M Bordival
email: bordival@mines-albi.fr

F M Schmidt

Y Le Maoult

B Cosson
email: cosson@ensm-douai.fr

B Plantamura
email: bernard.plantamura@sidel.com

A RAY TRACING METHOD TO SIMULATE THE INFRARED HEATING OF SEMI-TRANSPARENT THERMOPLASTICS

Keywords: Ray Tracing, Heat transfer modelling, Stretch blow moulding, Infrared heating, PET absorption

This paper focuses on RAYHEAT, a software based on the ray tracing method, developed in order to simulate the InfraRed (IR) heating of semi-transparent polymers. In this study, RAYHEAT is used to simulate the IR heating step of a PET preform for the Stretch-Blow Moulding (SBM) process. The general principle of the method is to discretize, into a set of rays, the radiative heat flux emitted by halogen lamps, then to follow these rays inside the oven while they are not fully absorbed. The ray tracer computes the optical path of each ray -accounting for specular or diffuse reflections, refractions, etc… -from its emission point, and throughout the preform thickness. PET is assumed to behave like a non-scattering cold medium. Thus, the radiative heat flux absorption inside the preform is computed according to the Beer-Lambert law. Finally, the distribution of the radiative source term is calculated in the preform. In a second step, the radiative source term is applied as an input data in the commercial finite element software ABAQUS ® , in order to calculate the 3D temperature distribution in the preform. The source term is assumed to be time dependent in order to account for the preform movement throughout the IR oven. This method provides relatively small computation times, while keeping the memory requirements down to a minimum. Numerical results have been compared with temperature measurements performed on an in-lab IR oven. The model simulates suitably the infrared heating stage, and provides accurate predictions of the temperature distribution in the preform. The relative error between the temperature calculated by RAYHEAT, and the measured temperature, is less than 5%.

INTRODUCTION

A large part of bottles intended to beverage market is manufactured using Stretch-Blow Moulding (SBM) process. SBM involves an InfraRed (IR) heating stage to condition the PET preform to the appropriate temperature distribution (around 100°C). Then, the preform is stretched-blown inside a mould to get the final bottle. The heating conditions, which control the preform temperature distribution, strongly affect the final properties of the bottle (mechanical, optical and barrier properties). Within the last ten years, significant researches have been carried out on the simulation of the IR heating stage. A recent literature review on this subject has been presented by Bordival et al. [START_REF] Bordival | Optimization of Preform Temperature Distribution for the Stretch-Blow Molding of PET Bottles: Infrared Heating and Blowing Modeling[END_REF]. PET behaves like a semi-transparent body over the IR spectral range, resulting in major challenges with regard to radiative heat transfers modelling. Different approaches are presented in the literature in order to compute the radiative absorption through the preform thickness. The most classical is the view factors method [START_REF] Monteix | Experimental study and numerical simulation of preform sheet exposed to infrared radiative heating[END_REF][START_REF] Huang | Temperature profiles within reheated preform in stretch blow molding[END_REF][START_REF] Yousefi | Simulation of the mobile preform reheat in injection stretch blow molding process[END_REF]. This method consists in calculating the radiation on the preform outside surface, and then to assume that the radiative heat flux is unidirectional through the preform thickness. More recent works investigated the efficiency of the zone method [START_REF] Michaeli | FE-Analysis of the Two-Step Stretch Blow Molding Process[END_REF] or the ray tracing method [START_REF] Champin | 3D Finite Element Modeling of the Blow Molding Process[END_REF]. The last one presents the advantage to account for a while variety of optical effects, such as specular reflection or refraction. In addition, ray tracing enables to take into account most of constitutive elements of an IR oven like multiple lamps (various geometries) and reflectors. In this work, we propose a numerical modelling of the IR heating stage. The temperature distribution of the preform is calculated using a two-step simulation. First, in-lab software, called RAYHEAT, is used in order to compute the absorption of the radiative heat flux inside the preform. Then, the results are applied as an input data in the finite element commercial package ABAQUS ® . The method is applied in order to simulate the IR heating stage of a rotating preform using the inlab blowing machine set-up. A numerical validation is performed using temperature measurements.

HEAT TRANSFER MODELLING

HEAT BALANCE EQUATION

The evolution versus time of the preform temperature is governed by the transient heat balance equation:

( ) r p q T k t T c ⋅ ∇ - ∇ ⋅ ∇ = ∂ ∂ ρ (1)
where T = temperature, t = time, ρ = density, c p = specific heat, k = thermal conductivity, q r = radiative heat flux density. The inside surface of the preform is assumed to be adiabatic, while the following boundary condition is applied to the outside surface:

( ) ( ) 4 4 ∞ ∞ - + - = ∂ ∂ - T T T T h n T k P PET P c P σ ε (2)
where h c = convection heat transfer coefficient, ε PET = PET's mean emissivity, σ = Stefan-Boltzman constant, T P = outside surface temperature, T ∞ = air temperature inside the oven. The boundary condition given by Equation (2) accounts for two types of heat transfer. The first one is due to the cooling by natural or forced convection if oven is ventilated, the second one to the preform own emission. These heat losses have a critical effect, especially throughout the cooling stage.

RADIATIVE SOURCE TERM

Over the spectral range corresponding to the emission of IR lamps (0.35-8 µm), PET behaves like a semitransparent body. This involves that the radiative heat flux is partly absorbed inside the wall thickness of the preform, and cannot be simply applied as a boundary condition. The radiation absorption must be taken into account through the divergence of the radiative heat flux (Equation 1), also called "radiative source term". The radiative heat flux is required to compute the radiative source term. Per definition, the radiative heat flux density is given by:

( ) ( ) ∫ ∫ ∞ Ω Ω Ω = 0 4 , π λ λ d d s I s q r (3)
where I λ = spectral intensity at the point s, along the direction Ω. The variation of the spectral intensity is governed by the Radiative Transfer Equation (RTC), given for a non-scattering medium by [START_REF] Modest | Radiative heat Transfer[END_REF]:

[ ] ) , ( ) ) ( ( ) , ( Ω - = Ω s I s T I ds s d I λ λ λ λ κ o (4)
where I°λ = Planck's spectral intensity at the preform temperature T, ds = optical path, κ λ = PET spectral absorption coefficient. Equation (4) has got an exact solution, given by [START_REF] Viskanta | Heat Transfer in Semitransparent Solids[END_REF]: In Equation ( 5), the first term of the right hand side member represents the fraction of the incident intensity (emitted by the IR oven) which is absorbed by PET. The second term represents the fraction of the intensity, emitted by PET, which is absorbed by PET (due to the own emission). Throughout the heating stage, the preform temperature (less than 400 K at the end of heating) is very low compared to the IR heater temperature (typically around 2400 K). In this case, the second term of Equation ( 5) can be neglected. This assumption leads to:

( ) ( ) ( ) ( ) ( )
( ) ( ) s e I s I λ κ λ λ - Ω = Ω , 0 , (6)
Equation ( 6) is called Beer-Lambert law [START_REF] Modest | Radiative heat Transfer[END_REF]. Then, by assuming that lamps emission is Lambertian, the radiative source term is given by [START_REF] Erchiqui | Modélisation par éléments finis du chauffage infrarouge des membranes thermoplastiques semi-transparentes[END_REF]: [START_REF] Modest | Radiative heat Transfer[END_REF] where M λ (0) = incident spectral emissive power. The radiative source term distribution is assumed to be temperature-independent. Therefore, it can be computed and then applied as an input data in the finite element software ABAQUS ® in order to solve the heat balance equation.

( ) ∫ ∞ - - = ⋅ ∇ 0 . ) 0 ( λ κ λ κ λ λ d e M s q s r

RAY TRACING METHOD

Ray tracing is widely acknowledged as one of the most accurate method used to simulate radiative heat transfers in semi-transparent media. The general principle of the method is to discretize, into a set of rays, the radiative heat flux emitted by halogen lamps. Then, each ray is followed inside the oven, from its emission point and throughout the preform thickness. The ray tracer computes the optical path of each ray, accounting for specular or diffuse reflections, refractions, etc… In our ray tracing software, assumptions are made for the different optical properties of lamps, reflectors and preforms. Those assumptions are referenced in table 1 for the following properties: emission, absorption, reflection and refraction. Reflection and refraction are averaged in order to reduce the number of ray stored in each calculation. In fact, for one ray that comes from the lamp and contained all the spectral information, if spectral reflection (or refraction) is computed, one ray by spectral band (an infinity for an exact model) has to be created for each air-PET interface crossing. Only the lamp filament is taken into account. Filaments are modelled by equivalent cylinders, the spiral form is neglected. Tungsten filaments are assumed to be Lambertian grey bodies. This assumption provides the definition of ray direction vectors (Figure 1) for rays coming from the filament. Determinist discretization of the emission space could lead to errors due to the ray effect [START_REF] Li | Ray effect in ray tracing method for radiative heat transfer[END_REF]. To avoid that, we have chosen to compute θ and φ according to stochastic variables:

2 1 2 ; ) ( arcsin R R π ϕ θ = = (8)
where R 1 and R 2 are independent uniform stochastic variables in the range [0; 1]. The direction change of a ray that cross a PET-air interface is given by the Snell-Descartes law: where n air (= 1) is the refractive index of air. Notations are illustrated on Figure 2. In order to validate numerical simulations, IR heating trials have been performed on an in-lab IR oven. The oven is composed of five halogen lamps (2 kW nominal power), with ceramic and back aluminium reflectors (see Figure 3). Percentages of nominal power of lamp are reported in Table 2. After 25 s heating, the preform is cooled down by natural convection during 10 s. No ventilation system is used. The natural convection coefficient was calculated using the empirical correlation of Churchill and Chu [START_REF] Churchill | Correlating Equations for Laminar and Turbulent Free Convection from a Vertical Plate[END_REF]. Its value was estimated to be 7 .5 W.m -2 .K -1 . The preform used is 27 g weight, 3.85 mm thickness. Its rotational speed is equal to 1.1 rev/s. The reference of polymer is PET INVISTA 1101 (grade VI = 0.8).

