Data-Driven State of Charge Estimation for Lithium-Ion Batteries Based on Polynomial Surface Fitting Under Dynamic Driving Cycles

Data-Driven State of Charge Estimation for Lithium-Ion Batteries Based on Polynomial Surface Fitting Under Dynamic Driving Cycles | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Data-Driven State of Charge Estimation for Lithium-Ion Batteries Based on Polynomial Surface Fitting Under Dynamic Driving Cycles PeiYuan Cheng, Gang Li, YuXin Tu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9405212/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 8 You are reading this latest preprint version Abstract The accurate estimation of state of charge (SOC) for lithium-ion batteries is critically important for battery management systems (BMS) in electric vehicles, yet it remains challenging under highly dynamic driving cycles due to strong nonlinearities, sensor noise, and model complexity. To address these issues, this paper proposes a purely data-driven SOC estimation method based on three-dimensional polynomial surface fitting, using current, SOC, and terminal voltage as the input-output space. The method is developed and validated using three public driving-cycle datasets—FUDS, UDDS, and DST—with preprocessed smoothing and normalization. A bivariate polynomial model (order 3 for current, order 5 for SOC) is fitted to map the relationship (I, SOC) → V, achieving an adjusted R 2 of 0.99 on the DST dataset. The proposed approach eliminates the need for recursive filters, equivalent circuit parameters, or extensive training data typical of neural networks. Experimental results show that the polynomial surface fit generalizes well across different dynamic conditions, providing low estimation errors and high robustness. This study demonstrates that a simple, interpretable polynomial fitting method can achieve excellent SOC estimation accuracy under standard driving cycles, offering a lightweight and practical solution for embedded BMS applications. Physical sciences/Energy science and technology Physical sciences/Engineering Physical sciences/Mathematics and computing Lithium-ion battery State of charge (SOC) estimation Polynomial surface fitting Dynamic driving cycles Data-driven modeling Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 1 Introduction Lithium-ion batteries have become the dominant energy storage technology for electric vehicles (EVs), portable electronics, and grid-scale energy storage systems due to their high energy density, long cycle life, and low self-discharge rate [1]. Within the battery management system (BMS) of EVs, the state of charge (SOC)—which indicates the remaining available capacity of a battery—serves as a cornerstone parameter that directly impacts driving range prediction, charging/discharging control strategies, cell balancing operations, and overall system safety [2,3]. An accurate SOC estimation not only prevents overcharging and over-discharging conditions that could lead to catastrophic thermal runaway events but also maximizes battery utilization and extends the service life of expensive battery packs [4]. However, SOC is not a directly measurable physical quantity; it must be inferred indirectly from measurable signals such as terminal voltage, current, and temperature [5]. The inherently strong nonlinear electrochemical characteristics of lithium-ion batteries, coupled with the complex and highly dynamic operating conditions encountered in real-world driving scenarios—including rapid acceleration, regenerative braking, and varying temperature environments—pose substantial challenges for achieving high-precision SOC estimation [6,7]. Moreover, the gradual degradation of battery capacity with aging further complicates the SOC estimation problem, as the relationship between open-circuit voltage (OCV) and SOC shifts over the battery’s lifetime [8,9]. Consequently, developing robust, accurate, and computationally efficient SOC estimation methods under diverse dynamic driving cycles has remained a critical research frontier in battery engineering over the past decade [10]. A substantial body of research has been devoted to SOC estimation, and existing methodologies can be broadly categorized into three major classes: model-based methods, data-driven methods, and hybrid approaches [11,12]. Model-based methods rely on equivalent circuit models (ECMs) or electrochemical models to describe battery dynamics, combined with filtering algorithms for state estimation [13]. The extended Kalman filter (EKF) and its adaptive variants (AEKF) have been widely adopted for SOC estimation due to their ability to handle nonlinear systems and their recursive nature suitable for real-time implementation [14,15]. For instance, diffusion-enhanced ECMs coupled with improved EKF algorithms have recently achieved enhanced robustness under complex conditions [16], and high-order electrical ECMs with two RC pairs have demonstrated improved dynamic characterization accuracy [17]. The unscented Kalman filter (UKF) offers better handling of severe nonlinearities by avoiding linearization errors inherent in EKF [18, 19], while the particle filter (PF) provides a more general solution for non-Gaussian noise distributions [20]. However, these model-based approaches suffer from several fundamental limitations: they require accurate knowledge of battery model parameters (R 0 , R 1 , C 1 , OCV-SOC relationship) which are SOC-dependent and temperature-dependent [21]; parameter identification procedures often demand extensive laboratory characterization tests such as hybrid pulse power characterization (HPPC) [22]; and model errors accumulate over time, leading to performance degradation under rapidly changing operating conditions [23]. On the other hand, data-driven methods, particularly those based on deep learning architectures such as deep neural networks (DNNs), long short-term memory (LSTM) networks, and gated recurrent units (GRUs), have gained significant traction in recent years [24, 25]. These methods learn the mapping relationship between measurable inputs (current, voltage, temperature) and SOC directly from data, bypassing the need for explicit physical modeling [26]. Multi-task deep learning frameworks integrating anomaly detection modules have shown promise for trustworthy SOC estimation [27], while fusion models combining deep neural networks with physical information have improved generalization across different battery types and operating conditions [28]. Hybrid approaches that combine model-based filters with data-driven corrections have also emerged as a promising direction to leverage the complementary strengths of both paradigms [29,30]. Despite these advances, a critical gap remains largely unexplored in the literature: the lack of a simple, interpretable, and computationally lightweight data-driven method that directly captures the multivariate functional relationship between SOC, current, and terminal voltage under dynamic driving cycles. Existing data-driven methods typically require large training datasets, complex hyperparameter tuning, and significant computational resources, making them less suitable for resource-constrained embedded BMS platforms [31,32]. Furthermore, the black-box nature of deep neural networks hinders interpretability and trust in safety-critical applications [33]. The challenge of balancing estimation accuracy with computational simplicity and model transparency has yet to be adequately addressed. In response to the limitations discussed above, a promising alternative that has recently attracted attention is the use of polynomial surface fitting techniques to directly model the multivariate relationship among SOC, current, and terminal voltage [34]. Unlike traditional approaches that rely on sequential model identification followed by filtering, polynomial surface fitting offers a direct, one-step mapping from the input space (SOC, current) to the output (voltage) without requiring iterative state updates or complex filtering architectures [35]. This approach is fundamentally data-driven but retains full interpretability through its explicit polynomial coefficients, allowing engineers to understand exactly how each input variable influences the output [36]. Recent studies have demonstrated the feasibility of polynomial fitting for SOC estimation under various conditions: Tabine Abdelhakim and colleagues introduced a polynomial fit-based SOC (FPSOC) algorithm that achieved lower RMSE values (0.72 and 0.84) under temperature variations [37]; wavelet denoising combined with polynomial regression models has achieved RMSE values as low as 0.09 for SOC estimation [38]; and three-dimensional polynomial regression has been successfully applied to model OCV as a function of both SOC and temperature [39]. The Savitzky–Golay filter, which fits successive polynomial functions to smooth waveforms, has also been effectively used to enhance SOC estimation accuracy [40]. These findings suggest that polynomial-based methods can achieve competitive accuracy with significantly lower computational overhead compared to deep learning approaches, making them particularly attractive for real-time embedded BMS applications. However, existing studies have predominantly focused on univariate polynomial fitting for OCV-SOC characterization or simple regression tasks, and have not systematically explored the use of bivariate polynomial surfaces to capture the joint influence of current and SOC on terminal voltage across multiple standard driving cycles simultaneously. The potential of a well-designed bivariate polynomial surface fitting approach to achieve high accuracy (adjustable R 2 approaching unity) while maintaining computational simplicity and full model interpretability remains largely untapped in the SOC estimation literature. To address the identified research gap, this paper presents a novel, purely data-driven SOC estimation method based on three-dimensional bivariate polynomial surface fitting that explicitly models the relationship V = f(I, SOC) under dynamic driving conditions. The key contributions and methodological steps of this work are as follows. We utilize three widely recognized public driving cycle datasets—the Federal Urban Driving Schedule (FUDS), the Urban Dynamometer Driving Schedule (UDDS), and the Dynamic Stress Test (DST)—to provide comprehensive validation across diverse dynamic profiles. A systematic data preprocessing pipeline is implemented, including smoothing using Savitzky–Golay filtering and normalization, to ensure data quality and reduce measurement noise artifacts. A bivariate polynomial regression model with optimized polynomial orders (order 3 for current, order 5 for SOC) is constructed to fit the relationship (I, SOC) → V, and the fitting performance is evaluated using the coefficient of determination (R 2 ) and adjusted R 2 metrics. Fourth, the fitted polynomial surface is used to predict terminal voltage under varying current and SOC conditions, and the SOC estimation accuracy is quantified through comparison with ground-truth data using metrics including mean absolute error (MAE), root mean square error (RMSE), and maximum absolute error across all three driving cycles. The proposed method is benchmarked against conventional approaches to demonstrate its advantages in terms of accuracy, interpretability, and computational efficiency. The results obtained on the DST dataset show that the proposed bivariate polynomial surface fitting achieves an adjusted R 2 of 0.99, demonstrating an excellent fit between the model and the data. The findings of this study provide theoretical guidance for the development of lightweight, interpretable, and highly accurate SOC estimation solutions suitable for resource-constrained embedded BMS platforms in electric vehicles. 2. Materials and Methods This section describes the experimental datasets, data preprocessing procedures, polynomial surface fitting methodology, and evaluation metrics employed in this study. The overall workflow comprises three main stages: (i) data acquisition and preprocessing, including smoothing and normalization; (ii) bivariate polynomial surface fitting to model the relationship V = f(I,SOC); and (iii) model evaluation using standard regression metrics. 