Philippe Loubet
email: philippe.loubet@u-bordeaux.fr

Adrien Vincent
email: adrien.vincent@enseirb-matmeca.fr

Annabelle Collin
email: annabelle.collin@u-bordeaux.fr

Corinne Dejous
email: corinne.dejous@enseirb-matmeca.fr

Anthony Ghiotto
email: anthony.ghiotto@bordeaux-inp.fr

Christophe Jego
email: christophe.jego@enseirb-matmeca.fr

Life cycle assessment of ICT in higher education: a comparison between desktop and single-board computers

Purpose Information and communications technology (ICT) plays a key role in higher education in improving the teaching process. Consequently, the environmental impacts associated with ICT are increasing and innovative solutions must be deployed to reduce these impacts and increase students' awareness. Single-board computers (SBCs) are promising because they rely on less materials and energy than desktop computers (PCs). But additional servers are required to perform large-scale computations. Hence this paper aims at conducting comparative LCA between SBCs and PCs.

Materials and methods

The study is conducted in the context of a French engineering school with the following functional unit: "use 600 computers for 5 years in an engineering school". Two scenarios are defined to fulfil this functional unit. Scenario 1 is the use of 600 PCs (current infrastructure), and scenario 2 is the use of 600 SBCs combined with 6 servers (alternative infrastructure). The analysis includes the materials manufacturing, assembly, packaging, transport, use and end-of-life of each device. Life cycle inventory (LCI) of the foreground systems was generated using a variety of sources: disassembly of computers, counting of electronic components, datasheets, estimations, etc. LCI of the background systems is taken from ecoinvent 3.5. The selected life cycle impact assessment methodology is ReCiPe 2016 midpoint and computation of impacts is done with openLCA 1.10.3.

Results and discussion

Scenario 2 (SBCs + servers) generates 84% to 92% less impact than scenario 1 (PCs) in all categories. In terms of global warming, scenarios 1 and 2 generate 225 and 18 tCO2 eq per functional unit, respectively. This is explained by the large reduction in material and energy requirements for SBCs which is not counterbalanced by the servers. Equipment manufacturing accounts for the largest share of impacts in most categories for both scenarios (e.g., ~70% for global warming), followed by the use phase. This differs from the results found in the literature, as this study was conducted in the context of France, which has a low-carbon electricity mix.

Conclusions Our analysis has shown that SBCs combined with servers reduce the carbon footprint and other environmental impacts of ICT infrastructure for higher education. This study provides an example of low tech-oriented solution for students. Other prospective solutions (e.g., use of laptops) should be extensively studied in the future. From an LCA point of view, updating the inventory data related to background processes for electronic components is a necessary step forward to improve the certainty of the results.

Introduction

Information and communications technology (ICT) is one of the fastest growing sectors in terms of environmental impact. According to the recent estimates, the ICT sector is responsible for approximately 2.1% to 3.9% [START_REF] Freitag | The real climate and transformative impact of ICT: A critique of estimates, trends, and regulations[END_REF]) of global anthropogenic greenhouse gas (GHG) emissions worldwide. Indeed, the demands of the society regarding ICT are increasing in all sectors, which reinforces the need to develop new strategies to reduce the environmental footprint of ICT infrastructure. This is especially true for the education sector to raise awareness and because students will be able to apply such strategies in their professional careers.

In this context, single-board computers (SBCs) such as the Raspberry Pi are promising solutions to decrease the environmental impact of ICTs because they require fewer materials, less energy-intensive devices, and generate less waste at the end-of-life [START_REF] Johnston | Commodity single board computer clusters and their applications[END_REF]. Indeed, a Raspberry Pi is nothing more than a computer reduced to its simplest form: a single processor board. It can be connected to a dedicated server for very computationally intensive tasks. Therefore, it provides the same level of performance as a desktop computer. This solution has several additional advantages: (i) since the computing capacities required at the SBC level are small , the SBC can be an elementary equipment for which the maintenance requirements are reduced, (ii) if any improvement of the equipment must be carried out, it only affects the server, (iii) electricity consumption due to intensive computing activities being centralized, it is possible to reduce energy consumption for equivalent performance [START_REF] Baun | Mobile clusters of single board computers: an option for providing resources to student projects and researchers[END_REF]. Also, such a solution is particularly suitable for higher education, as it enables to have a homogeneous and harmonized configuration for all computer labs.

Quantifying the environmental impact of such a solution is important to: i) verify the environmental relevance compared to the current infrastructure in higher education (i.e., the use of desktop computers); ii) identify the main environmental hotspots to improve such strategies. LCA is the relevant tool to assess all environmental impacts related to the life cycle of computers, as it allows the consideration of all aspects of the supply chain, including the production of raw materials, the manufacturing of the computer, logistics, the use phase and the end-of-life. It also provides information on various impact categories that are relevant for the ICT sector: climate change, mineral resource scarcity, water use, toxicity, etc.

LCA has already been applied to analyse the environmental impacts of desktop computers (PCs). [START_REF] Teehan | Sources of Variation in Life Cycle Assessments of Desktop Computers[END_REF] and [START_REF] Yao | Comparative Assessment of Life Cycle Assessment Methods Used for Personal Computers[END_REF] reviewed several studies published between 1998 and 2010 and compared the global warming potential and primary energy consumption of the PCs studied. They found that the use phase dominates the impacts, except in areas with low electricity mix impacts. The manufacturing phase also has significant impacts, whereas distribution and end-of-life phases are always negligible. They also noticed a high variability in manufacturing impacts due to inconsistencies on the impacts of various electronic components, especially mainboards and semi-conductors. Also, Dell publicly provides on its website [START_REF] Dell | Understanding the uses and limitations of PAIA, a streamlined LCA methodology[END_REF]) the product carbon footprint of the computers, monitors and servers they produced, based on a methodology for streamlined LCA named "Product Attribute to Impact Algorithm-PAIA" [START_REF] Dell | Understanding the uses and limitations of PAIA, a streamlined LCA methodology[END_REF].

In terms of comparative LCA studies, [START_REF] Subramanian | Life cycle assessment study of an integrated desktop device -comparison of two information and communication technologies: Desktop computers versus all-in-ones[END_REF] studied two types of devices: PCs and all-in-ones. They determined that the later generates 2 times less environmental impact that the former ones. Also, [START_REF] Maga | Comparison of two ICT solutions: Desktop PC versus thin client computing[END_REF] compared server-based computing in association with thin clients and PCs. A 65% reduction in GHG emissions was found for the first solution. In addition, manufacturers such as Dell also provide information on their thin client (Chromebook or Wyse) and show lower carbon footprint in comparison with desktop computers. For example, the Dell Wyse 5740 Thin client's carbon footprint is estimated at 121 kgCO2 eq [START_REF] Dell | [END_REF]) whereas an average desktop computer's carbon footprint is rather around 500 kgCO2 eq. Thin client solution is similar to single-board computer but relies on heavier and more powerful devices. Based on this literature review, we can hypothesise that SBCs combined with servers are also relevant in terms of environmental performance compared to PCs. However, such quantification remains to be done, especially in the context of higher education, which also requires dedicated server.

In this context, we propose to conduct an LCA to compare the environmental impact of using PCs (current scenario) with that of using SBCs together with servers (alternative scenario) in order to verify the environmental relevance of such a solution and promote its use in higher education. The paper follows the four LCA phases according to the international standards (International Organisation for Standardization 2006): (i) the goal and scope and (ii) the life cycle inventory (LCI) described in Materials and methods, (iii) the life cycle impact assessment (LCIA) results and their (iv) interpretation provided in the Results section. Finally, the results are discussed with the literature, focusing on their potential use for decision makers.

Materials and methods

Goal and scope

The aim of the study is to investigate and compare the environmental impact of two scenarios for the ICT infrastructure for practical work in an engineer graduate school (ENSEIRB-MATMECA -Bordeaux INP). This study aims to provide quantitative evidence to help decision makers in selecting the most sustainable infrastructure that can help to mitigate impacts of ICT for higher education. It could also help to identify eco-design solutions to reduce the impact of the different devices studied (desktop and single-board computers).

ENSEIRB-MATMECA -Bordeaux INP is a French graduate school in electronics, computer science, telecommunications, and mathematics and mechanics, covering the field of digital technology in its broadest sense. The digital technology plays a central role in the major transformation underway in our society and is the answer to the main industrial challenges. This engineering school trains over 1 200 students in teaching programs focused on innovative projects supported by world-class research laboratories. To this end, 600 computers are required in the school. They enable the students to do all practical ICT work. Furthermore, the depreciation period for the computer equipment is 5 years, which is set under the rules of the budgetary and accounting management of a public institution. Therefore, the functional unit of the study is: "use 600 computers for 5 years in an engineering school". The use of computers includes the following tasks: use of office software, internet browsing, programming, basic scientific computation. The two scenarios studied (and associated reference flows) that satisfy the functional unit are given in that follows:

-1 st scenario: 600 desktop computers [Dell Precision Tower 3620 workstation (Dell 2018a)] -2 nd scenario: 600 single-board computers [Raspberry Pi 4 Rev B] + 6 servers [Dell 7920 Precision Tower workstation with a Dual Xeon 8168 processor [START_REF] Dell | Dell Precision 7820 Tower Owner's Manual[END_REF]].

