Morteza Tadbiri Nooshabadi

Shahrokh Farhangi
email: farhangi@ut.ac.ir

Jean-Luc Schanen
email: jean-luc.schanen@grenoble-inp.fr

Hossein Iman-Eini
email: imaneini@ut.ac.ir

Frequency Model for EMI Study of Three-Phase Grid Connected Photovoltaic Inverter on Both DC and AC Sides

Keywords: electromagnetic interference, photovoltaic inverter, frequency model. I

This paper presents an EMC model in the frequency domain for grid connected three-phase photovoltaic inverters using the conventional Boost-Inverter topology. The aim is to estimate the EMC noises on both DC and AC sides, in order to design for instance EMI filters, or for any sensitivity analysis or optimization process. The model is developed and validated on an experimental prototype setup. The results are discussed, focusing on the contribution of each converter to the noise produced, both on the DC and AC sides by considering the impedance path of noises.

INTRODUCTION

In recent years, by significant decrease in cost of solar system (including panels and inverters), the transformer-less photovoltaic inverter is also used on a large scale [START_REF]RENEWABLES 2021 GLOBAL STATUS REPORT[END_REF]. Fig. 1 shows a block diagram of a grid connected transformer-less photovoltaic system. It this kind of systems, there is a potential and significant problem -the leakage current and electromagnetic interference on the AC and DC sides of the photovoltaic system without galvanic isolation [START_REF] Meneses | Review and Comparison of Step-Up Transformerless Topologies for Photovoltaic AC-Module Application[END_REF]. This problem can disrupt the operation of the inverter and cause some undesirable faults. Adding electromagnetic interference filter on both AC and DC sides of the inverter is proposed in [START_REF] Zhang | EMI filter design and optimization for both AC and DC side in a DC-fed motor drive system[END_REF]. However, sizing those filters is time-consuming using time domain simulations, and necessitates several iterations, since the noise on both DC and AC sides are not independent. A method of designing two EMI filters simultaneously by considering the crosstalk between them was proposed in [START_REF] Zhang | EMI filter analysis for transformer-less photovoltaic inverter[END_REF], which can solve this problem in the conventional design.

DC EMI Filter

AC EMI Filter

Inverter Boost Grid PV Array Fig. 1. The block diagram of the PV inverter system.

When high power density or lower cost is needed, filter optimization is requested. In this case, the conventional design methodology is no more sufficient and EMI models dedicated to optimization must be proposed. The frequency domain analysis [START_REF] Revol | EMI Study of Three-Phase Inverter-Fed Motor Drives[END_REF] exhibits very low simulation times, in comparison to time domain and is well adapted to optimization approach. It replaces the switching devices by equivalent sources, and has already been used in the simple case of a DC-DC converter [START_REF] Delhommais | Design by optimization of power electronics converter including EMC constraints[END_REF], or drive system [START_REF] Touré | EMC Modeling of Drives for Aircraft Applications: Modeling Process, EMI Filter Optimization, and Technological Choice[END_REF]. In [START_REF] Xiang | A Fast and Precise Method for Modeling EMI Source in Two-Level Three-Phase Converter[END_REF], [START_REF] Zhou | A New EMI Modeling Method for Mixed-Mode Noise Analysis in Three-Phase Inverter System[END_REF], the same method is applied to a three-phase inverter with a resistive load at the output, where only one EMI filter is needed on the input side. Further studies have been carried out in [START_REF] Boillat | EMI Filter Volume Minimization of a Three-Phase, Three-Level T-Type PWM Converter System[END_REF]- [START_REF] Nidhal | Modelling of conducted EMI generated by a three-phase PWM rectifier[END_REF] on the modeling of power electronics converters from an EMC point of view, with the focus on a converter. Also, in [START_REF] Ohn | Three-terminal common-mode EMI model for EMI generation, propagation, and mitigation in a full-sic three-phase ups module[END_REF], a three-terminal CM EMI model for a UPS has been developed. The modeling of such researches is more restricted to a converter and an EMI filter. The purpose of this paper is to propose a frequency model for a three-phase photovoltaic inverter, composed of two cascaded converters, a Boost and a three-phase inverter. The novelty is to consider simultaneously both converters, and to account for the EMI generation on both DC and AC sides, what was not done in previous studies. The model is presented in section II, identified in section III, and validated in comparison with both experiment and time domain simulation in section IV. A circuit diagram with the parasitic parameters of the PV system is shown in Fig. 2. Each component including capacitors and inductors are modelled using high frequency model including stray elements. Common mode parasitic capacitors between the power layout and the heatsink (supposed at ground potential) are considered. The LISNs schematic and their components value are presented in Fig. 3 and Table I. Also, the used experimental LISN is shown in Fig. 4. It is worth nothing that in this study, the LISNs are used to fix the impedance in input and output and due to availability of the material for experimental validation, the DC LISN is not the 150 Ω one, which is defined in CISPR 16-1-2 or EN 55011. A 50 Ω single cell version of CISPR 16 standard is used on both DC and AC sides. This is not a serious issue, since the LISN are used to verify the model, not to measure the noise according to any kind of emission standard. The disturbance voltages measured on LISN which are specified in Fig. 3 (V1, V2, VA, VB and VC) must be evaluated thanks to the frequency model. To obtain the equivalent EMC model, the well-known method of equivalent sources has been chosen. It replaces the switching devices by equivalent voltage and current sources reproducing the same discontinuities [START_REF] Revol | EMI Study of Three-Phase Inverter-Fed Motor Drives[END_REF]. Fig. 5 shows the equivalent circuit, with the voltage sources being the voltage of the MOSFETs, and the current sources being the current of the diode in the boost converter and the equivalent current of MOSFETs in the inverter. These sources can be expressed in the frequency domain which makes the solving of this circuit quite easy. The value of each source is obtained by its corresponding voltage or current spectrum which are obtained in ideal condition simulation or with either theoretical or experimental wave forms.

III. MODEL IDENTIFICATION

As it mentioned in section II, the sources in Fig. 5 (voltage and current) are obtained by transferring time domain values to frequency domain. Among mentioned methods, analytical method is the fastest one which is explained for each converter and source in the following:

A. Inverter Sources

The method is to use the PWM switching angles and a transform to convert a time signal into a frequency signal. The inverter's output voltage source is alternating at switching times, and is equal to +VDC when the corresponding switch (the top switch in the leg) is on and 0 when it is off. Considering a sum of signals delayed by the switching angles, the spectrum of the voltage is written as follows:

, ℎ ℎ ∑ ℎ ℎ ′
(1) With h, the harmonic order, , the closing angle and ', the opening angle as ( 2) and (3).

! " #$% &'() & * (2) ′ + ! " #$% &'() & * (3)
Where mf, ma, Tsw and ω0 are frequency index, modulation index, inverter switching frequency and grid frequency respectively. This is worth nothing the other voltage sources of the inverter have the same spectrum with only 120 degrees of phase shift.

Using the same method, the switch current is obtained by multiplying the line current in time domain by the switching pattern and finally, the input current of the inverter is obtained by summing the current of switches in three different legs as proposed in [START_REF] Voldoire | Analytical calculation of dc-link current for N-Interleaved 3-Phase PWM inverters considering AC current ripple[END_REF].

B. Boost Converter Sources

The value of voltage source of boost converter is obtained by the spectrum of voltage of boost converter switch as (4):

,,--. ℎ ℎ 1 ℎ0 1 (4)
Where Db is the duty cycle of the boost converter. The current source of the boost converter is representative of the diode current in the converter that carries the ripple current in addition to the step one.

2 -3.,,--. ℎ ,1 5 67 #$ + 89 :1 ,1 0 1 ,1 89 :1 0 1 ,1
; < , = ,, >

ℎ &? @ ,1 A; (5)
Besides the analytical method, the spectrum can also be obtained by a FFT of a time domain simulation, or a FFT of measured signals. As an illustration, the spectrum of Vsw,boost for the case study for three different method is shown in Fig. 6. The results match very well till 3 MHz, what is sufficient in the objective of filter design, which are very often governed by the low frequency part of the EMI spectrum. Obviously, the model would be the most accurate with sources modeled by experimental data, but due to the number of sources (six sources) and the requirement of 50 Hz harmonic accuracy for the spectrum, a large amount of data with a high sampling rate is needed. So, the measurement way is in practice very difficult or even impossible. In addition, because of the effect of the phase shift on the system, the sources must have the same reference and be considered simultaneously, what leads to another difficulty. Analytical method or FFT on simulated waveforms are therefore to be used. For practical reasons, simulation data for identifying the sources have been chosen. Other components that must be identified are the passive elements and their stray behavior. The value of these components is measured and listed in Table II. In addition, Cpv is assumed 100 pF as in [START_REF] Yang | Improved Transformerless Inverter With Common-Mode Leakage Current Elimination for a Photovoltaic Grid-Connected Power System[END_REF]. However, due to the ability of LISNs in fixing the impedance, this arbitrary value does not impact significantly the results, since it is before the stabilization network. Four modules (one module in the boost converter and three in the inverter) are used in the system. As shown in Fig. 7, each module consists in a leg and a 470nF decoupling capacitor. The top MOSFET in the boost converter module is not driven and only its diode is used. To identify the stray capacitance of the power modules the voltage across the decoupling capacitor is supposed to be fixed by the DC link voltage and so these two points are considered short circuited during identification. The modified power module schematic is shown in Fig. 8. Because of existence of semiconductor and decoupling capacitors, the measurement of each capacitor separately is not possible but the total capacitance is available by shorting the P and M points and measuring these points through the ground. In this case, the Coss is much higher than stray capacitances and does not affect on their identification. The measured capacitance is equal to:

B C 3DCE B FG, + B HG 100 JK (6)
The layout of the module is shown in Fig. 9. The main contribution of capacitance is the drain of the MOSFET which is connected to the back side of the TO-247 package. The capacitances from the drain to the heatsink are dedicated to stray capacitance. Based on the layout symmetry of the modules, the value of each capacitance is considered equal and 50 pF. Layout of the power module.

As the same modules are used for the system, the stray capacitance for other modules are considered identical. The CMG,m of each module is in parallel with the others. So, the total CMG is equal to four times that of each module and 200 pF. The value of stray capacitances of power modules are listed at Table III.

