Raphaël Labrunye

Gauthier Bolle

Karim El Alami

Ignacio Requena

Daniel Siret

Interdisciplinary research on the heritage of housing complexes in France (1945-75)

Keywords: 20th century, France, Mass Housing, Renovation, Heritage

This paper presents the methodology of the research program "SMART FRENCH, collective housing of the second part of the twentieth century, through the prism of energy". Many energy renovation operations of this heritage are mostly carried out in the perspective of very heavy transformations, in particular by external thermal insulation. The aim of the research program i s to appraise and rank the architectural, technical, historical or constructive value of these post-war housing complexes. The purpose is to build a database as exhaustive as possible on nearly 700 operations, published at the time of their construction and then to evaluate the initial environmental qualities of the projects in terms of integration into the landscape, solar orientation, spatial articulations, thermal inputs. This set of information will then be the subject of an analytical treatment to emphasize the most current situations and those that are considered obsolete in view of the contemporary demands.

Introduction

The main purpose o f this research proj ect i s to build an in-depth knowledge of the stock of collective housing projects produced after the Second World War, with an operational focus. This heritage, built between 1949 and 1974, constituted nearly 20% of housing in France. In the Greater Paris area, the collective dwellings produced during this period represent 26% of energy consumption and 31% of regional gas emissions. Victims of a global negative judgment, these groups have undergone since the 1980s recurrent policies of major t r a n s f o r m a t i o n s , part i cul arl y in the fi e l d of social housing, accentuated by the PNRU (National Program f o r Urban Renewal) and the so-called Grenelle de l'environment, a series of environmental laws voted in France in 2007-2009. Regulatory and financial measures have favored a generic thermal approach, through standardized calculation e n g i n es , by passing on new construction methods, themselves in question [START_REF] Sidler | Rénovation à basse consommation d'énergie des logements en France. Projet RENAIS-SANCE, programme européen CONCERTO[END_REF]. Insulation from the outside is often handled b y a solution applied on a large scale, reinforced by the recent decree on facade remodeling ( Décret n° 2016-711 du 30 mai 2016). The lack of a global environmental assessment is accompanied by explosive pathologies in several reports (RAGE 2012). However, recent work has a strong interest in having a detailed knowledge of each operation in order t o find sustainable and fi n an ci ally balan ced solutions [START_REF] Graf | La cité du Lignon 1963-1971[END_REF]). The question is how to study such a mass of buildings, without a generic and reductive approach nor a monographic and too limited one. Our main hypothesis is that these housings had been conceived with an environmental approach, even if it was not with our contemporary criteria. Combining historical and engineering sciences, we try to bring out relevant characteristics in the fields of design, urbanism, solar exposition o r construction types. To manage this ambitious program, we have associated several researchers coming from seven graduate schools of architecture in France (Nantes, Rennes, Strasbourg, Bordeaux, Clermont-Ferrand, Rouen) and in Belgium (Vrije Universiteit Brussels).

2 Between global and specific approach: using database Knowledge of the post-war housing estate stock remains, however, still limited by its mass. Since the "hard French" of Bruno Vayssière (1988), first militant synthesis book, some works tried to pursue this global perspective (Collectif 1995[START_REF] Voldman | La reconstruction des villes françaises de 1940 à 1954: histoire d'une politique[END_REF], Bourillon et al. 2006). Others are situated in the fields of social and political histories (Le Goullon 2014) in the following works by Jean-Claude [START_REF] Croizé | Politique et configuration du logement en France (1900-1980[END_REF], Danièle Voldman (1997), Annie Fourcaut andFrédéric Dufaux (2004). Some thematic works have been published with a specific focus on constructive innovations [START_REF] Lambert | Ensembles urbains, 1940-1977, Les ressorts de l'innovation constructive[END_REF], landscapes [START_REF] Blanchon | Le paysage dans les ensembles urbains de logements de 1940 à 1980[END_REF] or assembled concrete [START_REF] Delemontey | Reconstruire la France. L'aventure du béton assemblé[END_REF]. Recently, mainly monographic researches have been published, about an architect [START_REF] Bolle | architecte. Les Trente Glorieuses et la réinvention des traditions[END_REF][START_REF] Guillerm | Le dessin à l'épreuve des usages[END_REF], a specific project [START_REF] Klein | La cité de l'étoile à Bobigny, Candilis, Josic, Woods, un modèle de logement social[END_REF][START_REF] Gruet | Le Mirail, mémoire d'une ville, histoire vécue du Mirail de sa conception à nos jours[END_REF], the SCIC owner, Société centrale immobilière de la Caisse des dépôts (Landauer 2010) and a SAE construction company, Société Auxiliaire d'Entreprises [START_REF] Jambard | Un constructeur de la France du XXe siècle. La Société Auxiliaire d'Entreprise (SAE) et la naissance de la grande entreprise française de bâtiment[END_REF]. Aside from our last work (Carrie & Labrunye 2013[START_REF] Siret | Rayonnement solaire et environnement urbain: de l'héliotropisme au désenchante-ment, histoire et enjeux d'une relation complexe[END_REF], not one has been oriented to operational issues or environmental questions.

The ambition of the study is to approach this mass production on a relevant scale, going beyond the two types of existing work: the case study that provides specific and non-reproducible responses and the generic approach that reduces this pro-duction to "heat sieves". It is therefore a question of elaborating a method of analysis of forms that goes beyond the typo-morphology, perfectly applicable to the plot and the islet, but incomplete on the scale of a land area of several hectares, and obsolete with the spread of construction systems throughout the country on a large scale. The aim is to identify recurrences or characteristic series of a large sample. The methods defined since the 1960s by the historian Pierre Chaunu under the name of "serial history" are one of the most appropriate methodological bases [START_REF] Chaunu | Le guide ABC -Amélioration thermique des Bâtiments Collectifs construits de 1850 à 1974[END_REF]. For a corpus such as the one we are proposing to study, that is to say, spread over a limited period of time, according to location criteria and a sampling of objects with relatively homogeneous characters (also delimited by criteria: economy, scales, materials on the one hand, geographical situations, societies, etc. on the other hand), the adoption of a serial method does not pose the problems which constitute its limits: temporal discontinuity, exceptionality of the object.

In fact, we have constituted our own database, by a systematic analysis of architectural journals published between 1944 and 1975. We have worked on the three major French journals: Architecture d'Aujourd'hui, Architecture Française and Technique et Architecture. The main idea is to concentrate our work on projects with architectural qualities that are necessary to save. Architectural journals do not ever have the monopoly of architectural qualities, but thanks to this method, we managed to assemble nearly 700 urban operations from the period. Then, we organized a database with the software File Maker Pro to collect all information available. The structure of the data-base is organized with five tables: Project/Article/ Advertising/Ensemble/Building. The first one is the most important, with all information on each project, identified with a specific code made by ARCHITECT-TOWN-YEAR. Each article or ad is linked to the table "Project" in this way. "Ensemble" is used to link operations built in several parts with several forms and construction system in the same site. "Building" is used to describe each building of each project. With such process, we can add new information throughout the whole process, new headings or new recordings according to the progress of the study. It is a way to pursue the using of the database after the end of the study, and the developing of items.

Then, the question is which indicators we have to consider as the characteristics of the projects. The objective is to revisit the historical knowledge of the period commonly and too easily referred to as the Trente Glorieuses. Literature on the subject, often the result of critics of modernity, generalizes it as a period without environmental design [START_REF] Alexandroff | Architectures et climats: soleil et énergies naturelles dans l'habitat[END_REF], while the post-crisis period following the oil crisis is seen as innovative (Bobette et al. 2007). The technical literature considers that between 1948 and 1974, all architectural production is obsolete (Collective 2011). However, this period is characterized by at least two phases, the first is an economic situation of shortage and an unstable political context, to which succeeds an unprecedented prosperity. The attention paid by the designers to the control of the ambiances and the architectural quality also seems to evolve very substantially. By collecting discourses from articles in the database, it appears that the lack of materials, which lasted until the end of the 1950s, promoted the search for economic devices such as the use of insulating lacunary materials. Production attests to a will and specific research regarding the qualities and functionality of housing, adapting the ideals originating from the Modern Movement to a singular historical, social and economic context. With the database, we can see how "preserving existing trees" is regularly present in the discourses of architects during the 1950s and completely disappears at the end of the 1960s. Insulation potential is mentioned at the first years of the period, in the continuity of modern theories.

