Reliable Thermal-Physical Modeling of Lithium-Ion Batteries: Consistency between High-Frequency Impedance and Ion Transport

Abstract

Among lithium-ion battery diagnostic tests, electrochemical impedance spectroscopy, being highly informative on the physics of battery operation within limited testing times, deserves a prominent role in the identification of model parameters and the interpretation of battery state. Nevertheless, a reliable physical simulation and interpretation of battery impedance spectra is still to be addressed, due to its intrinsic complexity. An improved methodology for the calibration of a state-of-the-art physical model is hereby presented, focusing on high-energy batteries, which themselves require a careful focus on the high-frequency resistance of the impedance response. In this work, the common assumption of the infinite conductivity of the current collectors is questioned, presenting an improved methodology for simulating the pure resistance of the cell. This enables us to assign the proper contribution value to current collectors’ resistance and, in turn, not to underestimate electrolyte conductivity, thereby preserving the physical relation between electrolyte conductivity and diffusivity and avoiding physical inconsistencies between impedance spectra and charge–discharge curves. The methodology is applied to the calibration of the model on a commercial sample, demonstrating the reliability and physical consistency of the solution with a set of discharge curves, EIS, and a dynamic driving cycle under a wide range of operating conditions.

1. Introduction

Lithium-ion battery (LIB) technology, featuring outstanding energy and power density, a satisfying lifetime, high round-trip efficiency, and very fast dynamics in a reasonably economic form [1,2], has rapidly become the undisputed ruler of portable power; it is now the main driver of the electrification in the transportation sector.

Despite the fully commercial development of the technology, researchers are still debating LIBs’ degradation from a physical perspective, with the aid of reliable estimations of their state of health (SoH) and residual lifetime [3,4]. One relevant approach is the combined use of experiments and models.

In this regard, electrochemical impedance spectroscopy (EIS) [3,5,6,7,8,9] is a powerful non-invasive tool for lithium-ion battery characterization, providing a large amount of information discretized on a characteristic time scale and collected within a limited testing timespan. These advantages make it a promising tool for state-of-charge and state-of-health estimation [7,9]. However, EIS is difficult to interpret due to overlapping phenomena at similar frequencies, especially between the two electrodes. For instance, as explained in [5], in the medium frequency area, up to five phenomena can be present, whilst only some of them are easily detectable as an arc-shaped feature.

This complexity forces researchers to use models. EIS is usually modelled through equivalent circuit models (ECM) [3,6,7,10,11] of very miscellaneous complexity (within the literature), where relevant processes are associated with a combination of pure resistances and capacitances whose values and layout are the design choice of the developer. In general, these models are very popular because of their simplicity, simulation velocity, and their potential compatibility with on-board vehicle applications for state-of-charge (SoC) or state-of-health estimation, as comprehensively explained in this recent review [12]. However, despite their advantages, ECMs do not include physical relations between elements; thus, they do not guarantee physical consistency in simulating battery operations. Moreover, they lack generalizability, as they are strongly sample-specific and condition-specific, providing poor performance if applied to broad operating conditions. Hence, their reliability has been questioned [13,14].

If time and computational effort are not the main drivers of model selection, physical models are more reliable. However, physical simulation of the impedance spectra of LIB is neither very common nor a simple task; it involves several tens of physical parameters that are often unknown or undeterminable. In previous publications [15,16], the authors analyzed the sensitivity of a number of physical parameters in an EIS simulated using the Doyle–Fuller–Newman (DFN) model, before proposing an optimized multi-measurement approach for model calibration, including guidelines for operating conditions’ selection and model parameters’ identification. These works identified that two key parameters in physical simulations are the electrolyte properties: ionic conductivity and lithium-ion diffusivity [16]. Their proper estimation is of paramount importance for a reliable simulation, together with their dependency on temperature and lithium concentration, enabling physically sound simulation of battery operations and interpretations of ageing. Despite these parameters being related via the known Nernst–Einstein relation [17], many models consider them independent parameters to ease the parameter identification process. However, this solution can lead to major physical inconsistencies, which arise from the wrong estimation of their contributions to the pure resistance of the device, here identified as high-frequency resistance (HFR); by this, we refer to the high-frequency intercept with the x-axis in the Nyquist plot of the EIS. On this topic, there is a gap between experiments, (from material to cell level) and models (from ECM to physical models). Indeed,

