Najoua Erroui

Guillaume Gateau

Nicolas Roux

Continuous-Caliber Semiconductor Components

Keywords: Semiconductor, switching losses, conduction losses, thermal resistance, power converter, efficiency I

The needs in terms of power electronics has evolved and for high power applications, the increase of efficiency mainly goes through the increase of the voltage of the system. However, this is not possible with a conventional two-level topology because the semiconductor ranges are limited to 6.5 kV which approximatively corresponds to a blocking voltage of 3.6 kV. To overcome this semi-conductor limitation, multilevel topologies can be used and allow to extend the range of the DC bus input voltage while using smaller and more efficient components. Yet, the sizing of an optimal power converter with high efficiency and power density is hard to realize due to the discretization of semiconductor components calibers. Moreover, if we want to find the optimal value of the DC bus voltage for a specific application using an optimization algorithm, the use of continuous caliber for components will give better results. This paper aims to explain the used method to overcome this discretization by creating virtual components with exact suitable caliber, using the database parameters available from manufacturers. To do so, the used power converter losses and thermal models are recalled in order to specify the needed parameters. These parameters will be generated as to create the needed virtual component.

INTRODUCTION

HASTECS (Hybrid Aircraft Academic reSearch on Thermal and Electrical Components) project studies the possibility of electric propulsion hybridization in an aircraft. Besides the global architecture represented in Fig. 1, the different work packages aim to increase the specific power of the different components of a high-power electric chain incorporating a high-voltage DC bus, its wiring and power electronics interfacing the bus and the electric motor in order to have a solution that can be fitted in an aircraft [START_REF]Laboratoire plasma et conversion d'énergie -UMR5213 -Projet HASTECS : vers l'avion plus électrique[END_REF]. This study will lead to an optimization of the complete chain that will propose the interesting structure (s) integrating all aeronautical constraints such as partial discharges for electrical equipment placed in non-pressurized zone or in case of high operating voltage even in pressurized zone. In this study, power electronics topologies, smart control and modulation strategies and also semiconductor technologies will be taken into account. HASTECS project has set itself the challenge of doubling the specific power of electric machines including their cooling, while the power electronics, with their cooling system, would evolve from 15 kW/kg for 2025 to 25 kW/kg in 2035. This would reduce the inverter and motor weights, resulting in a reduction in fuel consumption of about 3.5% on a regional flight.

One of the work package focus on the static converters and the DC bus. In order to guide the work of the various work packages of the project, a first step is to determine an optimal DC bus voltage range. This has a direct impact on the study of partial discharges but also in the design of motor windings. Then, it is necessary to compare the efficiency and power density of several multilevel topologies. Using the simulation tool and the available database detailed in the second and fourth parts of this paper, the efficiency is plotted for several studied inverter topologies (Fig. 2). The noticed jumps of efficiency are due to the change in the used component calibers. Therefore, it is made difficult to find the optimal DC bus voltage due to the discretization of available voltage ratings. To overcome this problem, we created a continuous components database that suits the needed voltage rating by fitting and extrapolating the existing database components. Of course, these components may never exist, but it allows to see the effect of the choice of the static conversion architecture on the efficiency over the voltage range considered, while removing a bias (discretization of ratings). Fig. 2: Efficiency for several topologies using available database components This solution will help us explore several possible solutions. Validation of the results will be done by simulation so as not to limit the number of proposed architectures. The experiment will be done in a second time, after this project, to verify the solutions that will come from the overall optimization.

II. SIMULATION TOOL

The research project [START_REF] Didelot | Hastecs : un projet pour développer la propulsion hybride des avions du futur[END_REF] aims to develop a simulation tool which allows to pre-size converters. It computes different results for different conversion architectures. Its entries are the design constraints (DC bus voltage, output power, …), the foreseen converter topologies, and the semiconductor family or manufacturer choice from the available components database. The results such as the efficiency, number of semiconductor devices (series, parallel, total), voltage rating selection, maximal junction temperature, losses (conduction and switching), voltage and current margins, switching and apparent frequencies, total losses, semiconductors, heat exchanger, DC bus capacitor and flying capacitor weights, and also the power density are shown as figures for different parameters of different architectures behaviors.

