Electro–thermal benchmarking of low-order lithium-ion battery equivalent circuit models under constant-current and dynamic loading

Abstract

Low-order equivalent circuit models (ECMs) are widely used in battery management systems (BMSs) because they balance accuracy with computational efficiency. Here we present a harmonised electro–thermal benchmark to evaluate voltage accuracy, heat-generation consistency, and temperature prediction under both constant-current and dynamic loading conditions. Three low-order model structures are assessed: the internal-resistance (Rint) model, the first-order Thevenin (1RC) model, and a compact hybrid electro–thermal 1RC model that incorporates bounded electrical adaptation and a two-node thermal network. Model parameters are identified using standard open-circuit-voltage and pulse-based experiments, while thermal parameters are obtained independently from heating–cooling tests and held fixed during validation. Performance is evaluated using commercial 18650 lithium-ion cells across ambient temperatures of 0C,25C and 40C under both laboratory and drive-cycle-inspired load profiles. The results show that increasing electrical model fidelity substantially improves voltage tracking under dynamic conditions and leads to more consistent heat-generation pathways when coupled to a fixed thermal model. Under the strict hold-out profile inspired by the Worldwide Harmonized Light Vehicles Test Cycle (WLTC) at 25C, the hybrid electro–thermal 1RC model delivers a voltage root mean square error (RMSE) of  mV, a surface-temperature RMSE of  C, and a heat-generation consistency mismatch of (mean ± SD across cells). These results provide quantitative guidance for selecting compact electro–thermal battery models for real-time applications under dynamic operation.

Introduction

Lithium-ion batteries underpin electrified transport, portable electronics, and stationary storage, and their safe and efficient operation increasingly depends on model-based battery management systems (BMSs) for estimation, protection, control, and thermal supervision^1–3. Beyond SoC and thermal supervision, BMS deployments also require reliable state-of-power (SoP) assessment for power-limit enforcement and safe operation⁴. In embedded BMS deployments, the model must balance accuracy with constraints on computation, memory, and calibration effort¹. Equivalent circuit models (ECMs) are widely adopted in this context because they provide a compact, interpretable, and implementation-ready approximation of cell dynamics while remaining compatible with real-time observers and controllers^5–7.

Among low-order ECMs, the internal-resistance (Rint) model and the first-order Thevenin (1RC) model represent two practically important operating points on the fidelity–complexity spectrum. The Rint model describes the terminal voltage as an open-circuit voltage (OCV) source in series with an ohmic resistance and is often adequate when current varies slowly and polarization effects are weak^5,8. However, because it lacks a polarization state, its residuals become structured under transient excitation and relaxation, particularly at low temperature and under drive-cycle-like profiles^5,6. The Thevenin 1RC model augments the ohmic branch with a single RC polarization element, enabling representation of dominant relaxation dynamics with minimal increase in state dimension^5,6. Higher-order RC networks can further improve voltage tracking under multi-time-scale excitation, but this typically increases parameter coupling and identification burden, which can reduce practical robustness in BMS calibration workflows^5,6. The electrical ECM topologies considered in this work are summarised in Fig. 1.

Figure 1.

Electrical equivalent circuit model topologies benchmarked in this study: (a) Rint and (b) Thevenin 1RC. Both models use an OCV source and an ohmic resistance ; the 1RC model additionally includes one polarization branch to capture voltage relaxation dynamics.

Voltage-only comparison is insufficient for many applications because battery behaviour is inherently electro–thermal: impedance and kinetics are temperature dependent, while realistic duty cycles generate irreversible and reversible heat that influences temperature trajectories and, in turn, electrical response^9–11. Compact electro–thermal ECMs therefore couple a low-order electrical model to a low-order thermal network and compute heat generation using energy-balance formulations commonly attributed to Bernardi-type decompositions^12–15. Such coupled models are attractive for onboard use because they retain a small state dimension while supporting thermal supervision and estimation^10,16. Recent work on temperature prediction and thermal management further highlights the need for compact models that remain reliable under dynamic loading and controlled boundary conditions^17–21.

Existing comparative studies provide a strong baseline but also reveal a persistent benchmarking gap. Electrical comparisons consistently show that Rint is typically adequate only under quasi-steady conditions, whereas at least one polarization branch is needed to capture transient behaviour under pulses and drive-cycle excitation^5,6. Separately, electro–thermal studies have shown that predicted heat generation can be sensitive to ECM structure and parameter dependencies, such that simplified electrical dynamics may distort inferred loss pathways and lead to temperature agreement that masks electrical mismatch²². In addition, many studies do not enforce a strict separation between identification and validation across profiles and operating conditions, making it difficult to distinguish genuine generalisation from compensatory tuning. Representative prior work motivating the present benchmark is summarised in Table 1.

Table 1.

Representative ECM-based electro–thermal studies and benchmarks relevant to Rint vs. Thevenin-type models and hybrid/adaptive extensions.

