Nicolas Damay
email: nicolas.damay@utc.fr

Karrick Mergo Mbeya

Guy Friedrich

Christophe Forgez

Separation of the charge transfers and solid electrolyte interphase contributions to a battery voltage by modeling their non-linearities regarding current and temperature

Keywords: Battery, modeling, electrical circuit model, charge transfer, SEI, non-linearities

One major aging phenomenon during the normal operation of a Li-ion battery is the thickening of its anode solid electrolyte interphase (SEI), which affects the battery state of health. However, the diagnosis of this SEI is difficult because its contribution to the battery voltage is mixed with those of other phenomena. In this article, a method is proposed to separate the contributions of charge transfers and anode SEI. It is based on EIS and GITT measurements and on a model of the charge transfers and SEI non-linearities regarding current and temperature, inspired by Butler-Volmer and Arrhenius equations. The proposed method is applied to a commercial LiFePO 4 /graphite cell. The activation energies of its SEI and charge transfers resistances are respectively 0.59 eV and 0.81 eV , which is in accordance with literature data. By construction, the method should be applicable to other battery chemistries, unless their cathode SEI has a significant resistive contribution. After an initial calibration, the proposed model can be embedded in a BMS. The latter may then be able to recalibrate the model online thanks to currents pulses at low temperatures and thus track the SEI growth of a battery during its entire operating life.

Introduction

Lithium-ion batteries (LIB) aging is a major concern because it impacts their lifetime and performance and one major aging phenomenon during normal operation is the thickening of the anode solid electrolyte interphase (SEI). The SEI is a passivation layer on the anode surface that is created at the end of a battery production and it is mandatory for the battery to work properly. However, this SEI grows endlessly during the battery life, because of severe operating conditions and even during storage [START_REF] Vetter | Ageing mechanisms in lithium-ion batteries[END_REF][START_REF] Lawder | Model-Based SEI Layer Growth and Capacity Fade Analysis for EV and PHEV Batteries and Drive Cycles[END_REF]. Some positive electrodes may also have a SEI that have a significant contribution such as Ni-rich cathodes [START_REF] Schipper | Review-Recent Advances and Remaining Challenges for Lithium Ion Battery Cathodes[END_REF][START_REF] Pritzl | Identifying Contact Resistances in High-Voltage Cathodes by Impedance Spectroscopy[END_REF] and high-voltage LiCoO 2 cathodes [START_REF]Dynamic evolution of cathode electrolyte interphase (CEI) on high voltage LiCoO2 cathode and its interaction with Li anode[END_REF]. This study does not apply to batteries using these cathodes and is focused on batteries with SEI at their anodes.

Tracking the growth of the anode SEI of a LIB over its lifetime would lead to a better follow-up of its state of health, whether it is for scientific studies or for end-users applications. The operating limits of the LIB could be adjusted accordingly as well as its management strategy by its BMS (battery management system). To do this tracking of the SEI growth, advanced characterization methods, non-invasive and suiting a BMS, have to be developed.

The BMS of a battery pack has generally a limited calculation power and basic measuring instruments. It has access to cells voltages, to the battery pack total current and sometimes to some cells surface temperatures. However, BMS sensors accuracy and resolution are often worse than in a laboratory. Also, the major difficulty linked to the SEI-growth tracking is its contribution to a cell voltage is mixed together with the contributions of numerous other phenomena that occur inside an operating battery [START_REF] Illig | Understanding the impedance spectrum of 18650 LiFePO 4 -cells[END_REF].

Separating the contributions to a battery voltage can be done thanks to an electrical model. To that end, electrochemical models, based on the Newman approach, are interesting because they are based on physical equations. They are made of a set of differential and algebraic equations (DAEs) that brings a good description of electrochemical phenomena at the microscopic scale [START_REF] Malik | A Critical Review of the Li Insertion Mechanisms in LiFePO 4 Electrodes[END_REF][START_REF] Farkhondeh | Mesoscopic Modeling of a LiFePO 4 Electrode: Experimental Validation under Continuous and Intermittent Operating Conditions[END_REF]. The resolution of these coupled DAEs may require a lot of computer-power that would hardly suit a BMS capabilities. Moreover, they contain up to tens of physical parameters that are determined by non-invasive and invasive methods, which can make electrochemical models hard to recalibrate for tracking the SEI-growth in a final-user application.