The tool makes it possible from a specification to size the converter according to the chosen topology. It also makes it possible to carry out parametric studies by varying the DC bus voltage, the requested power or the modulation index to determine the optimum operating point. It can also take into account a mission profile that allows checking the performances of the converter for a given mission [START_REF] Ibrahim | High Power Conversion System for Aircraft Hybrid Propulsion: Multilevel Converters Design Conception, Analysis Study, and Optimization[END_REF].

For each design point, it selects from the available database, the most appropriate semiconductor components that fit well the desired voltage and current. In order to do so, an adapted algorithm was developed (Fig. 3). The first step of this algorithm is selecting the manufacturer chosen by the user to extract the corresponding components from the database. It is then determined whether the requested voltage rating can be achieved by the available components. If none of the components is adapted and therefore the biggest component is too small, components are connected in series to obtain the desired size. Otherwise, if several component calibers are suitable, the smallest caliber is chosen among those which are suitable so to have the best performing component.

Then the tool computes the different results. It is realized in Matlab and is based on analytical calculations of the losses in the semiconductor components for the various integrated multilevel architectures considering mainly a sinusoidal PWM (Pulse Width Modulation) control. These losses result in heating up the semiconductors. The thermal rising in the semiconductor will affect its functionality and its lifetime. Moreover, the losses impact the size of the cooling system needed in the design.

III. POWER LOSSES AND THERMAL MODELING

A. Modeling principle

In order to model the losses in the semiconductor, we need to calculate the switching and conduction losses. Both losses depend on the circuit parameters and the device characteristics.

To calculate the conduction losses, the voltage between the collector and emitter of the transistor must be taken into account during its conduction phase. The transistor can be modeled with a voltage drop and an internal resistance connected in series [4] [5]. The voltage is obtained with the relation (1).

( 1 )

Then the conduction losses can be computed using ( 2):

(2)

Where and are respectively the average and RMS values of the current.

To calculate the switching losses, the curves of the energy losses versus the switched current given for a switched voltage, present in the datasheets, are used. They can be approximated by a second-order equation with three parameters Ax, Bx, and Cx with:

(3)

For a switching frequency and a modulation period , the switching losses are computed with: The previous equations for the conduction and switching losses are also valid for the diode, knowing that the diode has only recovery losses as switching losses.

The equations stated before are given for one switch (transistor + diode). For all switches in the converter topology, the total power losses will be factored then by the number of switches.

(

) 6
Because of the semiconductors losses, a thermal model is needed in order to determine and manage the chip temperature. As for the electrical circuit analysis, the thermal analysis integrates power sources which represent the losses, temperature at different nodes and thermal resistors depending on the thermal conductivity and chip sizes.

In our architecture, a single power switch contains a transistor with an antiparallel diode that ensures a path for the reverse current. So the thermal model circuit will include two thermal resistors as in Fig. 4. Each resistor has a heat energy flow caused by the losses of each component. The temperatures of the different points in the switch package can be determined using the following equations:

(7)

(8) (9)
Where is the junction to case thermal resistance, is the case to heatsink thermal resistance and is the heatsink to ambient thermal resistance in . Here, the case of the chips package is connected directly to the ambient (80°C or 90°C), so the heatsink to ambient thermal resistance (

) is forced to zero. Then the assembly will be cooled down by a heat exchanger whose specification is written in order to fix the case temperature to the ambient (80°C or 90°C in our example). These parameters are extracted from semiconductor manufacturer datasheets and used to create a database.

IV. CONTINUOUS-CALIBER COMPONENT

The simulation tool uses a created database that sums up the semiconductors technological parameters obtained from the manufacturer datasheets. The components are classified as families according to their manufacturer. In this case, only Silicon IGBT components are studied.

A. Components' families

Semiconductor manufacturers can offer relatively low IGBT devices rates but can reach as high as 6.5 kV/0.75 kA. As shown in Fig. 5, the used database has two families of silicon components. The first consists of ABB components with four available voltage ratings (1700 V, 3300 V, 4500 V and 6500 V). Current ratings vary from 250 to 1600 A.