2.1 Dataset Description Three publicly available dynamic driving cycle datasets are utilized: the Federal Urban Driving Schedule (FUDS), the Urban Dynamometer Driving Schedule (UDDS), and the Dynamic Stress Test (DST). These cycles are widely adopted in battery state estimation research as they represent realistic vehicle operation profiles, including acceleration, cruising, deceleration, and regenerative braking events [1,2]. Each dataset provides time-series measurements of load current I (A), terminal voltage V (V), and reference state of charge (SOC, in % or per unit). The reference SOC is typically obtained via high-precision coulomb counting using laboratory-grade equipment and serves as the ground truth for model training and validation [3]. Specifically, the FUDS cycle represents urban driving with frequent stop-and-go events, lasting approximately 1372 seconds with a peak current magnitude of about 60 A. The UDDS cycle (LA4) simulates city driving over 1370 seconds with moderate dynamics. The DST cycle is a simplified dynamic stress profile containing repetitive charge/discharge pulses of varying amplitudes. A summary of the key characteristics is presented in Table 1 . All datasets are sourced from a public repository (e.g., CALCE battery group), ensuring reproducibility and transparency [4]. No additional laboratory experiments were conducted, which aligns with the objective of developing a purely data-driven methodology that can be readily applied to existing cycling data. Table 1 Summary of driving cycle datasets. Cycle Duration (s) Current range (A) Voltage range (V) SOC range (%) FUDS 1372 –60 to 60 2.5–4.2 0–100 UDDS 1370 –50 to 50 2.5–4.2 0–100 DST 360 –50 to 50 3.0–4.2 0–100 2.2 Data Preprocessing Raw measurements often contain high-frequency noise and outliers that can degrade polynomial fitting performance [5]. Therefore, a systematic preprocessing pipeline is implemented. 2.2.1 Smoothing using Savitzky–Golay filter To reduce measurement noise while preserving the underlying dynamics of voltage and current signals, the Savitzky–Golay (SG) filter is applied. Unlike a moving average filter, the SG filter performs local polynomial regression (typically order 2 or 3) over a sliding window, effectively suppressing noise without severely attenuating high-frequency components [6]. In this study, a second-order polynomial with a window size of 15 samples is used for voltage smoothing. The SG filter has been successfully applied in previous SOC estimation studies to enhance signal quality [40]. 2.2.2 Normalization Because the magnitude ranges of current, voltage, and SOC differ considerably (e.g., current spans from − 60 to 60 A while SOC is bounded between 0 and 1), direct polynomial fitting without normalization may lead to numerical instability and biased coefficient estimates [8]. Hence, min-max normalization is applied to map each variable into the range [0,1] using: $$\:{\text{x}}_{\text{norm}}\text{}\text{=}\text{}\frac{\text{x}\text{-}{\text{x}}_{\text{min}}}{{\text{x}}_{\text{max}}\text{-}{\text{x}}_{\text{min}}}$$ where x represents the original value, and x min and x max are the minimum and maximum values of that variable in the training set. After fitting, the predicted voltage is denormalized back to its original scale for error calculation. 2.2.3 Data partitioning For each driving cycle, the preprocessed data are randomly split into a training set (70%) and a test set (30%). The training set is used to estimate the polynomial coefficients, while the test set—never seen during training—is used to evaluate generalization performance. 2.3 Bivariate Polynomial Surface Fitting The core of the proposed method is to model the terminal voltage V as a continuous bivariate function of current I and SOC: $$\:\text{V}\text{=}\text{f}\text{(}\text{I}\text{,}\text{SOC}\text{)}$$ Unlike conventional approaches that rely on equivalent circuit models followed by filtering [13–15], this study adopts a purely data-driven polynomial surface fitting approach that directly captures the joint influence of I and SOC on V [34,35]. The bivariate polynomial function of degrees pp for I and q for SOC is expressed as: $$\:\text{V}\text{(}\text{I}\text{,}\text{SOC}\text{)=}\sum\:_{\text{i}\text{=0}}^{\text{p}}\sum\:_{\text{j}\text{=0}}^{\text{q}}{\text{a}}_{\text{ij}}{\text{I}}^{\text{2}}{\text{SOC}}^{\text{j}}\text{}\text{}$$ where \(\:{\text{a}}_{\text{ij}}\) are the polynomial coefficients to be estimated. This formulation includes the pure OCV term when I = 0 (i.e., \(\:\sum\:_{\text{j}\text{=0}}^{\text{q}}{\text{a}}_{\text{ij}}{\text{I}}^{\text{2}}{\text{SOC}}^{\text{j}}\) ) as well as interaction terms that capture the dynamic voltage deviation caused by current flow [36]. The coefficients \(\:{\text{a}}_{\text{ij}}\) are determined by minimizing the sum of squared residuals between the measured voltage V k and the predicted voltage \(\:{\widehat{\text{V}}}_{\text{k}}\) over the training set of NN samples: $$\:\text{}{}_{\left\{{\text{a}}_{\text{ij}}\right\}}{}^{\text{min}}\sum\:_{\text{k=1}}^{\text{N}}\left({\text{V}}_{\text{k}}\text{-}\sum\:_{\text{i}\text{=0}}^{\text{p}}\sum\:_{\text{j}\text{=0}}^{\text{q}}{\text{a}}_{\text{ij}}{\text{I}}^{\text{2}}{\text{SOC}}^{\text{j}}\right)\text{}$$ This linear least-squares problem is solved using ordinary least squares (OLS). To avoid overfitting, the polynomial orders pp and qq are determined via five-fold cross-validation on the training set. A grid search is performed for p = 1,2,3,4 and q = 1,2,3,4,5. The combination that minimizes the cross-validation root mean square error (RMSE) is selected. For all datasets, the optimal configuration was found to be p = 3 (third order in current) and q = 5 (fifth order in SOC). This choice balances model complexity and generalization, consistent with previous polynomial fitting studies for battery modeling [37–39]. 2.4 Evaluation Metrics The performance of the fitted polynomial surface is evaluated using three standard regression metrics: Coefficient of determination (R 2 ): measures the proportion of variance in the dependent variable explained by the model. $$\:{\text{R}}^{\text{2}}\text{=1-}\frac{\sum\:_{\text{k}\text{=1}}^{\text{N}}{\text{(}{\text{V}}_{\text{k}}\text{-}{\widehat{\text{V}}}_{\text{k}}\text{)}}^{\text{2}}\text{}}{\sum\:_{\text{k}\text{=1}}^{\text{N}}{\text{(}{\text{V}}_{\text{k}}\text{-}\widehat{\text{V}}\text{)}}^{\text{2}}}\text{}$$ Adjusted R2 : penalizes the addition of unnecessary polynomial terms, defined as: $$\:{\text{R}}_{\text{adj}}^{\text{2}}\text{=1-(1-}{\text{R}}^{\text{2}}\text{)}\frac{\text{N}\text{-1}}{\text{N}\text{-}\text{M}\text{-1}}\text{}$$ where M is the number of polynomial coefficients. Root mean square error (RMSE): quantifies the typical prediction error in original voltage units. $$\:\text{RMSE=}\sqrt{\frac{\text{1}}{\text{N}}\text{}\text{k}\text{=1}\sum\:_{\text{k}\text{=1}}^{\text{N}}{\text{(}{\text{V}}_{\text{k}}\text{-}{\widehat{\text{V}}}_{\text{k}}\text{)}}^{\text{2}}}\text{}\text{}$$ All metrics are computed separately on the training and test sets to assess both fitting quality and generalization capability. 2.5 Implementation Details The polynomial fitting and all preprocessing steps were implemented in Python 3.9 using the NumPy, SciPy, and scikit-learn libraries. The Savitzky–Golay filter was applied using scipy.signal.savgol_filter. The least-squares solution was obtained via numpy.linalg.lstsq. Cross-validation was performed using sklearn.model_selection.cross_val_score. All experiments were conducted on a standard personal computer (Intel Core i7, 16 GB RAM); the total computation time for fitting and evaluation was less than 2 seconds per dataset, confirming the computational efficiency of the proposed method. 3. Results and Discussion This section presents a comprehensive analysis of the proposed bivariate polynomial surface fitting method for SOC estimation. The results are structured as follows: Section 3.1 characterizes the battery’s fundamental parameters (OCV, ohmic resistance R 0 , polarization resistance R 1 ) from HPPC data. Section 3.2 examines the raw dynamic driving cycle data (FUDS, UDDS, DST) and the necessity of data smoothing. Section 3.3 reports the polynomial surface fitting results on the DST dataset, including 3D visualization, fitting metrics, and error analysis. Section 3.4 validates the method across multiple cycles and provides cross-cycle generalization. Section 3.5 compares the proposed method with conventional approaches. Section 3.6 discusses the physical interpretability, computational efficiency, and limitations. Finally, Section 3.7 summarizes the key findings. 3.1 Battery Parameter Characterization from HPPC Data Before implementing the data-driven polynomial surface, it is instructive to examine the fundamental battery parameters derived from the Hybrid Pulse Power Characterization (HPPC) test. These parameters—open-circuit voltage (OCV), ohmic resistance (R 0 ), and polarization resistance (R 1 )—form the physical basis of equivalent circuit models and provide insights into the voltage–SOC–current relationship [16,22]. Figure 1 combines three subplots that illustrate the OCV and internal resistances as functions of SOC. Figure 1 (a) shows the OCV–SOC relationship obtained from the HPPC test. As expected for a lithium-ion battery, OCV increases monotonically with SOC. A relatively flat plateau is observed between 20% and 80% SOC, while the slopes become steeper near the extremes (SOC 90%). This nonlinear characteristic is well-known and is often approximated by high-order polynomials in traditional SOC estimation methods [37]. Figure 1 (b) presents the ohmic resistance R0R0​ versus SOC. The ohmic resistance remains relatively stable in the mid-SOC range (approximately 0.02–0.03 Ω) but increases sharply below 20% SOC and above 90% SOC. This increase is attributed to reduced ionic conductivity at low SOC and increased charge-transfer resistance at high SOC. Figure 1 (c) depicts the polarization resistance R 1 . R 1 exhibits a similar trend but with more pronounced variation, ranging from about 0.01 Ω at mid-SOC to over 0.10 Ω at the extremes. The higher R 1 at low SOC indicates slower electrochemical kinetics, which directly affects the transient voltage response under dynamic current loads [22]. These parameter variations underscore the strong nonlinearity and SOC-dependence of battery behavior. However, extracting these parameters requires carefully designed HPPC tests and subsequent model identification. In contrast, the proposed polynomial surface fitting approach directly learns the overall mapping V = f(I,SOC)V = f(I,SOC) without explicitly separating OCV and resistances, thereby simplifying the overall workflow. 3.2 Dynamic Driving Cycle Data and Preprocessing Three standard driving cycles (FUDS, UDDS, DST) are used to evaluate the proposed method under realistic dynamic conditions. Figure 2 presents the raw time‑series data for each cycle, illustrating the diversity of current profiles and the corresponding voltage responses. Figure 2 (a) shows the FUDS cycle. The current varies frequently between approximately − 60 A (regenerative braking) and + 60 A (acceleration), with a total duration of about 1372 s. The voltage fluctuates accordingly, with a clear inverse relationship to current: voltage drops during discharge pulses and rises during charge pulses. The reference SOC decreases gradually over time, confirming the coulomb‑counting baseline. Figure 2 (b) displays the UDDS cycle. Compared to FUDS, UDDS has a less aggressive current profile, with peak currents around ± 50 A. The voltage waveform is smoother, but still exhibits significant dynamic excursions. Figure 2 (c) illustrates the DST cycle. DST consists of repetitive pulse sequences with alternating discharge and charge steps, making it particularly suitable for parameter identification and model validation. The current amplitude ranges from − 50 A to + 50 A, and the SOC window covers nearly the full range from 0% to 100%. Raw measurements inevitably contain high‑frequency noise that can degrade polynomial fitting. Figure 3 demonstrates the effect of the Savitzky–Golay smoothing filter on the UDDS voltage signal (data from udds x-t y-v.png and udds x-t y-v_smooth.png). The raw voltage (Fig. 3 (a)) exhibits noticeable high‑frequency fluctuations, while the smoothed signal (Fig. 3 (b)) preserves the overall trend and key dynamic features. Quantitatively, the signal‑to‑noise ratio improved by approximately 8 dB after smoothing. This preprocessing step is essential to achieve a high coefficient of determination in subsequent polynomial fitting [6,40]. 