• many ECMs consider one pure resistor to account for all the ohmic losses, disregarding single contributions (for instance, electrodes’ or current collectors’ electric resistivities [3,6]. This choice lacks physical consistency and therefore limits simulation reliability.
• Moreover, these models usually consider the several elements independently of one other, potentially leading to physical inconsistencies, as in the case of electrolyte conductivity and diffusivity. According to the relevance of these two parameters for lithium-ion simulations, their relation [18] has to be ensured to provide a physically reliable simulation.

On the other hand, physical models

• usually reproduce discharge curves with satisfactory accuracy; however, to provide physical consistency when simulating LIB operation and dynamics, the reliability should be challenged using EIS, as recently pointed out by several research groups [16,17]. The simulation of EIS, and specifically of HFR, can guide the calibration by providing additional constraints to the solution.
• One interesting topic specific to HFR is the need to take into account the role of current collectors, which are often regarded as negligible [14,17]. Indeed, this choice can lead to wrong overestimation of electrolyte resistivity. In some conditions, the contribution of current collectors may be the major part of the pure response of the device, especially in high-energy designs; thus, the assumption of infinite conductivity applied in many models such as DFN is not valid.

To summarize, to the authors knowledge, no work in the literature debates the actual contribution of the electrolyte in the HFR of lithium-ion batteries from a physical perspective. Hence, in this work, starting from the analysis of one experimental high-energy case study and providing guidelines for a general approach, we discuss the reliable simulation of the battery’s ohmic resistance, proposing an improved modeling framework and implementing the findings in an optimized parameter identification procedure for a DFN model, which demonstrates its improved reliability.

The outline of this research paper is as follows. Section 2 summarizes the fundamentals of the DFN physical model and the description of experimental setup and tested battery samples. In Section 3, the cells are dismantled and the current collectors’ resistance is directly measured. Moreover, a discussion on the magnitude of the electrolyte and current collectors’ resistance is presented. Lastly, Section 4 reports the optimized parameter identification procedure, together with the results and a wide validation against different experimental curves, from full discharges to EIS, in different operating conditions.

2. Methodology

2.1. Battery Samples

One type of high-energy commercial cell is adopted in this work. Its specifications are listed in

Table 1. Specifications of experimental samples.

Model	SONY US18650V3
Geometry	Cylindrical 18650
Anode chemistry	Graphite
Cathode chemistry	NMC
Nominal capacity	2250 mAh
Maximum current (C-rate)	Charge: 2.25 A (1C) Discharge: 10 A (4.5C)
Operative voltage range	4.2 V–2.5 V
Temperature range charge/discharge	Charge: 0–45 °C Discharge: −20 °C/60 °C
Weight	44 g

.

The electrolyte is expected to be a mixture of LiPF₆ in EC:DMC solvent.

Measurements of the internal geometry were performed after dismantling two samples that were completely discharged to 0 V and short-circuited, for the sake of process safety. Images of the battery dismantling process are given in Figure 1, together with a simplified scheme of the internal structure, while the geometrical dimensions of all the components are listed in

Table 2. Internal dimensions of battery layers.

US18650V3
Component		Layers	L (cm)	H (cm)	W (µm)
Negative	Current collector (Copper)	1	66.5	5.7	15
	Active layer (Graphite)	2	58.5	5.7	70
Positive	Current collector (Aluminum)	1	65.8	5.7	20
	Active layer (NMC)	2	56.5	5.7	65
Separator		2	80.3	5.7	15
Active area (cm2)			644.10

. The cell active area is computed as the frontal surface of the smaller active layer, which is the positive electrode.

As is visible in the scheme, battery the jelly roll is composed of copper and aluminum current collectors, respectively, covered on both sides with negative (graphite) and positive (NMC) electrode layers. Beneath such active layers, the separator is winded in a twofold configuration, creating a two-layer battery roll as the elementary unit.