The second family is made up of Infineon components whose voltage ratings are: 600 V, 1200 V, 1700 V, 3300 V and 6500 V. For current calibers, they are smaller than those of the previous family and range from 200 to 800 A.

B. Generated components

There are discontinuities in the resulted curves due to the discretization of the components. These jumps are due to the change in component size used since the selected family (ABB components for example, has only four calibers (1700 V, 3300 V, 4000 V and 6500 V)).

The manufacturer does not make a customized component. However, using discrete components in our simulation brings up jumps in the resulted performance curves. Since it is an exploratory study, it is interesting to have components for all voltage and current ratings which will involve determining component parameters for sizes that vary continuously. In order to have a continuous database, we have to create components for the different desired ranges. The user will still have the choice of using real components instead of the created ones.

To create a component that does not exist in our database, we first select the family or manufacturer to which the created component will belong. For this purpose, all the parameters of the components belonging to this family are extracted from the Excel file. The desired current and voltage ratings are then defined to identify the variation law of this family components' parameters by selecting the closest existing current caliber components.

Once the various components have been selected, they will be used to generate the parameters of the desired component. The laws of variation of all the parameters are then identified as a function of the voltage caliber or current-voltage calibers product and then applied to the desired one.

For the switching energies Eon, Eoff and Erec, we can model the variation of their parameters (Ax, Bx, and Cx seen in equation ( 3)) using a polynomial function which is the function that fits the best our data. We chose second-degree polynomial functions because we do not have enough points in order to pick higher degree polynomial functions.

For the conduction losses, for both the IGBT and the diode, we need the on-state voltage drop at zero current condition and resistive elements which variations are also approximated by first-degree polynomial functions. We search a function that fits the variation of these parameters depending on the voltage and current calibers. The same method is used to generate the thermal resistors (heatsink-case, junction-case for both transistor and diode) as a function of the voltage and current calibers.

In order to generate the surface of the component, we search a function that fits the variation of this parameter depending on the current-voltage calibers. We can approximate this parameter also with a polynomial function. The value of the created component surface will be very helpful to design the cooling system allowing the computation of the thermal density.

Table I summarizes the order of the polynomial functions for the different parameters modeled. The generated semiconductor components will make it possible to compare the different parameters of the studied topologies in order to choose the optimum voltage range which will minimize the losses and increase the power density.

C. Comparison of real and generated components

The first studies will verify that the results with the generated components are coherent with the ones obtained with the real components. A DC voltage sweep is carried out using Infineon components and the component choice is fixed manually in the simulation tool. Due to the limitation of the existing voltage calibers, the sweep range is limited to 4000 V due to the limitation of the existing components' voltage calibers which corresponds to 6500 V so a definition voltage of 3600 V. The power is fixed to 1 per unit.

1) Fixed component choice

For this study case, the voltage and current calibers are fixed for both real and generated components in order to check the validity of the generated components and also compare the impact of component calibers on the inverter performances. Fig. 7 represents the efficiency for fixed components allowing direct series association.

In order to have the adapted component, series and parallel connection are made if the needed component ratings are beyond the available current and voltage rating. The series connection is used to adapt the input voltage while the parallel connection is needed to adapt the current of the transferred power. For the tested components, the generated components have the same properties as the real ones. The difference between components conduction losses remains low compared to switching losses which is higher for larger voltage caliber components.

When comparing the properties of the resulted design based on the voltage calibers of used components, we notice that using small component is better than higher voltage ones which represent more switching losses even if we use more components in series in order to withstand the required voltage. The direct series association is hard to realize but seems to be the best option, efficiency-wise.

2) Fixed choice of the components' voltage caliber and adapted current

For the next simulation, the voltage calibers are fixed however the current ones are chosen in order to respond to the needed current with both real and generated components, so the last ones will always have a good use of the Silicon surface. In this case, the real components have smaller thermal density due to the surface of the component which is bigger than the generated components ones. The bigger surface is, the lowest the current density is so the component junction temperature will be lower.

The jumps noticed for the temperature of the generated components are mainly due to the thermal resistor which does not correspond to the optimal value. The generated components have a higher thermal resistance as shown in Fig. 9. This is linked to the distribution of the thermal resistances that do not follow an obvious mathematical law. This data spread is related to the gap between the selected components which result in noise, sampling errors and nonlinearity of the distribution law.