Figure 3: In-lab infrared oven

Temperature measurements have been performed using an AGEMA 880 LW IR camera, functioning within the long wave spectral range 8-12 µm. Over this spectral band, PET behaves like an opaque body [START_REF] Monteix | Quantitative infrared thermography applied to blow moulding process : measurement of a heat transfer coefficient[END_REF]. Thus, the camera measures a surface temperature.

PET THERMAL/RADIATIVE PROPERTIES

PET radiative properties were measured according to the protocols defined by Monteix et al. [START_REF] Monteix | Quantitative infrared thermography applied to blow moulding process : measurement of a heat transfer coefficient[END_REF]. Measurements were performed on PET INVISTA 1101 samples using a Perkin Elmer 950 spectrometer over the range 0.25-2.5 µm, and a Perkin Elmer FTIR spectrometer over the range 2.5-25 µm. PET thermal properties are assumed to be temperature-dependent, which is particularly true for the heat capacity that sharply increases above the glass transition temperature.

NUMERICAL PARAMETERS

The preform is meshed into 5220 rectangular linear elements (6000 nodes), with ten nodes in the thickness direction. For each lamp, 1.5 million rays are followed, for a total of 7.5 millions rays for the entire oven. The computational time of the source term is equal to 90 min CPU (T9500 2.6 GHz 3Go RAM). Temperature calculation was performed using ABAQUS ® standard 6.7, with a fixed time increment equal to 0.25 s. The computation time is equal to 12 min CPU.

RESULTS AND DISCUSSION

Figure 4 illustrates the radiative source term distribution versus time. We can observe that the source term has been defined as a function of time in order to account for the preform rotation. Figure 5 illustrates the external temperature distribution along the preform height, at the end of the cooling step. We can observe a good agreement between simulations and measurements, since the mean relative error is equal to 1.7 %. Figure 6 illustrates the variation of external temperature versus time on a single point, located at 40 mm from the neck of the preform (middle-height). The agreement is fair, especially during the cooling stage, which indicates the convection heat transfer coefficient is well estimated.

CONCLUSIONS

We have proposed in this work a modelling of the IR heating stage for the SBM process. The ray tracing software RAYHEAT provides accurate predictions of the 3D radiative source term distribution in the preform. The coupling between RAYHEAT and ABAQUS ® allows the computation of the preform temperature, accounting for the preform movement throughout the oven. A numerical validation has shown the efficiency of the method, since the relative error between simulations and experiments is less than 2%. Future work will aim to further improve the ray tracer in order to decrease the computation times and enlarge the validation on larger oven of industrial blow moulding machine.

Figure 1 :

 1 Figure 1: Ray definition for ray tracing

Figure 2 :

 2 Figure 2: Ray refraction at air-PET interface crossing

Figure 4 :Figure 5 :

 45 Figure 4: Source term distribution versus time

Figure 6 :

 6 Figure 6: External temperature versus time (mid-height)

Table 1 :

 1 Assumptions on optical properties

		Emission Absorption Reflection	Refraction
	Lamps	Spectral isotropic	none	none	none
	PET	Averaged	Spectral	Specular and averaged	averaged
	Ceramic reflector	none	opaque	diffuse and averaged	none
	Metallic reflector	none	opaque	Specular and averaged	None

Table 2 :

 2 Lamps setting for IR heating trials

		L 1	L 2	L 3	L 4	L 5
	Power (%)	100 100 20	5	60

ACKNOWLEDGEMENT

Special thanks to SIDEL (http://www.sidel.com) for their collaboration, inputs in some application range of these works, and for their supply of material specifications and preforms.


Benoit Cosson
email: cosson@ensm-douai.fr

Fabrice Schmidt

Yannick Le Maoult

Maxime Bordival

Infrared heating stage simulation of semi-transparent media (PET) using ray tracing method

Keywords: Infrared heating, Ray tracing, Stretch blow molding, Thermoforming processes, Semi-transparent media, PET

Stretch blow molding or thermoforming processes includes an infrared heating stage of the thermoplastic preform by infrared heaters. The knowledge of the temperature distribution on the surface and through the thickness of the preform is important to make good prediction of thickness and properties of the manufactured parts. Currently in industry, the fitting of the process parameters is given by experience and is expensive. Our objective is to provide tools that are able to simulate the heat transfers between infrared heaters and preforms in order to reduce the fitting cost and to control the qualities of the end products. The optical method called "ray tracing" is used to simulate the radiative transfer. First, we compare the ray tracing method with the view factor method on a simple example: the heating of a square sheet by one infrared lamp. Then, we perform 3D heating stage simulations and compare with experiments. The ray tracing method allows to compute a source term in the transient heat balance equation. Then commercial finite element method softwares can be used to solve the heat balance equation.

Introduction

The beverage market is a worldwide market. A great part of bottles used for the packaging are produced using the process of injection stretch-blow molding. For example only one industrial blower can produce up to 50,000 bottles per hour. So, the number of bottles produced per day, for only one machine, is around 1.2 million. This process is a two-step process. The first one is the injection of a preform and the second one, that can be performed in another place, is the stretch blow molding. Before the blowing stage, preforms have to be reheated. The technical solution used to heat preform is infrared (IR) oven (made of different modules of less than ten lamps). The cost of the energy being more and more expensive, the optimization of the heating stage of preform becomes an important challenge for industry.

In this paper, we propose numerical tools in order to perform temperature calculation. In previous works [START_REF] Cosson | Optimization of the thickness of PET bottles during stretch blow molding by using a mesh-free (numerical) method[END_REF][START_REF] Schmidt | Experimental study and numerical simulation of the injection stretch/blow molding process[END_REF] the stretch blow molding simulations were carried out by applying constant temperature through the thickness. But in fact, there is a temperature gradient and that makes important material property variations.

Two principal methods are used is this kind of problem. The first one, usually used in 3D image rendering and called view factor method, is already used in IR heating problems. This method is efficient for problem with opaque materials, where surface information is needed, but for problems with semi-transparent materials, the view factor method cannot provide information inside the material. In order to avoid this problem, authors [4,[START_REF] Monteix | Experimental study and numerical simulation of preform or sheet exposed to infrared radiative heating[END_REF] assume a normal propagation of the heat flux from the inner surface. This assumption provide good results for thin and plane parts. But, in the case of the stretch blow molding, preforms are often thick and have cylindrical shapes. Taking into account these geometrical constraints, we choose the ray tracing method, that is also used in 3D image rendering. The ray tracing method is more time consuming than the view factor method, but it is close to the physical description of infrared heating problems.

The heat balance equation (Eq. 1) of the heating stage of preform by IR lamps is solved in two steps. First, we compute the radiative source term ∇.q r (Eq. 5) by using the ray tracing method. Then the results are used in a commercial finite elements software (Com-sol®) as an input data. In order to validate the proposed method, the ray tracing method is compared to the view factor method, then numerical results and experimental results [START_REF] Monteix | Modélisation du chauffage convectoradiatif de préformes en pet pour la réalisation de corps creux[END_REF] are confronted in two representative cases of thermoforming and stretch blow molding processes, the heating stages of a PET sheet and a cylindrical PET preform. Sheets and preforms used in this study are made with the PET T74F9 from Tergal industries.

Heat transfer modeling

The temperature of the preform (sheet-shape or tubeshape) during the heating stage can be calculated by solving the following equation:

ρC p dT dt = ∇ • (k∇ ¯T) -∇.q r (1)
where ρ, C p , k are respectively the specific mass, the specific heat and the thermal conductivity of PET. C p is assumed to be temperature-dependent: strong increase above the glass transition temperature (T g ≈ 350 K) can be seen on Fig. 1. The thermo-optical parameters See on Fig. 1 [13] k 0.25 W.m -1 .K -1 [START_REF] Schmidt | Experimental study and numerical simulation of the injection stretch/blow molding process[END_REF] κ λ See on Fig.

2 [START_REF] Monteix | Modélisation du chauffage convectoradiatif de préformes en pet pour la réalisation de corps creux[END_REF] are referenced in Table 1. The radiative heat flux q r , due to the IR lamps, is given by Eq. 2 according to the Beer-Lambert's law (Eq. 3) under the assumption of the non-scattering cold medium [START_REF] Modest | Radiative heat transfer balbla[END_REF]. IR lamps are assumed to behave like isothermal grey bodies with an emissivity ε.

∇.q r (x) = - ∞ 0 S 2 κ λ I λ (x, s)dsdλ ( 2
)
I λ (x, s) = I λ (x p , s)e -κ λ (x-x p ) (3)
∇.q r (x) = - ∞ 0 S 2 κ λ I λ (x p , s)e -κ λ (x-x p ) dsdλ (4) ∇.q r (x) = - ∞ 0 κ λ M λ (x p )e -κ λ (x-x p ) dλ ( 5
)
κ λ is the spectral absorption coefficient of PET (Fig. 2), M λ (x p ) is the incident spectral emissive power (from the lamps to the preform). M λ (x p ) (W.m -2 .µm -1 ) is given by the Planck's law [START_REF] Modest | Radiative heat transfer balbla[END_REF] and represents the power received by the preform skin from the lamp. x p is the vector that represents the path of the ray come from the lamp and arrived on the preform skin. x is the current position of the ray. So, xx p is the path of the ray in the preform. The tungsten emissivity ε is 0.26 is an integrated coefficient and is assumed to be constant. PET radiative properties were measured Fig. 2 Spectral absorption coefficient κ λ [START_REF] Monteix | Quantitative infrared thermography applied to blow moulding process: measurement of a heat transfer coefficient[END_REF] according to the protocols defined in [START_REF] Monteix | Quantitative infrared thermography applied to blow moulding process: measurement of a heat transfer coefficient[END_REF]. Measurements were performed on PET T74F9 samples using a Perkin Elmer FTIR spectrometer (spectrum 2000) over the range 1-25 µm (Fig. 2). It covers 80% of the spectral emission of IR lamps, for a filament temperature close to 2400 K. κ λ is assumed temperature independent until the PET is crystallized. In this study, the source term ∇.q r in Eq. 5 is computed by using the ray tracing method.