Environmental assessment

These environmental qualities, however, go beyond the constructive aspect of the building envelope. One of the major characteristics of this built heritage is the skillful articulation of urban, architectural and landscape scales, specific to each operation. Considering the large area of the dwelling's lands, the architects were able to organize the volumes and to distribute the apartments relatively freely, according to the most favorable orientations on site; likewise, they could also structure the whole in an independent constructive plot. Accordingly, these works exemplify an excellent symbiosis of the industrialization of constructive processes de-signed aboveground, the modern urban forms arising from a free arrangement of independent objects in the space, and the attempt to rationalize the interior ways of living.

To assess the appropriateness of these hypothesis, we have conducted a pre-statistical analysis on 13 examples taken in the corpus, looking for design parameters related to sunlight. Based on the hypothesis that sunlight was a main design parameter in the urban theories of the early twentieth century [START_REF] Siret | Rayonnement solaire et environnement urbain: de l'héliotropisme au désenchante-ment, histoire et enjeux d'une relation complexe[END_REF], which were widely spread among architects, this work aims to better understand the role of sunlight in the organization of blocks, apartments and architectural solutions. This work started with a study of historic and current sources describing the overall shape in order to enrich the database in a progressive plan: project > building > facade. From the total corpus, the research methodology has been already tested in 13 projects, for a total of 112 buildings and 492 facades.

For this analysis, each facade constituted an item that was characterized by parameters regarding the building shape (length, height, orientation), apartments (crossing typology, main rooms functions), architectural devices (presence of balconies, loggias, passageways, other openings). We completed the characterization with sunlight indicators resulting from simple analysis of surfaces ratios, sun path diagrams and elemental digital models. These indicators include: facade length ratio compared to the building perimeter, percent-age of glazed surface, facade openness to sunlight according to facade length and the percentage of glazed surface, percentage of facade in the building shading mask at midday December 21 and sunshine hours potential. Such indicators allowed us to understand the coherence between the architects' discourse and the projects materialization. For instance, by multiplying the "facade length ratio" and the "percentage of glazed surface", we obtain the indicator of "facade openness" that highlights the prevalent facade of a building in terms of sunlight exposure. Lastly, we conducted detailed simulations of the 13 projects aiming at analyzing the impact of shadow casting for the public space (see Figure 1). The results of this study show three main conclusions. Firstly, the analysis of the orientations reveals a diversity of approaches that do not correspond to the more purist urban theories of sunlight in the early twentieth century (e.g., all facing South, East-West); indeed, the cardinal points are scarcely represented in the statistics.

The main trend shown in the analysis is a preference for diagonal orientations (SE-NW or NE-SW) that represent around 70% of the cases. More surprisingly, such distribution remains stable when we pay attention to the facades with a high "openness to sunlight" indicator. These facades rarely face the cardinal points, and, again, they mainly face diagonal orientations, including those facing northern orientations. This result means that compromise approaches to sunlight with diagonal orientations were preferred to the purist urban theories on sunlight for mass housing construction in the analyzed period.

Secondly, the study of architectural devices theoretically related to sunlight control (loggias, balconies) shows that they do not mainly face orientations where they are supposed to operate correctly, which is to say close to the South. Paradoxically, loggias and balconies mainly face to the North, NNW for loggias and WNW for balconies, even if orientations close to the South are also possible. This fact reveals that sunlight control was not the primary goal of such devices; rather, they were mostly designed like outside extensions of indoor living spaces and as a part of the modernist architectural expression.

Lastly, through the study of the function of the rooms located against the facade, we notice that the orientation of living rooms is almost identical to those of the bedrooms, whereas the orientation of kitchens is close to those of the bathrooms. However, none of these functional groups shows a prevailing orientation, all of them being distributed in the four diagonal orientations aforementioned for the facades. Solar orientation is not the only question to design the ground plan. Consequently, we have to reintroduce these problems into a larger historical context.

Mass housing values in an historical perspective

Our state-of-the-art shows that existing studies on the grands ensembles in France have mainly emphasized the industrial and technical aspects of this architecture. Considered only as a "statistical architecture" [START_REF] Vayssiere | Reconstruction-Déconstruction. Le hard french ou l'architecture française des trente glorieuses[END_REF]) in its urban and constructive sides, several research studies tend to show that the forms, the organization of blocks and the aesthetics developed are mainly at the service of a statistical necessity, without considering the implementation site (Fortin 2011). Other works highlight more the link between mass housing and the major steps of normative and political evolutions during this period (Le Goullon 2014). For a long time discarded for ideological reasons from the general history of architecture [START_REF] Canteux | Filmer les grands ensembles: villes rêvées, villes introuvables[END_REF], the double process of material transformation and public appropriation leads to reconsider these objects in new perspectives. The SMART FRENCH research project makes it possible to cross-historical and social analyses with technical and material studies. Thus, our database enables us to identify and highlight a certain number of actors and to also consider their cultural backgrounds, giving some unpublished keys to understanding this production. Indeed, these architects have all gone through a prestigious academic training in the workshops at the famous Ecole des beaux-arts, often during the inter-war period (Crosnier-Leconte 2017). They are therefore often imbued with a specific culture related to the classical art of composition [START_REF] Lucan | Retours d'expériences dans les bâtiments à basse consommation et risques de non-qualité, résultats 2011. Programme d'accompagnement des professionnels Règles de l'Art Grenelle Environnement (RAGE)[END_REF] applied to theoretical projects of a very different nature and fueled by a classical culture.

Among these actors, it is interesting to note that some choose to complete their training outside the school, notably at the Institut d'Urbanisme de Paris, especially with Pierre Lavedan (1885-1982) as a professor. Besides, despite the distance taken from the Modern Movement by the School of Fine Arts, many students seek to know and appropriate the theories of the great masters, including Le Corbusier in addition to their academic training. This has been clearly demonstrated concerning the career of the important architect Jean Dubuisson (1914Dubuisson ( -2011) ) [START_REF] Guillerm | Le dessin à l'épreuve des usages[END_REF]. However, this is also the case of the architect Xavier Arsène-Henry (Arsene-Henry 1999), whose career and production, particularly in Bordeaux, attest to this double anchoring in academic culture and modern influences. With our database, we can analyze this kind of aspects more broadly, through the biographical aspects as discourses published in specialized journals by architects, and also through the forms themselves.

To conceive and construct mass housing estates after World War II, the art of academic composition is associated with an opening, perhaps sometimes quite limited, to the theories and forms advocated by the modern movement. Thus, Gustave Stoskopf (1907Stoskopf ( -2004) ) [START_REF] Bolle | architecte. Les Trente Glorieuses et la réinvention des traditions[END_REF]) re-injects mainly his academic and classical culture in some aspects of the master plans of housing estates he designed. He particularly insisted on the creation of central squares and major visual axes. However, a certain degree of openness to modern theories and innovative technical devices is attested both by historical investigations and by the fine analysis of urban dispositions, and it confirms the reading made by the environmental and technical analyses.

This historical look at the cultural background of the actors can be extended to other important stakeholders of this period. The work carried out about SCIC company (Landauer 2010), a subsidiary of Caisse des Dépôts, the public bank, at the origin of important housing estates such as Sarcelles and Créteil, shows that these managers have imported many concepts and technical devices developed in the USA or in England, such as neighborhood unit's principle [START_REF] Janniere | Planifier le quotidien. Voisinage et unité de voisinage dans la conception des quartiers d'habitation en France (1945-1965)[END_REF]. The SCIC and its leader, Léon Paul-Leroy (1915[START_REF] Fortin | Grands ensembles, l'espace et ses raisons[END_REF], an ingénieur des Ponts et Chaussées, organized many study trips in search of alternatives forms of urbanization to oppose it to the model of classical suburbs with individual housing. SCIC-related architects travelled to the USA and also in the eastern and northern countries of Europe [START_REF] Bolle | architecte. Les Trente Glorieuses et la réinvention des traditions[END_REF], and they imported exogenous and innovative models to renew French suburbs. Some architects borrowed discoveries from trips across the Atlantic like the architect Marcel Lods [START_REF] Uyttenhove | Marcel Lods: action, architecture, histoire[END_REF]. Therefore, if the mass housing in France is a very specific form, it is also the result of exchanges and cultural transfers crossing national borders.

The cultural history, the architectural and urban analysis of the forms themselves and, finally, the technical analysis of objects, seem to indicate a much more complex and nuanced reality about mass housing. Our method of research in the SMART FRENCH project reveals new aspects, and specifically an existing consideration of environmental and contextual issues by architects and stakeholders of this important period of urban and construction history.

