Paolo Stringari

Marco Campestrini

Christophe Coquelet

An Equation of State for Solid-Liquid-Vapor Equilibrium Applied to Gas Processing and Liquefied Natural Gas

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Conclusion and Perspectives

The

Results for Pure Components

A pressure-temperature range has been fixed for each pure compound; each range extends from a minimum to a maximum, located on the SVE and the SLE branches, respectively. These values are indicated in Table 1. The VLE branch is instead studied between the triple and the critical points for all the pure compounds. For ethane and propane the lower limits correspond to the triple point coordinates.

The quantitative comparison of the phase equilibrium values obtained with the SLV PR EoS and those from auxiliary equations [START_REF] Huber | NIST Standard Reference Database 23, Reference Fluid Thermodynamic and Transport Properties REFPROP, Version 8.0[END_REF] has been evaluated with reference to the common statistical indexes: AAD%, Bias% and MAD%.

P [MPa]

  Calculated values: -CH4; -Kr. Isothermal P-x cross sections: ■ VLE; ■ SLE; ■ SVE. Thermodynamic model Yokozeki [1] proposed the first EoS introducing a discontinuity in the solid-fluid transition region. This EoS presents a fourth power of volume, which allows representing all the three phases of matter with a single functional form. The fourth root is the common unphysical volume placed under the S-loop connecting the liquid and the vapor regions. The SLV PR EoS has been obtained starting from the Yokozeki EoS in which the Peng-Robinson's attractive term has been considered; this EoS is defined as follow: P: pressure T: temperature R: gas constant v: volume a,d: parameters b: liquid covolume c: solid covolume The proposed EoS has been used for representing phase equilibria of some pure compounds (Kr, CO 2 , CH 4 , C 2 H 6 , and C 3 H 8 ) and binary mixtures usually involved in gas processing and LNG production.

  SLV PR EoS allows a good representation of the SLE, SVE, and VLE in a wide range of temperature and pressure for different compounds of interest in the gas processing and LNG production. The intention of the authors is to extend the study to other pure compounds and to reduce the number of the parameters included in the SLV PR EoS for an industrial application of this equation. The EoS proposed can be extended to mixtures, and allows evaluating their thermodynamic behavior in the cryogenic field. This should be an important aid for designing and optimizing industrial cryogenic processes; knowing at which conditions in a mixture the solid-fluid equilibria appears (where fluid means either the liquid or the vapor phase) represents an opportunity of great interest, because it could be used for improving the process design and for avoiding some drawbacks as solidification of the impurities. The phase diagrams, as Px and Tx cross sections at cryogenic conditions, can give important information for evaluating new separation techniques, exploiting the capability of the SLV PR EoS in representing phase equilibrium values in which also the solid phase is involved.

Introduction

  Cryogenic processes are widely used in natural gas industry for the production of liquefied natural gas. Many efforts are made by the Liquefied Natural Gas (LNG) producing companies to ensure the safety of their plants. Despite the use of purification units, traces of pollutants can remain. These unwanted substances can form a mass of solid deposit causing dysfunctioning of the gas processing units. In order to avoid these problems in the industrial cryogenic processes involving natural gas treatment, the accurate knowledge of thermodynamic properties like solid-liquid, liquid-vapor and solid-vapor equilibria is of a great importance. We present a thermodynamic model, the Solid-Liquid-Vapor Peng-Robinson Equation of State (SLV PR EoS), suitable for being applied in the simulation of gas processing and LNG production thanks to its capability of accounting for the solid phase and the solid-fluid equilibrium. This model has been used for representing phase equilibria of some pure compounds and binary mixtures of interest in the gas processing and LNG production.

Figure 2 :

 2 Figure 2: Isobaric T-x SLE behavior. Calculated values: -SLE. Data: ○ Veith and Schrö der [4].

Figure 3 :

 3 Figure 3: Isothermal P-x VLE behaviors. 116 K ≤ T ≤ 151 K. Calculated values: -VLE. Data: ○ Holcomb and Zollweg [5]; × Calado and Staveley [6].

Figure 1 :

 1 Figure 1: Isothermal P-x VLE behaviors. 161 K ≤ T ≤ 205 K. Calculated values: -VLE; -critical line. Data [3]: VLE (□), critical line (∆).

Figure 5 Figure 4 :

 54 Figure 5: P-T-x equilibrium behaviors.

Table 2 represents these errors obtained calculating the equilibrium temperature at fixed pressure, for the compounds reported in Table 1.

			VLE			SVE			SLE
	Fluid	AAD	BIAS	MAD	AAD	BIAS	MAD	AAD	BIAS	MAD
		%	%	%	%	%	%	%	%	%
	Kr	0.07 -0.02 0.16	0.61 -0.61 2.02	0.19 -0.04 0.69
	CO2	0.04 -0.01 0.09	1.19 -1.19 4.29	0.24 -0.09 0.62
	CH4	0.07 -0.02 0.16	0.36 0.36	0.49	0.13 -0.03 0.32
	C2H6	0.08 -0.01 0.22				0.03 0.01	0.07
	C3H8	0.11 -0.02 0.24				1.41 -0.30 3.65

Table 1 : P-T ranges for the pure compounds studied.

 1

Table 2 : Summary of the statistical errors in calculating SVE, SLE, and VLE with the SLV PR EoS.

 2


David Bluck
email: david.bluck@aveva.com

Seiya Hirohama
email: seiya.hirohama@aveva.com

Nevin Gerek Ince

Marco Campestrini

Paolo Stringari

Freddy Garcia

Jean-Jacques Bartuel

Improving the Capability of Process Simulators to Represent Solid-Fluid Equilibria Applied to Natural Gas Liquefaction: The Methane + Neopentane System at Low Temperature

Natural gas is mainly methane, but it contains a number of other components, such as linear, branched, and cyclic hydrocarbons, aromatics, sulphur compounds, carbon dioxide, water, nitrogen, etc. Natural gas can be transported in pipelines or in its liquefied form, LNG, depending on the distance between the production site and the place of its utilization, the existence of geographical barriers, geopolitical considerations, or the need of a more flexible market. In order to be liquefied, natural gas must be cooled to about 110 K. This temperature

is achieved after the natural gas is cooled in the Main Cryogenic Heat Exchanger (MCHE) at high pressure, usually higher than 50 bar, and then expanded down to the atmospheric pressure. The triple point temperatures of most of the natural gas components are higher than the temperatures that natural gas achieves in the MCHE. Consequently, if the mole fraction of these components in natural gas is higher than their solubility limits at the temperature and pressure in the MCHE, they can form solid phases in the heat exchanger. The solid formation increases the pressure drop in the MCHE and reduces the heat transfer efficiency. This means more frequent maintenance operations are required, that can have a very important negative impact on the profitability of the liquefaction plant. In order to avoid solid formation in the MCHE, an accurate knowledge of the solid-liquid, solid-vapor, and solid-liquid-vapor equilibria is mandatory to design purification units allowing reducing the content of the potential solid formers below their solubility limits before entering the MCHE. For such a purpose, the capability of the process simulators of representing accurately the phase diagram of natural gas is very important to avoid designing oversized purification units or to limit the energy and material consumption in these units.

This work is the result of a collaboration among a natural gas producer, an EPC (Engineering, Procurement, and Construction) company, a simulation software company and a research laboratory with the aim of improving the knowledge of the solubility limits of some key impurities in LNG. In this perspective, neopentane has been selected as a potential solid former because of its high triple point temperature, 256.6 K. In addition, the only two articles in the literature reporting solubility data of neopentane in methane (the main natural gas component) are not in agreement each other. A wide experimental investigation of the liquidvapor, solid-vapor, solid-liquid, and solid-liquid-vapor equilibria for the methane + neopentane system at low temperature has been carried out and presented in Campestrini et al.

2022. This work shows how the data produced by the research laboratory are made available to industries thanks to the implementation of dedicated models and algorithms in a process simulator. In particular, an accurate model, based on the Soave-Redlich-Kwong (SRK) cubic Equation of State (EoS), has been developed for the phase equilibria of the methane + neopentane system including solid phases. In addition, a robust multi-phase flash algorithm has been developed for the solid-liquid(-liquid)-vapor equilibrium calculation. Rigorous thermodynamic phase stability analysis is used to determine the correct number of phases that will coexist at any given temperature and pressure. The new model and the phase equilibrium algorithm are integrated in the process simulator to improve the accuracy in the representation of the phase boundaries in natural gas and to allow an optimized design of the liquefaction units.

Keywords: Process Design and Development, Liquefied Natural Gas (LNG) and Liquefied Petroleum Gas (LPG), Crystallization


Léo Thiercelin
email: leo.thiercelin@ensam.eu

Sophie Cazottes
email: sophie.cazottes@insa-lyon.fr

Aurélien Saulot

Frédéric Lebon

Florian Mercier

Christophe Le Bourlot

Sylvain Dancette

Damien Fabrègue

Development of Temperature-Controlled Shear Tests to Reproduce White-Etching-Layer Formation in Pearlitic Rail Steel

Keywords: hat-shaped specimen, shear stress, pearlitic steel, thermomechanical test, dynamic recovery, white etching layer

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The increase in rail traffic and cumulative tonnage over the past decades has led to an increase in rolling-contact fatigue defects on the rail surface [START_REF] Zhu | Understanding and Treatment of Squat Defects in a Railway Network[END_REF]. The emergence of this increasing number of defects is generally associated with microstructural changes on the rail surface, such as severe plastic deformation and the formation of new, harder, and more brittle phases commonly known as brown and white etching layers (BEL and WEL, respectively) [START_REF] Kumar | Microstructural Evolution of White and Brown Etching Layers in Pearlitic Rail Steels[END_REF].

Understanding the kinetics of WEL formation is complex, as it depends on contact conditions such as thermomechanical, cyclic, multiaxial, and dynamic loading. Therefore, the thermomechanical path leading to WEL formation is not unique [START_REF] Al-Juboori | Squat Formation and the Occurrence of Two Distinct Classes of White Etching Layer on the Surface of Rail Steel[END_REF][START_REF] Al-Juboori | Characterisation of White Etching Layers Formed on Rails Subjected to Different Traffic Conditions[END_REF]. Nevertheless, microstructural characterization of the rail surface has provided evidence that the temperature and/or mechanical-stress field are the driving forces in WEL formation.

