Aurélien Quelin

Nicolas Damay
email: nicolas.damay@utc.fr

Coupling electrical parameters of a battery equivalent circuit model to electrodes dimensions

Keywords: Battery, Electrical modeling, Equivalent circuit model, Electrodes dimensions

The design of a battery-powered system ideally requires the simultaneous sizing of all its components. To contribute to this purpose, we propose a physical approach to couple the electrical parameters of a battery equivalent-circuit-model (ECM) to the electrodes dimensions of the battery. Because it only requires non-invasive measurements, it can be easily used by system integrators. To test the proposed approach, we choose three commercial coin cells with different sizes and nominal capacities (25 mA h, 60 mA h and 120 mA h). Only the electrodes length of these cells varies, and this study is therefore focused on creating dependency-models that predict the ECM parameters values with respect to this specific dimension. The proposed ECM brings accurate voltage simulations and the dependency-models predictions are satisfactory for the three cells, with a mean accuracy of 6.3%. Thanks to the proposed approach, the parameters of any cell size and capacity can be predicted in the characterization range (between 25 mA h and 120 mA h here). It is thus a promising tool for developing custom-made cells.

Introduction

The persons designing a battery-powered microsystem are often forced to choose among several commercial batteries that are already existing. They have fixed mechanical and electrical properties and choosing one often requires to make some trade-offs at the system scale. To address this problem, Chen et al. [START_REF] Chen | Microbattery Development for Juvenile Salmon Acoustic Telemetry System Applications[END_REF] have, for example, fabricated a cell featuring a custom form factor more adapted to their device than the two commercial coin cells previously used. Solutions of this type are even proposed commercially [2].

These techniques are mainly aimed at increasing the volume of active material in a small available volume. However, they rarely take into account the effects of dimensional changes on the electrical behavior of the cells. For instance, Humble and Harb [START_REF] Humble | Optimization of nickel-zinc microbatteries for hybrid powered microsensor systems[END_REF] have determined a maximum operating power for a certain electrode thickness and width. Changing the dimensions of a battery thus changes its electrical performance, which may then become unsuitable for the system to be powered. In order to get a proper sizing of a cell for its integration into a system, it is therefore important to consider the coupled dimensional, mechanical and electrical constraints.

The possibility to get custom-made batteries from some manufacturers and the progress in the fabrication of printed thin-film batteries [START_REF] Lyu | Design and manufacture of 3D-printed batteries[END_REF] suggest that cell-size customization will become more common. Optimally sized batteries could be lighter and use less material while still guaranteeing sufficient electrical and thermal performance.

The optimization of the cell design can be done by trial and error, but a model-based optimization algorithm can lead to better -and sometimes faster -results, given that the model is well-adapted to a system-design situation. Mei et al. [START_REF] Mei | The effect of electrode design parameters on battery performance and optimization of electrode thickness based on the electrochemical-thermal coupling model[END_REF] used an electrochemical model coupled with a thermal model to compute the effects of a variable electrode thickness on the electrochemical and thermal behaviors of a 18.5 Ah prismatic cell. This approach is meant to improve the design of a cell, but the knowledge of the cell electrochemical parameters is mandatory to calibrate the model. It is hence hard to use by a system integrator, unless they have a good collaboration with the cell manufacturer.

Merla et al. [START_REF] Merla | An easy-to-parameterise physics-informed battery model and its application towards lithium-ion battery cell design, diagnosis, and degradation[END_REF] proposed a physic-based equivalent circuit model (ECM) created thanks to a pseudo-2D electrochemical model [START_REF] Von Srbik | Offer, A physically meaningful equivalent circuit network model of a lithium-ion battery accounting for local electrochemical and thermal behaviour, variable double layer capacitance and degradation[END_REF]. This model contains numerous parameters and their determination requires a few non-invasive tests, along with a substantial knowledge of the studied cell internal characteristics. Hence, less knowledge of the cell electrochemical parameters is required, but a good understanding of electrochemical phenomena is mandatory. Besides, there are no evidence that this model can easily predict the performance of a cell with different dimensions. Therefore, it appears to be more adapted for cell manufacturers than for system integrators.

Li et al. [START_REF] Li | Optimal cell tab design and cooling strategy for cylindrical lithium-ion batteries[END_REF] developed an 3D electro-thermal model based on two coupled equivalent circuit networks to model the electrical and thermal behaviors. The electrical model is non-physical and is aimed at modeling the electrical behavior at a local scale. It only requires non-invasive measurements and it is interesting for studying and improving the design of an existing cell, but this model also cannot predict the behavior of a cell with different dimensions.

To help design a battery-powered system, a model that can predict a cell performance as a function of its dimensions would be very useful. It has to be simple enough to be integrated in the calculation of the whole system behavior. It also has to be accurate enough to study the interactions between the battery and other components. A valuable feature is to use a model that can be built with very little knowledge of the studied cell, as it is often the case for system integrators.

This study contributes to the creation of such a model, with an approach that links each parameter of an ECM to the dimensions of the cell electrodes. We have chosen to model the electrical behavior of the batteries with an ECM because it is adapted to a user with no access to electrochemical parameters and because its parameters can be related to the electrochemical phenomena [START_REF] Von Srbik | Offer, A physically meaningful equivalent circuit network model of a lithium-ion battery accounting for local electrochemical and thermal behaviour, variable double layer capacitance and degradation[END_REF][START_REF] Damay | Separation of the charge transfers and solid electrolyte interphase contributions to a battery voltage by modeling their non-linearities regarding current and temperature[END_REF]. By nature, this approach can be applied to any insertion battery in which the same phenomena occur (lithium-ion, sodium-ion...).

In line with the context of integrating a battery into a microsystem, we chose to study three commercial lithiumion coin cells of the same design and chemistry (NMCgraphite, 3.7 V for nominal voltage, but of different dimensions and nominal capacities (25, 60 and 120 mAh). They have already been studied by Nehru Bandla [START_REF] Bandla | Modeling the internal inhomogeneous aging behavior in large-format commercial Li-ion batteries[END_REF] with an electrochemical modeling approach. They are cylindrical cells with coiled electrodes (Figure 1). Their height is the same, but their diameter changes. Their electrodes therefore have the same characteristics, except for their lengths. Thus, the proposed study is focused only on modeling and validating the effects of electrodes length on the parameters of an ECM. Nevertheless, the fact that the proposed approach is based on physics makes it relevant for modeling the dependencies of the parameters to other electrodes dimensions.

In the following section, we describe the detailed ECM that we used to model the cells. For each of its parameters, we then propose a dependency-model which predicts the evolution of the parameter with respect to the electrodes length. Because the detailed ECM parameters cannot be determined experimentally in a non-invasive way, we explain in section 3 how the detailed ECM is reduced into a simplified ECM that can be used at a system scale. We then develop the dependencies of the simplified-ECM parameters on the electrodes length. In section 4, the determination of the simplified ECM parameters is detailed and is validated for operating conditions that are typical of a micro-system. In section 5, we evaluate our dependencymodel on the electrodes length thanks to the previously determined ECM parameters.

Constructing an equivalent circuit model depending on electrodes dimensions

Defining a detailed ECM at electrodes scale

In order to define the relationships between electrodes dimensions and the parameters of an ECM, the cell is subdivided into its main active components, namely the conductive assemblies that can bring the electrons from the cell terminals to the interfaces between the active materials and the electrolyte (Figure 2). The solid electrolyte interphase (SEI) is represented as part of the negative electrode, between the active material and the electrolyte. The electrodes height and length are also represented on Figures 1 and 2 and the thickness corresponds to the third dimension (along the vertical axis). Please note that this cell diagram corresponds to the cell electrodes we used to experimentally validate the model, that is a long ribbon with only one tab at each end (Figure 1), and must therefore be adapted for cells with a different electrodes design.

On Figure 2, a detailed ECM of this cell is superimposed on the different cell elements. Because we chose a localized constant model, it is assumed that the composition and behavior of each element are homogeneous . Consequently, the current that is naturally spread out over the entire electrodes surfaces (Figure 1c) can be reduced to an equivalent mean current that corresponds to the average path of the electrical charges (ions and electrons). Along this path, and at the boundary between each of the cell elements, is placed a node (black dot on the figure). The electrical components between two nodes model the relationship between the flow of electrical charges and their potential difference. These components values are nonlinear and may vary with the current, the temperature, the state of charge, the state of health and the dimensions of the electrodes [START_REF] Eddahech | Ageing monitoring of lithium-ion cell during power cycling tests[END_REF]. Each node is at the potential of the interface between two elements of the cell. This interface is then assumed to be equipotential. A consequence of this assumption is that the voltage drop in the direction of the electrodes length L must be low, which will be verified in the experimental part (section 5). The resistors R connec,n and R connec,p model the electronic conduction phenomena in the electrical connection between each of the current collectors and the terminals of the cell. This electrical connection can contain soldered elements and include contact resistors.

Mean current 𝑰

The current collector and the active material of each electrode are grouped in a "conductive assembly". The two resistors in one assembly model the electronic conduction in the collector and the active material, the contact resistance between the collector and the active material, and the contact resistances between the particles of the active material [START_REF] Farkhondeh | Full-range simulation of a commercial LiFePO 4 electrode accounting for bulk and surface effects: A comparative analysis[END_REF][START_REF] Illig | Modeling graphite anodes with serial and transmission line models[END_REF]. The resistors R CA:L,n and R CA:L,p model the electronic transport in the "conductive assemblies" in the direction of electrodes length. The resistors R CA:T,n and R CA:T,p model the electronic transports in the conductive assembly along the thickness of the electrodes.