An equivalent circuit model (ECM) only contains a few parameters that can be determined in a non-invasive way, making them suitable to model commercial batteries with little knowledge of their internal compositions. Besides, they can easily be used with low calculation capabilities like in a BMS. However, the drawback of their simplicity is that the physical phenomena are lumped and are often mixed together inside "macroscopic" parameters, making their interpretation uncertain.

Nevertheless, many studies have proved that it is possible to find a physical interpretation for ECM parameters. Mauracher et al. [START_REF] Mauracher | Dynamic modelling of lead/acid batteries using impedance spectroscopy for parameter identification[END_REF] build a lead-acid battery model with a strong physical background and obtained a good correspondence with experimental data. Von Srbik et al. [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], Raël and Hinaje [START_REF] Raël | Using electrical analogy to describe mass and charge transport in lithium-ion batteries[END_REF] have successfully constructed ECM based on equations that are used classically in electrochemical models.

It is relatively easy to separate the contributions of diffusion phenomena from of other phenomena contributions, because their dynamics is significantly slower [START_REF] Mauracher | Dynamic modelling of lead/acid batteries using impedance spectroscopy for parameter identification[END_REF]. Concerning the contributions of ohmic drops, charge transfers at both electrodes and SEI, they are much harder to separate. More particularly, the charge transfers, associated with the corresponding double layer effects, have dynamics that are very close to the SEI dynamic and they often cannot be distinguished within an impedance spectrum [START_REF] Illig | Understanding the impedance spectrum of 18650 LiFePO 4 -cells[END_REF]. Illig at al. [START_REF] Illig | Separation of Charge Transfer and Contact Resistance in LiFePO 4 -Cathodes by Impedance Modeling[END_REF] proposed an interesting approach to separate the contributions of several phenomena measured by EIS (electrochemical impedance spectroscopy) thanks to the study of their activation energies (related to their non-linearities regarding temperature), but they could not separate the SEI and charge transfer contributions for the anode.

In this article, we propose a method to separate the contributions of charge transfers and anode SEI that are mixed inside an ECM parameter: the so-called "surface resistance". This method requires EIS and GITT (galvanostatic intermittent titration technique) tests to measure the surface resistance for different temperatures and currents [START_REF] Onda | Thermal behavior of small lithium-ion battery during rapid charge and discharge cycles[END_REF][START_REF] Gholizadeh | Estimation of State of Charge, Unknown Nonlinearities, and State of Health of a Lithium-Ion Battery Based on a Comprehensive Unobservable Model[END_REF]. Then, the non-linearities regarding current and temperature of the charge transfers and SEI contributions are modeled in order to separate them.

The proposed method is applied to a commercial LiFePO 4 /graphite cell (40 Ah, maximum continuous discharge is 2C), for which the separated contributions of charge transfers and SEI are presented. The non-linearities regarding state of charge (SoC) have been studied in a previous work for this cell [START_REF] Damay | Heterogeneous behavior modeling of a LiFePO 4 -graphite cell using an equivalent electrical circuit[END_REF]. More particularly, we found that its surface resistance does not significantly change with SoC. Consequently, the SoC influence on the proposed method is not studied in this article. The interested reader may refer to the study of Kremer et al. [START_REF] Kremer | 1D electrochemical model of lithium-ion battery for a sizing methodology of thermal power plant integrated storage system[END_REF] which is about SoC-dependency of the charge transfer process, modeled by a SoC-dependency of the exchange current.

In the next section is presented the ECM construction, along with the modeling of the non-linearities regarding current and temperature for the parameter that represents charge transfers and SEI contributions. The experimental determination of this parameter, thanks to EIS and GITT tests, is presented and applied to the studied cell in section 3. In section 4, the separation of the charge transfers and SEI contributions is done for the studied battery and discussed. The eventual use of the proposed non-linearity model with a BMS is also discussed.

Battery and surface-resistance non-linearities models

Electrical circuit model presentation

The ECM used in this study, presented on Figure 1, is composed of several elements that represent the main dynamics and the associated phenomena occurring during a cell operation [START_REF] Kuhn | Modeling diffusive phenomena using non integer derivatives[END_REF]:

• a voltage source U oc stands for the OCV (open circuit voltage), which is defined as the battery voltage in open circuit and at equilibrium ;

• a resistance R s represents all the purely resistive contributions (electrolyte, current collectors, contact resistances) ;

• a resistance R surf which is related to the potential drops between the surface of active materials and the electrolyte (mainly due to charge transfers at both electrodes and to the SEI) ;

