C Da

Silva Tousch

J Martin

G Marcos

T Czerwiec

Gérard Henrion

Evidence of in-depth incorporation of carbon nanotubes in alumina layers grown by plasma electrolytic oxidation

Keywords: Aluminum, Carbon nanotubes (CNTs), Plasma Electrolytic Oxidation (PEO), Micro-Arc Oxidation (MAO), Alumina coating, Incorporation mechanism

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

Introduction

Plasma electrolytic oxidation (PEO), sometimes referred to as micro-arc oxidation (MAO), is a plasma-assisted electrochemical surface treatment particularly suited to convert aluminum into a protective alumina layer. PEO differs significantly from acid anodizing processes since it is conducted under higher voltages/currents and oxidation results from the apparition of micro-discharges issued from the electric breakdown of the growing oxide layer. The resulting coating provides increased hardness, wear protection and corrosion resistance [START_REF] Yerokhin | Plasma electrolysis for surface engineering[END_REF]. Although PEO coatings are usually porous [START_REF] Curran | Porosity in plasma electrolytic oxide coatings[END_REF], this intrinsic feature can be seen as providing new opportunities for incorporating various kinds of elements or particles in the course of the process or post-process [START_REF] Fattah-Alhosseini | The effects of nano-and micro-particles on properties of plasma electrolytic oxidation (PEO) coatings applied on titanium substrates: A review[END_REF]. Indeed, it is now well established that dissolved species in the electrolyte such as Si, Na and K penetrate throughout the coating [START_REF] Martin | Formation of a metastable nanostructured mullite during Plasma Electrolytic Oxidation of aluminium in 'soft' regime condition[END_REF]. Several process parameters, especially the electrolyte chemistry and the applied electrical parameters [START_REF] Dehnavi | Effect of duty cycle and applied current frequency on plasma electrolytic oxidation (PEO) coating growth behavior[END_REF], can be tuned in order to promote the formation of porous PEO coatings with high specific surface area, thus amplifying the adsorption mechanism of these dissolved species.

More interestingly, the porous nature of the formed oxide layer stemmed an interest in incorporating solid particles to enhance its protective properties or even introduce new emerging properties. To achieve this goal, micro-or nano-particles are dispersed in the electrolyte before carrying out the PEO process with the intent of continually incorporating them during the growth of the coating. It results in the formation of an oxide-based composite coating whose microstructure is directly influenced by the composition, the size and the concentration of the dispersed particles. For example, several studies have investigated PEO treatments in electrolytes containing metallic or metal-oxide particles [START_REF] Lu | Plasma electrolytic oxidation coatings with particle additions -A review[END_REF][START_REF] Fattah-Alhosseini | Effect of particles addition to solution of plasma electrolytic oxidation (PEO) on the properties of PEO coatings formed on magnesium and its alloys: A review[END_REF]. From those studies, three main conclusions can be drawn.

(i) The incorporation of solid particles can be distinguished as either reactive or inert. In the case of reactive incorporation, the size, the shape, the chemical composition and the microstructure of the incorporated particles are severely altered. This is more pronounced for nano-particles [START_REF] Lu | Plasma electrolytic oxidation coatings on Mg alloy with addition of SiO2 particles[END_REF][START_REF] Lu | Insights into plasma electrolytic oxidation treatment with particle addition[END_REF] and for particles having a low melting temperature [START_REF] Lee | Evaluation of plasma temperature during plasma oxidation processing of AZ91 Mg alloy through analysis of the melting behavior of incorporated particles[END_REF][START_REF] Blawert | Role of sintering and clay particle additions on coating formation during PEO processing of AM50 magnesium alloy[END_REF][START_REF] Rapheal | Effect of current density on the microstructure and corrosion properties of plasma electrolytic oxidation (PEO) coatings on AM50 Mg alloy produced in an electrolyte containing clay additives[END_REF]. The generally accepted assumption is that the solid particles melt when locally exposed to the high temperature induced by the micro-discharges, and react chemically with the elements from the electrolyte and the substrate. Conversely, in the case of inert incorporation, the particles found in the pores of the coating remain majorly unaltered. This is mainly observed for particles exhibiting high melting temperatures [START_REF] Liang | Preparation and characterization of oxide films containing crystalline TiO2 on magnesium alloy by plasma electrolytic oxidation[END_REF][START_REF] Lee | Electrochemical response of ZrO2-incorporated oxide layer on AZ91 Mg alloy processed by plasma electrolytic oxidation[END_REF][START_REF] Daroonparvar | Preparation and corrosion resistance of a nanocomposite plasma electrolytic oxidation coating on Mg-1%Ca alloy formed in aluminate electrolyte containing titania nano-additives[END_REF] and high mass [START_REF] Lu | Plasma electrolytic oxidation coatings with particle additions -A review[END_REF].

(ii) The mechanisms of incorporation generally agreed upon are electrophoresis and mechanical insertion. Electromigration particularly concerns the colloidal suspensions, inside which particles are characterized by their zeta potential. When subjected to an electric field, particles of the colloidal suspension migrate towards either electrode depending on the sign of their zeta potential [START_REF] Spragg | Electrophoresis[END_REF]. In the context of the PEO process, particles exhibiting a negative zeta potential will be preferentially drawn towards the anode where the oxidation reaction occurs, and therefore promote their incorporation into the growing PEO oxide coating. On the other hand, the specific mechanism of the mechanical insertion results from the dynamics of apparition and extinction of micro-discharges on the surface of the substrate that induces repetitive expansion and cavitation of the plasma bubbles. This is accompanied with variations of the fluid pressure that are suspected to drag and entrap particles into the pores of the coating [START_REF] O'hara | The incorporation of particles suspended in the electrolyte into plasma electrolytic oxidation coatings on Ti and Al substrates[END_REF]. It implies that the efficiency of this mechanical insertion is directly dependent on the size of the dispersed particles; their dimensions must thus be smaller than that of the pores. Given that pores are typically of the micrometer scale, particles in the micron size range cannot effectively penetrate into pores and have a tendency to deposit on the extreme surface only, obstructing the outer pores [START_REF] Fattah-Alhosseini | The effects of nano-and micro-particles on properties of plasma electrolytic oxidation (PEO) coatings applied on titanium substrates: A review[END_REF][START_REF] Lu | Insights into plasma electrolytic oxidation treatment with particle addition[END_REF].

(iii) The presence of dispersed particles into the electrolyte as well as those already incorporated into the oxide layer alters the electrical response of the PEO process. Indeed, depending on their nature, it has been shown that particles modify the dielectric breakdown conditions of the growing coating, the characteristics of the micro-discharges and finally the morphology of the produced oxide coating [START_REF] Fattah-Alhosseini | Effect of particles addition to solution of plasma electrolytic oxidation (PEO) on the properties of PEO coatings formed on magnesium and its alloys: A review[END_REF]. Nevertheless, no general conclusions can be drawn due to the wide range of different PEO processing conditions reported in the literature. On the other hand, the applied electrical conditions themselves have an impact on the particles. PEO treatment conducted using high power promotes the reactive incorporation of nano-particles with low melting temperature since more energy is provided to heat up and melt them. Inversely, when diminishing the applied power to the PEO electrolyser system, the inert incorporation is favored given that not enough energy is available to melt the particles [START_REF] Chen | Formation of self-lubricating PEO coating via in-situ incorporation of PTFE particles[END_REF].

The present study focuses on carbon-based nano-particles and more specifically carbon nanotubes (CNTs). Due to their exceptional mechanical and electrical properties [START_REF] Popov | Carbon nanotubes: Properties and application[END_REF], CNTs have numerous applications as structural reinforcements in various composite structures [START_REF] Costa | Fabrication of all-solid-state textile supercapacitors based on industrial-grade multi-walled carbon nanotubes for enhanced energy storage[END_REF][START_REF] Metz | Carbon nanotube-epoxy composites: The role of acid treatment in thermal and electrical conductivity[END_REF]. Most notably, polymer-based composites reinforced with CNTs have had a noteworthy development resulting in materials with tunable properties [START_REF] Metz | Carbon nanotube-epoxy composites: The role of acid treatment in thermal and electrical conductivity[END_REF]. Metallic matrix composites as well as reinforced ceramics have also been reported, although to a lesser extent [START_REF] Nieto | Carbon Nanotubes[END_REF][START_REF] Flahaut | Carbon nanotubesceramic composites[END_REF].

Although the tribological properties and corrosion behavior of the formed oxide were evaluated, little evidence was provided to finely characterize the presence of CNTs throughout the oxide layer [START_REF] Isaza | Mechanical and Corrosion Behavior of Plasma Electrolytic Oxidation Coatings on AZ31B Mg Alloy Reinforced with Multiwalled Carbon Nanotubes[END_REF][START_REF] Hwang | Effects of a carbon nanotube additive on the corrosionresistance and heat-dissipation properties of plasma electrolytic oxidation on AZ31 magnesium alloy[END_REF][START_REF] Kara | Tribological and Electrochemical Corrosion Properties of CNT-Incorporated Plasma Electrolytic Oxidation (PEO) Coatings on AZ80 Magnesium Alloy[END_REF][START_REF] Aydin | Role of graphene additive on wear and electrochemical corrosion behaviour of plasma electrolytic oxidation (PEO) coatings on Mg-MWCNT nanocomposite[END_REF][START_REF] Guo | Effect of carbon nanotubes additive on tribocorrosion performance of micro-arc oxidized coatings on Ti6Al4V alloy[END_REF][START_REF] Daavari | Biotribology and biocorrosion of MWCNTs-reinforced PEO coating on AZ31B Mg alloy[END_REF]. As a matter of fact, few studies involving the PEO treatment of aluminum in an electrolyte containing CNTs were published. Yürektürk et al. reported a decrease in coating thickness, a denser surface and a decrease in surface pores leading to higher hardness and better wear resistance for coatings processed with CNTs [START_REF] Yürektürk | Characterization of micro arc oxidized 6082 aluminum alloy in an electrolyte containing carbon nanotubes[END_REF]. A lower porosity and a denser layer allowing better corrosion resistance were reported by Lee et al. [START_REF] Lee | Incorporation of multi-walled carbon nanotubes into the oxide layer on a 7075 Al alloy coated by plasma electrolytic oxidation: Coating structure and corrosion properties[END_REF]. On the other hand, Sabouri et al. observed a detrimental effect of MWCNT incorporation on the compactness of the coating, leading to a decrease in the protective properties of the coating [START_REF] Sabouri | Plasma electrolytic oxidation in the presence of multiwall carbon nanotubes on aluminum substrate: Morphological and corrosion studies[END_REF]. Lee et al. also observed that an increase in current density led to an increase in incorporated MWCNT and a decrease in surface micro-pores [START_REF] Lee | Incorporation of carbon nanotubes into microcoatings film formed on aluminum alloy via plasma electrolytic oxidation[END_REF]. In some cases, the surface morphology of the PEO coatings formed in the presence of MWCNTs did not display the typical pancake-like structures of PEO coatings [START_REF] Hyun | Incorporation of multi-walled carbon nanotubes into oxide layer formed on al alloy by plasma electrolytic oxidation[END_REF]. All these works also mention a decrease in breakdown voltage in the presence of CNTs attributed to the high electrochemical capacity of the CNTs. Finally, the main driving force for CNT incorporation suggested in all studies is electrophoresis, based on Zeta potential measurements of the CNTs dispersions indicating a negative surface charge, which would thus imply that the CNTs are drawn towards the anode during the PEO process. Also, those studies mainly reported the presence of CNTs from the top surface examination of the samples without much evidence of an in-depth incorporation of the CNTs. If up to now, no strong evidence of an in-depth incorporation was demonstrated, it might be explained by the fact that the porous nature of the PEO alumina, combined with limitations regarding reliably detecting carbon, pose major obstacles to finely characterize this incorporation. In the present work, complementary characterization techniques, including high resolution scanning electron microscopy (SEM), micro-Raman spectroscopy, glow discharge optical emission spectroscopy (GDOES) and transmission electron microscopy (TEM) equipped with an energy dispersive Xray (EDX) facility, were used to finely detect the presence of CNTs throughout the PEO oxide layer to get new insights into the mechanism of their incorporation.