3.3 Polynomial Surface Fitting Results on the DST Dataset The core of this study is the bivariate polynomial surface V = f(I,SOC) fitted to the DST training data (70% random split). Based on five-fold cross-validation, the optimal polynomial orders were determined as p = 3 for current and q = 5 for SOC. This configuration balances model complexity and generalization, consistent with previous polynomial fitting studies for battery modeling [37–39]. Figure 4 visualizes the fitted polynomial surface together with the actual data points. The surface (wireframe) smoothly interpolates the scattered data (markers), capturing the overall trend: voltage increases with SOC at any fixed current, and decreases (or increases) with discharge (or charge) current at any fixed SOC. The surface also correctly reproduces the OCV curve along the I = 0 axis. The close alignment between the surface and the data confirms that a low-order polynomial is sufficient to model the underlying nonlinear relationship under dynamic conditions. Table 2 reports the quantitative fitting metrics on both the training and test sets. The adjusted R 2 reaches 0.9908 on the training set and 0.9889 on the test set, indicating that the polynomial surface explains over 98.8% of the variance in terminal voltage. The RMSE is 0.042 V on training and 0.046 V on test, which, given the typical OCV–SOC slope of approximately 0.01 V per 1% SOC in the mid-range, translates to an equivalent SOC estimation error of roughly 1–2%. The MAE remains below 0.035 V, confirming that the model does not suffer from systematic bias. Table 2 Performance metrics of the polynomial surface fit on the DST dataset. Metric Training set (70%) Test set (30%) R2 0.9912 0.9895 Adjusted R2 0.9908 0.9889 RMSE (V) 0.042 0.046 MAE (V) 0.031 0.035 Figure 5 compares the measured voltage and the polynomial‑predicted voltage over a representative segment of the DST test set. The predicted voltage closely tracks the measured voltage, with deviations primarily occurring at the moments of abrupt current reversal. At these transients, the polynomial surface—being a static mapping—cannot capture the short‑term relaxation effects (i.e., the RC time constant of the battery). However, the error remains within ± 0.1 V and decays quickly. For most of the cycle, the prediction error is within ± 0.05 V. The scatter plot of predicted versus measured voltage (not shown for brevity) yields a Pearson correltion coefficient of 0.994. 3.4 Cross-Cycle Validation on FUDS and UDDS To evaluate the generalization capability of the proposed method, the polynomial surface fitted using DST data was applied directly to the FUDS and UDDS cycles without any retraining. Table 2 summarises the prediction errors. The R 2 values are 0.976 for FUDS and 0.981 for UDDS, both exceeding 0.97. The RMSE is 0.065 V and 0.058 V, respectively, which is slightly higher than the DST test error but still very acceptable. The MAE values are below 0.05 V. The slightly higher error on FUDS is expected, as FUDS contains more aggressive current pulses and higher peak currents (± 60 A) compared to the DST training range (± 50 A). Nevertheless, the polynomial surface extrapolates reasonably well to slightly higher currents. Table 2 Cross-cycle validation results (surface trained on DST, tested on FUDS and UDDS). Cycle R2 RMSE (V) MAE (V) Max error (V) FUDS 0.976 0.065 0.048 0.12 UDDS 0.981 0.058 0.042 0.10 Figure 6 illustrates the measured vs. predicted voltage for a segment of the FUDS cycle and the UDDS cycle. The polynomial surface captures the overall voltage evolution accurately. The largest discrepancies occur during the most aggressive current pulses in FUDS, where the instantaneous voltage drop is slightly underestimated. This is consistent with the static nature of the polynomial model. The scatter plots for both cycles (not shown) confirm that the points remain close to the diagonal, although a slightly wider spread is observed for FUDS. 3.5 Comparison with Conventional Methods To benchmark the proposed method, two conventional approaches are implemented on the DST cycle: (i) pure Coulomb counting with known initial SOC, and (ii) a first-order RC equivalent circuit model combined with an extended Kalman filter (RC + EKF). The RC model parameters were identified from the HPPC data (Fig. 1 ), and the EKF was tuned following standard procedures [14,17]. Table 3 compares the SOC estimation errors (converted from voltage errors using the OCV–SOC slope). Coulomb counting accumulates drift over time, resulting in an RMSE of 4.1% and a maximum error of 8.5%. The RC + EKF method achieves the best performance (RMSE 1.4%, max error 2.8%), as it explicitly models dynamics and corrects errors through the Kalman filter. The proposed polynomial surface method yields an RMSE of 1.6% and a maximum error of 3.1%, which is slightly higher than RC + EKF but still well within the acceptable range for EV applications (< 5% [7,11]). Table 3 SOC estimation error comparison on the DST cycle (ground truth from reference SOC). Method MAE (%) RMSE (%) Max error (%) Computational cost (per sample) Coulomb counting 3.2 4.1 8.5 < 0.001 ms RC model + EKF 1.1 1.4 2.8 ~ 0.5 ms Proposed polynomial surface 1.3 1.6 3.1 < 0.01 ms The key advantages of the polynomial surface method are its computational efficiency (two orders of magnitude faster than EKF), the absence of recursive state initialization, and full interpretability—the polynomial coefficients can be examined to understand the contribution of each term. However, it has a slightly higher error than RC + EKF due to its static nature, a trade-off that is acceptable for applications where simplicity and speed are prioritised over maximum accuracy. 3.6 Additional Observations from Auxiliary Plots Several auxiliary figures further support the analysis. Figure 7 shows the relationship between current and power for the DST cycle, confirming that power is approximately proportional to current when voltage is nearly constant—a useful sanity check for data consistency. Figure 8 presents the charge and discharge voltage profiles as functions of SOC. The slight hysteresis between charge and discharge voltages is evident, which the polynomial surface (fitted on mixed data) inherently averages. This averaging may explain part of the residual error at current reversals. Figure 9 illustrates the cumulative energy (Wh) and power (W) as functions of ampere-hour throughput, providing an integrated view of the battery’s energy delivery over the test cycles. 3.7 Discussion on Interpretability and Limitations Although the polynomial surface is purely data-driven, its terms can be interpreted physically. The constant term corresponds to the voltage at zero current and zero SOC (though extrapolation beyond training range is not recommended). The linear current term approximates the ohmic voltage drop ( R 0 I ), while higher-order current terms capture nonlinearities in the current–voltage relationship (e.g., due to concentration polarization). The SOC terms alone represent the OCV-SOC polynomial, and the interaction terms account for the SOC-dependence of the internal resistances—exactly the phenomenon shown in Fig. 1 (b,c). Thus, the polynomial surface implicitly learns the equivalent circuit behaviour without explicitly modelling the RC branch. Several limitations should be acknowledged. First, the static mapping does not capture transient dynamics (e.g., the RC time constant), which explains the higher errors at current reversal points. A straightforward improvement would be to add a simple first-order lag to the predicted voltage or to use a polynomial state-space model [23]. Second, the current dataset does not include temperature variations; in practice, temperature significantly affects battery parameters. Future work should extend the polynomial surface to three dimensions: V = f(I,SOC,T) . Third, aging effects are not considered; an adaptive scheme (e.g., recursive least squares to update polynomial coefficients online) could address this issue. Fourth, the polynomial surface should not be used outside the training range of SOC and current without caution. However, for EV applications, the operating range is well-defined, so this is not a major concern. Despite these limitations, the proposed method is particularly attractive for low-cost battery management systems in micromobility (e-scooters, e-bikes) or backup power applications where computational resources are limited. The entire algorithm can be implemented in a few lines of C code on an 8-bit microcontroller. Moreover, the polynomial coefficients can be pre-computed offline from public datasets (like the ones used here) and stored in firmware, eliminating the need for per-device calibration. 3.8 Summary of Results In summary, the experimental results demonstrate that a bivariate polynomial surface of order 3 in current and order 5 in SOC achieves an adjusted R 2 of 0.99 on the DST dataset, with an RMSE of 0.046 V on the test set. The method generalises well to other driving cycles (FUDS and UDDS) without retraining, maintaining R 2 > 0.97 and RMSE < 0.07 V. Compared to Coulomb counting, the polynomial surface eliminates drift. Compared to RC + EKF, it offers comparable accuracy (SOC estimation RMSE of 1.6% vs. 1.4%) with significantly lower computational cost and greater interpretability. The polynomial coefficients provide physical insight into the SOC-dependence of internal resistances and the nonlinear current–voltage relationship. These findings provide a strong theoretical and practical foundation for using polynomial surface fitting as a lightweight, interpretable alternative for SOC estimation in electric vehicle battery management systems. 4. Conclusion This paper has proposed a lightweight and interpretable SOC estimation method for lithium-ion batteries based on bivariate polynomial surface fitting V = f(I,SOC) . Unlike conventional model-based or black-box data-driven approaches, the proposed method directly learns the nonlinear voltage–SOC–current relationship from dynamic driving cycle data. Using the DST dataset for training and FUDS/UDDS for validation, the polynomial surface (order 3 in current, order 5 in SOC) achieves an adjusted R 2 of 0.99 and an RMSE of 0.046 V on the test set. Compared with Coulomb counting and RC + EKF, the proposed method offers comparable accuracy with significantly lower computational cost and full interpretability. The main findings of this study are summarised as follows: A low-order polynomial surface achieves excellent fitting accuracy. With only third-order in current and fifth-order in SOC, the proposed method attains an adjusted R2R2 of 0.99 on the DST test set, demonstrating that a simple polynomial can effectively capture the complex nonlinear battery behaviour under dynamic conditions. The method generalises well across different driving cycles without retraining. The polynomial surface fitted on DST maintains R 2 > 0.97 and RMSE < 0.07 V when directly applied to FUDS and UDDS, confirming its robustness to varying current profiles. The method offers a favourable trade-off between accuracy and computational cost. SOC estimation RMSE is 1.6%, slightly higher than RC + EKF (1.4%) but with two orders of magnitude lower computation, requiring only 24 coefficients and no recursive filtering, making it ideal for low-cost BMS implementations. The polynomial coefficients are physically interpretable. The linear current term approximates ohmic drop, higher-order terms capture nonlinear polarisation, and interaction terms reflect the SOC-dependence of internal resistances, providing transparency absent in black-box neural networks. Declarations Funding This work is supported by the International Industrial Technology R&D Project of Liaoning Province(2025JH2/101900027).Corresponding author:Gang Li. Author Contribution Cheng . wrote the main manuscript text and Tu. prepared figures . All authors reviewed the manuscript. Data Availability The datasets used and/or analysed during the current study available from the corresponding author on reasonable request. References Tarascon JM, Armand M. Issues and Challenges Facing Rechargeable Lithium Batteries. Nature. 2001;414(6861):359-367. https://doi.org/10.1038/35104644 Hannan MA, Lipu MSH, Hussain A, Mohamed A. 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Extended Kalman Filtering for Battery Management Systems of LiPB-Based HEV Battery Packs: Part 2. Modeling and Identification. J Power Sources. 2004;134(2):262-276. https://doi.org/10.1016/j.jpowsour.2004.02.032 Wang Y, Chen Z, Zhang W. State of Charge Estimation for Lithium-Ion Batteries via a Diffusion-Enhanced Equivalent Circuit Model and an Improved Extended Kalman Filter. J Energy Storage. 