IV. EXPERIMENTAL RESULTS

The configuration of the experimental test prototype is shown in Fig. 10. This is a very low power prototype just for demonstration purpose, using very low voltage. The input voltage is considered to be 25V and the boost converter operates with a duty cycle of 0.5 with a 35 kHz switching frequency. The 50W three-phase inverter switches at 40 kHz and the modulation index of the SPWM (Sine Pulse Width Modulation) control is 0.8. The two converters are assumed to operate at different switching frequencies to distinguish between noise generated by the two converters on each DC and AC sides. The reference voltage and output AC current are shown in Fig. 11, a simple resistive load on the AC side replacing the grid.

The disturbance voltages are measured and compared with both time domain simulation and the frequency model. The frequency range used for comparison is focused on the low frequency part of the spectrum (below 3MHz), since it is the most impacting range for EMI filter design regarding weight, volume or cost. Higher frequencies are linked with stray elements and layout. Fig. 12 and Fig. 13 show the disturbance voltages on the DC and AC side respectively. As presented, the model fits the measured noises as well as the simulation results and reproduces the effect of different switching frequencies of the two converters (the peaks amplitude are matching well). It is worth noting that the EMI noise on the DC side is not only due to the boost converter, but also contains part of the three-phase inverter (This can be clearly identified since the two switching frequencies have been chosen different). However, in the considered specific case, it is found that the noise created by the inverter is dominant in the AC side, and much larger than boost converter in the DC side. This is due to the different impedances in the DC and AC noise path where the AC noise passes through both LISNs and the DC noise passes through almost the DC LISN. As it is clear in Fig. 5, the current sources do not have any effect on measured noises on both sides, since they are looping in the DC capacitor. This is of course a first order approximation, but in the frequency range of interest, it is valid. The noises are thus generated by the voltage sources only. Based on superposition principle, the equivalent circuit in each voltage source consideration on DC and AC sides thanks to common mode are shown in Fig. 14 and Fig. 15 respectively. The Vcm,ac is the representative of the common mode voltage which is generated by the inverter. As it is clear in the equivalent circuits, the noises can pass through both DC and AC sides. In Fig. 14, the generated noise after passing through CPG is divided into two paths in the DC and AC sides. The amount of divided noise depends on the impedance path where the AC impedance is higher than DC one in the frequency range of interest as presented in Fig. 16. Thus, most of the DC noise passes through the DC side and has no significant effect on the AC side. Unlike the DC side, the generated noise by the inverter does not pass-through stray capacitances. This is illustrated in Fig. 17, where Zc, which represents the impedance of CAG+CBG+CCG by [START_REF] Xiang | A Fast and Precise Method for Modeling EMI Source in Two-Level Three-Phase Converter[END_REF], is higher than the impedance of the line inductor and AC LISN path and most of the noise passes through the inductor and AC LISN path. Also, from Fig. 15, it is deduced that the majority of noise passes through the DC LISN. On the other hand, due to the existence of CPG in the noise path, the impedance in the noise path of the boost converter (Fig. 14) is higher than that of the inverter (Fig. 15). Therefore, regardless of the value of voltage sources (even though the envelope of the boost converter voltage source is higher than that in the inverter (Fig. 18)), the noise generated by the inverter is greater than that of the boost converter, and since most of the noise generated by the inverter passes through the DC side, it is dominant in the DC side.

L , L " ! 1 ! M N (7)

V. CONCLUSIONS

In this paper, the frequency model for a three-phase photovoltaic system with two converters at different switching frequencies is explored. The identification methods of the sources and stray elements is explained. The results are then compared to experimental and time domain simulation. The contribution of each converter to the produced noise is discussed and shown by impedance path of noises.

Fig. 2 .

 2 Fig.2. The schematic of the grid connevted PV inverter system with high frequency parasitic parameters.

Fig. 3 .

 3 Fig. 3. Line voltage measurement in a known configuration using, (a) DC LISN on the DC side (b) three phase AC LISN on the AC side

Fig. 4 .

 4 Fig. 4. The experimental DC LISN.

Fig. 5 .

 5 Fig. 5. Equivalent circuit for EMC studies.

Fig. 6 .

 6 Fig. 6. Comparison between measurement, simulation and analytical methods in voltage source of the boost converter.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Power module schematic.P

Fig. 9 .

 9 Fig. 9.Layout of the power module.

Fig. 12 .

 12 Fig. 12. DC side disturbance voltages.

Fig. 10 .

 10 Fig. 10. Experimental prototype setup for EMC study.

Fig. 11 .

 11 Fig. 11. The reference voltage and output line current.

Fig. 13 .

 13 Fig. 13. AC side disturbance voltages. Simulation is obtained from a time domain simulation, Freqtool is the proposed frequency model.

Fig. 14 .Fig. 15 .

 1415 Fig. 14. The equivalent circuit of the system in the case of the DC voltage sources consideration.

Fig. 16 .

 16 Fig. 16. Impedance curve in different paths in the case of DC voltage source consideration.

Fig. 17 .

 17 Fig. 17. Impedance curve in different paths in the case of AC voltage source consideration.

Fig. 18 .

 18 Fig. 18. Voltage sources spectrum.

TABLE II

 II

	.	CONVERTER MODEL PARAMETERS

TABLE III .

 III POWER MODULES STRAY ELEMEMTNS

CMG [pF] 200 CFG, CAG, CBG, CCG [pF] 50


domain: phys spi

text:
S Gousset

D Picone

E Le Coarer

J M Rodrigo

D Voisin

L Croize

Y Ferrec

M Dalla Mura

IMSPOC: A NOVEL COMPACT HYPERSPECTRAL CAMERA FOR THE MONITORING OF ATMOSPHERIC GASES

Keywords: Hyperspectral imaging sensors, Fourier transform imaging spectrometer, Miniaturized sensors, Gas monitoring

There is an increasing industrial and scientific demand for large scale monitoring of gas concentration, e.g., to abide with constraints on pollutants for environmental safety. This paper aims to show the potential of the very recent ImSPOC technology for gas monitoring. We present the ImSPOC operating principle, some of the developed prototypes and two illustrative examples in the monitoring of methane and carbon dioxide in controlled environments.

INTRODUCTION

Gas monitoring in the atmosphere is a major concern for health and climate, as greenhouse gas emissions and pollutant released from anthropogenic activities are increasing every year. To be able to envision constraint policies on emissions of the main greenhouse gases (e.g., CO 2 and CH 4 ) and atmospheric pollutants (e.g., NO 2 and O 3 ), a monitoring at large scale is required, combining both high spatial resolution and frequent temporal coverage. Atmospheric probing both from the ground and remotely sensed from the space are complementary [START_REF] Ciais | Atmospheric inversions for estimating CO2 fluxes: methods and perspectives[END_REF]. The former allows for local reference measurements at very high sensitivity and high time frequency, whereas the latter ensures global coverage. Absorption spectroscopy is the mainstream technology of ground and remote sensing instruments to trace gas concentration in the atmosphere through dispersive [START_REF] Pasternak | The microcarb instrument[END_REF] or Fourier-based spectrometers [START_REF] Hamazaki | Fourier transform spectrometer for Greenhouse Gases Observing Satellite (GOSAT)[END_REF]. Hyperspectral imaging is a key to improve the spatial coverage of space missions, by combining together This work has been partly supported by a grant from Labex OSUG (ANR10 LABX56), the European Union's H2020 research and innovation program (grant n. 769032) and the ANR FuMultiSPOC (ANR-20-ASTR-0006). imagery of large field of views and spectroscopic measurements. The future Copernicus CO 2 mission [START_REF] Bézy | The European Copernicus Anthropogenic CO2 Monitoring Mission[END_REF] will integrate such a device. Nevertheless, the revisit period of the satellite is still too high to ensure improved precision with respect to emission surveys. Payload miniaturization may reduce the cost of a single satellite, allowing to launch a dedicated constellation complementary to the reference missions.

Imaging SPectrometer On Chip (ImSPOC) is a patented concept for a compact snapshot hyperspectral sensor [START_REF] Guerineau | [END_REF] develop for a variety of ground and on-board applications. The aim of this paper is to give an overview on the potential of the ImSPOC technology for gas monitoring. The paper is organized in 3 sections. In Section 2 the concept of ImSPOC is presented. Then, Section 3 describes some integrated prototypes dedicated to gas monitoring. In Section 4, we present some example of experiments we conducted in order to assess gas detection capability of these prototypes. We conclude with a discussion on the open challenges.

THE IMSPOC CONCEPT

ImSPOC [START_REF] Guerineau | [END_REF] is an innovative concept for a non-conventional compact (i.e., 0.2 to 1 kg) hyperspectral image spectrometer of low-cost and high spectral resolution. As shown in Figure 1 (a), the instrument is composed of a matrix of adjacent interferometers (i.e., Fabry-Pérot etalons) of different thickness, disposed in front of a focal plan array (FPA). Each interferometer is associated with a microlens which focuses onto the FPA an image of the same scene modulated by interferometric fringes depending on the interferometer thickness. In greater detail, the transfer function of an interferometer depends on the wavelength λ and the incident angle θ of the incoming rays. As shown in Figure 1(b) for a fixed angle of incidence θ, after two consecutive inner reflections within the etalon, the interferometer introduces a round trip phase shift φ = 2π δ λ , where δ = 2nL cos n n0 θ denotes the optical path difference (OPD). Here, L is the thickness of the interferometer, while n and n 0 denote the refraction index within and outside the etalon, respectively. The energy associated to the packet of rays that emerge from the etalon is then ideally focused on a single pixel of the FPA by the associated microlens. The transfer function A(λ, θ) between the transmitted energy and the first emerging ray, for a given interferometer, is given by the following relation, known as Airy's distribution [START_REF] Nur | Fabry-pérot resonator: spectral line shapes, generic and related airy distributions, linewidths, finesses, and performance at low or frequency-dependent reflectivity[END_REF]:

A(λ, θ) = +∞ k=0 Re -jkφ = 1 (1 -R) 2 + 4R sin 2 φ 2 , (1)
where R denotes the internal reflectivity of the etalon. The Airy's distribution (1) represents a domain transformation of the incident ray from the domain of the wavelengths to that of the OPD, which, in the continuous domain, is commonly known as interferogram. The availability of multiple interferometers allows to sample the interferograms at different OPD; this discretized representation can be inverted to recover the input spectrum, which can then be processed to recover the concentration of the gases to monitor.