Materials and methods

PEO was performed on pure aluminum alloy (series 1050) whose elemental composition is given in Table I. Prior to processing, samples were cut into a rectangular cuboid of dimensions 5 × 15 × 50 mm 3 corresponding to a processed surface of about 21.5 cm². Samples were then polished to 1200 grit and thoroughly rinsed with acetone then ethanol, and dried before any experiment.

A dry powder of multiwalled carbon nanotubes (MWCNTs) was provided by Arkema (Graphistrength C100). MWCNTs were produced using a catalytic chemical vapor deposition (CCVD) route. As described in Table II, their initial characteristics were estimated from SEM, thermogravimetric analysis (TGA) and TEM measurements. The powder consists of large spherical aggregates of about 200 to 1000 µm in diameter in which MWCNTs are intertwined.

The average external diameter of the MWCNTs is between 12 nm and 20 nm and their length extends from 0.1 to 10 µm. Their purity was assessed by TGA under air flow from 20 °C up to 1100 °C with a temperature ramp of 3 °C/min. From TGA measurements, the total carbon content was estimated at 93 %wt. The remaining weight only corresponds to iron catalysis particles that are required for the CCVD synthesis route.

Regarding the electrolyte, a standard base solution containing 1 g.L -1 KOH and 1.65 g.L -1 Na2SiO3 diluted in deionized water was prepared. Its electrical conductivity and pH were 8.5 mS.cm -1 and 13, respectively. From this stock solution, and as described in Table III, various solutions were then prepared by dispersing MWCNTs in concentrations of 0, 1, 2 and 3 g.L -1 .

In the following, samples processed within these solutions are referred to as S0, S1, S2 and S3, respectively. A special care was required to properly disperse and stabilize the suspension of MWCNTs in the electrolyte. This was achieved by submitting the suspension to probe sonication. The propagation of energetic acoustic shockwaves allows breaking aggregates into detangled CNTs. However, given their hydrophobic nature, non-functionalized carbon nanoparticles have a strong tendency to reaggregate when dispersed in water [START_REF] Geckeler | Carbon nanotubes: Are they dispersed or dissolved in liquids?[END_REF]. In order to maintain a stable suspension, an amphiphilic compound was added to the solutions. As noted in Table III, sodium dodecyl sulfate (SDS) was used and dissolved in various concentrations to maintain a constant ratio between the concentration of MWCNT and that of SDS. The suspension was evaluated by Zeta potential measurement. A mean value of -46.3mV with standard deviation or 1.9 mV was obtained. As the absolute value of Zeta potential is over 30 mV, the suspension is considered physically stable [START_REF] Mahbubul | Effect of Ultrasonication Duration on Colloidal Structure and Viscosity of Alumina-Water Nanofluid[END_REF]. Finally, to avoid sedimentation of the dispersed MWCNTs during the PEO process, PEO treatments were carried out in a 2 L beaker that was itself positioned in an ultrasonic tank.

PEO treatments were performed in a home-made experimental set-up. The power unit consists of a current generator that supplies the electrode with a bipolar current waveform. In the present study, a symmetrical current waveform was applied to the electrodes resulting in an anodic to cathodic charge quantity ratio equal to 1 [START_REF] Jaspard-Mécuson | Tailored aluminium oxide layers by bipolar current adjustment in the Plasma Electrolytic Oxidation (PEO) process[END_REF]. The anodic amplitude of the current was set at 10 A in order to provide an anodic current density of 50 A.dm -² on the treated sample.

The current pulse frequency was maintained at 100 Hz and the positive and negative pulses were identical quasi-square pulses. The detailed parameters of the applied electrical parameters are summarized in Fig. 1. The counter electrodes consisted of titanium plates with 100 × 50 mm 2 dimensions. The temperature of the electrolyte was maintained at 65 °C by a copper coil surrounding the processed sample in which water flowed at 17 °C. All samples were processed for 20 minutes. After PEO treatment, samples were immediately removed from the electrolyte, thoroughly rinsed with distilled water and finally dried under a warm air stream.

Morphology of the produced PEO coatings was observed from top-surface and cross-section views using a field emission gun scanning electron microscope (ZEISS GeminiSEM 500) working in the conventional secondary electron mode (SE) as well as in the InLens mode.

Contrary to the standard SE detector positioned at an angle relative to the electron source to obtain topographical information of the sample surface, the InLens detector is located in line with the field emission gun, perpendicular to the surface of the sample [START_REF] Griffin | A Comparison of Conventional Everhart-Thornley Style and In-Lens Secondary Electron Detectors-A Further Variable in Scanning Electron Microscopy[END_REF]. This allows high resolution observations at high magnification within the pores both in top-views and crosssections. Moreover, it was observed that with InLens imaging mode, the interaction between the electrons and the MWCNTs induces a very intense signal collected by the InLens detector thus making it easy to detect them even at low magnification. Prior to cross-section SEM observations, samples were cut and gradually polished to achieve a mirror-like surface aspect.

To avoid any measurement artifacts induced by a polymer mounting resin in the detection of the carbon element, cut samples were not mounted in resin. Unless mentioned otherwise, the average thickness of coatings was evaluated by considering ten measurements over the cross-section. Roughness measurements were performed at the center of each processed sample. The average mean roughness parameter of the coating (Ra) was assessed using a Bruker DektakXT stylus profilometer with lateral and depth resolutions of 0.2 µm and 1 nm, respectively.

Raman spectroscopy was performed on a confocal micro-Raman spectrometer (HR800 Horiba) using a 1800 grooves/mm grating and a laser wavelength of 532 nm. Within the considered wavenumber range from 1000 to 3500 cm -1 , this excitation wavelength induces a photoluminescent response from the PEO alumina that does not directly interfere with the Raman bands from the MWCNTs. MWCNTs being concentric cylindrical graphene sheets, they present: (i) a strong peak around 1580 cm -1 (G band) from the high frequency first-order mode (E2g), (ii) another strong peak around 1350 cm -1 (D band) resulting from a hybridized vibrational mode associated with edges and disorder in the structure, and (iii) a weaker band around 1620 cm -1 (D' band). The second order part of the Raman spectra of MWCNTs includes the scattering bands around 2450 cm -1 (D+D"), 2700 cm -1 (2D or G'), 2945 cm -1 (D+D') and 3175 cm -1 (2G) [START_REF] Antunes | Comparative study of first-and second-order Raman spectra of MWCNT at visible and infrared laser excitation[END_REF]. The intensity of the D band is considered as structural defect dependent.

According to the double resonance theory, the crystal defects and the crystal edges scatter the excited electrons, which explain that the more defects are present in the MWCNTs, the more intense the D band gets [START_REF] Li | Raman characterization of aligned carbon nanotubes produced by thermal decomposition of hydrocarbon vapor[END_REF]. The intensity ratio of the D band ~1350 cm -1 and the G band ~1590 cm -1 thus allows one to comparatively evaluate the structural integrity of the MWCNTs [START_REF] Ivanova | Pharmaceutical characterization of solid and dispersed carbon nanotubes as nanoexcipients[END_REF].

Two other bands on the typical spectrum of MWCNTs are the D" band (also known as D4 band) and the Gamorphous band (also known as D3 band), which are respectively located at ~1150 cm -1 and ~1500 cm -1 . The D" band is a shoulder to the D band that has been attributed to the presence of carbon soot and other amorphous carbon [START_REF] Merlen | A guide to and review of the use of multiwavelength Raman spectroscopy for characterizing defective aromatic carbon solids: From graphene to amorphous carbons[END_REF]. Regarding the Gamorphousband, it originates from the presence of disordered carbons [START_REF] Jawhari | Raman spectroscopic characterization of some commercially available carbon black materials[END_REF]. This peak is extrapolated from the mathematical spectral decomposition of the Raman spectrum and a direct impact of its intensity is illustrated on the spectrum in the region between the D and G peaks: the elevation compared to the baseline is partially influenced by the intensity of the Gamorphous peak. The main information regarding the band name, position and interpretation of MWCNT Raman peaks are summarized in Table IV.

Glow discharge optical emission spectroscopy (GDOES) was performed on a Horiba Profiler 1000RF. The characterized samples were placed in an enclosing bell in order to ensure stable analysis conditions. It is worth mentioning that the intrinsic roughness and porosity of the PEO oxide layers imply that the signal collected over time is not purely linked to a particular depth of erosion. For this reason, the GDOES element profiles are presented as a function of the sputtering time rather than the erosion depth. Thus, the elemental profile is a qualitative representation of the presence of the different elements through the coating depth.