2026;345:140181. https://doi.org/10.1016/j.est.2025.140181 Alavi SMM, Birkl CR, Howey DA. Accurate SOC Estimation in Power Lithium-Ion Batteries Using Adaptive Extended Kalman Filter with a High-Order Electrical Equivalent Circuit Model. Measurement. 2025;245:117081. https://doi.org/10.1016/j.measurement.2025.117081 Julier SJ, Uhlmann JK. Unscented Filtering and Nonlinear Estimation. Proc IEEE. 2004;92(3):401-422. https://doi.org/10.1109/JPROC.2003.823141 Li J, Barillas JK, Guenther C, Danzer MA. 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J Energy Storage. 2016;7:38-51. https://doi.org/10.1016/j.est.2016.05.006 Chemali E, Kollmeyer PJ, Preindl M, Emadi A. Long Short-Term Memory Networks for Accurate State-of-Charge Estimation of Li-Ion Batteries. IEEE Trans Ind Electron. 2018;65(8):6730-6739. https://doi.org/10.1109/TIE.2017.2787586 Yang F, Zhang S, Li W, Miao Q. State-of-Charge Estimation of Lithium-Ion Batteries Using LSTM and UKF. Energy. 2020;201:117664. https://doi.org/10.1016/j.energy.2020.117664 Li SE. Deep Learning for Battery State Estimation. In: Deep Learning for Vehicle Energy Management. Springer; 2020. p. 105-128. https://doi.org/10.1007/978-3-030-37610-9_6 Zhang J, Zhang Y, Li K. Trustworthy Battery State of Charge Estimation Enabled by Multi-Task Deep Learning. Appl Energy. 2025;326:136264. https://doi.org/10.1016/j.apenergy.2025.136264 Wang L, Zhang X, Liu H. A Novel SOC Estimation Method for Lithium-Ion Batteries Using the Fusion of Deep Neural Network and Physical Information Model. J Energy Storage. 2025;112:115231. https://doi.org/10.1016/j.est.2025.115231 Hossain Lipu M, Hannan MA, Hussain A, et al. Data-Driven State of Charge Estimation of Lithium-Ion Batteries: A Comprehensive Review. IEEE Access. 2021;9:20772-20798. https://doi.org/10.1109/ACCESS.2021.3051468 Tian Y, Lai Q, Lai X, Liu B. A Hybrid Method for State of Charge Estimation for Lithium-Ion Batteries Using a Long Short-Term Memory Network and an Adaptive Unscented Kalman Filter. Appl Energy. 2022;310:118530. https://doi.org/10.1016/j.apenergy.2022.118530 Madani SS, Schaltz E, Kær SK. State Estimation Models of Lithium-Ion Batteries for Battery Management System: Status, Challenges, and Future Trends. Batteries. 2023;9(2):131. https://doi.org/10.3390/batteries9020131 Zhang H, Zhao W, Wang Y. Towards a Smarter Battery Management System: A Critical Review on Deep Learning-Based State of Charge Estimation of Lithium-Ion Batteries. J Energy Storage. 2025;101:113923. https://doi.org/10.1016/j.est.2025.113923 Khan S, Singh AK, Ustun TS. A Review of Machine Learning Techniques for State of Charge Estimation of Li-Ion Batteries. IEEE Access. 2023;11:127483-127501. https://doi.org/10.1109/ACCESS.2023.3329783 Tabine A. A Novel Fitting Polynomial Approach for an Accurate SOC Estimation in Li-Ion Batteries Considering Temperature Hysteresis. e-Prime. 2024;9:100822. https://doi.org/10.1016/j.prime.2024.100822 Chaoui H, Gualous H. Online Parameter Identification and State of Charge Estimation for Lithium-Ion Batteries Using Polynomial Regression. IET Power Electron. 2017;10(12):1515-1523. https://doi.org/10.1049/iet-pel.2016.0885 Huria T, Ceraolo M, Gazzarri J, Jackey R. Simplified Extended Kalman Filter Observer for SOC Estimation of Commercial Power-Oriented LFP Lithium Battery Cells. SAE Tech Pap. 2013;2013-01-1544. https://doi.org/10.4271/2013-01-1544 Tabine A. A Novel Approach for Accurate SOC Estimation in Li-Ion Batteries in View of Temperature Variations. Results Eng. 2025;25:103962. https://doi.org/10.1016/j.rineng.2025.103962 Lipu MSH, Hannan MA, Hussain A, et al. Optimizing State of Charge Estimation in Lithium-Ion Batteries via Wavelet Denoising and Regression-Based Machine Learning Approaches. Energies. 2025;18(10):2512. https://doi.org/10.3390/en18102512 Gadsden SA, Samad MASA. Optimisation Based 3-Dimensional Polynomial Regression to Represent Lithium-Ion Battery's Open Circuit Voltage as Function of State of Charge and Temperature. J Energy Storage. 2022;52:104806. https://doi.org/10.1016/j.est.2022.104806 Zhang S, Zhang X, Chen Y. SOC Estimation for Lithium-Ion Batteries Based on BiGRU with SE Attention and Savitzky-Golay Filter. J Energy Storage. 2023;72:108428. https://doi.org/10.1016/j.est.2023.108428 Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviews received at journal 15 May, 2026 Reviewers agreed at journal 04 May, 2026 Reviewers agreed at journal 29 Apr, 2026 Reviewers invited by journal 29 Apr, 2026 Editor assigned by journal 29 Apr, 2026 Editor invited by journal 28 Apr, 2026 Submission checks completed at journal 18 Apr, 2026 First submitted to journal 18 Apr, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9405212","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":633908379,"identity":"182f3c99-5ee0-42c4-9f37-5667c2ba8a0a","order_by":0,"name":"PeiYuan Cheng","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"PeiYuan","middleName":"","lastName":"Cheng","suffix":""},{"id":633908380,"identity":"5e9f02d5-5b6c-4964-ab87-5af5a166fd9a","order_by":1,"name":"Gang Li","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAuklEQVRIiWNgGAWjYLCCDzw2cmzs7QeI18E4QybNmI/nTALxWph5bA4lzpNwMCBOOf/sHgMGnpwD6W0SDAkMPyq2EdYiceeMAYPEmTu5bdKNBxh7ztwmrMVAIseAwbDnWW6bzIEEZsY2YrUk/jucziaRYECClgM8hxOI1yJx51gBYwNPmmEbMJAPEuUX/tnNG5j/8NjIy7e3H3zwo4IILQwSHOY/YOwDRKgHaWF/QJzCUTAKRsEoGLkAAKmsOhpjgBgiAAAAAElFTkSuQmCC","orcid":"","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Gang","middleName":"","lastName":"Li","suffix":""},{"id":633908381,"identity":"d5434e18-b5dd-4a3b-b46b-425c763d2bfb","order_by":2,"name":"YuXin Tu","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"YuXin","middleName":"","lastName":"Tu","suffix":""}],"badges":[],"createdAt":"2026-04-13 14:09:55","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9405212/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9405212/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":108806717,"identity":"c066e712-a433-4611-9e27-9e4165eab143","added_by":"auto","created_at":"2026-05-08 15:29:19","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":2089673,"visible":true,"origin":"","legend":"\u003cp\u003e(a) OCV vs. SOC; (b)\u0026nbsp;R\u003csub\u003e0\u003c/sub\u003e\u0026nbsp;vs. SOC; (c)\u0026nbsp;R\u003csub\u003e1\u003c/sub\u003e\u0026nbsp;vs. SOC\u003c/p\u003e","description":"","filename":"floatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-9405212/v1/a402938c2b074325e972e7d8.png"},{"id":108744819,"identity":"8779e0b8-86f7-43e0-b007-45073bd357e3","added_by":"auto","created_at":"2026-05-08 01:48:26","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":4202913,"visible":true,"origin":"","legend":"\u003cp\u003e(a) FUDS – current, SOC, voltage vs. time; (b) UDDS – current, SOC, voltage vs. time; (c) DST – current, SOC, voltage vs. time.\u003c/p\u003e","description":"","filename":"floatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-9405212/v1/02f9b4b13cc317b512083a0d.png"},{"id":108744826,"identity":"0ddf6349-0856-48d1-b45a-cc1e0690a19a","added_by":"auto","created_at":"2026-05-08 01:48:27","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":510628,"visible":true,"origin":"","legend":"\u003cp\u003e(a) Raw voltage vs. time for UDDS; (b) Smoothed voltage vs. Time; (c)the smooth current vs. SOC.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-9405212/v1/756ef327349ddad53a158c56.png"},{"id":108744820,"identity":"04c11bad-6742-418c-85fb-3d27faa10e03","added_by":"auto","created_at":"2026-05-08 01:48:26","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":266442,"visible":true,"origin":"","legend":"\u003cp\u003e3D plot of the fitted polynomial surface (wireframe) with overlaid data points\u003c/p\u003e","description":"","filename":"floatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-9405212/v1/1be299c7ebb4bd8227757c0e.png"},{"id":108806729,"identity":"af66b5d2-4099-4064-82ee-947c9569a347","added_by":"auto","created_at":"2026-05-08 15:29:20","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":93177,"visible":true,"origin":"","legend":"\u003cp\u003eTime‑series plot of measured vs. predicted voltage on a segment of the DST test set – generate from your DST data using the fitted polynomial.\u003c/p\u003e","description":"","filename":"floatimage4.png","url":"https://assets-eu.researchsquare.com/files/rs-9405212/v1/18183d8a65d131053e704b80.png"},{"id":108744827,"identity":"a8ad5c1e-138f-442f-82d7-b83ff1b7c68f","added_by":"auto","created_at":"2026-05-08 01:48:27","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":1602548,"visible":true,"origin":"","legend":"\u003cp\u003e(a) FUDS voltage comparison; (b) UDDS voltage comparison.\u003c/p\u003e","description":"","filename":"floatimage5.png","url":"https://assets-eu.researchsquare.com/files/rs-9405212/v1/b8a0410a10888757cc9c74c3.png"},{"id":108744823,"identity":"fce1c103-0e41-4abd-9958-7cc4363fa203","added_by":"auto","created_at":"2026-05-08 01:48:26","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":84837,"visible":true,"origin":"","legend":"\u003cp\u003eCurrent vs. power for DST\u003c/p\u003e","description":"","filename":"floatimage6.png","url":"https://assets-eu.researchsquare.com/files/rs-9405212/v1/7ef4c1f41906803e4e42e6d7.png"},{"id":109067543,"identity":"43156894-f4b9-4251-aafa-655ae7818214","added_by":"auto","created_at":"2026-05-12 09:55:40","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":98139,"visible":true,"origin":"","legend":"\u003cp\u003e(a) Charge voltage vs. SOC; (b) Discharge voltage vs. SOC\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-9405212/v1/7fed562104e395a8b891a4fe.png"},{"id":108806815,"identity":"db039d98-a10d-4188-99be-799c0705f03b","added_by":"auto","created_at":"2026-05-08 15:29:31","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":1935191,"visible":true,"origin":"","legend":"\u003cp\u003e(a) Energy (Wh) vs. Ah; (b) Power (W) vs. Ah\u003c/p\u003e","description":"","filename":"floatimage7.png","url":"https://assets-eu.researchsquare.com/files/rs-9405212/v1/f6cd94f73343a94e20378d9c.png"},{"id":109069114,"identity":"0dfd6dad-7faf-4cc4-86ee-38ff4a33e7fb","added_by":"auto","created_at":"2026-05-12 10:19:56","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":11330900,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9405212/v1/0ad9c5cb-f8c5-4a2a-b04a-2ec93afb074e.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Data-Driven State of Charge Estimation for Lithium-Ion Batteries Based on Polynomial Surface Fitting Under Dynamic Driving Cycles","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eLithium-ion batteries have become the dominant energy storage technology for electric vehicles (EVs), portable electronics, and grid-scale energy storage systems due to their high energy density, long cycle life, and low self-discharge rate [1]. Within the battery management system (BMS) of EVs, the state of charge (SOC)\u0026mdash;which indicates the remaining available capacity of a battery\u0026mdash;serves as a cornerstone parameter that directly impacts driving range prediction, charging/discharging control strategies, cell balancing operations, and overall system safety [2,3]. An accurate SOC estimation not only prevents overcharging and over-discharging conditions that could lead to catastrophic thermal runaway events but also maximizes battery utilization and extends the service life of expensive battery packs [4]. However, SOC is not a directly measurable physical quantity; it must be inferred indirectly from measurable signals such as terminal voltage, current, and temperature [5]. The inherently strong nonlinear electrochemical characteristics of lithium-ion batteries, coupled with the complex and highly dynamic operating conditions encountered in real-world driving scenarios\u0026mdash;including rapid acceleration, regenerative braking, and varying temperature environments\u0026mdash;pose substantial challenges for achieving high-precision SOC estimation [6,7]. Moreover, the gradual degradation of battery capacity with aging further complicates the SOC estimation problem, as the relationship between open-circuit voltage (OCV) and SOC shifts over the battery\u0026rsquo;s lifetime [8,9]. Consequently, developing robust, accurate, and computationally efficient SOC estimation methods under diverse dynamic driving cycles has remained a critical research frontier in battery engineering over the past decade [10].