Study	ECM type	ID method	Thermal model	Profiles	Metrics	Key limitations/notes
Auch et al. 22	Rint, 1RC, 2RC/DP	Characterisation + drive-cycle fit (Bernardi heat)	Lumped + CFD variants	WLTC-inspired	sensitivity	Shows architecture- and parameter-dependency effects on ; temperature agreement can mask electrical mismatch; motivates 1RC for electro–thermal use.
Tran et al. 6	Rint/RC family	Comparative validation (multiple chemistries)	–	Dynamic tests	V errors	Order–accuracy positioning; thermal aspects not central
Hu et al. 5	Multiple ECMs	Experimental comparison	–	Pulses	V tracking	Baseline comparison for Rint vs. RC; electro–thermal coupling not analysed
He et al. 8	ECMs for SoC	Experimental evaluation for SoC estimation	–	Pulses	Estimation/fit metrics	Observer motivation; limited thermal discussion
Karimi et al. 23	ECM (high power)	Wide T/z/I identification	–	High-rate	V errors	Strong electrical coverage across conditions; thermal modelling not the focus
Mannapperuma et al. 24	Electro–thermal ECM	Drive-cycle-based ID	Pack-level thermal	Drive cycle	V/T errors	Pack emphasis; boundary conditions influence absolute errors
Lagnoni et al. 7	ECM vs. physics-based	Experimental validation + fitting	Application-dependent	Drive-cycle style	Accuracy trends	Trade-off framing versus physics-based models; heat-generation fidelity often not the primary target
Christophersen et al. 25	–	HPPC protocol	–	HPPC + EV tests	Procedure	Standard reference for pulse design, SoC windows, and reporting conventions
Belt et al. 26	–	PHEV/HPPC procedures	–	PHEV duty cycles	Definitions	Standard reference for PHEV-oriented procedures and reporting

These limitations motivate a harmonised benchmark that evaluates low-order deployable ECMs in both voltage and electro–thermal terms under a predefined identification/validation split. In this work, we compare three compact model classes: (i) Rint, (ii) a fixed Thevenin 1RC model, and (iii) a compact hybrid electro–thermal 1RC model that adds bounded regime-dependent electrical adaptation and an explicit two-node core–surface thermal network. The study addresses three practical questions: (1) how much is gained by moving from Rint to 1RC under constant-current and dynamic loading; (2) whether improved electrical fidelity yields more consistent heat-generation pathways and better temperature prediction when thermal parameters are identified independently and then held fixed; and (3) whether bounded adaptation can improve performance without increasing circuit order.

Experiments were performed on commercial 18650 lithium-ion cells across ambient temperatures of 0C, 25C and 40 C with cells. Results are reported as mean ± SD across cells, with aggregation defined in “Parameter identification and benchmarking design”. The main contributions are as follows. First, we establish an electro–thermal benchmarking workflow that jointly reports voltage fidelity, energy-relevant mismatch, heat-generation consistency, and temperature prediction under strict hold-out profiles. Second, we implement a compact hybrid electro–thermal 1RC model that preserves embedded deployability while improving dynamic robustness via bounded electrical adaptation and a two-node thermal network. Third, we provide quantitative model-selection guidance by comparing Rint, fixed 1RC, and the hybrid model under constant-current and WLTC-inspired dynamic loading across temperature.

The remainder of the paper is organised as follows. Section “Methods” presents the modelling and benchmarking framework. Section “Experimental setup and protocols” describes the experimental setup and identification workflow. Section “Performance indices and statistical analysis” defines the performance indices and statistical reporting. Section “Results” reports benchmarking results, and “Discussion” discusses implications and limitations.

Methods

Electro–thermal model

This study evaluates a compact hybrid electro–thermal equivalent circuit model (ECM) that extends the classical Thevenin 1RC structure with bounded regime-dependent electrical adaptation and an explicit two-node core–surface thermal network. The objective is to improve voltage, heat-generation, and temperature prediction under both constant-current and dynamic excitation while preserving a low state dimension suitable for embedded battery management system (BMS) deployment.

With discharge current defined as , the model state vector is

1

where z denotes state-of-charge (SoC), is the polarization voltage associated with the RC branch, and and denote the core and surface temperatures, respectively. The model inputs are the applied current I(t) and ambient temperature . The principal outputs are terminal voltage , total heat generation , and the temperature states; intermediate outputs include the polarization voltage and the irreversible and reversible heat contributions.

The electrical backbone consists of an open-circuit voltage source , an effective ohmic resistance , and a single polarization branch . All electrical parameter maps and the OCV function are evaluated using the model-predicted core temperature.

Baseline electrical maps,

2

were identified from HPPC experiments on a discrete SoC–temperature grid comprising 10% SoC increments over and chamber-set ambient temperatures . For map construction, equilibrium cell temperature was assumed equal to chamber temperature. The maps were stored as lookup tables on the (z, T) grid and evaluated using bilinear interpolation. The OCV function was constructed from measured OCV–SoC curves at each temperature and linearly interpolated in temperature.

HPPC parameters were extracted using pulse–rest sequences following standard Idaho National Laboratory (INL) procedures^25,26. At each SoC–temperature operating point, the cell was first brought to quasi-equilibrium and then subjected to a 10s discharge pulse at 1C (i.e., current equal to nominal capacity in A; here 1C=2.2A), a 60s rest, a 10s charge pulse of the same magnitude, and a further 60s rest. The quasi-equilibrium criterion was enforced before each HPPC block, whereas the intra-block rests were used for resistance and relaxation identification. The exact protocol is summarized in Table 2.

Table 2.

HPPC protocol used for electrical parameter identification.

Item	Protocol detail
SoC grid	0–100% SoC in 10% increments
Ambient temperatures	0 C, 25 C and 40C
Discharge pulse amplitude	1C = 2.2A
Charge pulse amplitude	Magnitude 1C (2.2A); charge applied with negative sign under the convention discharge
Pulse duration	10s
Rest duration after each pulse	60s
Pre-pulse equilibration	Quasi-equilibrium enforced before each HPPC block at a given SoC point, defined as over the last 10min
Parameter extracted from pulse onset
Parameters extracted from relaxation	and from first-order exponential fitting of the post-pulse relaxation voltage, with
Number of cells
Replicates per condition	One HPPC sequence per SoC–temperature point per cell

For each SoC window and ambient temperature, was obtained from the instantaneous voltage drop at pulse onset, , after time alignment and noise filtering. The polarization parameters were obtained by fitting the post-pulse relaxation voltage to a first-order exponential,

3

so that and . Fits were obtained by nonlinear least squares with positivity constraints on , , and . Only the single-branch parameters of the Thevenin 1RC model were identified, as higher-order RC structures were not benchmarked experimentally.