Ray tracing method

Physical background

The ray tracing method is usually used in 3D image rendering. This method is very close to the physics of light propagation, a ray can represent the path of a photon. Ray tracing allows the simulation of a wide variety of optical effects, such as reflection, refraction and absorption. In addition, this method enables to take into account most of constitutive elements of an IR oven like multiple lamps (various geometries) and reflectors (ceramic or metallic).

In our ray tracing software, assumptions are made for the different optical properties of lamps, reflectors and preforms. Those assumptions are referenced in Table 2 for the following properties: emission, absorption, reflection and refraction. Reflection and refraction are averaged in order to reduce the number of rays stored in each calculation. In fact, for one ray that comes from the lamp and contained all the spectral information, if spectral reflection (or refraction) is computed, one ray by spectral band (an infinity for an exact model) has to be created for each air-PET interface crossing.

The spectral PET reflection is plotted in Fig. 3. The variation of the reflectance r λ PET in the 0.2-20 µm spectral band is between 2% and 12%. Due to this small variation, we assume a constant reflectance r PET and we take the mean value in the spectral band 0.2-20 µm (r PET = 7%). The PET refractive index n PET (Fig. 4) is computed by Eq. 6 that gives the relation between reflection and refraction. The calculated value of the refractive index is taken equal to 1.7. The direction change of a ray that crosses a PET-air interface is given by the Snell-Descartes law (Eq. 7), where n Air (= 1) is the refractive index of air.

r PET = (n PET -1) 2 (n PET + 1) 2 (6)
n PET sin θ PET = n Air sin θ Air [START_REF] Modest | Radiative heat transfer balbla[END_REF] Optical properties of metallic and ceramic reflectors are given in [START_REF] Monteix | Modélisation du chauffage convectoradiatif de préformes en pet pour la réalisation de corps creux[END_REF]. The metallic reflector reflectance r MR is around 80% and depend of the state of conservation of the reflector: corrosion, cleanliness. The ceramic reflector reflectance cannot be easily obtained and it is computed, by inverse identification, by comparing two lamps: with and without ceramic reflector. The result of this comparison gives a global reflectance that takes into account the multiple reflections on the ceramic reflector (Fig. 5). The value of this global reflectance r CR is 35%.

In our software, only the lamp filament is taken into account. Filaments are modeled by equivalent cylinders, the spiral form is neglected. Tungsten filaments are assumed to be Lambertian grey bodies. This assumption provides the definition of ray direction vectors (Fig. 6) for rays coming from the filament. The direction vector -→ d is defined by 2 parameters: θ ∈ [0, π/2] and φ ∈ [0, 2π ]. In our first simulations, we used a discretization with constant step, as described on Fig. 6b, and the results of the ray tracing software was very discretization dependent.

In order to avoid this problem, parameters θ and φ are represented by stochastic variables and respectively. Both variables are independent and are defined by Eq. 8, the use of this representation, for a Lambertian surface, is given in [START_REF] Pharr | Physically based rendering: From theory to implementation[END_REF]. In the next section, the benefit of this description will be highlighted.

= arcsin( X 1 ) = 2π X 2 (8)
Where X 1 and X 2 are independent uniform stochastic variables in the range [0; 1].

Comparison with view factor method

In order to validate our software of ray tracing, more exactly our choice of direction definition (Eq. 8), we compare its results with an analytical solution (Eq. 9), given by view factor [START_REF] Leuenberger | Compilation of radiation shape factors for cylindrical assemblies[END_REF], in a representative case.

We choose the configuration of a squared sheet heated by one lamp (Fig. 7) of same length. This case is close to the infrared heating stage of polymer sheets used for thermoforming. The lamp is modeled by a cylinder and the sheet is the sum of infinitesimal surfaces (Fig. 8). The irradiation (W.m -2 ) exchanged between lamp and sheet is computed with ray tracing and view factor respectively.

F d1-2 = P S B -S 2Bπ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ cos Y 2 -B+1 A-1 + cos -1 C-B+1 C+B-1 -Y A+1 √ (A-1) 2 +4Y 2 cos -1 Y 2 -B+1 √ B(A-1) - √ C C+B+1 √ (C+B-1) 2 +4C cos -1 C-B+1 √ B(C+B-1 +H cos -1 1 √ B ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (9)
Fig. 7 Geometrical configuration of the lamp and the sheet

A = X 2 + Y 2 + S 2 ; B = S 2 + X 2 ; C = (H -Y) 2 S = s/r ; X = x/r ; Y = y/r ; H = h/r ( 10
)
geometrical variables (s, x, y, h) are defined on the Fig. 8. The power P of the lamp is taken equal to 1,000 W. Equation 9gives the irradiation (W.m -2 ) of a differential element of plan irradiated by a cylinder. The view factor computation is on the left side of the Fig. 9 for the conditions described in Fig. 7. The right side of Fig. 9 represents the irradiation of the sheet given by ray tracing. In this case, we use ten million rays. The little fluctuations, that can be observed, are due to the stochastic discretization.

The error between numerical solution and analytical solution is given by Eq. 11, where S is the surface of

With the stochastic discretization method, we can control the error (Fig. 10) at any time, so the computation can be stopped when the error is acceptable. In addition, the control of the error cannot be done with a fixed step discretization. Moreover, in a log-log scale the error decreases linearly when the number of rays increases. The slope of the error is equal to -1/2. So, the linearity (in a log-log graph) of the error shows Fig. 10 Error between view factor and ray tracing methods that our direction definition is the good one. For one million rays, the error between the solutions given by view factor and ray tracing method is under 5%. The computational time is less than a minute with an Intel Centrino CPU (2.6 GHz).

In more general case, when you simulate the infrared heating step of thermoforming process, you have not access to an exact solution. So, the error cannot be directly computed. An other advantage of stochastic method, is that you can also control the convergence of the computation. Figure 11 shows the convergence of the mean irradiation of the sheet. It can be seen that for three million rays, the convergence is obtained (2.88 kW.m -2 ). This type of figure allows a simple control of the computation quality, even in the general 3D case. In 3D, the simplest indicator is the mean of the source term ∇.q r .

The ray tracing method is validated in the case of a sheet irradiated by one lamp. In the next subsection, we compare numerical results to experimental results in order to validate some physical assumptions before testing the ray tracing method in 3D complex cases of preform heating stage.

Comparison with experiment: sheet irradiation

In the previous sub-section, we have validated the direction definition of rays, in this new sub-section, we compare the results given by our ray tracing software, coupled with software, to experimental results performed in [START_REF] Champin | Modélisation 3d du chauffage par rayonnement infrarouge et de l'étirage soufflage de corps creux en[END_REF][START_REF] Monteix | Modélisation du chauffage convectoradiatif de préformes en pet pour la réalisation de corps creux[END_REF] for the infrared heating stage of a rectangular sheet by IR lamps (Fig. 13b). The lamp length is 280 mm and the nominal temperature Fig. 11 Convergence of the ray tracing method at 1,000 W is 2,400 K. This comparison is used to validate the assumption done on the ceramic reflector reflectance. Two cases of IR oven (Fig. 12) are studied: the first one is the most simple oven composed by one single naked lamp (the single lamp oven) and the second one is an oven composed by three lamps with ceramic reflectors (the three lamps oven). Figure 13b shows lamps with and without ceramic reflector. In Fig. 12 is sketched the geometrical configuration of the ovens. In the single lamp case, only the central lamp is kept. We choose the example of the 3-mm thickness sheet because it is close to the thickness of the preform used for beverage bottles. The temperature measurement is done by IR camera. The calculation of the temperature in the sheet is a 2-steps computation. In a first step, we compute the radiative source term (Eq. 2), by using the ray tracing method, in all the sheet (3D). The radiative source term is then implemented into a commercial FEM software in order to solve the heat balance equation (Eq. 5).

The boundary conditions used for the simulations are radiative and convective (Eq. 12). The sheet is a rectangular prism, so there are six faces: the face that is in front of the lamps is called the front face, the face that is at the opposite is called the rear face and the four other faces are called the border faces. The boundary condition for the border faces is thermal insulation. In order to simplify the calculation, the air temperature is assumed to be constant T Air = 22 • C and equal to the initial temperature.

n(k∇T) = h i (T Air -T) + ε PET σ (T 4 Air -T 4 ) i = front, rear (12)
Where ε PET is the PET emissivity and it is equal to 0.93 [START_REF] Monteix | Modélisation du chauffage convectoradiatif de préformes en pet pour la réalisation de corps creux[END_REF] and σ = 5.67.10 -8 W.m -2 .K -4 is the Stefan-Boltzmann coefficient. The heat transfert coefficient (natural convection) for the front face and the rear face are respectively h front = 10 W.m -2 .K -1 and h rear = 7 W.m -2 .K -1 [START_REF] Sacadura | Initiation aux transferts thermiques[END_REF]. The heating stage of the sheet is in two steps. During a time t heat the preform is submitted to the IR radiation of the oven, then oven is switched off and there is a cooling period during a time t cool . So, the total time of the experiments is t tot = t heat + t cool .