IMSPOC PROTOTYPES AND APPLICATIONS

The ImSPOC concept can be specialized to address gas monitoring but also a wide range of other applications in remote sensing and planetary science such as the monitoring of vegetation, land/surfaces and solar activity [START_REF] Coarer | Optimization of a compact static interferometer based on ImSPOC technology for a wide field polar lights monitoring[END_REF]. The characteristics of the components of an ImSPOC device are dependent on the application. For example, the focal plane array used will determine the spatial resolution of the instrument, the spectral filter the spectral sensitivity range and the set of interferometers the frequency filtering and sampling rate. Two spectral domains are currently investigated, leading to different technological components and applications: i) UV-VIS: 300-1000 nm, from the near ultra violet to the cut-off frequency of the silicon based FPA (fully including the visible interval); and ii) SWIR: 800-1700 nm, in the short wave infrared domain.

There are two main modes of operation for the ImSPOC devices: i) the Hyperspectral mode, whose target is to capture a finely sampled interferogram in order to reconstruct with high resolution the full spectrum in the sensitivity range of the instrument making this a general purpose device; and ii) the Specialized mode, where the sampling of the interferogram is optimized with respect to a target of interest in the Fourier domain. This mode of operation is application-oriented and allows, for instance to correlate some CO 2 and CH 4 absorption spectral patterns in dedicated ImSPOC cameras [START_REF] Gousset | NanoCarb hyperspectral sensor: on performance optimization and analysis for greenhouse gas monitoring from a constellation of small satellites[END_REF], as presented in the experiments of the next section. Figure 2 presents some gas dedicated ImSPOC prototypes, and their associated characteristics.

GAS MONITORING USING IMSPOC

We present in this section some preliminary experimental results to illustrate the gas detection capability of the ImSPOC prototypes.

A first experiment aims at measuring the signal quality of a methane-dedicated ImSPOC-based camera in a controlled in-lab acquisition. A wide spectrum source is focused within a 10 cm long gas cell before entering into an integrated sphere. The camera is oriented towards the output port of the sphere, so that the whole field of view (FoV) is illuminated homogeneously. The methane flow in the gas cell is controlled; as the cell is not hermetically sealed, the pressure in the cell is around 1 atm for the whole duration of the experiment. At the beginning, the cell is filled with air then the methane starts flowing and stops once the cell gets fully saturated. The acquisitions after the methane flow has stopped allow to study the release of the methane from the cell. Figure 3(a) shows a comparison between the raw acquisitions before and after the methane injection: the image initially does not show any pattern (left), while interference rings clearly appear when the methane is present (right). in Eq. 1. When the cell is filled by air (blue curve) the signal follows a Poisson distribution showing a uniform dependency with the phase. When methane is present (black curve), an almost-sinusoidal fringe appears as expected and formalized by Eq. 1. We then analyse the temporal evolution of the acquisitions while the cell fills up with methane. Figure 4 shows the mean signal level for each acquired frame with respect to the ratio between the modulation amplitude and the mean level of the interferogram (i.e. fringe visibility). As expected, the two parameters have an inverse trend: when the methane concentration increases, light is absorbed and the mean of the signal with interference patterns of increasingly higher contrast (increased visibility) appearing. This first experiment shows that the designed camera is responsive to the methane spectroscopic signature, showing an encouraging capability to detect atmospheric methane.

In the second experiment, we study the evolution of the integrated slant column of CO 2 during a day using a CO 2specialized ImSPOC camera pointing at the Sun and taking acquisitions every ten seconds.

Figure 5 presents the recorded signal visibility with respect to the time of the acquisition during the day. For each acquisition, we plot on the Y-axis the visibility histogram for all the pixels of the image. The white curve stands for aver- aged visibility over all the pixels. Several time sequences of acquisitions are missing, and a sensor malfunction occurred in the segment before 15:00. As a consequence data are not continuous.

By using formula of the airmass factor (AMF) described in [START_REF] Pickering | The southern limits of the Ancient Star Catalog and the Commentary of Hipparchos[END_REF], we are able to estimate that the airmass is approximately multiplied by a factor 1.4 from 14:00 UTC (z = 62.5 • , AMF ≈ 2.16) to 15:00 (z = 70.79 • , AMF ≈ 3.01). By analyzing the obtained results, the visibility measured from 14:00 to 15:00 increases from about 0.024 to 0.030, leading to a factor of 0.030/0.024 = 1.25 which is quite close to the estimated AMF multiplication value of 1.4. The relative error of 10% is related to the airmass approximation we used while z becomes close to 80 • . We must integrate some additional atmospheric features as refraction and layered species distribution, in a more complex model. This simple proof of concept shows that the ImSPOC technology has a potential for the characterization of the atmospheric absorption, representing a first step towards an operational gas sensor for space applications.

In addition to these two preliminary experiments, some results about CO 2 and CH 4 monitoring from an aircraft us-Fig. 4. Temporal evolution of the mean signal and the mean interferometric visibility. Frame numbers are reported on the horizontal axis, one tick interval corresponds to 150 ms. The cell is filled with air until frame-16. A constant flow of methane then fills the cell until frame-24 when it stops and pressure decreases.

ing the devices we presented in this paper have been reported in [START_REF] Gousset | NanoCarb spaceborne miniaturized GHG sensor: first experimental results[END_REF] for a smallsat constellation feasibility study. Furthermore, a non-specialized ImSPOC prototype operating in the visible domain has been considered for NO 2 detection in [START_REF] Bourdin | No2 vertical column density estimation from interferograms captured by a snapshot interferometric imaging spectrometer[END_REF]. While this recent technology shows an enormous potential thanks to its high spectral resolution, compactness and low-cost, there are several open challenges. The most relevant problems lie in the signal processing, as the acquisitions are taken in a domain different to the one that is desired for the final applications, hence requiring a carefully designed pipeline of algorithms of data processing to recover the desired information. Such pipeline typically includes, image coregistration, characterization of the transfer function and inversion of the interferograms [START_REF] Picone | Model based signal processing techniques for nonconventional optical imaging systems[END_REF]. Moreover, in comparison with the controlled scenarios presented here, further complications are expected when the FoV is not illuminated homogeneously (i.e., when considering real scenes with complex structures) and when the interferometric signal is dominated by the scene dynamics, which can be typically two orders of magnitude greater than the target absorption patterns.

Fig. 1 .

 1 Fig. 1. The ImSPOC concept. (a) Optical design of a generic ImSPOC device and (b) the operating principle of a single interferometer of thickness L. The OPD can be determined by the difference between the green and the red paths.

  Figure 3(b) compares the histogram of the intensity of these two images as a function of the phase term (modulo 2π) expressed (a) CO 2 and CH 4 dedicated ImSPOC cameras. (b) SWIR ImSPOC cam. (c) Visible ImSPOC cam.

Fig. 2 .

 2 Fig. 2. Some ImSPOC prototypes dedicated to gas monitoring. (a) CO 2 and CH 4 dedicated ImSPOC SWIR cameras, designed for airborne detection of gas plumes, 64×64 pixels in the field of view (15 • ), 2 kg each. (b) SWIR ImSPOC hyperspectral camera designed for UAV: 40 spectral bands in range 1200-1700 nm, 64×64 pixels fov (15 • ), < 0.5 kg. (c) Visible ImSPOC hyperspectral camera (without front baffle) for sky monitoring: ∼100 spectral bands in 380-1000 nm, 96×96 fov (15 • ), 0.5 kg.

Fig. 3 .

 3 Fig. 3. Laboratory experiment of methane detection in a gas cell with a specialized ImSPOC-based camera. (a) Acquisition frame before (left) and after (right) the introduction of methane. (b) Signal histogram as a function of the interferometric state (phase). In blue, when filled with air, and in black, with methane saturated at atmospheric pressure.

Fig. 5 .

 5 Fig. 5. Temporal histogram of interference visibility during a solar view with a CO2-specialized prototypes.

5 .

 5 DISCUSSION AND CONCLUSIONThis work presented an overview of the ImSPOC technology. The preliminary results obtained in two controlled experiments illustrate the potential of the ImSPOC concept for the monitoring of atmospheric gases.


Fontaine Richard

Courdier Rémy

Payet Denis

A step towards more eco-responsible computing

Keywords: Agent, Ubiquitous computing, Eco-computing

On a global scale, the environmental footprint of digital technology represents a continent two to three times the size of France and five times the size of the French car fleet. In order to limit the negative impact of this overabundance and to optimize the emerging computing potential of our environment, we propose an architecture model and its implementation in order to allow the mutualization of the components embedded in our connected devices.

Introduction

We are at the beginning of a new computing era which is manifested by a trend to integrate computing into the physical objects of everyday life [START_REF] Weiser | Some computer science issues in ubiquitous computing[END_REF][START_REF] Böhlen | Ambient intelligence in the city overview and new perspectives[END_REF][START_REF] Ramos | Smart offices and intelligent decision rooms[END_REF]. While providing a lot of benefits, this technological advance brings with it a set of environmental issues [START_REF] Bordage | Study -The environmental footprint of the digital world[END_REF][START_REF] Cailloce | Numérique: le grand gâchis énergétique[END_REF].

According to the European Union [START_REF] Commission | Eu commission, critical raw materials, internal market[END_REF], in 2020, there were 30 critical raw materials (against 14 in 2011) and among them, many elements are directly linked to new technologies: Cobalt (lithium-ion batteries); Germanium (optical fibers); Hafnium (processor); Indium (touch screen); Tantalum (liquid crystal displays, dynamic random access memory (DRAM) chips), or Lithium (battery). These elements were generally related to the manufacture of computers, printers and other common digital objects. But since 2015, we notice that a shift has taken place [START_REF] Bordage | Study -The environmental footprint of the digital world[END_REF] :

-Televisions represented 5 to 15 % of the impacts in 2010 against 9 % to 26 % by 2025; -Smartphones represented 2 to 6 % of the impacts in 2010 against 4 to 16 % by 2025; -Connected objects represented 1 % of the impacts in 2010 against 18 % to 23 % by 2025.

Based on these statistics, we can see that the depletion of our global reserves is fast approaching. We therefore need a new vision of eco-design that focuses not only on the energy impact, but on a direct safeguard of our raw materials.

To do this, we believe it is necessary to see in our connected objects more than their initial functionalities. These hidden and currently unexploited functionalities, which we will call dispositions, represent the core of our approach. We propose, through an architecture model, to reveal them and to mutualize them virtually in order to allow the emergence of new services without systematically adding new physical connected objects. This model has been implemented within an Android framework named Agent Framework For Omnipresent Real Device (AFFORD).

Before highlighting the technical solutions that are used to make this mutualization of dispositions, we will illustrate our concept in order to demonstrate its advantages.