• a time constant τ surf accounts for the quick dynamic (often inferior to 1 s) associated with charge transfers, double layers and SEI [START_REF] Illig | Separation of Charge Transfer and Contact Resistance in LiFePO 4 -Cathodes by Impedance Modeling[END_REF][START_REF] Lawder | Model-Based SEI Layer Growth and Capacity Fade Analysis for EV and PHEV Batteries and Drive Cycles[END_REF]. It is assumed to be a first order dynamic that corresponds to the product R surf × C surf ;

• an equivalent diffusion impedance Z d that reproduces the apparent electrical behavior due to the charges diffusion in the electrolyte and lithium atoms diffusion in the active materials [START_REF] Mauracher | Dynamic modelling of lead/acid batteries using impedance spectroscopy for parameter identification[END_REF][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] Huang | An Analytical Three-Scale Impedance Model for Porous Electrode with Agglomerates in Lithium-Ion Batteries[END_REF].

It has been demonstrated in a previous work that the "slow polarization process" of the studied cell, which is associated with Z d , behaves like a one-dimensional limited diffusion phenomenon [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 corresponds to a Warburg-like behavior that evolves into a purely resistive contribution after a few minutes. Three different diffusion processes can be considered, namely the diffusion of lithium atoms in the active materials of both electrodes [START_REF] Huang | An Analytical Three-Scale Impedance Model for Porous Electrode with Agglomerates in Lithium-Ion Batteries[END_REF] and the diffusion of lithium ions in the electrolyte [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] Raël | Using electrical analogy to describe mass and charge transport in lithium-ion batteries[END_REF]. Nevertheless, grouping them into a single equivalent "limited-diffusion impedance" Z d gives a satisfactory representation of the studied cell behavior. In the Laplace domain, this leads to a Nernst impedance (Equation ( 1)) in which Z d is a function of a "diffusion resistance" R d and a "diffusion time constant" τ d [START_REF] Devillers | Caractérisation et modélisation de composants de stockage électrochimique et électrostatique[END_REF].

Z d (s) = R d . tanh( √ τ d .s) √ τ d .s (1)
Equation ( 1) can be rewritten thanks to Mittag-Leffler theorem to bring Equation (2) [START_REF] Kuhn | Modelling Ni-MH battery using Cauer and Foster structures[END_REF]. It allows a decomposition of Z d into an infinite number of RC circuits in series (Figure 1).

Z d (s) = R d √ τ d .s ∞ p=1 2 √ τ d .s τ d .s + (pπ -π/2) 2 (2)
This relation can now be rewritten to find the values of the RC circuits parameters : R d,p and

C d,p (τ d,p = R d,p × C d,p ). Z d (s) = ∞ p=1 8 R d 4 τ d .s + π 2 (2p -1) 2 = ∞ p=1 R d,p 1 + τ d,p .s (3)
Which finally yields:

R d,p = 8 R d π 2 (2p -1) 2 τ d,p = 4 τ d π 2 (2p -1) 2 C d,p = C d = τ d,p R d,p = τ d 2 R d (4)
Please note that Equation (3) requires an infinite number of RC cells to perfectly match Equation (1). However, because of calculation constraints, the number of RC cells n has to be reduced [START_REF] Kuhn | Modelling Ni-MH battery using Cauer and Foster structures[END_REF]. The removed RC cells are the ones with a high p value, that is with a short time constant τ d,p (Equation ( 4)). Consequently, the precision of the diffusion model is reduced for high frequencies (for very fast current changes), which is acceptable for this study. The number of RC cells n has been set to 10 in this study.

The focus of this study is on the R surf parameter and on the physical information that can be extracted from it, but the determination of the other parameters is important because their contributions are combined together within a battery behavior and all of them have to be considered in order to isolate the one from R surf .

Modeling the surface resistance R surf non-linearities regarding current

The surface resistance is related to three main contributions, namely the charge transfer processes for both electrodes and the SEI (see section 2.1). The charge transfer processes behaviors are assumed to be indistinguishable from each other. They can be modeled by Equation [START_REF]Dynamic evolution of cathode electrolyte interphase (CEI) on high voltage LiCoO2 cathode and its interaction with Li anode[END_REF] which is inspired by the Butler-Volmer equation, with I the current flowing through the active materials surfaces (this current is the cell current here), I 0 the exchange current, β the charge-transfer coefficient, V ct the charge transfers overpotential, F the Faraday's constant, R the gas constant and T the absolute temperature [START_REF] Farkhondeh | Mathematical Modeling of Commercial LiFePO 4 Electrodes Based on Variable Solid-State Diffusivity[END_REF][START_REF] Hariharan | A coupled nonlinear equivalent circuit -Thermal model for lithium ion cells[END_REF]. It should be noted that the Butler-Volmer equation is originally defined as the description of a local phenomenon. It is here applied to the whole electrode surface, which implies that the current density and the charge transfers overpotential are assumed to be uniform.