Moreover, we assume that our method could be developed on a bigger scale. The difficulty comes from data, which are not as available in architecture as in demography or economy. But nowadays, databases on open source have been developed. For example, ground plans can be found in vector drawing format on Openstreetmaps. With a short programming, we could imagine testing solar orientations on hundreds of situations. In history, we should be also able to mobilize data of the French Mérimée base (heritage preservation) or biography of architects (Archipedie or Agorha). Hypotheses on designing ground plans could be verified in many situations.

Figure 1 .

 1 Figure 1. Esplanade Quarter in Strasbourg, France, Charles Gustave Stoskopf architect, 1951. Cumulative shading analysis in June, September and December. Values computed with T4SU Plug-in for SketchUp. © AAU/CRENAU.

Graduate School of Architecture of Bretagne, Rennes, France

Graduate School of Architecture and Landscape of Bordeaux, Bordeaux, France

Graduate School of Architecture of Nantes, Nantes, France


Mohammed Yazid Lyazidi

Nadjib Aitsaadi

Rami Langar

A Dynamic Resource Allocation Framework in LTE Downlink for Cloud-Radio Access Network

Keywords: Cloud-RAN, LTE, Resource allocation, Power minimization, BBU-RRH assignment, Simulated annealing

One main asset of Cloud-Radio Access Network (C-RAN) lies in its centralized architecture that allows network operators to serve dynamic flows of mobile traffic with efficient utilization of baseband resources and lesser operation costs than the distributed RAN architecture. For this very reason, the implementation of online resource allocation algorithms in the BaseBand Unit (BBU) pool for handling loads of multiple Remote Radio Heads (RRHs) is one of the most motivating challenges in C-RAN. Those centralized algorithms must be able to handle efficiently interference between users, as well as to dynamically select RRHs that can be turned on/off based on traffic variation. By doing so, the total RRHs transmission power can be minimized and the number of active BBUs within the cloud can also be reduced. In this paper, the issues of dynamic wireless resource allocation, transmission power minimization and BBU-RRH assignment in downlink C-RAN are addressed in one framework. We have previously attempted to address these problems by proposing a approach based on the branch-and-cut algorithm to solve small instances of the problem to optimality. However, due to the combinatorial complexity of the problem, finding optimal solutions for a large-scale network may take a fair amount of time and will not be suitable for online optimization. Towards this end, we propose a novel two-stage approach to address these issues for a large-scale problem. The first stage is a new proposal that addresses the problems of dynamic resource allocation and power minimization in C-RAN using a simulated annealing approach with a specific neighborhood search program. The BBU-RRH assignment is handled in the second stage using a multiple knapsack formulation. Through extensive eventbased simulations, our proposal achieves significant reduction in time complexity and yields near optimal performance compared to state-of-the-art methods.

I. INTRODUCTION

Cloud Radio Access Network (C-RAN) has been recently introduced by China Mobile Research Institute as a novel cloud architecture for Long Term Evolution (LTE) and upcoming cellular standards (5G) [START_REF]The Road Towards Green RAN, China Mobile Research Institute[END_REF]. It is a new RAN paradigm that can address the challenges mobile network operators are faced with and meet their requirements in terms of capital and operational expenditure costs reduction. The C-RAN architecture is illustrated in Fig. 1. It is based on a central cloud pool composed of BaseBand Units (BBUs) that perform Physical M. Y. Lyazidi is with the LIP6, University of Pierre and Marie Curie (UPMC), Paris 75005, France (E-mail: yazid.lyazidi@lip6.fr).

N. Aitsaadi is with University of Paris-Est, LIGM-CNRS UMR 8049, ESIEE Paris, 93162 Noisy-le-Grand, France (E-mail: nadjib.aitsaadi@esiee.fr) R. Langar is with University of Paris-Est, LIGM-CNRS UMR 8049, University Paris Est Marne-la-Vallée (UPEM), 77454 Marne-la-Vallée, France (E-mail: Rami.Langar@u-pem.fr).

A preliminary version of this paper appeared in the proceedings of the 2016 IEEE International Conference on Communications (ICC 2016) [START_REF] Lyazidi | Resource Allocation in Cloud-RAN with Real-Time RRH-BBU Assignment[END_REF]. (PHY) and Medium Access (MAC) functions processing. The BBUs are connected to the Remote Radio Heads (RRHs) in the cell sites by means of a low-latency and high bandwidth fronthaul network. A cloud controller is situated in the BBU pool and performs resource and load balancing between BBUs that are interconnected through a high-speed backhauling network [START_REF] Haberland | Radio Base Stations in the Cloud[END_REF]. By replacing "hard" wireless network equipments by "soft" BBUs, the C-RAN capabilities can be dynamically adjusted based on the traffic load variations [START_REF] Checko | Cloud RAN for mobile networks -a technology overview[END_REF]. This not only fosters efficient resource utilization, but also allows the C-RAN to handle more areas than standalone clusters of base stations and facilitates service deployment on the edge [START_REF] Paolini | Benefits of C-RAN and adoption trends[END_REF].

However, the design of dynamic schemes for C-RAN's radio resource management constitutes a major challenge that hinders its commercial expansion. In fact, the optimization of C-RAN baseband resource allocation needs methods to cater to time-varying traffic demands at different RRHs [START_REF] Park | Robust and efficient distributed compression for cloud radio access networks[END_REF]. A centralized algorithm can help optimize the resource demands of mobile users located in different cells and with different bandwidth requests. Besides, such centralized approach will help network operators select the RRHs that can be dynamically turned on/off, based on their traffic loads patterns during the day. Consequently, the total RRHs transmission power can be minimized and the baseband resources can be efficiently utilized for handling traffic demands during the day. Moreover, lessening the number of active RRHs would help reduce the number of instantiated BBUs associated to them in the cloud and realize more power and cost savings. Therefore, for all these reasons, a careful C-RAN resource allocation strategy must be planned regarding users traffic demands, RRHs transmission power minimization and BBU pool capacity in terms of handled RRHs.

In [START_REF] Lyazidi | Resource Allocation in Cloud-RAN with Real-Time RRH-BBU Assignment[END_REF], we presented two optimization models for the i) resource allocation and power minimization problem and ii) the BBU-RRH assignment problem in C-RAN. The proposed scheme based on the branch-and-cut algorithm [START_REF] Mitchell | Branch-and-cut algorithms for combinatorial optimization problems[END_REF] has permitted to achieve reasonable gain in throughput satisfaction rate and transmission power minimization over state-of-the-art algorithms and for small instances of the problem. However, due to the combinatorial nature of the first problem (NPhard), the computational complexity is exponential if an exact optimal solution is to be calculated for a large-scale system. In this paper, a meta-heuristic algorithm, known as Simulated Annealing (SA), is used in providing fast and close-tooptimal solutions to the first-stage problem at a much reduced complexity. The near-optimality gap will be emphasized by comparison to solving the problem to optimality by the offline branch-and-cut algorithm used in [START_REF] Lyazidi | Resource Allocation in Cloud-RAN with Real-Time RRH-BBU Assignment[END_REF].

In summary, our key contributions are the following:

• We express in the first stage the Centralized Resource Allocation and Power Minimization (C-RAPM) problem, which is formulated as an Mixed Integer Linear Programming (MILP) problem. A reformulation is proposed using the framework of the well-known big-M method [START_REF] Lee | Mixed integer nonlinear programming[END_REF]. A novelty in this paper compared to our previous approach is we consider here a power allocation model based on static transmission instead of continuous. We compare our approach's results from event-based simulations to our previous approach DRAC in [START_REF] Lyazidi | Resource Allocation in Cloud-RAN with Real-Time RRH-BBU Assignment[END_REF] and to different literature schemes. We also discuss the associated performance gains. The remainder of the paper is organized as follows: Section II presents a review of related works regarding resource allocation, power minimization and BBU-RRH management in C-RAN. In Section III, we describe the two-stage system model for the C-RAPM and MKP problems, which is followed by Section IV that details our proposed SA approach. Discussion and analysis of simulation results are exposed in Section V. Finally, Section VI concludes the paper.