Example of a mutualisation of components: a monitoring agent

In this example; we will consider a monitoring agent A which has the objective Ω of alerting in case of a fall. This objective can be divided into three required objectives ω which are:

ω 1 : to detect a fall ω 2 : to alert the user by a sound ω 3 : alert the user with a flashing light In our case, the agent has access to a set of connected devices including: a light bulb, a computer, and an accelerometer. So, we can describe the Obj objects of the environment through the n dispositions they present:

-Obj 1 : The accelerometer

• n 1 detect a fall -Obj 2 : The computer

• n 2 produce sound • n 3 display a picture -Obj 3 : The connected light bulb

• n 4 make light

In a usual context, the service would be managed by a single entity, whereas in our case, it is possible to dissociate the monitoring service, the capture of information and the response of the system. When the system is started, agent A seeks to accomplish the Ω objective. Because of the presence of the accelerometer Obj 1 in its environment and its disposition n 1 to detect a fall, the objective ω 1 is made permanently active. If a fall is detected, the agent reacts and its ω 2 and ω 3 objectives become active. In an optimal case, where the light bulb Obj 3 and the computer Obj 2 are accessible, the agent has the possibility to activate the behaviours ω 2 and ω 3 . In this way, attention is focused on the problem of falling. In the same way, it is interesting to note that this same light bulb Obj 3 and this same computer Obj 2 can be used as an alert source for other types of monitoring (such as intrusion, temperature, or hydrometry monitoring). This vision of our environment thus allows us to reduce redundancies within a system by privileging a more intelligent and reasoned use of the objects that surround us.

An ecological challenge, but also a technical challenge

To make this vision real and to start this change of habit, three points must be taken into account:

From a conceptual point of view , we believe that it is necessary to propose a high-level solution, which would allow a quick deployment on a maximum of existing devices.

From a technological point of view , considering the ubiquitous aspect, the solution must be light enough to allow its deployment on devices with low computing capacity.

From a human point of view , its use must be as natural as possible. To achieve this, we will emphasize the presence of autonomous entities that can decide individually on the provisions they wish to share.

Current technological possibilities

Before presenting our solution, we will examine the existing technical possibilities that can be used to try to solve our problem. In order to do this, we will look at three different levels of abstraction : middleware, agents, and artifact concept.

Middleware. In ubiquitous systems, the middleware is generally considered as a generic layer which provides basic functions [START_REF] Baldauf | A survey on context-aware systems[END_REF][START_REF]Middleware and application adaptation requirements and their support in pervasive computing[END_REF]. This approach is commonly used [START_REF] Gateau | Ontology-based smart iot engine for personal comfort management[END_REF][START_REF] Laleci | Saphire: A multi-agent system for remote healthcare monitoring through computerized clinical guidelines[END_REF], however a limit to the use of middleware comes with the fact that, usually, a system has as many middlewares as there are communication problems [START_REF] Kjaer | A survey of context-aware middleware[END_REF][START_REF] Mohamed | Middleware for robotics: A survey[END_REF]. Consequently, the more heterogeneous entities are involved in a system, the more it may contain a large number of middleware in order to hide the communication problems.

Even if some authors propose the implementation of a "middleware of middlewares" in order to provide a unique interface to the applications [START_REF] Seinturier | A Component-Based Middleware Platform for Reconfigurable Service-Oriented Architectures[END_REF], the system will be unable to adapt as new technologies appear.

Agents. In an ambient intelligence context, an option to overcome the heterogeneity of the environment is the use of agents ([9, 13, 22, 25]). However, if we want to scale up, it is inconceivable to put the responsibility of describing all the connected objects and their possibilities on the level of the agents in charge of the services. Based on this observation, some proposals try to overcome the problems of making connected objects available by linking them to specialized "agent-objects" [START_REF] Maamar | How to agentify the internet-of-things?[END_REF][START_REF] Kwan | A agentified use of the internet of things[END_REF][START_REF] Pantoja | An architecture for the development of ambient intelligence systems managed by embedded agents[END_REF]. Unfortunately, this option is confronted with a conceptual problem. Indeed, it is not advisable to model an object using the agent concept, because it does not have its characteristics [START_REF] Cetnarowicz | From algorithm to agent[END_REF][START_REF] Deloach | Multiagent systems engineering: A methodology and language for designing agent systems[END_REF][START_REF] Jennings | On agent-based software engineering[END_REF][START_REF] Wooldridge | Agent-based software engineering[END_REF].

Artifacts. The A&A (Agent & Artifact) meta-model is characterized by three main abstractions [START_REF] Omicini | Artifacts in the a&a meta-model for multi-agent systems[END_REF]:

-Agents: proactive components of systems, encapsulating the execution of activities in a given environment; -Artifacts: passive components of systems, such as resources and media, intended to be used by agents to support their behaviour; -Workspaces: conceptual containers of agents and artefacts, useful for defining the topologies of the environment and notions of the locality.

The concept of artefact, due to its abstract nature and not being conceived for a specific purpose, unfortunately does not directly solve the problem of the mutualisation of the dispositions of objects present in our environment. However, its coherence in terms of agent/object modelling and its polymorphic nature make it an interesting theoretical tool for our model proposal.

Concept description

From the constraints and elements raised above, we can deduce that the current solutions do not meet the requirements of our proposal. From this observation, we propose to modify our vision of the environment by proposing a virtual decomposition of our objects [START_REF] Fontaine | Vers une utilisation éco responsable des objets connectés par la mutualisation de leurs composants physiques: Une approche basée sur le concept d'artefact[END_REF] in the form of a dispositional artifact that we will define as follows:

Definition 1 (Decomposition into dispositional artifacts). Each object Obj, can be represented by a set of artifacts Art according to the usage such as :

Obj-> {Art n }
with Obj a connected object, Art a virtual artifact, n the disposition of the object encapsulated by the artifact.

From this decomposition, it is then possible for the agent to perceive new possibilities within its environment that we will call dispositional opportunities.

Definition 2 (Emergence of dispositional opportunities ). Let W (an agentobject system) = (A, Obj) and W p (an agent-artifact system) = (A, Art p )composed of an artifact Art with a disposition p. A dispositional opportunity exists if and only if there exists an opportunity τ , such that : Based on these definitions, we propose an architecture model that involves three types of entities based on the A&A concept: agent, workspace and artifact. Within this model, the layouts are encapsulated in artifacts that will be used as virtual media to make the dispositions of connected objects accessible to an agent (Fig. 1).

Because the roles of the agent and the workspace are globally unchanged, we will directly focus on the conceptual additions related to the decomposition of the objects into a dispositional artifact tree.

Architecture model

We will therefore focus on the seven main artifact classes: Artifact, Composite Artifact, Main Artifact, Final Artifact, Sensor Artifact, Actuator Artifact, Service Artifact.

-Artifact

This class is the primary component of the layout decomposition. Globally, it defines all the primitives useful for programming the observable behaviours of artefacts in accordance with the frameworks or implementation environments. This first class allows the establishment of the dependency link between artefacts, as well as its observable state within workspaces.

-Composite Artifact

The CompositeArtifact class is mainly an artifact container. Its role is to keep a trace of the structure of the original connected object. It is also possible that it is itself under the management of an artifact if it is also a component element.

-Main Artifact

The Main Artifact class virtually represents the object and contains information about it (nature, state, mac address, etc.). It is also responsible for accessing the artifacts and only this class is directly linked to the workspaces.

-Final Artifact

The elements of this class represent the artifacts that are directly accessible in the workspace. All classes that inherit from it (Sensor Artifact, Actuator Artifact, Service Artifact) can be seen as the leaves of the decomposition tree of the original object and therefore cannot have artifacts that compose them.

Properties

Thanks to the artifact classes seen previously, it becomes possible to virtually describe the dispositions of an object in the form of a tree. Within this description, the leaves represent the object's dispositions and the internal nodes, the structure of its decomposition.

The Figure 2 illustrates a virtual decomposition of a smartphone into a tree where the root represents the original object (Main Artifact), the intermediate nodes the structure of the decomposition (Composite Artifact) and the leaves the dispositions that can be made accessible (Final Artifact).

Fig. 2. Example of decomposition

As a result, the object is no more represented in a monolithic way, but is decomposed into a set of dispositional artifacts. It then becomes possible to mutualize them within workspaces in order to propose services that fully use the possibilities of our connected environment. From a conceptual point of view, unlike the use of agents, mutualization is based on a coherent paradigm, and their uniformity and high level of abstraction avoid the overabundance of middleware. The use of artifacts also has the advantage of solving two key problems related to the Publish/Subscribe concept, which is generally used to define the modalities of transmission of sensor information. First, the problem of efficiently linking an event with many subscribers on one type of event can be solved by "relocating" the associated Sensor object to another device. The connected object will only have to manage one Subscriber per sensor. Broadcasting to interested entities will be done through the Sensor artifacts. Second, the problem of efficient multicasting of events within a network of event providers is solved by having the agent connected directly to the desired sensor. Thus, interested entities will only have access to information that is relevant to them. An implementation of this model, named Agent Framework For Omnipresent Real Device (AFFORD), has been developed on Android through a framework in order to show the feasibility of the model. AFFORD is based on the "Software Kit for Ubiquitous Agent Development" (SKUAD) platform, which is a platform for creating ambient agents, being able to manage sensors and effectors, whose performances and design allow an embedded use [START_REF] Denis | Skuad, software kit for ubiquitous agent development[END_REF]. The application we made, in APK format (9.13MB), can detect all sensors embedded in an Android device and currently offers seven effectors and two services (Fig. 3).

Android implementation

At runtime, if a component is available and the authorizations allow it, the system proceeds to the creation of artifacts as an instance of the associated class. The user can then choose, via the user interface, which artifacts to make available and which agents to activate within the workspace.

During the test phases, AFFORD was tested on different Android devices (smartphone and tablet) in order to highlight the presence of provisions. This first step confirmed that many possibilities were hidden in our smartphones. In addition to the effectors and services that can be implemented (e.g. ringtone, voice recognition, buzzer.), we can see in the table (Fig. 4) that natively they were able to propose at least twenty artifacts. To perform this test, we used the artifact encapsulating the accelerometer, present in any smartphone and usually used to determine the orientation of the screen or to stabilize photographs, as a shock sensor. It should be known that a smartphone accelerometer can generally detect movement in the micrometer range. Moreover, in order to show that the possibilities are not limited to the components of the device, we have created an artifact encapsulating synthesis and voice recognition to perform a simple dialogue with a user.