The interface model is an adaptation of the model proposed by Kuhn et al. [START_REF] Kuhn | Modeling diffusive phenomena using non integer derivatives[END_REF]. It models the relationship between the current and the potential difference between the electrode surface and the electrolyte. The voltage sources U n and U p correspond to the equilibrium potentials of the electrodes with respect to a reference potential (generally a lithium metal electrode). These equilibrium potentials depend on the lithium ion concentration of each of the electrodes. The equivalent resistances R ct,n and R ct,p model the potential drop caused by the charge transfer, a phenomenon linked to the insertion and deinsertion of ions in the active material. The capacitors C dl,n and C dl,p model the accumulation of charges on both sides of the electrolyte/active material interface called double layer. The components Z d,n and Z d,p model the overvoltage due to the effect of solid diffusion in the active material particles [START_REF] Huang | An analytical three-scale impedance model for porous electrode with agglomerates in lithium-ion batteries[END_REF]. This overvoltage does not rise instantaneously and its dynamics is characterized by a time constant τ d . When the current is constant, the overvoltage tends to a value that no longer depends on time and can therefore be considered as equal to the product R d I, where I represents the current and R d is a quantity that is homogeneous to a resistance and is dependent on the state of charge, the current, and the temperature [START_REF] Damay | Thermal modeling of large prismatic LiFePO 4 /graphite battery. Coupled thermal and heat generation models for characterization and simulation[END_REF].

The R SEI and C SEI components model the phenomena associated with ion transport in the SEI present on the negative electrode. This models the accumulation of charges on both sides of the SEI (capacitive phenomenon) and the migration of ions (resistive phenomenon) [START_REF] Meyers | The impedance response of a porous electrode composed of intercalation particles[END_REF][START_REF] Delacourt | Analysis of lithium deinsertion/insertion in LiyFePO 4 with a simple mathematical model[END_REF], or it would be both the diffusion and the migration of ions [START_REF] Dees | Alternating current impedance electrochemical modeling of lithiumion positive electrodes[END_REF][START_REF] Huang | An analytical three-scale impedance model for porous electrode with agglomerates in lithium-ion batteries[END_REF].

The resistance R e models the ionic conduction in the electrolyte. The impedance Z d,e models ion diffusion in the electrolyte [START_REF] Mauracher | Dynamic modelling of lead/acid batteries using impedance spectroscopy for parameter identification[END_REF].

Some models include an inductance in the electrical circuit modeling the inductive phenomena in the cell [START_REF] Esser | Electrochemical impedance spectroscopy setup based on standard measurement equipment[END_REF]. However, the inductive effects are often negligible, unless the cell is used at high frequencies. We will demonstrate in part 4.3 that the model without inductance describes satisfactorily the electrical behavior of the studied cells.

Dependencies of the detailed model parameters to the electrodes length

In this section, the relationships between the electrodes length and the detailed ECM parameters are justified and detailed. The summary of these relationships can be found in Table 1. Please note that we used the time constants τ of the RC circuits instead of the C parameters.

It is assumed that the electrode potentials U n and U p depend on the charge state only, expressed as a percentage. They are therefore independent of the electrodes length.

The time constants τ d,n , τ d,p , and τ d,e of the impedances modeling the diffusion effects are intensive quantities at electrodes scale if the electrodes have the same composition and thickness, which is the case here. Consequently, their values are assumed to be independent of electrodes length.

In steady state, the impedances Z d,e , Z d,n and Z d,p modeling each of the three diffusion phenomena are equal Parameter Relationship to the electrodes length L

Modified property Equation

U i none constant R CA:L,i length of the path ∝ L R CA:T,i cross section ∝ 1/L R e cross section ∝ 1/L R SEI cross section ∝ 1/L τ SEI none constant R connec,i connection configuration constant R ct,i exchange surface ∝ 1/L τ ct,i none constant R d,i exchange surface ∝ 1/L τ d,i
none constant to a resistance R d,i (i replacing any of the three indices n, p and e). From Ohm's relation we have Equation (1):

U d,i = R d,i Sj (1)
with U d,i the voltage across the considered resistance R d,i , S the exchange surface, and j the current density normal to the surface S. In the case of the model shown in Figure 2, if we multiply the electrodes length by a factor α, the area of the electrodes is multiplied by the same factor. The voltage U d,i is assumed to be an intensive quantity that does not depend on the electrodes length L. Keeping the current density j unchanged, the voltage U d,i therefore does not change and the resistance R d,i must be divided by the factor α to maintain equality. This brings Equation (2):

U d,i = R d,i α (Sα)j (2)
It can be seen here that a multiplication of the electrodes length by a factor α therefore results in a division of the resistance R d,i by the same factor. The resistances R d,n , R d,p and R d,e are therefore inversely proportional to the electrodes length L.

The reasoning presented above for the impedances modeling the diffusion effects can be used with the circuits R ct,n C dl,n , R ct,p C dl,p and R SEI C SEI because their impedances tend to the value of the corresponding resistance in steady state. Besides, the voltage at their terminals and their time constants are assumed to be intensive quantities. We concluded that these resistances are inversely proportional to the electrodes length L and that these time constants do not depend on L as presented in Table 1.

Assuming a linear behavior, the value of the resistances modeling the phenomena of electronic and ionic conduction can be calculated by the Equation (3) in which l is the length of material traversed by the charges flow (ions or electrons), S is the cross-section of the material traversed by this same flow, and ρ is the conductivity (electronic or ionic) of the traversed material.

R conduction = ρ l S (3)
For the resistance R e , the cross-sectional area S is proportional to the electrodes length L and the dimension l corresponds to an equivalent distance of electrolyte between the two electrodes. This dimension and the conductivity ρ are constant with respect to the electrodes length L. The resistance R e is then assumed to be inversely proportional to the electrodes length L.

In addition to conduction phenomena, the resistances R CA:T and R CA:L also model contributions from contact resistances. The contact resistance is a conduction phenomenon located at the interface of two materials. It depends on the contact area S between the materials and a coefficient ρ * related to the contact characteristics (Equation ( 4)). This modeling assumes that the current through the contact surface is uniform, which is an assumption already made by using localized constant modeling (see section 2.1).

R contact = ρ * S ( 4
)
For the resistances R CA:T,n and R CA:T,p , the area S of Equations ( 3) and ( 4) is proportionnal to the electrodes length L. The other quantities in these equations were assumed to be independent of L, so the resistances would be inversely proportional to the electrodes length L.

For the resistances R CA:L,n and R CA:L,p , the electrodes length L is proportional to the length l of Equation (3). The contact resistance also increases proportionally to the electrodes length L because the particles and the number of contacts between them increases linearly with the electrodes length. The other quantities in these equations were assumed to be independent of the electrodes length L, so these resistances would be proportional to it.

The dimensions of the elements modeled by the connection resistances R connec,n and R connec,p may not have a simple relationship with electrodes length because the cell manufacturer may change the electrodes tab from one cell version to another. For the sake of simplicity, we assumed that the sum of these resistances where constant according to the electrodes length. These resistances indeed include electronic conduction phenomena, which we supposed negligible, and contact resistance phenomena, which we considered constant regarding electrodes length. This is summed up by Equation ( 5): [START_REF] Mei | The effect of electrode design parameters on battery performance and optimization of electrode thickness based on the electrochemical-thermal coupling model[END_REF] with R cond an equivalent resistance grouping the resistances related to electronic conduction, and R cont an equiv-alent resistance grouping the contact resistances.

R connec,n (L) + R connec,p (L) = R cond (L) ≈ 0 + R cont (L) = constant
To estimate the value of R cond , we assumed that the conduction resistances were mainly due to the resistance of the electrical protection element of the cells, located between a collector and the case which is used as a terminal. The picture of this protection element can be found in the manufacturer's technical handbook [22, p. 42]. Based on its geometry and assuming it is made of aluminum, we estimated its resistance to be 2.2 mΩ. As this value is small compared to the value of R s (several hundred mΩ, see part 4.2.1), it can be neglected.

For the resistance R cont , we assumed that its value does not vary from one cell to another. The manufacturing process should indeed be the same for the three cells, and the dimensions of the cases do not change much. We therefore considered R cont to be constant with respect to the electrodes length (Equation 5).

Electrical circuit model simplifications for characterization and simulation

The detailed ECM described in section 2.1 contains parameters that cannot be determined using only the cell voltage and current measurements. Indeed, these are global parameters that make it difficult to distinguish the contributions of the different phenomena characterizing the electrical behavior of the cell. We therefore simplified the detailed ECM as explained in the following subsections.

Grouping resistances in series

Contact resistances and electronic and ionic conduction phenomena are modeled by resistances. Each each of these resistors causes an instantaneous voltage drop. However, it is not possible to distinguish their individual contribution by measuring the voltage across the cell. We have therefore grouped them into a single resistor noted R s (Equation ( 6)).

R s = R connec,n + R connec,p + R CA:L,n + R CA:L,p + R e + R CA:T,n + R CA:T,p (6)

Circuit topology simplification

The phenomena modeled by the interface model do not have the same dynamics: the time constant of the R ct C dl circuits is generally lower than a second, the time constant of the diffusion is of the order of a minute, and the electrode potentials U n and U p are instantaneously deduced from the charge state. As the dynamics of these elements are significantly different, they can be put in series as shown in Figure 3a [START_REF] Kuhn | Modeling diffusive phenomena using non integer derivatives[END_REF].