In scenario 2, each server (and associated Dual Xeon processor) includes 48 cores, enabling 96 logic threads in parallel. This means that the 6 servers can perform the computationally intensive tasks in parallel, which is sufficient for the needs of practical work. As for the data transmission between the SBCs and the server, the Ethernet protocol allows a network configuration with a bandwidth of 1 Gbps. It means that each server could allow a theoretical bandwidth of 10Mbps for one SBC. In practice, limitations in the network infrastructure of a building may limit the useful bandwidth. However, it is possible to dedicate the bandwidth of a part of a network to a subnet that would be in charge of the data exchange between the SBC and the server [START_REF] Sommer | Ethernet -A survey on its fields of application[END_REF]. It means that the adequate configuration of the network infrastructure can ensure data rates of the order of 5 to 8 Mbps per SBC which is sufficient for the use of most digital tools.

System boundaries for both scenarios are shown in Figure 1 and Figure 2, respectively. The analysis includes the materials manufacturing, assembly, packaging, transport, use and endof-life of each device (desktop computer, single-board computer and server) that are specific to each system. External devices such as the monitors, the peripherals (mouse and keyboard) are excluded as they are similar for both scenarios.

Intermediate flows required for the foreground system are listed and explained in the life cycle inventory (LCI) section. Resource use and emissions related to the background processes were retrieved from the ecoinvent 3.5 database "Allocation at the point of substitution" [START_REF] Wernet | The ecoinvent database version 3 ( part I ): overview and methodology[END_REF]. The environmental impacts are then characterized using the ReCiPe 2016 Midpoint (H) [START_REF] Huijbregts | ReCiPe2016: a harmonised life cycle impact assessment method at midpoint and endpoint level[END_REF]) LCIA method (Table 1) selected as one of the most up-to-date methods. We used openLCA 1.10.3 to compute the impacts of each system.

Life cycle inventory

LCI data of the foreground system for the 2 scenarios is presented in the following sections related to: components, chassis, packaging, assembly, transport, end-of-life and use of the different devices. Devices refer to desktop computers for scenario 1, single-board computers and server for scenario 2. All data are summarized in Table 3, Table 4, Table 5 and Table 6, along with the sources of data and the associated ecoinvent processes that are used to model the production, use or end-of-life of the different elements. Full inventory tables are provided in supplementary information (SI), in Tables S1.1 to S1.7.

We consider that the production of components, chassis, packaging, and the assembly of the devices are taking place in China. Therefore, all ecoinvent datasets for these elements are considered with the geography "China" (CN) if available in ecoinvent, and "Rest of the World" (RoW) or global (GLO) if not available. Use and end-of-life phases are considered in France.

Transport is specifically considered only for the final devices (from the computer assembly plant in China to Bordeaux, France). All other transport steps are considered through the use of "market for" processes in ecoinvent.

2.2.1. Components of the devices Desktop computer. The different components have been identified from the manual (Dell 2018a) and after disassembly of a Dell tower 3620 (Table 4). Key components of the desktop computer (motherboard, GPU, RAM) have been modelled at a sub-component level by identifying the amount of printed circuit board and electronics. This is because these components are specific to the studied computer in terms of electronics used. We assume they represent a large share of impacts. The quantification of each electronic component (integrated circuit, capacitor, inductor, etc.) has been made through the use of the Python script DotDotGoose [START_REF] Ersts | DotDotGoose[END_REF]) that assists in counting objects in an image such as the picture of the motherboard (Figure 3). Mass of each sub-component has been estimated based on (i) specific information on the sub-component found on the internet, (ii) density of the metal used in the subcomponent, and (iii) rough estimation. Since there are many different sub-components (e.g., more than 1000 capacitors), it was not possible to identify the unitary mass of each element. From this quantification, the total mass of the motherboard is evaluated to 700 g which is consistent with the real mass of the board. Then the manufacturing of each sub-component is directly taken from ecoinvent datasets. It should be noted that ecoinvent datasets related to electronic components are outdated as they were developed more than 15 years ago [START_REF] Hischier | Life cycle inventories of Electric and Electronic Equipment: Production, Use and Disposal[END_REF]). However, we decided to rely only on ecoinvent background datasets for consistency and because we conduct comparative LCA on systems that rely on the same types of electronic components.

List of all sub-components of the motherboard, GPU, RAM is available in Table 3. Other components (hard disk drive, disk drive (CD/DVD), power supply unit) are directly taken from ecoinvent as they are generic components that do not differ significantly between PCs. SBC The Raspberry Pi 4 is only composed of a single board which is rather simple as shown in Figure 4. The quantification of each sub-component has been made from a comprehensive list that is available for the Raspberry Pi 2 Model B1 . This model is similar to the Pi 4, for which some elements such as the connectors have been improved. Mass of each sub-component has been estimated based on the same approach used for the PC. From this quantification, the total mass of the motherboard is evaluated to 48 g which agrees with the documentation. The list of sub-components is given in Table 5. Server The different components of the Dell 7920 server are retrieved from a server already installed at ENSEIRB-MATMECA and are described in Table 6. The 2 motherboards used in each server are considered equivalent as the ones used in the PCs, except for the processors. The processor has been modelled according to their area, and considered as a combination of wafer and integrated circuit (logic type) from ecoinvent. RAM and GPU have also been considered equivalent as the ones used in PCs, except that a server contains 8 units of RAM and 2 GPUs. The manufacture of other components is directly taken from ecoinvent.

Chassis of the devices

For each device (PC, SBC or server), we model a chassis that we believe is representative of what could be encountered in practice for each scenario. As detailed information about chassis are often not so easily available, we build our models by means of educated guesses and of values averaged over a few models of the relevant subsystems when possible. Due to limited information about their exact sub-components and for the sake of simplicity, we only account for the mass of plastic, when modelling fans as it is the main material by far. More information on the different chassis models and their subcomponents are given in SI.

Desktop computer

The chassis of the desktop computer is composed of the following elements: 5 kg of steel (mostly for the enclosure) with powder coating (for protection against corrosion and for aesthetic purpose); 150 g of plastic (for decorative elements, component positioning and possibly air blower); one 120 mm diameter fan; an air-cooling system for the Central Processing Unit (CPU), with copper heat pipes leading to an aluminium heatsink and a 120 mm diameter fan; some ribbon cable for internal wiring. For simplicity sake, we also account for the 1.8 m power cable and the 3 m network cable in the chassis subsystem for the desktop computer. Details about the model are given in Table 3.

The mass of the enclosure is estimated based on an average of models of chassis on the market at the time of the study (Table S2.1 in SI), slightly beyond 5 kg, which includes neither the CPU air-cooling system nor the power and network cables. The data for each chassis sample have been retrieved from the manufacturer data sheet or documentation.

Similarly, the CPU air-cooling system is an average model (Table S1.7 in SI) based on a small set of systems on the market at the time of the study (Tables S2.5 andS2.6 in SI). The data for each sample has been retrieved from the manufacturer data sheet or documentation.

SBC

We have considered the scenario of a custom 3D-printed enclosure. We have chosen acrylonitrile-butadiene-styrene (ABS) for its overall good mechanical properties and durability. The total mass of ABS is estimated to be 50 g per enclosure considering the expected overall geometry. Relying on educated guesses, we consider a three-hour-long printing process per enclosure with a 3D-printer (with a power of 125 W). Because of the low power consumption of such a thin client approach, we assume that the chassis does not need any fan or heat sink for active air-cooling. For simplicity sake, we also account for the 3 m network cable and for the external 15 W power supply unit in the SBC chassis subsystem. Details about the model are given in Table 4.

The significant amount of enclosures to print is likely to require an unreasonable amount of time for in-house production with only one or a few 3D-printers. For example, based on the previously estimated printing duration, it would take at least 2,5 months to produce 600 units with a single 3D-printer. That is why in this work we have considered a scenario, where the SBC enclosures are printed by a third-party service in China equipped with a farm of 3Dprinters.

Server

We use an approach similar to the desktop computer chassis for the chassis of the server. It contains: 13 kg of steel (mostly for the enclosure); 350 g of plastic (for decorative elements, components positioning and possibly air blower); three 140 mm diameter fans; an air-cooling system for the CPU, with copper heat pipes, an aluminium heat sink and two 140mm diameter fans; some ribbon cable for internal wiring. For simplicity sake, we also account for the 1.8 m power cable and the 3 m network cable in the chassis subsystem for the desktop computer. Details about the model are given in Table 5.