I = I 0 exp (1 -β)F RT V ct -exp -βF RT V ct (5)
The contribution of the SEI can be related to the V ct expression with respect to Equation ( 6) [START_REF] Farkhondeh | Mathematical Modeling of Commercial LiFePO 4 Electrodes Based on Variable Solid-State Diffusivity[END_REF]. V surf is thus the potential difference between the solid phase (active materials) and the liquid phase (electrolyte) and R sei is the resistive contribution of the SEI.

V ct = V surf -I × R sei (6)
The charge transfer coefficient β is here assumed to be equal to 0.5 [START_REF] Farkhondeh | Mathematical Modeling of Commercial LiFePO 4 Electrodes Based on Variable Solid-State Diffusivity[END_REF][START_REF] Hariharan | A coupled nonlinear equivalent circuit -Thermal model for lithium ion cells[END_REF], leading to a more compact form of Equation [START_REF]Dynamic evolution of cathode electrolyte interphase (CEI) on high voltage LiCoO2 cathode and its interaction with Li anode[END_REF].

I = 2 I 0 sinh F 2 RT (V surf -I × R sei ) (7)
Equation ( 7) is inverted and divided by the current I to compute the surface resistance R surf , which is the sum of the SEI resistance R sei and an expression corresponding to the charge transfers. For the sake of simplicity, the expression corresponding to charge transfers is called "charge transfers resistance" R ct . The latter is thus nonlinear with respect to the current.

R surf (I) = V surf I = R sei + 2 R T F I asinh I 2 I 0 R surf (I) = R sei + R ct (I) (8)
Equation ( 8) is the foundation of the model proposed in this study. In the next subsections, the temperature dependencies of R sei and R ct are detailed and justified.

Modeling R surf non-linearities regarding temperature 2.3.1. Presentation of the Arrhenius law

As it is explained in the next subsections, the thermal dependencies within Equation ( 8) are related to thermally activated processes. Consequently, the Arrhenius law can be used (Equation ( 9)). k corresponds to a reaction rate at the absolute temperature T (expressed in Kelvin), A is a pre-exponential factor, E a is the activation energy of the considered reaction (expressed in eV ) and k B is the Boltzmann constant (expressed in eV /K). The universal gas constant R can be used instead of k B , thus changing the unit of the activation energy E a .

k = A × e -Ea/k B T (9)
Equation ( 9) has been originally created to describe the temperature dependence of a reaction rate, but it has been found that a lot of thermally activated processes evolves with respect to this formula [START_REF] Illig | Separation of Charge Transfer and Contact Resistance in LiFePO 4 -Cathodes by Impedance Modeling[END_REF][START_REF] Chung | Electronically conductive phospho-olivines as lithium storage electrodes[END_REF].

Thermal dependency of the charge transfers resistance R ct

The charge transfers resistance depends on the exchange current I 0 (see Equation ( 8)), which is considered as constant in many works [START_REF] Mauracher | Dynamic modelling of lead/acid batteries using impedance spectroscopy for parameter identification[END_REF][START_REF] Farkhondeh | Mathematical Modeling of Commercial LiFePO 4 Electrodes Based on Variable Solid-State Diffusivity[END_REF][START_REF] Hariharan | A coupled nonlinear equivalent circuit -Thermal model for lithium ion cells[END_REF]. However, its detailed expression exhibits a dependence to a reaction rate, which is known to evolve with respect to the Arrhenius law (see section 2.3.1) [START_REF] Mastali | Electrochemical Modeling of Commercial LiFePO 4 and Graphite Electrodes: Kinetic and Transport Properties and Their Temperature Dependence[END_REF]. This was confirmed by measurements made by Takahashi et al. [START_REF] Takahashi | Reaction behavior of LiFePO 4 as a cathode material for rechargeable lithium batteries[END_REF] on a LiFePO 4 electrode. Therefore, I 0 can be replaced by the following expression, with I 0 (25°C) the exchange current value at 25 °C and E a,I0 its activation energy. Equation ( 10) is derived from Equation ( 9), but it has been normalized so the exponential term equals one at 25 °C.