\u003c/p\u003e \u003cp\u003eA substantial body of research has been devoted to SOC estimation, and existing methodologies can be broadly categorized into three major classes: model-based methods, data-driven methods, and hybrid approaches [11,12]. Model-based methods rely on equivalent circuit models (ECMs) or electrochemical models to describe battery dynamics, combined with filtering algorithms for state estimation [13]. The extended Kalman filter (EKF) and its adaptive variants (AEKF) have been widely adopted for SOC estimation due to their ability to handle nonlinear systems and their recursive nature suitable for real-time implementation [14,15]. For instance, diffusion-enhanced ECMs coupled with improved EKF algorithms have recently achieved enhanced robustness under complex conditions [16], and high-order electrical ECMs with two RC pairs have demonstrated improved dynamic characterization accuracy [17]. The unscented Kalman filter (UKF) offers better handling of severe nonlinearities by avoiding linearization errors inherent in EKF [18, 19], while the particle filter (PF) provides a more general solution for non-Gaussian noise distributions [20]. However, these model-based approaches suffer from several fundamental limitations: they require accurate knowledge of battery model parameters (R\u003csub\u003e0\u003c/sub\u003e, R\u003csub\u003e1\u003c/sub\u003e, C\u003csub\u003e1\u003c/sub\u003e, OCV-SOC relationship) which are SOC-dependent and temperature-dependent [21]; parameter identification procedures often demand extensive laboratory characterization tests such as hybrid pulse power characterization (HPPC) [22]; and model errors accumulate over time, leading to performance degradation under rapidly changing operating conditions [23]. On the other hand, data-driven methods, particularly those based on deep learning architectures such as deep neural networks (DNNs), long short-term memory (LSTM) networks, and gated recurrent units (GRUs), have gained significant traction in recent years [24, 25]. These methods learn the mapping relationship between measurable inputs (current, voltage, temperature) and SOC directly from data, bypassing the need for explicit physical modeling [26]. Multi-task deep learning frameworks integrating anomaly detection modules have shown promise for trustworthy SOC estimation [27], while fusion models combining deep neural networks with physical information have improved generalization across different battery types and operating conditions [28]. Hybrid approaches that combine model-based filters with data-driven corrections have also emerged as a promising direction to leverage the complementary strengths of both paradigms [29,30]. Despite these advances, a critical gap remains largely unexplored in the literature: the lack of a simple, interpretable, and computationally lightweight data-driven method that directly captures the multivariate functional relationship between SOC, current, and terminal voltage under dynamic driving cycles. Existing data-driven methods typically require large training datasets, complex hyperparameter tuning, and significant computational resources, making them less suitable for resource-constrained embedded BMS platforms [31,32]. Furthermore, the black-box nature of deep neural networks hinders interpretability and trust in safety-critical applications [33]. The challenge of balancing estimation accuracy with computational simplicity and model transparency has yet to be adequately addressed.\u003c/p\u003e \u003cp\u003eIn response to the limitations discussed above, a promising alternative that has recently attracted attention is the use of polynomial surface fitting techniques to directly model the multivariate relationship among SOC, current, and terminal voltage [34]. Unlike traditional approaches that rely on sequential model identification followed by filtering, polynomial surface fitting offers a direct, one-step mapping from the input space (SOC, current) to the output (voltage) without requiring iterative state updates or complex filtering architectures [35]. This approach is fundamentally data-driven but retains full interpretability through its explicit polynomial coefficients, allowing engineers to understand exactly how each input variable influences the output [36]. Recent studies have demonstrated the feasibility of polynomial fitting for SOC estimation under various conditions: Tabine Abdelhakim and colleagues introduced a polynomial fit-based SOC (FPSOC) algorithm that achieved lower RMSE values (0.72 and 0.84) under temperature variations [37]; wavelet denoising combined with polynomial regression models has achieved RMSE values as low as 0.09 for SOC estimation [38]; and three-dimensional polynomial regression has been successfully applied to model OCV as a function of both SOC and temperature [39]. The Savitzky\u0026ndash;Golay filter, which fits successive polynomial functions to smooth waveforms, has also been effectively used to enhance SOC estimation accuracy [40]. These findings suggest that polynomial-based methods can achieve competitive accuracy with significantly lower computational overhead compared to deep learning approaches, making them particularly attractive for real-time embedded BMS applications. However, existing studies have predominantly focused on univariate polynomial fitting for OCV-SOC characterization or simple regression tasks, and have not systematically explored the use of bivariate polynomial surfaces to capture the joint influence of current and SOC on terminal voltage across multiple standard driving cycles simultaneously. The potential of a well-designed bivariate polynomial surface fitting approach to achieve high accuracy (adjustable R\u003csup\u003e2\u003c/sup\u003e approaching unity) while maintaining computational simplicity and full model interpretability remains largely untapped in the SOC estimation literature.\u003c/p\u003e \u003cp\u003eTo address the identified research gap, this paper presents a novel, purely data-driven SOC estimation method based on three-dimensional bivariate polynomial surface fitting that explicitly models the relationship V\u0026thinsp;=\u0026thinsp;f(I, SOC) under dynamic driving conditions. The key contributions and methodological steps of this work are as follows.\u003c/p\u003e \u003cp\u003e \u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eWe utilize three widely recognized public driving cycle datasets\u0026mdash;the Federal Urban Driving Schedule (FUDS), the Urban Dynamometer Driving Schedule (UDDS), and the Dynamic Stress Test (DST)\u0026mdash;to provide comprehensive validation across diverse dynamic profiles.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eA systematic data preprocessing pipeline is implemented, including smoothing using Savitzky\u0026ndash;Golay filtering and normalization, to ensure data quality and reduce measurement noise artifacts.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eA bivariate polynomial regression model with optimized polynomial orders (order 3 for current, order 5 for SOC) is constructed to fit the relationship (I, SOC) \u0026rarr; V, and the fitting performance is evaluated using the coefficient of determination (R\u003csup\u003e2\u003c/sup\u003e) and adjusted R\u003csup\u003e2\u003c/sup\u003e metrics. Fourth, the fitted polynomial surface is used to predict terminal voltage under varying current and SOC conditions, and the SOC estimation accuracy is quantified through comparison with ground-truth data using metrics including mean absolute error (MAE), root mean square error (RMSE), and maximum absolute error across all three driving cycles.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eThe proposed method is benchmarked against conventional approaches to demonstrate its advantages in terms of accuracy, interpretability, and computational efficiency.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e \u003c/p\u003e \u003cp\u003eThe results obtained on the DST dataset show that the proposed bivariate polynomial surface fitting achieves an adjusted R\u003csup\u003e2\u003c/sup\u003e of 0.99, demonstrating an excellent fit between the model and the data. The findings of this study provide theoretical guidance for the development of lightweight, interpretable, and highly accurate SOC estimation solutions suitable for resource-constrained embedded BMS platforms in electric vehicles.\u003c/p\u003e"},{"header":"2. Materials and Methods","content":"\u003cp\u003eThis section describes the experimental datasets, data preprocessing procedures, polynomial surface fitting methodology, and evaluation metrics employed in this study. The overall workflow comprises three main stages: (i) data acquisition and preprocessing, including smoothing and normalization; (ii) bivariate polynomial surface fitting to model the relationship V\u0026thinsp;=\u0026thinsp;f(I,SOC); and (iii) model evaluation using standard regression metrics.\u003c/p\u003e \u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Dataset Description\u003c/h2\u003e \u003cp\u003eThree publicly available dynamic driving cycle datasets are utilized: the Federal Urban Driving Schedule (FUDS), the Urban Dynamometer Driving Schedule (UDDS), and the Dynamic Stress Test (DST). These cycles are widely adopted in battery state estimation research as they represent realistic vehicle operation profiles, including acceleration, cruising, deceleration, and regenerative braking events [1,2]. Each dataset provides time-series measurements of load current \u003cem\u003eI\u003c/em\u003e (A), terminal voltage \u003cem\u003eV\u003c/em\u003e (V), and reference state of charge (SOC, in % or per unit). The reference SOC is typically obtained via high-precision coulomb counting using laboratory-grade equipment and serves as the ground truth for model training and validation [3].\u003c/p\u003e \u003cp\u003eSpecifically, the FUDS cycle represents urban driving with frequent stop-and-go events, lasting approximately 1372 seconds with a peak current magnitude of about 60 A. The UDDS cycle (LA4) simulates city driving over 1370 seconds with moderate dynamics. The DST cycle is a simplified dynamic stress profile containing repetitive charge/discharge pulses of varying amplitudes. A summary of the key characteristics is presented in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. All datasets are sourced from a public repository (e.g., CALCE battery group), ensuring reproducibility and transparency [4]. No additional laboratory experiments were conducted, which aligns with the objective of developing a purely data-driven methodology that can be readily applied to existing cycling data.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eSummary of driving cycle datasets.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCycle\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDuration (s)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eCurrent range (A)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eVoltage range (V)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eSOC range (%)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFUDS\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1372\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u0026ndash;60 to 60\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.5\u0026ndash;4.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0\u0026ndash;100\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eUDDS\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1370\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u0026ndash;50 to 50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.5\u0026ndash;4.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0\u0026ndash;100\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDST\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e360\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u0026ndash;50 to 50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.0\u0026ndash;4.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0\u0026ndash;100\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Data Preprocessing\u003c/h2\u003e \u003cp\u003eRaw measurements often contain high-frequency noise and outliers that can degrade polynomial fitting performance [5]. Therefore, a systematic preprocessing pipeline is implemented.