Thus, the intelligent entity, in this implementation a SKUAD agent, becomes able, by adjusting the sensitivity parameters, to discriminate different types of shocks (fall, heights, direction, presence of a final shock). In case of shock, the agent will try to attract the user's attention via the flash or the vibrator of Android devices or by making the computer screen blink. If the agent does not get a response, he can then use voice synthesis and voice recognition to talk to a user. In this case, the agent asks a series of closed questions giving the choice to the user to answer among a set of predefined answers. Each answer pronounced by the user will be transformed into text and then analyzed by the agent in order to make a report of the situation.

Encouraging results

Through the previous example, however simple, we find ourselves virtually and schematically in a system with 3 processors, 3 batteries, 12 GB of Ram and 580 GB of storage. This power is much higher than the one required in a current use. This particularity shows us that the sensing and broadcasting entities can be externalized without the necessity of adding components with a processing capacity. We have also shown that the current dynamics seeking to make the environment intelligent could pass, not by a systematic integration of computing power within each object, but rather by a more reasoned and mutualized use of the possibilities of our devices. Moreover, due to the fact that processing can be done at the peripheral level, this solution offers an alternative to cloud computing. It also allows to reduce the energy footprint, since it does not require the internet network or the datacenter, while limiting the leakage of personal data.

Conclusion

The current trend is to systematically integrate computing capabilities into each object. However, an alternative would be to propose a more reasoned and mutualised use of the possibilities of our devices. Thanks to an architecture model inspired by the concept of artefact, we propose to virtually put forward the dispositions of our connected devices in order to reduce the redundancies within an environment. We implemented the model on Android which allowed us to highlight the multitude of unused opportunities present in our environments. From the results, we found that a single smartphone or tablet could power many services. This collective power allows us to reduce the number of batteries, screens and computer components (RAM, processor, storage) and thus contribute to a step towards more eco-responsible computing.

1 .

 1 W = (A, Obj) does not possess τ 2. W p = (A, Art p ) possess τ and Art p ∈ Obj

Fig. 1 .

 1 Fig. 1. Mutualisation of components within a workspace

Fig. 3 .

 3 Fig. 3. An application on the Android platform

  Example in real environmentA real-world example of this implementation has been proposed through a falling detector. The devices used in this test are: a Samsung Galaxy S8 smartphone, a Samsung Tab A (2016) tablet and an Acer TravelMate P257 laptop. At the end of the layout analysis step, the smartphone offers twenty-six Sensors and the tablet offers five.

	Brands	Samsung Motorola Samsung Samsung	Xiaomi
	Model	Galaxy s8 One zoom SM-T870 SM-N960F Redmi Note 8T
	RAM (Go)	4	4	8	6	4
	SensorArtifact	26	44	20	27	22
		Fig. 4. Some examples of tested smartphones
	4.1


François Joseph Chatelon

Jacques Henri Balbi

Dominique Morvan

Jean Louis Rossi

Thierry Marcelli
email: marcelli@univ-corse.fr

A convective model for laboratory fires with well-ordered vertically-oriented fuel beds

Several studies in the literature explore the connection between rate of spread (ROS) and wind in wildland fires. These studies show very different positions about the role of radiation and convection as heat transfer mechanisms. In the case when the fuel bed is well-ordered and vertically-oriented, there seems to be a consensus leading to suggest that convective heating is the dominant heat transfer mode in that case. The purpose of this work is to propose a convective semi-physical model for the behaviour of the rate of spread in wind, when the fuel bed is vertically-oriented. Due to a specific fuel bed arrangement, flame radiation -i.e. radiation from the part of the flame above the vegetal stratum-is neglected. Only horizontal radiation from the fuel burning particles area and convective heating are taken into account. Convective heat transfer is assumed to be the primary heat transfer mechanism. The proposed model is confronted to 172 laboratory fires with a wide range of fuel characteristics. The predicted results are also compared with two simplified models from the literature. Statistical tools are used to check the agreement between the predicted ROS and the observed one where a strong agreement is generally observed, irrespective of fuel bed characteristics.

Introduction

Wind is commonly accepted [START_REF] Byram | Forest fire behaviour[END_REF] as one of the major factors that affects a fire is rate of spread. Generally, fire burning aided by wind expresses higher rates of spread than in 'no-wind' cases. Several studies have described the relationship between the ROS R and wind velocity U, where a power function of the wind velocity (R ∝ U n ) is commonly fitted to ROS data [START_REF] Pitts | Wind effects on fires[END_REF][START_REF] Morvan | Wind Effects, Unsteady Behaviors, and Regimes of Propagation of Surface Fires in Open Field[END_REF]. The exponent n derived from these studies, however, is inconsistent. For instance, Thomas and Pickard [START_REF] Thomas | Fire spread in forest and heathland materials[END_REF], Wolff et al. [START_REF] Wolff | Wind-aided firespread across arrays of discrete fuel elements. II[END_REF] and Catchpole et al. [START_REF] Catchpole | Rate of Spread of Free-Burning Fires in Woody Fuels in a Wind Tunnel[END_REF], observed n < 1 whereas the results provided by Rothermel and Anderson [START_REF] Rothermel | Fire spread characteristics determined in the laboratory[END_REF], Rothermel [START_REF] Rothermel | A mathematical model for predicting fire spread in wildland fuels[END_REF], or Mendes-Lopes et al. [START_REF] Mendes-Lopes | Flame characteristics, temperature-time curves, and rate of spread in fires propagating in a bed of Pinus pinaster needles[END_REF] suggested n > 1. All these cases are part of what Rothermel and Anderson [START_REF] Rothermel | Fire spread characteristics determined in the laboratory[END_REF] presented as the three possible curve shapes for ROS in wind conditions.

The main heat transfer mechanism induced by fuel bed geometries seems to be important in order to explain those different curve shapes. Indeed, fire spread models commonly assume a steady spread [START_REF] Rothermel | A mathematical model for predicting fire spread in wildland fuels[END_REF][START_REF] Albini | A Model for Fire Spread in Wildland Fuels by-Radiation[END_REF][START_REF] Catchpole | A model for the steady spread of fire through a homogeneous fuel bed[END_REF], and the interface between burning and preheating fuel related to fire spread has been widely studied, especially for shallow and continuous fuel beds [START_REF] Rothermel | A mathematical model for predicting fire spread in wildland fuels[END_REF][START_REF] Fons | Analysis of fire spread in light forest fuels[END_REF][START_REF] Albini | A physical model for firespread in Brush[END_REF]. Radiant heat transfer from the flame to the unburnt fuel has been widely considered as the primary mechanism controlling fire spread, the heating and ignition of fuel particles by flame contact being largely neglected [START_REF] Finney | On the need for a theory of wildland fire spread[END_REF]. However, Anderson [START_REF] Anderson | Heat transfer and fire spread[END_REF], Fang [START_REF] Fang | An investigation of the effect of controlled wind on the rate of fire spread[END_REF] for surface fires, and Van Wagner [START_REF] Van Wagner | Conditions for the start and spread of crown fire[END_REF] for crown fires, indicated that the radiant heat transfer could only account for a part of the total heat flux necessary to sustain a spreading fire. Some authors [START_REF] Baines | Physical mechanisms for the propagation of surface fires[END_REF][START_REF] Weber | Modelling fire spread through fuel beds[END_REF] also found that radiation heat transfer was not sufficient to ignite fuel bed particles due to the too low fuel particles temperature at the flame's arrival. Note also that some recent results on the wildland fire flame spread and ignition mechanisms detail the location where the local heat flux received by downstream surface in wind-driven fires is maximum [START_REF] Tang | Local flame attachment and heat fluxes in winddriven line fires[END_REF].

At the field scale, some fuels display discontinuities between individual plants. In order to investigate the fire behaviour in these fuel types, some authors have proceeded to laboratory experiments where fuel beds are made of well-ordered, vertically oriented particles with regular spacing. These fuel beds are usually constituted by matchsticks [START_REF] Fons | Analysis of fire spread in light forest fuels[END_REF][START_REF] Vogel | Flame propagation along matchstick arrays[END_REF][START_REF] Prahl | Preliminary investigations of forced convection on flame propagation along Paper and Matchstick Arrays[END_REF], toothpicks [START_REF] Wolff | Wind-aided firespread across arrays of discrete fuel elements. II[END_REF] or laser-cut cardboard [START_REF] Finney | A study of flame spread in engineered cardboard fuelbeds: Part I: Correlations and observations[END_REF]. For those discontinuous fuel beds, i.e. fuel beds with significant gaps between individual fuel elements, it has been suggested that convective heat transfer is necessary to correctly understand the fire spread mechanisms [START_REF] Pitts | Wind effects on fires[END_REF][START_REF] Morvan | Wind Effects, Unsteady Behaviors, and Regimes of Propagation of Surface Fires in Open Field[END_REF][START_REF] Weber | Modelling fire spread through fuel beds[END_REF][START_REF] Beer | Fire propagation in vertical Stick Arrays -the Effects of Wind[END_REF][START_REF] Finney | Experiments on Fire Spread in Discontinuous Fuelbeds[END_REF]. Finney et al. [START_REF] Finney | On the need for a theory of wildland fire spread[END_REF] also conclude that the ignition of fuel particles after direct contact is the main mechanism thanks to which the fire propagates.

In a recent work [START_REF] Finney | Role of buoyant flame dynamics in wildfire spread[END_REF], Finney et al. focused on the role of convective and radiative heating on fire spread. Their measurements performed on a set of experiments conducted on those discontinuous fuel beds show that radiation causes a slow increase of the fuel temperature whose value remains below 100 °C. Moreover, when bursts of flame and hot gases contact intermittently the unburnt fuel particles, fuel temperature increases fastly. In this type of fuel bed arrangement, convective heating seems to be the main heat transfer mechanism.

The main purpose of this paper is to present a simplified semiphysical model which is able to correctly reproduce the ROS in wind conditions, when the fuel bed is well-ordered and vertically-oriented. The main phenomena in fire spreading are represented by some physical laws, but nevertheless some empirical laws could be added. By using some approximations, this model tries to solve, the equations governing heat transfer and combustion, and as such can be classified as a simplified semi-physical model. Particularly, these simplifying assumptions avoid into account taking the gas transport equations explicitly, which leads to a model constituted by algebraic equations with two advantages: a computational time close to zero and an explicit analytic relationship giving the ROS as a function of the wind velocity and the main characteristics of the vegetal stratum and the environment. Among the three usual heat transfer modes, flame radiation -the radiation from the part of the flame located above the vegetal stratumis neglected because it is weakly received by the fuel particles due to their verticality -following the work of Finney et al. [START_REF] Finney | Role of buoyant flame dynamics in wildfire spread[END_REF]. Radiation from the flame base -fuel burning particles area-is obtained thanks to the assumption of an equivalent radiant panel. Because of the weak packing ratio and the well-ordered geometry of the fuel bed, the air flow easily enters the vegetal stratum and a part of this flow is going to go out through the flame base-unburnt fuel bed interface. Then the convective effects with direct flame contact are assumed to be important and represent the main heat transfer mechanism. This assumption is supported by the work of Finney et al. [START_REF] Finney | Role of buoyant flame dynamics in wildfire spread[END_REF]. Obviously, in the case of a continuous fuel bed, radiation cannot be neglected and may often control the flame spread, especially with large fires.