3) Adapted voltage and current calibers

For the third case, the voltage and current calibers are chosen in order to respond to the needed voltage and current to both real and generated components. The very high switching losses and maximal junction temperature for the first points of the simulation are due to the very low voltage calibers (100 to 500 V) that are lower than the smallest available caliber which is 600 V.

The efficiency curves overlap for the points that match a used Silicon ratio of 50% for the real components. This ratio is computed as follows:

(10)

Where

is the DC bus voltage, I is the load current, is the number of the used switches depending on the studied topology.

The generated components are optimized in order to use 50% of the installed Si all the time. The same remarks about the junction temperature can be made again.

V. USE OF CONTINUOUS-CALIBER COMPONENTS

In this approach, evaluation of converters specifications is studied for different DC bus voltages. The converters under study are the 2-level topology, 3-level FC, 5-level FC [START_REF] Meynard | Multicell converters: derived topologies[END_REF] and 5level ANPC [START_REF] Kieferndorf | ANPC-5L technology applied to medium voltage variable speed drives applications[END_REF]. A parametric sweep of the DC bus voltage is implemented starting from 100 V to 4 kV for an imposed power of 1 per unit. The junction temperature is calculated using equations ( 7), ( 8) and ( 9) with a fixed ambient temperature of 90°C. The high voltage components that are used for the 2-level topology have more switching losses due to their properties and also to the high switching frequency which is lower for 3 and 5-level topologies.

The 2-level topology maximal temperature is higher than the fixed thermal limit of 150°C for almost all the design point so it should not be taken into consideration. If we want to use the 2-level topology for a DC bus voltage higher than 1 kV keeping the temperature lower than the fixed thermal limit, we should reduce the switching frequency from 4 kHz to 800 Hz.

In the other hand, High-level topologies have better efficiencies than 2-level one and still have an acceptable junction temperature.

VI. CONCLUSION

The first step of this project was to define the DC bus voltage. The discretization of voltage and current ratings has introduced a bias in the study. To avoid this phenomenon and to be able to design an optimal system, adapted components are generated and multilevel topologies are used. The generated components have the same properties as the real ones. For both generated and components, the optimal response is noticed for used Silicon voltage ratio of 50%.

The generation method has its limits and still to be developed in order to get a better fitting of the thermal resistors. This could be done by taking into account not only the voltage and current calibers but also the semiconductor module packaging that is different within a component's family.

Based on the voltage calibers, the small components are better than the high-voltage ones in order to reduce switching losses even if more components should be used in series in order to withstand the required voltage. The direct series association is hard to realize but seems to be the best option. That is where the multilevel architectures seem interesting.

Fig. 1 :

 1 Fig. 1: Global Electrical point of view for Hybrid Electric Aircraft

Fig. 3 :

 3 Fig. 3: Available component selection algorithm It uses Matlab script and object programming to model converters architectures. It allows a great flexibility and a large degree of freedom for different designs and their behaviors during studies. This adds many advantages regarding multiple methods implementations and desired parametric sweep modeling.

  average and RMS values of the switched load current, is the switched voltage of the semiconductor that depends on the topology, is determined from the datasheet ( ) and the switching time interval.

Fig. 4 :

 4 Fig. 4: Equivalent thermal model of an IGBT component with antiparallel diode [4]B. Needed parametersIn order to evaluate losses and components temperature, we need the following parameters:• Thermal resistance: , , • Conduction parameters for IGBT and diode: Vce0 Rdson, Vd, Rd. • Switching losses parameters (Eon, Eoff et Erec): Ax, Bx and Cx • Component surface: Surface ( )

Fig. 5 :

 5 Fig. 5: Voltage and current ratings of high-power semiconductor devices

Fig. 6 :

 6 Fig. 6: Variation of Aon, Bon et Con versus voltage rating and Eon versus the current

Fig. 7 :

 7 Fig. 7: Real versus generated components efficiencies and maximal junction temperatures for fixed calibers

Fig. 8 :