On the one hand, a purely thermal mechanism of WEL formation has been proposed by several authors [START_REF] Wang | Microstructure Features on Rolling Surfaces of Railway Rails Subjected to Heavy Loading[END_REF][START_REF] Wild | Microstructure Alterations at the Surface of a Heavily Corrugated Rail with Strong Ripple Formation[END_REF][START_REF] Pyzalla | Changes in Microstructure, Texture and Residual Stresses on the Surface of a Rail Resulting from Friction and Wear[END_REF][START_REF] Österle | Investigation of White Etching Layers on Rails by Optical Microscopy, Electron Microscopy, X-ray and Synchrotron X-ray Diffraction[END_REF][START_REF] Takahashi | Atom Probe Tomography Analysis of the White Etching Layer in a Rail Track Surface[END_REF][START_REF] Daniel | Final Report on the Rail Squat Project R3-105[END_REF]. Indeed, under high-braking conditions with high wheel sliding [START_REF] Al-Juboori | Squat Formation and the Occurrence of Two Distinct Classes of White Etching Layer on the Surface of Rail Steel[END_REF], the rail surface undergoes thermal cycles of heating with temperatures above 700 • C followed by rapid cooling that could progressively produce WEL spots. In addition, Nakkalil [START_REF] Nakkalil | Formation of Adiabatic Shear Bands in Eutectoid Steels in High Strain Rate Compression[END_REF] explained that dynamic loading applied to the rail surface induces localized plasticity and would lead to the formation of multiple adiabatic shear bands in the rail surface. These shear bands would then accumulate to form a homogeneous WEL, as also explained by Baumann [START_REF] Baumann | Formation of White-Etching Layers on Rail Treads[END_REF]. This process would explain the presence of martensite and residual austenite within the thermal WEL [START_REF] Zhang | Microstructural Investigation of White Etching Layer on Pearlite Steel Rail[END_REF].

On the other hand, under conventional traffic conditions on aligned tracks, the relative sliding ratio between the wheel and rail is generally limited to 2% by anti-skid devices [START_REF] Simon | De la Dynamique Ferroviaire à L'Accommodation Microstructurale du Rail-Contribution des TTS à La réponse Tribologique des Aciers-Cas du Défaut de Squat[END_REF], which limits the contact temperature to only several hundred degrees [START_REF] Knothe | Determination of Temperatures for Sliding Contact with Applications for Wheel-Rail Systems[END_REF][START_REF] Lian | Thermo-Mechanical Coupled Finite Element Analysis of Rolling Contact Fatigue and Wear Properties of a Rail Steel under Different Slip Ratios[END_REF]. Therefore, wheel-rail contact tests performed at a sliding ratio above 2% are not representative of the wheel-rail contact and highlight the thermal WEL formation [START_REF] Zhou | Phase and Microstructural Evolution in White Etching Layer of a Pearlitic Steel during Rolling-Sliding Friction[END_REF][START_REF] Bernsteiner | Development of White Etching Layers on Rails: Simulations and Experiments[END_REF]. WELs observed in areas subjected to low temperature rise are composed of highly deformed grains associated with an accumulation of plastic deformation [START_REF] Al-Juboori | Squat Formation and the Occurrence of Two Distinct Classes of White Etching Layer on the Surface of Rail Steel[END_REF][START_REF] Al-Juboori | Characterisation of White Etching Layers Formed on Rails Subjected to Different Traffic Conditions[END_REF][START_REF] Newcomb | A Transmission Electron Microscopy Study of the White-Etching Layer on a Rail Head[END_REF][START_REF] Lojkowski | The Mechanical Properties of the Nanocrystalline Layer on the Surface of Railway Tracks[END_REF]. The severe plastic deformation (SPD) progressively transforms pearlitic grains of several dozen of micrometers into nanograins of dislocated ferrite supersaturated in carbon [START_REF] Kumar | Microstructural Evolution of White and Brown Etching Layers in Pearlitic Rail Steels[END_REF][START_REF] Takahashi | Nanoscale Characterisation of Rolling Contact Wear Surface of Pearlitic Steel[END_REF]. In fact, the increase of the dislocation density would facilitate the fragmentation of the grains [START_REF] Tao | An Investigation of Surface Nanocrystallization Mechanism in Fe Induced by Surface Mechanical Attrition Treatment[END_REF][START_REF] Sauvage | The Role of Carbon Segregation on Nanocrystallisation of Pearlitic Steels Processed by Severe Plastic Deformation[END_REF][START_REF] Pan | The Microstructure Analysis of White Etching Layer on Treads of Rails[END_REF][START_REF] Pan | Formation of Nanocrystalline Structure in Pearlitic Steels by Dry Sliding Wear[END_REF][START_REF] Li | Brown Etching Layer": A Possible New Insight into the Crack Initiation of Rolling Contact Fatigue in Rail Steels?[END_REF][START_REF] He | On the Microstructure Evolution and Nanocrystalline Formation of Pearlitic Wheel Material in a Rolling-Sliding Contact[END_REF][START_REF] Pan | Investigation into the Evolution of Tribological White Etching Layers[END_REF], increase the kinetics of cementite dissolution [START_REF] Languillaume | Cementite Dissolution in Heavily Cold Drawn Pearlitic Steel Wires[END_REF][START_REF] Sauvage | Atomic-Scale Observation and Modelling of Cementite Dissolution in Heavily Deformed Pearlitic Steels[END_REF], and favor the carbon-atom mobility [START_REF] Gavriljuk | Decomposition of Cementite in Pearlitic Steel Due to Plastic Deformation[END_REF]. In addition, such a level of deformation would be linked to the intrinsic conditions of the wheel-rail contact that combines high compressive stresses with high shear stresses. Under these severe conditions, the material could locally reach a level of severe deformation that would be analogous to the severely deformed specimens in the case of cold drawing [START_REF] Zhang | Microstructure and Strengthening Mechanisms in Cold-Drawn Pearlitic Steel Wire[END_REF][START_REF] Lamontagne | Comparative Study and Quantification of Cementite Decomposition in Heavily Drawn Pearlitic Steel Wires[END_REF][START_REF] Djaziri | Deformation-Induced Martensite: A New Paradigm for Exceptional Steels[END_REF] or SPD experiments [START_REF] Sauvage | The Role of Carbon Segregation on Nanocrystallisation of Pearlitic Steels Processed by Severe Plastic Deformation[END_REF][START_REF] Lojkowski | Nanostructure Formation on the Surface of Railway Tracks[END_REF][START_REF] Ivanisenko | The Mechanism of Formation of Nanostructure and Dissolution of Cementite in a Pearlitic Steel during High Pressure Torsion[END_REF]. Indeed, high-pressure torsion tests [START_REF] Ivanisenko | The Mechanism of Formation of Nanostructure and Dissolution of Cementite in a Pearlitic Steel during High Pressure Torsion[END_REF] performed for pearlitic steels at room temperature confirmed the mechanically induced WEL formation under hydrostatic pressure and shear coupling.

These two scenarios cannot really be distinguished, and the hypothesis of a thermomechanical coupling is the most probable to explain and predict WEL formation [START_REF] Kumar | Microstructural Evolution of White and Brown Etching Layers in Pearlitic Rail Steels[END_REF][START_REF] Steenbergen | On the Mechanism of Squat Formation on Train Rails -Part I: Origination[END_REF]. A macroscopic model developed by Antoni et al. [START_REF] Antoni | A Combined Thermo-Mechanical Model for Tribological Surface Transformations[END_REF] was then proposed to simulate the mechanical-stress effect on WEL formation. In this model, a coupling between the hydrostatic pressure and temperature was proposed. Nevertheless, this model does not take into account the shearing of the grains at the extreme surface of the rails. To overcome these limits, Thiercelin et al. [START_REF] Thiercelin | Thermomechanical Modelling of the Tribological Surface Transformations in the Railroad Network (White Eching Layer)[END_REF] improved this model by adding a shear contribution to the WEL formation criterion. The WEL formation would result from a coupling between the hydrostatic pressure and the shear stress, which would be enhanced by the temperature. The tendencies of WEL formation kinetics have already been confirmed in the literature [START_REF] Ivanisenko | The Mechanism of Formation of Nanostructure and Dissolution of Cementite in a Pearlitic Steel during High Pressure Torsion[END_REF][START_REF] Hilliard | Iron-Carbon Phase Diagram Isobaric Sections of the Eutectoid Region at 35, 50 and 65 Kilobars[END_REF][START_REF] Wu | Laboratory Simulation of Martensite Formation of White Etching Layer in Rail Steel[END_REF][START_REF] Vargolici | Influence of the Initial Surface State of Bodies in Contact on the Formation of White Etching Layers under Dry Sliding Conditions[END_REF]; however, the model needs to be more accurately experimentally identified. For this purpose, this coupling has been de-correlated to separately quantify the effects of temperature, hydrostatic pressure, and shear.

First, Merino [START_REF] Merino | An Attempt to Generate Mechanical White Etching Layer on Rail Surface on a New Rolling Contact Test Bench[END_REF], Lafilé [START_REF] Lafilé | Characterization of the Microstructure Changes Induced by a Rolling Contact Bench Reproducing Wheel/Rail Contact on a Pearlitic Steel[END_REF], and Thiercelin [START_REF] Thiercelin | Modélisation Multi-Physique Des Mécanismes de Formation de La Phase Blanche[END_REF] successfully reproduced the kinetics of WEL formation under representative wheel-rail contact conditions at 1/15th scale. These tests demonstrated the predominantly mechanical formation of the WEL, as the slip level was less than 2%. Moreover, the combined effect of contact pressure and sliding was confirmed, validating one part of the model proposed by Thiercelin et al. [START_REF] Thiercelin | Thermomechanical Modelling of the Tribological Surface Transformations in the Railroad Network (White Eching Layer)[END_REF], i.e., the pressure-shear coupling.

Second, temperature-controlled shear tests have demonstrated the formation of adiabatic shear bands in steels [START_REF] Habak | Etude de L'Influence de la Microstructure et des Paramètres de Coupe sur le Comportement en Tournage dur de L'Acier à Roulement 100[END_REF][START_REF] Dougherty | Mechanical Behavior and Microstructural Development of Low-Carbon Steel and Microcomposite Steel Reinforcement Bars Deformed under Quasi-Static and Dynamic Shear Loading[END_REF]. In addition, tests performed by Lins et al. [START_REF] Lins | A Microstructural Investigation of Adiabatic Shear Bands in an Interstitial Free Steel[END_REF] on low-carbon interstitial-free steels (IF steel) revealed shear-band formation kinetics, called "progressive subgrain misorientation", which is similar to the mechanically induced WEL kinetics. Such tests conducted on pearlitic steels are still lacking and would constitute a major breakthrough for the identification of the model proposed by Thiercelin et al. and more generally for the formation of shear bands in pearlitic steels.

The objective of the current study was to reproduce shear tests under controlled temperatures representative of wheel-rail contact. Monotonic and cyclic shear tests were conducted for different temperature ranges (from 20 • C to 400 • C). Optical and scanning microscopy observations were performed to characterize each test condition. First, static monotonic tests with microstructure analysis are presented for each temperature tested (20 • C, 200 • C, 300 • C, and 400 • C). Next, the results of the cyclic tests are presented for each tested temperature. Finally, the effect of temperature and loading path are discussed with respect to the microstructural evolution eventually leading to WEL formation.