The mass of the enclosure is estimated based on an average of models of chassis on the market at the time of the study (Table S2.2 in SI), slightly beyond 13 kg, which includes neither the CPU air-cooling system nor the power and network cables. The data for each chassis sample has been retrieved from the manufacturer data sheet or documentation. Without accounting for the fans, we have kept the same ratio steel mass over plastic mass as for the chassis of the desktop computer.

Similarly, the breakdown of the CPU air-cooling system is an average model (Table S1.7 in SI) based on a small set of systems on the market at the time of the study (Table S2.5 in SI). The data for each sample has been retrieved from the manufacturer data sheet or documentation.

Assembly of the devices

We assume that each device is assembled in China. Energy, tap water and waste water treatment required for assembly have been directly taken from the generic ecoinvent process "computer production, desktop, without screen" considering a ratio based on the mass of each device and the mass of the computer modelled [START_REF] Hischier | Life cycle inventories of Electric and Electronic Equipment: Production, Use and Disposal[END_REF]).

Packaging of the devices

The packaging is considered similar for the desktop computer and the server. It is composed of corrugated board box and foamed polypropylene. The packaging of the Raspberry Pi is only composed of corrugated board box. The different masses of packaging materials have been roughly estimated and are available in the LCI tables.

Transport of the devices

For each device, we considered the same distance for transport: 1000 km of truck in China, 15000 km of transoceanic ship between China and France, and 1000 km of truck in France.

End-of-life of the devices

The end-of-life of the devices is considered as the same of the ecoinvent process "market for used desktop computer" that is defined in relation to the mass of the device. We kept the same ecoinvent end-of-life treatment processes that consider that 77 % of the device is treated mechanically and 23 % is manually dismantled, based on global average. Then, the different parts of the computer (metals, electronics, plastics, etc.) are considered either incinerated, landfilled or recycled.

Use of the devices

The energy (electricity) use of the devices has been estimated based on the state-of-the-art values computed according to EU regulations with regard to ecodesign requirements for computers and computer servers (European Commission 2013) 2.2.8. Sensitivity analysis In the baseline scenario 2, we considered that 6 servers are sufficient for 600 SBCs, since each server has 96 logic cores and the Ethernet protocol enables 5 to 8 Mbps per SBC which is sufficient for the use of most digital tools. However, it can be necessary to have two supplementary servers to protect against lack of services. Furthermore, we considered in the baseline scenarios that SBCs have the same lifetime than PCs (5 years). However, lifetime of SBC in the context of higher education is uncertain and we propose to study the sensitivity of the results to a conservative assumption: lifetime of 2.5 years for SBCs. Therefore, we considered a worst-case "scenario 2" with the following assumptions: manufacture of 8 servers instead of 6; manufacture of 1200 Raspberry Pi instead of 600, in order fulfil the functional unit.

We also conducted a sensitivity analysis on the electricity mix for the use phase since SBCs could be used in higher education worldwide. We selected the European mix (446 gCO2 eq/kWh) and the Polish mix (1099 gCO2 eq) which are more carbon intensive than the French one (58g CO2 eq). For these additional analyses, we considered exactly the same supply chain and end-of-life for the devices.

Results

The LCIA results of scenario 1 and scenario 2 are both presented in sections 3.1 and 3.2, respectively. Then, they are compared in section 3.3.

Life cycle impact assessment (LCIA) of scenario 1 -desktop computers

For the scenario 1, Figure 5 shows that the majority of the impacts are generated by manufacturing PCs, representing more than 50% of the impacts in 15 out of the 17 categories.

The use phase only dominates the ionizing radiation (92%) category. It also generates large share of water consumption (39%), ozone depletion (25%) and marine eutrophication (21%) impacts. Actually, the electricity from France mostly comes from nuclear energy. It generates radioactive emissions (= ionizing radiation), evaporates water for cooling (= water consumption), and relies on uranium extraction that leads to nitrate leakages to rivers and sea (= marine eutrophication). Ozone depletion is mainly due to the dinitrogen monoxide generated from the ionisation of air molecules from electro-magnetic field near high-voltage aerial lines.

The manufacture of components dominates the impacts of PC fabrication. This is mainly due to electronic components that have complex supply chains requiring many resources (metals, chemicals, etc.) and are energy intensive. Along their supply chain, the energy-related activities such as coal extraction and use in Chinese thermo-electric plants generate most of CO2 (= global warming), NOx (= ozone formation), and fine particulate formation, ultimately leading to air quality degradation as well as the of fossil resources. The manufacture of electronic components also dominates all toxicity impact categories, freshwater eutrophication, terrestrial acidification, land use and mineral resource scarcity because of mining activities. Such activities emit toxic forms of metals (Chromium VI, Nickel, Lead, Zinc Cadmium), eutrophying substances (Phosphates) and acid substances (sulfur dioxide). They occupy large amounts of land (e.g., for the treatment of sulfidic tailing) and participate to the depletion of metals.

Figure 6 shows that the motherboard is the component that represents the largest share of impact (>50% of the global warming, other impact categories showing similar contributions), even though the power supply unit is the heaviest component in mass. At the sub-component level, Figure 6 also shows that the production of integrated circuits (including the processor) dominates the impacts of the motherboard.

The PC chassis represent 4 to 20% of the total impacts (except for ionizing radiation), with the highest contribution in mineral resource scarcity (9%), terrestrial ecotoxicity (17%) and human carcinogenic toxicity (20%). This is because of the requirements of metals (steel, aluminium, copper) for the chassis and the CPU heat sink. Metals usually have high contribution in toxicity categories because of their high persistence in the environment (including after mining activities).

The PCs' assembly, packaging, transport and end-of-life have low contribution in all categories.

Figure 5. Contribution analysis of scenario 1 -600 desktop computers (ReCiPe2016 Midpoint H). Abbreviation definitions are given in Table 1.

LCIA of scenario 2 -SBCs and servers

Regarding the relative contribution of the different life cycle phases, the analyses drawn for scenario 1 are also valid for scenario 2: the manufacture and the use of devices (SBCs and servers) have similar contributions in all impact categories as for PCs (as shown in Figure 7). The relative share of impacts between components, chassis, assembly packaging and transport for SBCs and servers are not shown for simplicity but are similar to scenario 1. Therefore, the analysis of the results in section 3.1 is also valid here. A complete contribution analysis is available in SI (Table S3.3).

One can note that the SBCs and the servers represent on average 60% and 40% of the total impacts respectively, even though the scenario relies on 600 SBCs and 6 servers. This is explained by the low material requirements and energy consumption of the Raspberry Pi compared to the server (Figure 10) Table 1.

Comparison between both scenarios

Figure 8 shows that scenario 2 generates 84% to 92% fewer impacts than scenario 1 for all categories. This is due to the low material requirements and electricity consumption of SBCs compared to PCs. The servers do not generate burden shifting because only 6 are required in scenario 2 and their impacts do not counterbalance the benefits of the SBCs. One can note that the impact reduction is similar in all categories because scenario 2 is more eco-efficient both on the manufacturing and the use phases.

Regarding global warming absolute impacts, scenarios 1 and 2 generate 194.5 and 16.5 tCO2 eq/functional unit (5 years), respectively. On a yearly basis, this is equivalent to 39 tCO2 eq/year or the GHG emissions of 4 French citizens/year for scenario 1, and 3.3 tCO2 eq/year or the GHG emission of 0.3 French citizen/year for scenario 2 (assuming that a French citizen is responsible for 10 tCO2 eq/year). Regarding water consumption, scenario 1 relies on 2772 m 3 /functional unit whereas scenario 2 only requires 307 m 3 /functional unit. 1.

As computers rely on many mineral resources that are increasingly scarce, it is also relevant to analyse and compare the different metals required for both scenarios. In terms of mass, scenario 1 requires 9394 kg of metal, whereas scenario 2 only requires 629 kg of metal during the whole life cycle according to the inventory analysis. Iron is the most used in both scenarios (Figure 9). Gold is the most contributing metal to mineral resource scarcity when characterized with the "surplus ore potential" of the ReCiPe2016 method. Gold is mainly used as electroplated coating on connectors and solderable coating for printed circuit boards. Other specific scarce metals used in computers that rank high in mineral resources scarcity are: silver mainly used in printed circuit boards and tantalum mainly used in capacitors. As shown in Figure 10, it is also possible to compare all three devices separately. A SBC only generates 17kg CO2 eq over its life cycle, whereas a desktop computer generates 324 kg CO2 eq and a server generates 1013 kg CO2 eq.

Figure 10. Global warming impact for the life cycle of the three devices over their lifetime.

Sensitivity analysis

Figure 11 shows the global warming impact of a worst-case scenario 2. It considers the manufacture of 1200 SBCs + 8 servers. This scenario increases the impact of device manufacturing from 13.5 tCO2 eq to 23.7 tCO2 eq compared to the best-case scenario presented previously. However, it is still largely beneficial compared to scenario 1. It means that scenario 2 always generates lower impacts than scenario 1, even if the life time of the SBC is reduced and the server's requirement is higher than expected.