 8 Fig. 8: Real versus generated components efficiencies and maximal junction temperatures for fixed voltage calibers and adapted current calibers

Fig. 9 :

 9 Fig. 9: Thermal resistance for real and generated used components

Fig. 10 :

 10 Fig. 10: Real versus generated components efficiencies and used Silicon ratio for adapted voltage and current calibers 0 1000 2000 3000 4000 DC Bus (V)

Fig. 11 :

 11 Fig. 11: Efficiencies and maximal junction temperature of 2-level, 3L FC, 5L FC and 5L ANPC topologies for DC voltage parametric sweeps

TABLE I

 I

		.		SUMMARY OF GENERATED PARAMETERS
	Generated parameter	Depending parameter	Approximation function
	Switching	losses	Second-degree
	parameters			Voltage rating	polynomial
	Ax, Bx and Cx
	Conduction parameters Vce0 Rdson, Vd, Rd. for IGBT and diode	Current rating Voltage rating *	First-degree polynomial
	Thermal resistance ,	Current rating and Voltage rating *	Second-degree power
	Surface			Current rating Voltage rating *	First-degree polynomial

ACKNOWLEDGMENT This project has received funding from the [European Union's Horizon 2020 (cleansky 2 JTI) research and innovation program, 2014-2024] under grant agreement No 715483


L Brès

N Gherardi

N Naudé

B Rives

Remote atmospheric pressure plasma for improving acid-base surface properties of PEEK polymer: relevance to coating adhesion

Keywords: APPJ, polymer, acid-base interactions, wettability, adhesion improvement

Our study focuses on a remote atmospheric pressure plasma for the improvement of polymer surface reactivity. From the perspective of linking surface modifications to adhesives performances, wettability measurements are made using two models (Owens-Wendt-Rabel-Kaelble and Van Oss-Good-Chaudhury). This study is a contribution to correlate wettability with adhesion results obtained with three-point bending test. Acid-base properties of the treated polymeric surfaces are discussed according to the Lewis's theory.

Introduction

The increasingly widespread use of Carbon Fiber Reinforced Polymers (CFRP) in structural engineering could be explained by their low weight coupled with high mechanical properties [1]. Poly-EtherEtherKetone (PEEK) matrix composites are particularly appreciate in aeronautic applications because of their high thermal stability and chemical resistance associated to a good ability to withstand high mechanical loads [START_REF] Collyer | A Practical Guide to the Selection of High-Temperature Engineering Thermoplastics[END_REF]. However, literature shows several studies aiming to reduce surface energy to obtain high and durable adhesive properties [START_REF] Comyn | [END_REF][4][5]. In this context, knowledge about adhesion mechanisms is a fundamental point for the control of the interface and the future paint interphase which depends on the surface quality of materials and the bonding conditions.

In this study, industrial PEEK composites are characterized prior to and after activation by a remote atmospheric pressure cold plasma. Wettability modifications are studied using two different models as OWRK (Owens-Wendt-Rabel-Kaelble) and VOG (Van Oss-Good-Chaudhury). Physicochemical properties of plasma-treated substrates are discussed with respect to adhesion performances of a PEEK/coating interface. The aim is to characterize effect of process parameters on the acid-base properties and hence, to forecast the strength of the interface.

Experimental part

The substrate studied is a CFRP with a PEEK matrix. The composite shows a visually planar surface, slightly rough and is around 2 mm thick. The samples are cleaned three times by moistened wipes before any treatment or characterization. The coating used in this study is an industrial polyurethane paint employed in external aircrafts. It is a bicomponent product applied with traditional spraying techniques. Dry thickness measured after complete polymerization is around 40 ± 7 µm. A remote atmospheric pressure cold plasma (UL-SCAN) from AcXys Technologies is used to activate the CFRP substrates [START_REF]AcXys Technologies[END_REF]. The process is operated with either dry air or pure nitrogen (Alphagaz 1 from Air Liquide) at a flow rate ranging from 30 to 60 slm. The system is driven by an alternating current power supply with frequency ranging from 80 to 200 kHz. Created arc between the inner and the coaxial outer electrode is blowing out from the torch. An afterglow region over a few tens of millimetres, namely the post-discharge, is created and contains reactive and excited species (metastables, neutrals, and radicals). Displacement of the torch is ensured by a directional nozzle mounted on a 3axes robot that scans the surface to be treated line by line. Activation performances can be promoted by tuning different parameters listed in Fig. 1. They allow operator maximizing the effect of the treatment while preventing some degradations on the sample surface.