Regarding TEM observations, structure of the elaborated oxide layers was investigated using a JEOL ARM 200Fcold FEG UHR-transmission electron microscope operating at 200 kV and equipped with a Cs corrector. In scanning mode (STEM), EDX analyses were obtained using a JEOL CENTURIO energy-dispersive X-ray spectrometer. EDX element distribution maps recorded over an area of 512 nm × 512 nm (one measurement point per nm), were produced by using an acquisition time of 1 ms per measurement point and by repeating 20 scans over the same area. The specimen for TEM was extracted from a MWCNTs-containing PEO alumina coating using the micro-cleaving procedure. According to this procedure, the surface of the PEO processed sample was scratched with a diamond knife in order to break off micrograins from the oxide coating that were then transferred onto a TEM grid for examination. TEM allowed observing the aggregation state and the dimensions of the MWCNTs. Also, HRTEM revealed the integrity of the carbon nanotube walls and extremities. Finally, it is worth noting that for comparison reasons, the as-received MWCNTs were also observed with TEM using a PHILIPS CM 200 FEG microscope. To achieve this, they were dispersed at low concentration in pure ethanol, a drop of which was deposited on a TEM grid and was left to dry until total evaporation of ethanol before observation.

Results and discussion

Degradation of the MWCNTs

The potential degradation of the MWCNTs dispersed in the electrolyte is a concern, especially when successive samples are PEO processed within the same electrolyte bath.

Indeed, several factors such as the PEO processing time and parameters may irreversibly affect the electrolyte and dispersed particles, thus making reproducibility and comparison between samples difficult.

Probe sonication has been shown to induce mechanical degradation of the particles [START_REF] Wang | Raman spectroscopy enabled investigation of carbon nanotubes quality upon dispersion in aqueous environments[END_REF]. The ultrasonic vibration generates cavitation of bubbles that rapidly expand and implode. It induces variations of the pressure stresses on the MWCNTs aggregates that allow dispersing the intertwined MWCNTs. However, the major drawback of the probe sonication technique is that the high energy transferred to the particles could also lead to their structural degradation by fracture and fragmentation.

TEM examination (Fig. 2) confirms the efficiency of the probe sonication technique regarding disaggregation. Indeed, the intertwined nanotube aggregates are effectively separated without any significant degradation regarding their length or diameter. Raman spectroscopy revealed no major alteration regarding the spectral response of the dispersed MWCNTs after ultrasonic dispersion. However, the PEO treatment may affect the MWCNTs. Indeed, Fig. 3 displays the Raman spectra of the as-received MWCNTs and the MWCNTs collected from the electrolyte after 20 min of PEO treatment and complete drying. The calculated average ID/IG intensity ratio over five measures in each case gives 1.19 for the as-received MWCNTs and 1.21 for the MWCNTs collected from the electrolyte after 20 min of PEO treatment. The minor variation can be attributed to statistical variance. However, it can be mentioned that the D' band that is a shoulder of the G band, is noticeably more detectable on the spectra collected after sonication dispersion and after PEO treatment. This observation indicates a slightly higher defect density within the particles after those steps [START_REF] Bai | Defect density in multiwalled carbon nanotubes influences ovalbumin adsorption and promotes macrophage activation and CD4+ T-cell proliferation[END_REF]. Additionally, after the PEO treatment, the D" and the Gamorphous bands increase in intensity indicating an increase in the proportion of soot and amorphous carbons. It suggests that a significant amount of the MWCNTs initially introduced in the electrolyte became amorphous during the PEO process [START_REF] Choi | Novel method of evaluating the purity of multiwall carbon nanotubes using raman spectroscopy[END_REF]. One explanation consists in considering the local temperature nearby micro-discharges that was estimated up to 3200 K, in the range of the sublimation temperature of MWCNTs. After extinction of the microdischarge, the melted MWCNTs particles are rapidly quenched by the cooled electrolyte and solidified into an amorphous structure [START_REF] Hussein | Coating growth behavior during the plasma electrolytic oxidation process[END_REF].

These observations indicate that the dispersed MWCNTs are affected by the different processing steps. It is thus questionable whether the electrolyte with dispersed MWCNTs could be re-used multiple times. Consequently, a new electrolyte was prepared from the stock solution for each experiment.

Top-surface characterization

Fig. 4 shows the visual aspect of the samples processed with various concentrations of MWCNTs dispersed in the electrolyte. Sample S0 treated without MWCNTs appears white.

The increase in MWCNTs concentration results in a significantly darker aspect suggesting a higher proportion of carbon on the processed surface. Sample S1 treated with 1 g.L -1 has a gray aspect with a few localized dark spots while samples S2 and S3 processed with higher concentrations of MWCNTs present increasingly larger dark regions randomly distributed over the entire surface. For S3, dark regions dominate throughout the surface.

In Fig. 5, top-surface SEM micrographs also evidence differences regarding the morphology of the processed surfaces. The surface of sample S0 treated without MWCNTs exhibits an arrangement of "pancake-like" structures uniformly distributed over the surface, as usually encountered for PEO alumina coatings elaborated in silicate-based electrolyte [START_REF] Sundararajan | Mechanisms underlying the formation of thick alumina coatings through the MAO coating technology[END_REF]. In contrast, the surface of samples treated with MWCNTs reveals an alternance of smoother and rougher regions next to each other. For these latter, numerous large and deep jagged structures are visible (evidenced in red in Fig. 5 for S2 sample). The presence of MWCNTs in the electrolyte locally induces the formation of a more rugged surface. These preliminary observations were confirmed by roughness measurements carried out on each sample. Average roughness value Ra for samples S0, S1, S2 and S3 is given in Table V. This roughness parameter increases by a factor of 3 between the reference sample S0 treated without MWCNTs and the sample S3 treated with a MWCNTs concentration of 3 g.L -1 . Fig. 6 shows SEM micrographs recorded using the InLens mode on the top-surface of the samples S1 and S3. Using the InLens mode, the presence of MWCNTs on the surface can more easily be observed and is indicated by red circles and arrows in Fig. 6. Despite the efforts made to efficiently disperse MWCNTs in the electrolyte, their spatial distribution over the surface of the PEO processed samples is not homogeneous. Indeed, some locations are fully lined with MWCNTs scaffolds while the surface coverage in other regions is more scarce. In that regard, Raman spectra recorded on the surface of the processed samples (Fig. 7) confirm that the nano-particles observed from top-surface of the samples S1, S2 and S3 definitively consists of MWCNTs since their distinctive Raman scattering bands are clearly identified. The two most distinguishable peaks D and G, as well as the 2D peak are detected, especially in the darker regions where particles are more concentrated. For the sample S0, treated without MWCNTs, the distinctive Raman bands of the MWCNTS are absent. In addition, when analyzing a MWCNTs scaffold on the surface of sample S1 (Fig. 7B), the average ID/IG ratio is about 1.53.

That is higher than the ratio calculated from MWCNTs collected in the electrolyte after 20 min PEO processing time (1.20). This means, to some extent, that the incorporation causes structural damages to the incorporated MWCNTs. Moreover, the relative intensity of the 2D (G') band is also diminished when compared to the as-received MWCNTs. Since that band results from the two-phonon second order scattering with creation of an inelastic phonon, it is indicative of longrange order [START_REF] Dileo | Purity assessment of multiwalled carbon nanotubes by Raman spectroscopy[END_REF]. Hence a lower intensity once again supports the hypothesis that disorder was introduced in the MWCNTs during their incorporation. The D" and G bands are also of great interest. Their respective intensities are comparable to those of the MWCNTs recovered from the electrolyte indicating that the formation of disorganized carbon structures is the same for incorporated particles and those still in the electrolyte.

In-depth characterization

TEM was used to detect and characterize the incorporated MWCNTs. As previously explained, TEM characterizations were carried out on alumina fragments collected after cleaving the surface of the processed samples. TEM observations were randomly carried out on several fragments in order to have a overview of the incorporation of MWCNTs throughout the overall coating thickness, from the interface aluminum/alumina to the top-surface of the alumina coating. Fig. 8 displays representative TEM pictures of the collected fragments as well as the associated elemental mapping for aluminum, oxygen and carbon elements. From elemental mapping, it appears that the carbon element (Fig. 8 B) covers homogeneously fragments enriched in aluminum and oxygen (Figs. 8C and8D). It suggests that MWCNTs scaffolds are embedded with or in grains of aluminum oxide. High resolution TEM pictures (Figs. 8E and8F) show isolated extremities of a few MWCNTs with their walls clearly visible.

The observed entangled MWCNTs appear to be highly curved with abnormally low curvature radii. It suggests a potential alteration of their structural integrity resulting from their incorporation. This observation is in good agreement with the previous micro-Raman analysis that evidence an increase in the ID/IG ratio for MWCNTs incorporated on the top-surface of the coating.

Even though the TEM observations confirm that MWCNTs are effectively incorporated through the structure of the PEO coating, no information regarding their specific location can be obtained. So, a close examination of the cross-section of the samples was performed in order to ascertain the penetration and repartition of MWCNTs inside the PEO oxide layer. Crosssection SEM micrographs of the processed samples are given in Fig. 9. The average thickness of the formed PEO coatings are tabulated in Table V. It appears that the overall thickness of the PEO oxide layer increases as the concentration of the dispersed MWCNTs increases. For sample S0, the average thickness approximates 35 µm while it is above 60 µm for samples S1, S2 and S3 processed in the presence of dispersed MWCNTs. However, although the introduction of MWCNTs into the electrolyte seems to enhance the growth kinetics of the PEO coating, their presence results in a more porous morphology. Considering sample S0, Fig. 9 shows that the produced oxide layer consists in a quite dense coating with fine pores at the topsurface. In contrast, the morphology of the PEO coatings formed in the presence of MWCNTs is clearly more porous and can be divided into two distinct sublayers: a porous outer one with large connecting pores and a dense inner sublayer located close to the substrate. In addition, the morphology of the S1, S2 and S3 coatings alternate between sections with very distinctive outer porous and inner dense sublayers and regions with only a porous layer composed of large and deep pores with a lack or loss of an inner dense sublayer. This juxtaposition can be seen in Fig. 9 for samples S2 and S3. Those regions depicting wide open pores throughout the whole thickness of the coating have been attributed to the more jagged regions observed during the top-surface examinations (Fig. 5 for S2, evidenced in red). Moreover, as the concentration of the dispersed MWCNTs increases, the thickness of the outer sublayer relative to the total thickness increases. As reported in Table V, for samples S1, S2 and S3, the part of the porous outer sublayer represents ~36 %, ~51 % and ~62 % of the total coating thickness, respectively. Moreover, the volume proportion of regions with and without inner dense sublayer was estimated by calculating their linear ratio over the observed cross-section of the samples (assuming that the ratio along a line is representative of the volume proportion). For samples S1, S2 and S3, the volume proportion of regions with inner dense sublayer are respectively 49 %, 33 % and 32 %. Fig. 10 shows cross-section SEM micrographs of sample S3 treated with 3 g.L -1 of dispersed MWCNTs. It evidences the presence of nanotubes deep within the porous outer sublayer. As for top-surface observations, the nanotubes appear in the form of scaffolds that line the internal walls of pores. It is worth noting however, that no MWCNTs was observed through the dense inner sublayer of the PEO coatings. Comparing S1, S2 and S3, as was done for the top-surface SEM characterization, the nanotubes are observed more uniformly as the concentration of MWCNTs increases. For sample S1, many pores do not contain any nanotubes while for the sample S3, nanotubes can readily be found on the walls of almost any pore.