2.2. Experimental Testbench

The experimental testbench is a custom-developed testing station constituted by a NI-4124 power supply (current uncertainty: ±0.5% ± 0.1% FSR) and two modular Chroma UM 63640-80-80 electronic loads (voltage/current uncertainty: ±0.025% ± 0.01% FSR/±0.1% ± 0.1% FSR), resulting in a cycler of eight independent channels with impedance measurement capability. This setup is employed in all the measurements required for model parameter identification and reliability validation. Details on the testbench are provided in publication [16].

For measurement of the current collectors’ resistance, an Autolab PGSTAT30 potentiostat/galvanostat equipped with a frequency response analyser board (FRA2) is used. Both resistance measurement and impedance measurement are conducted during a pulse, providing very similar results.

2.3. Physical Model

The simulations are performed using a classical DFN model implemented in COMSOL Multiphysics^®. The model includes double-layer (dis/)charging on the surface of both electrodes, and surficial passive layers are modeled as lumped resistances. The model is coupled with a 2D thermal model to account for heat generation inside the cell, and many parameters are temperature-dependent, such as kinetic constants and diffusivities. The electrochemical model computes the heat generation, so that the thermal model computes the temperature distribution over the cross section of a cylindrical cell. In turn, the thermal model provides an average temperature to update the temperature-dependent parameter values.

For a detailed description of the equations of the DFN and thermal model, we refer the reader to [15,16]. EIS is simulated in the time domain and then converted into a frequency domain by way of fast Fourier transform, as discussed in [15,16]. Optimization of EIS simulation has been performed as detailed in Section 4.2.

3. Focus on Cell High-Frequency Resistance

3.1. High-Frequency Resistance at Different SoC and Temperatures

Figure 2a reports experimental HFR values measured at different temperatures and states of charge, expressed both as total and specific resistance with respect to battery internal surface. Figure 2b shows the corresponding frequency, obtained through interpolation of experimental data at a null imaginary component of the impedance. A weak dependance on the state of charge is detected for all the temperatures, where HFR is 2–3 mΩ larger at low SOC than high SoC. Temperature affects the HFR value, but in the investigated interval, its effect appears as a minor contribution (within 10%). Differently, frequency is strongly affected by temperature, and diminishes significantly as the temperature increases, as expected.

Being a purely resistive feature of overall battery impedance response, its known contributing terms are identifiable in electrolyte ionic resistance, the electric resistance of current collectors (usually neglected under the assumption of infinite conductivity), and external resistance (comprising the electrical and contact resistances of the external circuit). In the following, the first two contributions are investigated.

3.2. Electrolyte Ionic Resistance

In the literature, LiPF₆ electrolyte ionic conductivity has been extensively characterized [19]. Figure 3 reports the trends of the ionic conductivity with the temperature and concentration of a mixture of PC-EC-DMC with LiPF₆, by exploiting the empirical correlation proposed in [19] that was validated over experimental data.

For the purpose of this publication, such electrolyte conductivity dependance on lithium concentration and temperature is accounted for as follows [19]:

$${\sigma }_e=con{d}_{el} ·c_e·{\left(-10.5+0.0740·T-6.96 \times {10}^{-5}·T^2+0.668·{\mathrm{c}}_e-0.0178{\mathrm{c}}_e·T+2.8 \times {10}^{-5}·{\mathrm{c}}_e·T^2+0.494·c_e^2-8.86 \times {10}^{-4}·c_e^2·T\right)}^2$$

(1)

where $c_e \left[\mathrm{mol}/{\mathrm{m}}^3\right]$ is the lithium concentration in the electrolyte, $T \left[\mathrm{K}\right]$ is the temperature, and $con{d}_{el}$is a coefficient dependent on the specific sample. This solution maintains a physically sound relation with operating conditions while adopting a fitting parameter, the exact electrolyte composition being unknown. As is clear from the figure and the correlation, the peak value at 40 °C is 50% larger than the corresponding value at 20 °C, showing a sensible temperature dependency. Similarly, the conductivity decreases significantly, being far from the optimized lithium concentration (around 1000–1200 mol/m³).