Grouping electrodes equilibrium potentials

Due to the change in topology of the interface model, the voltage sources U n and U p are in series. They can therefore be grouped into a single voltage source that we will note U c . It represents the open circuit voltage of the cell. The voltage sources being defined in opposite directions (Figure 2), the voltage U c is equal to their difference (Equation ( 7)).

U c = U p -U n (7)

Grouping similar dynamics

The contributions of the three RC circuits modeling the transport of ions through the SEI and the electrode/electrolyte interface are difficult to identify individually by a non-invasive approach. Kuhn et al. [START_REF] Kuhn | Modeling diffusive phenomena using non integer derivatives[END_REF] showed that a single RC circuit can satisfactorily model these three RC circuits. We have therefore chosen to use only one RC circuit characterized by a time constant τ surf and a resistance R surf (Figure 3b), which is equal to the sum of the resistances of the three RC circuits (Equation ( 8)).

R surf = R ct,n + R ct,p + R SEI (8)
As for the RC circuits, the contributions of the impedances Z d,n , Z d,p and Z d,e cannot be differentiated. We have therefore grouped them in a single impedance Z d , which can be represented by a limited diffusion impedance, noted Z d in Figure 3c [START_REF] Jossen | Fundamentals of battery dynamics[END_REF][START_REF] Damay | Modélisation des non-linéarités en courant et en température d'un circuit électrique équivalent de batterie en vue de sa caractérisation rapide[END_REF]. It is described in the Laplace domain by Equation ( 9) involving a so-called "diffusion time constant" τ d and a so-called "diffusion resistance" R d equal to the sum of the diffusion resistances

R d,n , R d,p and R d,e . Z d (s) = R d tanh √ τ d s √ τ d s ≈ Z d,n + Z d,p + Z d,e (9)

Discretization of the diffusion impedance

Equation ( 9) can be approximated by a finite sum of RC circuit impedances described by equation [START_REF] Bandla | Modeling the internal inhomogeneous aging behavior in large-format commercial Li-ion batteries[END_REF] and shown in Figure 3d [START_REF] Damay | Modélisation des non-linéarités en courant et en température d'un circuit électrique équivalent de batterie en vue de sa caractérisation rapide[END_REF].

Z d (s) ≈ N ∈N * p=1 R d,p 1 + sτ d,p (10)
In Equation [START_REF] Bandla | Modeling the internal inhomogeneous aging behavior in large-format commercial Li-ion batteries[END_REF], R d,p is calculated from R d using equation [START_REF] Eddahech | Ageing monitoring of lithium-ion cell during power cycling tests[END_REF], and τ d,p is calculated from τ d using [START_REF] Farkhondeh | Full-range simulation of a commercial LiFePO 4 electrode accounting for bulk and surface effects: A comparative analysis[END_REF].

R d,p = 8R d π 2 (2p -1) 2 (11) τ d,p = 4τ d π 2 (2p -1) 2 (12)
Please note that the diffusion resistance R d is also equal to the sum of the R d,p in Figure 3d, as stated by Equation [START_REF] Illig | Modeling graphite anodes with serial and transmission line models[END_REF].

R d = N ∈N * p R d,p (13)

Result of the simplifications

The simplifications made to the detailed model reduce the number of parameters from 21 to 6: U c , R s , τ surf , R surf , τ d and R d . These parameters can be non-invasively determined as presented in section 4.2. The simplified ECM is shown in Figure 4.

Dependency to electrodes length

Using Table 1 and the definition of R s in equation ( 6), the resistance R s would be the sum of terms whose values (i) are independent of the electrodes length (R connec,n and R connec,p ), (ii) are proportional to the electrodes length (R CA:L,n and R CA:L,p ), and (iii) are inversely proportional to the electrodes length (R e , R CA:T,n , R CA:T,p ). The resistance R s would thus be related to the length L of the electrodes by equation ( 14) in which k s1 , k s2 and k s3 are coefficients expressed in Ω m -1 , Ω m and Ω, respectively.

R s (L) = k s1 L + k s2 L + k s3 (14)
We have defined the resistances R surf and R d as the sum of resistances whose values were supposed to be inversely proportional to the electrodes length L. They would then themselves be inversely proportional to L, as shown in Equations ( 15) and ( 16), in which k surf and k d are coefficients expressed in Ω m.

R surf (L) = k surf L (15) R d (L) = k d L ( 16
)
The time constant τ surf represents the dynamic of the R surf C surf circuit modeling the R ct,n C dl,n , R ct,p C dl,p , and R SEI C SEI circuits (Figure 3b). We have made the assumption that the dynamics of these circuits are independent of the electrodes length, so the dynamics of the equivalent RC circuit should be independent of the electrodes length. The time constant τ surf would therefore be equal to a constant that we note τ * surf (Equation ( 17)).

τ surf (L) = τ * surf ( 17
)
Using the same reasoning and remembering that the time constants τ d,n , τ d,p , and τ d,e were assumed to be independent of the length of the electrodes, the time constant τ d should be independent of the length of the electrodes. It would therefore be equal to a constant that we will note τ * d (Equation 18).

τ d (L) = τ * d ( 18
)
The voltage source U c depends on the relative SOC (state of charge), expressed as a percentage of the cell capacity. Consequently, this term scales naturally with the electrodes length and can be considered to be constant regarding L.

In order to validate the proposed modeling of dependencies to the electrodes length described in this section, the simplified ECM parameters have to be determined for the three studied cells. This is described in the next section. Then, in section 5, the dependency-models are fitted to these ECM parameters to demonstrate the validity of Equations ( 14) to [START_REF] Delacourt | Analysis of lithium deinsertion/insertion in LiyFePO 4 with a simple mathematical model[END_REF].

Experimental determination of the ECM parameters

The experimental setup used for all the experimental work is presented in subsection 4.1. The experimental determination of the ECM parameters for the three studied cells is made thanks to EIS (electrochemical impedance spectroscopy) and GITT (galvanostatic intermittent titration technique) tests, as explained in subsection 4.2. These parameters are validated in subsection 4.3 by demonstrating that they accurately predict the cells electrical behaviors.

For this experimental work, we limited the influence of the SOC non-linearities affecting the electrical parameters by restraining the range of state of charge studied between 20 and 26.7 % of depth of discharge. In this range, we could consider that only the parameters U c and R d vary significantly with the state of charge, which is justified in subsection 4.2 below. In addition, even though the temperature influences the ECM parameters [START_REF] Tran | A comprehensive equivalent circuit model for lithium-ion batteries, incorporating the effects of state of health, state of charge, and temperature on model parameters[END_REF], discharges at 2 C in a climate chamber set at 20 °C showed that the cells temperature increased by at most 2 °C in these conditions. This temperature rise does not significantly change the electrical properties of the cells, so we considered their temperature to be constant during the tests. Furthermore, the studied cells had only had a few preliminary tests before their characterization, and we therefore considered them as new cells.

Experimental setup

The cells used for the experimental validation are commercial lithium-ion coin cells of the Varta brand. The materials of their negative and positive electrodes are graphite and LiNi 0.6 Mn 0.2 Co 0.2 O 2 respectively [START_REF] Bandla | Modeling the internal inhomogeneous aging behavior in large-format commercial Li-ion batteries[END_REF]. The names of the models used, their dimensions and their nominal capacities are given in Table 2. The nominal voltage of the cells is 3.7 V, and their operating voltage is between 3 and 4.2 V. We performed the electrical measurements and controls, including impedancemetry, with a Biologic system (VSP-300 with 2 A and 20 A boosters). A Biologic BH-1i battery holder allowed a 4-wire measurement of the cells voltages. For all tests, the current was defined as a current rate (in C), which is the measurement of the charge and discharge current with respect to its nominal capacity. The depths of discharge are expressed as a percentage and are also based on the nominal capacities because the measured capacities were close to them (difference of less than 3 %). Lastly, during all the experiments, the cells were placed in a climate chamber set at 20 °C.

ECM parameters determination

The EIS and GITT tests used to determine the ECM parameters are presented below.

EIS test: determination of R s

The series resistance R s is assumed not to depend on the current or on the depth of discharge. It is extracted from an impedance spectrum because it is difficult to distinguish its contribution from the one of R surf with the GITT tests (described in part 4.2.2). The impedance measurements used for the determination of R s were all performed with a sinusoidal signal of amplitude 0.1 C and a frequency between 0.01 Hz and 100 kHz with 5 points per decade in logarithmic spacing (Figure 5). The parameter R s is measured as the minimum value of the real part of the spectrum because the other parameters of the model can only increase the real part of the impedance [START_REF] Skoog | Parameterization of linear equivalent circuit models over wide temperature and SOC spans for automotive lithium-ion cells using electrochemical impedance spectroscopy[END_REF].

GITT test: experimental protocol

All ECM parameters except R s are determined using the pulsed discharge protocol below, called GITT. GITT tests are complementary to EIS tests because they can characterize a cell behavior at higher currents and because they are more efficient to characterize the slow dynamics [START_REF] Schmidt | Pulse-fitting -A novel method for the evaluation of pulse measurements, demonstrated for the low frequency behavior of lithium-ion cells[END_REF]. The test protocol is as follows:

1. CC/CV charge at 0.5 C to 4.2 V, 0.02 C; 2. open circuit for 3 h; 3. discharge at 0.5 C for 8 min (6.7 % of the nominal cell capacity); 4. open circuit for 3 h; 5. return to step 3 until the low voltage limit of the cell (3 V) is reached (13 complete pulses were completed).