Micro-Raman spectroscopy measurements were performed along the cross-section of the investigated samples at the same location previously observed by SEM (Fig. 10). Fig. 11 shows the Raman spectra recorded at different depths over the thickness of sample S3 from the topsurface of the PEO coating down to the aluminum substrate. It confirms the presence of MWCNTs in depth through the porous layer. Indeed, the D and G bands are clearly observed throughout the whole porous layer of the coating. In contrast, no scattering band is detected on the Raman spectrum recorded in the dense inner sublayer at the interface with the aluminum, suggesting the absence of any significant quantity of MWCNTs in this specific region of the coating. This is in good agreement with previous SEM observations. The distribution of carbon throughout the oxide layer was also evaluated using glow discharge emission spectroscopy (GDOES). As previously explained in the experimental section, and in order to take into account the heterogeneous morphology of the PEO coatings inducing variations in the erosion speed, the GDOES elements profiles in Fig. 12 are qualitatively displayed as a function of the sputtering time. Fig. 12A shows the elemental profiles in aluminum, oxygen and carbon through the oxide layer for sample S1. More specifically, the profiles in carbon for samples S0 and S1 are compared in Fig. 12B. It is also worth mentioning that for sample S0, the GDOES analysis was stopped after 900 s compared to 2000 s for sample S1. It is explained by considering the fact that the oxide layer for sample S0 is much thinner (~30 µm) than that for sample S1 (~80 µm), the aluminum substrate being therefore reached after a shorter erosion time. As displayed in Fig. 12A, the variations of the oxygen and aluminum element profile can be used to indicate the transition from the oxide layer to the substrate. Indeed, after about 1000 s erosion time, the signal of oxygen gradually decreases while the signal of aluminum inversely increases, indicating the transition from the oxide coating to the aluminum substrate. Regarding Fig. 12B, within the first few seconds, the raw signal for the carbon element starts high and sharply decreases for both the samples S0 and S1. One can legitimately attribute this behavior to the presence of organic contaminants on the extreme surface. Beyond the very first few seconds, the carbon profile significantly differs between the two samples. Sample S0 presents a profile in carbon that rapidly falls down below the detection limit (10 -3 V) in less than 200 s with no further variations. By contrast, for sample S1, carbon remains still detected until approximately 1500 s sputtering time, suggesting that carbon has been introduced deep within the coating.

Discussion about incorporation mechanisms

Regarding all the obtained results, a few propositions regarding the incorporation mechanisms can be suggested. Indeed, it is thought that the main driving force for particle accumulation within the coating is from particles being migrated into discharge pores left after the extinction of a micro discharge on the surface of the growing oxide. The collapse of the discharge channel leaves a local void that is filled with electrolyte and provided that the dispersed particles are smaller than the opening left after the micro discharge, they are dragged in with the electrolyte and deposited on the walls of the porosity network. This theory implies that: (i) bigger micro discharges generate larger channels, meaning that a greater volume of electrolyte may rush in and drag along a greater amount of particles, and (ii) the mechanism by which particles accumulate in the coating is rather violent and unequal throughout the process duration given that the repartition, lifetime and size of the surface micro discharges changes with time as evidenced in previous studies [START_REF] Martin | Delay in micro-discharges appearance during PEO of Al: Evidence of a mechanism of charge accumulation at the electrolyte/oxide interface[END_REF]. In this study, the opaqueness of the electrolyte with dispersed MWCNTs prevents the optical characterization from either fast imaging or optical emission spectroscopy. However, the observed morphology with wide open porosities throughout the whole coating thickness for samples processed with MWCNTs dispersed in the electrolyte are thought to result from high energy discharges traversing the whole oxide layer and melting-redepositing a substantial proportion of the growing coating resulting in a highly irregular network of large open porosities. Those regions were shown to preferentially accumulate MWCNTs. This observation is in accordance with the aforementioned mechanism such that the regions almost exclusively composed of an outer porous sublayer would result from the combination of high energy micro discharging events followed by infiltration of MWCNTs containing electrolyte. Also, the presence of particles in the electrolyte impacts the micro discharges. Indeed, the reference sample S0 did not display the same morphology as S1, S2 and S3 despite being processed in identical conditions if not for the absence of MWCNTs in the electrolyte. In addition, the observations regarding the degradation of the incorporated MWCNTs can be linked to this mechanism. Indeed, the incorporation mechanism requires the dispersed particles to be in close vicinity of a micro discharge and thus to be subjected to a succession of "severe" events (high local temperature, radical species, shockwaves from cavitation, mechanical compressive stress). This explains why the sharp increase in defects in the MWCNTs was observed for the incorporated particles and not those remaining in the electrolyte after the PEO treatment. Concomitantly, the aforementioned incorporation mechanism explains why the presence of MWCNTs was mostly observed in the outer porous layer that is more readily infiltrated by the electrolyte compared with the inner dense layer.

Although most of the MWCNTs found throughout the layer were present as scaffolds of different sizes, some individual MWCNTs are also scattered in the pores of the coating. Finally, some bridging MWCNTs were observed connecting across separate sections of the oxide as displayed in Fig. 13. It is however not readily possible to definitely confirm the nature of the bridging structures. Indeed, given that their dimensions are of the nanoscale in thickness, any available characterization technique would have either too low spatial resolution (Raman) or too low sampling size to get a chance to collect such a structure without destroying it (TEM).

However, given that the dimensions and shape match those of the MWCNTs, and that this observation was not made for samples treated without MWCNTs, it is still highly likely that those bridges are indeed MWCNTs properly embedded in the matrix joining two separate sections. The importance of this information resides in the fact that the bridging MWCNTs may very well indicate that some of the incorporated particles are not visible for they are embedded in the oxide. This probably results from particles being trapped inside molten alumina resulting from a discharge event. As the molten alumina is quenched after the extinction of the microdischarges, it can crack from thermomechanical stresses and reveal the entrapped MWCNTs thus appearing as bridging structures.

Conclusions

The present study detailed the incorporation of multi-walled carbon nanotubes (MWCNTs) into the plasma electrolytic oxidation alumina coating formed on an aluminum alloy when the process is carried out in a silicate-based electrolyte wherein various amounts of MWCNTs were dispersed. In particular, the main objective was to take advantage of a combination of characterization techniques to definitely assess the presence of MWCNTs both on the topsurface but also throughout the thickness of the PEO alumina coating. The following main conclusions can be drawn:

 Cross-checked results from various analysis ensure proper characterization of the indepth incorporation of MWCNTs. Most notably, it was evidenced that MWCNTs preferentially incorporate throughout the porous outer sublayer of the coating rather than in the denser inner sublayer.

 MWCNTs were also shown to have a significant impact on the final morphology of the PEO coating. Indeed, higher amounts of MWCNTs dispersed in the electrolyte results in a thicker yet more porous alumina layer. To add to that, the coating is composed of regions of high irregularity and porosity throughout the whole thickness of the coating and regions with an outer porous layer and an inner denser layer.

 For the first time, it was also established that the MWCNTs experienced some degree of degradation during their incorporation into PEO alumina coating. Indeed, both STEM/EDX imaging and micro-Raman spectroscopy measurements revealed a decrease in the structural integrity of the MWCNTs that could reasonably be assigned to the severe mechanical and thermodynamic conditions imposed by the PEO discharge regime.

Finally, based on the collected results, some precisions regarding the incorporation mechanisms were proposed. The heterogeneous distribution of MWCNTs throughout the PEO coating along with their presence were exclusively detected in the pores of the coating. This suggests that the main driving force for MWCNT incorporation during PEO treatments is electrolyte infiltration/deposition.

• This work was supported by the French Government through the programme "Investissements d'avenir" operated by the French National Research Agency (ANR) and referenced to as ANR-11-LABX-0008-01 ('LabEx DAMAS').

• This work was also partly supported by the Conseil Régional de la Région Grand Est for granting C. Da Silva Tousch PhD work under decision 19_GE8_049.

• The authors would like to acknowledge contributions of the following:

-The competence cluster on electron microscopy (CC 3M) at Institut Jean Lamour for providing advice in SEM observations as well as performing all TEM and EDX analyses.

-Dr. C. Herold, Dr. S. Cahen and Dr. S. Fontana for providing valuable advice in CNTs analyses.

-The competence cluster on optics and lasers (CC COL) at Institut Jean Lamour for providing advice in micro-Raman spectroscopy.

Fig. 6 A

 6 Fig.6A, B illustrate differences in the MWCNTs distribution between the samples S1 and S3.

Figure 1 :

 1 Figure 1: Schematic representation of the applied current waveform.

Figure 2 :

 2 Figure 2: Bright-field TEM micrographs recorded (A) on the as-received MWCNTs powders

Figure 3 :

 3 Figure 3: Raman spectra (and the fitted Raman scattering bands of MWCNTs) recorded (A)

Figure 4 :

 4 Figure 4: Top-surface photographs of the samples S0, S1, S2 and S3.

Figure 5 :

 5 Figure 5 : Top-surface (SE mode) SEM micrographs recorded on samples S0 and S2. Sample

Figure 6 :

 6 Figure 6: Top-surface (InLens mode) SEM micrographs recorded at different magnifications

Figure 7 :

 7 Figure 7: (A) Raman spectra recorded on the top-surface of samples S0, S1, S2 and S3. (B)

Figure 8 :

 8 Figure 8: High-resolution TEM micrographs showing entangled MWCNTs interlocked in

Figure 9 :

 9 Figure 9: Cross-section (SE mode) SEM micrographs recorded at different magnifications on

Figure 10 :

 10 Figure 10: Cross-section SEM micrographs of sample S3 recorded particularly in a very porous

Figure 11 :

 11 Figure 11: Raman spectra recorded at different depths along a cross-section of sample S3 at

Figure 12 :

 12 Figure 12: (A) GDOES chemical profiles in elements C, Al and O recorded on sample S1 and

Figure 13 :

 13 Figure 13: SEM micrographs (InLens mode) of polished cross-section recorded on sample S3

Figure 14 :

 14 Figure 14: Schematic of the proposed incorporation mechanism. (A) Coating being processed

Figure 14

 14 Figure 14

Table III :

 III MWCNTs and sodium dodecyl sulfate (SDS) surfactant content in electrolyte for the different samples labeled S0, S1, S2 and S3.