For the single lamp oven, the heating time is t heat = 65 s and the total time of the experiment is t tot = 110 s. In Fig. 14 the temperature versus time is plotted on two points of the sheet: the centers of the front and rear faces. There is a good agreement with the numerical simulation and experimental data. The mean square error, during the total time, is less than 5%.

For the temperature measurement on the front face, the IR camera has to take the place of the oven, in order to be in face-to-face with the front face. That is why experimental data on the front face begin after t heat . In order to validate the numerical model of the ceramic reflector, we now compare our ray tracing software to the experimental data obtained for the three lamps oven.

The Fig. 15 shows the temperature versus time in the center of the front and rear faces for the three lamps oven. In this case, the heating time is t heat = 15 s and the total time is t tot = 40 s. The agreement on the front face is very good, the averaged error for t heat ≤ t ≤ t tot is less than 2%. However, there is an overestimation of the temperature on the rear face. The error on the rear face is around 15%. Nevertheless, it is acceptable in light of the large number of parameters used in the simulation. A possible source of error can be the lamp modeling. Indeed, the emissivity of the tungsten is assumed to be constant. This assumption changes the way that the PET absorbs the energy emitted by IR lamps. By comparing the two previous results, we can validate the modeling of the ceramic reflector proposed in [START_REF] Monteix | Modélisation du chauffage convectoradiatif de préformes en pet pour la réalisation de corps creux[END_REF]. In order to confirm the capabilities of the ray tracing method, we present, in the next section, simulations of the infrared heating stage of stretch blow molding process and we compare this results with experimental measurements.

Infrared heating stage of PET preform

Trials used in this paper have been performed by Monteix [START_REF] Monteix | Modélisation du chauffage convectoradiatif de préformes en pet pour la réalisation de corps creux[END_REF] on the stretch blow molding prototype of the CROMeP laboratory (Fig. 13). Before the presentation of a complex heating case, we propose to study the case of a static preform. Figure 16 illustrates the configuration of the oven used in the CROMeP prototype. The aluminum reflectors shown on the Fig. 13 have been removed for the trials used in this study.

The oven is constituted of six halogen lamps (each had been set to a power of 1,000 W) with ceramic reflectors on the back. The use of a static preform allows us to study the propagation of rays through the two thicknesses of the preform. In fact, the preform having a hollow body, a ray may pass through two thicknesses of material before leaving the preform. The radiative source term ∇.q r obtained by the ray tracing method is shown on Fig. 17. The computational time of the source term is 90 min for the six lamps (6 × 2 million rays). This time can be decomposed into sub-time: the first one is the time of the geometrical calculation (60 min) and the second one is the time of power calculation (30 min). If Fig. 16 Oven for the infrared heating of PET preform Fig. 17 Source term computed in our software then used in Comsol only powers of the lamps are changed, the geometrical definition of rays can be kept and the computational time for the source term is reduced to 30 min. The source term will be used for the two next simulations.

The Fig. 18 shows the temperature variation versus time for two points on the preform. The points are at mid-height of the preform. The first one is inside the preform in front of the lamps and the second one is outside the preform at the opposite of the lamps. So, we can evaluate the absorption in the two thicknesses of the preform.

Like in the heating stage of sheets, in the previous section, the heating stage of the preform is in two times: an heating time t heat following by a cooling time t cool . In order to measure the temperature inside and outside the preform, the experimental temperature acquisition is made in two steps. The first one is the acquisition of the inside temperature. During the heating time, the acquisition is made on a semi-preform: preform are beforehand cut in two equal parts along the preform axis. On Fig. 16, only the right part of the preform is kept. The second step of the temperature acquisition is made on an other preform, this is the outside temperature measurement.

As in the experimental protocol, the numerical simulations are made on a semi preform and on an entire preform. The biggest difference between the two simulations are the boundary conditions on the inside face. In the first case, radiative and convective boundary conditions are applied. In the second case, thermal insulation is applied on the inside face. In both cases, the outside face is submitted to radiative and convective boundary conditions. The convections coefficients are the same as in the previous section. On the outside face, the convection coefficient is equal to 10 W.m -2 and on the inside face, when is it applied, the convection coefficient is equal to 7 W.m -2 . The radiative flux boundary condition is the same on the inside (when it is applied) and on the outside faces and it is equal to ε PET σ (T 4

Air -T 4 ) with T Air = 27 • C. Results of the simulations are shown on Fig. 18. The computational time in Comsol® is 1 min for 12,000 degrees of freedom. A good agreement is obtained for the inside temperature: the error is around 5% during the heating period and the cooling period. In addition, the error between the experimental temperature and the simulated temperature is around 25% on the outside face. This significant error can be explained by the results obtained for the sheet in the previous section. Indeed, for the sheet, the temperature on the rear face is overestimated, it means that the sheet had absorbed too much energy. If there was a second sheet after the first one, it will have less energy that expected and the temperature in the second sheet will be underestimated. So, if the two thicknesses of the preform are compared to two parallel sheets with the same thickness, the error between experiment and simulation is not really amazing. As in the sheet example, the constant emissivity of tungsten seems to be the principal error source. Now, we present the case of a rotating preform, this case is close to the industrial case. The biggest difference between the CROMeP prototype and industrial ovens is the translation of the preform across multiple ovens. There is only one oven and no translation in the CROMeP prototype. The boundary conditions of this example are radiative and convective as in Eq. 12) on the outside face, with h out = 10 W.m -2 and thermal insulation on the inside face. The rotational speed is 1.2 round per second, the heating time is t heat = 20 s and the cooling time is t cool = 20 s.

The source term used for the rotative preform is the same as the one used for the static preform (Fig. 17). An easy way to compute the rotation is not to consider a rotative preform in an oven, but it is to consider a rotative oven round a preform. So, only one calculation of source term is needed and the rotation of the source term shown on Fig. 17 is used for our simulations.

In Fig. 19 the temperature versus time is plotted on one point of the outside face at mid-height of the preform. The computational time in Comsol is 3 min. The agreement between the experiment and the simulation is better as in the previous case, the error is around 5%. The rotation of the source term works as an averaging. So, if there is an overestimation on one side and an underestimation on the other side, the error vanishes by rotation.

Conclusions and prospects

In this work, we have developed the ray tracing method in order to compute the radiative source term in IR heating problem. We first compared the ray tracing method with view factor in in case where exact solution is known for view factor and we proved the efficiency of the ray tracing method. In a second time, the ray tracing method is compared to basic experiments in order to validate some physical assumptions. Then, the simulation of a complex oven and the rotation of a preform is computed. This final comparison shows the capability of the ray tracing method to predict temperature distribution for stretch blow molding and thermoforming processes of semi-transparent materials. We have shown that the good knowledge of the radiative properties of emitter (emittance) and receiver (absorption) is needed to provide good estimation of the temperature field.

The next evolution of the software is the introduction of our own spectral data for the emissivity of tungsten. New experiments will be performed in order to measure the emissivity for all the emission range. Some work (experimental or theoretical) exist, but the emissivity is just given for a part of the emission spectrum of tungsten. Desvignes [START_REF] Desvignes | Modélisation par éléments finis du chauffage infrarouge des membranes thermoplastiques semi-transparentes[END_REF] gives the emissivity for wavelengths to 2.6 µm and [START_REF] Siegel | Thermal radiation heat transfer[END_REF] gives the emissivity for wavelengths from 2.6 µm.

The next steps of this study is the coupling of the thermal calculation with mechanical calculation in order to provide a tool which is able to predict the final bottle properties. This tool would be able to help engineers to find the good process conditions in order to make good bottles (thickness, mechanical properties).

1 )Fig. 1

 11 Fig.1PET specific heat versus temperature[START_REF] Schmidt | Experimental study and numerical simulation of the injection stretch/blow molding process[END_REF]

Fig. 3

 3 Fig.3Spectral reflection r λ of PET[START_REF] Monteix | Modélisation du chauffage convectoradiatif de préformes en pet pour la réalisation de corps creux[END_REF]

nFig. 4 aFig. 5 Fig. 6

 456 Fig. 4 a Snell-Descartes law representation, b reflection types

Fig. 8

 8 Fig.8Geometrical configuration for Eq. 10

2 Fig. 9

 29 Fig. 9 Irradiation (W.m -2 ) of a squared plate: view factor (left), ray tracing (right)

Fig. 12 Fig. 13 a

 1213 Fig.[START_REF] Sacadura | Initiation aux transferts thermiques[END_REF] Experimental configuration of the sheet heating[START_REF] Monteix | Modélisation du chauffage convectoradiatif de préformes en pet pour la réalisation de corps creux[END_REF]

Fig. 14

 14 Fig. 14 Temperature variation in the center of the front and rear faces versus time for the one lamp configuration

Fig. 15

 15 Fig. 15 Temperature versus time in the center of the front and rear faces for the three lamps configuration

Fig. 18

 18 Fig. 18 Temperature versus time inside and outside of the PET preform for the oven configuration

Fig. 19

 19 Fig. 19Temperature versus time at mid-height of preform (outside face)

Table 1

 1 Thermo-optical parameters of PET

	Parameter	Value	Reference
	ρ	1,335 kg.m -3	[6]
	C p

Table 2

 2

	Assumptions on optical properties	Lamps	Emission Spectral and isotropic	Absorption None	Reflection None	Refraction None
		PET	Averaged	Spectral	Specular and averaged	Averaged
		Ceramic reflector	None	Opaque	Diffuse and averaged	None
		Metallic reflector	None	Opaque	Specular and averaged	None

Acknowledgement This work could not be possible without our industrial partner: O. Demangeon from Toshiba Lighting.