For the purpose of the experimental validation presented in subsection 4.3 and because the parameters may vary with respect to the current, we have also realized GITT pulses of 4 min at 1 C in addition to the 8 min pulses at 0.5 C described above. The ECM parameters have hence been determined at 0.5 and 1 C. The test protocol was the same, except for the duration and intensity of the pulses. The variation in depth of discharge was the same in both cases.

The duration of the pulses was chosen so that the overvoltage reaches a quasi-steady state for each pulse [START_REF] Damay | Thermal modeling of large prismatic LiFePO 4 /graphite battery. Coupled thermal and heat generation models for characterization and simulation[END_REF]. This state is reached when the voltage response of a GITT test at a given current becomes equal to that of a steady state discharge at the same current, state of charge and temperature.

The GITT discharge was performed on all the capacity range of the battery, but we only used pulses 3 and 4 to extract the parameters at the studied depth of discharge (20-26.7 %). The parameters are extracted in two steps: U c is determined first, and then the others.

GITT test: determination of the OCV U c

The open circuit voltage U c depends on the depth of discharge and it is determined by measuring the cell voltage at the end of the 3-hour-long open circuit periods during the GITT tests [START_REF] Roscher | Dynamic electric behavior and open-circuit-voltage modeling of LiFePO 4 -based lithium-ion secondary batteries[END_REF]. These tests only allows the determination of OCV values at certain depths of discharge. To determine the value of U c between the measured values, we assumed that this parameter varied linearly with respect to the state of charge between two measured values. The validity of this assumption can be appreciated in Figure 6 in which the measured and interpolated OCV is compared to a pseudo-OCV measured at 0.02 C.

The discharge voltage at 0.02 C is close to the equilibrium voltage and highlights its variations, which are not always visible at higher current rates. The pseudo-OCV at 0.02 C is linear for pulses 3 and 4, so the linearity assumption is correct for these pulses. Concerning the other responses to GITT pulses, this assumption may not be true enough and the ECM parameters determination would be then less accurate. Because the scope of this study is to test the modeling of the parameters dependencies to electrodes length, we only used the pulses 3 and 4 to have the most accurate parameters values.

GITT test: determination

of R surf , τ surf , R d , τ d
With the exception of U c , the parameters of the ECM characterize the overvoltage η, which is the difference between the actual cell voltage and its equilibrium voltage U c . For a constant current I and an initial value of zero volts, the overvoltage η sim predicted by the ECM is given by Equation [START_REF] Dees | Alternating current impedance electrochemical modeling of lithiumion positive electrodes[END_REF]. The diffusion impedance Z d is described by N = 10 RC circuits (see Equation [START_REF] Bandla | Modeling the internal inhomogeneous aging behavior in large-format commercial Li-ion batteries[END_REF] and Figure 4). The overvoltage η sim was calculated using the MATLAB software.

η sim (t) = I R s +R surf exp -t τ surf + N =10 i=1 R d,i exp -t τ d,i (19)
The parameters in Equation ( 19) can vary with the current and the depth of discharge. For the studied cell, we assumed that the parameters R surf , τ surf and τ d do not depend on the depth of discharge, but depend on the current I. Their value is thus different at 0.5 C and 1 C, but they are assumed to be the same for the two studied pulses (3 and 4). The semi-ellipse of the impedance spectra in Figure 5 are indeed characteristic of the R surf and τ surf parameters and are very close over the considered depth of discharge range [START_REF] Illig | Understanding the impedance spectrum of 18650 LiFePO 4 -cells[END_REF]. The value of the diffusion resistance R d is assumed to be dependent on the current and on the depth of discharge.

Because the value of R d can only be accurately determined at the end of the GITT pulses, we chose to fit three values that represent the value of this parameter at different depths of discharge [START_REF] Damay | Thermal modeling of large prismatic LiFePO 4 /graphite battery. Coupled thermal and heat generation models for characterization and simulation[END_REF]. Hence, R 0 d , R 1 d , R 2 d represent respectively the value of R d at 13.3 % (beginning of pulse 3), 20 % (end of pulse 3 and beginning of pulse 4), and 26.7 % (end of pulse 4). The value of R d is then calculated as a linear interpolation between R 0 d , R 1 d and R 2 d . In this context, the ECM parameters are obtained by fitting Equation [START_REF] Dees | Alternating current impedance electrochemical modeling of lithiumion positive electrodes[END_REF] to the measurements from GITT. More specifically, the optimal parameter set θ * is determined by adjusting the parameter set θ = (R surf , τ surf , R 0 d , R 1 d , R 2 d , τ d ) using the ordinary least squares method as described in Equation [START_REF] Mauracher | Dynamic modelling of lead/acid batteries using impedance spectroscopy for parameter identification[END_REF]. The adjustments were performed with the MATLAB software and the fmincon function.

θ * = arg min θ M3 i=1 η meas 3 (t i ) -η sim 3 (t i , θ) 2 error of pulse 3 + M4 i=1 η meas 4 (t i ) -η sim 4 (t i , θ) 2 error of pulse 4 subject to θ > 0 τ surf < τ d 0.95R 0 qs < R s + R surf + R 0 d < 1.05R 0 qs 0.95R 1 qs < R s + R surf + R 1 d < 1.05R 1 qs 0.95R 2 qs < R s + R surf + R 2 d < 1.05R 2 qs ( 20
)
In Equation ( 20), η meas 3 (t i ) indicates the measured overvoltage value of pulse 3 at time t i , which is calculated by subtracting the estimated OCV U c at this time to the measured voltage ("GITT" in Figure 6). The term η meas the same, but for pulse 4. The adjustment was made on two pulses since the value of the parameter R d is not well estimated at the beginning of pulse 4 fitted alone. The fitting with both pulses 3 and 4 allows to obtain a correct value by imposing the continuity of the parameter R d with respect to the state of charge. It was then necessary to include pulse 3 in the fitting. Please note that the value of R 0 d (parameter R d at the beginning of pulse 3) is only here for mathematical consistency and that it is not meant to be used. It was thus discarded after the adjustment and only R 1 d and R 2 d are kept. To guide the optimization process represented by Equation [START_REF] Mauracher | Dynamic modelling of lead/acid batteries using impedance spectroscopy for parameter identification[END_REF], a constraint is added to the sum R s + R surf + R d which we call quasi-steady state resistance R qs . R qs represents the equivalent resistance of the cell when it reaches a quasi-steady state (see Equations ( 19) and ( 13)). The resistance R qs is measured by dividing the value of the overvoltage at the ends of the pulses by the current. As stated in Equation ( 20), the optimization algorithm only allowed a variation of 5 % around this value.

ECM parameters validation 4.3.1. Protocol

To validate the ECM and the values of its parameters, we used a discharge made of short pulses which is typical for microsystems powering [START_REF] Quelin | Experimental performance analysis of an electromagnetic impact-drive microrobot[END_REF][START_REF] Zhang | Characterization of battery behavior for battery-aware design of wireless sensing networks[END_REF][START_REF] Zhang | SolarPede: a stick-and-slip, light-powered, mobile micro-crawler[END_REF]. The discharge protocol is represented on Figure 7(a): a 40-ms-long current discharge at 1 C is followed by a 10-ms-long rest period, repeated 7200 times (6 min), which covers a depth of discharge range of 8 %. This protocol is also well adapted for the validation because it highlights the fast phenomena thanks to the short discharge pulses and the slow phenomena (diffusion) thanks to the prolonged repetition of the short discharges creating a significant mean discharge current. Before the cells were discharged with this protocol, they were charged according to the charging protocol specified in section 4.1 and then discharged for 24 min at 0.5 C to start with a depth of discharge of 20 %. A 3 h open circuit pause was performed at the beginning and at the end of the test to estimate the equilibrium voltage at those points using the method described in part 4.2.2. This protocol was used on the three studied cells.

Results and discussion

The equilibrium voltages determined with the GITT tests and those determined with the above-mentioned protocol are close, as their difference is 5 mV at most, which is small compared to the measured voltage (≈ 4 V, Figure 6). We can thus consider that the OCV U c is well estimated on this depth of discharge.

To validate the five remaining parameters (R s , R surf , τ surf , R d , τ d ), we compared the measured and simulated overvoltages (Figure 7(b) and (c)). The curves are similar for the three cells, so only the curves for the 60 mAh cell are presented. The simulations were stopped at the time corresponding to 26.7 % of depth of discharge, because the validation tests cover a depth of discharge range between 20 and 28 % and the parameters for the simulations were determined for the range between 20 and 26.7 % (pulse 4).

The overvoltages on Figure 7 has been simulated using the following parameters values. The value of the series resistance R s has been taken from the EIS test (see part 4.2.1). The values of the surface parameters τ surf and R surf are dependent on the current, which is 1 C during the discharge pulses. We have therefore taken for the simulation the values from the GITT test at 1 C (see part 4.2.4).