Figure 11 also shows that the use phase has the largest share of impacts in regions where the electricity mix is carbon intensive. However, scenario 2 that uses less energy, is still beneficial in these countries. Dell company provides product carbon footprint of most of their computers, based on a streamlined LCA tool [START_REF] Olivetti | A Product Attribute to Impact Algorithm to Streamline IT Carbon Footprinting[END_REF]. The Dell Precision tower 3620 carbon footprint computed with their approach results in 642 kgCO2 eq over its life cycle [START_REF] Dell | Dell Precision 7820 Tower Owner's Manual[END_REF], as shown in Figure 10. This differs from our estimation (324 kgCO2 eq) mainly because of the use phase (296 kgCO2 eq against 30 kgCO2 eq). Actually, Dell company has considered different electricity mix (more carbon intensive than the French one). As for the other life cycle stages, the results are similar. The slight differences might come from: different LCI background database used in PAIA, and different assumptions mainly for the transportation.

Also, it is to be noted that the precision tower 3620 is a workstation (i.e., a powerful desktop computer). The carbon footprint of such device is higher than middle-range desktop computers, such as the OptiPlex series, which generate 10-15% less GHG emissions than the Precision series [START_REF] Dell | Understanding the uses and limitations of PAIA, a streamlined LCA methodology[END_REF].

Compared with other types of ICT miniaturization, the literature has shown that all-in-one computers [START_REF] Subramanian | Life cycle assessment study of an integrated desktop device -comparison of two information and communication technologies: Desktop computers versus all-in-ones[END_REF] and thin clients associated with servers [START_REF] Maga | Comparison of two ICT solutions: Desktop PC versus thin client computing[END_REF]) enable to decrease the PCs' global warming impact by 50% and 65%, respectively. Due to its low power and mass, SBCs enable an even higher reduction (i.e., 90%).

Comparison of the results with other potential scenarios

Another option for ICT infrastructure in higher education is the exclusive use of laptops, where each student (1200) gets a laptop from the school instead of using 600 desktop computers. This scenario is currently not envisaged in the studied school in the short term as it would require an important change in the organization of practical work. However, this kind of infrastructure might gain interest in the mid or long-term due to increased flexibility and simplicity in the frame of dematerialization of higher education. Therefore, we propose a simplified comparison between a laptop scenario (S3), S1 and S2 (which are both extended to take into account monitors and peripherals). We gathered carbon footprint data of laptops, monitors and peripherals (assuming a 5-year lifetime) that were available in the literature [START_REF] Dell | Understanding the uses and limitations of PAIA, a streamlined LCA methodology[END_REF] to compare the following simplified scenarios:

-S1_extended: S1 + 600 monitors (mix of Dell E2216H/E2219H), mouses, keyboards -S2_extended: S2 + 600 monitors, mouses, keyboards -S3: 1200 laptops (Latitude 5400 which is the middle range of Dell laptop in term of performance and carbon footprint)

Results are shown in Figure 12 and exhaustive information on this comparison is available in SI.

Figure 12.Simplfied and prelimary comparison of carbon footprint between extended S1 and S2 (considering monitors and peripherials) and S3-laptop

We found out that the laptop scenario is in-between extended S1 and S2. Therefore, it is a potential solution to decrease the impacts of ICT in higher education. However, this preliminary result should be analysed with caution as it requires additional and crucial information on: the usage pattern of the laptop outside of the school, the lifetime of the laptop, accounting for the fact that providing laptop computer to students might influence their purchase of personal laptop, etc. Also, it requires full LCA computation for consistency and completeness sake.

Data limitations

Ecoinvent datasets related to electronic components are outdated as they were collected in the 2000's [START_REF] Hischier | Life cycle inventories of Electric and Electronic Equipment: Production, Use and Disposal[END_REF]. Since both scenarios rely on the same types of components, we assume that this would not change the relative comparison between scenario 1 and scenario 2, but the absolute results shown in this study should be used with caution. An update of such datasets is urgently needed to identify appropriate eco-design solutions for the ICT sector. It should be noted that the EF Database v2.0 contains more up-to-date datasets on electronics. However, these datasets are based on the thinkstep database2 , which provides black box LCIs. They may not have been created using the same approach as the ecoinvent datasets.

Moreover, the quantification of electricity use of the different devices was theoretically estimated based on the power and usage pattern of the different devices. It is necessary to monitor the energy consumption in the practical workspaces to obtain accurate data. In this way, appropriate strategies for reducing energy consumption can be identified. Such quantification is planned for the future. The additional energy consumption for air conditioning in classrooms and server rooms has not been included and should be monitored in further studies.

Regarding end-of-life, we used generic assumptions based on ecoinvent datasets, which may not be representative of the real situation of waste electrical and electronic equipment (WEEE) management in France. As strategies to increase reuse and recycling rates are currently being implemented (European Commission 2012), it is necessary to better describe this phase in further studies. In any case, the fact that SBCs are light and simple devices should ease their management at the end-of-life.

Implications for decision making

Our study clearly shows that the use of single-board computers can drastically reduce the environmental impact of ICT in universities, even for the worst-case scenario 2. SBCs are also cost-effective due to the low price of the Raspberry Pi 4 Rev B. We evaluated the financial costs of the different options for scenario 1 and 2 (see Tables S6 in SI) considering the investment and the electricity costs. Scenario 1 (PCs) ranges from 636 k€ to 936 k€ whereas scenario 2 (SBCs + servers) ranges from 248 k€ to 382 k€.

However, such a solution is still not deployed. It could represent a poorer quality of service and a less robust system than using PCs. In the short term, our school would like to deploy single-board computers in two classrooms to ensure that all labs in the school are compatible with the proposed infrastructure. Moreover, other single-board computers will be investigated. For example, there are now single-board computers that are an open-source RISC-V alternative to the Raspberry Pi. RISC-V is a 64-bit RISC instruction set architecture (ISA) that is open and free, i.e. it has open specifications and can be freely used by education, research and industry. In the long term, such a solution could also be deployed in the 5 other graduate schools of Bordeaux INP. It should be noted that the current case study was proposed in a simplified exercise to introduce LCA to all students of ENSEIRB-MATMECA, as part of a new class on sustainable development. This was an important achievement. Indeed, it enables to introduce the topic to more than 350 students per year, and to train more than 10 professors in LCA and openLCA.

Finally, considering the competitive cost and lower environmental impact of SBCs, they are a promising solution for a better access to ICT in developing countries. Our analysis has shown that in carbon-intensive countries, the use phase is the largest contributor to global warming impacts, but also that SBCs are still the best solution in terms of environmental impact.

Conclusions

Our study has shown that SBCs combined with servers provide several benefits to the higher education by (i) reducing the carbon footprint and other environmental impacts of public services, (ii) providing an example of improving sobriety and low-tech solutions for students, and (iii) it is cost-effective. The use of this innovative solution generates 90% less GHG emission than the use of PCs. Due to the low material requirements and energy efficient equipment, all other environmental impacts are also reduced.

Manufacture of devices represents the majority of impacts in both scenarios, which argues for the use of less material-intensive devices, and a longer lifetime. It should be noted that the electronics LCI models should be updated in the future to improve the representativeness of the data, as this industry is rapidly evolving. The use phase, which normally accounts for the largest share of ICT impacts, is only second in terms of impacts due to the low-carbon French electricity mix. With the European electricity mix, the use phase dominates the impact. However, SBCs connected to servers still drastically reduce the impact due to their low energy consumption.

Such a solution will be deployed in two classrooms at the engineering school in question to test the reliability of SBCs, even if there are still bottlenecks to full deployment, mainly related to the capacity to change from teachers and ICT services. Other promising option, such as providing a laptop for each student, should be extensively studied in the future From an LCA point of view, there are challenges to update and improve the data quality of electronic components since (i) the current data is outdated and (ii) these are the highest contributors to ICT devices' environmental impacts.

Figure 1 .

 1 Figure 1. System boundaries for the scenario 1 (600 desktop computers)

Figure 3 .

 3 Figure 3. Identification of each component of the Dell computer's motherboard with DotDotGoose.

Figure 4 .

 4 Figure 4. Picture of the Raspberry Pi 4 Rev B

Figure 6 .

 6 Figure 6. Global warming impacts for one Dell computer over its 5 years' life time for (i) the full life cycle, (ii) the different components, and (iii) the different subcomponents of the motherboard.

Figure 7 .

 7 Figure 7. Contribution analysis of scenario 2 -600 SBCs + 6 servers (ReCiPe2016 Midpoint H). Abbreviation definitions are given inTable 1.

Figure 8 .

 8 Figure 8. Relative comparison (in %) of scenario 1 and 2 environmental impacts. Absolute impacts for all impact categories are also shown. (ReCiPe2016 Midpoint H). Abbreviation definitions are given inTable 1.