	Sample	S0	S1	S2	S3
	[MWCNTs] (g.L -1 )	0	1	2	3
	[SDS] (g.L -1 )	0	1	2	3

Table IV :

 IV Band position, name and interpretation of the Raman spectra of MWCNTs:

	Band	Band name Interpretation	Ref
	position
	1150 cm-1	D" or D3	Presence of soot and amorphous carbon	43
	1350 cm -1	D	Defect dependent, associated with edges and	40-42
			disorder in the structure
	1500 cm -1	Gamorphous	Presence of disordered and amorphous carbons	44
		or D4
	1580 cm -1	G	High frequency first order mode (E2g) from	43
			graphene-like structures
	1620 cm -1	D'	Defect dependent	40
	2450 cm -1	D+D"	Second order scattering band	43
	2700 cm -1	2D or G'	Second order scattering band	40
	2945 cm -1	D+D'	Second order scattering band	40
	3175 cm -1	2G	Second order scattering band	40

Table V :

 V Average roughness parameter Ra, average coating thickness measurement and corresponding standard deviation for all PEO samples. MWCNT+PEO sample measurements were differentiated in two categories given the large difference in thickness between regions with and without an inner dense sublayer. For samples S1, S2 and S3 the average thickness of the outer porous sublayer relative to the total oxide thickness is also given.

	Sample	S0	S1	S2	S3
	Roughness parameter Ra (µm)	3.2 ± 1.0	8.4 ± 1.1	8.0 ± 2.0	10.5 ± 2.1
	Thickness in regions with a visible inner	35 ± 4	72 ± 11	76 ± 15	60 ± 11
	dense sublayer (µm)
	Thickness in regions without inner dense	n.a.	100 ± 20	127 ± 21	100 ± 12
	sublayer (µm)
	Average proportion of the outer sublayer	n.a.	32	51	62
	(%)

34

List of figure captions:

Acknowledgments

Data availability

The data that support the findings of this study are available from the corresponding author, J. Martin, upon reasonable request.

Additional information

The authors declare no competing financial interests.

List of tables


Fabio D ' Andreagiovanni
email: d.andreagiovanni@hds.utc.fr

Hicham Lakhlef
email: hicham.lakhlef@hds.utc.fr

Antonella Nardin
email: antonella.nardin@uniroma3.it

Green and robust optimal design of Single Frequency Networks by min-max regret and ACO-based learning

Keywords: Single Frequency Networks, DVB-T, Network Design, Robust Optimization, Metaheuristics

Notwithstanding the introduction of brand new 5Gbased wireless services, single frequency networks supporting digital television and radio broadcasting still represent a major source of telecommunications services in modern smart cities. In this work, we propose a robust optimization model for the green design of second generation single frequency networks based on the digital television DVB-T standard, whose ongoing adoption requires to reconfigure and redesign existing networks. Our robust model aims at protecting design solutions against the data uncertainty that naturally affect propagation of signals in a real environment. For reducing conservatism of solutions, we refer to a heuristic min-max regret paradigm and to solve the resulting problem we propose to adopt a hybrid exact-heuristic algorithm based on the combination of an Ant Colony Optimizationlike learning procedure, exploiting tight formulations of the optimization model, with an exact large neighborhood search. Results of computational tests considering realistic instances show that the heuristic min-max regret approach can produce solutions characterized by a substantially lower price of robustness without sacrificing protection against data uncertainty.

I. INTRODUCTION

Smart cities are nowadays enjoying a wide range of brand new telecommunication services enabled by last generation 5G networks [START_REF] Chiaraviglio | Algorithms for the design of 5G networks with VNF-based Reusable Functional Blocks[END_REF], [START_REF] Meshram | An effective mobile-healthcare emerging emergency medical system using conformable chaotic maps[END_REF], [START_REF] Priya | 5GAuNetS: an autonomous 5G network selection framework for Industry 4.0[END_REF], [START_REF] Richter | A 5G New Radio-Based Terrestrial Broadcast Mode: System Design and Field Trial[END_REF], [START_REF] Shehab | 5G Networks Towards Smart and Sustainable Cities: A Review of Recent Developments, Applications and Future Perspectives[END_REF], [START_REF] Yang | Using 5G in smart cities: A systematic mapping study[END_REF]. However, within this ultramodern context, older and more traditional broadcasting services like television and radio are still considered critical by many governments, since they support a cheap, easy and democratic access to information to all the population. A very important advancement in offering broadcasting services of higher quality has been represented by the switch from analogue to digital transmission, which has allowed to implement the paradigm of Single Frequency

Networks (SFNs

). An SFN is a broadcasting network in which all transmitting stations emit the same data on the same frequency. This allows for improving spectrum usage, a crucial goal considering that radio resources are scarce and that the expansion of telecommunications services has rapidly led to the congestion of frequencies (see e.g., [START_REF]DVB Project[END_REF]- [START_REF]Frequency and Network Planning Aspects of DVB-T2[END_REF], [START_REF]Terrestrial and satellite digital sound broadcasting to vehicular, portable and fixed receivers in the VHF/UHF bands[END_REF]). Major examples of SFNs are represented by DVB-T (Digital Video Broadcasting-Terrestrial) networks, which broadcast television content in a digital format and are based on the DVB-T standard adopted in more than 140 countries worldwide [22]. In the last years, SFNs have gained new attention because of the release of a second generation of standards like DVB-T2 [22]. The new standards supports improved modulation scheme and offer higher performance. In addition, thanks to an improved spectrum efficiency, they enable a higher number of broadcasting operators to operate simultaneously. The entrance of new operators and the need for reconfiguring older networks have renewed the interest in software for designing SFN networks, in particular those exploiting mathematical optimization models and algorithms.

In this work, we focus on developing a robust optimization model and algorithm for designing SFN networks taking into account the uncertainty that naturally affects propagation of signals in a real environment. Neglecting such uncertainty may lead to critical issues, such as infeasibility and suboptimality of design solutions, due to unexpected deviations of the input data representing signal propagation. We remark that, while the deterministic optimal design (i.e., not including the modelling of uncertainty) of SFN networks has received substantial attention (e.g., [START_REF] Anedda | Heuristic optimization of DVB-T/H SFN coverage using PSO and SA algorithms[END_REF], [START_REF] D'andreagiovanni | A Matheuristic for Joint Optimal Power and Scheduling Assignment in DVB-T2 Networks[END_REF], [START_REF] D'andreagiovanni | GUB Covers and Power-Indexed Formulations for Wireless Network Design[END_REF], [START_REF] Denkovska | Power optimization of LTE-800 and coexistence with DVB-T services[END_REF], [START_REF] Koutitas | Green Network Planning of Single Frequency Networks[END_REF], [START_REF] Lanza | Coverage Optimization and Power Reduction in SFN Using Simulated Annealing[END_REF], [START_REF] Mannino | The network packing problem in terrestrial broadcasting[END_REF], [START_REF] Nepal | Optimization of Multi-Frequency Network With DVB-T2 Services for Regions With Complex Geographies: A Case Study of Nepal[END_REF]), the robust design based on using optimization under uncertainty techniques has received less coverage. In this work, we thus propose a new optimization under data uncertainty approach based on min-max regret. Specifically, our main original contributions are:

1) we propose a mixed integer linear programming problem for representing a green SFN optimal design problem with the objective of minimizing the total power emission of the transmitters, while serving a target fraction of the population with a fixed level of quality of service; 2) we deal with the question of modelling the uncertainty of propagation of wireless signals, proposing a minmax regret model which offers less conservative design solutions than a classical robust optimization techniques; 3) since min-max regret reduces solution conservatism at the price of computational tractability, requiring the solution of more complex optimization problems, we propose to adopt a heuristic min-max regret solution approach, based on the identification of a peculiar median scenario that attempts at capturing the features of all the possible (potentially exponential) scenarios of realizations of the uncertain data. The resulting model is solved with the support of a metaheuristic that combines an Ant Colony Optimization-inspired learning algorithm, based on exploiting stronger formulations available for the model, with an exact large neighborhood search; 4) we test the performance of the heuristic min-max regret approach on realistic DVB-T instances, showing that we obtain a substantial reduction in the price of robustness without reducing protection against data uncertainty. The remainder of the paper is organized as follows: the optimal DVB-T network design model is presented in Sect.II; the heuristic min-max regret model is derived in Sect.III, while Sect.IV presents the hybrid exact-heuristic ACO-inspired learning algorithm. Finally, results of computational tests are presented in Sect.V and conclusions are presented in Sect.VI.

II. OPTIMAL SFN DESIGN

As reference SFN network, we consider a DVB-T network in which all the transmitters constituting the network transmit the same data on the same frequency in a synchronized way, by means of Orthogonal Frequency Division Multiplexing (OFDM) [22]. The transmitters offer broadcasting services to a region in which a number of users are located. The users constitute the receivers of the DVB-T telecommunication service. The task of designing the network consists of a) setting the power emitted by each transmitter and b) choosing the serving transmitter of each user/receiver, with the goal of minimizing the total power emission of transmitters, while guaranteeing service coverage to a target percentage of the population above a given level of quality of service.

For modelling purposes, we adopt the recommendation of international and national regulatory bodies like ITU and AGCOM to model the region where DVB-T services are provided through a rasterization process [START_REF]Specifications for a DVB-T planning software tool[END_REF]: the region is discretized into a grid of sufficiently small pixels and the strength of the DVB-T signal picked up in the point at the center of each pixel (the testpoint) is considered representative of the strength of signal picked up in all the other points of the pixel. Given such rasterization of the region to be covered with DVB-T services, we identify the set of transmitters T , which correspond to the DVB-T transmitting stations, and the set of receivers R, which correspond to the testpoints at the center of the pixels identified in the region.

The optimal design of an SFN network based on the DVB-T standard can be traced back to a family of optimization problems known as Wireless Network Design Problems (WNDPs), in which the configuration of a set of wireless transmitting stations must be chosen in order to provide services to users located in a target region, (see e.g., [START_REF] D'andreagiovanni | GUB Covers and Power-Indexed Formulations for Wireless Network Design[END_REF], [START_REF] Dely | Fair optimization of meshconnected WLAN hotspots[END_REF], [START_REF] Kennington | Wireless Network Design: Optimization Models and Solution Procedures[END_REF], [START_REF] Mannino | The network packing problem in terrestrial broadcasting[END_REF]). Specifically, referring to the hierarchy of WNDPs, the problem that we consider corresponds to a variant of a Scheduling and Power Assignment Problem, which has been proved to be NP-Hard [START_REF] Mannino | The network packing problem in terrestrial broadcasting[END_REF].