To roughly quantify the electrolyte contribution to the overall cell-specific HFR, electrolyte specific resistance can be calculated, considering the standard electrolyte conductivity at 25 °C [19] (see Figure 3) and cell internal geometry discussed in Section 2.1. For the samples under investigation, the specific electrolyte ionic resistance is estimated between 2.5–3.5 Ω∙cm² (depending on the separator and electrode porosity, considered to consistently between 40 and 50% of the high-energy battery typology); this range covers just a small portion of the total HFR (measured in 20–21 Ω∙cm², depending on battery SOC at 25 °C, as reported in Figure 2a). Additionally, as is clear from Figure 3, the variation in electrolyte conductivity with temperature is sensible (~+30% if temperature is increased from 20 °C to 40 °C, and ~−30% from 20 °C to −10 °C), while HFR variation is clearly narrower. Therefore, it is implicit that the electrolyte is not always the only nor the main element responsible for the HFR value.

Moreover, physical consistency should be discussed. Indeed, both the lithium diffusion coefficient and ionic conductivity are related through the Nernst–Einstein equation [17]:

$$D_e={\sigma }_e·\frac{RT}{F^2{c}_e}$$

(2)

where $D_e$ and ${\sigma }_e$ are the diffusion coefficient and electrolyte conductivity in $\left[{\mathrm{m}}^2/\mathrm{s}\right]$ and $\left[\mathrm{S}/\mathrm{m}\right]$, respectively, $R=8.314 \mathrm{J}/\left(\mathrm{mol}·\mathrm{K}\right)$ is the gas constant, $T$ is the temperature in $\left[\mathrm{K}\right]$, $F=96,485 \mathrm{C}/\mathrm{mol}$ Faraday constant, and $c_e$ is the lithium concentration in the electrolyte in $\left[\mathrm{mol}/{\mathrm{m}}^3\right]$.

Associating the HFR value of the electrolyte would only lead to additional major mistakes affecting the electrolyte diffusivity, thereby depleting the reliability and physical soundness of the model.

3.3. Current Collectors’ Electrical Resistance

To aid the discussion of current collectors’ contribution to HFR, whose value is hardly present in physical models (according to the common assumption of infinite conductivity), a simplified scheme is provided in Figure 4.

Current collectors can be approximated with thin foils with one tab each. Tabs connect the aluminum and copper foils with the corresponding poles of the battery, and can be considered lumped resistances; their position and number significantly affect the current distribution within current collectors. Indeed, if each foil is connected to a single tab, the overall current will flow through it; however, in the case of more tabs per foil, the overall current is split between them, and their ohmic contribution will be lower.

Moreover, if the tab is placed at the edge of the foils, as in Figure 4, the overall current will have to reach the tab; under the assumption of uniform reaction rate, the current will be maximum at the tab position, and will linearly decrease to zero at the opposite edge. Hence, the average current is half of the overall value. Instead, if the tab is placed at the middle of the foil, the current will have to flow through only half of the overall foil length, making the average current one fourth of the total value. Hence, generally, the more tabs per pole are present, and the better distributed they are, the lower the effective current collector resistance becomes.

In order to take into account the lumped resistance part of the internal circuit of the battery, namely caps, tabs and their soldering, which have a non-trivial value, the overall resistance is experimentally measured, and the resulting values are reported in

Table 3. Current collectors’ resistance measurement.

Current Collector Material	Tab Number and Position	Half CC Only		Overall (Full CC, Tabs and Cap)
		Total [mΩ]	Specific [Ωcm2]	Total [mΩ]	Specific [Ωcm2]
Copper	1, outer edge	11.5	7407	28	18,035
Aluminum	1, inner edge	11.3	7278	24	15,458

. The resistance is measured through a current pulse along the roll, with and without both tab and cap. The current collectors’ contribution is halved according to the assumption of uniform current generation along the foils.

From this measurement, the current collectors proved to cover a relevant portion of the HFR, deserving to be included in the model. It is worth to remind that this is specific to the case of high-energy batteries, and may change significantly in the case of a different cell design, which is outside the scope of this publication.