Materials and Methods

Material

The material studied was a pearlitic rail steel grade R260 used in most of the French railway network. The chemical composition of the material is given in Table 1. In its as-received state, the microstructure consists of pearlitic grains with a diameter measured to be approximately 9.6 ± 8.1 µm. The grains initially have a rather globular shape with an aspect ratio of 2.0 ± 0.9 and are weakly disoriented. There are initially 30% low-angle grain boundaries (LAGBs), a majority of medium-angle grain boundaries (MAGBs) (40%), and 30% high-angle grain boundaries (HAGBs) (Figure 1a). The group of grains having a similar crystallographic orientation constitutes a pearlitic colony whose average size is generally several tens of micrometers [START_REF] Dylewski | The Tridimensional Gradient of Microstructure in Worn Rails -Experimental Characterization of Plastic Deformation Accumulated by RCF[END_REF] (Figure 1b).

Thermomechanical Test Bench

Thermomechanical tests were performed using the Gleeble 3800 device, a test bench that can simulate a complex thermomechanical loading path by performing mechanical cycles and thermal cycles simultaneously (Figure 2). The Gleeble thermomechanical test bench can perform compression tests at a rate up to 2000 mm/s. The measurement and control of the strain was performed by the displacement of the mobile die previously calibrated. In addition, the specimen temperature was measured with type K thermocouple welded on the sample. The specimen was heated by conduction, and the temperature was continuously monitored with a control loop adapting the amount of current in the sample in order to get the desired thermal cycle. For more information, please refer to the Gleeble company website [51].

Hat-Shaped Specimen

The shear tests under controlled temperature were performed with the help of a hatshaped specimen, which, under the effect of a compression, generated a localized shear in a crown of width L = 2 mm and thickness d = 0.2 mm (Figure 3). The thickness d corresponded to the size of the deformation gradient observed at the rail surface. The other dimensions were chosen from previous work on similar alloys [START_REF] Hor | Simulation Physique des Conditions Thermomécaniques de Forgeage et d'Usinage: Caractérisation et Modélisation de la Rhéologie et de l'Endommagement[END_REF][START_REF] Hor | An Experimental Investigation of the Behaviour of Steels over Large Temperature and Strain Rate Ranges[END_REF] and are summarized in Table 2.

Figure 3. Scheme of an axisymmetric section of a hat-shaped specimen subjected to a compressive load with r i and r e representing the internal and external diameters of the specimen, respectively, and d = r ir e and L representing the thickness and width of the sheared zone, respectively. The sheared zone is then a crown of length L and thickness d. To analyze the experimental data, some theoretical assumptions about the stress and strain field in the sheared zone are required. The hat specimens are assumed to undergo pure shear, which is confined to the theoretical shear zone (green area in Figure 3). The stress and strain tensors are then expressed as follows:

σ =   0 τ 0 τ 0 0 0 0 0   with τ = F πL(r i + r e ) (1)
ε =   0 γ/2 0 γ/2 0 0 0 0 0   with γ = tan(α) = u d (2)
with τ representing the shear stress induced in the sheared zone, which depends on F, the applied compressive load; L, the length of the sheared zone; and r i and r e , the inner and outer radius of the specimen, respectively (Figure 3). In addition, γ is the shear strain, which depends on u, the displacement in the compression direction, and d, the theoretical thickness of the sheared zone.

It must be pointed out that the heterogeneous shape of the specimen induces temperature heterogeneity in the specimen. Preliminary heat-conduction tests showed that there was a difference of approximately 10 • C between the lower and upper part of the specimen. Nevertheless, the current study was focused on the effect of the temperature in the shear zone; therefore, only one thermosensor was welded in the vicinity of the shear stress (Figure 4), and the temperature was considered uniform in the specimen during all the mechanical tests.

Microstructural Characterization

Optical microscopy characterization was performed after mechanical polishing down to 1 µm and subsequent Nital etching to observe cementite lamellae and the possible WEL.

The samples were also prepared for observation in secondary electron (SE) mode and electron backscatter diffraction (EBSD) mode in the scanning electron microscope. For the EBSD mode, the samples were mechanically polished down to 1 µm using a conventional grinding machine. The final preparation step consisted of vibratory polishing for approximately 1 h using a colloidal silica suspension (Struers OP-S) with a grain size of 0.05 µm. For some cross-section samples, it was possible to perform an additional ionic polishing step with a GATAN ILLION 2 device with an accelerating voltage of 4 kV. The observations were performed using a FEG Zeiss Supra 55VP microscope equipped with an Oxford EBSD Symmetry detector. The accelerating voltage was set to 12 or 15 kV, depending on the size of the microstructural elements to be analyzed. The EBSD data treatment was performed using Atex software [START_REF] Beausir | Analysis Tools for Electron and X-ray Diffraction[END_REF]. For the data treatment of the EBSD maps, the disorientation angle selected for grain detection was set to 5 • , and only grains with at least 10 pixels were considered, as the detection is not reliable below this level. The statistical distributions of the grain size and the aspect ratio in each EBSD map were characterized from mean values followed by the the standard deviation noted "±". Figure 5 indicates the color code used to define the crystallographic textures of all the inverse pole figures (IPF) presented in this study.

The sheared areas of each specimen were then analyzed after having been axially cut and prepared as previously described. It is worth noting that some sheared zones were analyzed after the failure of the specimen and some were analyzed before. Depending on the final state of the specimen, an associated scheme is included to facilitate understanding of the microstructural analyses (Figure 6). The sheared zones were systematically compared to the stages of microstructural evolution leading to the WEL formation observed on the rail surface. Based on a mechanism proposed by several authors [START_REF] Pan | Formation of Nanocrystalline Structure in Pearlitic Steels by Dry Sliding Wear[END_REF][START_REF] Pan | Investigation into the Evolution of Tribological White Etching Layers[END_REF] and the formalism initially proposed by Thiercelin et al. [START_REF] Thiercelin | Modélisation Multi-Physique Des Mécanismes de Formation de La Phase Blanche[END_REF], three microstructural indicators were used to quantitatively describe the progressive evolution of the microstructure: mean grain size, aspect ratio, and grain-boundary disorientation. For the latter, three intervals of grain disorientation were considered:

• LAGBs for angles between 5 • and 15 Figure 7 is a longitudinal cross-section of worn pearlitic rail with evidence of a microstructural gradient from the pearlitic stage to the WEL stage over approximately 60 µm from the contact surface. The microstructural gradient analysis provides insights into the kinetics of WEL transformation. This gradient can be divided into several successive stages of transformation located at different depths. First, the grains begin to fiber and remain mainly unfragmented (stage 2) from 40-µm depth. The average grain size starts to decrease, the aspect ratio is very high, and the grain disorientation remains quite low. Next, between 20-and 40-µm depth, the grains fragment but still keep a rather elongated shape (stage 3). At 10-µm depth, there is a fine area of some micrometers composed of strongly disoriented nanograins. This is the non-fibrous and nanostructured stage (stage 4). Finally, in the first 10 µm from the contact surface, the grains appear white after Nital etching, which corresponds to a WEL spot. At this stage, the nanograins are spherical and much more disoriented than in the previous stages. This stage is considered the final stage in WEL formation (stage 5). The stages of transformation leading to WEL formation and the associated microstructural indicators proposed by [START_REF] Thiercelin | Modélisation Multi-Physique Des Mécanismes de Formation de La Phase Blanche[END_REF] are presented in Table 3. Longitudinal cross section of a rail extracted from region with a high WEL frequency [START_REF] Simon | Tribological Characterization of Rail Squat Defects[END_REF]; the microstructural gradient consists of four areas representing the differents evolution stages leading to WEL formation. Stage 2 is the fibrous state of the grains without fragmentation; stage 3 is a state where the grains are still fibrous but fragmented; stage 4 is a state where the grains have no particular orientation and are very fragmented; and stage 5 is the state that appears white in optical microscopy after Nital etching. Table 3. Evolution of indicators and the stages of evolution from the pearlitic state to the final WEL state according to [START_REF] Thiercelin | Modélisation Multi-Physique Des Mécanismes de Formation de La Phase Blanche[END_REF].

Stage

Grain

Results

Monotonic Tests

As the temperature in the wheel-rail contact does not exceed several hundred degrees, testing temperatures of less than 400 • C were selected. First, monotonic shear tests were performed at four temperatures until failure (20 • C, 200 • C, 300 • C, and 400 • C) at a quasi-static loading rate ( γ = 0.5 s -1 ). The conditions of the monotonic tests are given in Table 4. A macroscopic analysis of the experimental curves combined with microstructural characterization of each condition tested was conducted. The macroscopic behavior of the material was studied by plotting the stress-strain curve (Figure 8) obtained with the stress and the associated shear strain as theoretically observed by the material in the sheared zone (Equations ( 1) and ( 2)). For each temperature, at least two experiments were performed and showed similar stress-strain response, confirming the repeatability of the experiments. For temperatures below 300 • C, the ductility of the material decreased with temperature. The failure strain was on the order of 5.8 at room temperature, whereas it was 5.1 at 200 • and 4.9 at 300 • C. Moreover, in this temperature range, the stress-strain curves at 20 • C and 300 • C followed the same trend, differing from that at 200 • C. Indeed, for the latter, the material became significantly less resistant than at 20 • C and 300 • C as its apparent yield stress (stress beyond which the behavior becomes non-linear) and its critical shear stress before failure were much lower. The observations at these three temperatures reveal three different behaviors, which do not allow conclusions to be drawn on the trend of behavior in this temperature range.

In addition, the curves at 200 • C and 300 • C exhibit serrations characterized by microstress drops during monotonic loading. This mechanism can likely be attributed to the Portevin-Le Chatelier effect (PLC) already encountered for pearlitic steels in this temperature range [START_REF] Inoue | Microstructures of Deformation and Fracture of Cementite in Pearlitic Carbon Steels Strained at Various Temperatures[END_REF][START_REF] Tsuzaki | Fatigue Deformation Accompanying Dynamic Strain Aging in a Pearlitic Eutectoid Steel[END_REF][START_REF] Gonzalez | Measurement of Dynamic Strain Aging in Pearlitic Steels by Tensile Test[END_REF]. This mechanism, also known as dynamic strain aging (DSA), corresponds to an instability of the plastic flow in metals when dislocations interact with atoms in solid solutions. During plastic deformation, the dislocations are blocked by the atoms in solid solutions until a critical force is reached. Subsequently, the stress falls until the next obstacle, explaining the serrations.

It is, however, quite surprising that similar serrations were observed regardless of the temperature. Indeed, previous studies have shown that the PLC is favored at higher temperatures in pearlitic steels. Therefore, we consider that the serrations observed could also be experimental artifacts.

For a temperature of 400 • C, the material response differed completely from that at the previous temperatures tested. It became softer and more ductile compared with the material at the other temperatures, with a failure strain above 7. It must be pointed out that for this experiment, the specimen did not fail. Moreover, the apparent yield strength estimated at 300 MPa drastically decreased compared with that at 20 • C (approximately 450 MPa).