I 0 (T ) = I 0 (25°C) × exp -E a,I0 k B 1 T - 1 298 (10)

Thermal dependency of the SEI resistance R sei

The SEI resistance R sei is also temperature dependent. The diffusivity in a SEI layer depends on temperature in accordance with the Arrhenius law [START_REF] Pinson | Theory of SEI Formation in Rechargeable Batteries: Capacity Fade, Accelerated Aging and Lifetime Prediction[END_REF], making its conductivity temperature dependent [START_REF] Park | A review of conduction phenomena in Li-ion batteries[END_REF][START_REF] Ecker | Parameterization of a Physico-Chemical Model of a Lithium-Ion Battery: I. Determination of Parameters[END_REF]. In this study, the SEI resistance is thus modeled by Equation [START_REF] Raël | Using electrical analogy to describe mass and charge transport in lithium-ion batteries[END_REF] with R sei (25°C) being the resistance value at 25 °C and E a,sei its activation energy.

R sei (T ) = R sei (25°C) × exp E a,sei k B 1 T - 1 298 (11)

Complete non-linearities model for R surf

As a result, we propose Equation ( 12) to model the current and temperature non-linearities of the surface resistance R surf . The latter is considered as the sum of two contributions, namely the SEI resistance R sei and the charge transfers resistance R ct . It uses four parameters: two for the SEI contribution and two for the charge transfers (which are related to the exchange current I 0 ).

R surf (I, T ) = R sei (T ) + R ct (I, T ) R surf (I, T ) = R sei (25°C) × exp E a,sei k B 1 T - 1 298 + 2 R T F I asinh I 2 I 0 (25°C) exp E a,I0 k B 1 T - 1 298 (12)

Characterization of the surface-resistance nonlinearities

Experimental conditions

A prismatic commercial LiFePO 4 /graphite cell (40 Ah, maximum 2C for continuous discharge and 1C for continuous charge) was put into a climatic chamber to control its equilibrium temperature. A Bio-Logic system was used for tests (VSP-300), with a 20 A booster for current values of 20 A and below. For higher currents, we used a 100 A booster. The precision of these devices is 0.1% of their full scales. The presented model parameters have been extracted from EIS (Electrochemical Impedance Spectroscopy) and GITT (Galvanostatic Intermittent Titration Technique) tests for different operating conditions.

During an EIS test, a sine wave is applied to the battery and the variation of the voltage is divided by the variation of the current in order to compute the impedance of the battery. This process is repeated for different frequencies in order to get the impedance spectrum of the battery. For this study, EIS tests have been performed at -5 °C, 5 °C, 15 °C, 25 °C and 45 °C, with a C/10 current, at 50% SoC and from 1 kHz to 10 mHz, with 6 frequencies measured per decades.

A GITT test is a repetition of constant-current pulses separated by rest periods (1 h was used or this study). The electrical parameters can be extracted from each pulse, allowing their determination as a function of the SoC. By repeating this process for different currents and temperatures, a characterization of the cell on its whole operating range can be achieved.

GITT tests have been performed at the same temperatures and for the following currents : -100 A (2,5C discharge), -80 A, -60 A, -40 A, -20 A, -10 A, -4 A, -2 A, -1 A, 1 A, 2 A, 4 A, 10 A, 20 A and 40 A (1C charge). They were made between SoC 62,5% and 87,5%, because the OCV curve is flat there, that is on a plateau that allows a better separation of OCV and overvoltage.

It is reminded that the SoC dependency has been already studied for this LiFePO 4 /graphite cell (40 Ah) and is not considered here [START_REF] Damay | Heterogeneous behavior modeling of a LiFePO 4 -graphite cell using an equivalent electrical circuit[END_REF].

R surf determination based on EIS for near-zero currents

The Nyquist plots of the obtained impedance spectra are represented on Figure 2a and there is a zoom on the high frequencies values area on Figure 2b.

The sum R s + R surf is determined in the middle frequencies range (around 1 Hz and 10 HZ), where the arc of a circle returns toward the real axis and the imaginary part of the impedance reaches a local minimal value. There, the real part of the impedance value is considered to be equal to R s + R surf . This process is used for each temperature.

In order to isolate R surf for a given temperature, the R s parameter value is required. It is assumed to correspond to the minimal real value of the impedance spectrum as represented on Figure 2b [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]. This method has been satisfactory for the temperatures between 5 °C and 45 °C. By considering the shapes of the spectra for positive temperatures as references, we chose to consider three EIS values as outliers for the highest frequencies at -5 °C. Consequently, we used the fourth EIS value at -5 °C to determine R s as represented on Figure 2b.