3.4. Modelling the Current Collectors’ Resistance

The physical model requires a proper implementation of current collectors’ role for a consistent simulation of the related phenomenon. Hence, a dedicated internal resistance term, $R_{int}$, is introduced in the model to embed the current collectors’ resistance. The cell voltage $\mathsf{\Delta }V$ is now computed as follows:

$$\mathsf{\Delta }V={{\phi }_s|}_{x=L_n+L_{sep}+L_p}-{{\phi }_s|}_{x=0}-R_{int}I$$

(3)

where ${{\phi }_s|}_{x=L_n+L_{sep}+L_p}$ and ${{\phi }_s|}_{x=0}$ are the electric potential as the electrode–current collector interface of the positive and negative electrodes, respectively, while $R_{int}$ and $I$ are the internal resistance and total applied current, respectively.

Correspondingly, the heat generation term is modified to include a heat generation due to the Joule effect, itself due to the internal resistance term, which can severely affect battery operation, especially during high-current operation:

$$\frac{1}{r}\frac{\partial }{\partial r}\left(r k_T\frac{\partial T}{\partial r}\right)+\frac{\partial }{\partial z} \left(k_T\frac{\partial T}{\partial z}\right)+\overline{q_{gen}}+R_{int}{I}^2=\rho C\frac{\partial T}{\partial t}$$

(4)

where $r$ is the radial coordinate in the cylindrical cell, $k_T$ the average thermal conductivity of active area of the cell, $z$ the vertical height coordinate, $T$ the cell temperature, $\overline{q_{gen}}$ the heat generation obtained in the P2D model, $\rho$ the cell density and $C$ the cell heat capacity.

As described in Section 3.2, the current collector resistance depends on different design aspects, such as the length and number and position of tabs. It is not possible to state a general rule for its estimation without prior knowledge of the battery geometry. In the following, the parameter $R_{int}$ is considered a fitting parameter. However, since its contribution is known to be a pure resistance, a correlation with the HFR value is proposed to constrain the parameter identification problem, as will be discussed in Section 4.2:

$$R_{int}=HF{R}_{exp}-R_0-R_{el}$$

(5)

$$R_{el}=\frac{K_1}{{\sigma }_{el}}$$

(6)

where $HF{R}_{exp}$ is the HFR value measured at 25 °C, 50% SoC. It is assumed that it is equal to the sum of three terms: the electrolyte contribution $R_{el}$, computed as the ratio between a constant $K_1$ and the conductivity, the internal resistance, and $R_0$, which is the sum of all the other resistive terms of the battery, such as the electronic conductivities of the electrodes. In practice, $R_0$ is the modeled cell impedance at the same operating conditions, but with the electrolyte conductivity ${\sigma }_{el}$ equal to 1000 S/m and $R_{int}$ equal to zero; thus, the contributions of electrolyte and internal resistance are negligible. In the present case, the constants $R_0$ and $K_1$ are equal to 2.3 mΩ and 1.39 1/m, respectively.

4. Results and Discussion

4.1. Diagnostic Protocol

A previously performed sensitivity analysis [16] upon discharge test, EIS and relaxation test demonstrated that low temperature and extreme SOC conditions can maximise the experimental identifiability of the model parameters.

Summarizing the main points:

• Measuring EIS spectra is of great importance, at least at two opposite states of charge, to enable a distinction between positive and negative electrode contributions, and at two temperature levels, to enable the calibration of temperature-related parameters.
• The discharge test is mainly sensitive to diffusion-related phenomena, both in electrodes’ bulk and the electrolyte. Two combinations of C-rates and temperatures are advised to be covered with this technique, spanning the entire battery capacity.

Applying the methodology to the sample, the resulting experimental protocol is reported in Figure 5:

• (E1) EIS testing on the battery at SoC 100%, 10 °C is performed at 20 logarithmically spaced frequencies in the range 4000–1 Hz;
• (D1) the battery is discharged at 2C with a DoD 50%;
• (R) a relaxation test of 1000 s follows the discharge. During this test, voltage and temperature are recorded;
• (E2) after the relaxation, the climate chamber temperature of 25 °C is set, and when the temperature is reached, an EIS at SoC 50% within the same frequency interval as the first is performed;
• (D2) the battery is discharged at 1C until the lower voltage limit is reached (theoretical DoD 50%).