The two kind of decisions that must be taken in the optimal network design problem that we consider, namely choosing a power emission for the transmitters and choosing the serving transmitter of a receiver can be modelled by continuous and binary decision variables, as follows:

• for each transmitter t ∈ T , a continuous variables p t ∈ [0, P max ] is defined to represent the power emission of the transmitter; • for each transmitter t ∈ T and receiver r ∈ R, a binary variable x tr ∈ {0, 1} is defined to represent the fact that receiver r is assigned to transmitter t (specifically, x tr = 1 if r is assigned to t -we say in this case that t is the serving transmitter of r -and x tr = 0 otherwise). The power P tr that a receiver r ∈ R obtains from any transmitter t ∈ T is proportional to the power emitted by t according to a coefficient a tr ∈ [0, 1], i.e.: P tr = a tr • p t . The coefficient a tr sums up the reduction in power that a signal suffers when propagating from t to r in a real-world environment and commonly goes under the name of fading coefficient (see e.g., [START_REF] Rappaport | Wireless Communications: Principles and Practices[END_REF]). Once clarified how power behaves, we can proceed to discuss how signals combine when picked up by a receiver: in an OFDM-based DVB-T system, each receiver picks up signals from all the transmitters and their distinction in useful signals, which contribute to increase the quality of service, and interfering signals, which contribute to decrease the quality of service, depends upon whether they fall in a detection time window (see e.g., [START_REF] Ligeti | Minimal cost coverage planning for single frequency networks[END_REF], [START_REF] Mannino | The network packing problem in terrestrial broadcasting[END_REF] for details). According to a common design practice, in our model the detection window of a receiver starts in the instant in which the signal of the serving station is received. This allows identifying the useful and interfering transmitters easily once the serving transmitter is chosen: if t is the serving transmitter of r, then we denote by U (t, r) the set of useful transmitters and by I(t, r) the set of interfering transmitters for receiver r (note that in this setting a transmitter is either useful or interfering for a receiver).

The quality of service for receiver r served by transmitter t depends upon the Signal-to-Interference Ratio (SIR), namely the ratio of the sum of the power of the useful signals to the sum of the power of the interfering signals [START_REF] Rappaport | Wireless Communications: Principles and Practices[END_REF]:

SIR tr = τ ∈U (t,r) a τ r • p τ N + τ ∈I(t,r) a τ r • p τ ≥ δ . (1)
which involves the power emissions p τ of all transmitters and in which N > 0 and δ > 0 are the noise of the system and the minimum SIR value that guarantees the desired quality of service, respectively. It is easy to rewrite SIR inequality (1) as

τ ∈U (t,r) a τ r • p τ -δ τ ∈I(t,r) a τ r • p τ ≥ δ • N (2)
which allows to highlight the linear nature of the constraint with respect to the power decision variables p τ . The next step in the modelling procedure is to express the other decision of identifying the serving transmitter of each receiver. To do so, we link the binary variable x tr , denoting whether receiver r is served by transmitter t, to the corresponding SIR inequality (2) defining the following modified SIR constraint:

τ ∈U (t,r) a τ r • p τ -δ τ ∈I(t,r) a τ r • p τ + M • (1 -x tr ) ≥ δ • N (3)
that constitutes a so-called big-M constraint, in which the inclusion of the binary variable x tr multiplied by a suitably large coefficient M ≥ 0 (the big-M) is needed to activate or deactivate the constraints. Indeed, when x tr = 0, r is not served by t and it is not necessary to satisfy the corresponding SIR inequality: the term M (1 -x tr ) is then equal to the large value M , which is added to the left-hand-side of the constraint (3) and the constraint is thus satisfied by any combination of the decision variables p τ . If instead x tr = 1, r is served by t and the corresponding SIR constraint must be satisfied -in this case M (1 -x tr ) = 0 and thus (3) reduces to ( 2). The complete model can then be written as:

min t∈T p t (4)
τ ∈U (t,r)

a τ r • p τ -δ τ ∈I(t,r) a τ r • p τ + + M (1 -x tr ) ≥ δ • N t ∈ T, r ∈ R (5) t∈T x tr ≤ 1 r ∈ R (6)
r∈R t∈T

π r • x tr ≥ α • r∈R π r (7) 0 ≤ p t ≤ P max t ∈ T (8) x tr ∈ {0, 1} t ∈ T, r ∈ R . ( 9
)
in which, a) the objective function ( 4) pursues the minimization of the total power emission; b) the quality-of-service conditions are expressed by the SIR constraints (5) accompanied by constraints [START_REF] Bertsimas | Theory and Applications of Robust Optimization[END_REF] imposing that at most one transmitter can act as server of each receiver. The constraints [START_REF] Bertsimas | The Price of Robustness[END_REF] impose that at least a fraction α ∈ [0, 1] of the total population of the region must be covered with service (in the left-hand-side, each service decision variable is multiplied by a coefficient π r that equals the population of the pixel associated with receiver r, while in the right-hand-side the total sum of the population over all the pixels is multiplied by the value α). Finally, ( 8) and ( 9) are the decision variables previously defined.

III. PROTECTING FROM PROPAGATION UNCERTAINTY

The optimization problem (4)-( 9) is a deterministic model in which it is neglected the fact that the fading coefficients a tr constitute uncertain data: propagation of signals in a real-world environment is indeed affected by many factors (presence of obstacles, complexity of the lay of the land, etc.) and it is thus very complicated to accurately estimate the values of a ts over large territories like those involved in DVB-T services. The coefficients are then commonly estimated through empirical propagation models [START_REF] Rappaport | Wireless Communications: Principles and Practices[END_REF], which specify formulas to compute the coefficients. However, as telecommunication professionals know well, the actual values of the propagation coefficients may (even largely) vary with respect to those provided by the models. In our design problem, issues may arise both in cases of over-and under-estimation: if the fading coefficient of an interfering signal is under-estimated, the actual interference will result higher than expected, thus decreasing the quality of service; in contrast, in the case of a useful signal, issues arise in case of over-estimation, since the actual useful signal will result lower than expected. Consequently, the fading coefficients represent uncertain data in an optimization problem, i.e. data whose value is not exactly known when the problem is solved. The presence of uncertain data cannot be neglected, since the occurrence of unexpected deviations in the value of coefficients may cause infeasibility and suboptimality of solutions recognized as feasible and optimal (e.g., [START_REF] Baumgartner | Towards Robust Network Slice Design under Correlated Demand Uncertainties[END_REF]- [START_REF] Bertsimas | The Price of Robustness[END_REF]).

For tackling the uncertainty of fading coefficients, we assume that for each transmitter-receiver (t, r) we know a set of possible values that the coefficient may assume: we denote such set by A tr = {a 1 tr , a 2 tr , . . . , a k tr } and we assume that the k possible values are sorted increasingly. We then introduce the set

A = A 11 × • • • × A tr × • • • × A |T ||R|
to denote all the possible |T ||R|-tuples, which represent the set of all possible combinations of fading coefficients when considering all the transmitter-receiver couples.

If we follow the paradigm of Robust Optimization (e.g., [START_REF] Bertsimas | Theory and Applications of Robust Optimization[END_REF]), the aim is to find a robust optimal solution, namely a solution which guarantees the best objective value for the worst realization of the uncertain coefficients. If we denote the feasible set of solutions of problem ( 4)-( 9) by F and its objective function by f (x, p, a) , highlighting that it does not only depends upon the value of the service assignment variables x and power variables p, but also on the realization of the fading coefficients a ∈ A, the robust version of the optimal design problem can be wrttien as: min (x,p)∈F max a∈A f (x, p, a), which pursues the identification of an optimal solution (x * , p * ) that offers the minimum power emission while protecting against the worst fading coefficient realization of A [START_REF] Aissi | Min-max and min-max regret versions of combinatorial optimization problems: A survey[END_REF]. It is important to stress that such an optimal solution remains feasible for all the realizations of data of A and is thus fully protected against data uncertainty. However, at the same time, this full protection may entail a too high price of robustness, i.e. a too high reduction in optimality due to excluding non-robust solutions that are not feasible for all fading realizations in A.

Since the price of robustness may result too high and look-ing for full protection may result excessive, we investigate the adoption of an alternative robust methodology, min-max regret, which is less conservative [START_REF] Aissi | Min-max and min-max regret versions of combinatorial optimization problems: A survey[END_REF]. This methodology is based on evaluating the effects of taking a non-optimal decision through a regret function R(x, p, a), which in our case is defined as the following difference of the objective value of a solution (x, p) ∈ F with respect to the value of the optimal solution (x * , y * ) for a fading realization a ∈ A:

R(x, p, a) = f (x, p, a) -f (x * , y * , a) (10)
The maximum regret R max (x, p) for a solution (x, p) is then equal to the maximum of the regret function for all a ∈ A:

R max (x, p) = max a∈A R(x, p, a) (11)
and the min-max regret version of ( 4)-( 9) can be written as the problem of finding the feasible solution (x, p) that minimizes the maximum regret for all the fading realizations:

min (x,p)∈F max a∈A R(x, p, a) . (12)
Since the min-max regret problem needs to identify the maximum regret of every solution (x, y), therefore requesting to execute the impractical task of potentially verifying exponentially many realizations of the fading coefficients and optimally solving the associated optimization problems, we rely on a heuristic min-max regret method introduced in [START_REF] Flushing | Relay Node Placement for Performance Enhancement with Uncertain Demand: A Robust Optimization Approach[END_REF] for a wireless sensor network application. This method is centered around what we call a pessimistic median fading coefficient realization scenario: we first define a median fading coefficient scenario a med that, for every transmitter-receiver couple (t, s), includes the fading value ātr ∈ A tr that is closest to the median value of the values in the set A tr . Solving the design problem (4)- [START_REF] Chandra Mohan | A survey: Ant Colony Optimization based recent research and implementation on several engineering domain[END_REF] for the median scenario a med would allow to identify a solution that is on average of good quality and reliable. However, we could still take a chance on facing infeasibility in case of substantial deviations from the median values. Consequently, for reducing this risk, we identify a more robust fading realization scenario by introducing the concept of super-transmitters, namely transmitters that, favoured by their location with respect to the population to be covered, tend to contribute more decisively to the coverage of a target region (see [START_REF] D'andreagiovanni | Integrating LPguided variable fixing with MIP heuristics in the robust design of hybrid wired-wireless FTTx access networks[END_REF] for more details) and the setting of their power emission is more critical. For these super-transmitters, denoted by S ⊆ T the fading coefficient is set to the lowest value when act as serving transmitter and to the highest value when acting as interfering transmitter. We call Pessimistic Median (PM) scenario the resulting fading realization scenario and we denote the corresponding fading realization by a P M . A robust solution for problem (4)-( 9) is then identified by optimally solving the problem for the pessimistic median scenario a P M .