R surf determination based on GITT

Extracting R surf from a GITT test requires to separate different contributions to the battery voltage, as stated in the introduction. A typical voltage response of a LiFePO 4 /graphite cell to a GITT current-pulse has been reproduced on Figure 3 as a function of the time. The OCV and all of the five parameters, including R surf can be extracted from this curve by fitting the ECM of Figure 1 to the voltage response [START_REF] Damay | Thermal modeling of large prismatic LiFePO 4 /graphite battery . Coupled thermal and heat generation models for characterization and simulation[END_REF].

• U oc is the voltage value before the current pulse, assuming the cell is at equilibrium (measured) ; • R s is the instantaneous voltage change at the very beginning of the pulse (this value is extracted from EIS measurements, see section 3.2) ;

• R surf is the overvoltage related to the fast dynamic that occurs during the first tenths of second of the pulse (obtained by fitting);

• τ surf corresponds to the same fast dynamic at the beginning of the pulse (obtained by fitting) ;

• R d is measured indirectly from the quasi-steady state resistance R QS (Equation ( 13)), the latter being calculated using the overvoltage value at the end of the pulse (assuming it was long enough for a quasisteady state to be reached) [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 result is used as an initial value for the model fitting.

• τ d is determined from the voltage response dynamic (obtained by fitting) ;

R QS (SoC, T ) = R s (SoC, T ) + R surf (SoC, T ) + R d (SoC, T ) (13)
For this study, the focus is to determine R surf in the most accurate way. It has to be noted that a long GITT pulse voltage-response may not correspond exactly to the diffusion dynamic of the ECM and, thus, that the determination of the diffusion part may be biased. This can lead to a biased determination of R surf and τ surf . To avoid this potential issue, we chose to fit the ECM only on the first 20 seconds of the pulse voltage-response (Figure 4). We used this time frame for the diffusion process to be sufficiently long for it to be fitted easily, but short enough for the voltage response not to be "deformed". Considering that the fit is done only on the first 20 s of the pulse, it is assumed that the cell temperature remains constant and is equal to the climatic chamber temperature. The example of the Figure 4 represents the surface resistance determination for a discharging pulse at C/20 (2 A), 5°C. The resistance R s that has been extracted from EIS is reported on the curve, along with the value R surf determined by fitting the ECM to the measurements. Unsurprisingly, the latter is close to the value obtained thanks to EIS, that is 9.65 mΩ. The τ surf value of 27 ms corresponds to a characteristic frequency of 6 Hz that is in accordance with the corresponding frequency range in the EIS data. The fast dynamic of diffusion at the beginning of the pulse is in accordance with theory (see the end of subsection 2.1). Some operating points of the GITT tests lead to an overvoltage inferior to 10 mV for first few seconds of the pulse and the ECM could not be fitted properly to these measurements. This corresponds to the following operating conditions: 1 A at 5 °C, 2 A and below at 15 °C, 4 A and below at 25 °C, 4 A and below at 45 °C. These points from GITT cannot be used, but this is compensated by the EIS data that provide a good characterization at low current.

Separating charge transfers an SEI contributions of a LiFePO 4 /graphite cell

Fitting the proposed model to the experimental data

The results of R surf determination (by EIS and GITT) are presented Figure 5 for different temperatures and as functions of the current (positive current values are for charging conditions). The experimental values extracted from GITT are represented by circles and the ones from EIS are represented by triangles. There is a good correspondence between EIS and GITT results for near-zero currents. These data have been fitted to the model proposed in this study (Equation ( 12)) and the optimized model predictions have been represented as solid lines on Figure 5.

There is a good agreement between the data and the proposed model (root mean squared error is 0.33 mΩ). The fitting parameters have been reported in Table 1. The values of the parameters at 25 °C are 0.47 mΩ for R sei and 32.5 A for the exchange current I 0 . The corresponding "charge transfers resistance" R ct,0 at 25 °C, as it is defined for EIS analysis for a near-zero currents (see Equation ( 14)), is 0.79 mΩ [START_REF] Ecker | Parameterization of a Physico-Chemical Model of a Lithium-Ion Battery: I. Determination of Parameters[END_REF]. Considering that the studied battery is already several years old, it is not surprising to find a SEI resistance that is significant compared to the charge transfers resistance. Finally, the sum of R sei and R ct,0 is 1. [START_REF] Takahashi | Reaction behavior of LiFePO 4 as a cathode material for rechargeable lithium batteries[END_REF] 5a, corresponding to Equation [START_REF] Illig | Separation of Charge Transfer and Contact Resistance in LiFePO 4 -Cathodes by Impedance Modeling[END_REF]. The calculus of R ct,0 , corresponding to Equation 14, is included.