II. RELATED WORK

C-RAN has received a considerable amount of research attention after its introduction by China Mobile Institue. Authors in [START_REF] Checko | Cloud RAN for mobile networks -a technology overview[END_REF] highlighted C-RAN's advantages for operators and vendors compared to distributed RAN. In fact, traditional base stations are often under-utilized during certain hours of the day, which results in wasteful use of radio resources and baseband capacity. The authors showcased C-RAN's ability to handle this issue by dynamically instantiating BBUs and allocating the baseband resources to RRHs depending on traffic volumes [START_REF] Checko | Optimizing small cell deployment by the use of C-RANs[END_REF]. Furthermore, authors in [START_REF] Cai | Dynamic operations of cloud radio access networks (C-RAN) for mobile cloud computing systems[END_REF] introduced the concept of coupling C-RAN with mobile cloud computing systems to enhance end-to-end cloud services for future 5G networks. In their work, the authors proposed a novel topology framework and rate-allocation configuration in C-RAN to improve end-to-end traffic performance of mobile cloud computing users.

Regarding the transmission power minimization issue, authors in [START_REF] Shi | Group sparse beamforming for green Cloud-RAN[END_REF] described a Group Sparse-based Beamforming approach (GSB), that can minimize the C-RAN RRHs transmission and fronthaul links power consumption in downlink. The authors outlined the problem as a joint RRH selection and transmit plus fronthaul links power minimization problem, with a Signal-to-Interference-plus-Noise Ratio (SINR) constraint at each user. Their proposed GSB algorithm solves the problem by starting to sort all RRHs following their transmitting power gains. The algorithm then iteratively turns off RRHs with minimum power gain, until the power minimization problem becomes infeasible. However, the GSB approach was not a C-RAN-specific solution for power minimization, since it can also be applied to traditional base station networks, with an extension of fronthaul links. Furthermore, the GSB scheme could not measure the number of necessary BBUs in the cloud that can handle the system.

Our paper, to the best of our knowledge, is one of the precursory attempts to present a high-level centralized approach combining dynamic resource allocation, transmission power minimization and BBU-RRH assignment in one framework. Other attempts regarding centralized resource allocation have been previously tackled under rate constraint such as [START_REF] Hatoum | Qosbased power control and resource allocation in OFDMA femtocell networks[END_REF] [14] [START_REF] Peng | Energy-efficient resource assignment and power allocation in heterogeneous cloud radio access networks[END_REF]. Authors in [START_REF] Hatoum | Qosbased power control and resource allocation in OFDMA femtocell networks[END_REF] presented a QoS-based Power Control and Resource Allocation in LTE Femtocell network (QP-FCRA). Although their approach is mainly within the context of femtocell networks, it can be applied to C-RAN thanks to its centralization nature. In their proposal, a joint resource allocation and power minimization algorithm is implemented at a central level of each clustering cells. The QP-FCRA algorithm then exploits cooperation between neighboring RRHs to periodically optimize the throughput satisfaction rates of users. However, their optimization scheme was run in offline mode and the algorithm's computational time was fairly big. In [START_REF] Lyazidi | Resource allocation and admission control in OFDMA-based Cloud-RAN[END_REF], we have addressed the problem of admission control considering individual UEs Quality of Service (QoS) requirement for guaranteed-service users but the transmission power aspect was however not considered. In this paper we encompass jointly maximizing the throughput of best-effort users while minimizing the total transmission power.

Although solutions for resource BBU-RRH assignment procedures in C-RAN have received some notable attention, the number of contributions for this problematic remains nonetheless very limited. Authors in [START_REF] Namba | Colony-RAN Architecture for Future Cellular Network[END_REF] described a Colony RAN design that can lessen the number of BBUs by roughly 75% compared to distributed RAN. In [START_REF] Namba | BBU-RRH Switching Schemes for Centralized Ran[END_REF], the same authors carried out their Colony RAN framework by proposing two mapping schemes for BBU-RRH assignment: Semi-Static (SS) and adaptive. The SS approach fixes the dichotomies of BBU-RRH subject to traffic peak hours of all network's RRHs in a large time window (one day). On the other hand, the Adaptive scheme dynamically maps BBUs with RRHs based on BBUs resource capacity and neighboring RRHs loads within a short time interval (one hour). For a given business office area traffic distribution, the authors demonstrated that the SS and Adaptive schemes can help reduce the number of BBUs by 26% and 47%, respectively.

III. PROBLEM FORMULATION

We detail in this section our two optimization models: C-RAPM and MKP. We consider a C-RAN system composed by a number of S RRHs within the set S = {i|1 i S}.

The BBU pool jointly assigns to each RRH in S a number of K Physical Resource Blocks (PRBs) from the set K = {k|1 ≤ k ≤ K}. We assume that the fronthaul network has sufficient links capacity.

A. Centralized Resource Allocation and Power Minimization (C-RAPM) Problem Formulation

In our first optimization model, we consider N (N ≥ 1) number of User Equipments (UEs) entering the system at a given epoch and connecting to a certain RRH from S. Each UE u ∈ {1, .., N } requests from its serving RRH a number of PRBs N u to run its applications [START_REF] Kaddour | Power control in opportunistic and efficient resource block allocation algorithms for green lte uplink networks[END_REF]. We suppose that each RRH i handles one cell in a delimited area, and that a UE u can only be served by the RRH covering the area it is positioned within. We consider a static transmission power from RRH i to UE u on each allocated PRB k. We suppose that the transmission power is quantized into L ≥ 2 discrete power levels: p min = p 1 < p 2 < ... < p L = p max , where p min is the minimum power that can be transmitted to a UE u and p max is the maximum transmitted power for each RRH. An increase in the number of power levels L pushes the discrete domain to be closer to a continuous one, but undoubtedly increases the problem's computational complexity [START_REF] Jang | Qos-constrained resource allocation scheduling for LTE network[END_REF]. Each resolution can lead to different transmission powers. We define our UE-RRH attachment, PRB allocation and transmit power variables:

x u i = 1, if UE u is attached to RRH i, 0, otherwise. (1)
y u ik = 1, if PRB k is allocated to UE u on RRH i, 0, otherwise. (2)
p u ik = p ∈ {p 1 , ..., p L }, if y u ik = 1, 0, otherwise. (3)
The SINR achieved by UE u, attached to RRH i and on a given PRB k can be formulated as:

γ u ik = p u ik g u ik j =i v =u p v jk g u jk + σ 2 (4)
Where g u ik is the path gain between RRH i and UE u, and σ 2 is the noise power. The SINR is expressed per PRB, as both channel/fading and interference vary over PRBs due to multipath, frequency selectivity and domain scheduling [START_REF] Langar | An operations research game approach for resource and power allocation in cooperative femtocell networks[END_REF]. Our objective in this first stage is to find the best PRBs allocation to serve in a best effort way all existing UEs, while minimizing the total downlink RRHs transmission power. The C-RAPM optimization problem can be expressed as follows: minimize

x u ,y u ,p u N u=1 i∈S k∈K (ǫ p u ik P max -(1 -ǫ) x u i y u ik K ) (5)
subject to i∈S k∈K

x u i y u ik N u , ∀u (6)
i∈S

x u i 1, ∀u (7)
N u=1 k∈K p u ik p max , i ∈ S (8)
γ u ik y u ik Γ u k , i ∈ S, k ∈ K, ∀u (9)
p u ik y u ik p min , i ∈ S, k ∈ K, ∀u (10)
N u=1 y u ik 1, i ∈ S, k ∈ K (11)
y u ik x u i , i ∈ S, k ∈ K, ∀u (12)
x u i , y u ik ∈ {0, 1}, i ∈ S, k ∈ K, ∀u (13)
We outline in the objective function ( 5) that we target to minimize the total transmission power while maximizing all possible UEs-PRBs assignments. The objective function is standardized so as to return values in the same order of magnitude. ǫ is a constant optimization weight between 0 and 1. Constraint (6) imposes that the total number of allocated resources for UE u cannot surpass its original demand N u . Constraint [START_REF] Mitchell | Branch-and-cut algorithms for combinatorial optimization problems[END_REF] denotes that a UE can only be served by at most one RRH. This is already guaranteed by UEs' cell position, however, a decision should be made for edge UEs positioned in a coverage area overlapped with other RRHs. If so, the optimization should assign this UE to an RRH that satisfies the other problem constraints 1 . Conditions (8) and (10) are the power constraints on RRH and UE, respectively. Condition [START_REF]IBM ILOG CPLEX Optimizer Version l v12[END_REF] ensures that the received SINR is equal to the required one Γ u k when the PRB k is in use (i.e., y u ik = 1) [START_REF] Hatoum | Qosbased power control and resource allocation in OFDMA femtocell networks[END_REF]. Constraint [START_REF] Cai | Dynamic operations of cloud radio access networks (C-RAN) for mobile cloud computing systems[END_REF] stresses the fact that two users linked to the same RRH cannot be served with the same PRB. Constraint (12) enforces all y u ik = 0 if attachement variable x u i is equal to 0 (i.e., UE u is not receiving any PRBs from RRH i). Finally, constraint (13) refers that both y u ik and x u i are binary variables. The optimization problem formulated in (5) -( 13) is a Mixed Integer NonLinear Programming (MINLP), which is NP-hard [START_REF] Shi | Maximizing capacity in multihop cognitive radio networks under the sinr model[END_REF] due to the presence of the quadratic product in the objective function [START_REF] Paolini | Benefits of C-RAN and adoption trends[END_REF] and the non-linear SINR constraint [START_REF]IBM ILOG CPLEX Optimizer Version l v12[END_REF]. We propose next, to reformulate the problem in a Mixed Integer Linear Integer version (MILP) via the big-M method [START_REF] Lee | Mixed integer nonlinear programming[END_REF]. In fact, the product of two binary variables x u i and y u ik can be replaced with a single binary one z u ik that is defined in the following constraints:

z u ik y u ik , (14)
z u ik x u i , (15)
z u ik x u i + y u ik -1. (16)
regarding constraint [START_REF]IBM ILOG CPLEX Optimizer Version l v12[END_REF], we find it convenient to reformulate it as follows:

(1 + 1 Γ u k )p u ik g u ik y u ik Υ u k + y u ik σ 2 (17)
where

Υ u k is equal to j v p v jk g u jk .
Besides, the product between y u ik and Υ u k can be linearized using the big-M reformulation too; provided that Υ u k has explicit lower and upper bounds. From (3) and (10), we can deduce Lwr and U ppr, the respective lower and upper bounds of Υ u k . Hence, the binary-continuous product y u ik Υ u k can be substituted by a continuous variable w u ik and by including the following constraints:

y u ik Lwr w u ik y u ik U ppr (18) (1 -y u ik )Lwr Υ u k -w u ik (1 -y u ik )U ppr (19)
Hence, the ILP formulation of our C-RAPM problem can be expressed as follows: minimize

x u ,y u ,p u N u=1 i∈S k∈K ǫ p u ik P max -(1 -ǫ) z u ik K (20) subject to i∈S k∈K z u ik N u , ∀u (21)
(7), ( 8), ( 10), ( 11), ( 12), (13) ( 14), ( 15), ( 16)

(1 + 1 Γ u k )p u ik g u ik w u ik + y u ik σ 2 (22)
(18), (19) (23)

B. Multiple Knapsack Problem (MKP) Formulation for BBU-RRH Assignment

In a distributed RAN system, one BBU is entirely assigned to a single RRH in order to handle its total traffic load. Thanks to C-RAN's centralization and flexibility, the resources of one BBU can be shared across different RRHs that have few traffic loads [START_REF] Checko | Optimizing small cell deployment by the use of C-RANs[END_REF]. For instance, if a remote site is covered by 4 RRHs and each has 25% of traffic load, one BBU is enough to manage all four RRHs. In our study, we can compute the optimal number of needed BBUs B to manage the S loaded RRHs as follows:

B = ⌈ Sum of all RRHs traffic charges K ⌉ (25)
where ⌈.⌉ is the ceiling function and K is the number of PRBs. The total charge of active RRHs corresponds to the total number of assigned PRBs from transmitting RRHs to all users, that are returned after solving the C-RAPM problem. Our goal in this second stage consists of properly assigning RRHs to BBUs according to their traffic charges and the number of available B BBUs. Towards this end, we consider a MKP formulation of the BBU-RRH assignment problem [START_REF] Pisinger | An exact algorithm for large multiple knapsack problems[END_REF], where the objects and the knapsacks are represented by the RRHs and the BBUs, respectively. We introduce a new binary variable r ij , which is equal to 1 if RRH i is attached to BBU j and 0 otherwise. From the results of the C-RAPM problem, we can compute the weight c i of each RRH i as follows:

c i = k∈K y ⋆ ik /K (26)
where y ⋆ is the returned solution of y from the C-RAPM problem. The value of c i represents the percentage of traffic load RRH i handles. We suppose that each BBU j can handle 100% of a fully loaded RRH (i.e., all K PRBs are used). We then formulate our BBU-RRH MKP as follows:

maximize r B j=1 S i=1 r ij (27) subject to S i=1 c i r ij 1, j ∈ {1, ..., B}, (28)
B j=1 r ij 1, i ∈ {1, ..., S}, (29)
r ij ∈ {0, 1}, i ∈ {1, ..., S}, j ∈ {1, ..., B} (30)
where constraint (29) denotes that one RRH cannot be managed by more than one BBU. This formulated problem is an Integer Linear Program (ILP), which can be efficiently solved by standard ILP solvers such as CPLEX.

IV. PROPOSAL: DRAC-SA ALGORITHM

In this section, we present our Dynamic Resource Allocation in C-RAN based on Simulated Annealing (DRAC-SA) metaheuristic with defined neighborhood search to solve the C-RAPM problem formalized in (20) - [START_REF] Van Laarhoven | Simulated annealing[END_REF].

A. Algorithm Overview

The SA meta-heuristic [START_REF] Metropolis | Equation of state calculations by fast computing machines[END_REF] is a powerful stochastic algorithm used to solve many combinatorial optimization problems in a fixed amount of time. The framework is based on exploring the different states of the cooling process of a solid from an initial hot temperature to a fixed frozen one. Each state of the process corresponds to a solution of the optimization problem. From a given state, a subsequent one can be generated by performing a small perturbation mechanism. This corresponds to generating neighbors of the initial solution via some particular neighborhood structures. The acceptance rule of a new solution (or new state) to the initial one is defined by the Metropolis rule [START_REF] Van Laarhoven | Simulated annealing[END_REF], which imposes a probabilistic decision based on the varying temperature and the energy of both states. The energy refers to the cost function of the optimization problem. If the generated state has lesser energy, it is accepted as the current state. Otherwise, it is admitted with a probability exp( -∆E T ), where ∆E is the energy difference of the two states and T is the time varying temperature. It is worth noting that at high temperature exp( -∆E T ) is close to 1, therefore the majority of moves can be accepted. Whereas at low temperature, exp( -∆E T ) is close to 0, which severely limits the search process to only solutions decreasing the energy. Hereafter, we will define each function if the SA metaheuristic to resolve C-RAPM.

B. Initial Solution

We first start by employing a greedy search method to generate the initial solution of the C-RAPM problem. It is based on performing linear relaxation of the integer variables and limiting the local search at the first nodes containing feasible integer solutions. Moreover, by focusing the resolution on a limited optimization space generated by fewer variables, the local search can be further reduced. In fact, we can consider z u ik as the "core" variable of our problem; since variables x u i and y u ik can be derived from the big-M constraints ( 14), ( 15), [START_REF] Namba | Colony-RAN Architecture for Future Cellular Network[END_REF]. On the other hand, p u ik comes as a "subcore" optimization variable, which can be deduced from ( 8) and [START_REF] Checko | Optimizing small cell deployment by the use of C-RANs[END_REF]. We denote E 0 the cost function (or energy) of this initial solution and T max the maximum annealing temperature. In what follows, we define T SR u the throughput satisfaction rate of UE u, which is the ratio of its total allocated PRBs on its initial demand N u .

C. Neighborhood Search Structure

Here, we define our specific neighborhood search stage to generate the states. We initiate the neighborhood generation by selecting a uniformly random UE u from the outputs of the initial solution and by computing its T SR u . We define xu , ŷu and pu , the solution neighbors of x u , y u and p u for UE u, as follows:

• Step 1: UE u changes its RRH attachment following a discrete Bernoulli distribution with parameter (1 -T SR u ). A new RRH attachment vector xu is generated from this probability and by selecting the available RRHs to whom u can be linked to based on its geographical position. (y u ik = 0), we select the eligible ones that can be allocated to UE u based on the SINR constraint γ u ik ≥ Γ u k , while determining for each one the minimal power that satisfies this constraint.

• Step 3: For the eligible PRBs that satisfy γ u ik ≥ Γ u k , they are allocated to UE u following a Bernoulli distribution with parameter T SR u × γ u ik SN Rmax , where SN R max = p max g u ik /σ 2 represents the maximum Signal-to-Noise Ratio (SNR) achieved on UE u. This helps allocating PRBs to UE u, with respect to other users existing allocation and possible interference. After this, we set all allocated PRBs power levels to a unique one, corresponding to the highest level of the allocated PRBs (i.e., the maximum of all minimal powers that satisfy the SINR constraint or each PRB).