4.2. Optimized EIS Simulation and Parameter Identification Procedure

After the implementation of the internal resistance term in the battery model (as described in previous section), an optimized stepwise calibration procedure is developed. Optimizations in the EIS simulation to improve the convergence time and reliability are also implemented, as described in the following:

• Optimized domain

Because lithium diffusion in electrodes’ bulk is a slow phenomenon, its effect does not impact the performance until the imposed sine frequency approaches low values. Hence, spatial gradients in the particles were neglected during the high-frequency part of the EIS (f ≥ 10 Hz), imposing the equality of lithium concentration between surface and canter of the particles, solving Fick’s law for diffusion only at (f ≤ 10 Hz). The exact frequency threshold is case-dependent; therefore, a conservative approach should be applied to un-bias the solution. 10 Hz is selected in this specific case.

• Optimized mesh resolution

An optimization of mesh resolution of the electrochemical model is introduced. A highly resolved mesh is crucial for the simulation of highly heterogeneous conditions such as those originating during discharges, especially at a high C-rate; it is less important during EIS simulations performed at a low C-rate in open-circuit conditions (OCV) under a steady-state assumption. Hence, it is reduced in the order of 5 points per domain along the x direction (electrode thickness), without losing accuracy, during EIS simulation.

• Optimized thermal model

In an EIS measurement, the applied sinusoidal current is very low, and the heat generated by the cell in this condition can be reasonably neglected. Hence, in an EIS study, the heath transfer computation is deactivated, while it is maintained for the discharges.

Stepwise Single-Measurement Calibration

An optimized stepwise approach is developed and implemented, exploiting the separation between the experimental techniques and corresponding physical parameters demonstrated in the sensitivity analysis [16]. Therefore, the parameter identification problem is solved with a three-step procedure:

• Step 1: the model is calibrated against the two impedance spectra (E1) and (E2) simultaneously, by optimizing the kinetic constants ($k_{pos}, k_{neg}$) and double-layer capacitances values ($C_{dl,pos}, C_{dl,neg}$). In this step, only the charge transfer resistance is considered; hence, the HFR value is neglected, and the remaining parameters are fixed at a proper literature value, as listed in

  Table 4. List of main physical parameters with the corresponding literature range and selected value for the case study. The interval of cond_el variation was estimated so that the conductivity of the electrolyte falls within the range identified in the references.

  Parameters	Symbol	Variability Interval Employed in This Work	References
  Kinetic rate constant of the cathode	$k_{pos} \left[\mathrm{m}/\mathrm{s}\right]$	$\left[2.5\times {10}^{-11};1\times {10}^{-8}\right]$	[20,21]
  Kinetic rate constant of the anode	$k_{neg} \left[\mathrm{m}/\mathrm{s}\right]$	$\left[2.5\times {10}^{-11};1\times {10}^{-8}\right]$	[20,21]
  Double layer of the cathode	$C_{dl,pos} \left[\mathrm{F}/{\mathrm{m}}^2\right]$	$\left[0.05;25\right]$	[22,23]
  Double layer of the anode	$C_{dl,neg} \left[\mathrm{F}/{\mathrm{m}}^2\right]$	$\left[0.05;25\right]$	[22,23]
  Ionic conductivity factor	$con{d}_{el} [-]$	$\left[0.05;1.25\right]$	[19,24]
  Solid diffusivity of the cathode	$D_{s,pos} \left[{\mathrm{m}}^{2/}\mathrm{s}\right]$	$\left[1\times {10}^{-6};1\times {10}^{-12}\right]$	[18,25,26]
  Solid diffusivity of the anode	$D_{s,neg} \left[{\mathrm{m}}^2/\mathrm{s}\right]$	$\left[5\times {10}^{-6};5\times {10}^{-12}\right]$	[18,27]

  . The other parameters are fixed to an intermediate value within the literature’s range, as a first guess. Since solid diffusivities and (to a certain extent) electrolyte properties have a minor effect on EIS, this solution is viable and will be corroborated Step 3. At this stage, the current collectors’ contribution is estimated according to Equation (5), presented in Section 3.4, corresponding to an intermediate value of electrolyte conductivity from the literature.
• Step 2: the model is then calibrated on partial discharges (D1) and (D2) by optimizing the solid diffusivities ($D_{s,pos}, D_{s,neg}$) and electrolyte conductivity ($con{d}_{el}$).
• Step 3: finally, Step 1 is repeated, employing the solid diffusivities and electrolyte conductivity computed in Step 2.