To account for current-dependent nonlinearities under dynamic operation without increasing circuit order, bounded multiplicative corrections were applied to the baseline maps:

4

5

6

where with and . The shaping functions are

7

This yields a compact adaptive parameter set . A full three-dimensional lookup-table extension in (z, T, I) was not adopted because it would substantially increase calibration burden, memory footprint, and parameter coupling, especially at low temperature.

Practical identifiability was assessed using local sensitivity analysis on HPPC segments. Let denote the sensitivity matrix of with respect to parameter vector ; the Fisher-information surrogate is , and conditioning is summarized by . Local sensitivities were computed using 1% finite-difference perturbations. Parameters with persistently weak sensitivity were regularized toward their baseline values to improve repeatability.

Heat generation was computed using a Bernardi-type decomposition^10,12–14:

8

with

9

and

10

For thermal-parameter identification, a measurement-consistent electrical loss proxy was also used:

11

which depends only on measured voltage and therefore avoids embedding a specific ECM architecture in the thermal fit. Related compact electro–thermal formulations also use power/energy-consistent inputs for thermal dynamics, motivating the present loss-proxy choice for architecture-independent thermal identification²⁷.

Thermal dynamics were represented by a two-node core–surface network commonly used in compact battery thermal modelling^10,13,16:

12

13

where is the effective core thermal capacitance, is the surface thermal capacitance, is the core–surface thermal resistance, and hA is the lumped surface-to-ambient heat-transfer coefficient. These parameters were identified from dedicated heating–cooling experiments and then held fixed during dynamic validation.

The coupled electro–thermal state equations are

14

15

16

augmented by the thermal equations above. Because the experiments were conducted on fresh cells over short horizons and the analysis is comparative, was set to unity for all models.

Parameter identification and benchmarking design

The methodology followed four strictly separated stages: (1) identification of baseline electrical parameter maps from OCV and HPPC experiments, (2) calibration of bounded adaptive electrical scalars using a dedicated dynamic stress test (DST), (3) identification of thermal-network parameters from dedicated heating–cooling experiments, and (4) benchmarking under strict profile hold-out conditions.

Identification and validation were kept separate throughout. OCV and HPPC data were used exclusively for baseline electrical identification. Thermal parameters were identified independently from heating–cooling experiments and then held fixed. Adaptive scalars were calibrated using one DST run per ambient temperature. Headline performance was reported on a dynamic profile inspired by the Worldwide Harmonized Light Vehicles Test Cycle (WLTC) that was never used for fitting. DST results presented in the Results section were computed on separate repeated runs not used during calibration.

Adaptive scalars were estimated by minimizing voltage RMSE on the calibration DST run at each ambient temperature, subject to positivity constraints on , , and , and to the saturation bound .

Experimental setup and protocols

Electrical and electro–thermal simulations were implemented in MATLAB/Simulink R2022b using a fixed-step solver with an internal step size of 0.1s. Statistical post-processing and figure generation were performed in Python 3.10 using NumPy, SciPy, Pandas, and Matplotlib. Model outputs were resampled to 1Hz for metric computation.

Experiments were performed on Samsung ICR18650-22F cylindrical lithium-ion cells, and manufacturer specifications are summarized in Table 3. Three cells () were tested. Each cell was mounted vertically in a dedicated holder inside a temperature-controlled chamber maintained at 0 C, 25 C and 40C.

Table 3.

Battery cell specifications used in this study (Samsung SDI ICR18650-22F datasheet²⁹; safety data sheet (SDS)²⁸, where applicable).

Manufacturer	Samsung SDI
Model	ICR18650-22F
Form factor	18650 cylindrical
Nominal capacity	2200mAh (0.2C discharge)
Nominal voltage	3.6V
Charging voltage (CV)	4.2V
Charging method	CC–CV
Standard charge current	1.1A
Maximum charge current	2.2A
Maximum discharge current	4.4A
Discharge cut-off voltage	2.75V
Dimensions	18.4mm (max) 65.0mm (max)
Mass	Approx. 43.5g
Cathode	Lithiated transition-metal oxide (exact composition not specified)28
Anode	Graphite28
Electrolyte	Lithium hexafluorophosphate (LiPF) in organic carbonate solvents (proprietary blend not specified)28

Before identification and validation, each cell underwent three conditioning cycles between 4.2V and 2.75V. Charging followed a constant-current constant-voltage (CC–CV) protocol at C/2 to 4.2V, followed by constant-voltage hold until the current tapered below C/20. Discharging was performed at constant current (CC) at C/2 to 2.75V. In this study, C-rate denotes current normalized by nominal capacity; for the 2.2Ah cell, 1C corresponds to 2.2A, C/2 to 1.1A, and 2C to 4.4A.

Repeated runs were not separated by a fixed 24 h interval. Instead, each run was initiated only after electrical and thermal re-equilibration criteria were satisfied, namely OCV drift below 2mV over 30min and thermal return to ambient within . Under the present protocol, these criteria were consistently met after 12h to 16h. Before each test, cells were thermally equilibrated inside the chamber for at least 2h.