D. Equilibrium state

After generating the new solution neighbors, a new cost function E n is calculated. We increase the neighborhood search structure to other UEs if and only if the current solution does not improve the objective function and satisfies the following equation:

exp(- E n -E 0 T n ) ≥ δ ( 31
)
where δ is a random number in [0, 1], which refers to the random value of the equation to increase the neighborhood states in the SA meta-heuristic to see whether exp( -∆E T ) in equation ( 31) is close to 0 or 1, and thus accept increasing the neighborhood tree. Additionally, in each iteration n we use the following cooling equation to decrease the temperature:

T n ← T n ln(n) (32)

E. Stopping condition

Fig. 2 illustrates our DRAC-SA algorithm flow-chart. The algorithm converges as soon as the maximum number of iteration n max is elapsed, which corresponds to the maximum CPU time. Therefore, its value should be scalable based on the processing machine so as to not exceed the delays of mobile users resources requests during their stay time in the system.

F. MKP Resolution

Once the C-RAPM problem is solved and the resources are allocated to UEs, the next step consists in calculating the number of needed BBUs B to handle the total traffic demand (25). Since the MKP problem ( 27)-( 30) is a ILP, we make use of IBM's linear solver CPLEX to compute its solution using the solver's built-in algorithms. CPLEX's branch-andbound is able to return optimal results within a computation time n MKP very small compared to n max (n MKP ≪ n max ). Hence, by summing the two computation times, solutions for the C-RAPM and BBU-RRH associations can be dynamically found while respecting mobile users requests delays. [START_REF] Lyazidi | Resource Allocation in Cloud-RAN with Real-Time RRH-BBU Assignment[END_REF] 148.1 + 37.6log 10 (d), d in Km Shadowing standard deviation [START_REF] Haberland | Radio Base Stations in the Cloud[END_REF] 5 dB Fading model [START_REF] Haberland | Radio Base Stations in the Cloud[END_REF] Normal distribution N (0, I) Thermal noise [START_REF] Haberland | Radio Base Stations in the Cloud[END_REF] -174 dBm/Hz Transmit antenna power gain [START_REF] Haberland | Radio Base Stations in the Cloud[END_REF] 8 dBi Arrival rate of UEs

V. PERFORMANCE EVALUATION

In this section, we evaluate the benefits and performances of our proposed DRAC-SA algorithm, and compare the benefits of our solution with respect to state-of-the-art schemes: the QP-FCRA [START_REF] Hatoum | Qosbased power control and resource allocation in OFDMA femtocell networks[END_REF] and the Iterative GSB [START_REF] Shi | Group sparse beamforming for green Cloud-RAN[END_REF] algorithms for solving the C-RAPM problem. We also include comparisons to the greedy approach, which was used to generate the initial solution of the DRAC-SA algorithm, as well as to our previous DRAC approach in [START_REF] Lyazidi | Resource Allocation in Cloud-RAN with Real-Time RRH-BBU Assignment[END_REF]. The latter was run in offline mode due to its high computation time for the chosen system parameters. On another hand, we also compare the SS and Adaptive switching algorithms in [START_REF] Namba | BBU-RRH Switching Schemes for Centralized Ran[END_REF] to the returned solutions of our MKP regarding the BBU-RRH assignment problem. For our experimental environment, we simulated a wireless LTE environment consisting of 100 RRHs deployed in a 450 m×450 m square grid. Each RRH has a coverage radius of 35 m and the distance between two nearest RRHs is 50 m. We considered the following channel model [START_REF] Lyazidi | Resource Allocation in Cloud-RAN with Real-Time RRH-BBU Assignment[END_REF]: where L(d u i ) is the path-loss at distance d u i between RRH i and UE u, φ u i is the antenna gain, s u i is the shadowing coefficient, and g u i is the fading coefficient. We generate a fixed poisson arrival rate of mobile users of λ = 5 arrivals per time, and vary at each simulation run the users' stay time and service demand following an exponential and uniform laws, respectively. Each UE's geographical position is randomly generated at each run and remains fixed during its stay-time in the system. The service demand of each user is expressed in terms of number of PRBs from a downlink LTE frame of 100 PRBs and follows a uniform distribution from 1 to 25 PRBs. We run 30 simulations for each scenario of SINR threshold Γ: 10 and 25 dB, to reach a confidence level of 97%.

h u i = 10 -L(d u i )/20 φ u i s u i g u i ,
Table I reports the simulation parameters. In what follows, we present the corresponding simulation results in terms of Throughput Satisfaction Rate (TSR), computation time analysis, Spectrum Spatial Reuse (SSR), normalized throughput distribution, transmission power, and number of BBUs along with the active number of RRHs.

1) Throughput Satisfaction Rate (TSR):

We present in Fig. 3 the Cumulative Distributed Function (CDF) of the TSR. The latter represents the ratio of the number of allocated PRBs to the total initial demands N u . The CDFs of DRAC, GSB and QP-FCRA correspond to CDFs generated from offline resolutions, where we left the algorithms methods running until the end results. We emphasize the fact that they are not applicable in real-time context due to their high computational time, and we only added them for the sake of comparison. We can observe, by comparing the CDF of the offline methods and the online Greedy and DRAC-SA's ones, that for the latter, more than 50% of UEs have their TSR greater than 80% and 70% in SINR threshold equal to 10dB and 25dB, respectively. The TSR is lessened to 60% and 48% for QP-FCRA and GSB, respectively -as shown in Fig. 3(a) -at low SINR threshold, and to 47% and 35%, respectively, in Fig. 3(b), when the SINR threshold is high. Hence, our proposed DRAC-SA approach outperforms both QP-FCRA and GSB schemes, and approaches as well the highest throughput satisfaction rate given by DRAC, when the latter reaches the end of the resolution. However, we notice that the greedy online approach achieves better satisfaction rate at high SINR regime than the offline GSB scheme. In fact, the latter emphasizes on turning off as many as RRHs as possible to achieve maximum power savings, whereas the greedy approach turns a large number of RRHs on to find quick solutions for the C-RAPM problem.

2) CPU time vs network density: As shown in Fig. 4, the complexity evolution of DRAC-SA is polynomial in terms of network density, and is largely lower than that of DRAC. The time computation results indicate that the proposed DRAC-SA can solve the C-RAPM problem in less than 120 ms for a network with 100 mobile users. Besides, it can return solutions in a few ms when the number of active users is low (less than 20 mobile users). Overall, DRAC-SA achieves significantly high CPU time savings than DRAC, where the latter returns the optimum solutions after at least 1000 s for the highest dense network (150 users). This makes the DRAC approach unpractical for online optimization as it severely impacts the rate of served users and the global TSR, which will be described later on.

3) Spectrum Spatial Reuse (SSR): Table II reports the SSR of all aforementioned approaches. Note that SSR denotes the average portion of RRHs using identical PRB and can be expressed as follows:

SSR = 1 S × K N u=1 i∈S k∈K y u ik (33)
The more a PRB is reused, the better is the performance. Table II clearly shows that our proposal DRAC-SA increases PRBs reuse by a factor of 1.06 and 1.91 compared to QP-FCRA and GSB approaches, respectively, at low SINR threshold. When the SINR threshold is high, the reuse factor is enhanced by 1.03 and 2.13, respectively. We also notice that the gap between DRAC-SA's SSR and DRAC's is only of 5.71%, which exhibits the good performance of our algorithm. We further extend the analysis by investigating how each PRB is reused in the network compared to the greedy and the optimal solutions. Fig. 5 shows that DRAC-SA improves the reuse factor up to 32% for PRBs that are less re-used with the greedy method. In fact, DRAC-SA achieves globally 43.36% better performance in PRB reuse than the greedy approach. 4) Normalized throughput vs UEs demand: In order to illustrate how the allocated resources are affected by UEs' demand volume, Fig. 6 presents the normalized throughput evolution as a function of UEs demands N u , for both SINR regimes. Globally, QP-FCRA and GSB show a roughly constant behavior for SINR = 10 dB in Fig. 6(a), with an emphasis on low and high PRB demand, respectively. This implies that their resource allocation mechanisms are done independently of UEs' requested number of PRBs. On the other hand, DRAC-SA favors resource allocation of UEs with the highest demand N u in order to increase their total satisfaction rate. This is more clearly shown in the high SINR regime (Fig. 6(b)), where DRAC-SA favors high demands significantly more than DRAC and the other schemes. This may be interpreted as unfair to users with low PRB demands. However, from a network management perspective, it is a positive behavior as DRAC-SA can dismiss resource allocation to low user demands that would cause interference to high-demanding users with greedy resource applications, and eventually increase their total transmitted power.