The following section will focus on the microstructural characterization of the shear zone performed at the end of the monotonic test for each temperature. First, between 20 • C and 300 • C, a microstructural gradient similar to that of the rail surface was observed. At room temperature, the microstructural gradient extended over a hundred micrometers with grains that gradually transformed from the broken surface, in a fragmented and unfibered state (stage 4), to a fibrous stage at a depth of 100 µm (stage 2). In the intermediate area, between 30 and 60 µm, a mixture of grains making the transition between stage 3 and 4 was observed. These optical observations are confirmed by the EBSD map within this area, where the grain have a relatively globular shape with an aspect ratio of 2 ± 1 and are submicrometric in size (0.5 ± 0.2 µm). In addition, their disorientation is rather high as the proportion of MAGBs and HAGBs is predominant at 39% and 33%, respectively. The evaluation of all these indicators lead to the conclusion that the grains in the area studied by EBSD are in a transition stage between stages 3 and 4 (Table 3) as seen in Table 5.

Microstructural Characterization

Grain Size (µm)

Aspect Ratio LAGB-MAGB-HAGB (%) Stage Estimated 0.5 ± 0.2 1.9 ± 0.7 28-39-33 3 and 4

The same test performed at 200 • C revealed a similar microstructural gradient as that at room temperature. The sheared zone consisted of sheared grains that gradually became less fragmented and fibered to a depth of 60 µm. This sheared zone appears more confined than that for the test performed at 20 • C, which was approximately 100 µm thick. The effect of temperature on the strain localization appears to confirm the results at 300 • C, where the grains were slowly sheared. The analysis of the stress-strain curves combined with the microstructural gradient indicate that the plastic deformation was much less severe at 200 • C and 300 • C, where it is very confined.

As noted above, the material was much more ductile at 400 • C and did not fail. The micrograph shows the microstructural gradient of the specimen strained until a shear strain of 6.9. No WEL spots were observed; however, the grains were highly fibered over a large thickness of 200 µm, which corresponds to the theoretical sheared zone d (Figure 3). A main crack that initiated at the corner of the specimen propagated along the interface between the fibered structure and the as-received pearlitic grains. We also denote the presence of many secondary cracks within the sheared zone.

The EBSD map of the area close to the crack tip reveals a strong grain heterogeneity in the sheared zone. The grains on either side of the crack were deformed and reoriented in the shear direction without being fragmented (stage 2). Close to the most sheared zone, the grains were fibered and fragmented (stage 3). At the tip of the crack, the grains were not fibered and were very fragmented and randomly oriented, corresponding to a critical grain state close to the WEL state (stage 4). In addition, a network of secondary cracks in the presence of grains at the same critical stage (stage 4) nearby were observed.

Cyclic Tests

Macroscopic Analysis

Cyclic shear tests were then conducted for different temperatures to simulate the thermomechanical loading path undergone by the rail. Because WEL formation is related to the cementite dissolution by mechanical stresses and, in particular, by dislocations, a maximal cyclic stress slightly higher than the apparent elastic shear stress determined with the monotonic stress-strain curves was applied (Figure 8). In addition, to maintain contact and ensure good thermal conduction, a minimum cyclic shear stress of 100 MPa was imposed, which corresponds to the minimum contact force of 5 kN.

A maximum stress of 560 MPa was considered for 20 • C and 300 • C. The specimen failure occurred after 600 cycles compared to 1400 cycles at 300 • C.

For a temperature of 200 • C, only a few cycles would have led to the failure of the specimen. At this temperature, a lower maximum stress of 500 MPa was thus considered. A test of 10,000 cycles was conducted without breaking the specimen. The thermomechanical conditions for the cyclic tests are summarized in Table 6. These preliminary results highlight the significant effect of temperature on the fatigue limit of the material. In a similar manner as for the monotonic case, a different behavior was observed for each temperature. Indeed, between 20 • C and 200 • C, the fatigue limit of the material decreased as the temperature increased. In contrast, at 300 • C, the fatigue limit unexpectedly became greater than at 20 • C. Cyclic tests before specimen failure were conducted for all three temperatures. The following section will present the microstructural observations of the three tests before failure.

Microstructural Characterization

Figure 10 presents optical micrographs after Nital etching of the shear zone at the end of each cyclic test stopped before the specimen broke. Regardless of the test temperature, a crack usually started in the corner of the cap and propagated over a few hundred micrometers.

For the test at 20 • C, the grains downstream of the crack were sheared but appeared slightly fragmented (stage 2). Moreover, on both sides of the crack, a transformation gradient up to the WEL was visible (see the optical zoom in this area). It can be assumed that cracks might have propagated due to incompatibility of the microstructure. 6. The red dotted line denotes the start of cracking in the corner of the specimen. The red box corresponds to the area investigated by EBSD in Figure 11.

An EBSD map of the region around the crack (red box in the micrograph at room temperature of Figure 10) is presented to measure the microstructural indicators in this area (Figure 11). This map is divided into four distinct areas as follows: Region A, the lower part of the crack, was composed of fibered and elongated grains with an average size of 0.4 ± 0.2 µm and an aspect ratio of 2 ± 1, on average. In this region, there is a majority proportion of low disoriented grains (45%) and fairly close proportions of weakly and very disoriented grains (38% and 18%, respectively). All of these indicators are consistent with the identification of stage 3 in zone A.

Region B, corresponding to a WEL optically, was unindexed in EBSD, suggesting a significant grain-size reduction that is smaller than the indexation stepsize of 0.06 µm and a high level of grain disorientation.

Region C, which was optically dark (Figure 10, T = 20 • C), did not show evidence of any apparent grain fibering. This zone consisted of spherically shaped nanograins (aspect ratio of 1.6 ± 0.5), with a fairly uniform size (0.3 ± 0.1 µm). The grains had no particular crystallographic orientation and were much more disoriented than those in region A. Indeed, weakly disoriented grains occupied only 12% compared to 45% in zone A. There was then a large majority of large disorientation angles equally distributed between the moderately disoriented and highly disoriented grains (43% and 45%, respectively). All these observations are indicative of stage 4 in this region.

The set of indicator values in zones A and C of Figure 11 are summarized in Table 7.

Figure 11. T = 20 • C -N cycle = 500 with τ min /τ max = 100/560 MPa, γ final = 3.5. IPF X map of the microstructural gradient of the cracked zone that corresponds to the white box in Figure 10 for the case at room temperature, stepsize = 0.06 µm. This region is divided into four areas: zone A is composed of fibered and fragmented grains (stage 3). Zone B is a WEL area with very small grains and disoriented grains. Zone C is composed of very small and disoriented grains (stage 4), whereas zone D is a transition area with grains that begin to fiber and fragment themselves (stage 2 and 3). The micrographs at 200 • C and 300 • C exhibit the same microstructure (Figure 10). Contrary to the test performed at room temperature, the crack, being initiated at the corner, propagated normally to the shear direction. The grains were slowly deformed downstream to the crack, and no WEL was observed.

Discussion

Effect of the Thermomechanical Path on WEL Formation Kinetics

The monotonic tests conducted at the four temperatures from 20 • C to 400 • C did not lead to WEL formation. The monotonic tests revealed a strongly non-linear behavior of the material with temperature. Indeed, at 200 • C, the material was softer and more brittle than at room temperature. At 300 • C, behavior similar to that at 20 • C was observed, with the material being more resistant and ductile. At 400 • C, the stress-strain curve indicated a much more ductile behavior, which can be explained by the higher ductility of the grains [START_REF] Inoue | Microstructures of Deformation and Fracture of Cementite in Pearlitic Carbon Steels Strained at Various Temperatures[END_REF]. WEL formation was never observed after monotonic tests. However, the analysis of their microstructure indicates that the temperature does not seem to favor the deformation/fragmentation required for WEL formation. Indeed, an analysis of the microstructures using the WEL formation indicators defined in Table 3 reveals that the most advanced stages were found for the RT sample. Table 8 summarizes the different microstructures obtained at the end of the monotonic tests. Regarding the cyclic tests, a WEL was observed at 20 • C. However, at 200 • C and 300 • C, little deformation or transformation of the grains was observed. Table 9 summarizes the different microstructures obtained at the end of the cyclic tests.

It appears that WEL formation is possible under the effect of shear at room temperature via a mechanism of grain fragmentation and transformation for a sufficient level of deformation, as presented by Thiercelin [START_REF] Thiercelin | Modélisation Multi-Physique Des Mécanismes de Formation de La Phase Blanche[END_REF]. Indeed, the tests performed at higher temperatures (200 • C and 300 • C) resulted in less deformation of the microstructure for both the static and cyclic tests compared with that at room temperature. Localization of the stress was observed, which is likely related to the geometry of the specimens, which would crack before forming a WEL.

In addition, in the 200-300 • C temperature range, the kinetics of WEL formation could be inhibited by other microstructural mechanisms, such as dynamic recovery. The latter would enhance the annihilation of dislocations and therefore enhance the deformability of the microstructure.

Another physical mechanism could concern the cementite precipitation beginning at 300 • C [START_REF] Ivanisenko | The Mechanism of Formation of Nanostructure and Dissolution of Cementite in a Pearlitic Steel during High Pressure Torsion[END_REF], which would counterbalance its dissolution under the effect of mechanical stresses. Finally, the more ductile behavior at 400 • C can be attributed to a mechanism of deformation of each phase of the grains (ferrite and cementite) without fragmentation [START_REF] Tsuzaki | Fatigue Deformation Accompanying Dynamic Strain Aging in a Pearlitic Eutectoid Steel[END_REF].

Table 9. Synthesis for the cyclic shear tests. The same maximal stress of 560 MPa was considered for 20 • C and 300 • C. For the test at 200 • C, such a maximum stress would have been too close to the monotonic failure stress (Figure 8). A few cycles would have led to the failure of the specimen, which explains the choice of a lower maximum shear stress (500 MPa rather than 560 MPa). In this study, the effect of contact temperature on the fatigue strength of the rails was studied in order to anticipate the failure of the rails. At room temperature, the material was much more ductile, and failure occurred after WEL formation. For higher temperatures (200 • C and 300 • C), the material was less ductile and the failure occurred after a more localized deformation. Finally, the monotonic test conducted at 400 • C revealed crack initiation in highly nanostructured zones.

Temperature (

The failure mechanism clearly differs with temperature, and the deformation of the microstructure at stages less advanced than WEL formation can already be critical for the material depending on the contact temperature.