In a first section, the main equations of the model are set. Especially, horizontal and vertical velocities of the gaz flow in the vegetal stratum are estimated, which involves the divided streamline: gases get out of the flame base through its upper part or by the flame base-unburnt fuel interface. An assessment of the contact flame power and its absorbed part for the fuel preheating is done. Finally, according to Balbi et al. [START_REF] Balbi | A 3D physical real-time model of surface fires across fuel beds[END_REF], a thermal balance on the preheating fuel bed gives the expression of the ROS. This convective model is composed of three universal model parameters -set up on two experiments-and their value is the same whatever the experiment series.

In a second section, the model is confronted to several sets of laboratory experiments and is compared to two other empirical models found in the literature, namely the simplified models provided by Wolff et al. [START_REF] Wolff | Wind-aided firespread across arrays of discrete fuel elements. II[END_REF] and Catchpole et al. [START_REF] Catchpole | Rate of Spread of Free-Burning Fires in Woody Fuels in a Wind Tunnel[END_REF]. The effectiveness of the model is evaluated with usual statistical tools, such as the normalized mean square error (NMSE), the fractional bias (FB), and the Pearson correlation coefficient (r).

Main equations of the model

In order to obtain a simplified model, it is necessary to process complex phenomena in a simple way. Particularly, the gases movement in and around the flame is considered to be stochastic, but its effects on the rate of spread are completely deterministic. The proposed model is obtained by considering mean movements in time and space, so equivalent laminar flows are used in the place of turbulent flows and the main physical characteristics of the fire front (temperature, flame height…) are replaced with mean values. The fire front is considered to be a linear one.

Stream lines

Pyrolysis gases are emitted from the flame base, i.e. the fuel burning particles area -denoted by ABCD in Fig. 1. The temperature in the preheating zone (BB'C'C in Fig. 1), which is close to the one suggested by Pitts [START_REF] Pitts | Wind effects on fires[END_REF], ranges from ambient temperature to ignition temperature. The air stream which comes from the burnt area crosses the flame base and mixes with pyrolysis gases is subjected to progressive drag forces and driven to the top of the flame base, due to buoyancy. Due to the low fuel bed density and the vertical arrangement of the fuel particles, the drag forces are weak compared to a continuous fuel bed. So the stream lines will go out through the front panel of the flame base -denoted by BC in Fig. 1-or through the top of the flame base -denoted by AB in Fig. 1. Those two air streams are split by the line denoted by EB in Fig. 1. So, the flame can be divided in two parts, an external part above the vegetal stratum and an internal part which directly contacts the unburnt fuel bed and which will give the major part of energy transfer. The existence of this internal part of the flame has been emphasized by temperature measurements [START_REF] Korobeinichev | The velocity and structure flame front at spread of fire across the pine needle bed. Experiment[END_REF] or gas velocity measurements [START_REF] Anderson | Convective heat transfer in fire spread through fine fuel beds[END_REF] in the fuel bed.

Therefore, the flame contour is determined by the intersection of the flame base with stream lines number 1 and 3 (Fig. 1). The plume is

T v = T i T v = T a γ c BB ' E C x DF E A B u c u(z) U(L) U(x) z γ c γ c0
considered to be above the top of the flame (point I in Fig. 1). The stream lines above the flame body (number 4 in Fig. 1) follow the flame shape. In the same way, a fresh airflow (number 6 in Fig. 1) coming from the unburnt area, under the flame body also follows the flame shape. Indeed, in this area, the quasi-laminar flame acts as a barrier to the fresh wind stream. However, the flame is unsteady and discontinuous and external air movement can go through these discontinuities (number 5 in Fig. 1). The energy contribution of those stream lines is neglected because of their distance from the fuel bed and because of their temperature which is close to the ambient temperature.

On the other hand, all the stream lines located between number 1 and number 2 (Fig. 1) create an internal part of the flame -with the pyrolysis gases-(BCB' in Fig. 1) which will contribute to the unburnt fuel bed preheating. Then, it is necessary to quantify this heating energy which means to estimate the surface bounded by stream lines number 1 and 2. It is also necessary to quantify the part of the energy caught by the fuel bed. On the one hand, the external part of the flame (AIB') does not contribute to the preheating and on the other hand, there is no convective cooling coming from the unburnt area, but there might be a lateral convective cooling -coming from the face BB'C'C-in case of absent lateral walls.

Gas velocity modelling

In order to determine horizontal gas velocity, an empirical law is used to model drag forces. Indeed, as the air stream progresses through the fuel bed, drag forces -which are assumed to be proportional to the square of the normal wind velocity U 2 -become more and more important, thus resulting in the differential equation U(x) dU(x)/ dx =-K * U 2 (x), where K * is a coefficient and U(x) the stream velocity within the fuel bed at the fuel bed length x (Table 1).

Solving this differential equation leads to the following solution:

Ux U e ()= ( 0 ) Kx - (1)
Anderson et al. [START_REF] Anderson | Convective heat transfer in fire spread through fine fuel beds[END_REF] observed this decrease in U(x) at the top of the vegetal stratum in a region close to the flame front.

Upward gas velocity is due to buoyancy forces which result in a uniformly accelerated motion. An estimation -at mid-height vegetal stratum-is obtained with the simplified balance of the momentum equation in the vertical direction and is given by

uz g T T z ()=2 -1 m a 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ( 2
)
where z is the vertical coordinate, g is gravity acceleration, T a ambient temperature and T m represents mean gas temperature. Note that Thomas [START_REF] Thomas | The size of flames from natural fires[END_REF] proposed an equation similar to Eq. ( 2). The streamlines are determined by the velocity components U(x) and u(z) (Fig. 1) which are very different from one point of the fuel burning particles area to another. Two flows are splitted by the streamline denoted by EB (Fig. 1): the first one which leaves the area through the top (line AB) and the second one which leaves the area through the fire front side (line BC). In order to evaluate the amount of pyrolysis gases of the lower area, only the angle of the tangent in point B is needed. Its expression is then the following:

γ UL uh tan = () () c0 (3)
Considering that the flame time residence τ is inversely proportional to the surface area to volume ratio s (τ=τ 0 / s; τ 0 =75591 [START_REF] Anderson | Heat transfer and fire spread[END_REF]) and integrating this differential equation between x=0 -with U(0) =U, value of the wind velocity entering the flame base area-and x= L = R τ -when the internal stream leaves the flame base area, x is equal to the flame depth L, see Fig. 1)-yields:

UL U e ()= KR - (4) with K Kτ s = * 0 ( 5
)
where R is the rate of spread (ROS) and K is a drag coefficient which depends on the surface area to volume ratio s.

As the temperature of the fresh airstream entering into the flame is equal to ambient temperature and assuming that the pyrolysis gases temperature is close to the flame temperature, the expression of T m which depends on the air-pyrolysis gases mass ratio s tb in the flame base is obtained thanks to a weighted average formula:

T sT T s = + +1 m tb a tb ( 6
)
where T is the flame temperature. The value of the coefficient s tbcalled stoichiometric coefficient in Balbi et al. [START_REF] Balbi | A physical model for wildland fires[END_REF][START_REF] Balbi | Physical modeling of surface fire under nonparallel wind and slope conditions[END_REF]) is equal to 9. The expression of the mean flame temperature was set in [START_REF] Balbi | A 3D physical real-time model of surface fires across fuel beds[END_REF][START_REF] Balbi | A physical model for wildland fires[END_REF]:

T T Hχ sC =+ Δ (1-) (+ 1 ) a tc pa 0 (7)
T a Ambient temperature (K) =300 T i Ignition temperature (K) =600 T m Mean gas temperature (K) T v Mean fuel temperature (K) U Normal wind velocity (m s -1 ) U(x)
Air stream velocity within the burning fuel bed (m s -1 ) u c

Upward gas velocity at mid-height vegetal stratum (m s where ΔH is the heat of combustion of the pyrolysis gases (equal to 1.74×10 7 Jkg -1 [START_REF] Susott | Characterization of the thermal properties of forest fuels by combustible gas analysis[END_REF]), C pa is the specific heat of air (usually equal to 1150 J kg -1 K -1 ), χ 0 is a radiant factor (equal to 0.3 [START_REF] Mcgrattan | Thermal radiation from large pool fires[END_REF]) and s tc is the air-pyrolysis gases mass ratio in the flame. In presence of ambient wind, the amount of air dragged into the flame may be different from the stoichiometric requirement, due to different mixing efficiencyincrease of shear stress, flame stretching-. Indeed, Quintiere and Grove [START_REF] Quintiere | A unified analysis for fire plumes[END_REF] or Ma and Quintiere [START_REF] Ma | Numerical simulation of axi-symmetric fire plumes: accuracy and limitations[END_REF] suggested that the amount of air dragged into the flame may be much higher than the stoichiometric requirement. The value of the coefficient s tc is a universal coefficient which is fitted on one set of experiments (s tc =17).

Convective preheating intensity

The contact flame (BIB'C'C on Fig. 1) which gets out through the front face of the base is supplied with air/pyrolysis gases mixture coming from the area denoted by (BCDE) on Fig. 1. This surface area is difficult to assess because of its complex shape. As the proposed model has to be simple, this area is replaced by the surface area of the triangle (BCF) where the point F is defined thanks to the angle γ c (Fig. 1) where

γγ tan = 2 tan cc 0 (8)
Indeed, in order to have the two areas close enough to one another (triangle FBC and area DEBC), the angle γ c must be greater than γ c0 and fitting this angle leads to the value of the linear coefficient given in Eq. [START_REF] Rothermel | A mathematical model for predicting fire spread in wildland fuels[END_REF]. Note that the angle γ c does not represent the flame tilt angle. This error is minimized because the tilt angle of the streamline EB is lower than γ c .