R ct,0 (T ) = RT F I 0 (T ) (14)
The activation energy E a,sei related to R sei is 0.59 eV, which is close to the value found in literature (0.52 eV according to Pinson et al. [START_REF] Pinson | Theory of SEI Formation in Rechargeable Batteries: Capacity Fade, Accelerated Aging and Lifetime Prediction[END_REF]). The activation energy E a,I0 related to I 0 is 0.81 eV , which is higher than the values reported in the literature (between 0.52 to 0.73 eV [START_REF] Ecker | Parameterization of a Physico-Chemical Model of a Lithium-Ion Battery: I. Determination of Parameters[END_REF][START_REF] Xu | Charge-Transfer" Process at Graphite/Electrolyte Interface and the Solvation Sheath Structure of Li + in Nonaqueous Electrolytes[END_REF][START_REF] Jow | Distinguishing Li + Charge Transfer Kinetics at NCA/Electrolyte and Graphite/Electrolyte Interfaces, and NCA/Electrolyte and LFP/Electrolyte Interfaces in Li-Ion Cells[END_REF][START_REF] Liu | The Effect of Temperature on the Electrochemical Behavior of the V(IV)/V(V) Couple on a Graphite Electrode[END_REF]). This could be explained by the way the values found in the literature have been measured and we discuss this in the section 4.2. Based on the results, we consider that the model proposed in this study is experimentally validated.

The contributions of the charge transfers and the SEI to the battery voltage have been represented on Figure 6 by the corresponding non-linear resistances, respectively R ct and R sei (see Equation [START_REF] Illig | Separation of Charge Transfer and Contact Resistance in LiFePO 4 -Cathodes by Impedance Modeling[END_REF]). According to the proposed model, the charge transfers contribution R ct is the largest for near-zero currents. For the studied battery, R ct does not significantly depends on current at 25 °C and 45 °C. However, at 15 °C and for lower temperatures, it decreases quickly when the current increases. Consequently, the SEI contribution R sei becomes the largest at low temperatures (5 °C and below) and high currents (1C and above). These operating points are thus interesting for the SEI growth tracking.

Discussions

The authors would like to emphasize that the activation energies that are measured for the graphite-anode charge-transfer (same as E a,I0 in this study) in the cited references embed both charge transfer and SEI contributions to the anode impedance. In order to truly compare our results with the literature, we determined the activation energy related to the R surf values determined by EIS from -5 °C to 45 °C. By fitting these values to an Arrhenius law, we found a so-called surface-resistance activation-energy of 0.71 eV . This value lies in between E a,I0 (0.81 eV ) and E a,sei (0.59 eV ) and is more in accordance with the literature data (between 0.52 to 0.73 eV [START_REF] Ecker | Parameterization of a Physico-Chemical Model of a Lithium-Ion Battery: I. Determination of Parameters[END_REF][START_REF] Xu | Charge-Transfer" Process at Graphite/Electrolyte Interface and the Solvation Sheath Structure of Li + in Nonaqueous Electrolytes[END_REF][START_REF] Jow | Distinguishing Li + Charge Transfer Kinetics at NCA/Electrolyte and Graphite/Electrolyte Interfaces, and NCA/Electrolyte and LFP/Electrolyte Interfaces in Li-Ion Cells[END_REF][START_REF] Liu | The Effect of Temperature on the Electrochemical Behavior of the V(IV)/V(V) Couple on a Graphite Electrode[END_REF]). A proper separation of charge transfer and SEI activation energies might lead to higher values for the charge transfer of graphite anodes, like in this study.

Zoom on highest temperatures

As the balance between charge transfers and SEI contributions is likely to change during this battery lifetime because of SEI growth, an evolution of the "surface-resistance activation-energy" may occur.

It is noteworthy that the bell-shape of the surface resistance is narrower for lower temperature (Figure 5). At -5 °C, it is so narrow that even values at C/10 (4 A) are significantly smaller than the near-zero currents value of R surf . This makes the EIS measurements questionable, because it has been made with a current amplitude of C/10. Hence, the "small perturbation" hypothesis that is required for EIS measurements could be affected. This could be solved with an EIS test using a sufficiently reduced current for the lowest temperatures. Unfortunately, this could not have been done with the studied cell because his SEI resistance value had already significantly increased by the time we realized this issue. We consider that the possible underestimation of the EIS value at -5°C does not challenge this study conclusions.