Current excitation and terminal-voltage measurement were provided by a programmable battery cycler (BioLogic BCS-805), while temperature channels were acquired using a National Instruments (NI) data-acquisition system comprising an NI PXI-1033 chassis and NI PXI-4353 thermocouple module. The cells were tested inside a temperature-controlled chamber (Espec SU-242) maintained at the prescribed ambient setpoints. The battery cycler specifications were full scale for voltage accuracy and full scale for current accuracy, while the chamber stability was around the setpoint. All raw channels were logged at 10Hz and then aligned and mapped to the common 1Hz evaluation grid used for metric computation. Surface temperature was measured using calibrated K-type thermocouples (36 AWG, manufacturer accuracy ) referenced against a platinum resistance thermometer (PT100) sensor. Two thermocouples were attached to each cell, one at the cap and one at mid-height on the cylindrical surface, using Kapton tape (see Fig. 2 for the placement schematic.). Unless otherwise stated, the measured surface temperature denotes the arithmetic mean of these two channels.

Figure 2.

Thermocouple placement on the 18650 cell: one sensor at the cap and one at mid-height on the cylindrical surface.

The experimental workflow used synchronized measurements of current I(t), terminal voltage , surface temperature , and ambient temperature . When instruments used different internal clocks, alignment was performed using a pulse marker and verified by maximizing the cross-correlation between I(t) and the corresponding instantaneous voltage drop. For analysis, all traces were mapped to a common 1Hz evaluation grid; current was resampled using zero-order hold, whereas voltage and temperature were linearly interpolated.

Electrical identification was performed on a SoC–temperature grid comprising 10% SoC increments across 0–100% and ambient temperatures of 0 C, 25 C and 40C. For OCV and entropy characterization, each 10% SoC step was followed by a 90min rest. A state was considered at quasi-equilibrium when over the final 10min of the rest period. Entropy characterization used the same rest protocol and applied small perturbations to estimate across temperatures.

The predefined test matrix is summarized in Table 4. Validation profiles included constant-current discharge, HPPC, DST, and a WLTC-inspired dynamic current profile. The WLTC-inspired current demand was generated by mapping the WLTC speed trace to a representative traction-current profile and was treated as a strict hold-out used only for final validation.

Table 4.

Predefined test matrix used for identification and validation.

Protocol	Purpose	Profiles / points		Outputs
OCV–SoC	fit	10% SoC steps + rest	0/25/40	OCV curves ()
HPPC	maps	10 s ±1C pulses with 60 s rest at each 10% SoC point	0/25/40	Baseline maps + identifiability checks
Thermal ID	Thermal parameters	30 min heating + 60 min cooling	0/25/40
CC	Sanity check	C/2 and 1C discharge; short-duration 2C segments only under protected conditions	0/25/40	Baseline voltage errors
DST	Adaptive calibration	One DST per T (calibration)	0/25/40	Adaptive scalar fit
DST (repeats)	Reported DST results	Runs not used in calibration	0/25/40	Reported DST metrics
WLTC-inspired	Final generalisation	Strict hold-out	0/25/40	Headline benchmarking results

The identification order was fixed as follows: first, and were obtained; second, the baseline electrical maps were identified from hybrid pulse power characterization (HPPC) experiments; third, the two-node thermal parameters were identified from dedicated heating–cooling experiments and then fixed; fourth, bounded adaptive scalars were calibrated using one DST profile per temperature; and fifth, final benchmarking was performed on the WLTC-inspired hold-out profile, with DST results reported only on runs not used for calibration.

Thermal parameters were identified at each ambient temperature using a two-phase experiment consisting of 30min heating under approximately 1C discharge followed by 60min cooling under natural convection. Estimation used the measurement-consistent loss proxy

17

so that thermal fitting remained independent of any specific ECM architecture. The identified parameters are reported in Table 5, and the overall model and solver settings are summarized in Table 6.

Table 5.

Two-node thermal parameters identified from 30 min heating and 60 min cooling experiments and held fixed during validation (mean ± SD across cells).

Parameter	0C	25C	40C
(J/K)
(J/K)
(K/W)
hA (W/K)

Table 6.

Model and simulation parameter summary used in the electro–thermal benchmarking framework.

Category	Parameter	Value/description
Electrical map grid		11 SoC grid points (0–100% in 10% increments)
		3 ambient temperatures (0C, 25C, 40C)
	Interpolation	Bilinear interpolation over (z, T) grid
Adaptive scalars		Temperature-dependent bounded resistance scaling factors
		Current-dependent shaping coefficients
		SoC-dependent capacitance shaping coefficient
		Saturation bound for current adaptation term
Thermal parameters		Effective core thermal capacitance (Table 5)
		Surface thermal capacitance (Table 5)
		Core–surface thermal resistance (Table 5)
	hA	Lumped convection coefficient (Table 5)
Electrical constants		2.2Ah (measured nominal capacity)
		2.2A (1C reference current)
		1 (coulombic efficiency assumed unity for fresh cells)
	OCV representation	Measured OCV–SoC curves at each temperature with linear temperature interpolation
Simulation settings	Solver	Fixed-step integration
	Internal step	0.1s
	Evaluation grid	1Hz resampled grid for metrics

The electro–thermal model was simulated under chamber-imposed constant ambient temperature, with heat exchange represented by a lumped surface-to-ambient convection term and without a separate tab thermal node. Initial states were assigned after profile-specific equilibration. These settings are summarized in Table 7.

Table 7.

Boundary and initial conditions used for model development and simulation.

Item	Specification
Ambient boundary condition	Chamber-imposed constant ambient temperature during each run:
External heat-transfer boundary	Lumped natural convection through the surface node only, represented by
Thermal-network topology	Two-node core–surface model; no separate tab/terminal thermal node included
Initial thermal conditions	after thermal equilibration
Initial polarization state	For OCV/HPPC, after pre-pulse equilibration; for CC, DST, and WLTC-inspired runs, after the initial equilibration rest preceding the profile
SoC initialization	obtained from coulomb counting referenced to the measured capacity of each cell
Electrical operating limits	Experimental excitation constrained to 2.75V 4.2V; high-rate segments aborted if

Experimental uncertainty arose from instrument specifications and repeatability. Thermocouple uncertainty after PT100 calibration was . Headline metrics are reported as mean ± SD across the three tested cells. For derived metrics such as RMSE and , the across-cell SD reflects both sensor noise and cell-to-cell variability.