IV. AN ACO-BASED LEARNING SOLUTION APPROACH

Being a Mixed Integer Linear Program, problem (4)-( 9) could be solved through an optimization software like IBM ILOG CPLEX [START_REF]IBM ILOG CPLEX[END_REF]. However, since it is hard to optimally solve it within the limited time budget that is typically at disposal of network professionals, especially because of the presence of the complicating big-M SIR constraints, we propose to accelerate the convergence of a software like CPLEX by defining a warm-start heuristic (i.e., a heuristic algorithm that fast identifies good quality solutions to be passed as initial solutions to CPLEX). Specifically, we propose to adopt a hybrid exact-heuristic approach originally proposed in [START_REF] D'andreagiovanni | Towards the fast and robust optimal design of wireless body area networks[END_REF], [START_REF] D'andreagiovanni | Integrating LPguided variable fixing with MIP heuristics in the robust design of hybrid wired-wireless FTTx access networks[END_REF] that draws inspiration from Ant Colony Optimization (ACOsee e.g.,m [START_REF] Blum | Hybrid metaheuristics in combinatorial optimization: A survey[END_REF], [START_REF] Chandra Mohan | A survey: Ant Colony Optimization based recent research and implementation on several engineering domain[END_REF], [START_REF] Dorigo | Ant colony optimization theory: A survey[END_REF]), as a kind of reinforcement learning algorithm that is based on rewarding value settings of decision variables that have led to the definition of good quality feasible solutions and on penalizing value settings that have led to bad quality solutions (see e.g., [START_REF] Herrmann | Ant colony optimization and reinforcement learning[END_REF], [START_REF] Russell | Artificial intelligence: a modern approach[END_REF]). More precisely, the solution approach combines a probabilistic variable fixing procedure integrated with an exact large neighborhood search.

In the probabilistic fixing procedure, a number of solutions are built iteratively (according to the ACO metaphor, each of these solutions corresponds with an ant): at every iteration, we have at disposal a partial solution, namely a solution in which just a subset of decision variables has been assigned a value, and we can further complete the solution by setting the value of one or more additional variables. When all the variables have been assigned a value, we finally have a complete solution and its quality can be assessed using the corresponding value of the objective function.

In the case of our wireless network design problem, we can note that if we set the value of the power variable p t of each transmitter, we can then readily compute the coverage of receivers that is granted by checking which SIR constraints (5) are satisfied. We thus recognize the power variables as critical decision variables in our hybrid solution procedure. To this end, we identify equally spaced power values in the power emission range of each transmitter: we therefore characterize a set P = P 1 = 0, P 2 , . . . , P n = P max of discrete power values in the range [0, P max ] and each transmitter must be assigned a power value belonging to this discrete set. This allows to derive a power-indexed model of the network design problem, which provides stronger mathematical formulations (see [START_REF] D'andreagiovanni | GUB Covers and Power-Indexed Formulations for Wireless Network Design[END_REF]).

When we are at a generic iteration of the cycle that has the task of defining a feasible solution, we have a partial power solution in which a subset of transmitters T FIX ⊆ T has been assigned a power emission value of P. The next transmitter whose power is fixed is then chosen on the basis of probability formula that represents an improved formula of the classical ACO (see [START_REF] D'andreagiovanni | Towards the fast and robust optimal design of wireless body area networks[END_REF] for details):

P (p t = P ) = β • φ t + (1 -β) • η t τ ∈T \T FIX l∈L [β • φ τ l + (1 -β) • η τ l ] , (13)
that computes the probability of setting the power of transmitter t ∈ T \T FIX equal to P ∈ P, taking into account all the transmitters τ ∈ T \T FIX and power emission levels l ∈ L whose power has not yet been set.

In [START_REF] D'andreagiovanni | Negative cycle separation in wireless network design[END_REF], two measures of value fixing attractiveness are combined: φ τ l constitutes an a-priori attractiveness measure that resembles the pheromone trails of ACO and is obtained from a tight-relaxation of the model (in particular from model ( 4)-( 9) strengthened through the (strong) valid inequalities proposed in [START_REF] D'andreagiovanni | Negative cycle separation in wireless network design[END_REF], [START_REF] D'andreagiovanni | GUB Covers and Power-Indexed Formulations for Wireless Network Design[END_REF]), whereas η τ l is an a-posteriori attractiveness measure obtained from the linear relaxation of ( 4)-( 9) (i.e., a version of the model in which the integrality requirement of variables is removed). The two measures are combined by a parameter 0 ≤ β ≤ 1.

The probability measure is used in the parallel construction of γ > 0 solutions and, once that a round of solution construction is completed, we proceed to reward/penalize the single construction moves that have been operated, on the basis of the quality of the solutions that have been defined. Specifically, the update rule of the a-priori measure is:

φ τ l (i) = φ τ l (i -1) + γ SOL=1 ∆φ SOL τ l (14) ∆φ SOL τ l = φ τ l (0) • OG(v AVG , u) -OG(v SOL , u) OG(v AVG , u) (15)
in which φ τ l (i) is the a-priori attractiveness measure of setting the power of transmitter τ ∈ T equal to P in the i-th execution of the cycle and ∆φ SOL τ l

is the correction to this measure, computed considering a moving average over the last γ solutions that have been constructed. Furthermore, u is an upper bound on the optimal value, v SOL is the value of the SOLth feasible solution built in the last construction cycle, v AVG is the average of the values of the last γ solutions that have been constructed. The value OG(v, u) is an optimality gap and assesses the difference between the value v of a solution and an upper bound u (formally, OG(v, u) = (u -v)/v). Concretely, after having defined all its elements, formula ( 14) is introduced to update the a-priori measure with the aim of rewarding (penalizing) the setting of decision variables to values which have conducted to solutions associated with lower (higher) optimality gap, when compared to a moving average v AVG that involves the last γ solutions.

After the cycle of solution construction has ended, we conduct an improvement phase in which we rely on an exact large neighborhood search for improving the best solution found. Specifically, the strategy is to formulate the search in a large neighborhood of solutions of the best solution as an optimization problem which is solved by an optimization software like CPLEX [START_REF] Blum | Hybrid metaheuristics in combinatorial optimization: A survey[END_REF]. This is motivated by noticing that the software may be unable to find the optimal solution of the complete problem, but may instead efficiently identify optimal solutions of appropriate subproblems. In our specific case, we explore solution neighborhoods defined by allowing to modify the power emission of the transmitters: each transmitter can be either switched off (thus its power emission is set to zero) or its power emission can be increased to the closest higher discrete power level or decreased to the closest lower power level. This can be easily expressed through linear constraints involving binary variables representing the power modifications and that can be added to problem (4)-( 9), similarly to the procedure illustrated in [START_REF] Danna | Exploring relaxation induced neighborhoods to improve MIP solutions[END_REF]. The resulting model is then passed to CPLEX to be solved to an optimum. For not letting CPLEX waste too much time on closing the optimality gap, we impose a time limit on the solution of the problem corresponding to exploring the solution neighborhood (high quality solutions are typically fast obtained).

We provide the pseudocode of the hybrid ACO-based learning heuristic in Algorithm 1. The best solution found by the algorithm is used to warm-start the solver CPLEX, i.e. it is passed to CPLEX as starting feasible solution to accelerate its convergence to an optimum. In Algorithm 1, problem (4)-( 9) is solved referring to the pessimistic median scenario a P M and the first step corresponds with solving the strengthened powerindexed optimal design model and use it for initializing the values of the a-priori fixing measure. After this, the algorithm is centered around running a cycle for building solutions, with a time limit as stopping criterion. Each run of the cycle leads to the definition of γ solutions, built by first setting the value of the power decision variables by means of the probability measure [START_REF] D'andreagiovanni | Negative cycle separation in wireless network design[END_REF] and then by evaluating the obtained population coverage, checking which SIR constraints are satisfied by the power setting. Once a run of the cycle is completed, the next step is to update the a-priori attractiveness measure φ by means of formula [START_REF] D'andreagiovanni | GUB Covers and Power-Indexed Formulations for Wireless Network Design[END_REF], learning from the quality of the value fixing of decision variables that has been operated. Finally, once the construction time limit has been reached, the best solution found is used as initial solution in the conclusive exact large neighborhood search, to further improve it.

Algorithm 1 (Hybrid ACO-based learning heuristic) 1: define the pessimistic median scenario a P M 2: compute the linear relaxation of the strengthened power-indexed version of ( 4)-( 9) for a P M and initialize the values φ t (0) with the corresponding optimal values 3: let (p * , x * ) be the best solution found 4: while a time limit is not reached do 5:

for SOL := 1 to γ do 6:

iteratively set the power emission pt of every transmitter t ∈ T using formula [START_REF] D'andreagiovanni | Negative cycle separation in wireless network design[END_REF] and let p be the corresponding fixed power vector if the coverage satisfies constraint [START_REF] Bertsimas | The Price of Robustness[END_REF] and the total power emission of (p, x) is lower than that of (p * , x * ) then update the a-priori measure φ according to [START_REF] D'andreagiovanni | GUB Covers and Power-Indexed Formulations for Wireless Network Design[END_REF] 14: end while 15: run the exact large neighborhood search using (p * , x * ) as incumbent solution 16: return (p * , x * )

V. COMPUTATIONAL RESULTS

We test the performance of the heuristic min-max regret approach, based on considering the pessimistic media scenario, on a set of realistic DVB-T instances corresponding to regionwide networks deployed in Italy. The fading coefficients are computed according to propagation models defined taking into account recommendation of regulatory bodies [START_REF]Specifications for a DVB-T planning software tool[END_REF], [START_REF]Method for point-to-area predictions for terrestrial services in the frequency range 30 MHz to 4 000 MHz[END_REF]. These recommendations also specify how to model propagation stochasticity of DVB-T signals and we considered discussion held with professionals working in the field of single frequency networks for defining a range of deviation for the standard robust optimization model that does not result too conservative and include unlikely deviations. The instances that we consider refer to 20 Italian DVB-T regional networks that involve up to about 300 stations and 4000 pixels and for each pixel the population is derived from demographic data. In order to solve the considered mixed integer linear programming problem, we rely on the software IBM ILOG CPLEX [START_REF]IBM ILOG CPLEX[END_REF], running on a Windows machine with 2.70 GHz Intel i7 and 8 GB of RAM. In the warm-start hybrid heuristic presented in Algorithm 1, we allow the parallel construction of γ = 10 solutions and we assign a weight β = 0.5 so as to assign the same weight to the combination of the a-priori and a-posteriori measure. A time limit of 600 seconds is imposed to the final exact search.