The parameters τ d and R d are also dependent on the current, but here we used the average discharge current which is 0.8 C. As we did not have a GITT test at 0.8 C, we assumed that the evolution of the parameters was linear between 0.5 and 1 C currents rates. The contribution of the diffusion effects to the overvoltage would indeed be related to the concentration gradients of lithium ions in the electrolyte and of lithium atoms in the solid phase [START_REF] Raël | Using electrical analogy to describe mass and charge transport in lithium-ion batteries[END_REF]. The settling time of these gradients is of the order of ten seconds in our case (related to the diffusion time constant τ d ). We therefore assumed that the gradients were not directly influenced by the short (tens of ms) current pulses and that they were related to the mean current value (0.8 C).

In Figure 7(b) are presented the sliding averages (SA) of the overvoltages estimated from the measurements and two simulations: (i) all parameters determined at 1 C (sim 1 C) and (ii) all parameters determined at 1 C except diffusion parameters determined at the mean current: 0.8 C (sim 0.8 C). The sliding averages, calculated over a 1 s window, make it easier to see the evolution of the overvoltage over long times (Figure 7(b)). This allows a better evaluation of the diffusion impedance Z d contribution. As expected, the "sim 0.8 C" curve correctly follows the average evolution of the measurements. The "sim 1 C" curve correctly follows the average evolution of the measurements until about 100 s; beyond that, it diverges. This validates our hypothesis on using the mean current for the determination of the diffusion parameters values, and also validates the determination of the diffusion resistance R d and time constant τ d .

The parameters R s , R surf and τ surf are also well determined because the simulations correctly follow the trend of the measurements on the short pulses (Figure 7(c)). The sum R s + R surf is indeed related to the voltage amplitude during these short times, and τ surf describes the fast dynamic seen for these pulses.

Using the protocol described in part 4.3.1 and inspired by the powering of a microsystem, the ECM parameters are validated for the three studied cells on a specific operating range: temperature of 20 °C, limited depth of discharge (between 20 and 26,7 %) and current rate. Other operating points were not studied because this work is focused on the electrodes length dependency only. The validation was then carried for the three cell sizes (three electrodes lengths), leading to root mean square errors of 3.98, 4.71 and 5.13 mV for the 25, 60 and 120 mAh cells respectively. These errors are at most 5 % of the quasi-steady state overvoltage (about -120 mV for the sliding average), which is reasonable. Based on all the above results, we consider that the characterization method brings accurate parameters values. The ECM parameters are then valid at 1 C, and we assume that they are also valid at 0.5 C because the characterization method is the same.

Validation of the dependency-models to the electrodes length

The parameters validated in the previous section are now used to test the dependency-models of the parameters on the electrodes length using. The dependency-models were validated with the ECM parameters determined at 0.5 C and 1 C, but only the results at 1 C are detailed for the sake of clarity.

Determination of the electrodes lengths

We determined an electrodes length of 320 mm for the 60 mAh cell by measuring with an image analysis software (ImageJ) the length of the winding visible on the microtomography in Figure 1. As we did not have microtomographies for the two other cells, we had to estimate their electrodes lengths. For the same electrode material, the capacity is proportional to the amount of active material. Since the electrodes of the three cells have the same height and thickness, their capacitance is therefore proportional to the length of their electrodes. Using the electrodes length of the 60 mAh cell and the measured cell capacities (Table 2), we calculated an electrodes length of 133 mm for the 25 mAh cell and 656 mm for the 120 mAh.

Determination of the dependency-models parameters

Using the ECM parameters determined for each of the three Varta coin cells, we could fit the proposed dependencymodels to these ECM parameters. Hence, the dependencymodel parameters of R surf (Equation ( 15)), R d (Equation ( 16)), τ * surf (Equation ( 17)), τ * d (Equation ( 18)) and R s (Equation ( 14)) were determined using the ordinary least squares method presented in Equation [START_REF] Esser | Electrochemical impedance spectroscopy setup based on standard measurement equipment[END_REF].

γ * = arg min γ M L i=1 y meas (L i ) -y sim (L i , γ) 2 subject to γ > 0 (21)
In Equation ( 21), γ * is the optimal parameter set, y meas (L i ) is the measured electrical parameter for a cell with electrodes length L i , y sim (L i , γ) is the simulated electrical parameter parameterized by γ for the same cell, and M L is the number of electrodes lengths (M L = 3 in this study). The parameters have all been constrained to be positive in order to comply with physics laws.

Results and discussion

The graphs in the Figure 8 show the measured ECMparameters and the fitted dependency-models as functions of the electrodes length L. As the values of the ECMparameters change with the current rate, and because we determined them at 0.5 and 1 C, we show the results at these two current rates. The parameter values of each of the dependency-models are indicated in the graphs. The parameters k 1 d and k 2 d correspond to the k d parameter for R 1 d and R 2 d respectively. The value of these two resistors are close, and the parameters k 1 d and k 2 d are therefore close. The MAPE (mean absolute percentage errors) is also indicated on the graphs. The MAPE is defined by Equation ( 22) which uses the same conventions as in Equation [START_REF] Esser | Electrochemical impedance spectroscopy setup based on standard measurement equipment[END_REF]. The factor 100 gives a MAPE in percent. The MAPE is used in this article to compare the errors of parameters which are not of the same order of magnitude.

MAPE = 100 M L M L j=i y meas (L i ) -y sim (L i , γ) y meas (L i ) (22)
All dependency-models correctly describe the evolution of the ECM-parameters with respect to the electrodes length with only one parameter (except R s , three parameters). The R s dependency-model does not perfectly interpolate the measurements, although we have three measurements for three parameters to determine. The value of the parameter k s1 allowing a perfect interpolation would be slightly negative, which we have forbidden during optimization. However, the small value of k s1 found by fitting is in agreement with the assumption of equipotentiality of the surfaces implying that the voltage drop in the collectors is negligible. It would therefore be possible to remove the k s1 L term from Equation [START_REF] Kuhn | Modeling diffusive phenomena using non integer derivatives[END_REF]. Consequently, two cells would be sufficient to calibrate the dependency-models and use them to predict the electrical behavior of a cell with any electrodes length. Even though the evolution of the time constant τ surf is satisfactorily described, the value of this parameter slowly decreases linearly with the electrodes length for both current rates. This behavior is not predicted by the dependencymodel and is possibly due to modeling limitations. First, the R surf C surf circuit models the phenomena at the origin of the semi-ellipse visible on the impedance spectra (Figure 5), but this semi-ellipse is not perfect and presents a depression that is better modeled by one or more R-CPE circuits [START_REF] Illig | Physically based impedance modelling of lithium-ion cells[END_REF]. Second, the influence of electrodes curvature was neglected in the dependency-models. Wang et al. [START_REF] Wang | Fundamental understanding and rational design of high energy structural microbatteries[END_REF] showed that the curvature of the electrodes influenced the impedance of the cells they used. They attribute this to the mechanical constraints undergone by the electrodes due to their shaping. In our case, the radius of curvature at the periphery of the cells is not the same for the three cells, which could explain the different impedances and the evolution of τ surf .

Overall, the proposed approach allows for a satisfactory modeling of ECM parameters dependencies on the electrodes length. Even though the parameters may change with the current rate, they are well estimated with a mean error of 6.3 % (6.7 % at 0.5 C and 5.9 % at 1 C).

Conclusions and perspectives

The relationships between the equivalent circuit model (ECM) of a cell and its electrodes length have been detailed and a generic approach has been proposed to model these relationships. Because the proposed modeling approach is built from physics, it can theoretically be adapted to any insertion battery (lithium-ion, sodium-ion). Three coin cells with different electrodes lengths and capacities have been successfully modeled by an ECM. The latter brings accurate voltage simulation for fast and slow dynamics and its parameters have been made dependent on the electrodes length thanks to the proposed modeling approach. The error between the measured and estimated parameters equals 6.3 % on average, which is satisfactory.

We demonstrated that two cells would be sufficient to calibrate the parameters dependency-models and then use them to predict the electrical behavior of a cell with any electrodes length in the characterization range (between 25 mAh and 120 mAh here). Supplemented with dependencies on other dimensions (height and thickness of the electrodes), this model could be integrated into the simulation of a battery-powered system to determine the optimal sizing of the battery. For that purpose, the proposed approach should be extended to the operating range of the studied system (current, temperature, state of charge).

Acknowledgment

Part of this work has been founded by the french ministry of higher education and research. This article supports a study on the design of microrobots with embedded energy storage which would not have been possible without the help of Christine Prelle and Laurent Petit, whom we would like to thank. We also want to thank Pr. Guy Friedrich who helped us improve this article.

Abbreviations

Figure 1 :

 1 Figure 1: (a) Microtomography of a 60 mAh cell similar to the one studied here (top-view). (b) Diagram showing the unwinding of the electrodes. (c) Diagram of the unrolled electrodes with the average path of the electrical charges as a bold line.

Figure 2 :

 2 Figure 2: Detailed ECM of a cell superimposed on a cell diagram representing its main elements. The interface model shown at the top right is that of the negative electrode. The interface model of the positive electrode is identical (indices n being replaced by indices p). The shape used for the interface model identifies the direction in which the voltage sources in the cell are conventionally oriented.

Figure 3 :

 3 Figure 3: (a) Simplification of the interface-model topology for the negative electrode; the simplification for the positive electrode is the same, indices n being replaced by p. (b) and (c) Grouping of similar dynamics. (d) Approximation of the diffusion impedance with a series of N RC-circuits.