Nevertheless, the stage of transformation of the microstructure combined with the contact temperature is a good indicator of the probability of material failure. The WEL stage is the most critical. This assumption can be related to the studies of Saxena et al. [START_REF] Saxena | Micro Fracture Investigations of White Etching Layers[END_REF], where the toughness of WELs and work-hardened zones of rail steels were studied. The authors concluded that in the case of wheel-rail contact, the toughness of the material is inversely proportional to its hardness. As hardness is linked to the dislocation density, grain size, and carbon content in solid solution [START_REF] Zhang | Microstructure and Strengthening Mechanisms in Cold-Drawn Pearlitic Steel Wire[END_REF][START_REF] Carroll | Surface Metallurgy and Rolling Contact Fatigue of Rail[END_REF][START_REF] Park | Quantitative Measurement of Cementite Dissociation in Drawn Pearlitic Steel[END_REF], the toughness would then be directly dependent on the evolution stages (Table 3).

Thermomechanical Model and Wheel-Rail Contact Conditions

The role of temperature in the kinetics of WEL formation is contestable, as temperatures above 200 • C will activate other microstructural mechanisms that inhibit grain fragmentation-transformation kinetics. The model proposed by Thiercelin [START_REF] Thiercelin | Thermomechanical Modelling of the Tribological Surface Transformations in the Railroad Network (White Eching Layer)[END_REF] would then be invalidated for temperatures of 200 • C, 300 • C, and 400 • C. Future tests at lower temperatures (below 200 • C) will have to be conducted to determine whether low-temperature elevations could still favor the WEL formation mechanism, as noted by Newcombs et al. [START_REF] Newcomb | A Transmission Electron Microscopy Study of the White-Etching Layer on a Rail Head[END_REF]. In addition, for temperatures close to the austenitization temperature, the stresses could facilitate WEL formation [START_REF] Zhu | Understanding and Treatment of Squat Defects in a Railway Network[END_REF][START_REF] Murugan | Study on White Etching Layer in R260Mn Rail Steel by Thermo-Mechanical Simulation[END_REF][START_REF] Al-Juboori | Thermomechanical Simulation of White Etching Layer Formation on Rail Steel[END_REF].

The second point concerns a noteworthy difference in the thermomechanical path of the tests and the reality of the wheel-rail contact. Indeed, contrary to the contact conditions, the temperature was kept constant during the entire duration of the applied cyclic mechanical loading. In reality, the cyclic thermal load is more complex than in those tests and could modify the kinetics of the microstructural transformations. Therefore, the WEL formation criteria would then depend explicitly on the temperature evolutions with time (heating and cooling rates) and on the accumulation process of deformation/temperature/shear rather than on the temperature itself.

Finally, the interactions of the different scales of the wheel-rail contact (train, wheelrail interaction, and the effect of wear particles) lead to a variability of the thermomechanical field applied to the rail surface. The tests performed in this study showed a strongly non-linear behavior of the material, which was explained by the activation of various microstructural mechanisms, such as dynamic recovery, cementite precipitation and or dissolution, grain fragmentation, phase transformation, and/or deformation. Depending on the temperature, the material will have different cyclic responses, which may lead to the development of rolling-contact fatigue defects or contribute to the wear rate if the contact temperature is sufficiently high, as observed during bi-disc tests for temperatures above 300 • C [START_REF] Lewis | Mapping Rail Wear Regimes and Transitions[END_REF].

Conclusions

Monotonic and cyclic shear tests under controlled temperatures were performed using hat-shaped specimens and the Gleeble thermomechanical simulator in an attempt to reproduce the WEL formation induced at the rail surface.

The monotonic tests conducted at four temperatures (20 • C, 200 • C, 300 • C, and 400 • C) did not lead to WEL formation but resulted in grain-transformation stages close to WEL formation. The cyclic tests conducted at 20 • C, 200 • C, and 300 • C in a plastic regime confirmed the monotonic trends and resulted in WEL formation at room temperature only. The effect of medium temperature appears to be unfavorable for WEL formation by fragmentation-transformation of grains as it activates other mechanisms, such as dynamic recovery or cementite precipitation.

The thermal coupling part of the model proposed by Thiercelin et al. [START_REF] Thiercelin | Thermomechanical Modelling of the Tribological Surface Transformations in the Railroad Network (White Eching Layer)[END_REF] was then invalidated for temperatures between 200 • C and 400 • C; however, the formation of WEL by pure shear was confirmed. Further shear tests at temperatures between 20 • C and 200 • C must be considered to validate the model at low temperature.

The residual stresses in the sheared zone will be further estimated to improve the constitutive model of WEL formation. Moreover, these measurements will be compared to the residual stresses measured on the real rail surface in presence of WEL [START_REF] Wild | Microstructure Alterations at the Surface of a Heavily Corrugated Rail with Strong Ripple Formation[END_REF][START_REF] Pyzalla | Changes in Microstructure, Texture and Residual Stresses on the Surface of a Rail Resulting from Friction and Wear[END_REF].

The tests confirmed that the evolution stages leading to WEL formation could reflect a probabilistic criterion of crack initiation. The probability of cracking of the material increases inversely with decreasing grain size, which is related to the stage of evolution of the microstructure. In the railroad, the crack initiation could then be explained by an advanced stage of evolution of the microstructure (stage 4 or 5).

These tests showed limitations due to premature cracking, which could be overcome by performing identical tests under hydrostatic pressure to limit the damage [START_REF] Eleöd | Deformability of the Near Surface Layer of the First Body[END_REF][START_REF] Eleöd | Transient and Stationary Changes of the Mechanical Properties of the First Body Governed by the Hydrostatic Pressure Component of the Local Stress State during Dry Friction[END_REF] and to be more representative of the multiaxial stresses undergone by the rail. Moreover, the achievement of thermal cycles simultaneously with the mechanical cycles constitutes future perspectives to simulate more faithfully the thermomechanical path undergone by the rail.

Figure 1 .

 1 Figure 1. (a) SEM microstructure of the as-received structure and (b) EBSD IPFZ map of the asreceived microstructure.

Figure 2 .

 2 Figure 2. Gleeble 3800 test bench with the positioning of the hat-test specimen: the force (or displacement) is applied by a mobile die, and the temperature is induced by thermal conduction and is controlled with a type K thermocouple. For more information please see [51].

Figure 4 .

 4 Figure 4. Welding of the thermosensor. The control of the temperature was achieved with the sensor labelled "TC1".

Figure 5 .Figure 6 .

 56 Figure 5. Inverse pole figure color coding of orientation maps presented in this study.

Figure 7 .

 7 Figure 7.Longitudinal cross section of a rail extracted from region with a high WEL frequency[START_REF] Simon | Tribological Characterization of Rail Squat Defects[END_REF]; the microstructural gradient consists of four areas representing the differents evolution stages leading to WEL formation. Stage 2 is the fibrous state of the grains without fragmentation; stage 3 is a state where the grains are still fibrous but fragmented; stage 4 is a state where the grains have no particular orientation and are very fragmented; and stage 5 is the state that appears white in optical microscopy after Nital etching.

Figure 8 .

 8 Figure 8. Shear stress τ vs. shear strain γ curve using (1) and (2).

Figure 9

 9 Figure 9 presents optical and EBSD micrographs of the sheared area obtained at the end of the test for each temperature. Similar to the macroscopic analyses, the microstructural behavior in the range between 20 • C and 300 • C differed slightly from that at 400 • C. The observations are thus given separately.

Figure 9 .

 9 Figure 9. Effect of temperature on the microstructural gradient in the sheared zone of the specimen at the end of the monotonic tests. The specimen failed at the tests performed at T = 20 • C, 200 • C, and 300 • C but did not fail at 400 • C. IPF maps were constructed for the tests performed at room temperature (step size = 0.08 µm) and 400 • C (step size = 0.1 µm).

Figure 10 .

 10 Figure 10. Effect of temperature on the microstructural gradient in the sheared zone of the specimen at the end of the cyclic tests. The mechanical loading and the number of cycles at each temperature are shown in Table 6. The red dotted line denotes the start of cracking in the corner of the specimen. The red box corresponds to the area investigated by EBSD in Figure 11.

  • A fibered and fragmented zone in the lower part of the image (zone A); • A non-indexed zone in the upper part of the crack, which corresponds to the white zone observed optically (zone B); • A very fragmented zone without fibration above the very poorly indexed zone (zone C); • A transition zone where the material fibered and flowed until it fragmented (zone D).

Table 1 .

 1 Chemical composition of R260 pearlitic steel (weight%)[START_REF]Rail Steel Grades-Steel Compositions and Properties[END_REF].

	C	Si	Mn	P	S	Cr	Al
	0.62	0.80	0.15-0.58	0.70-1.20	<0.025	<0.15	<0.004

Table 2 .

 2 Hat-shaped specimen dimensions.

r i (mm) r e (mm) d = r i -r e (mm) L (mm)

	4	3.8	0.2	2

Table 4 .

 4 Monotonic test conditions until failure.

Table 5 .

 5 Evaluation of microstructural indicators for the EBSD map at room temperature.

Table 6 .

 6 Cyclic tests conditions.

	Temperature ( • C)	τ min (MPa)	τ max (MPa)	Number of Cycles
	20	100	560	500 and 600 (failure)
	200	100	500	10,000
	300	100	560	1000 and 1400 (failure)

Table 7 .

 7 Summary of measurements of indicators for the EBSD map in Figure11.

	Zone	Grain Size (µm)	Aspect Ratio	LAGB-MAGB-HAGB (%)
	A C	0.4 ± 0.2 0.3 ± 0.1	2.0 ± 1.0 1.6 ± 0.5	45-37-18 12-43-45

Table 8 .

 8 Synthesis for the monotonic tests.

	Temperature ( • C)	γ failure (-)	Presence of WEL	Final Stage
	20	5.3	no	4
	200	4.2	no	4
	300	4.4	no	2
	400	6.9	no	4

τ min /τ max (MPa) Number of Cycles γ final [-] Final Stage

	4.2. Effect of Temperature on Fatigue Strength
	20	100/560	500, 600 (failure)	3.5	5
	200	100/560	1 (monotonic case)	no data	no data
	200	100/500	10,000	1.8	2
	300	100/560	1000, 1400 (failure)	2.1	2

• C)

Funding: This research received no external funding.

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement: Not applicable.

Acknowledgments: This work is part of the multi-disciplinary project MOPHAB, which aims to improve our knowledge and understanding of the mechanisms leading to the formation of the white etching layer in the materials used to construct railways and to develop corresponding numerical models. This project was supported by IRT Railenium and other industrial partners (RATP: Régie Autonome des Transports Parisiens, France, SNCF: Société Nationale des Chemins de Fer Francais, France, SAARSTAHL rail).

Author Contributions: L.T.: conceptualization, methodology, software, investigation, writingoriginal draft. S.C.: conceptualization, methodology, investigation, resources, writing-original draft, supervision. A.S.: conceptualization, methodology, resources, writing-review and editing, supervision, project administration. F.L.: conceptualization, methodology, software, writing-review and editing, supervision, project administration. F.M.: investigation, resources, writing-review and editing. S.D.:investigation, resources, writing-review and editing. C.L.B.: investigation, resources, writing-review and editing. D.F.: investigation, resources, writing-review and editing. All authors have read and agreed to the published version of the manuscript.