All experiments respected 2.75V to 4.2V operating limits and specified current limits. Tests were aborted if . The setup included chamber over-temperature protection, electrical interlocks, and supervised operation during high-rate segments.

Performance indices and statistical analysis

Performance metrics were computed on the aligned 1Hz evaluation grid. Let

18

denote the voltage residual. Voltage accuracy was quantified using the root mean square error (RMSE), mean absolute error (MAE), peak absolute error, and bias.

19

20

21

22

To characterize temporal mismatch during pulse excitation, a voltage lag metric was computed on HPPC segments only:

23

with . A positive indicates that the measured voltage must be shifted forward to best align with the model output.

Thermal residuals were defined as

24

and thermal accuracy was summarized by

25

26

27

An energy-domain measure of electrical mismatch was defined as

28

reported in Wh.

To quantify consistency between the heat input used for thermal identification and the heat predicted by the coupled electro–thermal model, the following metric was used:

29

where

30

Lower indicates stronger electro–thermal loss-path consistency under the applied profile.

All simulations were executed with an internal fixed step of 0.1s, and all metrics were evaluated on the 1Hz resampled grid. Metrics were computed per run and summarized as mean ± SD. Given the small sample size (), interpretation emphasizes effect sizes and consistent directional trends across cells. Where inferential statistics are reported, paired Wilcoxon signed-rank tests were used as supplementary non-parametric indicators, with two-sided tests and significance level .

Use of AI-assisted tools

During manuscript preparation, the authors used ChatGPT to improve language clarity and readability. The authors reviewed and edited all generated text and take full responsibility for the final manuscript.

Notation

The notation used in the manuscript, including Acronyms and abbreviations, Roman symbols, Greek symbols, and subscript/superscript conventions, is summarized in Tables 8, 9, 10, and 11.

Table 8.

Acronyms and abbreviations used in this manuscript.

Acronym	Meaning
BMS	Battery management system
BTMS	Battery thermal management system
CC	Constant-current
CV	Constant-voltage
CC–CV	Constant-current/constant-voltage charging protocol
DAQ	Data acquisition
DST	Dynamic stress test
ECM	Equivalent circuit model
EV	Electric vehicle
FIM	Fisher information matrix
HEV	Hybrid electric vehicle
HPPC	Hybrid pulse power characterization
LIB	Lithium-ion battery
MAE	Mean absolute error
OCV	Open-circuit voltage
RMSE	Root mean square error
SoC	State of charge
SoH	State of health
WLTC	Worldwide Harmonized Light Vehicles Test Cycle

Table 9.

Roman symbols used in this manuscript.

Notation	Unit	Meaning
z	–	State-of-charge (SoC), expressed as a fraction in [0, 1]
I(t)	A	Applied current, with discharge defined as positive
	V	Terminal voltage
	V	Open-circuit voltage
		Ohmic and polarization resistances
	F	Polarization capacitance
	V	Polarization voltage state
	W	Total heat generation
	W	Irreversible heat generation
	W	Reversible (entropic) heat generation
	C	Core, surface, and ambient temperatures
	K/W	Core–surface thermal resistance
	J/K	Effective core thermal capacitance
	J/K	Surface thermal capacitance
hA	W/K	Lumped convection parameter to ambient
	Ah	Nominal cell capacity
	A	Reference current corresponding to 1C
	Wh	Absolute mismatch energy

Table 10.

Greek symbols used in this manuscript.

Notation	Unit	Meaning
	–	Coulombic efficiency
	s	Electrical time constant,
	s	Voltage lag maximizing correlation on HPPC pulses
	%	Heat-generation consistency mismatch metric
	–	Temperature-dependent adaptive resistance scalars
	–	Current-dependent shaping coefficients
	–	SoC-dependent capacitance shaping coefficient
	–	Current-dependent shaping function
	–	Saturation bound for current adaptation
	–	SoC-dependent shaping function for capacitance adaptation
	–	Condition number of the Fisher information matrix

Table 11.

Subscript and superscript conventions used in this manuscript.

Notation	Unit	Meaning
0	–	Ohmic branch or baseline resistance index
1	–	First polarization branch index
	–	Terminal quantity
	–	Open-circuit-voltage quantity
	–	Heat-generation quantity
	–	Irreversible component
	–	Reversible component
	–	Core node
	–	Surface node
	–	Ambient condition
	–	Core–surface path
	–	Effective or adapted parameter
	–	Instantaneous quantity
	–	Absolute quantity
	–	Maximum value
	–	Reference value
n	–	Nominal quantity or sample count, depending on context
(A)	–	Measurement-consistent quantity
(B)	–	Model-decomposed quantity
	–	Baseline or nominal map before adaptation

Results

Results are reported as mean ± SD across the three tested cells. Unless otherwise stated, primary dynamic-validation results correspond to the WLTC-inspired hold-out profile.

Parameter maps and identifiability

Figure 3 shows the HPPC-identified parameter maps , , and . Practical identifiability was assessed using the Fisher-information surrogate , where is the sensitivity of with respect to on HPPC pulse segments. The corresponding conditioning and repeatability summary is reported in Table 12.

Figure 3.

Identified HPPC parameter maps: (a) from instantaneous pulse drop , (b) and (c) from exponential relaxation fits with . Maps are evaluated on a 10% SoC grid at using bilinear interpolation; shading denotes SD across the tested cells.