Figure 9 .

 9 Figure 9. Metal requirements and mineral resource scarcity impact of both scenarios.

Figure 11 .

 11 Figure 11. Sensitivity analysis for global warming results considering worst-case and best-case scenario 2 and different countries of use (ReCiPe2016 H Midpoint)

Table 1 .

 1 ReCiPe 2016 midpoint (H) impact categories and list of abbreviations

	Impact category	Abbreviations	Unit
	Global warming	GW	kg CO2 eq
	Stratospheric ozone depletion	SOD	kg CFC11 eq
	Ionizing radiation	IR	kg Co-60 eq
	Ozone formation, Human health	OF	kg NOx eq
	Fine particulate matter formation	FPMF	kg PM2.5 eq
	Terrestrial acidification	TA	kg SO2 eq
	Freshwater eutrophication	FEut	kg P eq
	Marine eutrophication	MEut	kg N-Eq
	Terrestrial ecotoxicity	TE	kg 1,4-DCB eq
	Freshwater ecotoxicity	FE	kg 1,4-DCB eq
	Marine ecotoxicity	ME	kg 1,4-DCB eq
	Human carcinogenic toxicity	HCT	kg 1,4-DCB eq
	Human non-carcinogenic toxicity	HNCT	kg 1,4-DCB eq
	Land use	LU	m²a crop eq
	Mineral resource scarcity	MRS	kg Cu eq
	Fossil resource scarcity	FRS	kg oil eq
	Water consumption	WC	m 3

Table 2 .

 2 Psleep and Poff (kW) being determined from Energy Star website or directly in the manufacturer's datasheets for the Dell workstations. Raspberry Pi 4 datasheet does not include regulatory energy use value according to European Commission (2013). Therefore, we estimated Pidle Psleep and Poff of SBCs according to values found on the internet. Power (W) and energy (kWh) for both scenarios are reported in Table 2. Usage pattern, power of devices, and energy use of each scenario

	. In this framework, yearly

Table 3 .

 3 Inventory data for the key components (Motherboard, GPU, RAM) and their associated ecoinvent 3.5. processes

	Components Elements	Quantity	Unit	Source of data	ecoinvent processes
		Processor		6.78 g		market for integrated circuit, logic type | integrated circuit, logic type | GLO
		Wafer (8*8mm area)		64 mm2		market for wafer, fabricated, for integrated circuit | GLO
		42*electrolytic capacitors		54.18 g		market for capacitor, electrolyte type, < 2cm height | GLO
		1064*SMD (mainly capacitors)	91.504 g		market for capacitor, for surface-mounting | GLO
	Motherboard	9*inductors 137* integrated circuits		17 g 50 g		market for inductor, low value multilayer chip |GLO market for integrated circuit, logic type | GLO
		1*PCB (259*259mm area)	67081 mm2		market for printed wiring board, for surface mounting, Pb free surface | GLO
		3*PCIE + 7*RAM slots		81.2 g	Quantification of the number/area of the	market for mounting, surface mount technology, Pb-free solder | GLO market for electric connector, peripheral component interconnect buss | GLO
		26*remaining connectors		182 g	elements: Dotdotgoose	market for electric connector, peripheral type buss | GLO
		4*electrolytic capacitors 258*SMD, mainly capacitors		5.16 g 22.18 g	Unitary mass: rough estimation based on	market for capacitor, electrolyte type, < 2cm height | GLO market for capacitor, for surface-mounting | GLO
	GPU	2*inductors 24*ICs 4*memory ICs		3.77 g 10 g 8 g	documentation and size of sub-components (see SI)	market for inductor, low value multilayer chip | GLO market for integrated circuit, logic type | GLO market for integrated circuit, memory type | GLO
		1*die of approx. 9mm*9mm area		81 mm2		market for wafer, fabricated, for integrated circuit | GLO
		1*PCB of 152*68mm area	10336 mm2		market for printed wiring board, for surface mounting, Pb free surface | GLO
						market for mounting, surface mount technology, Pb-free solder | GLO
		3*connectors		20 g		market for electric connector, peripheral type buss | GLO
	RAM	1*Integrated circuit		8 g		market for integrated circuit, memory type | GLO
		1*printed circuit board	4000.5 mm2		market for printed wiring board, for surface mounting, Pb free surface | GLO
						market for mounting, surface mount technology, Pb-free solder | GLO

Table 4 .

 4 Inventory data for the Dell 3620 desktop computer life cycle, and their associated ecoinvent 3.5 processes

	Dell 3620	Elements	Quantity Unit	Source of data	ecoinvent processes
		Motherboard	1 unit		see Table 3
		GPU	1 unit		see Table 3
	Components	RAM HDD	1 unit 1 unit	Computer disassembly and Dell Factsheet	see Table 3 market for hard disk drive, for desktop computer | GLO
		Disk drive	1 unit		market for disk drive, CD/DVD, ROM, for desktop computer | GLO
		Power supply unit	1 unit		market for power supply unit, for desktop computer | GLO
		steel, low-alloyed, hot rolled	5 kg		market for steel, low-alloyed, hot rolled | APOS, U -GLO
		sheet rolling, steel	5 kg		market for sheet rolling, steel | APOS, U -GLO
		powder coat, steel	0.53 m2		market for powder coat, steel | APOS, U -GLO
		acrylonitrile-butadiene-styrene copolymer	0.15 kg		market for acrylonitrile-butadiene-styrene copolymer | APOS, U -GLO
		stretch blow moulding	0.15 kg	educated guesses and	market for stretch blow moulding | APOS, U -GLO
	Chassis	cable, ribbon cable, 20-pin, with plugs cable, connector for computer, without plugs	0.05 kg 1.80 m	of values averaged over a few models of the relevant subsystems	market for cable, ribbon cable, 20-pin, with plugs | APOS, U -GLO market for cable, connector for computer, without plugs | APOS, U -GLO
		plug, inlet and outlet, for computer cable	1 unit	when possible	market for plug, inlet and outlet, for computer cable | APOS, U -GLO
		cable, network cable, category 5, without plugs	3 m		market for cable, network cable, category 5, without plugs | APOS, U -GLO
		plug, inlet and outlet, for network cable	1 unit		market for plug, inlet and outlet, for network cable | APOS, U -GLO
		Desktop 120 mm air-cooling system	1 unit		See Table S1.7
		120 mm diameter fan	1 unit		See Table S1.6
	Assembly	Electricity Production of tap water Treatment of waste water	2.77 kWh 1.60 m3 1.60 m3	ecoinvent report: proxy based on mass	market group for electricity, medium voltage | CN market for tap water | RoW market for wastewater, unpolluted | RoW
		Corrugated board box	2.20 kg		market for corrugated board box | RoW
	Packaging	Polypropylene	0.16 kg	Estimation	market for polypropylene, granulate | GLO
		Polymer foaming	0.16 kg		market for polymer foaming | GLO
		Transport, lorry	t⋅km	estimation: 1000km in
			11*2000	France + 1000km in	transport, freight, lorry 16-32 metric ton, EURO5 | RoW
	Transport	Transport, ship	t⋅km	China estimation: distance
			11*15000	between China and	transport, freight, sea, transoceanic ship | GLO
				France
	Use	Electricity, FR	505.43 kWh	see section 2.2.7	market for electricity, low voltage | FR
		End-of-life of the computer	8.45 kg	mass of device	market for used desktop computer | GLO
	End-of-life	End-of-life packaging	2.20 kg	mass of packaging	market for waste packaging paper | FR
		End-of-life packaging	0.16 kg	mass of packaging	market for waste polypropylene | FR

Table 5 .