Conflicts of Interest:

The authors declare no conflict of interest.


Yassir Dahmane
email: yassir.dahmane@ec-nantes.fr

Raphael Chenouard

Malek Ghanes

Mario Alvarado-Ruiz

Optimized Time Step for Electric Vehicle Charging Optimization Considering Cost and Temperature

Keywords: Electric vehicle, lithium-ion batteries, smart charging, decentralized control, dynamic time step, temperature effect

An optimal decentralized scheduling strategy for charging one electric vehicle (EV) is proposed to minimize the customer charging cost. Moreover, the EVs can offers more profit when considering the vehicle to grid feature, by discharging the EV in the grid at high peak demand the EV' owner can earn money and reduce his charging bill. Compared to existing methods, the main advantages of the proposed strategy is the considerations of an optimized time step. By doing so, the optimization problem uses a minimum number of decision variables and constraints. Then, the problem can be solved by all optimization method to reach the global optimum in reduced time. To formulate and solve a non-linear constrained optimization problem, the scheduling process takes into consideration: the time of arrival and time departure of the EV, the daily energy prices, the initial state of charge (SOC) and the final SOC desired by the customer, the power limitations, and the temperature. The results obtained show a high impact of the optimal scheduling strategy and significant charging cost reduction compared to the uncontrolled charging and fixed time step algorithms. Moreover, the charging strategy only requires that each EV solves its optimization problem locally, therefore, its deployment requires a low computing capacity.

Introduction

EVs charging context

Lithium-ion batteries (LiBs) are widely used in electric and hybrid vehicles due to their high energy and power density [START_REF] Zhang | An overview on thermal safety issues of lithium-ion batteries for electric vehicle application[END_REF]. This type of battery is used as a power source for traction motor of electric vehicle (EV). Nowadays, charging EVs is mainly done simplistically. As soon as the vehicle is connected to the grid, the battery is charged by the minimal value between the available power on the grid, the maximal power accepted by the EV charger and the maximal power accepted by the battery, until it reaches a full charge. This type of charging named uncontrolled charging implementation is still in use today [START_REF] Nour | Smart charging of electric vehicles according to electricity price[END_REF].

The improvement of charging strategies for EVs is a challenge for the next decades. Power distribution systems can be optimized, but the uncontrolled strategy used in EVs creates high peaks of power demand when considering classical daily scenarii with home to work and work to home travels [START_REF] Lee | Acn-data: Analysis and applications of an open ev charging dataset[END_REF]. In such scenario, all EVs are plug-in at similar moment to the power network and they expect to start the charging immediately whereas in most cases the charging of vehicles can be delayed. Using electricity prices as a lever to control the charging of EVs is a possible solution in a decentralized charging strategy.

The user and the power network can directly obtain concrete benefits like less expensive charging, less load peaks and even network support in the case of vehicle to grid (V2G) feature.

Related Works

Several research studies have been conducted for developing new charging methods through centralized, and decentralized strategies. The centralized charging strategies carry out the charging of EVs from a system level viewpoint and consider EVs present on all the nodes of the distribution system collectively 2 [START_REF] Anand | A hierarchical incentive arbitration scheme for coordinated pev charging stations[END_REF]. These control strategies require extensive cyber-infrastructure, substantial communications, and processing resources [START_REF] Liu | Decentralized charging control of electric vehicles in residential distribution networks[END_REF]. Moreover, the centralized algorithms are more sensitive to privacy constraints, because all users data is saved in servers for real-time optimization or in the backup system for data recovery in the case of a system failure. Unlike the centralized implementation, the decentralized strategies operate at the nodal level and perform the charging of EVs present on each node locally [START_REF] Gan | Optimal Decentralized Protocol for Electric Vehicle Charging[END_REF]. One drawback of centralized control is the continuous change of the computed charging profile at any introduction of new EVs into the station or any demand from the transmission systems operator (TSO) or the distribution system operator (DSO). Thus, the aggregator may need to restart again the optimization to update control signals and sometimes this operation is very expensive in terms of computation time [START_REF] García-Villalobos | Plug-in electric vehicles in electric distribution networks: A review of smart charging approaches[END_REF]. The adoption of a decentralized control strategy can be a solution to previous drawbacks. It breaks the complexity of global optimization problem to local optimization problems at the EV level and it limits the communication between the aggregator and the EVs.

In general, the EV charging scheduling is almost focused on maximizing the aggregator profit without carefully addressing customers' needs. In [START_REF] Sortomme | Optimal Charging Strategies for Unidirectional Vehicle-to-Grid[END_REF],

an unidirectional smart charging algorithm has been formulated in order to maximize the aggregator profit. The aggregator uses the capacity of the EV' batteries to participate in the regulation energy markets. The study in [START_REF] Sortomme | Optimal scheduling of vehicle-to-grid energy and ancillary services[END_REF] proposes the bidirectional charging EVs while offering ancillary services (load regulation and spinning reserves) to the grid. The developed algorithm is used to maximize the annual revenues of the aggregator. Two stages of the EVs charging problem coordination have been proposed in [START_REF] Clement-Nyns | The impact of charging plug-in hybrid electric vehicles on a residential distribution grid[END_REF], deterministic and stochastic; aiming to minimize power losses and voltage deviation when the improvements in power quality was the purpose of the study. The authors in [START_REF] Mehta | Hybrid planning method based on cost-benefit analysis for smart charging of plug-in electric vehicles in distribution systems[END_REF] designed a smart charging strategies to operate in the unidirectional and the bidirectional mode with the consideration of the cost-benefit analysis in a distribution system. An EV charging that takes advantage of photovoltaic energy tariffs and provides ancillary services has been described in [START_REF] Ram | Integrated PV Charging of EV Fleet Based on Energy Prices , V2G , and Offer of Reserves[END_REF]; the mixed integer linear programming (MILP) is implemented to support the grid by the V2G feature and to offer ancillary services in the form of reserve. The main drawbacks of this type of scheduling is that the state of charge (SOC) desired by the EV' owner may not be reached in some cases at the departure time, therefore the customer would not have the required SOC to return home.

However, the EV charging scheduling from the customer's point of view is almost neglected, a few recent works studied the minimization of the EV owners charging cost. The concept of V2G and vehicle to home (V2H) have been exploited in the context of residential charging of one EV, [START_REF] Turker | Optimal Minimization of Plug-in Electric Vehicle Charging Cost with Vehicle-to-Home and Vehicle-to-Grid concepts[END_REF] proposed six decentralized smart charging algorithms with the aim to minimize the charging cost of the EV' owner. In [START_REF] Wang | Smart charging for electric vehicles to minimise charging cost[END_REF], a smart charging strategy has been proposed to minimize EV charging costs while maintaining acceptable distribution system voltages. The optimization decision is based on the time variations in electricity prices, the voltage variation throughout the distribution system and the cost of battery degradation caused by the charging. The authors in [START_REF] Ayyadi | Optimal charging of electric vehicles in residential area, Sustainable Energy[END_REF] formulated the smart charging approach as an optimization problem, aiming at minimizing the EVs charging cost considering the day-ahead electricity price, the battery degradation cost and uncertainties in EVs arrival and departure time. In [START_REF] Dahmane | Coordinated charging of large electric vehicle fleet in a charging station with limited transformer power[END_REF] a coordinated charging strategy has been proposed to manage the EV fleet charging considering the charging infrustructure and the EV' users satisfaction.

In the literature various charging strategies have been proposed. Nevertheless, all of them use a constant sampling period or fixed time step defined before starting the optimization [START_REF] Sortomme | Optimal Charging Strategies for Unidirectional Vehicle-to-Grid[END_REF][START_REF] Sortomme | Optimal scheduling of vehicle-to-grid energy and ancillary services[END_REF][START_REF] Clement-Nyns | The impact of charging plug-in hybrid electric vehicles on a residential distribution grid[END_REF][START_REF] Mehta | Hybrid planning method based on cost-benefit analysis for smart charging of plug-in electric vehicles in distribution systems[END_REF][START_REF] Ram | Integrated PV Charging of EV Fleet Based on Energy Prices , V2G , and Offer of Reserves[END_REF][START_REF] Turker | Optimal Minimization of Plug-in Electric Vehicle Charging Cost with Vehicle-to-Home and Vehicle-to-Grid concepts[END_REF][START_REF] Wang | Smart charging for electric vehicles to minimise charging cost[END_REF][START_REF] Ayyadi | Optimal charging of electric vehicles in residential area, Sustainable Energy[END_REF][START_REF] Dahmane | Coordinated charging of large electric vehicle fleet in a charging station with limited transformer power[END_REF]. In the event of a fluctuation of energy prices of a few seconds within a long planning optimization window, the charging scheduling will cause an important issue with a huge number of steps. Indeed, the calculation time step is defined by the minimal duration between two changes of energy price. Therefore, the size of the decision variable vector becomes very significant, so the computation time and the complexity of the problem increase. The classical scheduling algorithms in a low computational embedded system with constant calculation time steps may not be able to carry out this optimization task because of the large number of decision variables and constraints involved in this optimization.

The objective of the paper is to transform a complicated optimization problem into a simplified one by decreasing the number of decision variables and the number of constraints. Studies have proposed advanced methods for solving such problems using reinforcement learning [START_REF] Qiu | A deep reinforcement learning method for pricing electric vehicles with discrete charging levels[END_REF], and adaptive dynamic programming [START_REF] Xie | Fair energy scheduling for vehicle-to-grid networks using adaptive dynamic programming[END_REF]. In order to be distinguished from studies that use advanced optimization methods, the proposed strategy will directly affect the mathematical modeling of the optimization problem by significantly reducing the number of decision variables and constraints. With the constraint of our industrial partner to implement such a solution on an embedded system in the electric car, our choice is turned to decentralized algorithms. With the constraint of an embedded architecture with a low computational capacity. The use of a dynamic time step greatly simplifies the optimization problem, it can change a hard problem with hundreds of decision variables and constraints to an easy problem with dozens of decision variables and constraints. Therefore the problem could be solved with all optimization methods and the convergence to the optimal solution will be easily and fast. So, the use of advanced optimization methods will not add up to much to the convergence to the optimal solution due to the simplicity of the problem.

In addition to the studies mentioned in the introduction many other optimal algorithms and strategies are available in the literature to solve the cost minimization problem, but they are tested under different conditions and assumptions: battery capacity, customers or aggregators optimization point of view, energy prices applied for aggregators (e/MWh) or customers (e/kWh), etc. However, each study is conducted in its own context and therefore it is difficult to compare the performance of different strategies.