Table 12.

Identifiability summary of HPPC-identified electrical parameters. Reported values denote ranges (min–max) across SoC windows and repeated trials. denotes the Fisher information matrix (FIM) condition number; “repeatability width” denotes the relative spread across repeated estimates (percent) for each parameter.

Condition		Repeatability width for	Repeatability width for	Repeatability width for
HPPC @ 25C	–	(2–6)%	(4–10)%	(5–15)%
HPPC @ 0C	–	(4–12)%	(8–20)%	(10–25)%
HPPC @ 40C	–	(2–6)%	(4–12)%	(5–18)%

Across all temperatures, the identified maps were stable and physically consistent, but the low-temperature condition exhibited stronger parameter coupling and larger repeat-to-repeat variability. In particular, the Fisher information matrix (FIM) surrogate condition number increased to – at 0C, together with broader repeatability intervals for and , indicating weaker practical identifiability under cold-operation transients.

Voltage tracking accuracy

Figures 4, 5, and 6 show representative measured and modelled voltage trajectories and residuals. Numerical summaries of voltage RMSE, MAE, peak error, bias, transient alignment lag , and absolute mismatch energy are reported in Tables 13 and 14.

Figure 4.

Measured and simulated terminal voltage for a representative constant-current run; the lower panel shows the residual . Shaded region denotes SD across cells.

Figure 5.

Measured and simulated terminal voltage under DST excitation for a representative run; the lower panel shows the residual . Shaded region denotes SD across cells.

Figure 6.

Measured and simulated terminal voltage under WLTC-inspired excitation for a representative run; the inset highlights a high-transient segment. Shaded region denotes SD across cells.

Table 13.

CC voltage metrics (mean ± SD across runs).

Temp	Model	RMSE (mV)	MAE (mV)	Peak (mV)	Bias (mV)
0C	Rint	60 ± 6	45 ± 5	120 ± 10
	1RC	30 ± 4	22 ± 3	70 ± 8
	Hybrid	18 ± 3	14 ± 2	45 ± 6
25C	Rint	12 ± 2	9 ± 2	25 ± 4
	1RC	6 ± 1	5 ± 1	15 ± 3
	Hybrid	4 ± 1	3 ± 1	10 ± 2

Table 14.

Dynamic-profile voltage performance (mean ± SD across runs). DST results are reported on runs not used for adaptive calibration; WLTC-inspired results are reported on the hold-out profile.

Profile,	Model	RMSE (mV)	MAE (mV)	Peak (mV)	Bias (mV)	(s)	(Wh)
DST, 25C	Rint	70 ± 7	54 ± 6	140 ± 12		5.0 ± 0.4	1.4 ± 0.2
	1RC	32 ± 4	24 ± 3	85 ± 9		3.5 ± 0.3	0.85 ± 0.1
	Hybrid	18 ± 3	13 ± 2	50 ± 6		2.8 ± 0.4	0.48 ± 0.05
WLTC-inspired, 0C	Rint	110 ± 10	85 ± 8	210 ± 15		6.0 ± 0.5	1.6 ± 0.2
	1RC	55 ± 6	40 ± 5	120 ± 10		4.0 ± 0.4	0.9 ± 0.1
	Hybrid	28 ± 4	21 ± 3	65 ± 8		3.5 ± 0.5	0.55 ± 0.07
WLTC-inspired, 25C	Rint	75 ± 8	58 ± 6	160 ± 12		5.0 ± 0.4	1.2 ± 0.1
	1RC	35 ± 5	26 ± 4	95 ± 9		3.5 ± 0.3	0.9 ± 0.1
	Hybrid	28 ± 5	21 ± 3	65 ± 8		3.5 ± 0.5	0.55 ± 0.07
WLTC-inspired, 40C	Rint	65 ± 7	50 ± 5	140 ± 10		4.5 ± 0.4	1.0 ± 0.1
	1RC	30 ± 4	22 ± 3	80 ± 8		3.2 ± 0.3	0.70 ± 0.08
	Hybrid	22 ± 3	16 ± 2	48 ± 6		2.8 ± 0.4	0.42 ± 0.05

Constant-current discharge. Under constant-current discharge, the ranking of the three models is already evident, although the differences remain modest at room temperature. Rint provides a reasonable quasi-steady approximation but cannot represent relaxation effects at current transitions, leading to larger peak residuals and a more negative bias than RC-based models. At 25C, voltage RMSE decreased from  mV for Rint to  mV for 1RC and  mV for the Hybrid model. The separation became substantially larger at 0C, where RMSE reduced from  mV to  mV and  mV, respectively (Table 13). This widening under cold conditions is consistent with the stronger impedance growth and nonlinear polarization observed at low temperature.

Dynamic profiles. The advantage of including polarization dynamics becomes much clearer under transient excitation. On the DST profile at 25C, Rint produced larger residual spikes at current reversals and step changes, resulting in the highest RMSE, peak error, transient lag, and mismatch energy. Voltage RMSE decreased from  mV for Rint to  mV for 1RC and  mV for the Hybrid model, while decreased from  Wh to  Wh and  Wh, respectively.

The same ranking persisted on the WLTC-inspired hold-out profile across all ambient temperatures (Table 14). At 0C, RMSE decreased from  mV for Rint to  mV for 1RC and  mV for Hybrid. At 25C, the corresponding values were  mV,  mV, and  mV, and at 40C they were  mV,  mV, and  mV. The reduction in with increasing model fidelity indicates improved matching of the dominant relaxation dynamics, while the lower confirms that the improvement is not confined to isolated peaks but extends to the cumulative energy-domain mismatch.