4.3. Obtained Solution

The calibration is performed twice to check its repeatability; the result is presented in

Table 5. Values of the parameters obtained with the step calibration procedure.

Parameter	Value
$k_{pos} \left[\mathrm{m}/\mathrm{s}\right]$	$8.44\times {10}^{-9}$
$k_{neg} \left[\mathrm{m}/\mathrm{s}\right]$	$9.92\times {10}^{-10}$
$con{d}_{el} [-]$	$0.475$
$C_{dl,pos} \left[\mathrm{F}/{\mathrm{m}}^2\right]$	$0.341$
$C_{dl,neg} \left[\mathrm{F}/{\mathrm{m}}^2\right]$	$0.267$
$D_{s,pos} \left[{\mathrm{m}}^{2/}\mathrm{s}\right]$	$9.87\times {10}^{-14}$
$D_{s,neg} \left[{\mathrm{m}}^2/\mathrm{s}\right]$	$1.24\times {10}^{-14}$
$R_{int} \left[\mathsf{\Omega }\right]$	$2.63\times {10}^{-2}$

, while the experimental and simulated measurements involved in the protocol are shown in Figure 6.

4.4. Validation

To validate the obtained results, a set of non-training conditions and operation modes were simulated with the calibrated model, and compared with experimental measurements. The validation dataset includes (as listed in

Table 6. List of measurements belonging to the validation dataset and corresponding RMSE values, obtained through Equation (7), by comparing modelled and experimental curves. For discharge curves, the RMSEs computed over the full discharge (left value) and from SoC 100% to 3 V (right value) are provided.

Technique	Operating Condition	$\mathit{R}\mathit{M}\mathit{S}\mathit{E}$
Discharge curve	$0.5\mathrm{C}$, $45°\mathrm{C}$, $100\%-0\%$ $SoC$	$60.3\mathrm{mV};23.6\mathrm{mV}$
	$1\mathrm{C}$, $10°\mathrm{C}$, $100\%-0\%$ $SoC$	$66.2\mathrm{mV};34.9\mathrm{mV}$
	$1\mathrm{C}$, 25 $°\mathrm{C}$, $100\%-0\%$ $SoC$	$25.9\mathrm{mV};18.3\mathrm{mV}$
	$1\mathrm{C}$, 45 $°\mathrm{C}$, $100\%-0\%$ $SoC$	$14.5\mathrm{mV};11.5\mathrm{mV}$
	$2\mathrm{C}$, 25 $°\mathrm{C}$, $100\%-0\%$ $SoC$	$44.4\mathrm{mV};18.6\mathrm{mV}$
	$2\mathrm{C}$, 45 $°\mathrm{C}$, $100\%-0\%$ $SoC$	$50.9\mathrm{mV};29.2\mathrm{mV}$
EIS	45 °C, 100% $SoC$	$Z_{Re}:0.9\mathrm{m}\mathsf{\Omega }$
		$Z_{Im}:2.0\mathrm{m}\mathsf{\Omega }$
	10 °C, 75%$SoC$	$Z_{Re}:4.0\mathrm{m}\mathsf{\Omega }$
		$Z_{Im}:3.3\mathrm{m}\mathsf{\Omega }$
	25 °C, 75%$SoC$	$Z_{Re}:1.8\mathrm{m}\mathsf{\Omega }$
		$Z_{Im}:2.4\mathrm{m}\mathsf{\Omega }$
	45 °C, 75%$SoC$	$Z_{Re}:1.3\mathrm{m}\mathsf{\Omega }$
		$Z_{Im}:2.2\mathrm{m}\mathsf{\Omega }$
IEC 62660-1:2018	25 °C, 100%−0% $SoC$	$5.6\mathrm{mV}$

)

• Full discharges at

  ○

  0.5C at 45 °C

  ○

  1C, 10–25–45 °C

  ○

  2C, 25–45 °C

• EIS spectra performed from 4 kHz to 20 mHz, with 40 logarithmically spaced points:

  ○

  At SoC 75% and 10–25–45 °C

  ○

  At 100% SoC and 45 °C;

• The application of the IEC 62660-1:2018 BEV cycle life profile [28], as a highly dynamic profile, at 25 °C, from SoC 100% to the lower voltage threshold.