Figure 4 :

 4 Figure 4: Simplified ECM.

Figure 5 :

 5 Figure 5: Impedance spectra of the 60 mAh cell at different depths of discharge. Frequency between 0.01 Hz and 100 kHz with 5 points per decade in logarithmic spacing.

Figure 6 :

 6 Figure 6: Voltage responses of the 60 mAh cell to the GITT test presented in part 4.2.2, to a constant current at 0.5 C (CC 0.5 C) and to a constant current at 0.02 C (CC 0.02 C). The estimated OCV during the GITT test is also plotted (Uc estimated). The figure is a zoom on pulses 3, 4, 5 and 6.

Figure 7 :

 7 Figure 7: Overvoltage of the 60 mAh cell measured (meas) and simulated with diffusion parameters determined at 1 C (sim 1 C), and at 0.8 C (sim 0.8 (a): current profile. (b): evolution of the averaged overvoltage over all the simulated time, represented with sliding averages on a 1 s window noted SA(.). (c): zoom near 180 s.

kMAPEFigure 8 :

 8 Figure 8: Estimated and measured ECM-parameters as functions of the electrodes length (GITT 1 C 4 min and GITT 0.5 C 8 min).

C 1 d 2 d 1 d 2 d

 1212 capacitor C d,i capacitor of the i-th RC circuit modeling the discretization of the impedance Z d (i ∈ {1, 2, . . . , N }) C dl,n capacitor modeling the double layer effect at the negative electrode C dl,p capacitor modeling the double layer effect at the positive electrode C SEI capacitor of the RC circuit modeling the SEI H height of the electrodes I current flowing through the cell k d parameter of the dependency-model of R d k parameter of the dependency-model of R 1 d k parameter of the dependency-model of R 2 d ks1, ks2, ks3 parameters of the dependency-model of Rs k surf parameter of the dependency-model of R surf L length of the electrodes Mc number of measures used for pulse c ML number of electrodes lengths N number of RC circuits considered for the discretization of Z d R resistance R CA:L,n resistance modeling the electronic conduction in the conductive assembly along the length of the negative electrode R CA:L,p resistance modeling the electronic conduction in the conductive assembly along the length of the positive electrode R CA:T,n resistance modeling the electronic conduction in the conductive assembly along the thickness of the negative electrode R CA:T,p resistance modeling the electronic conduction in the conductive assembly along the thickness of the positive electrode Rconnec,n resistance modeling the electronic conduction in the electrical connection between the negative current collector and the terminal of the cell Rconnec,p resistance modeling the electronic conduction in the electrical connection between the positive current collector and the terminal of the cell Rct,n resistance modeling the charge transfer at the interface between the negative electrode and the electrolyte Rct,p resistance modeling the charge transfer at the interface between the positive electrode and the electrolyte R d equivalent resistance due to all diffusive phenomena in steady state R 0 d value of R d at 13.3 % of depth of discharge R value of R d at 20 % of depth of discharge R value of R d at 26.7 % of depth of discharge R d,i resistance of the i-th RC circuit modeling the discretization of the impedance Z d (i ∈ {1, 2, . . . , N }) R d,e equivalent resistance modeling the overvoltage across Z d,e in steady state R d,n equivalent resistance modeling the overvoltage across Z d,n in steady state R d,p equivalent resistance modeling the overvoltage across Z d,p in steady state Rqs quasi-steady state resistance Rs series resistance R SEI resistance of the RC circuit modeling the SEI R surf resistance of the RC circuit modeling the combined effects of the SEI, the charge transfer and the double layer at both electrodes t time ti time associated with the i-th measurement (i ∈ N * ) Un equilibrium potential of the negative electrode Up equilibrium potential of the positive electrode Uc equilibrium voltage of the cell (OCV) Z d impedance modeling the contributions of all diffusive phenomena in the cell Z d,e impedance modeling the diffusion effects in the electrolyte Z d,n impedance modeling the diffusion effects in the negative electrode Z d,p impedance modeling the diffusion effects in the positive electrode j current density S surface y any of the electrical parameters of the simplified ECM except Uc, that is: Rs, τ surf , R surf , τ d or R d Greek symbols γ parameter set used for the adjustment:γ = {(ks1, ks2, ks3), τ * surf , k surf , τ * d , k1 d , k 2 d } η meas measured overvoltage η sim simulated overvoltage ρ resistivity τct,n time constant modeling charge transfer associated with double layer effect at the negative electrode τct,p time constant modeling charge transfer associated with double layer effect at the positive electrode τ d time constant associated with Z d τ * d parameter of the dependency-model of τ d τ d,i time constant of the i-th RC circuit modeling the discretization of the impedance Z d (i ∈ {1, 2, . . . , N }) τ d,e time constant associated with the diffusion impedance in the electrolyte Z d,e τ d,n time constant associated with the diffusion impedance in the negative electrode Z d,n τ d,p time constant associated with the diffusion impedance in the positive electrode Z d,p τ SEI time constant associated with the R SEI C SEI circuit τ surf time constant associated with the surface phenomenon at the electrode/electrolyte interface τ * surf parameter of the dependency-model of τ surf θ electrical parameter set used for the adjustment: θ = (τ surf , R surf , τ d , R 0 d , R

Table 1 :

 1 Dependency of the parameters of the detailed model to the electrodes length. The subscripts n, p and e were not differentiated and were replaced by i. The operator ∝ indicates a proportional relationship.

Table 2 :

 2 Nominal dimensions and capacities of the cells tested.

	Model	CP0854 CP1254 CP1654
	Height (mm) Diameter (mm) Capacity (mAh)	5.4 8 25	5.4 12 60	5.4 16 120

(t i ) is


Camille Vatier

Alain Carrié

Marie-Christine Renaud

Noémie Simon-Tillaux

Alexandre Hertig

Isabelle Jéru
email: isabelle.jeru@aphp.fr

Lessons from the impact of COVID-19 on medical educational continuity and practices Running title: Reshaping medical education in light of COVID-19 experience

Keywords: COVID-19, undergraduate students, teachers, remote learning, digital resources

Purpose. The COVID-19 crisis necessitated abrupt transition to remote learning in medical schools. We aimed to assess the impact of COVID-19 on French undergraduate students and teachers, to identify practice changes, and to evaluate successes and areas for improvement of this remote learning experience.

Methods

. Data from two online questionnaires were analyzed with 509 participants among students and 189 among teachers from Sorbonne University. Responses to multiple choice, Likert response scale, and open-ended questions were evaluated.

Results. COVID-19 had negative impact on teaching continuity. 67% of students were in a dropout situation and many suffered from psychological stress, leading to set up of a psychological support unit. Although most teachers (81%) and students (72%) had limited knowledge of digital resources, distance learning was quickly implemented, with a predominant use of Zoom. The analysis of several parameters revealed that students were significantly more satisfied than teachers by remote learning. Nevertheless, both students and teachers agreed to replace classical lectures by digital media, and to promote in-person teaching in small interactive groups.

Conclusion.

This paper shares tips for faculty rapidly establishing remote learning. This comparative study of the students' and teachers' points of view underlines that new medical curricula should include more digital contents. We make recommendations regarding general university organization, equipment, and curricular development for long-term implementation of digital resources with reinforced relationships between faculty and students.

INTRODUCTION

The coronavirus disease 2019 (COVID- [START_REF] Soled | Medical Student Mobilization During a Crisis: Lessons From a COVID-19 Medical Student Response Team[END_REF], caused by coronavirus 2 (SARSCoV-2), disrupted countless aspects of society and economy. On March 16 th 2020, its high transmissibility led the French government to implement a confinement of the whole population for two months. Preventive measures included enforced social distancing, closure of educational and public institutions, and stay-at-home orders except for essential tasks and daily exercise.

This had a profound impact on the education sector. Universities halted campus-based teaching, and supported continuing courses through online resources. Simultaneously, COVID-19 brought the health care system to the limits of its capacities. Since most teachers in medical schools combine hospital, teaching, and research duties, they were foremost mobilized to fight the disease, and were less available for students. French medical students are also present in hospital wards from the second year of their curriculum. During confinement, many of them were encouraged to go into action and contributed in many ways to the care of patients affected by COVID-19. This brought in some anxiety, but also enthusiasm with respect to their vocation. This large-scale mobilization was however at the expense of their theoretical training. COVID-19 is not the first and will probably not be the last global health emergency that we will encounter [START_REF] Bloom | Infectious Disease Threats in the Twenty-First Century: Strengthening the Global Response[END_REF][START_REF] Lim | The challenges of "continuing medical education" in a pandemic era[END_REF], but the exceptional scale of its waves urged us to re-think medical training. In particular, this catalyzed implementation of remote medical education for a period that will probably outlast this pandemic. Numerous articles have discussed the impact of COVID-19 on medical education, but they mostly consisted of editorials or opinion pieces [START_REF] Dedeilia | Medical and Surgical Education Challenges and Innovations in the COVID-19 Era: A Systematic Review[END_REF][START_REF] Ferrel | The Impact of COVID-19 on Medical Education[END_REF][START_REF] Mukhopadhyay | Leveraging Technology for Remote Learning in the Era of COVID-19 and Social Distancing[END_REF][START_REF] Rose | Medical Student Education in the Time of COVID-19[END_REF], and as for studies, most of them focused on a single curricular discipline, offering a limited glimpse into the students' global experience. Lastly, few reports have evaluated the impact of COVID-19 on undergraduate medical students before residency.