B Cosson

F Schmidt

Y Le Maoult

NUMERICAL INFRARED HEATING OF SEMI TRANSPARENT THERMOPLASTIC USING RAY TRACING METHOD

Keywords: Heat transfer, Ray Tracing, Thermoforming

Thermoforming process include a heating stage of the thermoplastic specimen by infrared heaters. The knowledge of the temperature distribution on the surface and through the thickness of the preform is important to make good prediction of thickness and properties of the manufactured part. Currently in industry, the fitting of the process parameters is given by experience and it is expensive. Our objective is to provide tools that are able to simulate the heat transfers between the infrared heaters and the specimen in order to reduce the fitting cost and to control the qualities of the end product. We use the optical method called " ray tracing " (RTM) to simulate the radiative transfer. First, we compare the RTM with the shape factor method on a simple example: the heating of a square sheet by one infrared lamp. Then, we show the 3D heating stage of a rectangular specimen. In that case, we use 6 lamps to simulate an infrared oven. The RTM allows to compute a source term in the transient heat balance equation. We can use commercial codes to solve the heat balance equation.

INTRODUCTION

In this study we use two steps to solve the heat balance equation [START_REF] Cai | Développement et applications de modèles d'échanges radiatifs par suivi de rayons[END_REF] of the heating stage of a Poly Ethylene Terephthalate (PET) plate by IR lamps. We compute the radiative source term Q (Eq.2) by using the ray tracing method [START_REF] Cai | Développement et applications de modèles d'échanges radiatifs par suivi de rayons[END_REF], then we put the result in a finite elements software (here Comsol) as an input data. We choose ray tracing method because it is close to physics of IR heating. An other method, called radiosity, is often used in IR heating problem. This method is efficient for problem with opaque materials, but for problem with semi-transparent materials the radiosity method can not provide information inside the material.

ρC p dT dt = -∇ • (k∇ ¯T ) + Q (1)
where ρ, C p , k are respectively the density, the heat capacity and the thermal conductivity of PET, you can see F. Schmidt et al. for numerical values [START_REF] Bellet | Experimental study and numerical simulation of the injection stretch/blow molding process[END_REF]. The radiative term Q, due to the IR lamps, is given by the equation ( 2).

Q = -K(T )M e -K(T )s (2) M = ∞ 0 M λ dλ (3)
K(T ) = ∞ 0 K λ M λ dλ ∞ 0 M λ dλ (4)
K λ is the spectral linear absorption coefficient of PET (Fig. 1), K(T ) is the integrated linear absorption coeffi- * Corresponding author: bcosson@enstimac.fr, 0033 563493085 cient that depends on the IR source (Fig. 2), M λ is the spectral exitance of the IR source given by the Plank law. )

Figure 1: K λ [3]

VALIDATION OF RTM

RADIOSITY METHOD

In order to validate our software of ray tracing, we compare its results with an analytical solution (Eq.5), given by radiosity [START_REF] Person | Compilation of radiation shape factors for cylindrical assemblies[END_REF], in a representative case. We choose the configuration of a square sheet heated by one lamp (fig 3) of same length. The lamp is modelized by a cylinder. In both cases, we compute the power by unit of area (W.m -2 ) exchange between the lamp and the sheet.

Spectral linear absorption coefficient

F d1-2 = P S B -S 2Bπ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ cos( Y 2 -B+1 A-1 ) + cos -1 ( C-B+1 C+B-1 ) -Y A+1 √ (A-1) 2 +4Y 2 cos -1 ( Y 2 -B+1 √ B(A-1) ) - √ C C+B+1 √ (C+B-1) 2 +4C cos -1 ( C-B+1 √ B(C+B-1 ) +H cos -1 ( 1 √ B ) ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ (5) S = s/r ; X = x/r ; Y = y/r ; H = h/r (6) A = X 2 +Y 2 +S 2 ; B = X 2 +S 2 ; C = (H -Y ) 2 (7)
The geometrical variables (s, x, y, h) are defined on the figure 4. The equation 5 gives the illumination (W.m -2 ) of differential element of a plan illuminated by a cylinder. The cylinder have an isotropic luminance. The result is given on figure 5 for the conditions described in figure 3. The power P of the lamp is 1000 W. The lamp radiates in all the space. To use the ray tracing method, we have to discretize the radiation of the lamp and define ray properties : departure, direction and arrival. We use the Monte Carlo's method to define the ray departure and direction. We do not use regular discretization because the obtained result is discretization dependent. With the Monte Carlo's method we can control the error (Er F %, Eq.8) between the exact solution (Fig. Er F % = 100

F radiosity d1-2 -F raytracing d1-2 2 dΩ F radiosity d1-2 2 dΩ
(8) For 1.2 million of rays, the error between the solutions given by radiosity (Fig. 5) and ray tracing (Fig. 6) method is smaller than 5%. The calculating time is about 90s with an Intel Centrino CPU (2.6 GHz). The ray tracing method is validated in the case of a plate illuminated by one lamp. In the next section, we compare numerical results to ex-

COMPARISON WITH EXPERIMENT

In this section we compare the results given by our ray tracing software, coupled with Comsol (FEM), to experimental results given in [START_REF] Champin | Modélisation 3D du chauffage par rayonnement infrarouge et de l'étirage soufflage de corps creux en PET[END_REF] for the heating stage of a rectangular plate by an IR lamp (Fig. 8). In a first step, we compute the radiative source term (Eq.2), by using the ray tracing method, in all the plate (3D). Then, we use the result as an input data in a FEM software (here Comsol) to solve the heat balance equation (Eq.1). The figure 9 shows the source term for y= 0 in W.m -3 . The radiation of the IR lamp is absorbed in few µm in the plate. For the chosen example, the heating stage of the plate is in two steps: in first the face of the plate that is in front of the lamp is heated, during 65 s, then the lamp is turned off and the temperature, in the thickness, is homogenised by conduction, during 25 s. The figure 10 shows the evolution of the temperature in two points of the plate: one in the middle of the face that is in the front of the lamp (front face) and the second is at the opposite (rear face). On the same figure, there are the numerical and the experimental results. At the end of the heating stage (90 s) the difference between simulation and experimentation is lower than 5 %. The numerical results are the dotted line, the blue one is for the front face and the green one is for the rear face. The grey lines are for the experimental results. The boundary conditions used for the simulation are given in the table 1. The computation time for this simulation is about 15 mn (Ray Tracing) for the source term and 3 mn (Comsol) for the temperature with a 2.6 Ghz CPU. The computation time of the source term is proportional to the number of lamps. In industrial oven you can have over 6 lamps and for this configuration the computation time is up to 1.5 hour. This time is acceptable for one simulation, but it can be unacceptable if you want to do an optimisation of the heating stage. In the next section we propose a technique to reduce the computation time for a lot of simulations.

TIME OPTIMISATION

In the simulation of the heating stage of a plate (or preform in a most general case), the calculation of the radiative source term is very time costly. In order to reduce the computing time cost, we propose to use interpolation for the calculation of the radiative source term. To realize this interpolation, we do a fixed number of direct calculation of the source term (Fig. 11). In our example, we can change the x and z position of the lamps. We discretize the plan (x,z) in 25 (5 on x axis and 5 on z axis) lamp positions, and we create a data base of 25 radiative source terms . The calculation time is 5 hours for the 25 positions. Now if you want to know the radiative source term for any x,z position of a lamp, you can use interpolation and the calculation time is reduced to 3 s. We use the interp2 function in Matlab, with a cubic interpolation, to realize the interpolation of the radiative source term. In order to validate this technique, we compare the result of a direct calculation and the result given by interpolation for arbitrary selected positions, in all our tests the error (Er Q %, Eq.9) between both results (Q interp and Q direct ) is under 3 %. So, with this technique the calculation time of the radiative source term comes from 15 mn to 3 s.

Er Q % = 100 (Q interp -Q direct ) 2 dΩ (Q direct ) 2 dΩ (9)

CONCLUSIONS

In this study we use the ray tracing method to compute the radiative source term used in the heat balance equation. The ray tracing method is validated with two different ways : comparison with analytical solution and comparison with experimental data. In both cases the error is lower than 5 %. The error can be controlled by using the Monte Carlo's method.

We propose a simple technique to reduce the computation time cost of the ray tracing method in the case where we need a lot of iterations, like in optimisation of forming process. By using this technique, we introduce an error smaller than 3% and reduce the computation time from 15 mn to 3 s. Next steps of this study are : the consideration of reflectors, the application to the heating stage of preform for the stretch blow moulding process.

  Figure 2: K(T )

Figure 3 :

 3 Figure 3: Square plate with 1 lamp

Figure 4 :Figure 5 :

 45 Figure 4: Geometrical description of the cylinder-plate

  7.a) and thereby reduce calculating time. We can also control the convergence of the Monte Carlo's solution by looking the evolution of the mean of the illumination of the plate (Fig.7.b).

2 Figure 6 :

 26 Figure 6: Illumination of the square plate (W.m -2 , ray tracing)

Figure 7 :

 7 Figure 7: (a) Error between radiosity and ray tracing methods, (b) Convergence of the ray tracing method

Figure 2 Figure 9 :Figure 10 :

 2910 Figure 8: Experimental configuration

Figure 11 :

 11 Figure 11: Position of the lamps in the plan x,z

Table 1 :

 1 Simulated boundary conditionsperimental results and we propose a technique to reduce the calculating time in the case of 3D computation.

ACKNOWLEDGEMENT

This research was supported by Toshiba Lighting France.