The results of the preliminary computational tests that we have conducted are presented in Table I in which: 1) ID identifies the regional DVB-T network instance; 2) columns 2 and 3 report the Price of Robustness (PoR%) for the case of the standard robust optimization approach (SR) and for the case of the min-max regret approach (mMR) -we note that the PoR% is computed as the percentage increase in the total power that must be faced with respect to the deterministic version of the design problem which neglects the presence of uncertain fading coefficients; 3) column ∆PoR% reports the percentage decrease in the price of robustness that the use of the less conservative min-max regret approach grants with respect to the standard robust optimization approach; 4) the last three columns report measures Prot% about the protection granted by standard robustness SR and min-max regret mMR. More in detail, for each instance, we generate 1000 realizations of the uncertain fading coefficients and we then compute the percentage of realizations in which the robust solutions of SR and mMR are feasible. The percentage difference between the two measures of protection is expressed by ∆Prot%,

If we analyze the data reported in Table I, a first important remark that can be made is that granting protection against fading coefficient uncertainty leads to a remarkable increase in the total power needed to cover the population in the considered territory. Indeed, for both SR and mMR the total power emission of the transmitters increase of more than 10% on average. However, at the same time, it can be clearly seen that for all instances the increase in power and thus the price of robustness is substantially higher for the standard robust optimization model: SR on average requires 16% more of average power emission, while mMR entails a lower increase in power that allows to save 26% of power emission. This is of course a remarkable result, since a greener and less power consuming network does not only lead to lower operational costs, but also limits electromagnetic pollution, exposing people to lower electromagnetic fields. The reduction of the total power emission needed to handle uncertainty results particularly significant in the case of instances like I17 and I19, for which the price of robustness and thus the total power emission is reduced of more than 30% and even reaches more than 40%. A natural question that then arises is whether this reduction in the price of robustness granted by the min-max regret approach is accompanied also by a significant reduction in protection. However, this is not the case as it can be seen from the column P rot%: the min-max regret solutions are feasible for more than 93% of the 1000 scenarios of fading realizations that we generated, leading to a very small decrease in protection with respect to standard robustness (which offers on average protection for about 95% of the scenarios). The small reduction in protection is thus much more than compensated by the very relevant reduction in total power. This shows the expediency and advantages of adopting the min-max regret approach, which, for the considered wireless network design problem, is able to return optimal solutions that, at the same time, are wellprotected against uncertainty and grant sensibly lower price of robustness. We have considered the problem of optimally designing a single frequency network while taking into account the uncertainty that naturally affects the propagation of wireless signals in a real-world environment. Since neglecting such uncertainty may lead to wrong design solutions affected by unexpected holes in the coverage of users, we have proposed to adopt a robust optimization approach. Robust optimization is a major methodology adopted in optimization to deal with the presence of uncertain data, namely data whose value is not exactly known when the problem is solved. Since canonical robustness may result too conservative, leading to robust solutions that entails a too high price of robustness (i.e., a too high deterioration in the objective value to guarantee protection), we have investigated the adoption of an alternative way to guarantee protection, namely a (heuristic) min-max regret approach, which grants less conservatism and higher computational tractability. The solution of the resulting problem is accelerated by a hybrid heuristic that combines an Ant Colony Optimization (ACO)-inspired learning algorithm with exact large neighborhood search. Computational results considering realistic instances show that the heuristic min-max regret approach can identify substantially less conservative solutions, without sacrificing protection. This encourages to further develop the approach, in particular by investigating the definition of refined median scenarios that could better capture the stochastic behaviour of signal propagation and lead to robust optimal solutions associated with even lower price of robustness. Furthermore, we plan to investigate a multi-objective version of the problem, pursuing joint coverage maximization and power minimization, adopting an iterative solution procedure as in [START_REF] Zakrzewska | Biobjective optimization of radio access technology selection and resource allocation in heterogeneous wireless networks[END_REF].

  This work was partially funded by the Institute for Information Sciences (INS2I) of the French National Centre for Scientific Research (CNRS) through project "Secure Softwarization of Heterogeneous Internet of Things" (Appel à projets unique INS2I 2022, Ref: INS2I/D/2022/15).

7 :

 7 check which SIR constraints (3) are satisfied by p and let x be the corresponding service coverage vector 8: compute the total population coverage of solution (p, x) 9:

  * , x * ) with (p, x)


Kang Wang
email: kang.wang@universite-paris-saclay.fr

Propagation and transmission of a light beam in a graphene-like metallic photonic lattice

We study theoretically light beam propagation in a graphene-like metallic photonic lattice, associated to distorted Dirac cones, and discuss the beam transmission in terms of the coupling between the local resonance modes and incident fields.

, and ∓ corresponding respectively to the upper and lower Dirac cone.

JTu4B.

Introduction

Graphene-like photonic structures create Dirac cones in their frequency spectra and provide opportunities for optically probing Dirac cone related physical properties. Potential applications, such as valley photonics exploring optically the valley degree of freedom associated to Dirac cones [START_REF] Garcia-Pomar | Fully valley-polarized electron beams in graphene[END_REF], have been proposed for information processing [START_REF] Dong | Valley photonic crystals for control of spin and topology[END_REF]. Studies on K-valley dependent light beam propagation, associated to the lower Dirac cones in dielectric photonic crystals [START_REF] Garcia-Pomar | Fully valley-polarized electron beams in graphene[END_REF][START_REF] Deng | Observation of valley-dependent beams in photonic graphene[END_REF] and the upper cones in analogous situations of electron beams in atomic graphene [START_REF] Wang | Manipulation of electron beam propagation by hetero-dimensional graphene junctions[END_REF][START_REF] Park | Generation of valley-polarized electron beam in bilayer graphene[END_REF], are both carried out. Beam splitting and self-collimation are obtained for the zigzag and armchair edge incidences. We have shown previously that a light beam propagating in a metallic graphene-like photonic lattice displays drastically different behaviors as compared to the above results. Namely, at the armchair edge, the incident beam associated to the lower Dirac cone is collimated to one central beam, and that associated to the upper cone is split into two beams with a 2π/3 angle. The other expected beams are all strongly inhibited [START_REF] Wang | Abnormal K-point dependent electromagnetic beam propagation in a graphenelike triangular metallic photonic structure[END_REF]. Here we report a study of the propagation of narrow incident beams, of the widths of the crystal period sizes, allowing to probe the coupling between the incident field and the local resonance modes at elementary structure scales, as well as the consequence on the beam transmission, for different local structure configurations.

Light beam propagation

The metallic structure is constructed by infinite cylinders, with radius r = 0.25a and dielectric constant set to negative infinity, placed at the triangular lattice [Fig. 1 We can show that, in a tight-binding (TB) model, for wave vectors close to the edges of the quasi-triangular contours, the crystal frequency ω k can be expressed as

ω k 2 ≈ (ω 0 2 - β ∓ γ 1 ∓ α ) - ∓2[cos(
The first term in Eq.1 corresponds to the energy levels of two coupled modes in a unit cell, with ∓ corresponding, respectively, to the anti-bonding and bonding state. The band structure along a Γ-K direction close to the quasi-triangular contours can then be viewed as arising from a series of bonding or anti-bonding states on onedimensional chains, with the band frequencies derived from that of these states, adjusted by the second term in Eq. 1, and the wave propagation can be considered as being based on a series of such states formed in the unit cells.

Transmission of narrow beams, associated to both the upper and lower Dirac cones, is investigated for armchair and zigzag edges for different local configurations [AB, BA, A1 and A2 cases, see Fig. 1(a)] using the FDTD method. Strong excitation difference in the first unit cells, following their positions and orientations with respect to the incident fields, is observed. This in turn determines the beam transmission associated to the upper and lower Dirac cones for different K valleys. Large transmission is obtained for incident conditions allowing strong coupling between the incident fields and the resonance modes in the first unit cells at the incident edges, and leading to field distributions corresponding or close to bonding or anti-bonding states for these modes. As an example, the A 1 beam (associated to the lower Dirac cone) transmission in the AB and BA cases is shown in Fig. 2, that illustrates well that a strong field coupling in the impact zone leads to strong beam transmission, and confirms, besides, that the beam propagation is based on a series of resonance states in the unit cells.

Conclusion

This work allows understanding of the particularity of a metallic graphene-like photonic lattice in its interaction with the light waves at scales down to the size of local resonance states. It may find application in valley photonics in the designs of beam collimation and splitting devices for information processing, by providing information for tuning and optimizing the beam transmission in various incident conditions.

  (a)] nodes in an air background. The frequency band diagram for TM polarization [Fig. 1(b)] is obtained by solving Maxwell's equations using the finite-difference time-domain (FDTD) method. Dirac cones are formed by the first two bands. The iso-frequency contours of the upper and lower cones are depicted in the same figure [1(d) and 1(e)].

Fig. 1 .

 1 Fig. 1. Triangular photonic lattice (a) and the frequency band diagram (b), as well as a 3D view of the Dirac cone at K (4π/3a, 0) [(c)] and the iso-frequency contours of the upper [(d)] and lower part [(e)]. The frequency level (ω 1 and ω 2 ) and interval (∆ω) values are normalized to ωa/2πc.

Fig. 2 .

 2 Fig. 2. Transmission spectra for the armchair edge incidence for the AB [(a)] and BA [(c)] cases in the center direction. The electric field distributions of the A 1 beams are shown in (b) and (d), with the thick dashed lines at the sample left side symbolizing the line sources, and the thin dashed lines at the right side the outgoing regions. The upper part insets show a magnified region in the vicinity of the impact zone, with the arrows indicating the incident beam axes. The lower part insets show the electric field magnitude along the long diagonal of the first unit cell, passing by the A and B tiles, in the upper insets.

Frontiers in Optics + Laser Science (FiO/LS) @ Optica Publishing Group 2022 Disclaimer: Preliminary paper, subject to publisher revision