From the results of Figures 5 and6, we can infer that GITT measurements are sufficient to fit the proposed nonlinearities model and, thus, to separate charge transfers and SEI. Assuming that an initial calibration of the model is done for a pristine cell and that it is embedded into a BMS, it seems achievable to do an online recalibration of the model parameters based on current pulses at different temperatures during operation (the lowest temperatures being the best). Nevertheless, several factors may impact the extraction quality of R surf from the BMS measurements and would have to be carefully studied, namely: potentially low sampling frequency, sensors accuracy, measurement noise, inaccurate separation of R surf and the diffusion part (see Figure 4).

Conclusion

The contributions of both electrodes charge transfers and of the anode SEI have been associated with one of the parameters of a simple equivalent circuit model (ECM), the so-called surface resistance R surf . The experimental determination of this parameter has been presented. It is based on GITT tests made from C/40 to 2.5C) and on EIS tests for near-zero currents. This experimental characterization has been carried out on a commercial LiFePO 4 /graphite prismatic cell. The determination of its surface resistance has been done over most of its operating range, from -5 °C to 45 °C and from 2.5C in discharge to 1C in charge.

A method has been proposed to separate the contributions of charge transfers and SEI that are mixed inside the surface resistance parameter. This method is based on an analytical model of the surface-resistance non-linearities regarding current and temperature. It is inspired by the Butler-Volmer relation in which the SEI contribution has been included in accordance with the literature. The thermal dependencies of the current exchange and the SEI equivalent resistance, using the Arrhenius law, have been justified and included into the non-linearities model.

Using the proposed model, the charge transfers and SEI contributions to the studied cell voltage have been separated with a satisfactory accuracy. They are characterized by their respective values at 25 °C and activation energies, that is four parameters. The activation energies of the SEI and charge transfers resistances are respectively 0.59 eV and 0.81 eV , which is in accordance with literature data. We showed that the contribution of the charge transfers to the surface resistance is major for low currents and low temperatures. Concerning the SEI contribution to the surface resistance, we demonstrated that it is dominant at high current and low temperature for the studied cell.

By construction, the proposed method is not specific to LiFePO 4 /graphite batteries and should be applicable to other battery electrodes inside of which the same electrochemical phenomena occur. In particular, the proposed approach is based on the assumptions of a negligible cathode-SEI contribution and of charge transfers that do not depend on SoC. By allowing the separation of charge transfers and anode SEI contributions, it allows to track the SEI growth of a battery during its lifetime in a non-invasive way. It would be expected for the activation energies to remain the same but for the SEI resistance, at least, to increase when battery is aging. This article approach may thus be interesting for a finer analysis of a battery state of health during aging After an initial calibration, the proposed model could be embedded in a BMS and, based on high current pulses at low temperature during the normal operation of a battery, it may be possible to recalibrate the model and thus estimate the SEI or charge transfer resistance evolution.

Figure 2 :

 2 Figure 2: Examples of parameters Rs and R surf extraction from EIS results at different temperatures, from 10 kHz to 10 mHz with 6 points per decade. (a) All impedance spectra and (b) zoom on the high frequencies values area.

Figure 3 :

 3 Figure 3: Example of parameters extraction from a GITT result for the studied cell. Their determination is made for several current and temperature values.

Figure 4 :

 4 Figure 4: Example of parameters extraction from the beginning of a GITT result for the studied cell (discharging pulse at C/20, 5 °C, between SoC 62.5% and 87.5%).

Figure 5 :

 5 Figure 5: Fitting the experimental surface-resistance values to Equation (12). Solid lines correspond to the model values.

Figure 6 :

 6 Figure 6: Separated contributions of the charge transfers and the SEI to the battery voltage, represented by the corresponding non-linear resistances Rct and R sei , as functions of the current and for different temperatures.

Table 1 :

 1 mΩ, which is close to the measured value by EIS (5% lower than 1.33 mΩ). Fit parameters for the surface resistance R surf model of the Figure

		Value at 25 °C Activation energy
	R sei	0.47 mΩ	0.59 eV
	I 0	32.5 A	0.81 eV
	(R ct,0 )	(0.79 mΩ)

Acknowledgment

The authors would like to thank Dr. Charles Delacourt for the fruitful discussions about the physical interpretation of ECM.