Fatemeh Nasiri

Navid Mahmoudian

Bidgoli Frédéric

Thomas Maugey

A GEOMETRY-AWARE FRAMEWORK FOR COMPRESSING 3D MESH TEXTURES

Keywords: Intra prediction, 3D mesh, texture map

This paper proposes a novel prediction tool for improving the compression performance of texture atlases. This algorithm, called Geometry-Aware (GA) intra coding, takes advantage of the topology of the associated 3D meshes, in order to reduce the redundancies in the texture map. For texture processing, the concept of the conventional intra prediction, used in video compression, has been adapted to consider neighboring information on the 3D surface. We have also studied how this prediction tool can be integrated into a complete coding solution.

In particular, a block scanning strategy and a graph-based transform for residual coding have been proposed. Results show that the knowledge of the mesh topology significantly improves the compression efficiency of texture atlases 1 .

Introduction

With the recent development of 3D scanning technologies, 3D models have been finely detailed. This has made the 3D representation a trending media format for novel applications, such as architectural or historical artifacts presentation, virtual reality, etc [START_REF]Open heritage: Google art and culture[END_REF][START_REF] Yang | 3D mesh compression and transmission for mobile robotic applications[END_REF][START_REF] Gao | Realistic 3D mesh compression based on predicted angle-normal images[END_REF][START_REF] Peng | Technologies for 3D mesh compression: A survey[END_REF]. As a result, the ability to transmit them easily through networks has attracted great attention in both academia and industry. 3D models commonly consist of a mesh, with geometry information (3D vertices) and its corresponding topology (connectivity among elements), as well as optional attribute information such as normals, colors, etc. To make the rendering more realistic, usually, 2D texture maps such as the diffusion, specular and bump maps are applied on mesh surface to present material properties, high frequency details, etc.

The compression of 3D mesh has been widely studied. A survey of such methods can be found in [START_REF] Maglo | 3D mesh compression: Survey, comparisons, and emerging trends[END_REF]. While the compression of textures from the perspective of rendering systems has been thoroughly studied [START_REF] Beers | Rendering from compressed textures[END_REF][START_REF] Krajcevski | GST: GPU-decodable supercompressed textures[END_REF], the texture compression from the perspective of transmission or storage is almost overlooked. The former aims at compressing the texture maps to reduce the memory usage at the run-time and to provide the ability to render directly from the compressed textures loaded in rendering systems. The latter compresses the texture for storage and transmission, i.e. to reduce download or disk size. This paper proposes a new framework to compress the texture maps in order to reduce the storage size more efficiently than the traditional methods.

Traditional still image compression schemes, such as JPEG, are often used for storing texture maps [START_REF] Wallace | The JPEG still picture compression standard[END_REF]. Although such approaches may provide up to 50:1 compression rate [START_REF] Krajcevski | GST: GPU-decodable supercompressed textures[END_REF], they are still less efficient than the conventional intra coding used in video compression standards [START_REF] Nguyen | Performance analysis of HEVC-based intra coding for still image compression[END_REF]. To further exploit spatial redundancies, the conventional intra coding methods partition an image into blocks, and use parts of already coded adjacent blocks as references to predict the current block. Afterward, the residual signal between the original block and the predicted block is represented in the transform domain to remove the remaining correlation. Examples of intra coding algorithm integrated in video compression standards can be found in Advanced Video Coding (AVC) [START_REF] Saleh | Review for high efficiency video coding (HEVC)[END_REF] and High Efficiency Video Coding (HEVC) [START_REF] Sullivan | Video compressionfrom concepts to the H.264/AVC standard[END_REF].

Adjacent parts of the 3D mesh might not be adjacent in the associated 2D texture maps, even though the texture on the surface of an object seems continuous. An example of 3D model, decomposed into geometry and texture information, is shown in Fig. 1. As can be seen, the texture information can be split into several disconnected patches in a 2D atlas. Consequently, blocks on the patch borders have non-informative neighbors (i.e. black areas between the patches). For instance, for block C in Fig. 2, none of the conventional neighboring blocks, namely A and B contains proper texture information for its prediction. Therefore, traditional intra coding methods would result in a poor prediction accuracy.

In this paper, we assume that the mesh is already coded and available at the decoder. So, we exploit it for finding more relevant references for texture prediction. To this end, for the regions where the conventional reference block is noninformative, such as Fig. 2, the texture information of adjacent areas on the 3D surface is used (Section 2). Due to the fact that the texture consists of several disconnected regions, we also propose a block scanning method based on the knowledge of the mesh surface as an alternative to the regular raster scanning (Section 3). Moreover, a graph transformation is proposed to decorrelate the residual signal of blocks on the border of patches (Section 4).

2 Geometry-aware intra prediction

Conventional intra prediction

From high level point of view, intra prediction is a predefined modeling tool with several internal modes that are adjusted with its parameters. These parameters determine texture pattern to be produced by reference pixels from previously coded blocks. Depending on the parameter setting, the selected pattern can either be directional or plain. For example, HEVC uses one row and one column of references (i.e. from the left and to the above of a block) to apply a prediction model with 35 internal modes, including the DC, the Planar and 33 angular modes.

In our context, the conventional intra prediction is not able to distinguish between informative and non-informative parts of a texture atlas. Therefore, its use for texture compression produces high energy residual signals on blocks which are positioned on patch borders. This poor prediction performance is due to the inaccuracy of non-informative references. In order to address this problem, the topology of the mesh is proposed to be utilized for finding more correlated references for intra prediction. However, for the texture modeling step, we adopt the same set of 35 intra modes of HEVC.

Proposed intra prediction

Discrete patches on texture atlas can be mapped to continuous 3D texture using mesh triangles, as shown in Fig. 3-(a). The algorithm proposed in this paper uses this information to fill the empty set of references, when the block is positioned on a patch border (Fig. 3-(b)). For this purpose, first, the neighbors of a border block are detected using the mesh connectivity. According to this mapping, the block F7 can be used as the remote neighbor of F1, hence, its interior border is copied to the exterior border of F1 (c).

Then, if they are already coded, their border pixels are copied to fill the references of the current block. Such neighbors might be remote on the 2D texture atlas, yet, adjacent on the 3D mesh. Fig. 3-(c) shows how the proposed algorithm copies the inner border of remote triangles on the border of the current triangle. In some inevitable cases, the proposed reference copying is not possible, since the remote neighbor is not coded yet.

The references that were replaced on the border of the patches with such an approach are irregularly positioned. This is in contrary with the conventional intra prediction algorithm, where the references are regularly placed on the top row and left column. This makes the conventional block prediction challenging. For instance, the block prediction algorithm of HEVC performs a backward projection along with a linear interpolation on the reference pixels in order to produce the prediction values within the block [START_REF] Lainema | Intra coding of the HEVC standard[END_REF]. The formulation of this process requires the reference pixels to be aligned in one row and one column.

To address the above problem, our algorithm integrates a forward reference projection, instead of the regular backward one. This process is shown in Fig. 4 and consists of two steps: 1) forward projection of reference pixels on non-integer positions within a block, 2) interpolation of integer positions using up to four non-integer neighbors. For comparison, Fig. 4 also shows the resulting prediction with the conventional intra prediction algorithm on a border block.

Geometry-aware block scanning

The use of remote references on border blocks makes the regular raster scanning of blocks inefficient. Indeed, the scanning may require access to remote references that are not yet coded. As discussed, this problem is inevitable due to in- complete data access on the 2D atlas. However, its frequency of occurrence can significantly be reduced by using a smart block scanning that takes advantage of mesh connectivity.

The proposed scanning algorithm starts from an initial block that contains the geometric centroid of the largest patch on the 2D atlas. Once the initial block is coded, its neighboring blocks are checked to choose the next block to be coded. For this purpose, two rules are applied to prioritize neighboring blocks: Rule 1: Candidates with higher number of available neighboring blocks have a higher priority. Rule 2: Among candidates with the same number of available references, those with available reference blocks at above and left have higher priority.

Starting from the centroid, all blocks within the first patch (i.e. largest patch) can apply the conventional intra prediction algorithm. The only exception is that sometimes, we may have to use a reference row and a column at bottom and right, respectively. For these blocks, one can simply rotate the block as well as its references, in order to put the reference row and column on their right position. Then, the conventional algorithm can be applied to any block in the first patch. Even the border blocks.

Once all the blocks within the first patch are scanned, the next step is to scan the next patch. To do so, we first detect the triangles positioned on the border of the first patch. Then, their remote neighbor triangles are found in other patches, by using the 3D mesh. In the proposed scanning method, the next patch is selected as the one with the most neighboring pixels on the border.

Geometry-aware residual coding with graph transform

Conventional residual coding methods usually use the Discrete Cosine Transform (DCT) to compact the information for transmission. Despite its numerous benefits, applying the DCT on the residual signal of border blocks is not efficient. This is due to the fact that the informative part of border residual blocks is not square, and this would result in a large energy on the coefficients in higher frequencies. To address this problem, our geometry-aware residual coding method utilizes the mesh connectivity and detects border blocks to apply a graphbased transform on their residual, called the Graph Fourier Transform (GFT) [START_REF] Shuman | The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains[END_REF][START_REF] Rizkallah | Graph-based transforms for predictive light field compression based on super-pixels[END_REF][START_REF] Hu | Multiresolution graph fourier transform for compression of piecewise smooth images[END_REF]. Once a border block with the size of B S ×B S is predicted and its residual signal R is calculated, the GFT is performed with the following steps:

Step 1) Constructing a graph for each border block based on its residual data R. Moreover, a mask M is reconstructed from the geometry information of the block to contain its informative pixels coordinates.

Step 2) Forming an undirected connected graph G = {V, E, A} from the masked residual, which consists of a finite number of vertices corresponding to the residual data at position {i, j}, i, j = 1 . . . B S , such that M (i, j) = 1. The set E of edges is formed by using four neighbors around each residual value.

Step 3) Constructing an adjacency matrix A from the edges in G.