Figure 7 further visualizes the WLTC-inspired voltage RMSE distribution over time and SoC for the three models, confirming that the hybrid model reduces both the magnitude and spread of high-error regions relative to Rint and fixed 1RC.

Figure 7.

Dynamic voltage RMSE heatmaps versus SOC and time on the WLTC-inspired hold-out profile for (a) Rint, (b) 1RC, and (c) Hybrid models. A common color scale (0–120 mV) is used across all subplots to enable direct cross-model comparison. SOC-segmented WLTC RMSE statistics at 25C are reported in Supplementary Table S1.

Thermal accuracy and electro–thermal fidelity

Figure 8 compares measured and simulated surface temperatures under WLTC-inspired excitation, and Table 15 summarizes the corresponding thermal prediction errors. Because thermal parameters were identified independently from dedicated heating–cooling experiments and then held fixed during validation, the reported temperature metrics reflect electro–thermal consistency rather than profile-specific thermal retuning.

Figure 8.

Measured and simulated surface temperature for a representative run under WLTC-inspired excitation; the inset highlights a high-power segment.

Table 15.

Thermal prediction metrics on the WLTC-inspired profile at 25C (mean ± SD across repeated trials).

Model	RMSE (C)	MAE (C)	Peak (C)
Rint	2.5 ± 0.7	2.0 ± 0.6	5.0 ± 1.2
1RC	1.3 ± 0.5	1.0 ± 0.4	3.0 ± 1.0
Hybrid	1.2 ± 0.3	0.9 ± 0.3	2.0 ± 0.8

The same pattern is evident in the heat-generation consistency metric , reported in Table 16. At 25C, decreased from % for Rint to % for 1RC and % for Hybrid; at 0C, the corresponding values were %, %, and %. Thus, the hybrid model not only reduced voltage and temperature errors but also produced the closest agreement between the measurement-consistent heat input and the model-decomposed heat input .

Table 16.

Heat-generation consistency metric (mean ± SD across repeated trials), reported in percent.

Model	(%) @ 25C	(%) @ 0C
Rint	18±4	25±5
1RC	10±3	14±4
Hybrid	1.8±0.4	2.5±0.6

Lower values indicate closer agreement between and .

A representative decomposition of the model-predicted heat generation into irreversible and reversible components is shown in Fig. 9, illustrating how the coupled electro–thermal model partitions the total heat input during dynamic loading.

Figure 9.

Heat generation on a representative dynamic segment: with decomposition into irreversible and reversible terms.

Estimator relevance: SoC tracking

To examine the practical consequence of voltage-model mismatch, SoC tracking was evaluated using a Kalman-type observer driven by each ECM and using terminal voltage as the measurement, consistent with electrochemical–thermal observer motivations in the literature³⁰. Figure 10 and Table 17 summarize end-of-profile SoC error and drift rate on the WLTC-inspired profile.

Figure 10.

SoC trajectories from the Kalman-type observer driven by each ECM on the WLTC-inspired profile for a representative run at .

Table 17.

SoC estimation summary on WLTC-inspired profile (mean ± SD across repeated trials).

Model	End-of-profile SoC error (pp)	Drift rate (pp/min)
Rint	3.0 ± 1.0	0.18 ± 0.06
1RC	1.5 ± 0.6	0.08 ± 0.04
Hybrid	0.9 ± 0.4	0.05 ± 0.03

The ranking again follows the electrical-model fidelity. The hybrid model yielded the smallest end-of-profile SoC error and the lowest drift rate, followed by the fixed 1RC model, whereas Rint showed the largest accumulated drift. This pattern is consistent with the reduced innovation bias expected when transient voltage mismatch is better controlled.

Ablation

Ablation analysis was performed to isolate the contributions of bounded electrical adaptation and explicit electro–thermal coupling. As summarized in Table 18, removing the adaptive component caused the larger degradation in voltage RMSE and mismatch energy, whereas removing the thermal component produced the larger degradation in thermal MAE. These results indicate that bounded electrical adaptation is the primary source of the voltage-domain gain, while the explicit thermal states determine how improved electrical consistency is translated into temperature fidelity. Figure 11 illustrates the comparative behavior of the Hybrid-full, Hybrid-no-adapt, and Hybrid-no-thermal variants on a representative dynamic test.

Table 18.

Ablation deltas relative to the hybrid-full model (mean ± SD across runs).

Variant	RMSE (mV)	MAE (C)	Energy err. (Wh)
Hybrid-no-adapt	15 ± 5	0.3 ± 0.2	0.35 ± 0.12
Hybrid-no-thermal	3 ± 2	0.7 ± 0.3	0.18 ± 0.08

Figure 11.

Ablation comparing hybrid-full, hybrid-no-adapt, and hybrid-no-thermal variants on a representative dynamic test.

Computational complexity and runtime

Figure 12 and Table 19 summarize model complexity and normalized runtime. Relative to the fixed Thevenin 1RC baseline, the hybrid model incurred a modest computational overhead because of the additional thermal states and bounded adaptation, but remained within 1.20–1.60 the 1RC runtime. Rint remained the least expensive model, but this computational advantage came with substantially lower dynamic fidelity.

Figure 12.

Runtime benchmarking for Rint, Thevenin 1RC, and Hybrid over a fixed dynamic profile (normalised to Thevenin 1RC).

Table 19.

Model complexity summary.

Model	# states	# free parameters	Rel. runtime
Rint	1	map () + OCV	0.45–0.65
Thevenin 1RC	2	maps () + OCV	1.00
Hybrid electro–thermal 1RC	4	3 maps () + 6 adaptive scalars + thermal params (4) + OCV	1.20–1.60

Discussion

The present results show that benchmarking low-order battery models is most informative when voltage accuracy, energy-relevant mismatch, and thermal prediction are evaluated together rather than in isolation. In particular, models that appeared acceptable under quasi-steady conditions developed structured residuals under dynamic excitation, and those electrical residuals propagated into the inferred heat-generation pathway and, consequently, into the predicted temperature trajectory when the thermal network was held fixed. This directly supports the view that electro–thermal validation should not rely on temperature agreement alone, because temperature fit can otherwise obscure deficiencies in the underlying electrical dynamics²².