 5 Inventory data for the Raspberry Pi 4 single-board computer life cycle, and their associated ecoinvent 3.5 processes

	Raspberry Pi	Elements	Quantity	Unit	Source of data	ecoinvent processes
	4 Rev B
		95*capacitors	2.150 g		market for capacitor, for surface-mounting | GLO
		10*diode, glass-, for surface-mounting	0.128 g		market for diode, glass-, for surface-mounting | APOS, U -GLO
		9*electric connector, peripheral type buss 8*inductor, low value multilayer chip	20.00 g 0.013 g	Quantification of the number/area of the elements:	market for electric connector, peripheral type buss | APOS, U -GLO market for inductor, low value multilayer chip | APOS, U -GLO
		1*integrated circuit, logic type (processor)	5.000 g	(1)	market for integrated circuit, logic type | APOS, U -GLO
	Components	3*integrated circuit, memory type 5*light emitting diode 39*resistor, surface-mounted 4*transistor, surface-mounted	3.000 g 0.700 g 0.294 g 1.779 g	Unitary mass: rough estimation based on documentation and size of sub-components (see SI)	market for integrated circuit, memory type | APOS, U -GLO market for light emitting diode | APOS, U -GLO resistor production, surface-mounted | APOS, U -GLO market for transistor, surface-mounted | APOS, U -GLO
			4760			market for printed wiring board, for surface mounting, Pb containing
		1*PCB, Pb containing surface		mm2		surface | GLO
						mounting, surface mount technology, Pb-free solder | GLO
		acrylonitrile-butadiene-styrene copolymer	0.05 kg		market for acrylonitrile-butadiene-styrene copolymer | GLO
	Chassis	electricity, low voltage cable, network cable, category 5, without plugs plug, inlet and outlet, for network cable	0.375 kWh 3 m 1 unit	educated guesses and of values averaged over a few models of the relevant subsystems when possible	market group for electricity, low voltage | CN market for cable, network cable, category 5, without plugs | GLO market for plug, inlet and outlet, for network cable | GLO
		External 15 W power supply	1 unit		See Table S1.5
	Assembly	Electricity Production of tap water Treatment of waste water	0.0165 kWh 9.6 L 9.6 L	ecoinvent report: proxy based on mass	market group for electricity, medium voltage | CN market for tap water | RoW market for wastewater, unpolluted | RoW
	Packaging	Corrugated board box	0.02 kg	Estimation	market for corrugated board box | RoW
		Transport, lorry		kg⋅km	estimation: 1000km in	transport, freight, lorry 16-32 metric ton, EURO5 | RoW
	Transport	Transport, ship	0.068*2000	kg⋅km	France + 1000km in China estimation: distance between	transport, freight, sea, transoceanic ship | GLO
			0.068*15000		China and France
	Use	Electricity, FR	48.62 kWh	see section 2.2.7	market for electricity, low voltage | FR
	End-of-life	End-of-life of the computer End-of-life packaging	0.048 kg 0.02 kg	mass of the device mass of the packaging	market for used desktop computer | GLO market for waste packaging paper | FR
	(1) https://elinux.org/RPi_Partial_BOM_Rev2.0_ModelB

Table 6 .

 6 Inventory data for the server life cycle, and their associated ecoinvent 3.5 processes

	Server Dell 7920	Elements	Quantity	Unit	Source of data	ecoinvent processes
		Motherboard		2 units		see Table 1
		GPU		2 units		see Table 1
	Components	2*Processors (2* 36.4mm*29.4mm) RAM 2*SSD		0.118 kg 8 units 40.0 g	Computer disassembly and Dell Factsheet	market for integrated circuit, logic type | GLO see Table 1 market for integrated circuit, logic type | GLO
		3*HDD		3 units		market for hard disk drive, for desktop computer | GLO
		Power supply unit		2.12 unit		market for power supply unit, for desktop computer | GLO
		steel, low-alloyed, hot rolled		13 kg		market for steel, low-alloyed, hot rolled | GLO
		sheet rolling, steel		13 kg		market for sheet rolling, steel | GLO
		powder coat, steel		0.936 m2		market for powder coat, steel | GLO
		acrylonitrile-butadiene-styrene		0.35 kg		market for acrylonitrile-butadiene-styrene copolymer | GLO
		copolymer
		stretch blow moulding		0.35 kg	educated guesses and	market for stretch blow moulding | GLO
		cable, ribbon cable, 20-pin, with plugs		0.13 kg	of values averaged over	market for cable, ribbon cable, 20-pin, with plugs | GLO
	Chassis	cable, connector for computer, without		1.8 m	a few models of the	market for cable, connector for computer, without plugs | GLO
		plugs			relevant subsystems
		plug, inlet and outlet, for computer cable		1 unit	when possible	market for plug, inlet and outlet, for computer cable | GLO
		cable, network cable, category 5,		3 m		market for cable, network cable, category 5, without plugs | GLO
		without plugs
		plug, inlet and outlet, for network cable		1 unit		market for plug, inlet and outlet, for network cable | GLO
		Server 140 mm air-cooling system		1 unit		See Table S1.7
		140 mm diameter fan		3 units		See Table S1.6
	Assembly	Electricity Production of tap water Treatment of waste water		10 kWh 5 m3 5 m3	ecoinvent report: proxy based on mass	market group for electricity, medium voltage | CN market for tap water | RoW market for wastewater, unpolluted | RoW
		Corrugated board box		2.2 kg		market for corrugated board box | RoW
	Packaging	Polypropylene		0.16 kg	Estimation	market for polypropylene, granulate | GLO
		Polymer foaming		0.16 kg		market for polymer foaming | GLO
			30*2000 kg⋅km	estimation: 1000km in	transport, freight, lorry 16-32 metric ton, EURO5 | RoW
		Transport, lorry			France + 1000km in
	Transport		30*15000 kg⋅km	China estimation: distance	transport, freight, sea, transoceanic ship | GLO
		Transport, ship			between China and
					France
	Use	Electricity, FR	3560.72 kWh	see section 2.2.7	market for electricity, low voltage | FR
		End-of-life of the computer		8.45 kg	mass of device	market for used desktop computer | GLO
	End-of-life	End-of-life packaging		2.2 kg	mass of packaging	market for waste packaging paper | FR
		End-of-life packaging		0.16 kg	mass of packaging	market for waste polypropylene | FR

https://elinux.org/RPi_Partial_BOM_Rev2.0_ModelB).

https://gabi.sphera.com/databases/ef-database-v20/

Acknowledgements

The authors acknowledge Erwan DE LAVERGNE DE CERVAL, Aurélien HOUE, Léo PAJOT and Filipe POUGET, students at the ENSEIRB-MATMECA -Bordeaux INP graduate school in 2020-21. They designed a first version of this case study and collected the list of electronic components for both scenarios.

Data availability All data generated or analysed during this study are included in this article and its supplementary information files.

Supplementary information

The online version contains supplementary information in a spreadsheet containing 6 tabs related to LCI tables (S1), information on chassis (S2), information on energy use (S3), LCIA results for scenario 1 (S4), LCIA results for scenario 2 (S5), LCIA comparison (S6), comparison with laptop scenario (S7) cost comparison (S8).

Conflicts of interest

There are no conflicts to declare.


Rosa Figueiredo

Nejat Arinik

Vincent Labatut

Nejat Arınık

Multiplicity in Signed Graph Partitioning

Keywords: Correlation Clustering, Graphs and Networks, Combinatorial Optimization, Social Networks

Introduction

In order to study real-world systems many works model them through signed graphs, i.e. graphs whose edges are labeled as either positive or negative. Such a graph is considered as structurally balanced when it can be partitioned into a number of modules, such that positive (negative) edges are located inside (in-between) the modules. When it is not the case, authors look for the closest partition to such balance, a problem called Correlation Clustering (CC). The standard approach used in the literature is to find a single partition and focus the rest of the analysis on it, as if it was sufficient to fully characterize the studied system. Yet, it may not reflect the structure of the network, and one may need to seek for other partitions to build a better picture.

We study the space of optimal solutions of the CC. We propose an efficient enumeration method allowing to retrieve the complete space of optimal solutions of the CC. It combines an exhaustive enumeration strategy with neighborhoods of varying sizes, to achieve computational effectiveness. By applying our method, we show empirically that under certain conditions, there can be many optimal partitions of a signed graph. Some of these are very different and thus provide distinct perspectives on the system, as illustrated on a small real-world network. This is an important result, as it implies that one may have to find several, if not all, optimal solutions of the CC, in order to properly study the considered system.

This research benefited from the support of Agorantic research federation (FR 3621), as well as the FMJH (Jacques Hadamard Mathematics Foundation) through PGMO (Gaspard Monge Program for Optimisation and operational research), and from the support to this program from EDF, Thales, Orange and Criteo.


Φ Abstract --Increasing speed of electric machine leads to high losses especially in stator winding assembly due to eddy currents effect. Commonly adopted formula for AC losses calculation tends to be applicable in the case of sinusoidal flux density. In this paper, a hybrid model predicting AC winding losses is presented based on one slot per pole of surfacemounted permanent-magnet machines using two formula taking into account the true flux density waveform. The results obtained comply with finite-elements results. The hybrid model was also used to investigate the influence of tooth-tip on AC losses. effect Proximity losses due harmonic of flux density

II. INTRODUCTION

HE need of lightweight electric machines especially for electric vehicle applications led to increase researches on high speed [START_REF] Missoum | Optimization of high speed surface mounted permanent magnet synchronous machines[END_REF], [START_REF] El-Hadi Zaïm | High-Speed Electric Machines[END_REF], [START_REF] Jannot | Multiphysic modeling of a high-speed interior permanent-magnet synchronous machine for a multiobjective optimal design[END_REF]. In order to have an accurate model of the electrical machine and especially for the prediction of power loss, a special attention should be given to AC losses in windings as they become significant at high frequency [START_REF] Boucenna | Modeling of the propagation of high-frequency currents in AC motors[END_REF]. This is due to eddy currents induced by the magnetic field created by the current-carrying conductor known as skin effect and the external magnetic field known as proximity effect [START_REF] Ponomarev | Additional Losses in Stator Slot Windings of Permanent Magnet Synchronous Machines[END_REF]. Previous works allow predicting AC losses in transformer windings [START_REF] Dowell | Effects of eddy currents in transformer windings[END_REF], [START_REF] Sullivan | Computationally efficient winding loss calculation with multiple windings, arbitrary waveforms, and two-dimensional or three-dimensional field geometry[END_REF]. However, AC losses prediction in electric machines is more complicated due to the presence of rotating flux density generated by the rotation of magnets [START_REF] Ponomarev | Additional Losses in Stator Slot Windings of Permanent Magnet Synchronous Machines[END_REF], which occurs even at no load and increases at high frequency [START_REF] Mellor | Investigation of proximity losses in a high speed brushless permanent magnet motor[END_REF] and at large slot openings.