5) Transmitted Power per RRH: Fig. 7 illustrates the percentage of RRHs transmitting on each transmission power. We remark that at low SINR regime (see Fig. 7(a)), the majority of RRHs are transmitting on the lowest power levels: p min = 0.1 mW and p 2 = 1 mW , whereas for most RRHs the greedy method favors the highest power level p max = 20 mW , which results in a high total transmission power. What is more, DRAC-SA focuses mostly on the second power level p 1 = 1 mW . At high SINR regime (see Fig. 7(b)), most approaches emphasize on higher transmission powers such as p 4 = 15 mW and p max = 20 mW . By scattering the transmission powers on the lowest levels, our approach can achieve less energy consumption compared to the greedy resolution method and QP-FCRA, as shown in Fig. 8, which presents the total C-RAN transmitted power. We can remark that the GSB scheme realizes minimum transmission power thanks to its successive RRH switching algorithm; however, this is negatively reflected on the TSR of mobile users as seen in Fig. 3, since they are less satisfied by their allocated PRBs. The QP-FCRA approach, on the other hand, supposes that all RRHs are turned on, which results in a higher power consumption but to a good TSR. As can be observed in Fig. 3 and Fig. 8, our proposed DRAC-SA scheme performs a good tradeoff balance between UEs satisfaction rates and overall transmission power minimization in both SINR threshold levels.

6) Number of BBUs and on

RRHs: Fig. 9 shows the variations of the number of on RRHs and the number of instantiated BBUs B per time returned by DRAC-SA, when the SINR threshold is equal to 25 dB. As can be observed, the number of instantiated BBUs computed from the output of DRAC-SA can achieve important savings compared to a conventional RAN scenario. The latter follows a static variation due to the one-one mapping, which imposes as many BBUs as deployed RRHs. This consequently inflicts heavy investments from operators to manage their network and increase their total BBU equipment costs to handle the radio traffic loads of all cells. Our approach can realize up to 86% and 92% BBUs savings compared to a RRH-based RAN scenario, which will result in important OPEX savings for the operator.

For the BBU-RRH assignment problem, we solve the MKP in (26) using CPLEX, which is able to find optimal results [START_REF] Mitchell | Branch-and-cut algorithms for combinatorial optimization problems[END_REF] Global TSR vs arrival rates: We vary next the arrival rates of mobile users in the system, where λ takes values in [START_REF] Lyazidi | Resource Allocation in Cloud-RAN with Real-Time RRH-BBU Assignment[END_REF][START_REF] Checko | Optimizing small cell deployment by the use of C-RANs[END_REF]. Fig. 10 illustrates the evolution of the global TSR in both SINR regimes for each of DRAC-SA, DRAC and the greedy approaches. As λ increases, more users penetrate the system, which leads to less time intervals between each user arrival. As stated before, a large proportion of new arrived users are discarded by DRAC, since it is still solving the C-RAPM problem of the previous existing users. What is more, starting from λ = 4, the global TSR returned by DRAC is severely impacted and results in more than 70% of UEs not served by the system. This is depicted in Fig. 11, which illustrates the evolution of rejected users' rate with different arrival rates. DRAC-SA, on the other hand, clearly outperforms DRAC thanks to its reduced complexity and possible online optimization, which provides a very global TSR at high arrival rate (74% and 61% for low and high SINR threshold, respectively) as well as a low rate of rejected UEs (14% and 19% for low and high SINR threshold, respectively).

8) Number of BBUs vs arrival rates:

In the following, we present the variation of the number of instantiated BBUs as a function of the network density for each arrival rate. Fig. 12 presents the evolution of BBUs and the number of on RRHs in the system with the variation of the poisson arrival rate λ for SINR threshold equal to 25 dB. The plot illustrates the evolution for the DRAC-SA and Greedy methods as well as the number of BBUs required in case of the conventional RAN. The latter corresponds to the number of active RRHs, which has to be equal to the number of BBUs due to the one-one mapping in a distributed RAN deployment. As illustrated, the number of instantiated BBUs for the DRAC-SA solutions achieves important savings in BBUs compared to the conventional scenario. On another hand, we can remark that for the highest arrival rate, λ = 10, the number of on BBUs is at its maximum capacity for the conventional case, whereas DRAC-SA and the greedy methods are still at 55% and 40% of the total system's capacity, respectively. Therefore, it is up to the operator to manage its C-RAN deployment: wether is increasing the number of RRHs to satisfy maximum users, or turning them off to achieve energy efficiency is the better choice.

VI. CONCLUSION

In this paper, we have portrayed a novel approach based on simulated annealing to address the problem of resource allocation and transmission power minimization in C-RAN for a dynamic flow of UEs traffic. Specifically, our newly improved DRAC-SA framework can quickly find the best PRB allocation and transmission power strategy to cater the traffic demand, while satisfying individual SINR constraints and maximum power limitations. Besides, our approach can dynamically determine the best (if not optimal) number of active RRHs and BBUs to instantiate in order handle the C-RAN traffic. Through our extensive event-based simulations, we have demonstrated that our method finds several good balances regarding, firstly, throughput satisfaction rate and total transmitted power and, secondly, resolution time and global user satisfaction. In fact, DRAC-SA achieves 43.36% better performance in PRB distribution than a greedy approach, and only 5.71% of difference is between the global optimum offline approach and DRAC-SA in terms of throughput satisfaction. Besides, the number of BBUs calculated from DRAC-SA can help increase the BBUs savings to 85.6% compared to distributed RAN scenarios. We hence believe that this approach represents a promising solution for centralized resource allocation in future C-RAN deployments.

Fig. 1 :

 1 Fig. 1: Cloud Radio Access Network (C-RAN) Architecture

Fig. 2 :

 2 Fig. 2: DRAC-SA Flow Chart

Fig. 3 :

 3 Fig. 3: Throughput Cumulative Density Function

λ ∈ [ 1 , 10 ]

 110 (default 5) Departure rate of UEs µ = 0.1 UE's PRB demand Uniform distribution U (1, 25) BBU capacity[10] W 1 (100%) Initial hot temperature Tmax = 1000 Max. number of iterations 1000

Fig. 4 :

 4 Fig. 4: CPU time vs number of UEs

Fig. 5 :

 5 Fig. 5: SSR per PRB, SINR = 25 dB

Fig. 6 :Fig. 7 :

 67 Fig. 6: Throughput distribution as a function of user demands

Fig. 8 :Fig. 9 :

 89 Fig. 8: Total RRHs transmitted power in C-RAN

Fig. 10 :Fig. 11 :

 1011 Fig. 10: Global TSR vs arrival rate λ

Fig. 12 :

 12 Fig. 12: Number of BBUs vs arrival rate λ

Table I :

 I Simulation Parameters (p min) /p 2 /p 3 /p 4 /p 5 /p 6 (pmax)

	Parameters	Values
	Number of RRHs	100
	Bandwidth	20 MHz
	Total number of PRBs	100
	Power levels L	6
	p 1 0.1/1/5/10/15/20 mW
	Constant ǫ	0.5
	Path loss model

Table II :

 II Mean Spectrum Spatial Reuse

	SINR	DRAC-SA	DRAC	Greedy	QP-FCRA	GSB
	10 dB	4.29 ± 0.12	4.55	2.45 ± 0.5	4.05	2.25
	25 dB	4.23 ± 0.15	4.55	2.45 ± 0.5	4.10	2.01

Table III :

 III Number of BBUs and RRHs

	Scheme	Mean BBU	Mean on RRHs	Max RRHs/BBU
	MKP	7.97 ± 0.06	55.4 ± 2.2	39
	SS	15.1	59.7	27
	Adaptive	13.5	62.4	28

  TableIIIpresents the average number of BBUs and on RRHs as well as the minimum and maximum number of handled active RRHs per BBU. Clearly, DRAC-SA achieves more BBUs savings to cater to the same volume of traffic load with a reduced number of on RRHs. This not only improves the network capacity, since many RRHs can be handled by the same BBU, but also helps maximizing the efficiency of BBUs within the virtual pool.