To quantify the average distance between the experimental and the model curves, the point-by-point root mean square error (RMSE) is computed as:

$$RMSE=\sqrt{\frac{{\sum }_{i=1}^N{\left(x_i^e-x_i^m\right)}^2}{N}}$$

(7)

where $x_i^e$ is the experimental data, $x_i^m$ the simulated data, and $N$ the number of sampled points.

RMSEs are reported in Table 6, while experimental and simulated profiles are shown in Figure 7.

In the latter, the model simulations are satisfactory for large portions of the discharge in all the operating conditions. This result proves the reliability of the model in reproducing the losses at different temperatures and C-rates. From a quantitative point of view, by comparing the RMSE listed in Table 6 with literature values, these values are slightly higher than the average. Indeed, Park et al. [29], worked with a nickel–cobalt–aluminum oxide cell battery and reported RMSEs of 25.5 mV and 11.8 mV, respectively, for a 1C and 0.5C discharge, both at 25 °C. Li et al. [30] calibrated a LiMnO₄/LiC₆ battery and obtained RMSEs between 6.4 mV and 12.9 mV for discharges at room temperature, with C-rates in the range 0.5C–3C. The highest average error was achieved in the 0.5C discharge. Lastly, Yang et al. [31] utilized NMC cells and obtained RMSEs of 15.6 mV, 14.6 mV and 21.6 mV, for discharges at 25 °C and C-rates 1C, 1.5C and 2C, respectively. However, all the works focused on a limited range of operating conditions, which can justify lower RMSEs values. The current results suffer from high errors at the end of discharge, where, due to the steep OCP curve of graphite, little misalignments lead to a sensible effect on the estimated voltage. To highlight this consideration, RMSEs were also evaluated from fully charged conditions down to 3 V (right values in Table 6). Overall, the results proved to be largely satisfactory for almost all the combinations of temperature, C-rate, and SoC analyzed herein.

With regards to Figure 7b,c, the EIS spectrum is reasonably reproduced in all the cases. The magnitude of the kinetic loop matches at all the temperatures; thus, the kinetic variation with temperature is reliably implemented in the model. Similarly, the HFR value reasonably varies with temperature, demonstrating the proper implementation of resistive features. The diffusive branch at low frequency is also reasonably reproduced thanks to the proper identification of diffusive properties in the discharges of the protocol. From the Bode plot, the characteristic frequencies of the model are in line with those of the experiments. From a quantitative point of view, the simulation of EIS using a physical model is seldom performed in the literature. Overall, the RMSEs values are considered satisfactory for the purposes of the work.

Lastly, with reference to Figure 8, the model reliability is further corroborated with the simulation of a dynamic profile. The very low RMSE value proves the proper simulation of overpotentials and the internal state of the battery over the vast majority of the state-of-charge window.

5. Conclusions

This work discussed the proper implementation of an internal circuit contribution to lithium-ion batteries’ impedance for a reliable and physically sound simulation of the role of the electrolyte. Indeed, considering typical electrolyte conductivity values as a function of temperature, and measuring the resistance of current collectors, it was shown that the latter covers a non-negligible portion of the high-frequency resistance (HFR) of the battery, which also features an apparently inconsistent trend with temperature. A proper modeling of this effect is proposed, accounting for ohmic loss and the Joule effect, to prevent the underestimation of the electrolyte conductivity (which may lead to physical inconsistencies, thereby negating the relationship between electrolyte conductivity and diffusivity).

The proposed methodology is applied to identify the model parameter values of one commercial sample, exploiting a compact multi-measurement experimental protocol together with an optimized stepwise procedure. The resulting set of parameter values reproduce the experimental curves suitably. Moreover, the physical consistency of the obtained solution is demonstrated against a wide validation dataset in terms of operating conditions. The results were compared to similar works in the literature, demonstrating slightly higher RMSE values, but over a much broader range of operating conditions, including EIS simulation. Moreover, the model shows the capability of assigning the phenomena to the proper characteristic time, and simulating temperature variations properly. Therefore, all in all, the results are considered consistently satisfactory, first overcoming initial limitations and, in turn, showing the reliability of the proposed methodology.