Acid-base surface properties are explored by contact angle measurements using static sessile drop. In the OWRK theory [START_REF] Owens | [END_REF], surface free energy γS is expressed as follow:

𝛾 𝐿 (1 + cos 𝜃 𝑎 ) = 2√𝛾 𝑆 𝑑 𝛾 𝐿 𝑑 + 2√𝛾 𝑆 𝑝 𝛾 𝐿 𝑝 (1)
with γS D and γS P respectively the dispersive and the polar component.

In the VOGC theory [8], the surface free energy is decomposed as follow:

𝛾 𝐿 (1 + cos 𝜃 𝑎 ) = 2√𝛾 𝐿 𝐿𝑊 𝛾 𝑆 𝐿𝑊 + 2√𝛾 𝐿 + 𝛾 𝑆 -+ 2√𝛾 𝐿 -𝛾 𝑆 + (2)
with γS LW , γS AB , γS + and γS -respectively the Lifshitz-Van Der Waals (LW), Acid-Base (AB), acidic and basic components.

The surface free energy components in both theories are determined using several reference liquids of known γL D , γL LW Adhesion performances of the coating/substrate interface prior to and after plasma activation are evaluated by three-point bending test. According to the standard NFT 30-010 [START_REF][END_REF], test specimens are made up of PEEK composite coated with an industrial polyurethane external paint. An epoxy stiffener is formed on the latter in order to create a discontinuity of the strains and focus the failure initiation along the stiffener (see fig 2). Bold black arrow represents the bending stress of the specimens between supports (in blue). The strength of the tested system is achieved under the conditions of 0.5 mm/sec loading rate, 35mm distance between supports, and with a 5 kN load sensor. The maximum tensile strength value of each interface is calculated from the average of 8 specimens.

Results and discussions

At initial state, PEEK composites used in this study have a poor total surface free energy (53mN/m±) associated to a rather high water contact angle (68°±). This is in accordance with other authors who consider that PEEK matrix is quite inert when applying a coating without activation [11][12]. Surface free energy γS of PEEK substrates activated by air plasma according to the device power are shown in Fig. 3. It is calculated from the water and diiodomethane contact angle using the OWRK theory. This approach reflects both dispersive interactions as the Van Der Waals ones through γS D and polar interactions through γS P . One can see a significant increase of the polar component γS P after air plasma while no changes in the dispersive component γS D . This give rise to the total surface free energy γS which indicate a more wetting property of the PEEK surface. Previous chemical analyses made on the top layer of the PEEK matrix (not shown here) are in accordance with this results since they notice a greater oxygen grafting ratio. This tendency is also observed in the literature [12][13]. Despite the increase of the surface polarity after air plasma, Fig. 3 illustrates that it is impossible to provide a significant trend about the influence of the device power. Fig. 3. Surface free energy discriminates in polar and dispersive components according to OWRK theory.

Although the OWRK is a widespread method to calculate the surface free energy, it seems to present some shortcomings for this materials. Indeed, Lavielle also noticed that this method is not sufficient to explain the whole surface interactions when a droplet is deposited on a substrate [14]. She attributed these observations to the polar initial character of her substrate before treatment. In order to overcome this problem, she used the VOGC method that consider interactions as electron exchanges between an acidic substrate and a basic liquid or viceversa. The two components γS + and γS -in (2) express this acid-base potential.