Thus, substituting Eqs. ( 3) and (4) in Eq. ( 8) yields:

γ Ue u tan = 2 c KR c - (9)
where the expression of the upward gas velocity u c depends on the fuel depth (h):

uh gH χ ss C T = ∆ (1-) ( +1) ( +1) c tb tc pa a 1 2 0 1 2 ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ (10)
The convective heat flux received by the fuel is equal to the product of the heat of combustion of the pyrolysis gases, the pyrolysis gases flux and the absorption percentage. So, the expression of the convective heat flux is

ϕH σ τ chγ h h a =∆ ( 1 -) 1 2 tan + cc H 2 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ( 11
)
where σ is the fuel load, H the flame height and c is the char rate. The flame height modelling is given by Balbi et al. [START_REF] Balbi | A physical model for wildland fires[END_REF].

The coefficient a takes into account lateral heat losses. If the wind tunnel has lateral walls, there is no lateral heat losses and the coefficient a is equal to 1. Without lateral walls, the amount of lateral energy losses depends on the fire front width (W) in the following way:

aa W = min ( ; 1) 0 ( 12
)
where a 0 is a universal coefficient (a 0 =0.4).

Flame base radiation

If flame body radiation is negligible due to the fuel bed arrangement -vertically-oriented particles-, the radiation from the flame base remains significant. Its expression is obtained by considering the burnt/unburnt interface (BC on Fig. 1) as a radiant panel. The surface panel is equal to the surface of the projection of the fuel particles and its emissivity is then the projected surface/interface surface ratio. When the fuel bed is composed of vertically-oriented particles, this ratio can be calculated:

ε d D h ω D = min + sin ,1 b ⎡ ⎣ ⎢ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎤ ⎦ ⎥ (13)
where d, D and ω are respectively the particle diameter, the distance between two particles and the particle tilt angle (Fig. 2). Note that the particle tilt angle depends on the fuel particle and the experiments apparati; in most cases, it is practically equal to zero (e.g. [START_REF] Finney | A study of flame spread in engineered cardboard fuelbeds: Part I: Correlations and observations[END_REF]) or very small and then the fuel particles are vertically oriented. In some experiments (e.g. [START_REF] Wolff | Wind-aided firespread across arrays of discrete fuel elements. II[END_REF]) as the particle diameter is smaller than the diameter of the hole in which they are put in, a tilt angle due to the apparatus should not be close to zero and the fuel bed arrangement is practically vertical. The heat flux provided to the fuel is given by the usual Stefan-Boltzmann formulation for a grey radiant panel:

ϕε B T h = b b 4 ( 14
)
where B is the Stefan-Boltzmann constant.

Preheating thermal balance

Due to the fuel bed arrangement and geometry, flame radiation poorly impinges on the vegetal stratum. So, the thermal balance in a preheated fuel cell is the following:

σd xC dT dt φ dx φdx h dσ dt dx =+ -Δ p v bc H20 ( 15
)
where dx is the fuel cell thickness, C p the specific heat of fuel, T v the fuel temperature, Δh the heat of latent evaporation, φ b the radiative flux per length unit and φ c the convective flux per length unit. Integrating Eq. ( 15) over the preheated area and denoting dx/dt by R yields:

∫∫ σ C T T m h R φ dx φdx ϕ ϕ [(-) +∆ ]= + = + pi a bc b c ( 16
)
where m is fuel moisture content and T i is ignition temperature. Finally, rewriting Eq. ( 16) yields:

RR R =+ b c (17) with R σC T T m h ϕ εBT βρ q = 1 [(-) +∆ ] = b pi a b b v 4 ( 18
)
where q is ignition energy:

qC TT m h =( -) +∆ pi a ( 19
)
and

R σC T T m h ϕ H τq cs h h

h aγ = 1 [(-) +∆ ] = ∆ 2 (1-) + tan c pi a c H c 0 2 (20)

Model parameters

It is assumed that the drag coefficient K* is an increasing function of the porosity and a decreasing function of the fuel bed depth:

K sβ h * ~( 21
)
where β is the packing ratio. Combining Eq. ( 21) with Eq. ( 5) yields:

K β h ~ (22)
Using Eq. ( 12), the ratio coefficient is expressed in the following way:

K K aW β h = min ( ; 1) 1 0 ( 23
)
where K 1 expresses drag forces. As previously asserted, the value of the convective parameter a 0 , the air-pyrolysis gases mass ratio in the flame s tc and the drag forces coefficient K 1 have been fitted on the experiments carried out by Wolff et al. [START_REF] Wolff | Wind-aided firespread across arrays of discrete fuel elements. II[END_REF] (s tc =17, a 0 =0.4 and K 1 =15). These universal coefficients allow the proposed model to be a predictive model.

Note that due to Eq. ( 12) if the fire front width is wide enough (W 0 > 2.5 m) or if the experimental apparatus has lateral walls, the convective coefficient a is equal to one and the proposed model only has two universal model parameters (s tc and K 1 ).

Model equations

Finally, the rate of spread equation is obtained by rewriting Eq. ( 17) with Eqs. ( 12), ( 13), ( 18), ( 19) and ( 20):

R hB T βρ C T T m h H τCT T mh cs h h

h aW e u U = min + ,1 ((-) +∆ ) + ∆ 2( ( -) + ∆ ) (1-) + min( ; 1) d D ω D v pi a pi a H KR c sin 4 0 2 0 - ⎡ ⎣ ⎢ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎤ ⎦ ⎥ ⎛ ⎝ ⎜ ⎜ ⎜ ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎞ ⎠ ⎟ ⎟ ⎟ (24)
Note that obviously only the fuel bed characteristics change as well as wind velocity from one experiment to another.

According to Eq. ( 24), when the normal component of the wind velocity (U) is weak, then the ROS (R) is close to R b and the exponential function is close to 1. So, the trend of the ROS is practically linear. But when wind velocity increases, ROS increases and the exponential strongly decreases. The ROS behaviour becomes nonlinear and smaller than the linear straight line obtained when U is weak.

Numerical results

Many fuels, especially live vegetation canopies are not continuous because they contain gaps of a diameter similar to vegetation clumps. As well-defined fuel beds are less subject to random variations, several studies on wood cribs or well-ordered fuel arrays are reported. Predictions of the proposed model are confronted to the results of 172 laboratory experiments of fire spread carried out in five different references [START_REF] Wolff | Wind-aided firespread across arrays of discrete fuel elements. II[END_REF][START_REF] Catchpole | Rate of Spread of Free-Burning Fires in Woody Fuels in a Wind Tunnel[END_REF][START_REF] Fons | Analysis of fire spread in light forest fuels[END_REF][START_REF] Finney | A study of flame spread in engineered cardboard fuelbeds: Part I: Correlations and observations[END_REF][START_REF] Steward | Fire spread and individual particle burning rates for uniform fuel matrices[END_REF].

This model is also compared to two other simplified models, namely the models set out by Wolff et al. [START_REF] Wolff | Wind-aided firespread across arrays of discrete fuel elements. II[END_REF] and Catchpole et al. [START_REF] Catchpole | Rate of Spread of Free-Burning Fires in Woody Fuels in a Wind Tunnel[END_REF].

Confrontation to discontinuous fuel bed experiments

The experiments considered here are different from each another: both wind velocity and fuel bed characteristics vary -Table 2 briefly details the main fuel parameters and wind velocities. For instance, Wolff et al. [START_REF] Wolff | Wind-aided firespread across arrays of discrete fuel elements. II[END_REF] carried out a set of 29 fires with the same fuel bed -White pine toothpicks, pinus monticola-except the fuel load which was changed. Steward and Tennankore [START_REF] Steward | Fire spread and individual particle burning rates for uniform fuel matrices[END_REF] modified the fuel spacing or the particle diameter and consequently the fuel load in a birch dowels fuel bed. In the 49 fires conducted by Fons (Pinus ponderosa) [START_REF] Fons | Analysis of fire spread in light forest fuels[END_REF], a lot of fuel parameters were changed -Surface area to volume ratio, fuel moisture content, fuel load etc. Finney et al. [START_REF] Finney | A study of flame spread in engineered cardboard fuelbeds: Part I: Correlations and observations[END_REF] used a CO 2 laser system in order to cut cardboard fuel elements in an accurate way. This accuracy allows the authors to easily change some fuel parameters -surface-area-to-volume ratio, fuel load, particle diameter, fuel spacing etc. Catchpole et al. [START_REF] Catchpole | Rate of Spread of Free-Burning Fires in Woody Fuels in a Wind Tunnel[END_REF] have conducted 357 experimental fires over a range of particle sizes, fuel bed depths, packing ratios, moisture contents, and wind velocities. As the primary purpose of this article is to give a convection modelling and to study the trend of the ROS, a group of fires is selected from this substantial number of experiments. Four different fuels were used, of which only the Pinus ponderosa (heartwood sticks) experiments are simulated because of their vertically-oriented arrangement.

Error statistics on ROS observations versus predictions were calculated for the model performance study. These include the normalized mean square error (NMSE), an estimate of the overall deviations between predicted and measured values and the Pearson correlation coefficient which is a measure of the linear correlation between predicted and observed ROS. Finally, the fractional bias (FB) is used to estimate the proposed model's under-predictions or over-predictions.

The first results are presented in Fig. 3. The plain line in the scatter diagram represents the line of perfect agreement.

The scatter diagram shown in Fig. 3 indicates that predicted ROS matches measured ROS (NMSE=6.75%) with a bias close to zero (FB=0.05) which means that the proposed model neither overestimates nor underestimates the measured ROS. Note that the NMSE for each The mean deviation between the proposed model and the laboratory experiments is weak (NMSE=6.75%). Moreover, a major part of the error only comes from three fires conducted by Finney et al. [START_REF] Finney | A study of flame spread in engineered cardboard fuelbeds: Part I: Correlations and observations[END_REF]. Those three fires are not obviously in the general trend of all the other fires of this set of experiments. Removing those fires leads to an NMSE equal to 5.58% for the Finney et al. set of experiments and to 5.13% for all the experiments, which is in the margin of error measurements. Below this threshold, the exact agreement does not make any sense.

Comparison with other simplified models

The predictions of the proposed model are compared with those from Wolff et al. [START_REF] Wolff | Wind-aided firespread across arrays of discrete fuel elements. II[END_REF] and Catchpole et al. [START_REF] Catchpole | Rate of Spread of Free-Burning Fires in Woody Fuels in a Wind Tunnel[END_REF]. In these two simplified models a power law function matches the ROS dependence on wind velocity.