Step 4) Calculating the degree of each node and construct matrix D as a diagonal degree matrix whose i-th diagonal element is equal to the sum of the weights of all edges incident to node i.

The subtraction of matrix A from D form the Laplacian matrix L:

L = D -A, (1)
where L is a symmetric semi-definitive matrix and can be diagonalized as:

L = U T ΛU, (2)
where U is the matrix of eigen-vectors and the diagonal matrix Λ consists of eigenvalues. The matrix U is used to compute the unweighted GFT coefficients as:

R = UR, (3)
and the inverse is computed as:

R = U T R. (4)
One interest of GFT is that its number of coefficients is equal to the number of informative coordinates in the input residual R. These coefficients are normally quantized and coded using a proper arithmetic coding.

Experimental Results

We have implemented all the proposed tools in a complete coding solution. This section presents its compression performances in different conditions. The input data comes from [START_REF]Smithsonian institution 3D program[END_REF], which consists of three ancient objects with different maps ( normal, diffusion etc.). In this experiments, only the diffusion maps that represents the "natural" texture of object are tested.

The proposed Geometry-Aware (GA) algorithm is compared against the conventional intra prediction as the anchor, in terms of Bjontegaard Delta Rate (BD-R) [START_REF] Bjontegaard | Calculation of average PSNR differences between RD-curves[END_REF]. For this purpose, different configurations were considered for the anchor algorithms. The first configuration (Anc-1) uses the raster scanning and the conventional intra prediction, but applies the GFT on border blocks and the Peak Signal-to-Noise Ration (PSNR) has been calculated only on the informative pixels of the atlas. The second anchor (Anc-2) performs similarly to Anc-1, except that it uses the DCT instead of the GFT on the border blocks.

For simplicity and without loss of generality, the performance evaluation has been carried out under two conditions. First, a united entropy coder based on arithmetic coding has been shared between the proposed and the anchor methods. This entropy coder allows rate calculation of individual syntax elements as well as an entire sequence. Second, a fixed block size partitioning has been used in each configuration. In other words, for each comparison, the entire 2D atlas has been coded with a constant block size. However, the results are provided as the average BD-R gain over different block sizes, namely 16×16, 32×32 and 64×64.

Table 1 compares the defined configurations of the proposed algorithm GA against the two anchor methods, in terms of BD-R. Each negative value indicates the percentage of compression gain achieved by the "GA" configuration against the "Anc" configuration of the corresponding column, when applied on the 3D sample of the corresponding row. Moreover, Fig. 5 demonstrates the rate-distortion curves of different configurations on each test sample. As can be seen, a promising compression gain can be achieved by using the proposed geometry-aware intra coding tools.

Our GA encoding scheme is more complex than the anchors. However, it is not a problem as free view-point applications support asymmetrical compression-decompression scheme (compress once, decompress many).

Table 1. Performance comparison of the the proposed GA algorithm against, two anchors, in terms of BD-R (%).

Sample

Anc-1 Anc-2

Sample-1 -16.1% -29.6% Sample-2 -11.0% -23.2% Sample-3 -17.3% -19.4%

Conclusion and future work

In this work, we propose a framework for compressing texture atlases of 3D models. Conventional image compression methods treat atlas of 3D objects simply as 2D signals and do not benefit from their geometry information. In this paper, a pixel prediction method with a proper reference selection and block scanning strategy and an alternative residual transformation are integrated in the conventional intra coding algorithm. As a future work, one can integrate the proposed algorithm within existing codecs, such as HEVC test Model (HM), in order to evaluate its coding performances on texture maps. This would require normative changes in all three domains of prediction, scan and residual coding. Such experiment provides allows benefiting from highly optimized tools of HEVC, such as Context Adaptive Binary Arithmetic Coding (CABAC), residual coding, flexible block partitioning etc.

Fig. 1 .

 1 Fig. 1. (a) A realistic 3D model consists of (b) texture information, mapped onto a (c) triangular mesh.

Fig. 2 .

 2 Fig. 2. The blocks A and B around the border block C are non-informative references.

Fig. 3 .

 3 Fig. 3. Process of copying border references from remote neighbors. A 3D mesh in (a) is mapped into a 2D atlas in (b).According to this mapping, the block F7 can be used as the remote neighbor of F1, hence, its interior border is copied to the exterior border of F1 (c).

  a) Original block with surrounded pixels. c) Proposed intra prediction with irregular references. b) Conventional intra prediction.

Fig. 4 .

 4 Fig. 4. Block prediction of a border block. Left: set of reference samples. Middle: propagation of reference samples (reference projection). Right: intra prediction (interpolation).

Fig. 5 .

 5 Fig. 5. Rate-PSNR curves of the proposed algorithm (GA) against two configurations of the anchor (Anc) on 3 diffusion maps.

This work was partly supported by a GdR-Isis grant and the Cominlabs excellence laboratory and managed by the National Research Agency in the "Investing for the Future" program with reference ANR-10-LABX-07-01.


Imene Elfeki
email: ielfeki@lanpark.eu

Thierry Doligez

Ismail Aouichak

Jean-Charles Lebunetel

Yves Raingeaud
email: raingeaud@univ-tours.fr

A New Method For Input Installation Impedance Measurement

Keywords: installation impedance, access impedance, loads, short-time, LCL filter, classification, Narroband

The access impedance for Narrowband PLC communication is dependent of the global installation impedance. The interest of this paper is to measure the domestic installation impedance using a new setup based on an LCL filter for masking the network impedance. The short-time and the frequency variation of the global impedance in the frequency band between 30 and 500 kHz is measured in several scenarios. The domestic loads are classified according to their impact on the global installation impedance by connecting to the installation and changing the mode of operation. The access impedance is recalculated using the measured installation impedance in parallel to the network.

I. INTRODUCTION

The use of powerline distribution network for data communication is an old solution that interested many energy distributors in the world. The main advantage of narrowband powerline communication PLC technologies resides in the use of existing cables infrastructure between the transformer substation and the smart meter in the customer side. The PLC outdoor transmission is characterized by a low impedance specifically in the customer side [START_REF] Razazian | G3-PLC Specification for Powerline Communication: Overview, System Simulation and Field Trial Results[END_REF][START_REF] Bouassam | Influence of load impedances on the quality of Power Line Communication[END_REF]. Domestic installations in the same electrical network are physically linked to each other's so the access impedance measured in many works [START_REF] Chu | Narrow Band Power Line Channel Characteristics for Low Voltage Access Network in China[END_REF][START_REF] Hooijen | Channel Model for the Residential Power Circuit Used as a Digital Communications Medium[END_REF][START_REF] Sigle | On the Impedance of the Low-Voltage Distribution Grid At Frequencies Up to 500 kHz[END_REF] is impacted mainly by the input impedance of the customer installation and the impedance of the network. In this article, we have an interest to measure the impedance of one domestic installation.

Indoor PLC communications use the home grid as a transmission support. In France, the standard NFC 15-100 regulates the low voltage electrical installations [START_REF] Nizigiyimana | Characterizatio n and Modeling Breakers Effect on Power Line Communications[END_REF]. It guarantees the protection of the installation by the using of a general circuit breaker and protection device attached to the house electrical panel. From this panel every departure has the differential circuit breaker and an electrical cable with an adapted section to supply many devices like the lighting circuits, socket, heating, water heater, etc. Each of these loads can be connected on bypass, star, cascade or a combination of all these solutions.

Many studies have an interest to simulate a scenario of a global installation impedance [START_REF] Hallak | Impedance Measurement of Electrical Equipment Loads on the Power Line Network[END_REF][START_REF] Ou Kharraz | Characterization of the Input Impedance of Household Appliances in the FCC Frequency Band[END_REF] by the paralleling of some loads [START_REF] Hallak | Impedance Measurement of Electrical Equipment Loads on the Power Line Network[END_REF][START_REF] Ou Kharraz | Characterization of the Input Impedance of Household Appliances in the FCC Frequency Band[END_REF] but the complexity of the architecture in the customer installation make the simulation of the total impedance of the installation very difficult. It is impossible to predict the domestic architecture and the number of loads connected. Also, the simulation need the knowledge of the impedances of each domestic load, but they vary according to the brand and the type of the equipment.

In our study, we have an interest about the measurement of different scenarios of the impedance of a domestic installation as a function of loads connected. The goal is to identify the loads which has the major impact on the global impedance of the installation. To realize this measure, a new setup is developed to mask the network impedance which is time and frequency variable using an LCL filter.

The domestics load are time and frequency variable [START_REF] Arzbergerl | Fundamental Properties of the Low Voltage Power Distribution Grid[END_REF][START_REF] Bausch | Characteristics of Indoor Power Line Channels in the Frequency Range 50-500 klz[END_REF][START_REF] Antoniali | Measurement and characterization of load impedances in home power line grids[END_REF][START_REF] Corripio | Analysis of the cyclic short-term variation ofindoor power line channels[END_REF][START_REF] Antoniali | Measurement and characterization of load impedances in home power line grids[END_REF]], that's why, we have an interest to study the short-time variation of the installation impedance and the impact of the global impedance of installation on the access impedance measured in parallel to the meter. This paper is organized as follows: in the section II, the conception of the LCL filter is presented. In the section III, the filter installation and the global impedance measurement method are described. In the section IV, the measurement of the installation impedance is used to recalculate the access impedance. In the section V, the classification of the domestic loads according to its impact on the global impedance measurement of the installation is introduced. In the section VI, the impact of the loads on the access impedance is studied and finally in the section VII, the short-time variation of the installation impedance will be introduced.