Values of surface free energy included AB and LW components are summarized in Fig. 4(a) whereas basic γS - and acidic γS + components are illustrated in Fig. 4(b). As can be seen, untreated substrate present a basic monopolar character whereas plasma activation makes it an amphoteric character, i.e. both acidic and basic properties. At initial state, the basic monopolar character is due to the presence of oxygen atoms in the PEEK monomer which are Lewis-base [15]. The constant increase of γS -with the power arises from the constant oxygen incorporation onto the surface, not fully understood with the OWRK theory. These data predict a strong interaction with a Lewis-acid, as OH -molecules, NH3 or polyurethane product [START_REF] Joud | Physicochimie des surfaces et acido-basicité[END_REF]. Improvement of the surface reactivity through the oxidation previously highlighted results in a better strength of the interface between the polyurethane paint used in this study and PEEK substrates. Indeed, Fig. 5 show the fracture face of an untreated (a) and an air plasma-treated (b) specimen obtained after a three-points bending test. In the first case, failure is clearly adhesive which means that there is no adhesion between the untreated PEEK and the polyurethane coating. In contrast, after air plasma and whatever the conditions (all the failure faces not shown here), the breaking is of cohesive type in parallel of the improvement of the strength's interface.

The decomposition of the AB component indicates an

Conclusions

The study focuses on the influence of atmospheric pressure plasma process parameters on the wettability of a PEEK based CFRP. This paper is a contribution to the correlation of the surface reactivity according to the Lewis theory with mechanical behaviour of the new interphase, characterized by bending three points.

To do that, the VOGC coupled with the acid-base approach are chosen because they seem to give valuable chemical information about the surface interactions, attractive as repulsive ones, through both components γS - and γS + . Bending three-point test is also a meaningful way to get both qualitative and quantitative characterization of the interface through the maximal tensile strength and optical analysis of the failure face.

This approach seems to be suitable for better understanding of adhesion phenomena and interface ageing mechanisms.

Acknowledgments

This research is supported by Airbus Helicopters Company within the SURFINNOV IRT Saint-Exupéry contract project.

Fig. 1 .

 1 Fig. 1. Schematic of the experimental device UL-SCAN and process parameters.

Fig. 2 .

 2 Fig. 2. Diagram of the specimen's position during threepoint bending test.

  properties of treated surface through γS -.

Fig. 4 .

 4 Fig. 4. Surface free energy according to VOGC theory and decomposition of acid-base component into acidic and basic parts.

Fig. 5 .

 5 Fig. 5. Fracture faces of an untreated (a) and an air plasma-treated (b) specimen associated to their maximal tensile strength (N).

Table 1

 1 , γS + and γS -(see table1). For each individual sample, repeatability is checked by analysing 3 drops with each liquid at room temperature, 5 seconds after deposition of the drop.

		γL	γL LW	γ AB	γ +	γ -
	Water	72 8	21 8	51	25 5	25 5
	Diiodomethane	50.8	50.8	0	0	0
	Formamide	58	39	19	2.28	39.6

. Wettability characteristics of liquids used in the study in (mN/m)

[START_REF] Van Oss | Interfacial Forces in Aqueous Media[END_REF]

.


Ghita Bencheikh
email: ghita.bencheikh@enit.fr

Ayoub Tahiri
email: a.tahiri@cacg.fr

Pascale Chiron
email: pascale.chiron@enit.fr

Bernard Archimede
email: bernard.archimede@enit.fr

Francis Martignac
email: f.martignac@cacg.fr

Étude de l'apport de bassins contrôlés pour la diminution des inondations suite aux crues *

Keywords: Réseaux de transport, optimisation

Introduction

Ce travail concerne la minimisation des inondations par l'utilisation de réservoirs d'écrêtement de crues disposés le long d'une rivière. Au cours de l'évolution de la crue, il est nécessaire de trouver à quel instant quelle quantité stocker ou restituer de chaque réservoir afin de maintenir le débit de la rivière en dessous d'un seuil prédéfini. La théorie des graphes a été utilisée comme outil de modélisation afin de traiter ce problème comme un problème de flot.