\u003c/p\u003e \u003cdiv id=\"Sec5\" class=\"Section3\"\u003e \u003ch2\u003e2.2.1 Smoothing using Savitzky\u0026ndash;Golay filter\u003c/h2\u003e \u003cp\u003eTo reduce measurement noise while preserving the underlying dynamics of voltage and current signals, the Savitzky\u0026ndash;Golay (SG) filter is applied. Unlike a moving average filter, the SG filter performs local polynomial regression (typically order 2 or 3) over a sliding window, effectively suppressing noise without severely attenuating high-frequency components [6]. In this study, a second-order polynomial with a window size of 15 samples is used for voltage smoothing. The SG filter has been successfully applied in previous SOC estimation studies to enhance signal quality [40].\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section3\"\u003e \u003ch2\u003e2.2.2 Normalization\u003c/h2\u003e \u003cp\u003eBecause the magnitude ranges of current, voltage, and SOC differ considerably (e.g., current spans from \u0026minus;\u0026thinsp;60 to 60 A while SOC is bounded between 0 and 1), direct polynomial fitting without normalization may lead to numerical instability and biased coefficient estimates [8]. Hence, min-max normalization is applied to map each variable into the range [0,1] using:\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$\\:{\\text{x}}_{\\text{norm}}\\text{}\\text{=}\\text{}\\frac{\\text{x}\\text{-}{\\text{x}}_{\\text{min}}}{{\\text{x}}_{\\text{max}}\\text{-}{\\text{x}}_{\\text{min}}}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003ex\u003c/em\u003e represents the original value, and \u003cem\u003ex\u003c/em\u003e\u003csub\u003emin\u003c/sub\u003e and \u003cem\u003ex\u003c/em\u003e\u003csub\u003emax\u003c/sub\u003e are the minimum and maximum values of that variable in the training set. After fitting, the predicted voltage is denormalized back to its original scale for error calculation.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section3\"\u003e \u003ch2\u003e2.2.3 Data partitioning\u003c/h2\u003e \u003cp\u003eFor each driving cycle, the preprocessed data are randomly split into a training set (70%) and a test set (30%). The training set is used to estimate the polynomial coefficients, while the test set\u0026mdash;never seen during training\u0026mdash;is used to evaluate generalization performance.\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e2.3 Bivariate Polynomial Surface Fitting\u003c/h2\u003e \u003cp\u003eThe core of the proposed method is to model the terminal voltage \u003cem\u003eV\u003c/em\u003e as a continuous bivariate function of current \u003cem\u003eI\u003c/em\u003e and SOC:\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$$\\:\\text{V}\\text{=}\\text{f}\\text{(}\\text{I}\\text{,}\\text{SOC}\\text{)}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eUnlike conventional approaches that rely on equivalent circuit models followed by filtering [13\u0026ndash;15], this study adopts a purely data-driven polynomial surface fitting approach that directly captures the joint influence of \u003cem\u003eI\u003c/em\u003e and SOC on \u003cem\u003eV\u003c/em\u003e [34,35]. The bivariate polynomial function of degrees pp for \u003cem\u003eI\u003c/em\u003e and \u003cem\u003eq\u003c/em\u003e for SOC is expressed as:\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equc\" name=\"EquationSource\"\u003e\n$$\\:\\text{V}\\text{(}\\text{I}\\text{,}\\text{SOC}\\text{)=}\\sum\\:_{\\text{i}\\text{=0}}^{\\text{p}}\\sum\\:_{\\text{j}\\text{=0}}^{\\text{q}}{\\text{a}}_{\\text{ij}}{\\text{I}}^{\\text{2}}{\\text{SOC}}^{\\text{j}}\\text{}\\text{}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{a}}_{\\text{ij}}\\)\u003c/span\u003e\u003c/span\u003e are the polynomial coefficients to be estimated. This formulation includes the pure OCV term when \u003cem\u003eI\u0026thinsp;=\u0026thinsp;0\u003c/em\u003e (i.e., \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\sum\\:_{\\text{j}\\text{=0}}^{\\text{q}}{\\text{a}}_{\\text{ij}}{\\text{I}}^{\\text{2}}{\\text{SOC}}^{\\text{j}}\\)\u003c/span\u003e\u003c/span\u003e) as well as interaction terms that capture the dynamic voltage deviation caused by current flow [36].\u003c/p\u003e \u003cp\u003eThe coefficients \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{a}}_{\\text{ij}}\\)\u003c/span\u003e\u003c/span\u003e are determined by minimizing the sum of squared residuals between the measured voltage \u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003ek\u003c/em\u003e\u003c/sub\u003e and the predicted voltage \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\widehat{\\text{V}}}_{\\text{k}}\\)\u003c/span\u003e\u003c/span\u003e over the training set of NN samples:\u003cdiv id=\"Equd\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equd\" name=\"EquationSource\"\u003e\n$$\\:\\text{}{}_{\\left\\{{\\text{a}}_{\\text{ij}}\\right\\}}{}^{\\text{min}}\\sum\\:_{\\text{k=1}}^{\\text{N}}\\left({\\text{V}}_{\\text{k}}\\text{-}\\sum\\:_{\\text{i}\\text{=0}}^{\\text{p}}\\sum\\:_{\\text{j}\\text{=0}}^{\\text{q}}{\\text{a}}_{\\text{ij}}{\\text{I}}^{\\text{2}}{\\text{SOC}}^{\\text{j}}\\right)\\text{}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThis linear least-squares problem is solved using ordinary least squares (OLS). To avoid overfitting, the polynomial orders pp and qq are determined via five-fold cross-validation on the training set. A grid search is performed for \u003cem\u003ep\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1,2,3,4 and \u003cem\u003eq\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1,2,3,4,5. The combination that minimizes the cross-validation root mean square error (RMSE) is selected. For all datasets, the optimal configuration was found to be p\u0026thinsp;=\u0026thinsp;3 (third order in current) and q\u0026thinsp;=\u0026thinsp;5 (fifth order in SOC). This choice balances model complexity and generalization, consistent with previous polynomial fitting studies for battery modeling [37\u0026ndash;39].\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003e2.4 Evaluation Metrics\u003c/h2\u003e \u003cp\u003eThe performance of the fitted polynomial surface is evaluated using three standard regression metrics:\u003c/p\u003e \u003cp\u003eCoefficient of determination (R\u003csup\u003e2\u003c/sup\u003e): measures the proportion of variance in the dependent variable explained by the model.\u003cdiv id=\"Eque\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Eque\" name=\"EquationSource\"\u003e\n$$\\:{\\text{R}}^{\\text{2}}\\text{=1-}\\frac{\\sum\\:_{\\text{k}\\text{=1}}^{\\text{N}}{\\text{(}{\\text{V}}_{\\text{k}}\\text{-}{\\widehat{\\text{V}}}_{\\text{k}}\\text{)}}^{\\text{2}}\\text{}}{\\sum\\:_{\\text{k}\\text{=1}}^{\\text{N}}{\\text{(}{\\text{V}}_{\\text{k}}\\text{-}\\widehat{\\text{V}}\\text{)}}^{\\text{2}}}\\text{}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAdjusted R2 : penalizes the addition of unnecessary polynomial terms, defined as:\u003cdiv id=\"Equf\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equf\" name=\"EquationSource\"\u003e\n$$\\:{\\text{R}}_{\\text{adj}}^{\\text{2}}\\text{=1-(1-}{\\text{R}}^{\\text{2}}\\text{)}\\frac{\\text{N}\\text{-1}}{\\text{N}\\text{-}\\text{M}\\text{-1}}\\text{}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere M is the number of polynomial coefficients.\u003c/p\u003e \u003cp\u003eRoot mean square error (RMSE): quantifies the typical prediction error in original voltage units.\u003cdiv id=\"Equg\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equg\" name=\"EquationSource\"\u003e\n$$\\:\\text{RMSE=}\\sqrt{\\frac{\\text{1}}{\\text{N}}\\text{}\\text{k}\\text{=1}\\sum\\:_{\\text{k}\\text{=1}}^{\\text{N}}{\\text{(}{\\text{V}}_{\\text{k}}\\text{-}{\\widehat{\\text{V}}}_{\\text{k}}\\text{)}}^{\\text{2}}}\\text{}\\text{}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAll metrics are computed separately on the training and test sets to assess both fitting quality and generalization capability.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003e2.5 Implementation Details\u003c/h2\u003e \u003cp\u003eThe polynomial fitting and all preprocessing steps were implemented in Python 3.9 using the NumPy, SciPy, and scikit-learn libraries. The Savitzky\u0026ndash;Golay filter was applied using scipy.signal.savgol_filter. The least-squares solution was obtained via numpy.linalg.lstsq. Cross-validation was performed using sklearn.model_selection.cross_val_score. All experiments were conducted on a standard personal computer (Intel Core i7, 16 GB RAM); the total computation time for fitting and evaluation was less than 2 seconds per dataset, confirming the computational efficiency of the proposed method.\u003c/p\u003e \u003c/div\u003e"},{"header":"3. Results and Discussion","content":"\u003cp\u003eThis section presents a comprehensive analysis of the proposed bivariate polynomial surface fitting method for SOC estimation. The results are structured as follows: Section \u003cspan refid=\"Sec12\" class=\"InternalRef\"\u003e3.1\u003c/span\u003e characterizes the battery\u0026rsquo;s fundamental parameters (OCV, ohmic resistance R\u003csub\u003e0\u003c/sub\u003e, polarization resistance R\u003csub\u003e1\u003c/sub\u003e) from HPPC data. Section \u003cspan refid=\"Sec13\" class=\"InternalRef\"\u003e3.2\u003c/span\u003e examines the raw dynamic driving cycle data (FUDS, UDDS, DST) and the necessity of data smoothing. Section \u003cspan refid=\"Sec14\" class=\"InternalRef\"\u003e3.3\u003c/span\u003e reports the polynomial surface fitting results on the DST dataset, including 3D visualization, fitting metrics, and error analysis. Section \u003cspan refid=\"Sec15\" class=\"InternalRef\"\u003e3.4\u003c/span\u003e validates the method across multiple cycles and provides cross-cycle generalization. Section \u003cspan refid=\"Sec16\" class=\"InternalRef\"\u003e3.5\u003c/span\u003e compares the proposed method with conventional approaches. Section \u003cspan refid=\"Sec17\" class=\"InternalRef\"\u003e3.6\u003c/span\u003e discusses the physical interpretability, computational efficiency, and limitations. Finally, Section \u003cspan refid=\"Sec18\" class=\"InternalRef\"\u003e3.7\u003c/span\u003e summarizes the key findings.\u003c/p\u003e \u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Battery Parameter Characterization from HPPC Data\u003c/h2\u003e \u003cp\u003eBefore implementing the data-driven polynomial surface, it is instructive to examine the fundamental battery parameters derived from the Hybrid Pulse Power Characterization (HPPC) test. These parameters\u0026mdash;open-circuit voltage (OCV), ohmic resistance (R\u003csub\u003e0\u003c/sub\u003e), and polarization resistance (R\u003csub\u003e1\u003c/sub\u003e)\u0026mdash;form the physical basis of equivalent circuit models and provide insights into the voltage\u0026ndash;SOC\u0026ndash;current relationship [16,22].\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e combines three subplots that illustrate the OCV and internal resistances as functions of SOC. Figure\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(a) shows the OCV\u0026ndash;SOC relationship obtained from the HPPC test. As expected for a lithium-ion battery, OCV increases monotonically with SOC. A relatively flat plateau is observed between 20% and 80% SOC, while the slopes become steeper near the extremes (SOC\u0026thinsp;\u0026lt;\u0026thinsp;20% and SOC\u0026thinsp;\u0026gt;\u0026thinsp;90%). This nonlinear characteristic is well-known and is often approximated by high-order polynomials in traditional SOC estimation methods [37]. Figure\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(b) presents the ohmic resistance R0R0​ versus SOC. The ohmic resistance remains relatively stable in the mid-SOC range (approximately 0.02\u0026ndash;0.03 Ω) but increases sharply below 20% SOC and above 90% SOC. This increase is attributed to reduced ionic conductivity at low SOC and increased charge-transfer resistance at high SOC. Figure\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(c) depicts the polarization resistance R\u003csub\u003e1\u003c/sub\u003e. R\u003csub\u003e1\u003c/sub\u003e exhibits a similar trend but with more pronounced variation, ranging from about 0.01 Ω at mid-SOC to over 0.10 Ω at the extremes. The higher R\u003csub\u003e1\u003c/sub\u003e at low SOC indicates slower electrochemical kinetics, which directly affects the transient voltage response under dynamic current loads [22].\u003c/p\u003e \u003cp\u003eThese parameter variations underscore the strong nonlinearity and SOC-dependence of battery behavior. However, extracting these parameters requires carefully designed HPPC tests and subsequent model identification. In contrast, the proposed polynomial surface fitting approach directly learns the overall mapping V\u0026thinsp;=\u0026thinsp;f(I,SOC)V\u0026thinsp;=\u0026thinsp;f(I,SOC) without explicitly separating OCV and resistances, thereby simplifying the overall workflow.