II. LCL FILTER DEVELOPPEMENT

The LCL filter is realized to mask the network impedance which is time and frequency variable [START_REF] Bouassam | Influence of load impedances on the quality of Power Line Communication[END_REF][START_REF] Chu | Narrow Band Power Line Channel Characteristics for Low Voltage Access Network in China[END_REF][START_REF] Hooijen | Channel Model for the Residential Power Circuit Used as a Digital Communications Medium[END_REF]. The impedance of the filter must be high because the domestic installation impedance will be installed in parallel to the filter. The risk of measurement of the impedance of the filter can appear if its impedance is very low. The filter impedance must be also time invariant.

To respect this specification, an LCL filter is proposed with a first coil L1 equal to 3.6 mH, a capacitance equal to 5µF and a second coil L2 with a value equal to 0.5 mH. The electrical schematic of the filter is presented in the fig. 1. The impedance of the filter is measured using a network analyzer. The impedance result is depicted in the fig. 2.

Fig. 2. Impedance of the filter

The impedance of the filter is high and varied from 90Ω in 30 kHz to 60 kΩ in the resonance peak.

III. MEASUREMENT SETUP

A. Filter Installation

The installation of the filter is performed on the user side in the house downstream of the meter.

Fig. 3. Filter installation and measurement setup

The filter is set in serial between the main circuit breaker and the installation electrical panel as it presented in the fig. 3.

The measurement setup consists of the use of a capacitive coupler to filter the 50 Hz. The network analyzer role is the measuring of the temporal and frequency variation of the impedance by the measuring of the reflection coefficient S11 [START_REF] Ou Kharraz | Characterization of the Input Impedance of Household Appliances in the FCC Frequency Band[END_REF]. Two types of configuration are done for the two types of measurement:

• For the Frequency variation.

To measure the frequency variation of the impedance, we realized a sweep with a START frequency fixed to 30 kHz and STOP frequency equal to 500 kHz. The number of points is equal to 801 and the IFBW is set to 200 Hz.

• For the short-time variation

To measure the impedance variation during 20 milliseconds between 30 kHz and 500 kHz. We measured 201 points of frequency on 0 span mode. Every frequency is measured during 20 milliseconds which is the sweep time duration to measure the rapid variation of one frequency during one period of the network. Every measurement is synchronous with the frequency of the sector 50 Hz using a developed card for the zero crossing [START_REF] Amanci | Synchronization System with Zero-Crossing Peak Detection Algorithm for Power System Applications[END_REF]. The results will be depicted in Time Frequency graph.

B. Measurement Method

Two measures are performed leading to the measurement of the impedance:

• Measurement of the impedance of reference Zref This impedance of the reference represents the impedance of the network in serial with the impedance of the filter when the customer installation is disconnected to the network.

Fig. 4. Zref measurement

The measurement results of the frequency variation and shorttime variation of Zref are depicted in the fig. 5 and6. We conclude that Ztotal is equal to Zinstallation, so the measured impedance would be directly Zinstallation,

C. Input Impedance Installation Measurement Results

The impedance of the installation is measured in the same configuration of the loads connected. The list of the connected loads consists on a laptop, a washing machine, a dryer, a printer, a lamp, a fridge, an oven, a microwave, a coffee maker tassimo, a robot, an induction table, a dishwasher, a TV, an internet box, 2 PLC plugs, a boiler, a VMC, a phone charger, an electrical couch and electric shutters.

Zinstallation. is measured in two cases:

• when the installation is connected to the network. The measurement is done using the new setup based on the filter to mask the network impedance, • when the installation is disconnected from the electrical network the impedance is measured directly because the installation is unpowered.

The measure is done when the loads are in OFF operating mode. The results are depicted in the fig. 8.

Fig. 8. The installation impedance measurement

The results show similar variation of the Zinstallation in the two cases. The impact of domestic loads connected to the installation when they are powered in OFF operating mode is very close to their impact when they are unpowered.

IV. IMPACT OF THE INSTALLATION IMPEDANCE ON THE ACCES

IMPEDANCE

The access impedance measured in the meter side represents the network impedance in parallel to the customer installation impedance. The access impedance is also measured in the meter when the customer installation is connected to the network.

Fig. 11. Comparison of Zaccess calculated and measured

The access impedance measured and calculated has the same variation in the frequency band from 30 kHz to 500 kHz with a small difference (3Ω in 500 kHz). The network and the installation impedance aren't measured in the same time. The time variation of the impedance of the network can explain the difference between the calculated and the measured measurement.

V. IMPACT OF THE LOADS ON THE INSTALLATION IMPEDANCE

In this part, the setup is used to measure the Zinstallation in several scenarios to study the impact of:

• the impedance of the electrical cable and circuit breakers,

• the loads due to their connection to the installation and the operating mode of the load (ON or OFF).

A. The Impact of the Cables and The Circuit Breakers

To measure the impedance due to those elements, we measured Zinstallation when all the loads are disconnected.

Fig. 12. Zinstallation without loads

The results presented in the fig. 12 show a high impedance in the low frequency band. The impedance decreases in function of frequency. The negative phase represents the high impact of the impedance of the cables.

B. The Impact of The Loads

To study the impact profile of some domestic loads on global impedance of the installation, Zinstallation is measured in several scenarios.

1)

The impact of connecting of the loads Zinstallation is measured for when we connected loads to the installation. The operating mode is fixed to OFF for all the loads when they are connected to the installation. Many measures show two types of loads:

• Non-impacting load Many loads have a major impact on the global impedance of the installation. The impacting loads at the connection are the loads that have a low impedance in the frequency band from 30 to 500 kHz. Some loads like the washing machine and the phone charger and some type of lamp does not cause any frequency variation of the impedance, but they generate some types of noise such as the phone charger.

2) Impact of the operating mode (OFF to ON)

In general, the domestic loads are always connected to the installation and the user does not disconnect the equipment, but it changes their operating mode. This is why in this party we are interested to classify the loads according to their impact on the global impedance of installation when they change their operating mode from OFF mode to ON. Many measure are carried out in different scenarios shows two types of loads. Fig. 15. Non-impact of the lamp, the oven and the dryer We studied the impact of many domestic loads when we changed their operating mode from OFF to ON. Most loads have non-impact when the operating mode pass to ON except for the TV, robot and the coffee marker.

VI. IMPACT OF THE LOADS ON THE ACCES IMPEDANCE

The measurement of the access impedance when the customer installation is connected to the electrical network is carried out in several scenarios to study the impact of the connection of some loads on the access impedance measured in the meter. The results show that the impact of the loads on the global impedance of the installation can be seen in the access impedance. The loads that have a high impedance are masked by low impedance loads.

VII. TIME VARIATION OF A DOMESTIC INSTALLATION IMPEDANCE

The variation of Zinstallation. during 20 milliseconds is carried out in 201 frequencies between 30 kHz to 500 kHz.

Fig. 1 .

 1 Fig. 1. Electrical schematic for of the LCL filter

Fig. 5 .

 5 Fig. 5. Frequency variation of Zref

Fig. 6 .

 6 Fig. 6. Time-frequency variation of Zref Zref has a high impedance from 100 Ω to 10 kΩ on the resonance frequency . It represents a short time invariant behavior during the main period. This invariation masks the time variation of the impedance of the network. The reference measurement will be used to extract the input impedance of the installation Zinstallation. • Measurement of the total impedance Ztotal and deducing of Zinstallation This impedance Ztotal is equal to Zref in parallel with the Zinstallation.as shown in Fig. 7.

Fig. 7 .

 7 Fig. 7. Ztotal measurement The impedance of the customer installation Zinstallation is deduced from Zref and Ztotal according to the expression (1) and (2).

Fig. 9 .Fig. 10 .

 910 Fig. 9. Access impedance measurement point

Fig. 13 .

 13 Fig. 13. Impact of the phone charger

Fig. 16 .

 16 Fig. 16. Impact of some loads on the access impedance

Fig. 17 .

 17 Fig. 17. Short-Time behavior of the installation impedanceThe time varying behavior of a global impedance of installation show a low variation in the low frequencies. This time variation is the result of the time variation behavior of the domestic loads because the components of electrical devices depend on the instantaneous amplitude of the mains voltage[START_REF] Hallak | Impedance Measurement of Electrical Equipment Loads on the Power Line Network[END_REF][START_REF] Ou Kharraz | Characterization of the Input Impedance of Household Appliances in the FCC Frequency Band[END_REF][START_REF] Antoniali | Measurement and characterization of load impedances in home power line grids[END_REF][START_REF] Corripio | Analysis of the cyclic short-term variation ofindoor power line channels[END_REF][START_REF] Antoniali | Measurement and characterization of load impedances in home power line grids[END_REF]

TABLE I .

 I

CLASSIFICATION OF THE LOADS IMPACT WHEN CONNECTING TO THE INSTALLATION

Connection of

loads in OFF operating mode Domestic loads Impacted Non-impacted

	Waching machine	×
	Dryer	×
	Printer	×
	Laptop	×
	lamp	×
	Oven	×
	Microwave	×
	Caffee maker	×
	Robot	×
	Electrical couch	×
	Induction table	×
	Boiler	×
	Dishwasher	×
	VMC	×
	Fridge with open door	×
	TV	×
	Phone charger	×

TABLE II .

 II CLASSIFICATI:ON OF LOADS IMPACT ACCORFING TO THE

	OPERATING MODE
	Switch from OFF to ON
	Domestic loads	Impacted	Non-impacted
	Waching machine		×
	Dryer		×
	Printer		×
	Laptop		×
	lamp		×
	Oven		×
	Microwave	×
	Caffee maker	×
	Robot	×
	Electrical couch		×
	Induction table		×
	Boiler		×
	Dishwasher		×
	VMC		×
	Fridge open / closed door		×
	TV	×
	Phone charger		×

IEEE International Symposium on Power Line Communications and its Applications (ISPLC) 978-1-5386-1913-1/18/$31.00 ©2018 IEEE