The study of Cooperative Multipont Processing (CoMP) in C-RAN is out of the scope of this paper.

ACKNOWLEDGMENT

This work is partially supported by the ANR ABCD project (Grant no. ANR-13-INFR-0005-01) and the FUI ELASTIC Network project (Grant no. C16/0287).


Guillaume Worms
email: guillaume.worms@ensiacet.fr

Michel Meyer

David Rouzineau

Mathias Brehelin

Production Zone Method: a New Non-ideal Shortcut Method for Distillation Column Design

Keywords:

 the method uses production segment rather than completely specified product which eliminates any sensitivity to the composition of the minor products. It has two aims. The first is to determine if a specified separation respects the mass balance and the thermodynamic feasibility. The second one is to find the minimum reflux ratio and a preliminary design of the column. The following mixtures are investigated: an ideal mixture of ethanol, n-propanol, and n-butanol; a non-ideal mixture of acetone, water, and acetic acid; and an azeotropic mixture of acetone, isopropanol, and water. Designs obtained with this new method lead to purity and recovery rate close to specifications which is sometimes impossible with conventional ideal shortcut like the well-known Fenske-Underwood-Gilliland shortcut.

Introduction

Distillation is one of the most important fields in process engineering and attracts an important attention as reported by [START_REF] Yildirim | Dividing wall columns in chemical process industry: A review on current activities[END_REF]. Several distillation shortcut methods already exist including methods with graphical tools, but the latter have two major inconveniences. The first is that many are sensitive to product composition like the Boundary Value Method [START_REF] Levy | Design and synthesis of homogeneous azeotropic distillations 1. Problem formulation for a single column[END_REF] or the Zero-Volume Criterion [START_REF] Julka | Geometric behavior and minimum flows for nonideal multicomponent distillation[END_REF]. The second is that if composition areas are defined, most of the time the boundaries are straight lines which take not in good account the thermodynamic behavior of the systems. This is for example the case with the Rectification Body Method [START_REF] Bausa | Minimum energy demand for nonideal multicomponent distillations in complex columns[END_REF] and another method developed by [START_REF] Thong | Multicomponent homogeneous azeotropic distillation 2. Column design[END_REF]. Based on the operation leaves of [START_REF] Castillo | Homogeneous azeotropic distillation 1. Design procedure for single-feed columns at non-total reflux[END_REF], the aim of the presented method called Production Zone Method (PZ method), concerns the feasibility of the separation in the acquisition of a column pre-design. The method uses production segment rather than product with completely specified composition which avoid being sensitive to the composition of the minor product. Concerning phase equilibria, no restrictive assumptions are made.

Description of the Method

The PZ method is based on the use of operation leaves. Each leaf is the area of all reachable compositions in a single column section. Figure 1 presents three operation leaf. It is defined by two curves: the distillation line [START_REF] Stichlmair | Distillation and rectification[END_REF], which represents the liquid composition profile at total reflux and the pinch-point curve [START_REF] Wahnschafft | The product composition regions of single-feed azeotropic distillation columns[END_REF], which represents the liquid composition profile for an infinite number of stages. All composition points between these two curves can be reached depending of the choose reflux or reboiling ratio. Operation leaf advantage is the thermodynamic meaning of its boundaries. However, it is very sensitive to the product composition.

To overcome this constraint, PZ method requires only one specification for each product stream instead of both complete defined products. Graphically, for a ternary mixture, this is equivalent to defining a production segment. In Figure 2, a production segment is defined for a composition of 70 % in A at the distillate and 80 % in C at the residue. Both distillate and residue segments must respect the masse balance feasibility represented by the two dotted lines and also the thermodynamic feasibility of the separation. The latter, shown in Figure 1, corresponds to the overlap of the distillate and the residue leaves which provides an intersection between both section liquid profiles. Once the feasibility of the separation is established the method can be used to find which specific mass balance provides the smallest minimum reflux. This parameter is interesting to obtain an energy-optimized preliminary design. To do that, the method scans the production segments and determines the minimum reflux for each couple of distillate and residue points. The minimum reflux is the smallest reflux which provides an intersection of both liquid profilescalculated with stage to stage calculation and equilibrium stages model -and the respect of the relation between r, the reflux ratio, s, the reboil ratio, q, the feed thermal condition and the feed (zF), the distillate (xD) and the residue (xW) composition presented in Eq.(1). Then an actual reflux ratio R=a.Rmin is chosen. In this study the common value of a=1.3 is used. Thus, using Eq.( 1), the associated reboil ratio is found and the corresponding profiles are calculated. The number of stages in each column section is then the number of calculated points on each profile between the product and the intersection point. The feed stage corresponds to the latter.

(1)

Case studies

The proposed method was compared with the well-known FUG shortcut. The reflux ratio, the distillate flowrate, the feed stage and the number of theoretical stages obtained by both shortcut are used to initialize rigorous simulations in ProsimPlus software. Results are shown in Figure 3, 4, 5 and 6. Three mixtures were studied: an ideal mixture of Ethanol / n-Propanol / n-Butanol, a non-ideal mixture of Acetone / Water / Acetic Acid and an azeotropic mixture of Acetone / Isopropanol / Water. For each case, xF, the respective feed compositions, can be found in Figure 4, 5 and 6 as well as q, the feed thermal condition, and xD/xW, the distillate/residue specifications. The feed flowrate is 100 mol/s for all cases.

Relative gap study for the specifications

In this first study, rigorous simulations were made without specification on the product compositions: the outlet compositions of each case are then different depending of the shortcut method used. Figure 3 highlights six cases that are representative of the whole results. The figure shows the relative gap between the distillate light key composition results of the simulation and the input distillate specification of the shortcut. The relative gap is usually smaller than 1 % in absolute value when the simulation was initialized with PZ method whereas, in several cases, the gap is higher than 5 % in absolute value when the simulation was initialized with FUG shorcut. Relative gaps for the residue specification and the recovery specifications follow the same trend. These results show that the compositions and the recovery ratios are close to their input values with the PZ column design and the relative gaps are of the same order of magnitude. With the FUG column design, relative gap sizes varied considerably from case to case. PZ method seems to be more precise and reliable than FUG shortcut.

Cost study

In this second rigorous simulation study, the energy consumption and the total annual cost (TAC) of each designed column are compared more specifically. TAC calculations are based on the equations and the recommended values of [START_REF] Kiss | Design, control and economics of distillation, Advanced distillation technologies: design, control and applications[END_REF] and the packing properties are those of Mellapack 250Y. In these simulations, the feed stage and the number of theoretical plates are fixed while the reflux ratio and the reboil duty are adjusted to obtain the product specifications. Results in Figure 4, 5 and 6 show that PZ designs provide always the smallest TAC compared with FUG designs except for cases 09 and 10. In some case, like case 11, TAC of the FUG design is almost the double of the TAC of the PZ design. In case 13 and 14, non-convergence can be found with the FUG design. However, cases 09 and 10 show the limits at the moment of the method: in some configuration, the imprecision due to the resolution of the code in Matlab becomes significant.

Conclusion

A new shortcut method for distillation has been developed. This method provides the knowledge of the mass balance and thermodynamic feasibility of the separation and determines the minimum reflux ratio when the separation is possible. From the minimum reflux ratio, a first design can be obtained and used to initialize a rigorous simulation. Results show that the method gives a good approximation of the rigorous results and is more precise and reliable than the FUG shortcut. Moreover designs proposed by the method are more efficient than those proposed by the FUG shortcut in terms of economic. Required data for the method are the composition, the thermal condition and the flowrate of the feed and one specification for each outlet stream. These data were chosen because they will be obtained easier than others possible data: it will be easier to use it for industrial and engineering purposes. Future work will be the adaptation of this method to unconventional column design and the first one which will be investigated will be the design of a dividing wall column.

Figure 1 .

 1 Figure 1. Operation leaves in a A (light), B and C (heavy) ternary diagram -Separation of D and W1 is unfeasible whereas separation of D and W2 is feasible. Operation leaf of D shows some liquid composition profiles for some values of the reflux ratio

Figure 3 .

 3 Figure 3. Relative gap between the distillate composition results of the simulation and the desired distillate specification

Figure 4 .

 4 Figure 4. Total annual costs ($) A: an ideal mixture of Ethanol / n-Propanol / n-Butanol -B: a non-ideal mixture of Acetone / Water / Acetic Acid -C: an azeotropic mixture of Acetone / Isopropanol / Water