Moreover, the LiBs health issue is related to optimizing the power scheduling of charging/discharging sequence. More specifically, the implementation of an optimized bidirectional charging with charging power equal to the fast charging power based only on the user's needs. Fast charging has a negative impact on the main considerations related to battery life objectives [START_REF] Jaguemont | Thermal effect of fast-charging profiles on lithium-ion batteries[END_REF] and the power grid [START_REF] Steen | Impacts of fast charging of electric buses on electrical distribution systems[END_REF]. Charging LiBs involves many nuanced considerations and subtleties to consider conflicting objectives of maximizing the battery lifespan, reducing the charging time, and improving the charging performance. Furthermore, optimal charging strategies do not take into account the temperature. Due to the higher sensitivity to the temperature of the LiBs compared to other type of battery chemistry [START_REF] Guo | A novel echelon internal heating strategy of cold batteries for all-climate electric vehicles application[END_REF], extreme outside temperatures such as 40 • C and over or -20 • C and lower, accelerate the ageing capacity loss [START_REF] Pesaran | Battery Thermal Management in EVs and HEVs : Issues and Solutions, Advanced Automotive Battery Conference[END_REF].

High temperatures increase power acceptance of the battery but the battery lifetime decreases in a short time, causing premature ageing of the LiBs [START_REF] Liu | Thermal issues about li-ion batteries and recent progress in battery thermal management systems: A review[END_REF].

On the other hand, subzero temperatures decrease the power acceptance, increasing the internal resistance of the LiBs, causing the raising of the joules power losses, decreasing the efficiency of the charging, and affecting the state of health (SOH) of the LiBs [START_REF] Jaguemont | Lithium-ion battery aging experiments at subzero temperatures and model development for capacity fade estimation[END_REF][START_REF] Jaguemont | A comprehensive review of lithium-ion batteries used in hybrid and electric vehicles at cold temperatures[END_REF]. In case of cold weather when the temperature is not considered, the final SOC estimation maybe false and the battery does not reach the SOC target desired by the customer [START_REF] Dahmane | Decentralized control of electric vehicle smart charging for cost minimization considering temperature and battery health[END_REF]. Thus, charging the battery while considering temperature is a very important issue, to get a best estimation of SOC and to conserve the lifespan of EVs batteries. Considering the outside temperature and temperature of the LiBs is a big challenge to make the EVs suitable to any climate condition and to extend the batteries lifespan.

Contributions

This article is an extension of an earlier conference paper [START_REF] Dahmane | Decentralized control of electric vehicle smart charging for cost minimization considering temperature and battery health[END_REF], which has been significantly improved to take into account the constraint of low computational capacity of the embedded EV' system. The contribution of this paper has been capitalized in a patent [START_REF] Dahmane | Procédé d optimisation de la recharge et/ou de la décharge de batteries pour un véhicule automobile électrique[END_REF]. The main contributions of the paper can be summarized as follows:

A new decentralized algorithm for optimal charging of an EV; Low computation complexity with a dynamic time step; Minimization of the charging cost from the user perspective; Consideration of V2G feature to reduce the user cost;

Target SOC evaluation based on temperature to avoid lower estimations like in cold temperature conditions.

The organization of the paper is as follows: the modeling of the optimization problem is given in the Section II. Section III presents the simulation results of the optimal planning strategy. Finally, Section IV addresses the conclusion of this paper and the future work.

Charging scheduling strategy: optimization problem modeling

The charging strategy consists of three steps: pre-processing, optimization, and post-processing, as shown in Fig. 1. The device receives from the BMS the measurement of the initial temperature, the outdoor temperature, the initial SOC, and the battery capacity.

The charging station sends the data over a maximum availability period of 24 hours. The device receives from the charging station the following information shown in Fig. 2: The charging energy price for the next 24 hours: price G2V

The maximum charging power for the next 24 hours: P max G2V

The discharging fees from V2G for the next 24 hours: price V 2G

The maximum discharging power for the next 24 hours: P max V 2G

In this step, the pre-processing algorithm restricts the time base to the actual availability period of the EV on the charging station i.e. between the arrival time and departure time of the EV. The next step is to subdivide the horizon time into a sequence of time slots.

In each defined time slot, the four variables price G2V , P max G2V , price V 2G , and P max V 2G have a constant value. The subdivision to time slots is the conversion of data from time scale in Fig. 2 to slot scale in Table 1.

Thus, we can conclude the number of time slots corresponding to the size of the decision vectors N = 12 and the duration of each time slot dmax shown in Fig. 3. The subdivision on time slots of the price G2V , P max G2V , price V 2G , and P max V 2G , can be observed in figures 2-3 and in Table 1.

For current data with a constant time step for scheduling, the maximal sampling time that can be used with optimal charging strategies is ∆t = 1 hour corresponding to the minimal duration of d max . Thus the size of decision vector is 24. For our strategy, the size of decision vector is 12. The difference between these two numbers mainly depends on the minimal duration that can be smaller than 1 hour in many real cases. In case of a small variation of G2V energy prices for 10 minutes between 20:00 and 20:10 (case 2), the maximal sampling time that can be used with optimal charging strategies with constant time step is ∆t = 10 minutes corresponding to a size of 144 for the decision vector. For the proposed strategy, the size of decision vector is 14.

A single event of a few minutes duration can penalize the whole optimization problem. To sum up, a short fluctuation of prices or maximal power can make the optimization task very difficult and even impossible (time and memory constraints) for an embedded charging scheduling system.

Optimization

This section is devoted to modeling the EV charging problem with cost minimization and temperature consideration. The modeling is done as follows:

The time horizon vector is described by S = [1, ..., i, ..., N ] and it contains N non equal duration time slots as defined in the previous subsection and in Figure 3. The charging and the discharging of the EV can be expressed by four decision vectors as expressed in [START_REF] Zhang | An overview on thermal safety issues of lithium-ion batteries for electric vehicle application[END_REF].

X = P G2V P V 2G d G2V d V 2G (1)
With P G2V = P G2V 1 , ..., P G2V i , ..., P G2V N

P V 2G = P V 2G1 , ..., P V 2Gi , ..., P V 2G N d G2V = d G2V 1 , ..., d G2V i , ..., d G2V N d V 2G = d V 2G1 , ..., d V 2Gi , ..., d V 2G N (2)
For the i th time slot the vector X i can be defined as shown in (3):

X i = P G2V i P V 2Gi d G2V i d V 2Gi (3)
This algorithm have two aims:

Minimize the vehicle's charging cost through optimal grid to vehicle power flow taking into account G2V energy prices.

To maximize the profit from selling energy from vehicle to grid considering V2G energy prices.

The objective function is composed of two objectives, a positive one C 1 and a negative one C 2 . The optimization leads to minimize C 1 that refers to the EV charging cost and to maximize C 2 corresponding to the EV discharging remuneration or the economic profit of discharging EV's battery on the grid.

The two objectives are expressed in (4), ( 5):

C 1 = N i=1 price G2V i • P G2V i • d G2V i (4)
C 2 = N i=1 price V 2Gi • P V 2Gi • d V 2Gi (5)
Where price G2V i is the charging electricity price of the i th time slot in e/kWh, price V 2Gi is the the discharging electricity remuneration of the i th time slot in e/kWh, P G2V i , P V 2Gi is the charging and the discharging power of the i th time slot in kW respectively, d G2V i , d V 2Gi are the calculation step in hours, and i the time slot index.

The proposed optimization approach is formulated to select the optimum charging power P G2V for the period of time d G2V , and the discharging power

P V 2G
for the period of time d V 2G that minimize the weighted sum of the two criteria. The proposed formulation of the objective function to be minimized is given as follows:

F (X) = α 1 C 1 (X) + α 2 C 2 (X) ( 6
)
where X is the decision variable, α 1 , α 2 are constant positive values, given the weight for each criterion: α 1 enforces charging operation mode, α 2 leads the system to discharge the EV using the available battery power to support the grid.

The optimization problem includes linear and nonlinear constraints resulting from EV technical constraints and customer needs.

The charging power constraint related to the daily available power on the grid and the discharging power constraint related to the daily required power by the grid are expressed in (7):

0 ≤ P G2V i ≤ P max G2V i i = 1, ..., N -P max V 2Gi ≤ P V 2Gi ≤ 0 i = 1, ..., N (7)
where P max G2V i , P max V 2Gi are the maximum available power on the grid and the maximum required power by the grid in the i th time slot respectively.

The maximum duration of use for charging and discharging in the i th time slot is formulated in (8):

0 ≤ d G2V i ≤ dmax i i = 1, ..., N 0 ≤ d V 2Gi ≤ dmax i i = 1, ..., N (8)
To allow the charging and discharging operation in the same time slot, a constraint is expressed in (9):

d G2V i + d V 2Gi ≤ dmax i i = 1, ..., N (9)
The maximum power that can be accepted or delivered by the battery depends on the relation between the SOC and the battery's temperature. This power is set by the values obtained in the Powermap function:

P G2V i ≤ P i,Bat+ i = 1, ..., N P V 2Gi ≥ P i,Bat- i = 1, ..., N (10)
Where      P i,Bat+ = P owermap(SOC i , T i )

P i,Bat-= -P owermap(SOC i , T i ) (11)
The Powermap function is the internal dependence of battery power on temperature T i of the i th slot and SOC of the i th slot SOC i . It provides information on the maximum power that can be accepted by the battery or delivered to the battery. An example of Powermap is shown in Figure 5. In order to limit the battery cycling degradation in discharging mode and to avoid customer's range anxiety in case of an emergency use of the EV, we add a SOC constraint expressed as follows:

SOC i ≥ SOC mini i = 1, ..., N (12)
Where SOC mini is the minimum value of the SOC expected by the user, avoiding high battery Depth of Discharge (DOD) during V2G mode.

The case of overcharging is taken into account because it affects the lifetime of LiBs. Despite the fact that EV owners tend to prefer autonomy over battery life, because of the anxiety related to autonomy, which is considered one of the main obstacles to the large-scale adoption of EVs.

SOC i ≤ SOC max i = 1, ..., N (13)
The calculation of the required energy E required to reach the SOC desired by the costumer is expressed as follows:

E required = (SOC target -SOC 0 ) × E 0 ( 14
)
SOC 0 is the initial SOC of the EV, SOC target is the SOC desired by the customer, and E 0 is the capacity of the battery in kW h.

The constraint related to the final energy of the battery is:

E required ≤ E f inal ≤ E 0 ( 15
)
The evaluation of the energy variation of the battery at each time step can be computed as follows:

E i = η charger (P G2V i • d G2V i + P V 2Gi • d V 2Gi ) i = 1, ..., N (16)
Where η charger is the charger efficiency. Then, the final energy quantity in the battery can be expressed as:

E f inal = SOC 0 × E 0 + N i=1 E i ( 17
)
The dynamic monitoring of the SOC is given by ( 18):

SOC i+1 = SOC i + E i E 0 i = 1, ..., N (18)
The temperature calculation is a first order model expressed as follow:

mC p dT dt = P joule + P convective ( 19
)
The convective power is modeled by the Newton law showed in [START_REF] Steen | Impacts of fast charging of electric buses on electrical distribution systems[END_REF]:

P convectivei = T i -T out R th out i = 1, ..., N (20)
Where T out is the outside temperature and R th out is the heat transfer coefficient between the battery and the outside.