The aim of this study was thus to determine the impact of the COVID-19 outbreak on French undergraduate medical students from Sorbonne University (SU). SU, located in Paris, is a research-intensive, world-class university with three segments of education (medicine, science, and humanities), and a long tradition of academic excellence. Prior to COVID-19, the standard curriculum of medical students consisted of organ system block-based lectures, problem-based learning in small groups, laboratory-based classes, and apprenticeships in hospital wards. During lockdown, all theoretical classes and activities had to be conducted remotely and the possibility for students to be implicated in clinical services was very heterogeneous, going from complete cancellation of clerkship to full-involvement in the front line. This study assesses the impact of COVID-19 on the continuity of theoretical medical teaching, lists the changes in practices made during that period, and evaluates the relative successes and failures of this remote learning experience. Two questionnaires were designed, one sent to all teachers from SU medical school, and the other to undergraduate students before residency. Four major themes were addressed in the questionnaires: the impact of COVID-19 on the continuity of the educational program, the knowledge and experience of remote learning at the beginning of crisis, the changes in teaching formats made during that period, and the lessons drawn from remote learning experience. The originality of this work is to provide a comparative study of the students' and teachers' points of view and to establish recommendations for curriculum design in the future. Sharing this experience should benefit other medical schools or universities from different fields.

MATERIALS AND METHODS

Study populations

This observational study was conducted in SU School of Medicine and was approved by the Data Protection Department from SU. It was based on the analysis of data from two online surveys. The first survey was sent to all teachers from SU medical school (n=750) (Supplementary Table 1). The second survey was dedicated to undergraduate students from the second to the sixth year of the curriculum, before residency (n=2103) (Supplementary Table 2). First-year students were not implicated since they take a competitive exam and only a small proportion of them can continue the course of medical studies. The students of the second to the sixth year represent a homogeneous group of students who take theoretical courses at the university and follow part-time training in hospital departments (Supplementary Figure 1). At the end of the sixth year, there is a national classifying exam serving as an entrance examination for residency. Residents working full time at the hospital were not included in this study.

Survey design

Questionnaires were designed by two investigators and combined multiple choice and Likert scale questions, with the option to provide further free text responses for some of the questions, as well as an overall reflection. Prior to sending out the survey, a representative focus group of teachers (n=10) and students (n=10) was conducted to ensure that questions were clear and cohesive. The whole project was validated by three home-institution staff members from the educational supervisory board. Questions from the survey were aimed at evaluating the overall impact of COVID-19 on the continuity of theoretical training, and at establishing the main difficulties encountered by teachers and students. It collected the developments made on remote teaching. Participants rated their satisfaction regarding remote learning resources and expressed their views regarding distance teaching for the future.

Survey implementation

Responses to the survey were collected and managed using the Survey Monkey online platform. An invitation to participate in the survey via an electronic link was sent out by email, together with an introductory message from the educational supervisory board. Data were collected from June 10 th to July 21 st .

Data analyses

Statistical comparison was conducted in R version 4.0 using chi-square tests and chi-square tests for trends. Graphs were generated in GraphPad Prism 6 software. Frequency distributions were represented by pie charts for binary variables and histograms for multinomial variables. The number of responses for multiple choice and Likert scale questions was always greater than 115 for teachers, and always greater than 378 for students.

Differences were considered significant when p<0.05. Free text comments from teachers (n=271) and students (n=996) were studied by two independent investigators. Comments representative of major thematic contents were extracted for illustration. Word cloud representations were also used to evaluate the importance of specific issues in free responses.

RESULTS

Participation rate

A total of 189 teachers out of 750 responded to the survey with a response rate of 25%. All teaching departments were represented covering 40 medical disciplines (Supplementary Table 3). 509 students among 2103 responded to the survey, with a participation rate of 24%. All years from the second to the sixth of the medical curriculum were represented. Respondents consisted of 144 second-year students (28%), 99 third-year (19%), 120 fourth-year (24%), 86 fifth-year (17%), and 12 sixth-year students (12%).

Impact of COVID-19 on the continuity of the educational program

One of the most immediate changes induced by COVID-19 was the broad cancelling of inperson classes. The capacity of teachers to maintain distance teaching was very heterogeneous (Figure 1A). As many as 15% of teachers were not in a situation to do any of their usual teaching duties, 50% maintained less than half of their classes, and only 13% of teachers gave all of the scheduled lessons. When we now consider remote lessons available during that period of time, follow-up by students was also very uneven: 6% of students did not follow any lesson, 36% less than half of courses, and 21% used all resources available. In openended questions related to this issue, there were 104 comments from students reporting incomplete courses and lack of clarity in digital media. Forty students reported the poor quality of sound and image in videos, which was a hurdle for the follow-up of many courses.

The reasons accounting for this gap in teaching continuity were not the same for teachers and students (Figure 1B). Teachers reported a number of reasons, which were not mutually exclusive, to explain the absence of courses: cancellation by the education supervisory board (74%), hospital work overload (31%), difficulties in using new informatic tools (16%), lack of a receptive mind due to the stressful context (6%), difficulties in organization of family life (5%), lack of a quiet place to work (3%), medical reasons (3%), and lack of access to an internet connection at hospital (2%). Additional reasons were added by students, mostly related to mental health and well-being. They mentioned a lack of motivation because of isolation feeling (53%), psychological stress (46%), difficulties in personal organization of family life (33%), lack of a quiet place to work (27%), hospital work overload (26%), absence of internet access at home (7%), medical reasons (4%), lack of internet access at hospital (3%), technical difficulties in using new informatic tools (3%), and lack of access to a computer (2%).

This strong impact of COVID-19 on the continuity of educational program was further underlined by the very high proportion of students declaring to have fallen behind in their teaching with a partial dropout rate of 51% and a complete dropout in 16% of cases (Figure 1C).

Knowledge and experience of remote learning at the beginning of COVID-19 crisis

We were challenged by the abrupt transition to entirely remote learning and we evaluated if students and teachers felt prepared for this complete shift towards online education. Only half of the teachers (52%) declared being aware of technological tools and resources available for remote learning (Figure 2A). Most teachers (81%) and students (72%) had no experience in remote learning (Figure 2B). For most teachers (66%) and students (45%), their motivation to participate in remote learning was moderate, as compared to classical classes (Figure 2C).

Analysis of changes in teaching format during COVID-19 pandemic

We did not identify any major innovation in the educational tools developed during that period of time. The first objective of teachers was to quickly adapt to remote learning. A very high number of teachers (77%) used video conference tools, especially Zoom (Figure 3A).

Students were highly supportive of this type of change, 147 of them praising the use of Zoom in free comments. The crucial role of Zoom for distance teaching was further underlined by a word-cloud analysis of free responses relative to the digital resources most used by teachers and most appreciated by students during confinement (Figure 3B). Only 38% of teachers used e-learning platforms for submitting course materials, even though the Moodle online learning site of SU was already available before COVID-19. Lectures were also recorded with videos and audios in Panopto, allowing students to review contents at any time. During confinement, 27% of teachers used videos and 13% proposed new written course materials. Some teachers (5%) also tried to facilitate interactions with students using Wooclap, which was very appreciated. Additional initiatives were taken, such as the creation of practice exchange groups, forums, and video chats. Free comments from teachers revealed three main concerns regarding the implementation of these changes. Firstly, they didn't have professional licenses to use Zoom or Skype. Secondly, they expressed the need to reinforce the secretariat for digital resources in charge of technical issues, with allocation of a dedicated staff. Finally, they lacked computers equipped with good cameras and microphones. The presence of tight firewalls on hospital computers, where many physicians were stuck because of the pandemic, have also considerably limited the implementation of distance learning.

The perception of the quality of interaction during remote lessons was different between teachers and students (4.10 -16 ). Less than half of teachers (40%) ranked it as "good" or "very good" against 63% of students (Figure 3C). When the majority of students underlined the necessity of part-time face-to-face teaching for better interactivity, a small number of students reported feeling less shy about asking questions during remote courses. In free comments, many teachers emphasized the irreplaceable value of attending some sessions in person, lauding the real-time feedback and exchange that develop in class.

Teachers were less convinced than students by the overall quality of distance teaching, as compared to in-person classes (7.10 -6 ). More than half of the teachers (56%) considered remote learning less contributive and ranked it as "far less good" or "less good" than classical in-person teaching. Only 32% of students shared the same opinion, 41% evaluating it as equivalent, and 27% ranking it as "better" or "far better" (Figure 3D).

Lessons from remote learning during COVID-19 crisis

Regarding the overall satisfaction of distance teaching, students were more satisfied than teachers (5.10 -7 ) (Figure 4A). Indeed, 62% of students declared to be rather satisfied or very satisfied, as compared to 39% of teachers with a similar satisfaction level.

There was a crucial need for training on digital resources among teachers, 75% of them expressing their wish for additional training, when only 13% of students had the same need (Figure 4B).

The regret of not having face-to-face teaching was more present among teachers (70%) than among students (55%) (3.10 -3 ) (Figure 4C).

Students were also more prone than teachers to keep a high percentage of remote learning in the future (1.10 -13 ). Among them, 42% wished to have more than half of their classes online, whereas only 10% of teachers expressed the same wish (Figure 4D). The two main reasons reported by students in free comments to explain their attachment for remote learning were saving time in public transport, and a better organization in daily life. Nevertheless, the great majority of them considered that direct contact with teachers and classmates was a real necessity for their psychological well-being, social life, and for proper acquisition of knowledge and skills. The great majority of teachers considered that the curriculum should combine face-to-face and distance teaching. When considering all comments, the emerging consensus was to replace in-person lectures by recorded digital media combined with a higher number of interactive sessions in small groups.