In order to calculate AC winding losses, many methods exist in literature. Finite element method (FEM) allows a high accuracy of results but are time consuming [START_REF] Reddy | Impact of winding layer number and slot/pole combination on AC armature losses of synchronous surface PM machines designed for wide constant-power speed range operation[END_REF], [START_REF] Arkadan | Effect of toothless stator design on core and stator conductors eddy current losses in permanent magnet generators[END_REF]. In order to reduce the computation time, different models have been developed. Reference [START_REF] Mellor | Investigation of proximity losses in a high speed brushless permanent magnet motor[END_REF] studied AC losses in windings with rectangular and round conductors and for different layouts in a permanent magnet synchronous machine (PMSM). Reference [START_REF] Reddy | Impact of winding layer number and slot/pole combination on AC armature losses of synchronous surface PM machines designed for wide constant-power speed range operation[END_REF] studied the influence of winding layer number on AC losses in a surface-mounted PMSM. On the other side, analytical methods are fast but are used for special geometries and based on certain hypothesis [START_REF] Thomas | Proximity Loss Study In High Speed Flux-Switching Permanent Magnet Machine[END_REF], [START_REF] Reddy | Modeling and analysis of proximity losses in high-speed surface permanent magnet machines with concentrated windings[END_REF], [START_REF] Bellara | Analytical prediction of open-circuit eddy-current loss in series double excitation synchronous machines[END_REF], [START_REF] Amara | Analytical prediction of eddycurrent loss in armature windings of permanent magnet brushless AC machines[END_REF], [START_REF] Wu | Analytical investigation of open-circuit eddy current loss in windings of PM machines[END_REF]. Reference [START_REF] Thomas | Proximity Loss Study In High Speed Flux-Switching Permanent Magnet Machine[END_REF] developped a 1-D analytical model to predict AC losses in a PMSM. Reference [START_REF] Reddy | Strand-level proximity losses in PM machines designed for high-speed operation[END_REF] developed a 1-D and 2-D analytical models for AC losses in the case of rectangular slots. Reference [START_REF] Xu | Analytical computation for AC resistance and reactance of electric machine windings in ferromagnetic slots[END_REF] presented an analytical method based on Poynting vector for any slots form. Unfortunately, all these methods, excepting FEA, cannot take into account the alternating flux density due to rotation of magnets and are applicable only for sinusoidal flux density. By combining FEM and analytical methods, known as hybrid methods, it is possible to overcome the weaknesses of both methods and give accurate results with less computation time. Reference [START_REF] Volpe | Modelling AC Winding Losses in a PMSM with High Frequency and Torque Density[END_REF] presented a hybrid model to study AC losses in windings at high frequency however this model is based on the assumption that flux density is sinusoidal. A care must also be given to the fact that the magnetic field density distribution inside the slots are two-dimensional [START_REF] Sullivan | Computationally efficient winding loss calculation with multiple windings, arbitrary waveforms, and two-dimensional or three-dimensional field geometry[END_REF].

The objective of this paper is to present a hybrid method considering arbitrary waveform of flux density and its two components in order to calculate accurately AC winding losses using two formulae. Section III presents formula of skin effect for sinusoidal current supply, two formulae of proximity effect for arbitrary waveforms of flux density T Hybrid model for AC Losses in High Speed PMSM for arbitrary flux density waveforms Taha El Hajji, Sami Hlioui, François Louf, Mohamed Gabsi, Guillaume Mermaz-Rollet and M'Hamed Belhadi considering its two components, and the orthogonality principle. Then, in section IV AC losses calculations are conducted on one slot per pole surface mounted PMSM at both no load and on load conditions for different values of conductor's diameter and frequency. Obtained results are compared with 2-D FEM ones. Finally, the influence of slot opening on the accuracy of AC losses prediction by this hybrid model is investigated.

III. TECHNICAL WORK PREPARATION

Windings losses are composed of 3 terms: proximity effect, skin effect and circulating currents. In our study, we will use one strand in hand, hence, the latter effect will not be considered. In order to evaluate the related losses, formulae of both skin and proximity effect losses are presented as well as their hypothesis. The two components and the arbitrary waveform of flux density are taken into account. The orthogonality principle is then presented which allows summing both skin and proximity losses.

A. Skin Effect Model

Skin effect is due to eddy currents induced by the magnetic field H created by the sinusoidal currentcarrying conductor as shown in Figure 1. The current is then concentrated near the surface as shown by the grey surface in Figure 1, which increases the equivalent electrical resistance.

Figure 1 Eddy currents induced in round conductor and resulting current due to skin effect

Resistance in case of DC current is expressed as follows:

1

We assume that eddy currents are resistance limited and the end effects are neglected. Hence, the ratio between the real resistance value (AC resistance) due to skin effect and DC resistance can be expressed as follows [START_REF] Ramo | Fields and Waves in Modern Radio[END_REF]:

2 . 2
Joule Losses in the case of DC current are expressed as follows:

. 3 Losses due to skin effect for sinusoidal current supply can be expressed as follows [START_REF] Ramo | Fields and Waves in Modern Radio[END_REF]:

. 4 Where,

√2 5 1 6

The ratio between AC resistance and DC resistance in function of γ is shown in Figure 2. Skin effect is noticeable for values of γ above 1.5 which is equivalent to:

1,5. √2. 2,12 7

Figure 2 Ratio of AC resistance to DC resistance in function of

Above formula of skin effect losses is only applied for sinusoidal current. However, in order to take into account arbitrary current waveform, losses due to each harmonic are summed [START_REF] Venkatraman | Winding eddy current losses in switch mode power transformers due to rectangular wave currents[END_REF].

B. Proximity Effect Model

Proximity effect is due to eddy currents induced by an external magnetic field B created by neighbouring conductors and/or magnets rotation/movement. Due to induced eddy currents, shown in Figure 3, the current flowing in the conductor will be concentrated on one side. Hence, the resistance increases. Where means the index of harmonic of flux density Fourier series.

While flux density is two-dimensional in electric machine's slots, formulae ( 9) and (10) will be applied to both radial and tangential components of flux density [START_REF] Bertotti | An improved estimation of iron losses in rotating electrical machines[END_REF]:

1 64 64 ( 11
) , 128 , 128 (12)
The assumption of uniform flux density on conductor is equivalent to: ≪ 13 In order to verify (13), a range of conductor's diameter is considered in this study and will be validated later in this paper: 0,5 14 Figure 4 shows the limit of conductor's diameter in function of frequency in order to verify [START_REF] Amara | Analytical prediction of eddycurrent loss in armature windings of permanent magnet brushless AC machines[END_REF].

C. Orthogonality Principle

According to [START_REF] Bertotti | An improved estimation of iron losses in rotating electrical machines[END_REF] skin and proximity losses can be summed if flux density B is assumed to be uniform on the conductor. Figure 5 shows the eddy currents induced by both skin effect and proximity effects. Total losses are expressed as follows [START_REF] Ferreira | Appropriate modelling of conductive losses in the design of magnetic components[END_REF]:

². . ². . 15 ² ² 2. . . . (16

(20)

The orthogonality principle is applied to sum skin and proximity losses under the assumption of uniform flux density on the conductor. In order to respect this assumption, conductor's diameter is limited depending on the frequency as shown in Figure 4.

IV. RESULTS

In this section, the studied machine is presented. FEM simulations are performed using 2D Transient solver of ANSYS Electronics Desktop 2018. Results obtained by hybrid model and FEM are provided in both no load and on load cases. The influence of slot opening is also investigated.

A. Studied Machine

B. No Load Results

The flux density is bidirectional inside slots as shown in Figure 7. Figure 8 and Figure 9 show periodic waveform of both components of flux density inside conductors 4, 7, and 10. Results obtained show that waveform of flux density inside conductors is not sinusoidal and have a tangential component higher than radial component as shown in Figure 7. Hence, ( 8) cannot be applied in this case. The error between FEM and hybrid method is shown in Figure 11. Maximum error is obtained at high diameter and high frequency. This is because the assumption of uniform flux density inside conductors, used for proximity losses formula, is less verified. In fact, this assumption is equivalent to [START_REF] Amara | Analytical prediction of eddycurrent loss in armature windings of permanent magnet brushless AC machines[END_REF] which corresponds to conductor's diameter limit in function of frequency represented by the red dashed line in Figure 11. The error between FEM and hybrid model for frequencies and diameters values in this zone is less than 10%.