Carrier et al. [START_REF] Carrier | Wind-Aided Firespread across Arrays of Discrete Fuel Elements. I. Theory[END_REF] developed a simple model for the ROS of a fire propagating across an array of fuel particles. Wolff et al. [START_REF] Wolff | Wind-aided firespread across arrays of discrete fuel elements. II[END_REF] extended this work by taking into account the varying fuel depth. This model is based on the assumption that the dominant heat transfer process is convective heating and is also described by Eq. ( 25) where c w is a model parameter, in the present notations.

Rc U σ hs =( ) w 1 2 2 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ (25)
The value that best matches data sets and model performance results are respectively reported in Table 3. The numerical predictions from Eq. ( 25) are compared with the observed ROS in Fig. 4. Note that thanks to the square root dependence of the wind velocity, the predicted ROS fits the data well when the ROS curve presents a slow increase. The value of the Wolff model parameter which is the best fit for all data sets is equal to 0.07. But the model performances are poor (NMSE=41.57% with a bad correlation coefficient r=0.58).

Catchpole et al. [START_REF] Catchpole | Rate of Spread of Free-Burning Fires in Woody Fuels in a Wind Tunnel[END_REF] built a predictive model for the ROS based on energy conservation and experimental laboratory fires, which depend on the wind velocity and fuel bed properties. In the present notations, where k is the moisture damping coefficient:

The moisture damping coefficient (k) is the only parameter determined from the tuning. Catchpole et al. suggested that it might depend on packing ratio, fuel particle size, or both. If the values of k are known for each fuel type, the model is fully predictive. The value of k which best fits the five experimental data sets is equal to 11. This simplified model is then fully predictive but the results are poor (Table 3) with a large error (46.69%) and a bad correlation (r =0.56).

In comparison with these two simple models, the proposed model provides a better correlation between predicted and observed ROS over a wide range of fuel bed properties (see Figs. 345). Not only the precision, but the trend of the ROS is accurately reproduced. Examples of fires conducted by Wolff et al. [START_REF] Wolff | Wind-aided firespread across arrays of discrete fuel elements. II[END_REF] and by Steward and Tennakore [START_REF] Steward | Fire spread and individual particle burning rates for uniform fuel matrices[END_REF] are presented in Figs. 6 and7 respectively. In some Steward and Tennakore experiments, the ROS becomes faster as the wind velocity increases and the proposed model correctly expresses this acceleration of the ROS. The main difference between the three simplified models is the empirical side of the Wolff et al. and Catchpole et al. simplified models Moreover, the results given by the proposed model are obtained without any change in the values of the three model parameters whatever the experimental data set. The model is fully predictive with a very fast computational time -about 2 s to simulate all the 172 fires using a MacBook Pro with an i7 processor-.

Conclusion

This work deals with the development of a simplified propagation model in fuel beds composed of well-ordered, vertically oriented particles at the laboratory scale. Due to the specific fuel bed arrangement, the flame radiation is weakly caught by the vegetal stratum and thus is neglected. The model takes into account two heat transfer mechanisms, namely the horizontal radiation from the base of the flame -fuel burning particles area-and the convection when a part of the flame directly contacts the unburnt fuel.

The expression of these heat transfer mechanisms is partially obtained with usual physical laws and by using some usual empirical laws. The proposed model is constituted by an algebraic relationship of the ROS which depends on the wind velocity, fuel characteristicsmoisture content, packing ratio, thickness, width, surface area-tovolume ratio, density etc…-and three universal model parametersa drag coefficient K 1 , a convection coefficient a 0 , an air/pyrolysis gases mass ratio s tc -. The important fact is that the value of the three model parameters remains the same whatever the experiments, which means that the model is fully predictive.

The confrontation with 172 laboratory fires found in the literature leads to a small error (less than 7%), a small bias and a proper correlation.

The proposed model is easy to use because of its algebraic structure and then the computational time is close to zero -less than two seconds to simulate all the 172 fires using a MacBook Pro with an i7 processor.

In this work, only fires propagating in fuel beds composed of well ordered and vertically-oriented particles are taken into account. Indeed, these types of fuel bed emphasize the role of convective heating in fire spread. A modelling of this convective heating is given as a flame located into the vegetal stratum which contacts the unburnt fuel. The radiation from the fuel burning particles area is taken into account and due to the fuel bed arrangement, flame radiation is neglected.

Fig. 1 .

 1 Fig.1. Flaming zone combustion profile in presence of wind in the normal direction.

3 ΔH

 3 Heat of combustion of the pyrolysis gases (J kg -1 ) =1.74×107 ΔhHeat of latent evaporation (J kg -1 ) =2.3×10 6 ρ v fuel particle density (kg m -3 )σ Fuel load (kg m -2 ) φ b , φ cRadiative flux, convective flux per unit length (W m -2 ) ω Fuel particle tilt angle (°) τ 0 Flame residence time parameter =75591 τ Flame residence time (s)

Fig. 2 .

 2 Fig. 2. Main characteristics of the fuel bed arrangement.

Fig. 3 .

 3 Fig.3. Predicted ROS given by the proposed model versus observed ROS for the sets of fires spreading in well-ordered and vertically-oriented fuel beds (172 fires).

Fig. 5 .

 5 Fig. 5. Predicted ROS given by the Catchpole et al. model versus observed ROS for the sets of fires spreading in well-ordered and vertically-oriented fuel beds (172 fires).

Fig. 6 .

 6 Fig. 6. Predicted ROS given by the proposed model versus observed ROS for a set of experiments carried out by Wolff et al. with a 0.11 kg m -2 fuel load.

Fig. 7 .

 7 Fig. 7. Predicted ROS given by the proposed model versus observed ROS for a set of experiments carried out by Steward and Tennankore with a 4.88 kg m -2 fuel load.

Table 1

 1 Nomenclature of the model variables and fuel bed characteristics.

	Latin symbols
	a 0	Convection coefficient (m -1 )	=0.4
	B	Stefan-Boltzmann constant (W m -2 K -4 )	=5.6×10 -8
	c	Char rate	=0.15
	c w	Wolff et al. model parameter
	C p	Specific heat of fuel (J kg -1 K -1 )
	C pa	Specific heat of air (J kg -1 K -1 )	=1150
	d	Fuel particle diameter (m)
	D	Distance between two fuel particles (m)
	dx	Elementary fuel cell thickness (m)
	g	Gravity acceleration (m s -2 )	=9.81
	h	Fuel bed depth (m)
	H	Flame height (m)
	k	Catchpole et al. model parameter
	K	Law for drag forces
	K *	Intermediate drag coefficient
	K 1	Drag coefficient	=7
	L	flame depth (m)
	m	Fuel moisture content
	q	Ignition energy (kW m -1 )
	R	Rate of fire spread (m s -1 )
	R b	Contribution of radiation of burning fuel bed to the ROS (m s -1 )
	R c	Contribution of convection to the ROS (m s -1 )
	s	Surface area to volume ratio of fine fuel (m -1 )
	S	Leaf area by square meter (m 2 m -2 )
	s tb	Air-pyrolysis gases mass ratio in the flame base	=9
	s tc	Air-pyrolysis gases mass ratio in the flame body	=17
	T	Mean flame temperature (K)

Table 2

 2 Main wind and fuel bed characteristics of the different sources confronted to the proposed model. Table3is low. The value of Pearson's r close to 1 (r=0.94) also indicates a suitable correlation between predictions of the proposed model and experimental ROS. The experiments carried out by Steward and Tennankore[START_REF] Steward | Fire spread and individual particle burning rates for uniform fuel matrices[END_REF] are complex to simulate because the ROS trends are different from each another -fast increase, square root trend or very slow increase-. If the proposed model slightly overestimates the ROS (FB=0.11), the overall performances are satisfactory (NMSE=7.89%, r =0.96).

	Source	Fuel bed	Surface area to	FMC	Fuel load (kg m -2 ) Particle diameter	Fuel spacing	Wind (m s -1 ) Fuel density
		depth (cm)	volume ratio (m -1 )			or witdh (cm)	(cm)		(kg m -3 )
	Wolff et al. 1991 [5]	4.6	3120	0.08	0.11 to 0.88	0.13	1 to 2	0.4 to 4.6	721
	Steward and	6.7	329 to 1630	0.06	0.36 to 19	0.25 to 1.91	2.54	0.242 to 3.48	700
	Tennankore 1979	14					3.82
	[37]
	Catchpole et al. 1998	7.6	630	0.055 to	0.61.21	0.6	4.28	0.45 to 2.68	442
	[6]	15.2		0.089			4.38
	Fons 1946 [12]	14	618 to 1094	0.05 to	0.51 to 2	0.37 to 0.66	2.54 to 4.45	1.79 to 3.58	418
				0.13
	Finney et al. 2013 [23] 2.5 to 35.6	1590 to 3818	0.1	0.079 to 2.68	0.1 to 1.3	0.64 to 15.7	0.11 to 2.24

Table 3

 3 Comparison of normalized mean square error (NMSE), fractional bias (FB) and Pearson's correlation coefficient (r) obtained by three simplified ROS models when simulating the experiments carried out by various authors. Predicted ROS given by the Wolff et al. model versus observed ROS for the sets of fires spreading in well-ordered and vertically-oriented fuel beds (172 fires).

	Source	nb of	Main varying parameters	Proposed model		Wolff et al. model (model	Catchpole et al. model (model
		fires	(other than wind)				param. c w =0.07)		parameter k=11)
				NMSE (%) FB	r	NMSE (%)	FB	r	NMSE (%)	FB	r
	Wolff et al. 1991 [5]	29	Fuel load	4.15	-0.11 0.96 38.21	0.42	0.97	38.6	0.32	0.95
	Steward and	48	Fuel spacing Particle	7.89	0.11	0.96 18.19	0.26	0.91	15.47	0.35	0.94
	Tennankore 1979		diameter
	[37]
	Catchpole et al. 1998	20	Fuel depth	2.11	0.05	0.96 81.83	-0.76	0.88	33.46	-0.33	0.79
	[6]
	Fons 1946 [12]	49	Surface-to-volume ratio	1.95	0.05	0.91 4.53	0.09	0.82	19.22	0.13	0.59
	Finney et al. 2013 [23] 26	Surface-to-volume ratio	8.74	0.10	0.86 68.7	-0.36	0.78	107.8	-0.78	0.77
	Sticks experiments			6.75	0.05	0.94 41.57	0.03	0.58	46.69	0.03	0.56

Acknowledgments

The present work was supported in part by the French Ministry of Research, the Corsican Region and the National Center for Scientific Research, under Grant CPER 2007-2013. The authors also wish to thank Mr. Pierre Régis Gonsolin for his help in English reviewing.