DISCUSSION

Remote learning has gained popularity in higher education over the last decade, yet its integration into the French medical curricula has been relatively slow. While students had the possibility to use some digital resources, most courses were still in-person, a format that students seem more and more disinterested by. The abrupt transition to remote learning due to COVID-19 led to rapidly develop different means of delivering courses to meet complex learning objectives, and precious lessons can be learned to benefit medical education (Table 1). A number of findings from our study could be expected, such as the overall negative impact of COVID-19 on the continuity of teaching, the need for training in digital resources, or the pronounced interest of students in distance teaching. However, we had no idea of the relative perception that teachers and students might have of the large-scale development of distance teaching implemented simultaneously in all teaching departments and in all years of the curriculum. In addition, we did not expect to find so many students in a dropout situation, and we thought that there would be a greater diversification of teaching materials offered by teachers for distance learning.

A number of studies have shown that quarantine is associated with negative psychosocial outcomes, including depressive symptoms, anxiety, and stress [START_REF] Lei | Comparison of Prevalence and Associated Factors of Anxiety and Depression Among People Affected by versus People Unaffected by Quarantine During the COVID-19 Epidemic in Southwestern China[END_REF][START_REF] Meo | COVID-19 Pandemic: Impact of Quarantine on Medical Students' Mental Wellbeing and Learning Behaviors[END_REF][START_REF] Rohr | Psychosocial Impact of Quarantine Measures During Serious Coronavirus Outbreaks: A Rapid Review][END_REF]. Our study underlines the negative impact of confinement on medical students. They did show deterioration in their study performances, more than 65% of them declaring to have fallen behind in their curriculum. They especially reported isolation feeling and psychological stress. Many felt disconnected from medical school and missed the positive effect of socialization associated with studying and working together. To try to circumvent this phenomenon, a psychological support unit was set up in April 2020, each student receiving phone calls from a dedicated mentor to keep a link and identify potential difficulties. We think that such initiatives should be encouraged in the future. In addition, due to COVID-19, an exceptional measure was implemented in SU for grading of students in 2020. Usually, a student who fails an exam for a given subject in the first test in June and in the re-take exam in September, repeats the academic year. In 2020, students who failed in a single subject were able to pass into the next academic year with a debt, which consists of retaking that exam the following year. Only 15 students out of the 750 were affected by this exceptional measure.

Two difficulties shared by many teachers and students during the first wave of COVID-19 were hospital work overload and difficulty in organization of daily life. Hospital resources were stretched in efforts to contain the outbreak. Many teachers were called in to work and, with schools and daycares being closed, it was a challenge to work and take care of children at the same time. Many students were also directly involved in the management of the health crisis and could not work and study at the same time. This resulted in medical training taking a back seat. About one third of students felt that their living conditions were not conducive to remote learning. This was primarily attributed to a lack of a quiet space to study, bad internet connection, or lack of a computer. In this regard, a plan to help students to buy computers at reduced costs has been set up. A plan to make laptops equipped with good cameras and microphones available for teachers seems also necessary.

When confinement started in Paris, two-thirds of the school year had already passed, so that many students ended their year without too much disturbance. Thanks to the work of committed faculty and staff as well as the resilience and adaptability of students, medical education continued despite the circumstances, which significantly affected the delivery of planned courses. A huge effort has been made by teachers and students to develop and use distance learning tools, given the little experience they had. An additional improvement would be to create a coordination board, involving school leaders and department faculty, for remote learning integration into the medical curriculum. COVID-19 has been a trigger to leverage technology for medical education. Recent events have allowed us to take a step back and carefully examine our own teaching practices [START_REF] Vollbrecht | Lessons learned while creating an effective emergency remote learning environment for students during the COVID-19 pandemic[END_REF].

We have learned that many curricular elements can be delivered just as well virtually as in classrooms and amphitheaters. Videoconferencing applications such as Zoom offer live streaming functions, which are very useful for interactive and problem-based solving sessions [START_REF] Gaber | Online team-based learning sessions as interactive methodologies during the pandemic[END_REF][START_REF] Halpin | The use of Twitter and Zoom videoconferencing in healthcare professions seminar course benefits students at a commuter college[END_REF]. These changes, brought on by necessity, will probably have lasting impact on medical training (6, 9, 10), especially since digital resources have proven to increase student commitment to learning [START_REF] Kay | Using technology to increase student (and faculty satisfaction with) engagement in medical education[END_REF]. The younger generations are perhaps best equipped to integrate innovative online settings into learning practices. The current study reveals that most students will readily embrace this shift to e-learning, valuing the personalized and flexible learning experience. In contrast, it would be very useful for many teachers to get additional training for some of these tools, as done in other universities [START_REF] Veerapen | A scaffolded structured approach for efficient transition to online small group teaching[END_REF]. It seems now obvious that new curricula, which are being transformed (3), should combine remote learning and in-person teaching. The main concern is that the replacement of in-person classes with online equivalents might create a loss of interaction. Direct interactions with students are certainly decisive to develop critical thinking, lifelong learning engagement, and a collaborative state of mind. We propose to favor in-person small-sized group activities that focus on collaborative case-based or problem solving, simulation, and learning of technical skills. This should reinforce relationships between students and teachers in the spirit of companionship. Medical education is not only imparting of knowledge [START_REF] Sandars | The use of reflection in medical education: AMEE Guide No. 44[END_REF]. Involvement of students in the crisis management might have contributed to the development of noncognitive skills. Many students were in charge of unexpected tasks [START_REF] Collaborative | Medical student involvement in the COVID-19 response[END_REF][START_REF] Soled | Medical Student Mobilization During a Crisis: Lessons From a COVID-19 Medical Student Response Team[END_REF], such as development of telehealth medicine, which will be an essential new tool in medicine practice. Some students were implicated on the front line in the management of COVID-19 cases, thereby being taught important lessons in courage, empathy, and teamwork. Lessons have also been taught in the rational management of limited resources, such as personal protective equipment and isolation facilities. These learning opportunities can be hard to come by in classical training and are also important for future healthcare professionals.

This study, which summarizes how COVID-19 has reshaped education in SU, allows us to draw positive lessons. This should help medical schools to create new curricula including more virtual learning and more in-person small-sized group activities to reinforce relationships between faculty and students. Sharing experiences across the globe should lead to major pedagogical advances throughout educational systems and specifically medical education.

TABLES

General organization

Department for digital resources with dedicated staff Coordination board for remote learning integration, made of faculty leaders and administrative staff Psychological support unit for students, mobilizable at any time in situation of health crisis

Equipment

Professional licenses for videoconference tools (Zoom, Wooclap, Skype…) Laptops equipped with cameras and microphones available for teachers Financial support for precarious students in need of informatic equipment or internet access

Medical curriculum

Recorded digital media to replace traditional in-person lectures Higher number of interactive sessions dedicated to small groups of students

Continuing medical education

Training on digital resources, especially for teachers

Table 1. Resources needed for the development of remote teaching, based on the experience of COVID-19 crisis

Figure 1 .

 1 Figure 1. Impact of COVID-19 on the continuity of the educational program

Figure 2 .

 2 Figure 2. Situational analysis of remote learning at the beginning of COVID-19 crisis

Figure 3 .

 3 Figure 3. Implementation of distance teaching during COVID-19 outbreak

Figure 4 .

 4 Figure 4. Lessons from the remote learning experience during COVID-19 crisis

FIGURES Figure 1 1:Figure 2 A

 12 FIGURES

A 1 :

 1 Figure 4

ACKNOWLEDGEMENTS

We acknowledge heads of teaching departments, teachers, and students who took part in this study.

The authors have no funding to declare.

DECLARATION OF INTEREST

The authors report no conflict of interest

SUPPLEMENTS

The Sorbonne University Faculty of Medicine wishes to evaluate the choices you have made in each teaching department to maintain (or not, constrained by the epidemic) the training of our students, and the innovations that you would have set up in a hurry, in particular and above all thanks to digital tools, whatever the scale. This approach also aims to assess your satisfaction with the new tools implemented, in order to guide our choices in terms of medical education in the years to come. It is therefore very important that each of you can answer the questionnaire below. The response time should not exceed 15 minutes. We thank you very much for your participation.

Supplementary Table 1. Teacher survey

The Sorbonne University Faculty of Medicine wishes to make an assessment with the students of the courses followed during the period of the Covid-19 pandemic. This survey is particularly interested in the educational innovations that you have experienced, in particular and above all thanks to digital tools. This approach also aims to assess your satisfaction with the new tools implemented, in order to guide our choices in terms of medical education in the years to come. It is therefore very important that each of you can answer the questionnaire below. The response time should not exceed 15 minutes. In France, the medical studies curriculum begins with a year of theoretical courses common to all health professions. At the end of the year, a selective competition determines those who will be entitled to continue the curriculum. From the second to the sixth year, students attend theoretical courses at the university and have part-time training in hospital wards: 3 months in the second year, all year long between the third and the sixth year. At the end of the sixth year, there is a national classifying test that puts medical students from all French universities in competition to choose their medical specialty and serves as an entrance examination into residency.

High school

Clinical training

Master -Diploma of advanced training in medical sciences

Theoretical lessons

Medical Doctor Degree