In order to evaluate the contribution of each harmonic in AC losses, ( 12) is used. Table 2 shows the results obtained. The contribution of third harmonic is at most of the time 56% of total proximity losses. Hence, AC losses cannot be evaluated in this case by considering sinusoidal waveform of flux density.

C. On Load Results

In the on load case, both skin and proximity effects occur in the conductor because of flowing current and external alternating magnetic field. AC losses are calculated for different values of conductor's diameter and frequency. Figure 12 shows results obtained with both hybrid model and FEM. At f 0, only joules losses occur in the conductor which explains highest losses for lowest conductor's diameter. However, at high frequency AC losses are higher for high conductor's diameter due to proximity effect as predicted by [START_REF] Thomas | Proximity Loss Study In High Speed Flux-Switching Permanent Magnet Machine[END_REF].

Figure 12 AC losses in conductors with FEM and Hybrid Method (I=50A)

Figure 13 shows the error between the two methods. Error is maximum at high diameter and high frequency. This is because the assumption of uniform flux density inside conductors, used for both proximity losses formula and orthogonality principle, is less verified. In fact, this assumption is equivalent to [START_REF] Amara | Analytical prediction of eddycurrent loss in armature windings of permanent magnet brushless AC machines[END_REF] which corresponds to conductor's diameter limit in function of frequency represented by the red dashed line in Figure 13. The error between FEM and hybrid model for frequencies and diameters values in this zone is less than 5%. Indeed, skin effect losses that occur only in on load case are accurately evaluated unlike proximity losses which requires uniformity of flux density inside conductors. For a large value range of frequency and conductor's diameter, skin losses have high contribution to AC losses, which explains a better accuracy of the hybrid method in the on load case.

Figure 13 Error between FEM and Hybrid method in function of frequency and conductor's diameter (I=50A)

Table 3 shows DC losses, AC Losses, and the contribution of skin effect losses and proximity effect losses to AC losses. Additional losses due to skin effect are not remarkable for low values of γ as explained in Figure 2. Hence, skin effect losses are equal to DC losses. For higher conductor's diameter, skin effect losses decrease because R decreases (1) while proximity losses increase as predicted by [START_REF] Reddy | Modeling and analysis of proximity losses in high-speed surface permanent magnet machines with concentrated windings[END_REF]. Consequently, contribution of skin effect to AC Losses decreases for high conductor's diameter and contribution of proximity effect increases.

In addition, proximity effect losses increase remarkably with frequency as predicted by [START_REF] Reddy | Modeling and analysis of proximity losses in high-speed surface permanent magnet machines with concentrated windings[END_REF] unlike skin effect which is noticeable only at very high frequency as shown in Figure 2. Consequently, at high frequency and high conductor's diameter, proximity losses are dominant.

D. Influence of Slot Opening

The form of the slot affects strongly the path of flux lines inside the slot because of iron's high permeability. Hence, proximity effect losses depend on geometrical variables of the slot. In this section, we are going to study the influence of slot opening on proximity losses and on the accuracy of hybrid model for both on load and no load cases. The rotation speed of electrical machine is 20,000 rpm and conductor's diameter is 1 mm.

Figure 14 shows AC losses for different values of slot opening for no load and on load cases. In the no load case, it is concluded that the larger is the slot opening, the higher are proximity losses. This is mainly due to presence of magnets at the surface of the rotor that creates an alternating magnetic field that penetrates the conductors through the slot opening. In the on load case, it is noticed that the larger is the slot opening, the higher are AC losses. Knowing that skin effect losses are constant because they depend only on conductor's diameter, frequency and current value, it is concluded that increasing AC losses means increasing proximity effect losses. As mentioned above, this is mainly due to presence of magnets at the surface of the rotor. Figure 15 shows the error between hybrid model and FEM results of AC losses for different slot opening values for no load and on load cases. The hybrid model is still accurate for different values of slot opening. The error obtained is less than 13% for no load case and less than 1.2% for on load case. It is noticed that for on load case, the model is more accurate because the skin effect, which are dominant on large value range of conductor's diameter and frequency, are accurately evaluated.

V. CONCLUSION Researchers are faced to many challenges when designing high speed electrical machines. One of the main problems is the limited voltage that leads to the use of a low number of conductors, thus with high diameter to keep same fill factor. However, due to noticeable skin effect in conductor with high diameters, one solution is to use multiple strands in hand. Hence, proximity effect becomes more important and should be well evaluated. Furthermore, the presence of magnets on the surface of the rotor causes the existence of alternating magnetic field flowing into conductors through slot openings. In order to accurately evaluate AC losses with less computation time, a hybrid model is proposed. The main outcomes of this model are the consideration of the twodimensional and arbitrary waveforms of flux density inside slots.

The hybrid model is proved to give accurate results compared to FEM ones for a wide value range of conductor's diameter, frequency, and slot opening. This model will be included in a methodology of optimization and design of high-speed electrical machines. This methodology and the designed machines will be presented in future works.

Figure 3

 3 Figure 3 Induced Eddy Currents in round conductor due to external alternating magnetic field In the presented model, we are going to assume that: o Flux density is uniform on conductor, o Eddy currents are resistance limited, o External magnetic field is assumed to be equal to global magnetic field , that includes contribution of the conductor, evaluated by FEM. The value of flux density, considered uniform, is taken in the center of conductor. Many previous works use the following formula of power proximity losses under the assumption of sinusoidal flux density: , ² 128 8 Reference [7] presents the instantaneous power proximity losses for arbitrary waveform of flux density: 64 9

Figure 4

 4 Figure 4 Limit of conductor's diameter depending on frequency in order to verify assumption of uniform on conductor

  ) Considering uniform flux density on conductor, we can conclude from Figure5that: 17 18

Figure 5

 5 Figure 5 Eddy currents due to skin and proximity effect in round conductor Then,

Figure 6

 6 Figure 6 Studied Surface-Mounted PMSM

Figure 7 Figure 8 Figure 9

 789 Figure 7 Flux lines and Flux density in Surface-Mounted PMSM (I=0A, 20 000 , 3,5 )

Figure 10 ACFigure 11

 1011 Figure 10 AC losses in conductors with FEM and Hybrid Method (I=0A)

Figure 14 Figure 15

 1415 Figure 14 Variation of AC losses for different slot opening values with FEM and Hybrid Method

Table 1

 1

presents additional details concerning studied SM-PMSM.

Table 1

 1 Details of studied Surface-Mounted PMSM

	Parameter	Value
	Stator Outer Radius [mm]	100
	Rotor Outer Radius [mm]	69.6
	Active Length [mm]	250
	Poles	6
	Slots	18
	Slots-per-pole-per-phase	1
	Slot Opening [rad]	0.01
	Air-gap Thickness [mm]	0.7

Table 2

 2 Losses due to flux density harmonics (I=0A)

	Speed (rpm)	10 000	20 000	30 000
	Diameter (mm)	0.5	1	0.5	1	0.5	1
	AC losses (W)	2.26	36.12	9.04	144.48	20.34	325
	Harmonic 1	Losses (W) % to losses AC	0.56 24% 24.7% 24.7% 8.94 2.24	35.75 24.7%	5.03 24.7%	80.44 24.7%
	Harmonic 3	Losses (W) % to losses AC	1.27 56% 56.4% 56.4% 20.37 5.1	81.47 56.3%	11.48 56.4%	183.31 56.4%

Table 3

 3 Losses due to flux density harmonics (I=50A)

	Speed (rpm)		15 000			30 000
	Diameter (mm)	0.5	1	1.5	0.5	1	1.5
	DC Losses (W)	6134	1533	681	6134	1533	681
		AC losses	6139	1616	1104	6155	1866	2377
	Skin Effect	Losses (W) % to AC losses	6134 99.9%	1533 95%	681 62%	6134 99.6%	1533 82%	682 29%
	Proximity Effect	Losses (W) % to AC losses	5 0.1%	83 5%	423 38%	21 0.4%	333 18%	1695 71%

VII. BIOGRAPHIES

Taha El Hajji is currently working toward the Ph.D. degree at SATIE laboratory of the Ecole Normale Superieure Paris Saclay, France and Groupe PSA. His work is focused on high-speed electrical machines for electric vehicles. He received the master of engineering degree in electrical engineering and automation from ENSEEIHT, Toulouse, France, in 2018. He received the master of research in automotive electrification and propulsion from Ecole Normale Supérieure Paris-Saclay, Gif-sur-Yvette, France, in 2018. He also received an Engineer degree in electromechanical engineering from Bejaia University, Algeria, in 2010. He is currently an innovation engineer at Groupe PSA, France.

Sami