Gestion des réservoirs

Le système rivière-réservoirs est constitué d'un ensemble de n réservoirs de capacités fixes U i répartis géométriquement sur un tronçon de rivière. La résolution du problème est réalisée sur un horizon de temps prédéfini 1, ..., H. A chaque itération, t, il s'agit de déterminer la quantité d'eau à stocker/restituer de chaque réservoir, afin d'éviter, lorsque le débit d'entrée est supérieur à un seuil Q lam , le débordement au niveau de la zone à risques 1 . Ce problème est non déterministe, c'est pourquoi, sans tenir compte de l'évolution future, seul le débit arrivant à t est considéré [Nouasse, 2015]. Or, les réservoirs sont de capacités finies et ne peuvent couvrir n'importe quelle crue, il se peut qu'ils se saturent avant que la crue arrive à son pic. Pour remédier à ce problème non déterministe, et afin que les décisions prises à chaque itération soient pertinentes, nous supposons connaître à t, l'estimation des débits entrants durant tout l'intervalle I t = [t, t + k] (avec k <<< H). Dans ce cas, le sous-problème correspondant est déterministe, et peut ainsi être résolu par une méthode telle que celle que nous présentons dans la section 2.2.

Modélisation

Le système est modélisé à l'aide d'un graphe dynamique G = (V, A, τ , C, U ). Avec : V = W ∪ S ∪ P : l'ensemble des sommets du graphe, tels que S et P représentent l'amont et l'aval de la rivière et W l'ensemble des réservoirs. A : l'ensemble des arcs orientés représentant les débits. τ = τ ∪ ξ : l'ensemble des temps de transfert, avec τ le temps de transfert d'un réservoir au réservoir suivant et ξ le temps nécessaire pour que l'eau passe de la porte d'entrée à la porte de sortie d'un réservoir donné.

C = C x ∪ C s : l'ensemble des coûts de transport sur l'arc e ∈ A, avec C x : le coût de débordement d'une unité d'eau et C s le coût de stockage dans les réservoirs. U : l'ensemble des capacités. Dans ce graphe, le problème de minimisation des inondations se traduit par la recherche d'un flot maximal à coût minimum avec contraintes additionnelles. Nous proposons, après transformation du graphe dynamique en un graphe statique, [START_REF][END_REF], une formulation mathématique de ce problème sous forme d'un programme linéaire en nombres entiers mixtes ainsi qu'une méthode de résolution.

Méthode de recherche du seuil optimal

Lorsque les prévisions nous informent de l'arrivée d'une crue telle que les réservoirs ne peuvent y faire face, il n'est pas judicieux de commencer le stockage dès que le débit entrant dépasse le seuil Q lam . Ainsi, nous cherchons, à chaque itération, dans l'intervalle [Q lam ,Q max2 ], le seuil, Q lam , à partir duquel il faudrait commencer le stockage pour que les réservoirs puissent absorber le pic de crue. Afin d'utiliser au mieux les réservoirs, nous vérifions également si dans l'intervalle de temps nous pouvons effectuer la restitution d'une quantité d'eau rendant ainsi de l'espace disponible pour un futur stockage.

Emulation du comportement de la rivière

En réalité des phénomènes purement hydrauliques contribuent aux variations des débits dans les biefs3 et des temps de transfert. En l'absence de mesures, afin de disposer de valeurs réalistes, nous utilisons le logiciel hydraulique SIC [START_REF] Baume | [END_REF] pour émuler le système rivière-réservoirs. A chaque pas de temps, SIC évalue les valeurs des débits devant chaque porte et dans la rivière. Ces valeurs sont utilisées en entrée par nos algorithmes afin d'élaborer en sortie le niveau d'ouverture des portes que SIC appliquera à l'itération suivante. Plusieurs expériences basées sur la modélisation de la zone de Bastillac située dans la ville de Tarbes, pour laquelle un projet d'implantation de réservoir est à l'étude, ont été réalisées. Les instances générées correspondent aux données mesurées lors d'une précédente crue centennale. Afin de juger le comportement du modèle, d'autres instances générées aléatoirement ont été étudiées (pics de crue et capacité du réservoir multiples, etc.).

Conclusions et perspectivesLes résultats obtenus sur les instances générées ont mis en évidence l'intérêt et les performances de notre approche comparée à la modélisation proposée dans[Nouasse, 2015]. Des améliorations telles que l'étude de l'impact de la taille de la fenêtre temporelle de prédiction des débits, ou l'étude et la proposition de fonctions de coût de stockage dans le but de protéger les biens présents dans les réservoirs feront l'objet de futurs travaux.

le débit maximum sur la fenêtre temporelle It

portion de rivière