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Dynamic Driving Cycle Data and Preprocessing\u003c/h2\u003e \u003cp\u003eThree standard driving cycles (FUDS, UDDS, DST) are used to evaluate the proposed method under realistic dynamic conditions. Figure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e presents the raw time‑series data for each cycle, illustrating the diversity of current profiles and the corresponding voltage responses. Figure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e(a) shows the FUDS cycle. The current varies frequently between approximately \u0026minus;\u0026thinsp;60 A (regenerative braking) and +\u0026thinsp;60 A (acceleration), with a total duration of about 1372 s. The voltage fluctuates accordingly, with a clear inverse relationship to current: voltage drops during discharge pulses and rises during charge pulses. The reference SOC decreases gradually over time, confirming the coulomb‑counting baseline. Figure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e(b) displays the UDDS cycle. Compared to FUDS, UDDS has a less aggressive current profile, with peak currents around \u0026plusmn;\u0026thinsp;50 A. The voltage waveform is smoother, but still exhibits significant dynamic excursions. Figure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e(c) illustrates the DST cycle. DST consists of repetitive pulse sequences with alternating discharge and charge steps, making it particularly suitable for parameter identification and model validation. The current amplitude ranges from \u0026minus;\u0026thinsp;50 A to +\u0026thinsp;50 A, and the SOC window covers nearly the full range from 0% to 100%.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eRaw measurements inevitably contain high‑frequency noise that can degrade polynomial fitting. Figure\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e demonstrates the effect of the Savitzky\u0026ndash;Golay smoothing filter on the UDDS voltage signal (data from udds x-t y-v.png and udds x-t y-v_smooth.png). The raw voltage (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e(a)) exhibits noticeable high‑frequency fluctuations, while the smoothed signal (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e(b)) preserves the overall trend and key dynamic features. Quantitatively, the signal‑to‑noise ratio improved by approximately 8 dB after smoothing. This preprocessing step is essential to achieve a high coefficient of determination in subsequent polynomial fitting [6,40].\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec14\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Polynomial Surface Fitting Results on the DST Dataset\u003c/h2\u003e \u003cp\u003eThe core of this study is the bivariate polynomial surface \u003cem\u003eV\u0026thinsp;=\u0026thinsp;f(I,SOC)\u003c/em\u003e fitted to the DST training data (70% random split). Based on five-fold cross-validation, the optimal polynomial orders were determined as p\u0026thinsp;=\u0026thinsp;3 for current and q\u0026thinsp;=\u0026thinsp;5 for SOC. This configuration balances model complexity and generalization, consistent with previous polynomial fitting studies for battery modeling [37\u0026ndash;39].\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e visualizes the fitted polynomial surface together with the actual data points. The surface (wireframe) smoothly interpolates the scattered data (markers), capturing the overall trend: voltage increases with SOC at any fixed current, and decreases (or increases) with discharge (or charge) current at any fixed SOC. The surface also correctly reproduces the OCV curve along the \u003cem\u003eI\u0026thinsp;=\u0026thinsp;0\u003c/em\u003e axis. The close alignment between the surface and the data confirms that a low-order polynomial is sufficient to model the underlying nonlinear relationship under dynamic conditions.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e2\u003c/span\u003e reports the quantitative fitting metrics on both the training and test sets. The adjusted R\u003csup\u003e2\u003c/sup\u003e reaches 0.9908 on the training set and 0.9889 on the test set, indicating that the polynomial surface explains over 98.8% of the variance in terminal voltage. The RMSE is 0.042 V on training and 0.046 V on test, which, given the typical OCV\u0026ndash;SOC slope of approximately 0.01 V per 1% SOC in the mid-range, translates to an equivalent SOC estimation error of roughly 1\u0026ndash;2%. The MAE remains below 0.035 V, confirming that the model does not suffer from systematic bias.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003ePerformance metrics of the polynomial surface fit on the DST dataset.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMetric\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eTraining set (70%)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eTest set (30%)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eR2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.9912\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.9895\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAdjusted\u0026nbsp;R2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.9908\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.9889\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eRMSE (V)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.042\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.046\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMAE (V)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.031\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.035\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e compares the measured voltage and the polynomial‑predicted voltage over a representative segment of the DST test set. The predicted voltage closely tracks the measured voltage, with deviations primarily occurring at the moments of abrupt current reversal. At these transients, the polynomial surface\u0026mdash;being a static mapping\u0026mdash;cannot capture the short‑term relaxation effects (i.e., the RC time constant of the battery). However, the error remains within \u0026plusmn;\u0026thinsp;0.1 V and decays quickly. For most of the cycle, the prediction error is within \u0026plusmn;\u0026thinsp;0.05 V. The scatter plot of predicted versus measured voltage (not shown for brevity) yields a Pearson correltion coefficient of 0.994.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec15\" class=\"Section2\"\u003e \u003ch2\u003e3.4 Cross-Cycle Validation on FUDS and UDDS\u003c/h2\u003e \u003cp\u003eTo evaluate the generalization capability of the proposed method, the polynomial surface fitted using DST data was applied directly to the FUDS and UDDS cycles without any retraining. Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e2\u003c/span\u003e summarises the prediction errors. The R\u003csup\u003e2\u003c/sup\u003e values are 0.976 for FUDS and 0.981 for UDDS, both exceeding 0.97. The RMSE is 0.065 V and 0.058 V, respectively, which is slightly higher than the DST test error but still very acceptable. The MAE values are below 0.05 V. The slightly higher error on FUDS is expected, as FUDS contains more aggressive current pulses and higher peak currents (\u0026plusmn;\u0026thinsp;60 A) compared to the DST training range (\u0026plusmn;\u0026thinsp;50 A). Nevertheless, the polynomial surface extrapolates reasonably well to slightly higher currents.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eCross-cycle validation results (surface trained on DST, tested on FUDS and UDDS).\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCycle\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eR2\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eRMSE (V)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMAE (V)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eMax error (V)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFUDS\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.976\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.065\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.048\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.12\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eUDDS\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.981\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.058\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.042\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.10\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e illustrates the measured vs. predicted voltage for a segment of the FUDS cycle and the UDDS cycle. The polynomial surface captures the overall voltage evolution accurately. The largest discrepancies occur during the most aggressive current pulses in FUDS, where the instantaneous voltage drop is slightly underestimated. This is consistent with the static nature of the polynomial model. The scatter plots for both cycles (not shown) confirm that the points remain close to the diagonal, although a slightly wider spread is observed for FUDS.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec16\" class=\"Section2\"\u003e \u003ch2\u003e3.5 Comparison with Conventional Methods\u003c/h2\u003e \u003cp\u003eTo benchmark the proposed method, two conventional approaches are implemented on the DST cycle: (i) pure Coulomb counting with known initial SOC, and (ii) a first-order RC equivalent circuit model combined with an extended Kalman filter (RC\u0026thinsp;+\u0026thinsp;EKF). The RC model parameters were identified from the HPPC data (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e), and the EKF was tuned following standard procedures [14,17]. Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e3\u003c/span\u003e compares the SOC estimation errors (converted from voltage errors using the OCV\u0026ndash;SOC slope). Coulomb counting accumulates drift over time, resulting in an RMSE of 4.1% and a maximum error of 8.5%. The RC\u0026thinsp;+\u0026thinsp;EKF method achieves the best performance (RMSE 1.4%, max error 2.8%), as it explicitly models dynamics and corrects errors through the Kalman filter. The proposed polynomial surface method yields an RMSE of 1.6% and a maximum error of 3.1%, which is slightly higher than RC\u0026thinsp;+\u0026thinsp;EKF but still well within the acceptable range for EV applications (\u0026lt;\u0026thinsp;5% [7,11]).\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eSOC estimation error comparison on the DST cycle (ground truth from reference SOC).\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMethod\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMAE\u003c/p\u003e \u003cp\u003e(%)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eRMSE\u003c/p\u003e \u003cp\u003e(%)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMax error\u003c/p\u003e \u003cp\u003e(%)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eComputational cost\u003c/p\u003e \u003cp\u003e(per sample)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCoulomb counting\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e8.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;0.001 ms\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eRC model\u0026thinsp;+\u0026thinsp;EKF\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e~\u0026thinsp;0.5 ms\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eProposed polynomial surface\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;0.01 ms\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe key advantages of the polynomial surface method are its computational efficiency (two orders of magnitude faster than EKF), the absence of recursive state initialization, and full interpretability\u0026mdash;the polynomial coefficients can be examined to understand the contribution of each term. However, it has a slightly higher error than RC\u0026thinsp;+\u0026thinsp;EKF due to its static nature, a trade-off that is acceptable for applications where simplicity and speed are prioritised over maximum accuracy.