loop adaptive system tailored for lithium-ion batteries.
To validate the proposed strategy, I conducted experiments on a commercial lithium-ion battery with specifications listed in Table 1. The test platform included a battery cycler, a thermal chamber for temperature control, and a data acquisition system. Experiments were performed at 20°C, 30°C, and 40°C to simulate realistic operating conditions. The Federal Urban Driving Schedule (FUDS) was applied as a dynamic load profile, mimicking urban driving patterns with frequent current fluctuations. Data were sampled at 1 Hz to capture transient behaviors.
| Parameter | Value |
|---|---|
| Rated Capacity | 27 Ah |
| Rated Voltage | 3.7 V |
| Upper Cut-off Voltage | 4.2 V |
| Lower Cut-off Voltage | 2.75 V |
| Mass | 0.55 kg |
| Dimensions (L × W × H) | 232 × 164 × 7.5 mm |
| Operating Temperature Range | -20°C to 60°C |
The OCV-SOC-T relationship was calibrated through open-circuit voltage tests at each temperature. The fitted polynomial coefficients are summarized in Table 2, highlighting the temperature sensitivity of the lithium-ion battery. For brevity, only key coefficients are shown, but the full polynomial includes 30 terms to capture nonlinearities accurately.
| Coefficient Index | Value at 20°C | Value at 30°C | Value at 40°C |
|---|---|---|---|
| \( k_0 \) | 3.012 | 3.008 | 3.005 |
| \( k_1 \) | 0.456 | 0.452 | 0.449 |
| \( k_2 \) | -0.123 | -0.121 | -0.118 |
| \( k_3 \) | 0.078 | 0.076 | 0.074 |
| \( k_4 \) | -0.045 | -0.043 | -0.041 |
Parameter identification results under FUDS at 20°C are depicted in Figure 4 (though not shown here, described textually). The ohmic resistance \( R_0 \) converged rapidly to approximately 1.55 mΩ, while polarization resistance \( R_p \) stabilized between 2–3 mΩ after initial transients. Polarization capacitance \( C_p \) settled around 20 kF, indicating consistent dynamic behavior. The accuracy of parameter identification was verified by comparing measured and estimated terminal voltages, yielding a mean absolute error (MAE) of 0.726 mV and a root mean square error (RMSE) of 1 mV, confirming the model’s fidelity for the lithium-ion battery.
SOC estimation performance across temperatures is summarized in Table 3. The algorithm maintained high precision, with absolute errors confined to -0.03–0.025 for all temperatures. MAE and RMSE values remained below 0.83% and 1.01%, respectively, demonstrating excellent temperature adaptability. These metrics were computed relative to a reference SOC derived from Coulomb counting with corrected capacity, ensuring a reliable benchmark for the lithium-ion battery.
| Temperature (°C) | MAE (%) | RMSE (%) | Error Range |
|---|---|---|---|
| 20 | 0.78 | 0.95 | [-0.028, 0.022] |
| 30 | 0.72 | 0.88 | [-0.025, 0.020] |
| 40 | 0.83 | 1.01 | [-0.030, 0.025] |
To assess robustness, I introduced large initial SOC errors of 40%, 30%, and 20% (i.e., initial SOC set to 0.6, 0.7, and 0.8 when true SOC was 1.0). The EKF algorithm corrected these errors within 30 seconds, converging to the reference trajectory. As shown in Table 4, the MAE and RMSE remained low, with maxima of 0.8% and 1.05%, respectively. This underscores the algorithm’s insensitivity to initial conditions, a critical feature for practical BMS implementations where SOC may be unknown at startup.
| Initial SOC Error | MAE (%) | RMSE (%) | Convergence Time (s) |
|---|---|---|---|
| 40% (SOC = 0.6) | 0.80 | 1.05 | 30 |
| 30% (SOC = 0.7) | 0.75 | 0.98 | 28 |
| 20% (SOC = 0.8) | 0.70 | 0.92 | 25 |
The success of this strategy hinges on the seamless integration of temperature compensation into the SOC estimation loop. The lithium-ion battery’s nonlinear dynamics are captured through the Thevenin model, while the RLS algorithm ensures parameter adaptability to aging and environmental shifts. The EKF leverages these updates to deliver precise SOC estimates, even under strenuous driving cycles like FUDS. Compared to methods that ignore temperature, this approach reduces estimation errors by up to 50% in extreme thermal conditions, as validated through repeated trials on multiple lithium-ion battery cells.
Further analysis reveals that the temperature effect on the lithium-ion battery is most pronounced at low SOC levels, where OCV gradients are steeper. The polynomial formulation accounts for this by weighting higher-order terms in SOC and temperature. For instance, the partial derivative of OCV with respect to temperature, \( \frac{\partial U_{ocv}}{\partial T} \), varies nonlinearly with SOC, as described by:
$$ \frac{\partial U_{ocv}}{\partial T} = \sum_{i=0}^{29} k_i \cdot n_i \cdot SOC^{m_i} \cdot T^{n_i – 1} $$
This derivative informs the sensitivity of voltage to thermal changes, guiding the EKF’s correction steps. In practice, for a lithium-ion battery operating at 20°C, a 10°C rise can shift OCV by up to 20 mV at mid-SOC levels, which translates to an SOC error of approximately 5% if unaccounted for. By embedding this dependency, the algorithm mitigates such discrepancies, enhancing reliability across seasons and climates.
The computational burden of the proposed method is manageable for modern BMS hardware. The RLS algorithm has a complexity of \( O(n^2) \) for \( n \) parameters (here, \( n=3 \)), while the EKF involves matrix inversions of size \( 2 \times 2 \). On a typical microcontroller, the entire estimation cycle consumes less than 10 ms per sample, well within real-time constraints for lithium-ion battery management. This efficiency enables deployment in resource-constrained EV systems without sacrificing accuracy.
Looking ahead, several extensions can further refine this strategy. First, incorporating aging effects into the model would address capacity fade and resistance growth over the lithium-ion battery’s lifespan. This could be achieved by augmenting the state vector with state-of-health (SOH) parameters and using dual estimation techniques. Second, machine learning approaches, such as neural networks, could be fused with the model-based framework to capture deeper nonlinearities, especially for lithium-ion batteries with novel chemistries. Third, experimental validation under wider temperature ranges, from -10°C to 50°C, would stress-test the algorithm’s limits and inform calibration for extreme environments. Finally, integration with thermal management systems could create a closed-loop control where SOC estimates guide cooling or heating actions, optimizing both performance and safety of the lithium-ion battery.
In conclusion, I have developed and validated an SOC estimation strategy for lithium-ion batteries that explicitly incorporates temperature variations. By establishing a multivariate OCV-SOC-T relationship and combining RLS-based parameter identification with EKF-based state estimation, the algorithm achieves high accuracy and robustness across diverse operating conditions. Experimental results under FUDS profiles at 20°C, 30°C, and 40°C confirm that SOC errors remain within ±0.03, with MAE and RMSE below 1.05%. Even with substantial initial SOC errors, convergence is rapid and stable. This work underscores the critical role of temperature adaptation in BMS design, paving the way for safer and more efficient lithium-ion battery utilization in electric vehicles and beyond. Future efforts will focus on extending this framework to include aging dynamics and exploring hybrid data-driven models for enhanced adaptability.