The joule power is formulated as a linear model in terms of charging and discharging power:

P joule i = k × P G2V i + k × (-P V 2Gi ) i = 1, ..., N (21)
Where k is a thermal factor depending on the thermal inertia of the battery.

The optimization problem is solved with MATLAB optimization solver f mincon with an Intel Core i7 CPU @ 2.70GHz.

d V 2Gm ← round(d V 2G × 60) d maxm ← round(d max × 60) t maxm ← N i=1 d G2V mi + N i=1 d V 2Gmi P (1) ← P G2V (1) while t ≤ t maxm do for i ← 1, N do for j ← 1, d G2V m (i) do P (t) ← P G2V (i) t = t + 1 end for for j ← 1, d V 2Gm (i) do P (t) ← P V 2G (i) t = t + 1 end for R = d maxm (i) -d G2V m (i) -d V 2Gm (i) for j ← 1, R do P (t) ← 0 t = t + 1
end for end for end while

Post-processing

This step is used to generate the order for the charger to control the charging and discharging of the EV, respecting the scheduling carried out in the optimization stage. It consists in generating the control signal of the charge/discharge to charger by choosing a time step adapted to the communication between the CMS and the charger.

Firstly, the scheduling power obtained by the proposed strategy is transformed from the slot scale to the time scale using the Algorithm 1. The algorithm consists in processing each slot one by one separately to generate the power with the chosen time step. For example in the slot number 2, the power is 19kW (blue bar) and the duration of the slot is 1.5 hours (blue bar) as shown in Fig. 6. The output power is a constant power of 19kW that will be used from 2:00 AM to 3:30 AM and the value of zero for the remained duration until the end of the slot 5:00 AM. Secondly, when the charging power is generated then the SOC can be estimated using the equation ( 22)

SOC(t) = SOC 0 + 1 Q 0 t 0 η charger P (τ )dτ (22)
where SOC 0 is the initial value of the SOC, Q 0 is the nominal capacity in kWh and P is the charging power generated by the Algorithm 1.

Thirdly, the temperature can be estimated using the generated charging power by the equation ( 23):

T (t) = T 0 + 1 mC p t 0 (P joule (τ ) + P convective (τ ))dτ (23)
To summarize, in the post-processing step, the conversion of the optimization results from slot scale to time scale is performed, and the SOC estimation and the temperature estimation are carried out.

Simulations and results

The purpose of this section is to demonstrate the effectiveness of the proposed strategy. On the one hand, the results of the proposed strategy with an optimized time step have been compared to the classical approach using a fixed time step. On the other hand, the proposed strategy will be tested under an extreme outside temperature to show the effect of the temperature on the power scheduling and the final SOC. Finally, the optimized time step strategy will be tested on several daily energy price profiles to prove the effectiveness of the proposed strategy compared to the fixed time step strategy, in term of running time, the number of decision variables and the number of constraints.

The initial conditions are SOC 0 = 0.35, SOC target = 0.7, SOC mini = 0.1, E 0 = 60kW h, η charger = 0.9, P G2V M ax = 7kW , and The Fig. 7 shows the comparison between several charging strategies. The SOC target is achieved by all strategies, however, the charging cost is different from one strategy to another. Using the uncontrolled charging, the charging starts at the moment of plug-in, neglecting the high energy prices, so the charging cost will be high and it is estimated to 2.1e. Smart charging algorithm with only G2V can shift the charging to midnight, the EV is plugged-in at 18h00, but the charging effectively starts until midnight when the electricity price is more attractive so the estimated charging cost is 0.47e. Using the V2G feature, the scheduling strategy, begins the discharging in the period of high V2G remuneration to maximize the profit while the SOC decreases until it reaches the minimal value of 0.1 corresponding to the SOC mini . When the G2V energy price becomes cheaper, the charging begins to reach the desired SOC. The two strategies with optimized time step and fixed time step use the V2G feature, but with the proposed strategy the charging profit is 0.31e and the executing time is 0.25 second compared to 0.28e and 1.95 second for the classical strategy with fixed time step.

P V 2G M ax = -7kW . The
For the strategy using optimized time step, the number of decision variables is 12 and the number of constraint is 21. However, for the classical strategy using fixed time step of 10 minutes, the number of decision variables is 84 and the number of constraint is 420.

In brief, the classical strategy with fixed time step requires higher computing capacity because of the high number of decision variable and constraints compared to the proposed strategy with optimized time step that could be in-tegrated easily to EV onboard embedded system. as genetic algorithm and the locally method based on the gradient, the two charging power profiles are the same corresponding to the optimal solution presented in Fig. 7 with the blue color. Because of the low number of decision variables and constraints (12 decision variables and 18 constraints) the genetic algorithm and the gradient method converge to the same optimal solution. In brief, the proposed method does not require the use of an advanced optimization method to converge to the optimal solution. A local optimization method may be sufficient to solve the optimization problem. For the second case, the charging is done under an outside temperature of -20 • C. The comparison between the fixed time step strategy and optimized time step strategy shows the advantage of fixed time step to follow the temperature constraint. However, it highlights the fact that the constraint evaluating process in the scheduling process is done 48 times for the fixed time step against 2 times for the optimized time step. In the case of significant variations of system inputs (energy prices and maximum powers) and long planning period the most important thing is to find a sub optimal solution in the feasible area that satisfies the constraints as quickly as possible. Therefore, the proposed strategy with optimized time step performs perfectly this task.

By applying the two power profiles defined above to the battery model, the results show a slight error between the final SOC and the SOC target. Despite the consideration of temperature in the scheduling strategy, it remains important to note that the scheduling power profiles in Fig. 10 are distinct from the effective charging power profile in Fig. 11. Due to the rapid decrease of the battery power acceptance caused by the fast drop of the battery temperatures, the power profile is limited by the battery power restriction given by the powermap updated every minute. It is possible to overestimate the energy requirement by 5% to 15% to overcome this problem in extreme temperature.

Moreover, the proposed algorithm with optimized time step has been tested on five daily energy prices profiles and the results were compared to classical algorithms with 10 minutes and 1 minute fixed time step. The comparison has been done on several levels such as the number of the decision variables, the number of constraints, the running time and the charging cost/profit. The results of the study are presented in Table 2.

According to the results presented in Table 2, the algorithm with optimized time step performs the charging scheduling by using a minimal number of decision variables and the number of constraints. By subdividing the energy price profiles on optimal time slot the number of decision variables is minimized compared to the classical strategies with fixed time step. For an optimization window of 14 hours, the number of decision variables is 840, 84, for 1 minute and 10 minutes fixed time step respectively. Therefore, the executing time is very small for the proposed strategy with optimized time step compared to the strategies with fixed time step.

Moreover, the last column shows the charging cost (negative number) or the charging profit (positive number). The charging cost/profit value gives the information about the convergence to the optimal solution. The strategy with optimized time step has always the best value. In case of many variations in the energy price profile (profile 2 and 5) presented in Figure 12, the charging profit of the optimized time step strategy is almost double compared to the

Conclusion

In order to provide an economic benefit to EV users, an optimal decentralized smart charging strategy has been proposed in this work with the objective of reducing the charging cost for the EV user. When assuming the high variations in energy prices, the classic smart charging strategies can not perform the scheduling due to high number of decision variables and the high number of constraints. The added value of the proposed smart charging algorithm is the use of a dynamic optimized time step taken as a decision variable, contrary to

Figure 1 :

 1 Figure 1: Decentralized charging system synoptic

Fig. 1

 1 Fig. 1 presents the context of the contribution and the elements interacting with the Charging Management System (CMS).

Figure 2 :

 2 Figure 2: Subdivision on time slots case 1

Figure 3 :

 3 Figure 3: Duration of time slot: dmax

Figure 4 :

 4 Figure 4: Subdivision on time slots case 2

Figure 5 :

 5 Figure 5: Lithium-ion battery power map

Figure 6 :Algorithm 1

 61 Figure 6: Data conversion to time scale

  initial battery temperature is fixed to 20 • C. The simulation results for the proposed scheduling strategy are presented below. The algorithm has been tested for several scenarios to validate its performance. A real French energy price profile was chosen to illustrate the charging cases. The results show two cases, the first case is the charging under an outside temperature of 20 • C, and the second case is a charging under an extreme outside temperature of -20 • C. For the first case, which implies a charging of the EV at night under an outside temperature of 20 • C.

Figure 7 :

 7 Figure 7: Charging strategies comparison under Tout = 20 • C.

Figure 8 :

 8 Figure 8: The impact of optimization algorithm

Fig. 9

 9 Fig. 9 shows the impact of an extra event such as a football match. The energy prices are directly impacted by this extra event. In order to demonstrate the advantage of the proposed optimized time step strategy and the inconvenient of the use of fixed time step, a scenario of two perturbations of 15 minutes and

Figure 9 :

 9 Figure 9: The impact of an extra event on power scheduling power

Figure 10 :

 10 Figure 10: Charging strategies comparison under Tout = -20 • C.

Figure 11 :

 11 Figure 11: The simulation of battery charging with the obtained scheduling power, Tout = -20 • C.

Figure 12 :

 12 Figure 12: The used energy price' profiles

Table 1 :

 1 Slot price G2V P max G2V price V 2G P max V 2G d max Subdivision results to slot scale

	1	0.16	20	0.10	8	2
	2	0.12	20	0.10	8	3
	3	0.12	20	0.10	12	1
	4	0.12	14	0.10	12	1
	5	0.16	14	0.14	12	2
	6	0.16	14	0.14	6	1
	7	0.16	14	0.10	6	2
	8	0.16	20	0.10	6	2
	9	0.12	20	0.10	6	3
	10	0.16	20	0.14	14	1
	11	0.16	10	0.14	14	4
	12	0.16	10	0.14	8	2

This work was supported by the project Chair between Renault and Centrale Nantes about performances of electric vehicles propulsion.

the existing approaches that use a constant time step as a fixed parameter. The proposed algorithm can perform the power scheduling despite the high fluctuation of energy prices in a reduced time and low computational capacities such as those available on an onboard controller. Taking into account the TOU energy prices, the initial SOC, the final SOC desired by the EV user, the maximum power of the charging infrastructure, the power limitation the Li-ion battery, the initial battery temperature and the outside temperature, the smart charging strategy with optimized time step outperforms the classical strategies with a fixed time step in terms of computing time and the charging cost reduction.

Future work should include a degradation model (in scheduling) that consider the effect of the V2G feature on the degradation of Li-ion batteries, due to the large depth of discharge and long discharge time.