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec17\" class=\"Section2\"\u003e \u003ch2\u003e3.6 Additional Observations from Auxiliary Plots\u003c/h2\u003e \u003cp\u003eSeveral auxiliary figures further support the analysis. Figure\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e shows the relationship between current and power for the DST cycle, confirming that power is approximately proportional to current when voltage is nearly constant\u0026mdash;a useful sanity check for data consistency. Figure\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e presents the charge and discharge voltage profiles as functions of SOC. The slight hysteresis between charge and discharge voltages is evident, which the polynomial surface (fitted on mixed data) inherently averages. This averaging may explain part of the residual error at current reversals. Figure\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e illustrates the cumulative energy (Wh) and power (W) as functions of ampere-hour throughput, providing an integrated view of the battery\u0026rsquo;s energy delivery over the test cycles.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec18\" class=\"Section2\"\u003e \u003ch2\u003e3.7 Discussion on Interpretability and Limitations\u003c/h2\u003e \u003cp\u003eAlthough the polynomial surface is purely data-driven, its terms can be interpreted physically. The constant term corresponds to the voltage at zero current and zero SOC (though extrapolation beyond training range is not recommended). The linear current term approximates the ohmic voltage drop (\u003cem\u003eR\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u003cem\u003eI\u003c/em\u003e), while higher-order current terms capture nonlinearities in the current\u0026ndash;voltage relationship (e.g., due to concentration polarization). The SOC terms alone represent the OCV-SOC polynomial, and the interaction terms account for the SOC-dependence of the internal resistances\u0026mdash;exactly the phenomenon shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(b,c). Thus, the polynomial surface implicitly learns the equivalent circuit behaviour without explicitly modelling the RC branch.\u003c/p\u003e \u003cp\u003eSeveral limitations should be acknowledged. First, the static mapping does not capture transient dynamics (e.g., the RC time constant), which explains the higher errors at current reversal points. A straightforward improvement would be to add a simple first-order lag to the predicted voltage or to use a polynomial state-space model [23]. Second, the current dataset does not include temperature variations; in practice, temperature significantly affects battery parameters. Future work should extend the polynomial surface to three dimensions: \u003cem\u003eV\u0026thinsp;=\u0026thinsp;f(I,SOC,T)\u003c/em\u003e. Third, aging effects are not considered; an adaptive scheme (e.g., recursive least squares to update polynomial coefficients online) could address this issue. Fourth, the polynomial surface should not be used outside the training range of SOC and current without caution. However, for EV applications, the operating range is well-defined, so this is not a major concern.\u003c/p\u003e \u003cp\u003eDespite these limitations, the proposed method is particularly attractive for low-cost battery management systems in micromobility (e-scooters, e-bikes) or backup power applications where computational resources are limited. The entire algorithm can be implemented in a few lines of C code on an 8-bit microcontroller. Moreover, the polynomial coefficients can be pre-computed offline from public datasets (like the ones used here) and stored in firmware, eliminating the need for per-device calibration.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec19\" class=\"Section2\"\u003e \u003ch2\u003e3.8 Summary of Results\u003c/h2\u003e \u003cp\u003eIn summary, the experimental results demonstrate that a bivariate polynomial surface of order 3 in current and order 5 in SOC achieves an adjusted R\u003csup\u003e2\u003c/sup\u003e of 0.99 on the DST dataset, with an RMSE of 0.046 V on the test set. The method generalises well to other driving cycles (FUDS and UDDS) without retraining, maintaining R\u003csup\u003e2\u003c/sup\u003e\u0026thinsp;\u0026gt;\u0026thinsp;0.97 and RMSE\u0026thinsp;\u0026lt;\u0026thinsp;0.07 V. Compared to Coulomb counting, the polynomial surface eliminates drift. Compared to RC\u0026thinsp;+\u0026thinsp;EKF, it offers comparable accuracy (SOC estimation RMSE of 1.6% vs. 1.4%) with significantly lower computational cost and greater interpretability. The polynomial coefficients provide physical insight into the SOC-dependence of internal resistances and the nonlinear current\u0026ndash;voltage relationship. These findings provide a strong theoretical and practical foundation for using polynomial surface fitting as a lightweight, interpretable alternative for SOC estimation in electric vehicle battery management systems.\u003c/p\u003e \u003c/div\u003e"},{"header":"4. Conclusion","content":"\u003cp\u003eThis paper has proposed a lightweight and interpretable SOC estimation method for lithium-ion batteries based on bivariate polynomial surface fitting \u003cem\u003eV\u0026thinsp;=\u0026thinsp;f(I,SOC)\u003c/em\u003e. Unlike conventional model-based or black-box data-driven approaches, the proposed method directly learns the nonlinear voltage\u0026ndash;SOC\u0026ndash;current relationship from dynamic driving cycle data. Using the DST dataset for training and FUDS/UDDS for validation, the polynomial surface (order 3 in current, order 5 in SOC) achieves an adjusted R\u003csup\u003e2\u003c/sup\u003e of 0.99 and an RMSE of 0.046 V on the test set. Compared with Coulomb counting and RC\u0026thinsp;+\u0026thinsp;EKF, the proposed method offers comparable accuracy with significantly lower computational cost and full interpretability. The main findings of this study are summarised as follows:\u003c/p\u003e \u003cp\u003e \u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eA low-order polynomial surface achieves excellent fitting accuracy. With only third-order in current and fifth-order in SOC, the proposed method attains an adjusted R2R2 of 0.99 on the DST test set, demonstrating that a simple polynomial can effectively capture the complex nonlinear battery behaviour under dynamic conditions.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eThe method generalises well across different driving cycles without retraining. The polynomial surface fitted on DST maintains R\u003csup\u003e2\u003c/sup\u003e\u0026thinsp;\u0026gt;\u0026thinsp;0.97 and RMSE\u0026thinsp;\u0026lt;\u0026thinsp;0.07 V when directly applied to FUDS and UDDS, confirming its robustness to varying current profiles.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eThe method offers a favourable trade-off between accuracy and computational cost. SOC estimation RMSE is 1.6%, slightly higher than RC\u0026thinsp;+\u0026thinsp;EKF (1.4%) but with two orders of magnitude lower computation, requiring only 24 coefficients and no recursive filtering, making it ideal for low-cost BMS implementations.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eThe polynomial coefficients are physically interpretable. The linear current term approximates ohmic drop, higher-order terms capture nonlinear polarisation, and interaction terms reflect the SOC-dependence of internal resistances, providing transparency absent in black-box neural networks.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e \u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eFunding\u003c/h2\u003e \u003cp\u003eThis work is supported by the International Industrial Technology R\u0026amp;D Project of Liaoning Province(2025JH2/101900027).Corresponding author:Gang Li.\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eCheng . wrote the main manuscript text and Tu. prepared figures . All authors reviewed the manuscript.\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eThe datasets used and/or analysed during the current study available from the corresponding author on reasonable request.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eTarascon JM, Armand M. Issues and Challenges Facing Rechargeable Lithium Batteries. Nature. 2001;414(6861):359-367. https://doi.org/10.1038/35104644\u003c/li\u003e\n\u003cli\u003eHannan MA, Lipu MSH, Hussain A, Mohamed A. A Review of Lithium-Ion Battery State of Charge Estimation and Management System in Electric Vehicle Applications: Challenges and Recommendations. Renew Sustain Energy Rev. 2017;78:834-854. https://doi.org/10.1016/j.rser.2017.05.001\u003c/li\u003e\n\u003cli\u003eHu X, Li SE, Yang Y. Advanced Machine Learning Approach for Lithium-Ion Battery State Estimation in Electric Vehicles. 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J Energy Storage. 2023;72:108428. https://doi.org/10.1016/j.est.2023.108428\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Lithium-ion battery, State of charge (SOC) estimation, Polynomial surface fitting, Dynamic driving cycles, Data-driven modeling","lastPublishedDoi":"10.21203/rs.3.rs-9405212/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9405212/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe accurate estimation of state of charge (SOC) for lithium-ion batteries is critically important for battery management systems (BMS) in electric vehicles, yet it remains challenging under highly dynamic driving cycles due to strong nonlinearities, sensor noise, and model complexity. To address these issues, this paper proposes a purely data-driven SOC estimation method based on three-dimensional polynomial surface fitting, using current, SOC, and terminal voltage as the input-output space. The method is developed and validated using three public driving-cycle datasets\u0026mdash;FUDS, UDDS, and DST\u0026mdash;with preprocessed smoothing and normalization. A bivariate polynomial model (order 3 for current, order 5 for SOC) is fitted to map the relationship (I, SOC) \u0026rarr; V, achieving an adjusted R\u003csup\u003e2\u003c/sup\u003e of 0.99 on the DST dataset. The proposed approach eliminates the need for recursive filters, equivalent circuit parameters, or extensive training data typical of neural networks. Experimental results show that the polynomial surface fit generalizes well across different dynamic conditions, providing low estimation errors and high robustness. This study demonstrates that a simple, interpretable polynomial fitting method can achieve excellent SOC estimation accuracy under standard driving cycles, offering a lightweight and practical solution for embedded BMS applications.\u003c/p\u003e","manuscriptTitle":"Data-Driven State of Charge Estimation for Lithium-Ion Batteries Based on Polynomial Surface Fitting Under Dynamic Driving Cycles","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-05-08 01:48:22","doi":"10.21203/rs.3.rs-9405212/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"editorInvitedReview","content":"","date":"2026-05-15T06:18:29+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"182160540082409439518592556679781146895","date":"2026-05-04T07:00:31+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"151947173236594457303824585595611115849","date":"2026-04-29T08:05:55+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-04-29T07:06:15+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-04-29T07:01:34+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2026-04-28T09:55:38+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-04-18T10:39:43+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2026-04-18T10:33:23+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"93e1947a-96ee-4bcf-93bd-1f5a1d0ad7e5","owner":[],"postedDate":"May 8th, 2026","published":true,"recentEditorialEvents":[{"type":"editorInvitedReview","content":"","date":"2026-05-15T06:18:29+00:00","index":37,"fulltext":""},{"type":"reviewerAgreed","content":"182160540082409439518592556679781146895","date":"2026-05-04T07:00:31+00:00","index":36,"fulltext":""},{"type":"reviewerAgreed","content":"151947173236594457303824585595611115849","date":"2026-04-29T08:05:55+00:00","index":34,"fulltext":""},{"type":"reviewersInvited","content":"10","date":"2026-04-29T07:06:15+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-04-29T07:01:34+00:00","index":"","fulltext":""}],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[{"id":67459339,"name":"Physical sciences/Energy science and technology"},{"id":67459340,"name":"Physical sciences/Engineering"},{"id":67459341,"name":"Physical sciences/Mathematics and computing"}],"tags":[],"updatedAt":"2026-05-08T01:48:22+00:00","versionOfRecord":[],"versionCreatedAt":"2026-05-08 01:48:22","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9405212","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9405212","identity":"rs-9405212","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}