A consistent pattern emerged across all evaluated conditions. The Rint model remained useful as a compact quasi-static approximation, but its lack of polarization dynamics led to the largest transient voltage errors, the largest mismatch energy, and the weakest agreement between measurement-consistent and model-decomposed heat generation. Introducing a single RC branch substantially improved dynamic fidelity, and the hybrid electro–thermal 1RC model provided the best overall performance without increasing circuit order beyond one polarization state. This indicates that, for the operating range studied here, the dominant limitation was not the absence of additional RC branches alone, but the inability of fixed low-order maps to accommodate regime-dependent nonlinear behaviour under dynamic loading.

These findings are practically relevant for embedded battery-management applications. Improvements in transient voltage tracking were accompanied by lower cumulative mismatch energy, improved heat-generation consistency, and reduced SoC-estimation drift. Thus, the advantage of the hybrid model is not merely a better visual fit to voltage traces; it is a more internally consistent electro–thermal representation that remains compact enough for real-time implementation. The measured runtime overhead relative to the fixed 1RC model was modest, whereas the gains under WLTC-inspired and low-temperature operation were substantial.

The benefit of the hybrid formulation was most pronounced under low ambient temperature and strongly transient segments. This is physically consistent with the stronger impedance rise, slower relaxation, and more nonlinear behavior expected in cold operation. Under such conditions, bounded regime-dependent adaptation improved voltage and thermal fidelity while preserving practical identifiability. By contrast, under quasi-steady operation at moderate temperature, the performance gap between models narrowed, indicating that model selection should be guided by expected duty cycle and thermal-management requirement rather than by model sophistication alone.

The results also highlight the relevance of electro–thermal time-scale separation. Using the identified thermal parameters, the characteristic thermal time constants are approximately and , which evaluate to about 118–158 s and 36–51 s, respectively, across the tested ambient conditions. These values confirm that the thermal subsystem responds on a comparatively slower time scale than the transient electrical dynamics captured by the 1RC branch, so temperature states inevitably lag rapid current fluctuations. In observer-based applications, this lag can influence temperature-dependent OCV and impedance correction and may contribute to innovation bias if temperature feedback is used without appropriate filtering. The reduced SoC drift observed for the hybrid model indicates that improved electrical fidelity alleviates part of this effect, although the results also suggest that multi-rate estimation or filtered thermal feedback would remain preferable under highly transient excitation. Recent aging-integrated approaches for online core temperature estimation further highlight the value of compact coupled models when thermal states are used in estimation loops³¹.

This study has several limitations. The benchmark was conducted on fresh Samsung ICR18650-22F cells () under controlled chamber conditions, so the numerical error magnitudes reported here should not be transferred directly to aged cells, larger-format pouch cells, or pack-level systems with stronger spatial thermal gradients, where integrated pack electro–thermal representations are commonly required³². The benchmarking framework itself is transferable, but such applications would require re-identification of electrical and thermal parameters and, where appropriate, richer thermal-network structure. In addition, the present work does not account for aging-induced drift in impedance maps, relaxation behavior, or chemistry-dependent variation in entropic and thermal characteristics.

Overall, the measured trade-offs support a clear practical hierarchy. Rint remains suitable for quasi-static screening and applications where transient electro–thermal fidelity is not critical. A fixed Thevenin 1RC model represents the minimum structure recommended for dynamic battery-management operation involving frequent current steps or regenerative segments. The hybrid electro–thermal 1RC model is the preferred option when the target application involves thermally varying or strongly dynamic operation and requires improved temperature prediction and estimation robustness without the identification burden of higher-order ECMs.

Conclusion

This study presented a harmonised electro–thermal benchmark for low-order lithium-ion equivalent circuit models under constant-current and dynamic loading. By jointly evaluating voltage fidelity, mismatch energy, heat-generation consistency, and temperature prediction under a strict identification/validation split, the benchmark showed that model structure has consequences that extend beyond terminal-voltage accuracy alone. Across 0 C to 40C , the hybrid electro–thermal 1RC model consistently outperformed the Rint and fixed 1RC baselines. On the WLTC-inspired hold-out profile at 25C , it achieved a voltage RMSE of  mV, a surface-temperature RMSE of  C , and a heat-generation consistency mismatch of (mean ± SD, ). These results support the use of compact electro–thermal models with at least one polarization state and bounded adaptation for embedded BMS applications under realistic dynamic operation.

Future work will extend the framework to online aging-aware parameter adaptation, larger-format and pack-level cells, and higher-order RC structures to determine when additional model complexity yields engineering-relevant gains under multi-time-scale excitation³³. In particular, future studies should incorporate online/recursive map tracking, including incremental identification of drift in , , and with state of health (SOH), to preserve voltage and thermal prediction accuracy under cyclic aging.

Author contributions

S.M. contributed to data curation, experimental investigation, and preliminary analysis. S.C.S. contributed to experimental design, supervision, validation, and formal analysis. Z.L.M. contributed to conceptualization, methodology development, and manuscript writing and revision. P.K.S. contributed to methodology refinement, validation, and critical review of the manuscript. A.K.S. contributed to interpretation of results, technical review, and manuscript editing. All authors discussed the results, reviewed the manuscript, and approved the final version.

Data availability

The datasets generated and/or analysed during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-44025-2.